Research Article Correlation Measures of Dual Hesitant...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 593739 12 pageshttpdxdoiorg1011552013593739

Research ArticleCorrelation Measures of Dual Hesitant Fuzzy Sets

Lei Wang Mingfang Ni and Lei Zhu

Institute of Communication Engineering PLA University of Science and Technology Nanjing Jiangsu 210007 China

Correspondence should be addressed to Lei Wang iponly126com

Received 23 June 2013 Revised 9 September 2013 Accepted 9 September 2013

Academic Editor Toly Chen

Copyright copy 2013 Lei Wang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dual hesitant fuzzy sets (DHFSs) were proposed by Zhu et al (2012) which encompass fuzzy sets intuitionistic fuzzy setshesitant fuzzy sets and fuzzy multisets as special cases Correlation measures analysis is an important research topic In this paperwe define the correlation measures for dual hesitant fuzzy information and then discuss their properties in detail One numericalexample is provided to illustrate these correlationmeasuresThenwe present a direct transfer algorithmwith respect to the problemof complex operation ofmatrix synthesis when reconstructing an equivalent correlationmatrix for clustering DHFSs Furthermorewe prove that the direct transfer algorithm is equivalent to transfer closure algorithm but its asymptotic time complexity and spacecomplexity are superior to the latter Another real world example that is diamond evaluation and classification is employed toshow the effectiveness of the association coefficient and the algorithm for clustering DHFSs

1 Introduction

Correlation indicates how well two variables move togetherin a linear fashion In other words correlation reflects alinear relationship between two variables It is an importantmeasure in data analysis in particular in decision makingmedical diagnosis pattern recognition and other real worldproblems [1ndash7] Zadeh [8] introduced the concept of fuzzysets (FSs) whose basic component is only a membershipfunction with the nonmembership function being one minusthe membership function In fuzzy environments Hungand Wu [9] used the concept of ldquoexpected valuerdquo to definethe correlation coefficient of fuzzy numbers which liesin [minus1 1] Hong [10] considered the computational aspectof the 119879

120596-based extension principle when the principle is

applied to a correlation coefficient of 119871-119877 fuzzy numbersand gave the exact solution of a fuzzy correlation coefficientwithout programming or the aid of computer resourcesAtanassov [11 12] gave a generalized form of fuzzy set calledintuitionistic fuzzy set (IFS) which is characterized by amembership function and a non-membership function Inintuitionistic fuzzy environments Gerstenkorn and Manko[13] defined a function measuring the correlation of IFSsand introduced a coefficient of such a correlation Bustinceand Burillo [14] introduced the concepts of correlation andcorrelation coefficient of interval-valued intuitionistic fuzzy

sets (IVIFSs) [12] Hung [15] and Mitchell [16] derived thecorrelation coefficient of intuitionistic fuzzy sets from astatistical viewpoint by interpreting an intuitionistic fuzzy setas an ensemble of ordinary fuzzy sets Hung and Wu [17]proposed a method to calculate the correlation coefficientof intuitionistic fuzzy sets by means of ldquocentroidrdquo Xu [18]gave a detailed survey on association analysis of intuitionisticfuzzy sets and pointed out that most existing methodsderiving association coefficients cannot guarantee that theassociation coefficient of any two intuitionistic fuzzy setsequals one if and only if these two intuitionistic fuzzy setsare the same Szmidt and Kacprzyk [5] discussed a conceptof correlation for data represented as intuitionistic fuzzy setadopting the concepts from statistics and proposed a formulafor measuring the correlation coefficient (lying in [minus1 1])of intuitionistic fuzzy sets Robinson and Amirtharaj [19]defined the correlation coefficient of interval vague sets lyingin the interval [0 1] and proposed a new method for com-puting the correlation coefficient of interval vague sets lyingin the interval [minus1 1] using a-cuts over the vague degreesthrough statistical confidence intervals which is presentedby an example Instead of using point-based membership asin fuzzy sets interval-based membership is used in a vagueset In [20] Robinson and Amirtharaj presented a detailedcomparison between vague sets and intuitionistic fuzzy setsand defined the correlation coefficient of vague sets through

2 Journal of Applied Mathematics

simple examples Hesitant fuzzy sets (HFSs) were originallyintroduced by Torra [21 22] In hesitant fuzzy environmentsChen et al [23] derived some correlation coefficient formulasfor HFSs and applied them to two real world examples byusing clustering analysis under hesitant fuzzy environmentsXu and Xia [24] defined the correlationmeasures for hesitantfuzzy information and then discussed their properties indetail

Recently Dubois and Prade introduced the definition ofdual hesitant fuzzy set Dual hesitant fuzzy set can reflecthumanrsquos hesitance more objectively than the other classicalextensions of fuzzy set (intuitionistic fuzzy set type-2 fuzzyset (T-2FS) [25] hesitant fuzzy set etc) The motivation topropose the DHFSs is that when people make a decisionthey are usually hesitant and irresolute for one thing oranother which makes it difficult to reach a final agreementThey further indicated that DHFSs can better deal withthe situations that permit both the membership and thenonmembership of an element to a given set having a fewdifferent values which can arise in a group decision makingproblem For example in the organization some decisionmakers discuss the membership degree 06 and the non-membership 03 of an alternative 119860 that satisfies a criterion119909 Some possibly assign (08 02) while the others assign(07 02) No consistency is reached among these decisionmakers Accordingly the difficulty of establishing a commonmembership degree and a non-membership degree is notbecause we have a margin of error (intuitionistic fuzzy set)or some possibility distribution values (type-2 fuzzy set) butbecause we have a set of possible values (hesitant fuzzy set)For such a case the satisfactory degrees can be representedby a dual hesitant fuzzy element (06 08 07) (03 02)which is obviously different from intuitionistic fuzzy number(08 02)or (07 02) andhesitant fuzzy number 06 08 07The aforementioned measures however cannot be used todeal with the correlation measures of dual hesitant fuzzyinformation Thus it is very necessary to develop sometheories for dual hesitant fuzzy sets However little hasbeen done about this issue In this paper we mainly discussthe correlation measures of dual hesitant fuzzy informationTo do this the remainder of the paper is organized asfollows Section 2 presents some basic concepts related toDHFSs HFSs and IFSs In Section 3 we propose somecorrelation measures of dual hesitant fuzzy elements obtainseveral important conclusions and given an example toillustrate the correlation measures In Section 4 we proposea direct transfer clustering algorithm based on DHFSs andthen use a numerical example to illustrate our algorithmFinally Section 5 concludes the paper with some remarks andpresents future challenges

2 Preliminaries

21 DHFSs HFSs and IFSs

Definition 1 (see [26]) Let119883 be a fixed set then a dual hesitantfuzzy set (DHFS)119863 on119883 is described as

119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

in which ℎ(119909) and 119892(119909) are two sets of some values in[0 1] denoting the possible membership degrees and non-membership degrees of the element 119909 isin 119883 to the set 119863respectively with the conditions

