An multi-criteria decision-making approach based on Choquet integral-based TOPSIS with simplified...

8202019 An multi-criteria decision-making approach based on Choquet integral-based TOPSIS with simplified neutrosophic shellip

httpslidepdfcomreaderfullan-multi-criteria-decision-making-approach-based-on-choquet-integral-based 138

1

An multi-criteria decision-making approach based on Choquet

integral-based TOPSIS with simplified neutrosophic sets

Juan-juan Peng12

Jian-qiang Wang2 Chao Tian

1 Xiao-hui Wu

12 Xiao-hong Chen2

1 School of Economics and Management Hubei University of Automotive Technology Shiyan 442002 China

2 School of Business Central South University Changsha 410083 China

Corresponding author Tel+8673188830594 Fax +867318710006

E-mail address jqwangcsueducn

Abstract In this paper the new approach for multi-criteria decision-making (MCDM) problems is developed

based on Choquet integral in the context of simplified neutrosophic environment where the truth-membership

degree indeterminacy-membership degree and falsity-membership degree for each element are singleton

subsets in [01] Firstly the novel operations of simplified neutrosophic numbers (SNNs) and relational

properties are discussed and the comparison method and distance of SNNs are presented as well Then two

aggregation operators for SNNs namely the simplified neutrosophic Choquet integral weighted averaging

operator and the simplified neutrosophic Choquet integral weighted geometric operator are defined The

properties among two aggregation operators are further discussed in detail In addition based on aggregation

operators and TOPSIS (technique for order preference by similarity to ideal solution) a novel approach is

developed to handle MCDM problems Finally one practical example is provided to illustrate the practicality

and effectiveness of the proposed approach And the comparison analysis is also presented based on the same

example

Keywords Multi-criteria decision-making simplified neutrosophic sets Choquet integral Aggregation

operators

1 Introduction

In many cases it is difficult for decision-makers to definitely express preference in solving MCDM

problems with inaccurate uncertain or incomplete information Under these circumstances Zadehrsquos fuzzy sets

(FSs) [1] where the membership degree is represented by a real number between zero and one are regarded

as an important tool to solve MCDM problems [2 3] fuzzy logic and approximate reasoning [4] and pattern

anuscript

ck here to download Manuscript PJ11-simplified neutrosophic sets20140917pdf

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

3233

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



recognition [5]

However FSs can not handle some cases that the membership degree is hard to be defined by one specific

value In order to overcome the lack of knowledge of non-membership degrees Atanassov [6] introduced

intuitionistic fuzzy sets (IFSs) the extension of Zadehrsquos FSs In addition Gau and Buehrer [7] defined vague

sets Later Bustince [8] pointed out that the vague sets and IFSs are mathematically equivalent objects At

present IFSs have been widely applied in solving MCDM problems [9-16] neural networks [17 18] medical

diagnosis [19] color region extraction [20 21] and market prediction [22] IFSs take into account the

membership degree the non-membership degree and the degree of hesitation simultaneously So it is more

flexible and practical in addressing fuzziness and uncertainty than the traditional FSs Moreover in some

actual cases the membership degree the non-membership degree and the hesitation degree of an element in

IFSs may be not a specific number Hence it was extended to the interval-valued intuitionistic fuzzy sets

(IVIFSs) [23] To handle the situations that people are hesitant to express their preference over objects in a

decision-making process hesitant fuzzy sets (HFSs) were introduced by Torra and Narukawa [24 25] Zhu et

al [26 27] proposed dual HFSs and outlined their operations and properties Furthermore Chen et al [28]

proposed interval-valued hesitant fuzzy sets (IVHFSs) and applied it to MCDM problems Then Farhadinia

[29] discussed the correlation for dual IVHFSs and Peng et al [30] introduced a MCDM approach with

hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) which is an extension of dual IVHFSs

Although the FSs theory has been developed and generalized it can not deal with all sorts of uncertainties

in different real problems Some types of uncertainties such as the indeterminate information and inconsistent

information can not be handled For example when we ask the opinion of an expert about a certain statement

he or she may say the possibility that the statement is true is 05 the one that the statement is false is 06 and

the degree that he or she is not sure is 02 [31] This issue is beyond the scope of the FSs and IFSs Therefore

some new theories are required

Smarandache [32 33] proposed neutrosophic logic and neutrosophic sets (NSs) Rivieccio [34] pointed out

that an NS is a set where each element of the universe has a degree of truth indeterminacy and falsity

respectively and it lies in ]0 1 [ the non-standard unit interval Obviously it is the extension of the standard

interval of IFSs And the uncertainty presented here ie indeterminacy factor is dependent on of truth

and falsity values while the incorporated uncertainty is dependent of the degree of belongingness and degree

[0 1]

2

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



3

of non-belongingness of IFSs [35] And the aforementioned example of NSs can be expressed as x(05 02

06) However without specific description NSs are difficult to apply in real-life situations Hence a

single-valued neutrosophic sets (SVNSs) was proposed which is an instance of NSs [31 35] Majumdar et al

[35] introduced a measure of entropy of SVNSs Furthermore the correlation and correlation coefficient of

SVNSs as well as a decision-making method using SVNSs were presented [36] In addition Ye [37] also

introduced the concept of simplified neutrosophic sets (SNSs) which can be described by three real numbers

in the real unit interval [01] and proposed an MCDM method using the aggregation operators of SNSs Wang

et al [38] and Lupiaacutentildeez [39] proposed the concept of interval neutrosophic sets (INSs) and gave the

set-theoretic operators of INSs Broumi and Smarandache [40] discussed the correlation coefficient of INSs

Zhang et al [41] developed the MCDM method based on aggregation operators under interval neutrosophic

environment Furthermore Ye [42 43] proposed the similarity measures between SVNSs and INSs based on

the relationship between similarity measures and distances However in some cases the SNSs operations

provided by Ye [37] may be unreasonable For instance the sum of any element and the maximum value

should be equal to the maximum one while it does not hold using the operations [37] The similarity measures

and distances of SVNSs based on those operations also may be incredible Based on the operations in Ye [37]

Peng et al [44 45] developed some aggregation operators and outranking relations of SNSs and applied them

to MCDM and MCGDM problems

However in those decision-making methods mentioned above most of the criteria are assumed to be

independent of one another However in real life decision-making problems the criteria of the problems are

often interdependent or interactive This phenomenon is referred to as correlated criteria in this paper The

Choquet integral [46] is a powerful tool for solving MCDM and MCGDM problems with correlated criteria

and has been widely used for this purpose [47-54] For example Yager [47] extended the idea of order

induced aggregation to the Choquet aggregation and introduced the induced Choquet ordered averaging

(I-COA) operator Meyer and Roubens [48] proposed the fuzzy extension of the Choquet integral and applied

it to MCDM problems Yu et al [49] used the Choquet integral to propose a hesitant fuzzy aggregation

operator and applied it to MCDM problems within a hesitant fuzzy environment Tan and Chen [50]

introduced the intuitionistic fuzzy Choquet integral operator Tan [51] defined the Choquet integral-based

Hamming distance between interval-valued intuitionistic fuzzy values and applied it to MCGDM problems

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



Bustince et al [52] proposed a new MCDM method for interval-valued fuzzy preference relation which was

based on the definition of interval-valued Choquet integrals Wei et al [53] developed a generalized triangular

fuzzy correlated averaging (GTFCA) operator based on the Choquet integral and OWA operator Finally

Wang et al [54] developed some Choquet integral aggregation operators with interval 2-tuple linguistic

information and applied them to MCGDM problems

However TOPSIS (technique for order preference by similarity to ideal solution) which developed by

Hwang and Yoon [55] also plays an important role in solving MCGDM problems and successfully applied in

many fields [56-60] whilst the Choquet integral has a critical role in handing MCGDM problems with

correlated criteria Therefore developing a method of combining these two methods in order to solve

simplified neutrosophic MCGDM problems with correlated criteria is seen as a valuable research topic In this

paper the novel operations and comparison method of SNSs are developed and the distance of SNSs is

proposed Two aggregation operators are defined based on Choquet integral and the corresponding properties

are discussed Furthermore an approach for MCGDM problems with SNSs is developed which could

overcome the drawbacks as we discussed earlier

The paper is structured as follows Section 2 contains the definition and the operations of SNSs In Section

3 some novel operations comparison method and distance of SNS are defined In Section 4 we develop two

aggregation operators based on Choquet integral and discuss some properties as well In Section 5 an

approach of MCGDM problems with SNSs is developed One worked example appear in Section 6 In Section

7 is the conclusion

2 Preliminaries

In this section fuzzy measure the Choquet integral and the definition of HFSs are reviewed Some operations

and comparison laws of HFSs which will be utilized in the latter analysis are also presented

21 Fuzzy measure and the Choquet integral

Let 1 2 n

X x x x be the set of the criteria P X be the power set of X then the fuzzy measure

is defined as follows

Definition 1 [61] A fuzzy measure on the set X is a set function [01]P X and satisfies the

following axioms

4

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) 0 1 X

(2) if 1 2 B B X then 1 2 B B

(3) 1 2 1 2 1 2 B B B B B B for 1 2 B B X 1 2 B B where ( 1 )