0 le 120574 120578 le 1 0 le 120574+

+ 120578+

le 1 (2)

where 120574 isin ℎ(119909) 120578 isin 119892(119909) 120574+ isin ℎ+

(119909) = cup120574isinℎ(119909)

max120574 and120578+

isin 119892+

(119909) = cup120578isin119892(119909)

max120578 for all 119909 isin 119883 For conveniencethe pair 119889

119864(119909) = (ℎ

119864(119909) 119892119864(119909)) is called a dual hesitant fuzzy

element (DHFE) denoted by 119889 = (ℎ 119892) with the conditions120574 isin ℎ(119909) 120578 isin 119892(119909) 120574+ isin ℎ

+

(119909) = cup120574isinℎ(119909)

max120574 and 120578+

isin

119892+

(119909) = cup120578isin119892(119909)

max120578 0 le 120574 120578 le 1 and 0 le 120574+

+ 120578+

le 1

Definition 2 (see [21 22]) Let119883 be a fixed set a hesitant fuzzyset (HFS) 119860 on119883 is in terms of a function that when appliedto119883 returns a subset of [0 1] which can be represented as thefollowing mathematical symbol

119860 = ⟨119909 ℎ119860(119909)⟩ | 119909 isin 119883 (3)

where ℎ119860(119909) is a set of values in [0 1] denoting the possible

membership degrees of the element 119909 isin 119883 to the set 119860 Forconvenience we call ℎ

119860(119909) a hesitant fuzzy element (HFE)

We use ⟨119909 ℎ119860⟩ for all 119909 isin 119883 to represent HFSs

Definition 3 (see [11 12]) Let119883 be a fixed set an intuitionisticfuzzy set (IFS) 119860 on119883 is an object having the form

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (4)

which is characterized by a membership function 120583119860and a

non-membership function ]119860 where 120583

119860 119883 rarr [0 1] and

]119860 119883 rarr [0 1] with the condition 0 le 120583

119860(119909) + ]

119860(119909) le 1

for all 119909 isin 119883 We use ⟨119909 120583119860 ]119860⟩ for all 119909 isin 119883 to represent

IFSs considered in the rest of the paper without explicitlymentioning it Furthermore 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is

called a hesitancy degree or an intuitionistic index of 119909 in 119860In the special case 120587(119909) = 0 that is 120583

119860(119909) + ]

119860(119909) = 1 the

IFS 119860 reduces to an FS

22 Correlation Coefficients of HFSs and IFSs Manyapproaches [4 13 17 20 21] have been introduced to computethe correlation coefficients of IFSs Let 119883 = 119909

1 1199092 119909

119899

be a discrete universe of discourse for any two 119860 and 119861 on119883

The correlation of the IFSs 119860 and 119861 is defined as [13]

119862IFS1(119860 119861) =

119899

sum119894=1

(119906119860(119909119894) 119906119861(119909119894) + V119860(119909119894) 119906119861(119909119894)) (5)

Journal of Applied Mathematics 3

Then the correlation coefficient of the IFSs119860 and119861 is definedas

120588IFS1(119860 119861) =

119862IFS1(119860 119861)

(119862IFS1(119860 119860) sdot 119862IFS

1(119861 119861))

12

= (

119899

sum119894=1

(119906119860(119909119894) 119906119861(119909119894) + V119860(119909119894) 119906119861(119909119894)))

times (((

119899

sum119894=1

(1199062

119860(119909119894) + V2119860(119909119894)))

sdot(

119899

sum119894=1

(1199062

119861(119909119894) + V2119861(119909119894))))

12

)

minus1

(6)

In [23] Chen et al defined the correlation and correlationcoefficient for HFSs as follows respectively

119862HFS1(119860 119861) =

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894))

120588IFS1(119860 119861) =

119862HFS1(119860 119861)

(119862HFS1(119860 119860) sdot 119862HFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894)))

times(((

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119860120590(119895)

(119909119894)))

sdot(

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119861120590(119895)

(119909119894))))

12

)

minus1

(7)

where 119897119894= max119897(ℎ

119860(119909119894)) 119897(ℎ119861(119909119894)) for each 119909

119894in 119883 and

119897(ℎ119860(119909119894)) and 119897(ℎ

119861(119909119894)) represent the number of values in

ℎ119860(119909119894) and ℎ

119861(119909119894) respectively We will talk about 119897

119894in detail

in the next section

3 Correlation Measures of DHFEs

In this section we first introduce the concept of correlationand correlation coefficient for DHFSs and then proposeseveral correlation coefficient formulas and discuss theirproperties

We arrange the elements in 119889119864(119909) = (ℎ

119864(119909) 119892119864(119909)) in

decreasing order and let 120574120590(119894)119864

(119909) be the 119894th largest valuein ℎ119864(119909) and 120578

120590(119895)

119864(119909) the 119895th largest value in 119892

119864(119909) Let

119897ℎ(119889119864(119909119894)) the number of values in ℎ

119864(119909119894) and 119897

119892(119889119864(119909119894)) be

the number of values in 119892119864(119909119894) For convenience 119897(119889(119909

119894)) =

(119897ℎ(119889(119909119894)) 119897119892(119889(119909119894))) In most cases for two DHFSs 119860

and 119861 119897(119889119860(119909119894)) = 119897(119889

119861(119909119894)) that is 119897

ℎ(119889119860(119909119894)) = 119897ℎ(119889119861(119909119894))

119897119892(119889119860(119909119894)) = 119897119892(119889119861(119909119894)) To operate correctly we should

extend the shorter one until both of them have the samelength when we compare them In [24 27] Xu and Xiaextended the shorter one by adding different values inhesitant fuzzy environments Similarly Torra [21] also appliedthis ideal to derive some correlation coefficient formulas forHFSs In fact we can extend the shorter one by adding anyvalue in it The selection of this value mainly depends onthe decision makersrsquo risk preferences Optimists anticipatedesirable outcomes and may add the maximum value whilepessimists expect unfavorable outcomes and may add theminimum value The same situation can also be found inmany existing references [13 14]

We define several correlation coefficients for DHFEs

Definition 4 For two DHFSs 119860 and 119861 on 119883 the correlationof 119860 and 119861 denoted as 119862DHFS

1

(119860 119861) is defined by

119862DHFS1(119860 119861) =

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

(8)

Definition 5 For two DHFSs 119860 and 119861 on 119883 the correlationcoefficient of119860 and 119861 denoted as 120588DHFS

1

(119860 119861) is defined by

120588DHFS1(119860 119861)

=119862DHFS

1(119860 119861)

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(9)


Definition 6 For two DHFSs 119860 and 119861 on 119883 the correlationcoefficient of 119860 and 119861 denoted as 120588DHFS

2

(119860 119861) is defined by

120588DHFS2(119860 119861)

=119862DHFS

1(119860 119861)

max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(10)