In Definition 1 if 0 then the third condition is reduced to the additive measure

for 1 2 B B X and 1 2 B B 1 2 1 2 B B B B

If the elements of i B are independent then

fori

B X i i

i

x B

i B x

(1)

In Definition 1 if 0 then the fuzzy measure is a probability measure and the elements are independent

if 1 0 then a redundant relation exists among elements if 0 then a complementary relation

exists among elements

Definition 2 [46] Let be a fuzzy measure on X P X then the Choquet integral

on

[0 ) f X

f with respect to can be defined as follows

0 X fd x f x t

dt

where ( ) x f x t P X for Ift R 1 2 n

X x x x is a finite set then the discrete Choquet

integral can be described as

( ) ( ) ( 1)1

n

i i i X

i

fd f x B B

i

(2)

or

( ) ( 1) ( )1

n

i i X

i

fd f x f x B

(3)

Where (1) (2) ( )n is a permutation of 12 n such that

(1) ( )0n

f x x (2 ) f x f

(0) 0 f x

( ) ( ) ( 1)

i i i B x x

( )n

x

and ( 1) 0n

B

Example 1 Let 1 2 3 X x x x 1 2 3 x x x and 2 x f x then 1 2 3 f x f x f x so 1 1

2 2 3 3 1 1 A x 2 3 x x 2 2 A x 3 x 3 A 3 x Suppose 1 03 x 2 025 x

5

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



3 037 x 1 2 05 x x 2 1 3 0 x x 65 2 3 04 x x 5 1 2 3 1 x x x if they are calculated by

using Eq (3) then the following is obtained

2

6

1 (0) 1 2 1 2 3 3 X

fd

f x f x B f x f x B f x f x B

22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs

In this section the definitions of NSs and SNSs are introduced for the latter analysis

Definition 3 [32] Let X be a space of points (objects) with a generic element in X denoted by x An

NS in A X is characterized by a truth-membership function AT x a indeterminacy-membership

function and a falsity-membership function A I x A

F x x A

T A I x and A

F x are real standard or

nonstandard subsets of that is]0 1 [ ]0 1 [ A

T x X ]0 1 X [ A

I x and

There is no restriction on the sum of ] A

F x X 0 1 [ AT x and x

A I (

A)F x so

0 supT x sup sup A A

I x

1 A

3 A

F x

Definition 4 [32] An NS is contained in the another NS denoted by if and only if2 A 1 A A 2

1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and

2 A1sup Asup F x F x for any x X

Since it is difficult to apply NSs to practical problems Ye [37] reduced NSs of nonstandard intervals into

the SNSs of standard intervals that will preserve the operations of the NSs


NS in A X is characterized by AT x x

A I and A

F x

[01]

which are singleton subintervalssubsets in

the real standard [0 1] that is AT x X [01] X A

I x

and Then a

simplification of is denoted by

[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



which is called an SNS It is a subclass of NSs For convenience the SNSs is denoted by the simplified

symbol ( ) ( ) ( ) A A A A T x I x F x The set of all SNSs is represented as SNSS

The operations of SNSs are also defined by Ye [37]

Definition 6 [37] Let and be three SNSs For any A 1 A 2 A x X the following operations are true

1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)

A A A A A A A A A A A A

A A

T x T x T x T x I x I x I x I x F x F x F x F x

(2) 1 2 1 2 1 21 2

A A A A A A A A T x T x I x I x F x F x

(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2

There are some limitations related to Definition 6 and these are now outlined

(1) In some situations operations such as 1 A A and 1 2 A A might be impractical This can be

demonstrated in the example below

Example 2 Let and1 050505 A 2 100 A

1 2 1 A A

2

be two SNSs Clearly is the larger of

these SNSs Theoretically the sum of any number and the maximum number should be equal to the maximum

one However according to Definition 6

2 100 A

0505 A therefore the operation ldquo+rdquo cannot be

accepted Similar contradictions exist in other operations of Definition 6 and thus those defined above are

incorrect

(2) The correlation coefficient of SNSs [36] which is based on the operations of Definition 6 cannot be

accepted in some special cases

Example 3 Let and be two SNSs and1 0800 A 2 0700 A 100 A be the largest one of the

SNSs According to the correlation coefficient of SNSs [36] 1 2 2 W A AW A A 1

1 A A

can be obtained but

this does not indicate which one is the best However it is clear that is superior to 2

(3) In addition the cross-entropy measure for SNSs [42] which is based on the operations of Definition 6

cannot be accepted in special cases


7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



SNSs According to the cross-entropy measure for SNSs [42] 1 1 2 2 S A A S A A 1 can be obtained

which indicates that 1 A is equal to 2 A Yet it is not possible to discern which one is the best Since

and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A

F x F x for any x in X it is clear that is superior to2 A

1 A

8

(4) If for any 1 A

I x 2 A x I x in X then and are both reduced to two IFSs However the

operations presented in Definition 6 are not in accordance with the laws of two IFSs [9-22]

1 A 2 A

Definition 7 [37] Let 1 2 n X x x x and1 A 2 A be two SNSs then is contained in1 A 2 A

ie if and only if 1 2 A A T x x 2 A x

1 2 A AT 1 A I x I and

1 2 A AF x F x for any x X

Obviously if the equal is not accepted then we have 1 2 A A

3 The novel operations comparison method and distance of SNNs

Subsequently the novel operations the comparison method and distance of SNSs are defined

Definition 8 Let and A 21 A A be three SNNs Then the operations of SNNs can be defined as follows

(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1

Theorem 1 Let be three SNNs then the following equations are true3nd A A A

(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C

Example 5 Let and1 060102 A 2 050304 A be two SNNs and 2 then we have following

results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A

Definition 9 The complement of an SNN A is denoted by which defined byC A

1 1 1C A A A A T I F for any x X

Definition 10 Let and1 A 2 A be two SNNs then 1 A A2 if and only if and 1 A A 2 2 1 A A

Based on the score function and accuracy function of IFNs (Xu 2007 2008 2010 Yager 2009) the score

function accuracy function and certainty function of an SNN are defined as follows

Definition 11 Let A A A

A T I F be an SNN and then the score function s A accuracy function

and certainty function of an SNN are defined as follows a A c A

(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T

The score function is an important index in ranking SNNs For an SNN A the bigger the truth-membership

T A is the greater the SNN will be furthermore the smaller the indeterminacy-membership I A is the greater the

SNN will be similarly the smaller the false-membership F A is the greater the SNN will be For the accuracy

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



function the bigger the difference between truth and falsity the more affirmative the statement is As for the

certainty function the certainty of any SNN positively depends on the value of truth- membership T A

On the basis of Definition 11 the method for comparing SNNs can be defined as follows

Definition 12 Let 1 A and 2 A be two SNNs The comparison method can be defined as follows

(1) If 1 2s A s A then is greater than denoted by 1 A 2 A 1 2 A A

(2) If 1 2s A s A and 1a A a A 2 then is greater than denoted by 1 A 2 A 1 2 A A

(3) If 1 2s A s A 1a A a A 2 and 1c A c A 2 then 1 A is greater than 2 A denoted by

1 2 A A

(4) If 1 2s A s A 1a A a A 2 and 1c A c A 2 then is equal to denoted by1 A 2 A 1 2 A A

Example 6 Based on Example 3 and Definition 11 1

08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A

2can be obtained According to Definition 12 1s A s A therefore

1 2 A A ie 1 A is greater than 2 A which avoids the drawbacks discussed in Example 3

Example 6 Based on Example 4 and Definition 11 1 2s A s A then 2 1 A A ie 2 A is greater than

which also avoids the shortcomings discussed in Example 41 A

Definition 13 Let j j j j A A A

A T I F and 12 j j j

j A A A A T I F j n

be two collections of SNNs

then the generalized simplified netrosophic normalized distance between j

A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n

gsnn j j j A A A A A A

j

d A A w T T I I F F n

(5)

If 1 then the generalized weighted simplified netrosophic normalized distance is reduced to the weighted

simplified neutrosophic normalized Hamming distance

1

1( )

3 j j j j j j

n


j


(6)

If 2 then the generalized weighted simplified netrosophic normalized distance is reduced to the

weighted simplified neutrosophic normalized Euclidean distance

10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)

4 Generalized simplified neutrosophic operators based on Choquet integral

In this section the aggregation operators of SNNs are introduced the corresponding properties are

discussed as well


A T I F be a collection of SNNs and 12 j n be a fuzzy measure

on X Based on fuzzy measure a simplified neutrosophic Choquet integral weighted averaging (SNCIWA)

operator of dimension is a mapping SNCIWA such thatn SNN SNNn

1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)

Here (1) (2) ( )n is a permutation of 12 n and such that (1) (2) ( )n A A A

( ) ( )( )i B i n and ( 1)n B

Theorem 2 Let j j j j A A A A T I F be a collection of SNNs and 12 j n be a fuzzy measure on

X Then their aggregated result using the SNCIWA operator is also an SNN and

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



Proof For simplicity let ( ) ( ) ( 1) j jw B B

j in the process of proof By using the mathematical

induction on n

(1) If based on the operations (1) and (3) in Definition 82n

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)