Theorem 7 For two DHFSs 119860 and 119861 the correlation coeffi-cient of 119860 and 119861 denoted as 120588

119863119867119865119878119894

(119860 119861) (119894 = 1 2) shouldsatisfy the following properties

(1) 0 le 120588119863119867119865119878

119894

(119860 119861) le 1

(2) 119860 = 119861 rArr 120588119863119867119865119878

119894

(119860 119861) = 1

(3) 120588119863119867119865119878

119894

(119860 119861) = 120588119863119867119865119878

119894

(119861 119860) 119894 = 1 2

Proof (1) The inequality 0 le 120588DHFS1

(119860 119861) and 0 le

120588DHFS2

(119860 119861) is obvious Below let us prove that120588DHFS

1

(119860 119861) le 1 120588DHFS2

(119860 119861) le 1

119862DHFS1(119860 119861)

=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

=

119899

sum119894=1

(

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

119897ℎ(119894)

+

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)

119897119892(119894)

)

= (

119897ℎ(1)

sum119895=1

120574119860120590(119895)

(1199091)

radic119897ℎ(1)

sdot120574119861120590(119895)

(1199091)

radic119897ℎ(1)

+

119897ℎ(2)

sum119895=1

120574119860120590(119895)

(1199092)

radic119897ℎ(2)

sdot120574119861120590(119895)

(1199092)

radic119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

120574119860120590(119895)

(119909119899)

radic119897ℎ(119899)

sdot120574119861120590(119895)

(119909119899)

radic119897ℎ(119899)

)

+(

119897119892(1)

sum119896=1

120578119860120590(119896)

(1199091)

radic119897119892(1)

sdot120578119861120590(119896)

(1199091)

radic119897119892(1)

+

119897119892(2)

sum119896=1

120578119860120590(119896)

(1199092)

radic119897119892(2)

sdot120578119861120590(119896)

(1199092)

radic119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

120578119860120590(119896)

(119909119899)

radic119897119892(119899)

sdot120578119861120590(119896)

(119909119899)

radic119897119892(119899)

)

(11)

Using the Cauchy-Schwarz inequality

(11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899)2

le (1199092

1+ 1199092

2+ sdot sdot sdot + 119909

2

119899) sdot (1199102

1+ 1199102


2

119899)

(12)

where (1199091 1199092 119909

119899) isin 119877119899

(1199101 1199102 119910

119899) isin 119877119899 we obtain

119862DHFS1(119860 119861)

2

le (

119897ℎ(1)

sum119895=1

1205742

119860120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119860120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119860120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119860120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119860120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119860120590(119896)

(119909119899)

119897119892(119899)

)

sdot (

119897ℎ(1)

sum119895=1

1205742

119861120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119861120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119861120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119861120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119861120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119861120590(119896)

(119909119899)

119897119892(119899)

)


=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

= 119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861)

(13)

Therefore

119862DHFS1(119860 119861) le (119862DHFS

1

(119860 119860))12

sdot (119862DHFS1(119861 119861))

12

(14)

So 0 le 120588DHFS1

(119860 119861) le 1In fact we have

(1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

le (max ((11990921+ 1199092


2

119899) (1199102

1+ 1199102


2

119899)))2

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

le (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)+

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

2

(15)

Then

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

le max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

(16)

We also obtain 0 le 120588DHFS2

(119860 119861) le 1(2) and (3) are straightforward

Moreover from the proof of Theorem 7 we have Theo-rem 8 easily

Theorem 8 For two DHFSs 119860 and 119861 on 119883 then120588119863119867119865119878

1

(119860 119861) ge 120588119863119867119865119878

2

(119860 119861)

However from Theorem 7 we notice that all the abovecorrelation coefficients cannot guarantee that the correlationcoefficient of any two DHFSs equals one if and only if thesetwo DHFSs are the same Thus how to derive the correlationcoefficients of the DHFSs satisfying this desirable propertyis an interesting research topic To solve this issue in whatfollows we develop a newmethod to calculate the correlationcoefficient of the DHFSs 119860 and 119861


3

(119860 119861) is defined by

120588DHFS3(119860 119861)

= (1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(17)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(18)

Equation (17) is motivated by the generalized ideaprovided by Xu [18] Obviously the greater the value of120588DHFS

3

(119860 119861) the closer 119860 to 119861 By Definition 9 we haveTheorem 10

Theorem 10 The correlation coefficient 120588119863119867119865119878

3

(119860 119861) satisfiesthe following properties

(1) 0 le 120588119863119867119865119878

3

(119860 119861) le 1(2) 119860 = 119861 hArr 120588

1198631198671198651198783

(119860 119861) = 1(3) 120588119863119867119865119878

3

(119860 119861) = 120588119863119867119865119878

3

(119861 119860)


(119860 119861) is obvious Belowlet us prove that 120588DHFS

3

(119860 119861) le 1

Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

for 119894 = 1 2 119899

=(Δ120574120582

minΔ120574120582

max) + 1

(Δ120574120582119894Δ120574120582max) + 1

+(Δ120578120582

minΔ120578120582

max) + 1

(Δ120578120582119894Δ120578120582max) + 1

le 2

(19)

We obtain

120588DHFS3(119860 119861)

=1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

)

le1

2119899sdot 2119899

= 1

(20)

(2) and (3) are obvious


Usually in practical applications the weight of eachelement 119909

119894isin 119883 should be taken into account and so

we present the following weighted correlation coefficientAssume that the weight of the element 119909

119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend

the correlation coefficient formulas given

120588DHFSminus1199081

(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)

Note that all these formulas satisfy the properties inTheorem 7

In what follows we use a medical diagnosis problemin [28 29] to illustrate the developed correlation coefficientformulas Actually this is also a pattern recognition problem

Example 11 To make a proper diagnosis 119876 = 1198761(viral

fever) 1198762(malaria) 119876

3(typhoid) 119876

4(stomach problem)

and 1198765(chest problem) for a patient with the given values

of the symptoms 119878 = 1198781(temperature) 119878

2(headache)

1198783(cough) 119878

4(stomach pain) and 119878

5(chest pain) Xu

[18] considered all possible diagnoses and symptoms asHFEs Utilizing DHFSs can take much more informationinto account the more values we obtain from patients thegreater epistemic certainty we have So in this paper we useDHFEs to deal with such cases each symptom is describedby a DHFE which is described by two sets (120574

119894119895) and (120578

119894119895)

(120574119894119895) indicates the degree that symptoms characteristic 119878

119894

satisfies the considered diagnoses 119866119895and (120578

119894119895) indicates the

degree that the symptoms characteristic 119878119894does not satisfy

the considered diagnoses 119866119895 The data are given in Table 1

The set of patients is 119875 = AlBob JoeTed The symptomswhich can be also described by DHFEs are given in Table 2We need to seek a diagnosis for each patient