(2) If Eq (9) holds for thenn k

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F

If by the operations (1) and (3) in Definition 81n k

13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j

ie Eq (9) holds for Thus Eq (9) holds for all then1n k n

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

The proof is complete

Now some special cases of the SNCIWA operator is considered in the following

Let j j j j A A A

A T I F be a collection of SNNs and 12 j n be a fuzzy measure on X then

followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n

SNCIWA A A A A A A A n

(2) If for any 0 B B P X and B X then

1 2 1 2 (1) min n n

SNCIWA A A A A A A A

15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B

j 12 j Thus the SNCIWA operator is reduced to the

following simplified neutrosophic weighted averaging operator

n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w

for any B X where || || B is the number of the elements in B then

( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw

Thus the SNCIWA operator is reduced to the

following simplified neutrosophic ordered weighted averaging operator

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)

which was introduced by Peng et al [44]

Proposition 1 Let j j j j A A A


on X If all are equal ie 12 j

A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA

Proof Based on Theorem 2 if j A A

A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A


A T I F 12 j n be a collection of SNNs and be a fuzzy measure

on X If and

j j j

j A A A

A T I F j j

A A 12 j n then 1 2 nWA A A A

SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x

then it is a decreasing function If[01] x

( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T

n12 j Since ( 1) ( ) j j B B then

and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y

it is a decreasing function on [01] If(01] y

( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I

12 j n Since ( ) ( j j

B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F

According to Definition 7 can be obtained1 2( )nSNCIWA A A A

1 2( )

nSNCIWA A A A



on X If andmin max ma j j A A

j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x

and Then it is a decreasing function Since[01] x

( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B

( ) then and ( j B B

1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B Bn n n A A A j j

j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y

it is a decreasing function on [01] Since(01] y

( )min max

j j j A A A j j

I I I

12 j n then ( )max min

j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I

Since ( ) ( 1) j j B B 0 and

20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j

B B B B B Bn A A A j j

j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F

According to Definition 7 1 2 n

A SNCIWA A A A A



on X Based on fuzzy measure a simplified neutrosophic Choquet integral weighted geometric (SNCIWG)

operator of dimension is a mapping SNCIWG such thatn SNN SNNn

21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B

n nSNCI A A A A A AWG

(13)


( ) ( )( )i B i n and ( 1)n B

Theorem 3 Let j j j j A A A

A T I F

be a collection of SNNs and 12 j n be a fuzzy measure on

X Then their aggregated result using the SNCIWG operator is also an SNN and

(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B

Proof Theorem 3 can be proved by the mathematical induction method and the process is omitted here

Now letrsquos consider some special cases of the SNCIWG operator in the following

Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n

SNCIWG A A A A A A A n


1 2 1 2 (1) min n n

SNCIWG A A A A A A A

22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j

nand 12 j Thus the SNCIWG operator is reduced to the

following simplified neutrosophic geometric averaging operator

1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n

Here and 1 2 nw w w w 0iw 12 j n 1 1n

ii w Thus the SNCIWG operator is reduced to the

following simplified neutrosophic ordered geometric averaging operator

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A

A A T I F A 12 j n then

1 2 SNCI A A n

A AWG

Proof The proof is omitted here

23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F

j j A A 12 j n then 1 2

nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A

F max min min j j A A

j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S


5 Choquet integral-based TOPSIS approach of MCGDM with simplified neutrosophic information

Assume there are n alternatives 1 2 n A a a a and m criteria 1 2 mC c c c and the weight vector

of criteria is 1 2 m

w w w w where 0 12 j

w j m 1

1m

j

j

w

Suppose that there are

decision-makers

k

1 2 k D d d d whose corresponding weight is k 1 2 Let k ij R ak

n m

be the simplified neutrosophic decision matrix where I k k ij ij

k

ij a aa T k

ijaF

ia

is the value of a criterion denoted

by SNNs where indicates the truth-membership function that alternative satisfies criterionk ija

T jc

ia

for

the k-th decision-maker indicates the indeterminacy-membership function that alternative satisfies

criterion

k ija

I

jc for the k-th decision-maker and k

ijaF indicates the falsity-membership function that alternative

satisfies criterionia j

c for the k-th decision-maker This method is an integration of SNSs and aggregation

operators to solve MCGDM problems mentioned above

The method is an integration of SNSs and the TOPSIS method to handle MCGDM problems mentioned

above In general there are benefit criteria and cost criteria in MCGDM problems The cost-type criterion

values can be transformed into benefit-type criterion values as follows

for benefit criterion

for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



Here is the complement of as defined in Definition 7 c

ija

ija

In the following a procedure to rank and select the most desirable alternative(s) is given

Step 1 Transform the decision matrix

For each criterion can be divided into two types including benefit-type which means the lager the better

and cost-type which means the smaller the better For the benefit-type criteria nothing is done for the

cost-type criteria the criterion values can be transformed We can transform the SNS decision matrix

k k

ij n m R a

into a normalized SNS decision matrix k k

ij n m R b

based on Eq (18)

Step 2 Confirm the fuzzy measures and expert sets of D

Based on the fuzzy measures and expert sets of D the weight of criteria can be obtained as follows

( ) ( ) ( 1) j jw B B j

12 i m

Here (1) (2) ( )n is a permutation of 12 n

Step 3 Aggregate all the decision-makersrsquo values to get the collective simplified neutrosophic decision

matrix

Utilize the SNCIWA operator and SNCIWG operator to aggregate the SNNs of each decision-maker and

we can get the collective simplified neutrosophic decision matrix ijn m

R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27

1 2 12 i i i ima b b b i n and the simplified neutrosophic positive-ideal solution b can be

calculated respectively

( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)

Where ( ) ij ij ij j j j

i j ij j b b bb bT T I I F b

bF d b

and (1) (2i ij j i ij j

d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)

Where ( ) ij ij ij j j

i j ij j b b bb bd b b T T I I F F

jb and (1) (2i ij j i j j

d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

Step 7 Calculate the closeness coefficient of each alternative

Based on Step 6 the closeness coefficient of each alternative can be obtained as follows

12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)

Step 8 Rank the alternatives

According to the closeness coefficients iG a the smaller the value i

G a the better the alternative

ia

12 i n

6 Illustrative examples (adapted from [62])

In this section an example for the MCDM problem with simplified neutrosophic information is used as the

demonstration of the application of the proposed decision-making method as well as the comparison analysis

ABC Nonferrous Metals Holding Group Co Ltd is a large state-owned company whose main business is

producing and selling nonferrous metals It is also the largest manufacturer of multi-species nonferrous metals

in China with the exception of aluminum In order to expand its main business the company is always

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



engaged in overseas investment and a department which consists of executive managers and three experts in

the field has been established specifically to make decisions on global mineral investment Recently the

company has decided to select a pool of alternatives from several foreign countries based on preliminary

surveys In this survey the focus is on the first step in finding suitable candidate countries Four countries

(alternatives) are taken into consideration which are denoted by and During the assessment

four factors including politics and policy (such as the support of government) infrastructure (such as

railway and highway facilities) are considered according to previous investment examples from the

department resources (such as the suitability of the minerals and their exploration) economy (such

as development vitality and the stability) The decision-makers can provide their evaluations about the project

under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia

jc in the form of SNNs

ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which

represents their degrees of satisfaction indeterminacy and dissatisfaction regarding an alternative by using the

concept of ldquoexcellentrdquo against each criterion The simplified netrosophic decision matrix k

ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201

61 An illustration of the proposed approach

The procedures of obtaining the optimal alternative by using the developed method are shown as following

Step 1 Normalize the data in Table 1 Because all the criteria are of maximizing type and have the same

measurement unit there is no need for normalization and 4 4 4 4( ) ( )ij ij

R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



Step 2 Determine the fuzzy measure

Determine the fuzzy measure of expert of and expert sets ofK 1 2 3 K k k k Suppose that

1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9

Step 3 Aggregate all the decision-makersrsquo values to get the collective simplified neutrosophic deision matrix

Utilize the SNCIWA operator to aggregate the SNNs of each decision-maker According to Eq (20) the

collective simplified neutrosophic decision matrix can be obtained as follows

046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b

Take for example based on Definition 11 the detail compute process are as follows

1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085

Step 4 Confirm the simplified neutrosophic positive-ideal solution and the negative-ideal solution

29

Based on the collective simplified neutrosophic decision matrix R and Eq (21) the following result can be

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b

Step 5 Confirm the fuzzy measures of criteria of and the criteria sets of C C

Based on the fuzzy measures of criteria of and the criteria sets ofC 1 2 3 4 C c c c c suppose that

1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c

Step 6 Calculate the distance

Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b

Since 14 14 4 12 12 2 13 13 3 11 11 1 d b b d b b d b b d b b and

1 (1) 14 14 4 1 (2) 12 12 2 1 (3) 13 13 3 1 (4) 11 11 1 ij j ij j ij j ij jd b b d b b d b b d b b d b b d b b d b b d b b

so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3

gsnnd a b d b b B B d b b B B

d b b B B d b b B B

Similarly the following results can be obtained

11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450

gsnn gsnnd a b d a b

4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a


Since 4 2 3G a G a G a G a

1a

1 1 So the final ranking is and the best alternative is

while the worst alternative is

4 2 3a a a a

4a

If the SNCIWG operator is utilized in Step 3 then the collective simplified neutrosophic decision matrix can

be obtained as follows

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34

measures that have similar characteristics which always need to a large amount of computation Finally the

result using the method of Peng et al [45] is consistent with that of the proposed approach Although the

method was constructed based on reliable theories and may be robust to some degree they cannot consider the

correlation of criteria

From the analysis presented above it can be concluded that the main advantages of the approach developed

in this paper over the other methods are not only due to its ability to effectively overcome the shortcomings of

the compared methods but also due to its ability to consider the criteria are interactive This can avoid losing

and distorting the preference information provided which makes the final results better correspond with real

life decision-making problems

7 Conclusion

SNSs can be applied in solving problems with uncertain imprecise incomplete and inconsistent

information that exist in scientific and engineering situations Based on related research achievements in IFSs

the novel operations of SNSs were defined in this paper and two aggragation operators are developed based

on Choquet integral their desirable properties were discussed in detail Thus an MCDM method was

established based on Choquet integral aggregation operators and TOPSIS method By using the proposed

method the ranking of all alternatives can be determined and the best one can easily be identified One

illustrative example demonstrated the applicability of the proposed decision-making method The advantage

of this study is that an outranking approach for MCDM problems with SNSs could overcome the

shortcomings of existing methods as we discussed earlier Moreover the proposed approach could consider

the interactive of criteria with simplified neutrosophic formation The comparison analysis also showed that

the final result produced by the proposed method is more precise and reliable than the results produced by

existing methods In the further research we will continue to study the distance measures of SNNs

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Nos 71271218 and

71221061) and the Science Foundation for Doctors of Hubei University of Automotive Technology

(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35

[1] LA Zadeh Fuzzy sets Information and Control 8 (1965) 338-356

[2] R Bellman LA Zadeh Decision making in a fuzzy environment Management Science 17 (1970)

141-164

[3]

RR Yager Multiple objective decision-making using fuzzy sets International Journal of Man-MachineStudies 9 (1997) 375-382

[4] LA Zadeh Fuzzy logic and approximate reasoning Synthese 30 (1975) 407-428

[5] W Pedrycz Fuzzy sets in pattern recognition methodology and methods Pattern Recognition 23 (1990)

121-146

[6] K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems 20 (1986) 87-96

[7] WL Gau DJ Buehrer Vague sets IEEE Transactions on Systems Man and Cybernetics 23 (1993)

610-614

[8]

H Bustince P Burillo Vague sets are intuitionistic fuzzy sets Fuzzy Sets and Systems 79 (1996)

403-405

[9] HW Liu GJ Wang Multi-criteria methods based on intuitionistic fuzzy sets European Journal

Operational Research 179 (2007) 220-233

[10] Z Pei L Zheng A novel approach to multi-attribute decision making based on intuitionistc fuzzy sets

Expert Systems with Applications 39 (2012) 2560-2566

[11]

YT Chen A outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy

optimisticpessimistic operators Expert Systems with Applications 37 (2010) 7762-7774

[12]

SZ Zeng WH Su Intuitionistic fuzzy ordered weighted distance operator Knowledge-based Systems

24 (2011) 1224-1232

[13] ZS Xu Intuitionistic fuzzy multiattribute decision making an interactive method IEEE Transactions on

Fuzzy Systems 20 (2012) 514-525

[14] JQ Wang RR Nie HY Zhang XH Chen Intuitionistic fuzzy multi-criteria decision-making method

based on evidential reasoning Applied Soft Computing 13 (2013) 1823-1831

[15] JQ Wang HY Zhang Multi-criteria decision-making approach based on Atanassovs intuitionistic

fuzzy sets with incomplete certain information on weights IEEE Transactions on Fuzzy Systems 21 (3)

(2013) 510-515

[16]

JQ Wang RR Nie HY Zhang XH Chen New operators on triangular intuitionistic fuzzy numbers

and their applications in system fault analysis Information Sciences 251 (2013) 79-95

[17] L Li J Yang W Wu Intuitionistic fuzzy hopfield neural network and its stability Expert Systems

Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36

[18] S Sotirov E Sotirova D Orozova Neural network for defining intuitionistic fuzzy sets in e-learning

NIFS 15 (2009) 33-36

[19]

TK Shinoj JJ Sunil Intuitionistic fuzzy multisets and its application in medical fiagnosis International

Journal of Mathematical and Computational Sciences 6 (2012) 34-37

[20]

T Chaira Intuitionistic fuzzy set approach for color region extraction Journal of Scientific amp Industrial

Research 69 (2010) 426-432

[21] T Chaira A novel intuitionistic fuzzy C means clustering algorithm and its application to medical images

Applied Soft Computing 11 (2011) 1711-1717

[22] BP Joshi S Kumar Fuzzy time series model based on intuitionistic fuzzy sets for empirical research in

stock market International Journal of Applied Evolutionary Computation 3 (2012) 71-84

[23]

K T Atanassov G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems 31 (1989)343-349

[24] V Torra Hesitant fuzzy sets International Journal of Intelligent Systems 25 (2010) 529-539

[25]

V Torra Y Narukawa On hesitant fuzzy sets and decision The 18th IEEE International Conference on

Fuzzy Systems Jeju Island Korea pp 1378-1382 2009

[26] B Zhu ZS Xu MM Xia Dual hesitant fuzzy sets Journal of Applied Mathematics doi

org1011552012879629 2012

[27]

B Zhu ZS Xu Some results for dual hesitant fuzzy sets Journal of Intelligent and Fuzzy Systems 26(2014) 1657ndash1668

[28]

N Chen ZS Xu MM Xia Interval-valued hesitant preference relations and their applications to group

decision making Knowledge-Based Systems 37 (2013) 528ndash540

[29] B Farhadinia Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets

International Journal of Intelligent Systems 29 (2014) 184ndash205

[30]

JJ Peng JQ Wang J Wang XH Chen Multi-criteria decision-making approach with hesitant

interval-valued intuitionistic fuzzy set The Scientific World Journal 2014 Article ID 868515 22 pages

[31] H Wang F Smarandache YQ Zhang and R Sunderraman Single valued neutrosophic sets Multispace

and Multistructure 4 (2010) 410-413

[32] F Smarandache A unifying field in logics neutrosophy logic Philosophy pp 1-141 1999

[33] F Smarandache A unifying field in logics neutrosophic logic Neutrosophy neutrosophic set

neutrosophic probability neutrsophic logic Neutrosophy neutrosophic set neutrosophic probability

Infinite Study 2005

[34] U Rivieccio Neutrosophic logics prospects and problems Fuzzy Sets and Systems 159 (2008)

1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37

[35] P Majumdar SK Samant On similarity and entropy of neutrosophic sets Journal of Intelligent and

Fuzzy Systems 26 (3) (2014) 1245-1252

[36]

J Ye Multicriteria decision-making method using the correlation coefficient under single-value

neutrosophic environment International Journal of General Systems 42 (4) (2013) 386-394

[37]

J Ye A multicriteria decision-making method using aggregation operators for simplified neutrosophic

sets Journal of Intelligent and fuzzy Systems 26 (5) (2014) 2459-2466

[38] H Wang F Smarandache YQ Zhang and R Sunderraman Interval neutrosophic sets and logic Theory

and applications in computing Hexis Phoenix AZ 2005

[39] F G Lupiaacutentildeez Interval neutrosophic sets and topology Kybernetes 38(3-4) (2009) 621-624

[40] S Broumi F Smarandache Correlation coefficient of interval neutrosophic set Applied Mechanics and

Materials 436 (2013) 511-517

[41]

HY Zhang JQ Wang XH Chen Interval neutrosophic sets and their application in multicriteria

decision making problems The Scientific World Journal Volume 2014 Article ID 645953 15 pages

[42] J Ye Single valued neutrosophic cross-entropy for multicriteria decision making problems Applied

Mathematical Modeling 38 (3) (2014) 1170-1175

[43] J Ye Similarity measures between interval neutrosophic sets and their applications in multicriteria

decision-making Journal of Intelligent and Fuzzy Systems 26 (1) (2014) 165-172

[44]

JJ Peng JQ Wang J Wang HY Zhang XH Chen Simplified neutrosophic sets and their

applications in multi-criteria group decision-making problems International Journal of Systems Science

In Press 2015

[45] JJ Peng JQ Wang HY Zhang XH Chen An outranking approach for multi-criteria decision-making

problems with simplified neutrosophic sets Applied Soft Compution In Press 2014

[46] G Choquet Theory of capacities Annales de lrsquoinstitut Fourier 5 (1953) 131ndash295

[47] RR Yager Induced aggregation operators Fuzzy Sets and Systems 137(2003)59ndash69

[48] P Meyer M Roubens On the use of the Choquet integral with fuzzy numbers in multiple criteria

decision support Fuzzy Sets and Systems 157(2006) 927ndash938

[49]

DJ Yu YY Wu W Zhou Multi-criteria decision making based on Choquet integral under hesitant

fuzzy environment Journal of Computational Information Systems 7(12) (2011) 4506ndash4513