We utilize the correlation coefficient 120588DHFS1 to derive adiagnosis for each patient All the results for the consideredpatients are listed in Table 3 From the arguments in Table 3we can find that Ted suffers from viral fever Al and Joe frommalaria and Bob from stomach problem

If we utilize the correlation coefficient formulas 120588DHFS2and 120588DHFS3 to derive a diagnosis then the results are listedin Tables 4 and 5 respectively

From Tables 3ndash5 we know that the results obtained bydifferent correlation coefficient formulas are different Thatis because these correlation coefficient formulas are based ondifferent linear relationships

4 Clustering Method Based on Direct TransferAlgorithm for HFSs

Based on clustering algorithms for IFSs [30 31] and HFSs[23] and the correlation coefficient formulas developedpreviously for DHFSs in what follows we propose a direct


Table 1 Symptoms characteristic of the considered diagnoses

Temperature Headache Cough Stomach pain Chest pain

Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)

Stomach problem (05 04 02)(05 03)

(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)

Table 2 Symptoms characteristic of the considered patients


Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)

Table 3 Values of 120588DHFS1 for each patient to the considered set ofpossible diagnoses

Viral fever Malaria Typhoid Stomachproblem

Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644

transfer algorithm to clustering analysis with respect to theproblem of complex operation of matrix synthesis whenreconstructing analogical relation to equivalence relationclustering under hesitant fuzzy environments Before doingthis some concepts are introduced firstly

Definition 12 Let 119860119895(119895 = 1 2 119898) be 119898 DHFs then 119862 =

(120588119894119895)119898times119898

is called an associationmatrix where 120588119894119895= 120588(119860

119894 119860119895)

is the association coefficient of 119860119894and 119860

119895 which has the

following properties

(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898

Definition 13 (see [23 30]) Let119862 = (120588119894119895)119898times119898

be an associationmatrix if 1198622 = 119862 ∘ 119862 = (120588

119894119895)119898times119898

then 1198622 is called a

composition matrix of 119862 where 120588119894119895= maxmin120588

119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898

Based on Definition 13 we have the following theorem

Theorem 14 (see [23 30]) Let 119862 = (120588119894119895)119898times119898

be an associationmatrix then the composition matrix 1198622 is also an associationmatrix


be an associationmatrix then for any nonnegative integer 119896 the compositionmatrix 119862

2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896

is also anassociation matrix

Definition 16 (see [23 30]) Let 119862 = (120588119894119895)119898times119898

be anassociation matrix if 1198622 sube 119862 that is

max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)

then 119862 is called an equivalent association matrix

By the transitivity principle of equivalent matrix we caneasily prove the following theorem

Theorem 17 (see [23 30 32]) Let 119862 = (120588119894119895)119898times119898

be anassociation matrix then after the finite times of compositions119862 rarr 119862

2

rarr 1198624

rarr sdot sdot sdot rarr 1198622119896

rarr sdot sdot sdot there must exista positive integer 119896 such that 1198622

119896

= 1198622119896+1

and 1198622119896

is also anequivalent association matrix

Definition 18 (see [23 30 31]) Let 119862 = (120588119894119895)119898times119898

be anequivalent correlation matrix Then we call 119862

120582= (120582

120588119894119895)119898times119898

the 120582-cutting matrix of 119862 where

120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)

and 120582 is the confidence level with 120582 isin [0 1]


Table 4 Values of 120588DHFS2 for each patient to the considered set of possible diagnoses

Viral fever Malaria Typhoid Stomach problem Chest problemAl 08718 08888 07571 08286 07591Bob 07586 08451 07960 09852 08889Joe 09075 08607 08152 08712 06500Ted 08764 09140 08835 08678 07550



Next a traditional transfer closure algorithm is given asfollows

Step 1 Let 1198601 1198602 119860

119898 be a set of DHFSs in 119883 =

1199091 1199092 119909

119899 We can calculate the correlation coefficients

of the DHFSs and then construct a correlation matrix 119862 =

(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)

Step 2 Check whether 119862 = (120588119894119895)119898times119898

is an equivalentcorrelation matrix that is check whether it satisfies 1198622 sube

119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)

If it does not hold we construct the equivalent correlationmatrix 1198622

119896

119862 rarr 1198622

rarr 1198624


rarr sdot sdot sdot until1198622119896

= 1198622119896+1

Step 3 For a confidence level 120582 we construct a 120582-cuttingmatrix 119862

120582= (120582

120588119894119895)119898times119898

through Definition 18 in order toclassify the DHFSs 119860

119895(119895 = 1 2 119898) If all elements of the

119894th line (column) are the same as the corresponding elementsof the 119895th line (column) in119862

120582 then the DHFSs119860

119894and119860

119895are

of the same type By means of this principle we can classifyall these119898 119860

119895(119895 = 1 2 119898)

By analyzing the aforementioned transfer closure algo-rithm this algorithm has one drawback such as complexoperation of matrix synthesis when reconstructing the equiv-alent correlation matrix In this paper we have the followingtheorem of the correlation coefficients in dual hesitant fuzzyenvironment

Theorem 19 For all 119909 119910 isin 119860 for the confidence level 120582if exist1199091 1199092 1199093 119909

119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910

are of the same type

Proof we are motivated by the generalized idea based onthe transitivity principle of ordinary equivalent relation 119877for all 119909 119910 isin 119860 (here 119860 is an ordinary set not a fuzzyset) exist119909

1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1

) isin 119877 (119909119901 119910) isin 119877 we can have (119909 119910) isin 119877

And from Definition 18 we can see that the 120582-cuttingmatrix of 119862 is an ordinary correlation matrix which com-pletes the proof of Theorem 19

From the above theoretical analysis we propose a directtransfer algorithm for clustering DHFSs as follows

Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)

Step 2 By setting the threshold to the confidence level 120582 wecan construct a 120582-cutting matrix 119862

120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1

this means that the DHFSs119860119894and119860

119895are of the same type By

means of this principle we can classify all these 119898 119860119895(119895 =

1 2 119898)

We can see that the transfer closure algorithm mustconstruct the equivalent correlation matrix 119862 rarr 119862

2

rarr

1198624


rarr sdot sdot sdot until 1198622119896

= 1198622119896+1

and then construct a 120582-cutting matrix 119862120582

= (120582

120588119894119895)119898times119898

through Definition 18 in order to classify the DHFSs 119860119895(119895 =

1 2 119898) Simply the transfer algorithm only constructs a120582-cutting matrix 119862

120582= (120582

120588119894119895)119898times119898

by setting the threshold tothe confidence level 120582 and then classifies the DHFSs 119860

119895(119895 =

1 2 119898) directly In what follows we will talk about therelationship between the transfer closure algorithm and thedirect transfer algorithm

Theorem20 Theclustering results are the same by the transferclosure algorithm and the direct transfer algorithm at the sameconfidence level