[50] CQ Tan XH Chen Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making

Expert Systems with Applications 37 (2010)149ndash157

[51] CQ Tan A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet

integral-based TOPSIS Expert Systems with Applications 38 (2011) 3023ndash3033

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



[52] H Bustince J Fernandez J Sanz M Galar R Mesiar A Kolesaacuterovaacute Multicriteria decision making

by means of interval-valued Choquet integrals Advances in Intelligent and Soft Computing 107 (2012)

269ndash278

[53] GW Wei XF Zhao R Lin HJ Wang Generalized triangular fuzzy correlated averaging operator and

their application to multiple attribute decision making Applied Mathematical Modelling 36 (2012)

2975ndash2982

[54] J-Q Wang D-D Wang H-Y Zhang X-H Chen Multi-criteria group dcision making method based

on interval 2-tuple linguistic and Choquet integral aggregation operators Soft Computing

doi101007s00500-014-1259-z 2014

[55]

CL Hwang K Yoon Multiple attributes decision making methods and applications Berlin Heidelberg

Springer 1981

[56]

CT Chen Extensions of the TOPSIS for group decision-making under fuzzy environment Fuzzy Setsand Systems 114 (2000) 1ndash9

[57] I Mahdavi N Mahdavi-Amiri A Heidarzade R Nourifar Designing a model of fuzzy TOPSIS in

multiple criteria decision making Applied Mathematics and Computation 206 (2008) 607ndash617

[58]

TY Chen CY Tsao The interval-valued fuzzy TOPSIS method and experimental analysis Fuzzy Sets

and Systems 159 (2008) 1410ndash1428

[59] B Ashtiani F Haghighirad A Makui G Montazer Extension of fuzzy TOPSIS method based on

interval-valued fuzzy sets Applied Soft Computing 9 (2009) 457ndash461

[60]

F RL Junior L Osiro LCR Carpinetti A comparison between fuzzy AHP and fuzzy TOPSIS

methods to supplier selection Applied Soft Computing21 (2014) 194ndash209

[61]

Z Wang GJ Klir Fuzzy measure theory Plenum press New York 1992

[62] JQ Wang JJ Peng HY Zhang T Liu and X-H Chen An uncertain linguistic multi-criteria group

decision-making method based on a cloud model Group Decision and Negotiation 2014 doi

101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



recognition [5]

However FSs can not handle some cases that the membership degree is hard to be defined by one specific

value In order to overcome the lack of knowledge of non-membership degrees Atanassov [6] introduced

intuitionistic fuzzy sets (IFSs) the extension of Zadehrsquos FSs In addition Gau and Buehrer [7] defined vague

sets Later Bustince [8] pointed out that the vague sets and IFSs are mathematically equivalent objects At

present IFSs have been widely applied in solving MCDM problems [9-16] neural networks [17 18] medical

diagnosis [19] color region extraction [20 21] and market prediction [22] IFSs take into account the

membership degree the non-membership degree and the degree of hesitation simultaneously So it is more

flexible and practical in addressing fuzziness and uncertainty than the traditional FSs Moreover in some

actual cases the membership degree the non-membership degree and the hesitation degree of an element in

IFSs may be not a specific number Hence it was extended to the interval-valued intuitionistic fuzzy sets

(IVIFSs) [23] To handle the situations that people are hesitant to express their preference over objects in a

decision-making process hesitant fuzzy sets (HFSs) were introduced by Torra and Narukawa [24 25] Zhu et

al [26 27] proposed dual HFSs and outlined their operations and properties Furthermore Chen et al [28]

proposed interval-valued hesitant fuzzy sets (IVHFSs) and applied it to MCDM problems Then Farhadinia

[29] discussed the correlation for dual IVHFSs and Peng et al [30] introduced a MCDM approach with

hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) which is an extension of dual IVHFSs

Although the FSs theory has been developed and generalized it can not deal with all sorts of uncertainties

in different real problems Some types of uncertainties such as the indeterminate information and inconsistent

information can not be handled For example when we ask the opinion of an expert about a certain statement

he or she may say the possibility that the statement is true is 05 the one that the statement is false is 06 and

the degree that he or she is not sure is 02 [31] This issue is beyond the scope of the FSs and IFSs Therefore

some new theories are required

Smarandache [32 33] proposed neutrosophic logic and neutrosophic sets (NSs) Rivieccio [34] pointed out

that an NS is a set where each element of the universe has a degree of truth indeterminacy and falsity

respectively and it lies in ]0 1 [ the non-standard unit interval Obviously it is the extension of the standard

interval of IFSs And the uncertainty presented here ie indeterminacy factor is dependent on of truth

and falsity values while the incorporated uncertainty is dependent of the degree of belongingness and degree

[0 1]

2

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



3





























1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





















7 is the conclusion

2 Preliminaries




Let 1 2 n




following axioms

4

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) 0 1 X

(2) if 1 2 B B X then 1 2 B B





fori

B X i i

i

x B

i B x

(1)





on

[0 ) f X


0 X fd x f x t

dt




( ) ( ) ( 1)1

n

i i i X

i

fd f x B B

i

(2)

or

( ) ( 1) ( )1

n

i i X

i

fd f x f x B

(3)


(1) ( )0n

f x x (2 ) f x f

(0) 0 f x

( ) ( ) ( 1)

i i i B x x

( )n

x

and ( 1) 0n

B



5

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





2

6

1 (0) 1 2 1 2 3 3 X

fd


22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs





F x x A

T A I x and A



T x X ]0 1 X [ A

I x and



A I (

A)F x so


I x

1 A

3 A

F x


1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and






A I and A

F x

[01]



I x

and Then a


[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



3





























1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





















7 is the conclusion

2 Preliminaries




Let 1 2 n




following axioms

4

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) 0 1 X

(2) if 1 2 B B X then 1 2 B B





fori

B X i i

i

x B

i B x

(1)





on

[0 ) f X


0 X fd x f x t

dt




( ) ( ) ( 1)1

n

i i i X

i

fd f x B B

i

(2)

or

( ) ( 1) ( )1

n

i i X

i

fd f x f x B

(3)


(1) ( )0n

f x x (2 ) f x f

(0) 0 f x

( ) ( ) ( 1)

i i i B x x

( )n

x

and ( 1) 0n

B



5

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





2

6

1 (0) 1 2 1 2 3 3 X

fd


22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs





F x x A

T A I x and A



T x X ]0 1 X [ A

I x and



A I (

A)F x so


I x

1 A

3 A

F x


1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and






A I and A

F x

[01]



I x

and Then a


[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





















7 is the conclusion

2 Preliminaries




Let 1 2 n




following axioms

4

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) 0 1 X

(2) if 1 2 B B X then 1 2 B B





fori

B X i i

i

x B

i B x

(1)





on

[0 ) f X


0 X fd x f x t

dt




( ) ( ) ( 1)1

n

i i i X

i

fd f x B B

i

(2)

or

( ) ( 1) ( )1

n

i i X

i

fd f x f x B

(3)


(1) ( )0n

f x x (2 ) f x f

(0) 0 f x

( ) ( ) ( 1)

i i i B x x

( )n

x

and ( 1) 0n

B



5

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





2

6

1 (0) 1 2 1 2 3 3 X

fd


22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs





F x x A

T A I x and A



T x X ]0 1 X [ A

I x and



A I (

A)F x so


I x

1 A

3 A

F x


1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and






A I and A

F x

[01]



I x

and Then a


[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



(1) 0 1 X

(2) if 1 2 B B X then 1 2 B B





fori

B X i i

i

x B

i B x

(1)





on

[0 ) f X


0 X fd x f x t

dt




( ) ( ) ( 1)1

n

i i i X

i

fd f x B B

i

(2)

or

( ) ( 1) ( )1

n

i i X

i

fd f x f x B

(3)


(1) ( )0n

f x x (2 ) f x f

(0) 0 f x

( ) ( ) ( 1)

i i i B x x

( )n

x

and ( 1) 0n

B



5

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





2

6

1 (0) 1 2 1 2 3 3 X

fd


22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs





F x x A

T A I x and A



T x X ]0 1 X [ A

I x and



A I (

A)F x so


I x

1 A

3 A

F x


1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and






A I and A

F x

[01]



I x

and Then a


[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





2

6

1 (0) 1 2 1 2 3 3 X

fd


22 037 x

438

312 045 2 x x

X fd

1 20 1 2 x x

32 3 x x

2

1 21 x

If then we have

22 NSs and SNSs





F x x A

T A I x and A



T x X ]0 1 X [ A

I x and



A I (

A)F x so


I x

1 A

3 A

F x


1 2 A A

T xinf inf T x 1 2

sup A A

T x T sup x 1 2

inf A A

I xinf I x 1 2

sup A A

I xsup I x

inf i A 1 2

nf AF x F x and






A I and A

F x

[01]



I x

and Then a


[0 AF x X 1]

A

| A

x F x x A A

A x T x I X (4)

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59







1 2 1 2 1 2 1 2 1 2 1 2

1 2(1)