Proof (1) For a confidence level 120582 for all 119860119894 119860119895

isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860

119895are of the same type by

the direct transfer algorithmAssume that we construct the equivalent correlation

matrix 1198622119896

when we employ the transfer closure algorithmWe must prove that 120588

1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622

supe 119862 and for the same reason we have 1198622119896

supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622

supe 119862 Consider

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)

For a confidence level 120582 when we get that 119860119894and 119860

119895are

of the same type using the direct transfer algorithm we canalso have the same clustering results by the transfer closurealgorithm

(2) For a confidence level 120582 for all 119860119894 119860119895isin 119860 exist the

equivalent correlation matrix 1198622119896

1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894

and119860119895are of the same type by the transfer closure algorithm

Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860

119895are the same type in 119862

2119896minus1

by the directtransfer algorithm

For the same reason exist1199092 1205881198622119896minus2 (119860119894 1198601199092

) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860

119895are the same type in1198622

119896minus2

by the direct transferalgorithm

So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860

119895are the same type in 119862 by the direct transfer

algorithmFor a confidence level 120582 when we get 119860

119894and 119860

119895are of

the same type using the transfer closure algorithm we canalso have the same clustering results by the direct transferalgorithm which completes the proof

We assume 119860 = 1198601 1198602 119860

119898 to be a set of DHFSs

and we construct the equivalent correlation matrix1198622119896

119862 rarr

1198622

rarr 1198624



= 1198622119896+1

andthen construct a 120582-cutting matrix 119862

120582= (120582

120588119894119895)119898times119898

for thetransfer closure algorithm Consequently the running timeof the transfer closure algorithm is 119879tca = 119874(119896119898

3

+ 1198961198982

) bythe same arguments the direct transfer algorithm requires119879dta = 119874(119898

2

) time on the same example And we haveestablished 119878tca = 119874(119898

2

) space bound at least for the step ofconstructing the equivalent correlation matrix based on thetransfer closure algorithm while for the transfer algorithmit constructs a 120582-cutting matrix 119862

120582= (120582

120588119894119895)119898times119898

by settingthe threshold to the confidence level 120582 and needs 119878tca = 119874(119898)

space bound We can see that the computational complexityof both two algorithms ranges depends on the number of 119898and the direct transfer algorithm exhibits better behavior

Below we conduct experiments in order to demonstratethe effectiveness of the proposed clustering algorithm forDHFSs

Example 21 Every diamond is a miracle of time and placeand chance Like snowflakes no two are exactly alike Everyconsumer shopping for diamonds is faced with endlessdiamond combinations In addition to different diamondcombinations prices are also influenced by market supplyand demand conditions fashion trends and so forth While


Table 6 Diamond data set

ldquoDrdquo color ldquoFLrdquo clarity ldquo3 excellentrdquo cut ldquo1 caratrdquo weight1198601

(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)

consumersrsquo tastes and budgets change most seek to find afair price for the diamond they choose Until the middle ofthe twentieth century there was no agreed upon standard bywhich diamonds could be judged No matter how beautifula diamond may look you simply cannot see its true qualityGIA created the first and now globally accepted standardfor describing diamonds color clarity cut and carat weightConcerning color the less color in the stone there is themore desirable and valuable it is Grades run from ldquoDrdquoto ldquoXrdquo Clarity measures the amount size and placementof internal ldquoinclusionsrdquo and external ldquoblemishesrdquo Gradesrun from ldquoFlawlessrdquo to ldquoincludedrdquo Cut does not refer toa diamondrsquos shape but to the proportion and arrangementof its facets and the quality of workmanship Grades rangefrom ldquoexcellentrdquo to ldquopoorrdquo Carat refers to a diamondrsquos weightGenerally speaking the higher the carat weight the more

expensive the stone Two diamonds of equal carat weighthowever can have very different quality and price whenthe other three Cs are considered We choose a ldquoperfectrdquodiamond whose 4C is ldquoDrdquo color ldquoFLrdquo clarity ldquo3 excellentrdquocut and ldquo1caratrdquo weight For the convenience of analysis theweight vector of these attributes is119908 = (025 025 025 025)Here there are ten diamonds In order to better make theassessment several evaluation organizations are requestedThe normalized evaluation diamond data represented byDHFSs are displayed in Table 6

Now we utilize the direct transfer algorithm to cluster theten diamonds which involves the following steps

Step 1 Utilize (21) to calculate the association coefficientsand then construct an association matrix

119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)

Step 2 We give a detailed sensitivity analysis with respectto the confidence level and by (26) we get all the possibleclassifications of the ten diamonds see Table 7 and Figure 1

From the above numerical analysis under the groupsetting the expertsrsquo evaluation information usually does notreach an agreement for the objects that need to be classifiedExample 21 clearly shows that the clustering algorithm basedon DHFSs provides a proper way to resolve this issue

In the following a comparison is made among themethod proposed in this paper Chen et alrsquos method [23] andZhao et alrsquos method [31] in Table 8

Through Table 8 it is worthy of pointing out that theclustering results of the direct transfer clustering methodproposed in this paper are exactly the same with thoseof Chen et alrsquos transfer clustering method and Zhao etalrsquos Boole method but our method does not need to usethe transitive closure technique to calculate the equivalentmatrix of the association matrix and thus requires muchless computational effort than Chen et alrsquos method Thecomputational complexity of Chen et alrsquos method and Zhaoet alrsquos method has relatively high computational complexitywhich indeed motivates our clustering method proposed inthis paper Furthermore from Example 21 we can see that


Table 7 The clustering result of 10 diamonds

Class Confidence level Dual hesitant fuzzy clustering algorithm10 09853 lt 120582 le 1 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

Table 8 Comparisons of the derived results

Classes The results derived by the direct transferalgorithm method

The results derived by Chen et alrsquostransfer algorithm method

The results derived by Zhao et alrsquos Boolemethod

10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10

Figure 1 The clustering result of 10 diamonds

the clustering results have much to do with the thresholdthe smaller the confidence level is the more detailed theclustering will be

5 Conclusions

Dual hesitant fuzzy set as an extension of fuzzy set candescribe the situation that people have hesitancy when theymake a decision more objectively than other extensions offuzzy set (interval-valued fuzzy set intuitionistic fuzzy settype-2 fuzzy set and fuzzy multiset) In this paper thecorrelation coefficients for DHFSs have been studied Theirproperties have been discussed and the differences andcorrelations among them have been investigated in detailWe have made the clustering analysis under dual hesitantfuzzy environments with one typical real world example Tofurther extend the application range of the present clusteringalgorithm in particular for the case that needs to assignweights for different experts it will be necessary to generalizethe original definition of DHFSs

Given that DHFSs are a suitable technique of denotinguncertain information that is widely encountered in daily lifeand the latent applications of our algorithm in the field of data


mining information retrieval and pattern recognition and soforth may be the directions for future research

Acknowledgments

The authors are very grateful to the anonymous reviewers fortheir insightful and constructive comments and suggestionsthat have led to an improved version of this paper This workis supported by the National Nature Science Foundation ofChina (no 70971136)

References

[1] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010

[2] P Bonizzoni G D Vedova R Dondi and T Jiang ldquoCorrelationclustering and consensus clusteringrdquo inAlgorithms and Compu-tation vol 3827 of Lecture Notes in Computer Science pp 226ndash235 2005

[3] H P Kriegel P Kroger E Schubert and A Zimek ldquoA generalframework for increasing the robustness of PCA-based correla-tion clustering algorithmsrdquo in Scientific and Statistical DatabaseManagement vol 5069 of Lecture Notes in Computer Sciencepp 418ndash435 2008

[4] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[5] E Szmidt and J Kacprzyk ldquoCorrelation of intuitionistic fuzzysetsrdquo inComputational Intelligence for Knowledge-Based SystemsDesign vol 6178 of Lecture Notes in Computer Science pp 169ndash177 2010

[6] G W Wei H J Wang and R Lin ldquoApplication of correla-tion coefficient to interval-valued intuitionistic fuzzy multipleattribute decision-making with incomplete weight informa-tionrdquoKnowledge and Information Systems vol 26 no 2 pp 337ndash349 2011

[7] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[8] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[9] W L Hung and J W Wu ldquoA note on the correlation offuzzy numbers by expected intervalrdquo International Journal ofUncertainty Fuzziness and Knowledge-Based Systems vol 9 no4 pp 517ndash523 2001

[10] D H Hong ldquoFuzzy measures for a correlation coefficient offuzzy numbers under 119879

119882(the weakest t-norm)-based fuzzy

arithmetic operationsrdquo Information Sciences vol 176 no 2 pp150ndash160 2006

[11] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[12] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[13] T Gerstenkorn and J Manko ldquoCorrelation of intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 44 no 1 pp 39ndash43 1991

[14] H Bustince and P Burillo ldquoCorrelation of interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Sets and Systems vol 74 no 2pp 237ndash244 1995

[15] W L Hung ldquoUsing statistical viewpoint in developing cor-relation of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness and Knowledge-Based Systems vol 9 no4 pp 509ndash516 2001

[16] H B Mitchell ldquoA correlation coefficient for intuitionistic fuzzysetsrdquo International Journal of Intelligent Systems vol 19 no 5pp 483ndash490 2004

[17] W L Hung and J W Wu ldquoCorrelation of intuitionistic fuzzysets by centroid methodrdquo Information Sciences vol 144 no 1ndash4pp 219ndash225 2002

[18] Z Xu ldquoOn correlation measures of intuitionistic fuzzy setsrdquoIntelligent Data Engineering and Automated Learning vol 4224pp 16ndash24 2006

[19] P J Robinson and E C H Amirtharaj ldquoVague correlationcoefficient of interval vague setsrdquo International Journal of FuzzySystem Applications vol 2 no 1 pp 18ndash34 2012

[20] P J Robinson and E C H Amirtharaj ldquoA short primer on thecorrelation coefficient of vague setsrdquo International Journal ofFuzzy System Applications vol 1 no 2 pp 55ndash69 2011

[21] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[22] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Korea August 2009

[23] N Chen Z Xu andM Xia ldquoCorrelation coefficients of hesitantfuzzy sets and their applications to clustering analysisrdquo AppliedMathematical Modelling vol 37 no 4 pp 2197ndash2211 2013

[24] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[25] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 Academic Press New York NY USA1980

[26] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

[27] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[28] E Szmidt and J Kacprzyk ldquoA similaritymeasure for intuitionis-tic fuzzy sets and its application in supportingmedical diagnos-tic reasoningrdquo in Artificial Intelligence and Soft Computing vol3070 of Lecture Notes in Computer Science pp 388ndash393 2004

[29] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[30] Z Xu J Chen and J Wu ldquoClustering algorithm for intuition-istic fuzzy setsrdquo Information Sciences vol 178 no 19 pp 3775ndash3790 2008

[31] H Zhao Z Xu and Z Wang ldquoIntuitionistic fuzzy clusteringalgorithm based on boolean matrix and association measurerdquoInternational Journal of Information Technology amp DecisionMaking vol 12 no 1 pp 95ndash118 2013

[32] P Z Wang Fuzzy Set Theory and Applications ShanghaiScientific and Technical Publishers Shanghai China 1983

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


simple examples Hesitant fuzzy sets (HFSs) were originallyintroduced by Torra [21 22] In hesitant fuzzy environmentsChen et al [23] derived some correlation coefficient formulasfor HFSs and applied them to two real world examples byusing clustering analysis under hesitant fuzzy environmentsXu and Xia [24] defined the correlationmeasures for hesitantfuzzy information and then discussed their properties indetail

Recently Dubois and Prade introduced the definition ofdual hesitant fuzzy set Dual hesitant fuzzy set can reflecthumanrsquos hesitance more objectively than the other classicalextensions of fuzzy set (intuitionistic fuzzy set type-2 fuzzyset (T-2FS) [25] hesitant fuzzy set etc) The motivation topropose the DHFSs is that when people make a decisionthey are usually hesitant and irresolute for one thing oranother which makes it difficult to reach a final agreementThey further indicated that DHFSs can better deal withthe situations that permit both the membership and thenonmembership of an element to a given set having a fewdifferent values which can arise in a group decision makingproblem For example in the organization some decisionmakers discuss the membership degree 06 and the non-membership 03 of an alternative 119860 that satisfies a criterion119909 Some possibly assign (08 02) while the others assign(07 02) No consistency is reached among these decisionmakers Accordingly the difficulty of establishing a commonmembership degree and a non-membership degree is notbecause we have a margin of error (intuitionistic fuzzy set)or some possibility distribution values (type-2 fuzzy set) butbecause we have a set of possible values (hesitant fuzzy set)For such a case the satisfactory degrees can be representedby a dual hesitant fuzzy element (06 08 07) (03 02)which is obviously different from intuitionistic fuzzy number(08 02)or (07 02) andhesitant fuzzy number 06 08 07The aforementioned measures however cannot be used todeal with the correlation measures of dual hesitant fuzzyinformation Thus it is very necessary to develop sometheories for dual hesitant fuzzy sets However little hasbeen done about this issue In this paper we mainly discussthe correlation measures of dual hesitant fuzzy informationTo do this the remainder of the paper is organized asfollows Section 2 presents some basic concepts related toDHFSs HFSs and IFSs In Section 3 we propose somecorrelation measures of dual hesitant fuzzy elements obtainseveral important conclusions and given an example toillustrate the correlation measures In Section 4 we proposea direct transfer clustering algorithm based on DHFSs andthen use a numerical example to illustrate our algorithmFinally Section 5 concludes the paper with some remarks andpresents future challenges

2 Preliminaries

21 DHFSs HFSs and IFSs

Definition 1 (see [26]) Let119883 be a fixed set then a dual hesitantfuzzy set (DHFS)119863 on119883 is described as