A A


(2) 1 2 1 2 1 21 2


(3) 1 1 1 1 1 1 0 A A A

A T x I x F x

(4) A A A

A T x I x F x 0

2





1 2 1 A A

2




2 100 A



incorrect





1 A A

can be obtained but





7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





and 2 1 A A

T x T x 2 A

I x 1 A

I x 2 1 A A


1 A

8

(4) If for any 1 A



1 A 2 A









(1)

2 2 0

2

A A

e

A A A

I A

I I F F

1 1

1 1

A A

A A

T T

T T

2 A

F

(2)

2 1 1 1 1

2 1 1 1

e A A A

A A A A A

I F A

T T I I F F

A A

T I 0

1 A

F

(3)

1 2 2 1

1 2 1 2 2

1 2 1 1 1 1 1 1

A A A A

e

A A A A A

I F A A

T T I I F F

1 A

I

2

11

A

A

F

T T

(4)

1 2 1 2 1 2

1 2 1 21 2

1 2 1 11 1

1 2 a

1

A A A A A A

e

A A A A A A

T T I I F F A A

I I F F T T

1


(1) 1 2 2 A A A A

A

0 A

(2) 1 2 2 A A A 1

(3) A B B

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



(4) 0 A B A B

9

0

0

(5) 1 2 1 2 1 2 0 A A A

(6)

1 2 1 2( )

1 2 0 A A A

(7) A B C A B C

(8) A B C A B C


results

(1)

2 2 2

12 1 1 06 01 02 084001004 A

(2) 2 22 2

1 06 1 1 01 1 1 02 036019036 A

(3) 1 2 06 05 06 0501 03 02 04 080003 008 A A

(4) 1 2 06 0501 03 01 0302 04 02 04 030037052 A A









(1) 1 1 A A As A T I F 3

(2) A Aa A T F

(3) Ac A T




1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59










1 2 A A



08 1 0 1 0 28

3 3s A

and

2

07 1 0 1 0 27

3 3s A







j A A A A T I F j n



A and j

A can be defined as

follows

1

1

1( )

3 j j j j j j

n


j


(5)



1

1( )

3 j j j j j j

n


j


(6)



10

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



1

22 2

1

1( )

3 j j j j j j

n


j


2

(7)



discussed as well





1 2

(1) (2) (1) (2) (3) (2) ( ) ( 1) ( )

n

n n

SNCIWA A A A

B B A B B A B B A

n

(8)


( ) ( )( )i B i n and ( 1)n B



( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F

(9)


( ) ( )( ) j B j n and ( 1)n B

11

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





induction on n


(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1) (2) (2)

(1) (1)

(1) (1) (2

1 1 1 1

1 1 1 1

1 1 1 11

1 1 1

w w w w

A A A A

w w w w

A A A A

w w w w

A A A A

w w

A A A

T T T T

T T T T

T T T T

T T T

(2) (2)

) (2)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1) (2) (2) (2) (2) (1) (1)

(1) (1)

(1) (1) (2)

1

1 1 1 1 1 1 1 1

1 1 1

w w

A

w w w w w w w

A A A A A A A A

w w w

A A A

T

T T T T T T T T

T T T

w

(2) (2) (1) (1) (2) (2)

(2) (1) (1) (2) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1)

(1) (2) (1) (2

1 1 1 1 1

2 1 1 2 1 1

2 1 1 2 1 1

w w w w

A A A A A

w w w w

A A A A

w w w

A A A A

T T T T T

T T T T

T T T T

w

(2)

)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

1 1 1 1

1 1 1 1

w

w w w w

A A A A

w w w w

A A A A

T T T T

T T T T

and

(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

I I

I I I I

I I I

I I I I I

(2)

(2 )

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

(1) (2)

(1) (2)

(1) (2) (1)

(1) (2) (1) (2)

2

2

4

2 2 2 2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

w w

A A

w w w w

A A A A

I

I I I

I I

I I I I

I I

I I I I

w

(2)

Similarly

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



(1) (2)

(1) (2)

(1) (1) (2) (2)

(1) (1) (1) (2)

(1) (1) (1)

(1) (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

2 2

2 2

2 2 22 2

2 2 2

w w

A A

w w w w

A A A A

w w w

A A A

w w w w w

A A A A A

F F

F F F F

F F F

F F F F F

(2)

(2)

(1) (2) (2)

(1) (2) (2)

(1) (2)

(1) (2)

(1) (2) (1) (2)

(1) (2) (1) (2)

2

2

2

2 2

w

A

w w

A A A

w w

A A

w w w w

A A A A

F

F F F

F F

F F F F

w

So

(1) ( 2 ) (1) ( 2)

(

(1) ( 2 ) (1) ( 2)

(

1) ( 2) (1) ( 2)

(1) ( 2 )

(1) (

1) ( 2 ) (1) ( 2)

(1) ( 2 )

2 ) (1

(1) ( 2) (1) (

) ( 2)

2)

1 2

1 1 1 1

1 1 1 1

2 2

2

A A A A

A A A A

A A

A A A A

w w w w

w w w w

w w

w w w w

T T T T

T T

SNCIW

T T

I I

I I I

A A A

I

(1) ( 2)

(

(1) ( 2)

(1) ( 2 ) (

1) ( 2) (1

1) (

) (

2)

2

2 2

2w w

w w

A

w

A

A A A A

w

F F

F F F F

)


( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

1 2

1 1

1 1

1

1 1

1

1 1

1 1

1 1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j j

j j

k k

A A

j j

k k

A A

j j

k

A

j

w w

k w

k k

A A

j j

k

A

w

w

w w

w

w

i

k wk

A A

j j

T T

T

SNCIWA A A A

T

I

I I

F

F F


13

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( )( 1) ( 1)

( ) ( )( 1) ( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( 1) ( 1)

( 1) ( 1)

( 1)

( 1)

1 1

1 1

1 11 1

1 1 1 1

1 111

k k

k k

k k

k k

k k k k

k k k k

k k

k k

k

k

k k w ww w

A A A A j j

w w k k w w

A A A A

j j

w w

A A

w

A

T T T T

T T T T

T T

T

( ) ( )

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( )

( ) ( 1)

1 1

1 1

1 1

1

1 1

1 1 1

2 1 1 2 1 1

2 1 1

k k

k k

k k k

k k k

k k k

k k k k

k

k k

k k w w

A A

j j

w k k w w

A A A

j j

k k w w w

A A A A

j j

k w w

A A

j

T T

T T T

T T T T

T T

k w

( 1) ( ) ( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

2 1 1

1 1 1 1

1 1 1 1

k k

k k

k k k

k k k k

k k k

k k k k

k w w

A A

j

k k w w w w

A A A A

j j

k k w w w w

A A A A j j

T T

T T T T

T T T T

k

k

k

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

I I

I I I I

I I

I I I

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( ) ( 1)

( ) ( 1)

( ) (

( ) ( 1)

1

1 1

1

1

22

2 2

4

2 2 2

k

k

k

k

k k k k k

k k k k k

j k

j k

j k

j k

k ww

A A j

k w w k k w w w

A A A A A

j j

k

w w A A

j

k w w

A A

j

I I

I I I I

I I

I I

1 j

I

1) ( ) ( 1)

( ) ( 1)

( )

( )

( ) ( )

( ) ( )

1

1

1

1 1

1 1

2

2

2

j k

j k

j

j

j j

j j

k w w

A A

j

k w

A

j

k k w w

A A

j j

I I

I

I I

Similarly

( )( 1)

( )( 1)

( 1) ( 1) ( ) ( )

( 1) ( 1) ( ) ( )

( )( 1)

( )( 1)

( 1) ( 1) ( )

( 1) ( 1) ( )

1

1 1

1

1

22

2 2

222

2 2

k

k

k

k

k k k k

k k k k

k

k

k

k

k k k

k k k

k ww

A A j

w w k k w w

A A A A

j j

k ww

A A j

w w k w

A A A

j

F F

F F F F

F F

F F F

( )( 1)

( )( 1)

( 1) ( 1)( ) ( ) ( )

( 1) ( 1)( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

1

1 1

1

1

1 1

1 1

22

2 2

2

2

k

k

k

k

k k k k k

k k k k k

j

j

j j

j j

k ww

A A j

k w w k k w w w

A A A A A

j j

k w

A

j

k k w w

A A

j j

F F

F F F F

F

F F

1 j

F

14

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



So

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

1 1

1 1

1 1

1 1

1 2

1

1

1 1

1 1

1

1

1

1 1

1

1

2

2

2

2

j j

j j

j j

j j

j

j

j j

j j

j

j

j

j

k k

A A

j j

k k

A A j j

k

w w

k k w w

w

w w

w

A

j

k k

A A

j j

k

A

A

w

i

T T

T T

I

I

SNCIWA A

F

F

A

I

A A

F

( )

( )

1 1

1 1

j

j

k k

A

j

w

j


( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( )

(

) ( 1)

( ) ( 1)

( ) ( 1) )

) ( )

(

1 1

1 1

1

1 2

1 1

1

2

2

1

j j j j

j

j

j

j j

j j

j j

j j

j

j

j

j

Bn n

A A

j j

n n

A A

j j

B B

n B B B B

B B

B

n

B

A

B

j

B

A A

T T

T T

I

SNCIWA A A A

I I

j

B

( )