119863 = ⟨119909 ℎ (119909) 119892 (119909)⟩ | 119909 isin 119883 (1)

in which ℎ(119909) and 119892(119909) are two sets of some values in[0 1] denoting the possible membership degrees and non-membership degrees of the element 119909 isin 119883 to the set 119863respectively with the conditions

0 le 120574 120578 le 1 0 le 120574+

+ 120578+

le 1 (2)

where 120574 isin ℎ(119909) 120578 isin 119892(119909) 120574+ isin ℎ+

(119909) = cup120574isinℎ(119909)

max120574 and120578+

isin 119892+

(119909) = cup120578isin119892(119909)

max120578 for all 119909 isin 119883 For conveniencethe pair 119889

119864(119909) = (ℎ

119864(119909) 119892119864(119909)) is called a dual hesitant fuzzy

element (DHFE) denoted by 119889 = (ℎ 119892) with the conditions120574 isin ℎ(119909) 120578 isin 119892(119909) 120574+ isin ℎ

+

(119909) = cup120574isinℎ(119909)

max120574 and 120578+

isin

119892+

(119909) = cup120578isin119892(119909)

max120578 0 le 120574 120578 le 1 and 0 le 120574+

+ 120578+

le 1

Definition 2 (see [21 22]) Let119883 be a fixed set a hesitant fuzzyset (HFS) 119860 on119883 is in terms of a function that when appliedto119883 returns a subset of [0 1] which can be represented as thefollowing mathematical symbol

119860 = ⟨119909 ℎ119860(119909)⟩ | 119909 isin 119883 (3)

where ℎ119860(119909) is a set of values in [0 1] denoting the possible

membership degrees of the element 119909 isin 119883 to the set 119860 Forconvenience we call ℎ

119860(119909) a hesitant fuzzy element (HFE)

We use ⟨119909 ℎ119860⟩ for all 119909 isin 119883 to represent HFSs

Definition 3 (see [11 12]) Let119883 be a fixed set an intuitionisticfuzzy set (IFS) 119860 on119883 is an object having the form

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (4)

which is characterized by a membership function 120583119860and a

non-membership function ]119860 where 120583

119860 119883 rarr [0 1] and

]119860 119883 rarr [0 1] with the condition 0 le 120583

119860(119909) + ]

119860(119909) le 1

for all 119909 isin 119883 We use ⟨119909 120583119860 ]119860⟩ for all 119909 isin 119883 to represent

IFSs considered in the rest of the paper without explicitlymentioning it Furthermore 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is

called a hesitancy degree or an intuitionistic index of 119909 in 119860In the special case 120587(119909) = 0 that is 120583

119860(119909) + ]

119860(119909) = 1 the

IFS 119860 reduces to an FS

22 Correlation Coefficients of HFSs and IFSs Manyapproaches [4 13 17 20 21] have been introduced to computethe correlation coefficients of IFSs Let 119883 = 119909

1 1199092 119909

119899

be a discrete universe of discourse for any two 119860 and 119861 on119883

The correlation of the IFSs 119860 and 119861 is defined as [13]

119862IFS1(119860 119861) =

119899

sum119894=1

(119906119860(119909119894) 119906119861(119909119894) + V119860(119909119894) 119906119861(119909119894)) (5)



120588IFS1(119860 119861) =

119862IFS1(119860 119861)

(119862IFS1(119860 119860) sdot 119862IFS

1(119861 119861))

12

= (

119899

sum119894=1

(119906119860(119909119894) 119906119861(119909119894) + V119860(119909119894) 119906119861(119909119894)))

times (((

119899

sum119894=1

(1199062

119860(119909119894) + V2119860(119909119894)))

sdot(

119899

sum119894=1

(1199062

119861(119909119894) + V2119861(119909119894))))

12

)

minus1

(6)


119862HFS1(119860 119861) =

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894))

120588IFS1(119860 119861) =

119862HFS1(119860 119861)

(119862HFS1(119860 119860) sdot 119862HFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894)))

times(((

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119860120590(119895)

(119909119894)))

sdot(

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119861120590(119895)

(119909119894))))

12

)

minus1

(7)

where 119897119894= max119897(ℎ

119860(119909119894)) 119897(ℎ119861(119909119894)) for each 119909

119894in 119883 and

119897(ℎ119860(119909119894)) and 119897(ℎ


ℎ119860(119909119894) and ℎ


119894in detail

in the next section




119864(119909) 119892119864(119909)) in



120590(119895)


119864(119909) Let


119864(119909119894) and 119897

119892(119889119864(119909119894)) be


119894)) =


and 119861 119897(119889119860(119909119894)) = 119897(119889

119861(119909119894)) that is 119897

ℎ(119889119860(119909119894)) = 119897ℎ(119889119861(119909119894))





1

(119860 119861) is defined by

119862DHFS1(119860 119861) =

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

(8)


1

(119860 119861) is defined by

120588DHFS1(119860 119861)

=119862DHFS

1(119860 119861)

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(9)



2

(119860 119861) is defined by

120588DHFS2(119860 119861)

=119862DHFS

1(119860 119861)

max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(10)


119863119867119865119878119894


(1) 0 le 120588119863119867119865119878

119894

(119860 119861) le 1

(2) 119860 = 119861 rArr 120588119863119867119865119878

119894

(119860 119861) = 1

(3) 120588119863119867119865119878

119894

(119860 119861) = 120588119863119867119865119878

119894

(119861 119860) 119894 = 1 2


(119860 119861) and 0 le

120588DHFS2


1

(119860 119861) le 1 120588DHFS2

(119860 119861) le 1

119862DHFS1(119860 119861)

=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

=

119899

sum119894=1

(

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

119897ℎ(119894)

+

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)

119897119892(119894)

)

= (

119897ℎ(1)

sum119895=1

120574119860120590(119895)

(1199091)

radic119897ℎ(1)

sdot120574119861120590(119895)

(1199091)

radic119897ℎ(1)

+

119897ℎ(2)

sum119895=1

120574119860120590(119895)

(1199092)

radic119897ℎ(2)

sdot120574119861120590(119895)

(1199092)

radic119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

120574119860120590(119895)

(119909119899)

radic119897ℎ(119899)

sdot120574119861120590(119895)

(119909119899)

radic119897ℎ(119899)

)

+(

119897119892(1)

sum119896=1

120578119860120590(119896)

(1199091)

radic119897119892(1)

sdot120578119861120590(119896)

(1199091)

radic119897119892(1)

+

119897119892(2)

sum119896=1

120578119860120590(119896)

(1199092)

radic119897119892(2)

sdot120578119861120590(119896)

(1199092)

radic119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

120578119860120590(119896)

(119909119899)

radic119897119892(119899)

sdot120578119861120590(119896)

(119909119899)

radic119897119892(119899)

)

(11)


(11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899)2

le (1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

(12)

where (1199091 1199092 119909

119899) isin 119877119899

(1199101 1199102 119910

119899) isin 119877119899 we obtain

119862DHFS1(119860 119861)