( ) (

( 1)

( ) ( 1)

( ) ( 1) ( ) ( 1

)

)

1 1

1

1 1

2

2

j

j

j

j j

j j j j

j

n n

j j

n

A

i

n n

A A

j j

B B

B B B B

F

F F



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


15

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



16

(3) 1 2 B B P X 1 2| | | | B B if 1 2 B B and ( )

112

i

n i B i n

n

then

1 2

1

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1 1

1 1

21 1

1 1

2

2

2

j j j j

j j j j j

n

n

n n n n

A A An n n

n n n n n

A j j j i

n n n n n n

A A A A A A

j j j j j j

n

T T I F

T T I

SNCIWA A A A

I F F

j

(10)

(4) If ( ) ( ) ( 1) j j x B B



n

1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1

( )

1

1 1

1

2 2

1 2 2

j j j j

j j j j

j j j j

j

j

j j

j

j j

w n

x x x x

x x x x x

n n n n

A A A A

j j j j

n n n n n n

A A A A A

x

A

j j j j j j

S

T T I F

T T I

NWA A A A

I F F

j

(11)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B

12 j n

Here and 1 2 n

w w w w 0 12 j n i

w 1

1n

iiw



( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1 1 1

1 2

1 1

1 1

2

2

2

2

j j j j

j j j

j j j j

j j j j

j j

j

j

n n n n

A A A

w n

w w w w

w

A j j j i

n n n n n n

A A A A A A

j j j

w w w w

j j

w

j

T T I F

T

SNOW

T I I

A

F

A A A

F

j

(12)





A j n j A A

A A T I F A

12 j n then

1 2( SNCI A A )

n A AWA


A A T I F A

12 j n then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( 1) ( ) ( 1)

1 1

( ) ( 1) ( ) ( 1)

1 1

( ) ( 1)

1

( ) ( 1) ( ) ( 1)

1 1

( )

1 2

2

1 1

1 1

2

2

n n

j j j j

j j

n n

j j j j

j j

n

j j

j

n n

j j j j

j j

j

B B B B

n B B B B

B B

B B B B

B

A A

A A

A

A A

A

T T SNCIWA A A

T

F

A

T

I

I I

( 1)

1

( ) ( 1) ( ) ( 1)

1 1

2

n

j

j

n n

j j j j

j j

B

B B

A A

B BF F

Since ( ) ( 1)1

1n

j j

j

B B

So

1 2

2 2

1 1

1 1 2 2

A A A A

A A A A A A

A A A

w nSNCIWA A

T T I F

T T I I F A A

F

T I F A



on X If and

j j j

j A A A

A T I F j j


SNCI

1 2 n

A AS WA A NCI

Proof If j j

A A then 12 j n ( ) ( ) j j

A A ie

( ) ( ) j j A A

T T

( ) ( ) j j

A A I I

and ( ) ( ) j j

A AF F

Let 1

1

x f x

x


( ) ( ) j j A A

T T

n12 j then

( )( ) j j A A 12 f T f T j n ie

( ) ( )

( )( )

1 1

1 1 j j

j j

A A

A A

T T

T T


and ( ) ( j B B

1) 0

j 1

n

j

( ) ( 1) 1 j j

B B

So

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

j j j j

j j

j j

B B B B

A A

A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 1

j j j j

j j

j j

B B B B

n n A A

j j A A

T T

T T

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

1 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 2

1 11 11 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T T T

( ) ( 1) ( ) ( 1)

( ) ( )

( )( )

1 1

2 21 1

1 11 1

1 1

j j j j

j j

j j

B B B Bn n

A A

j A j A

T T

T T

ie

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( ) ( ) ( )

( ) ( 1)( ) ( 1) ( ) ( 1)

( ) (

)(

)

1 1 1 1

1 1

1 1 1 1

1 1 1

j j j j j j j j

j j j j

j j j j j j

j j j

B B B B B B B B

B B B B B B

n n n n

A A A A j j j j

n n

A A A j j j

T T T T

T T T

( ) ( )

1

( )1 1

1 j j

j

n B

j

Bn

AT

Let2

( ) y

g y y


( ) ( ) j j A A

I I

12 j n then

ie ( ) ( ) j j

A Ag I g I

( ) ( )

( ) ( )

2 2 j j

j j

A A

A A

I I

I I


B B

1) 0 12 j n

( 1) j j B B ( ) ( )

( ) ( )

( ) ( )

j j

j j

B B

A A

A A

I I

I I

( 1)

2 2 j j

Thus

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2 j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 21 1

j j j j

j j

j j

B B B B

n n A A

j j A A

I I

I I

18

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

1 1

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( )

1 1

2 2

2 21 1

j j j j

j j

j j

B B B Bn n

A A

j j A A

I I

I I

ie

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

( ) ( )(

) ( )

1 1

1 1 1 1

2 2

2 2

j j j j

j j

j j j j j j j j

j j j j

n n B B B B

B B

A A j j

n n n n

A A A A

j j j

B B B B B B

j

I I

I I I I

Similarly we have

( ) ( 1)( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)( ) ( 1) ( ) ( 1)

(

) ( )( ) ( )

1 1

1 1 1 1

2

2

2

2 j j j j

j j

j j j j j j j j

j j j j

n n

A A j j

n n n n

A A A

B B B B

B B B B B B

A j

B

j j j

B

F F

F F F F


1 2( )

nSNCIWA A A A




j j j A T I x

j AF max min min

j j A A j j j

A T I

j AF 12 j n then

1 2 n

A A A A A

A NCIW S

Proof Let 1

1

x f x

x


( )min max

j j j A A A j j

T T T

so

in j j A( )

max m j A A

j j f T f T f

T ie 12

j

j

A j

A

j nT

( )

( )

1 max 1 min1

1 max 1 1 min

j j

j j

A A j

A A j j

T T T

T T

Because

( 1) ( ) j j B B


1) 0 j ( ) ( 1)1

n

j j

j

B B

1 So 12 j n

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

T T T

T T T

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

1 max 1 min1

1 max 1 1 min

j j j j j j

j j j

j j j


j j j A A A j j

T T T

T T T

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

1 max 1 min1

1 max 1 1 min

n n

j j j j j j j j

j j j

j j j

B B B B

B Bn A A A j j

j A A A j j

T T T

T T T

( ) ( 1)

( )

( )1

1 max 1 min1

1 max 1 1 min

j j

j j j

j j j

B Bn A A A j j

j A A j j

T T T

T T

A

T

( ) ( 1)

( )

( )

1

12 21

1 max 1 1 min

j j

j

j j j

B Bn

A

j A A j j

T

T T

A

T

( ) ( 1)

( )

( )1

1 max1 min 1

2 211

1

j j

j j

j

j

A A j j

B Bn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

21 min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

( ) ( 1)

( )

( )1

2min 1 max

11

1

j j j j

j

j

A A B B j jn

A

j A

T T

T

T

ie

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1 1

1 1

1m x1in ma

j j j j

j j

j j j j j j

j j

B B B B

A A B B B B j

n n

A A

j j

n n

A A

j j

jT T

T T

T T

Let2

( ) y

g y y


( )min max

j j j A A A j j

I I I


j j A A j j j A

g I g I g I

ie

( )

( )

2 max 2 min

min

2

max

j j j

j j

A A

A A j

I I

I I

12 j n

j

A j j

A j

I

I


20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



( ) ( 1)1

1n

j j

j

B B

12 j n so

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )

2 max 2 min2

max min

j j j j j j

j j j

j j j

B B B B B B

A A A j j

A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)

( )

( )1 1 1

2 max 2 min2

max min

j j j j j j

j j j

j j j


j j j A A A j j

I I I

I I I

( ) ( 1) ( ) ( 1)( ) ( 1)1 1

( )

( )1

2 max 2 min2

max min

n n

j j j j j j

j j j j

j

j j j


j A A A j j

I I I

I I I

( ) ( 1)

( )

( )1

2 max 2 min2

max min

j j

j j j

j j

B Bn A A A j j

j A A j j

I I I

I I

j A

I

( ) ( 1)

( )

( )1

22 21

max min

j j

j

j j j

B Bn

A

j A A j j

I

I I

A

I

( ) ( 1)

( )

( )1

maxmin 1

2 221

j j

j j

j

j

A A j j

B Bn

A

j A

I I

I

I

( ) ( 1)

( )

( )1

2

min max21

j j j j

j

j

A A B B j jn

A

j A

I I I

I

Thus

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max

2

2

j j

j

j j j j j j

j j

B B

A A

n

A

j

n n

A A

j j

B B B B j j

I

I

I I

I

Similarly

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1

1 1

min max2

2

j j

j

j j j j j j

j j

B B

A A

n

A j

n n

A A

j j

B B B B j j

F

F

F F

F


A SNCIWA A A A A





21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



(1) ( 2) (1) ( 2) (1) ( 2)

1 2 (1) (2) ( ) B B B B B B


(13)


( ) ( )( )i B i n and ( 1)n B


A T I F



(

( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) (

( ) ( )

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) (

1

1 1

1

2

1

1

1

2

2

1 1

1 1

j j

j j j j

j j j j

j

j

j j

j j

j j

j j

n

A

j

n n

A A

j j

n n

A A

j j

n

A

B B

n B B B B

B B B B

B B

A

j

B

T

T T

I

WGSNCI

I

I I

A A A

( ) ( )

( ) ( )

) ( 1)

( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1)

1

1 1

1 1

1 1

1

1

j

j j j j

j

j j

j

j j

j

B

B B B B

B B B B

n

j

n n

A A

j j

n n

A A

j j

F F

F F

j

(14)


( ) ( )( ) j B j n and ( 1)n

B



Let j j j j A A A


followings is true

(1) B P X then 1 B

1 2 1 2 ( ) max n n



1 2 1 2 (1) min n n


22

(3) 1 2 B B P X 1 2| | | | B B if 1( ) ( )2 B B and ( )

112

j

n j B j n

n

then

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



1 1 1 1 1

1 1 1 1

1 2

1 1 1 1

1 1 1 1 1

1 1

1 1 1 1

2 1

2

1 1 1

j j j j

j j j j j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

n

n n n n

n

j j j j j

n n n n

T I I F

T T

SNCI A A A

F F

G

I

W

I

1

1

j An

n

F

(15)

(4) If ( ) ( ) ( 1) j j x B B j



1 2

( ) ( ) (

1 1 1 1 1

1 1 1 1 1 1

) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 1 1

2

1 1

j j j j

j j j j

j j j j

j

j j j j j j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A

n

x x x x

A

x

x x x

A A

x x

A A

j j j j j j

T I I F

SNCI A A A

F

T T I I F F

WG

( ) j

A

x

(16)

(5) If || ||

1

B

j

j

B w


( ) ( 1) j j jw B B 12 j n




( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1 1

1 2

2

1 1 1 1

2 1

1 1 1

j j j j

j j j j

j j

j

j j

j j j j

j

j j

n n n n n

A A A A

j j j j j

n n n n n n

A A A A A A

j

w n

w w

j j j j j

w w

w w w w w

T I I F

T T I I F

SNOWG A A A

F

j

j

A

w

w

F

(17)





A j n j A A


1 2 SNCI A A n

A AWG


23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





on X If and

j j j

j A A A A T I F


nWG A A A SNCI

1 2 n A AS WG A NCI





j j j A T I

x j A


j j j A T I

j AF 12 j n then

1 2 n

G A A A A

A NCIW S






w j m 1

1m

j

j

w


decision-makers

k


n m


k

ij a aa T k

ijaF

ia



T jc

ia

for


criterion

k ija

I










for cost criterion

ij j

cij

ij j

a cb

a c

12 12 i n j m (18)

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59




ija

ija






k k

ij n m R a


ij n m R b

based on Eq (18)



( ) ( ) ( 1) j jw B B j

12 i m



matrix



R b

Where

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1) ( ) ( 1)

( ) ( )

( ) ( 1)

( )

( )

( )

1

1

1

1 1

1

2

1 1

2

1 1

2

r r r r

r r ij ij

r r r r

r r ij ij

r r

r ij

r

r ij

k k

r r

B B B B

b bk

ij ij ij ij B B B B

b b

B B

b

B

n k

j r

k

r

b

b SNC

T T

T T

I

IWA b b

I

b

( 1) ( ) ( 1)

( )

( ) ( 1)

( )

( ) ( 1) ( ) ( 1)

( ) ( )

1 1

1

1 1

2

2

j r

r ij

r r

r ij

r r r r

r r ij ij

k k

r r

k

r

k k

r

B B B

b

B B

b

B B

b br

B B

I

F

F F

r (19)

or

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





27



( ) ( ) ( 1)1

1 3

m

nn i i j ij j j j

j

d a b d b b B B

igs

(22)



bF d b


d b b d b b

)

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B

( ) ( ) ( 1)1

1

3

m

i i i j ij j j j

j

d a b d b b B B

(23)




d b b d b b

) i

( ) i m ij jd b b

( ) ( ) ( 1) ( ) j j j

B c c c m

and ( 1)m B



12

i i

i

i i i i

d a bG a i n

d a b d a b

(24)




ia

12 i n







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59












under the criterion

1a 2a 3a 4a

2c1c

3c 4c

ia


ijaF k k

ija a k ij

k

ija T I 1 234k i j

23 1 which



ij R ak

n m can

be found as follows

1

040102 050201 030204 060202

070102 060203 040203 070202

040103 050201 040202 050103

06 0301 05 03 02 05 01 02 0

R

70102

2

060102 050202 040103 070201

050202 060201 050302 060202

050201 050103 050102 070302

0503 02 080202 0502 02 0

R

50201

3

040203 040203 070302 060102060102 050102 050201 070201

030203 050203 050303 070103

06 0001 060102 06 0201 0

R

80201





R a a

28

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





1 2 3 1k k k 1 2 3 1 2 1 305 03 02 0 08 k k k k k k k 9




046560107302085 049050200001483 043560165803121 063240163101631

063600115202000 057170163101931 046140235902065 068180200001747

042180132502187 R

050000152301758 044140170202085 062920157302558

05817001152 063950193102000 052130132501747 069110132501523

11b


1 2 311 11 1107000 07667 and 06333s b s b s b

Then 3 111 11 11

2s b s b s b So 3 111 11 11b b b 2 (1) 3 (2) 1

11 11 11 11b b b b and (3) 211 11b b

Thus (1) (2) 1 2 3 1 2( ) ( ) ( ) ( ) 1 09 01 B B k k k k k

(2) (3) 1 2 2

09 03 06 B B k k k

(3) (4) 2 03 B B k

So

01 06 03 01 06 03

01 06 03 01 06 03

01 0

1 2 311 11 11 11

6

1 04 1 04 1 06 1 04 1 04 1 06

1 04 1 04 1 06 1 04 1 04 1 06

2 02 01

040102 060102 040203

b SNCIWA b b b

SNCIWA

03

01 06 03 01 06 03

01 06 03

01 06 03 01 06 03

01

2 02 2 01 2 01 02 01 01

2 03 02 02

2 03 2 02 2 02 03 02 0

04656010

2

7302

085


29


1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



true

1 max 04656063600421805817 min 0107301152013250 min 02085020218701152

06360001152

b

2

3

4

063950152301483

052130132501747

069110132501523

b

b

b

Similarly

1

2

3

4

042180132502187

049050200002000

043560235903121

062920200002558

b

b

b

b



1( ) 030c 2( ) 025c 3( ) 037c 4( ) 020c 1 2( ) 052c c 1 3( ) 065c c

1 4( ) 0c c 50 2 3( ) 0c c 45 2 4( ) 0c c 34 3 4( ) 0c c 42 1 2 3( ) 085c c c 1 2 4( ) 068c c c

2 3 4( )c c c 057 1 3( c c c4 ) 076 1( c c2 3 4c ) 1c


Based on Eq (22)

11 11 1

11

046560107302085 06360001152

04656 06360 01073 0 02085 01152

03710

d b b

d

12 12 2 13 13 3 14 14 4 01967 02564 01001d b b d b b d b b



so

30

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



(1) (2) 1 2 3 4 2 3 1( ) ( ) 1 085 015 B B c c c c c c c

(2) (3) 1 2 3 3 1( ) ( ) 085 065 020 B B c c c c c

31

035 (3) (4) 1 3 1 065 030 B B c c c

(4) (5) 1 030 B B c

Thus

11 14 14 4 (1) (2) 12 12 2 (2) (3)

13 13 3 (3) (4) 11 11 1 (4) (5)

1

3

101001 015 01967 02 02564 035 0371 03 00851

3


d b b B B d b b B B


11 0027gsnn

d a b 0 2 22 2 00559 00553gsnn gsnn

d a b d a b

3 33 3 00842 00450


4 44 4 00126 00889gsnn gsnn

d a b d a b


11

1 1 11 1

0085107591

00851 00270

gsnn

gsnn gsnn

d a bG a

d a b d a b

2 3 405027 06517 01241G a G a G a



1a



4 2 3a a a a

4a



1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59







34










7 Conclusion













Acknowledgement



(BK201405)

References

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



35



141-164

[3]




121-146



610-614

[8]


403-405





[11]



[12]


24 (2011) 1224-1232


Fuzzy Systems 20 (2012) 514-525





(2013) 510-515

[16]




Applications 129 (2005) 589-597

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



36


NIFS 15 (2009) 33-36

[19]



[20]


Research 69 (2010) 426-432





[23]



[25]




org1011552012879629 2012

[27]


[28]





[30]








Infinite Study 2005


1860-1868

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59



37


Fuzzy Systems 26 (3) (2014) 1245-1252

[36]



[37]







Materials 436 (2013) 511-517

[41]







[44]



In Press 2015







[49]







1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59





269ndash278



2975ndash2982



doi101007s00500-014-1259-z 2014

[55]


Springer 1981

[56]




[58]





[60]



[61]




101007s10726-014-9385-7

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

An multi-criteria decision-making approach based on Choquet integral-based TOPSIS with simplified...

Documents

Transcript of An multi-criteria decision-making approach based on Choquet integral-based TOPSIS with simplified...