2

le (

119897ℎ(1)

sum119895=1

1205742

119860120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119860120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119860120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119860120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119860120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119860120590(119896)

(119909119899)

119897119892(119899)

)

sdot (

119897ℎ(1)

sum119895=1

1205742

119861120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119861120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119861120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119861120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119861120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119861120590(119896)

(119909119899)

119897119892(119899)

)


=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

= 119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861)

(13)

Therefore

119862DHFS1(119860 119861) le (119862DHFS

1

(119860 119860))12

sdot (119862DHFS1(119861 119861))

12

(14)

So 0 le 120588DHFS1


(1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

le (max ((11990921+ 1199092


2

119899) (1199102

1+ 1199102


2

119899)))2

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

le (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)+

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

2

(15)

Then

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

le max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

(16)





1

(119860 119861) ge 120588119863119867119865119878

2

(119860 119861)



3

(119860 119861) is defined by

120588DHFS3(119860 119861)

= (1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(17)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(18)


3



3


(1) 0 le 120588119863119867119865119878

3

(119860 119861) le 1(2) 119860 = 119861 hArr 120588

1198631198671198651198783

(119860 119861) = 1(3) 120588119863119867119865119878

3

(119860 119861) = 120588119863119867119865119878

3

(119861 119860)



3

(119860 119861) le 1

Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

for 119894 = 1 2 119899

=(Δ120574120582

minΔ120574120582

max) + 1

(Δ120574120582119894Δ120574120582max) + 1

+(Δ120578120582

minΔ120578120582

max) + 1

(Δ120578120582119894Δ120578120582max) + 1

le 2

(19)

We obtain

120588DHFS3(119860 119861)

=1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

)

le1

2119899sdot 2119899

= 1

(20)






119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend



(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)





3(typhoid) 119876

4(stomach problem)



2(headache)

1198783(cough) 119878


5(chest pain) Xu


119894119895) and (120578

119894119895)


119894














Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588IFS1(119860 119861) =

119862IFS1(119860 119861)

(119862IFS1(119860 119860) sdot 119862IFS

1(119861 119861))

12

= (

119899

sum119894=1

(119906119860(119909119894) 119906119861(119909119894) + V119860(119909119894) 119906119861(119909119894)))

times (((

119899

sum119894=1

(1199062

119860(119909119894) + V2119860(119909119894)))

sdot(

119899

sum119894=1

(1199062

119861(119909119894) + V2119861(119909119894))))

12

)

minus1

(6)


119862HFS1(119860 119861) =

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894))

120588IFS1(119860 119861) =

119862HFS1(119860 119861)

(119862HFS1(119860 119860) sdot 119862HFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ119860120590(119895)

(119909119894) ℎ119861120590(119895)

(119909119894)))

times(((

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119860120590(119895)

(119909119894)))

sdot(

119899

sum119894=1

(1

119897119894

119897119894

sum119895=1

ℎ2

119861120590(119895)

(119909119894))))

12

)

minus1

(7)

where 119897119894= max119897(ℎ

119860(119909119894)) 119897(ℎ119861(119909119894)) for each 119909

119894in 119883 and

119897(ℎ119860(119909119894)) and 119897(ℎ


ℎ119860(119909119894) and ℎ


119894in detail

in the next section




119864(119909) 119892119864(119909)) in



120590(119895)


119864(119909) Let


119864(119909119894) and 119897

119892(119889119864(119909119894)) be


119894)) =


and 119861 119897(119889119860(119909119894)) = 119897(119889

119861(119909119894)) that is 119897

ℎ(119889119860(119909119894)) = 119897ℎ(119889119861(119909119894))





1

(119860 119861) is defined by

119862DHFS1(119860 119861) =

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

(8)


1

(119860 119861) is defined by

120588DHFS1(119860 119861)

=119862DHFS

1(119860 119861)

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(9)



2

(119860 119861) is defined by

120588DHFS2(119860 119861)

=119862DHFS

1(119860 119861)

max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(10)


119863119867119865119878119894


(1) 0 le 120588119863119867119865119878

119894

(119860 119861) le 1

(2) 119860 = 119861 rArr 120588119863119867119865119878

119894

(119860 119861) = 1

(3) 120588119863119867119865119878

119894

(119860 119861) = 120588119863119867119865119878

119894

(119861 119860) 119894 = 1 2


(119860 119861) and 0 le

120588DHFS2


1

(119860 119861) le 1 120588DHFS2

(119860 119861) le 1

119862DHFS1(119860 119861)

=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

=

119899

sum119894=1

(

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

119897ℎ(119894)

+

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)

119897119892(119894)

)

= (

119897ℎ(1)

sum119895=1

120574119860120590(119895)

(1199091)

radic119897ℎ(1)

sdot120574119861120590(119895)

(1199091)

radic119897ℎ(1)

+

119897ℎ(2)

sum119895=1

120574119860120590(119895)

(1199092)

radic119897ℎ(2)

sdot120574119861120590(119895)

(1199092)

radic119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

120574119860120590(119895)

(119909119899)

radic119897ℎ(119899)

sdot120574119861120590(119895)

(119909119899)

radic119897ℎ(119899)

)

+(

119897119892(1)

sum119896=1

120578119860120590(119896)

(1199091)

radic119897119892(1)

sdot120578119861120590(119896)

(1199091)

radic119897119892(1)

+

119897119892(2)

sum119896=1

120578119860120590(119896)

(1199092)

radic119897119892(2)

sdot120578119861120590(119896)

(1199092)

radic119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

120578119860120590(119896)

(119909119899)

radic119897119892(119899)

sdot120578119861120590(119896)

(119909119899)

radic119897119892(119899)

)

(11)


(11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899)2

le (1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

(12)

where (1199091 1199092 119909

119899) isin 119877119899

(1199101 1199102 119910

119899) isin 119877119899 we obtain

119862DHFS1(119860 119861)

2

le (

119897ℎ(1)

sum119895=1

1205742

119860120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119860120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119860120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119860120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119860120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119860120590(119896)

(119909119899)

119897119892(119899)

)

sdot (

119897ℎ(1)

sum119895=1

1205742

119861120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119861120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119861120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119861120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119861120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119861120590(119896)

(119909119899)

119897119892(119899)

)


=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

= 119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861)

(13)

Therefore

119862DHFS1(119860 119861) le (119862DHFS

1

(119860 119860))12

sdot (119862DHFS1(119861 119861))

12

(14)

So 0 le 120588DHFS1


(1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

le (max ((11990921+ 1199092


2

119899) (1199102

1+ 1199102


2

119899)))2

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

le (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)+

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

2

(15)

Then

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

le max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

(16)





1

(119860 119861) ge 120588119863119867119865119878

2

(119860 119861)



3

(119860 119861) is defined by

120588DHFS3(119860 119861)

= (1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(17)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(18)


3



3


(1) 0 le 120588119863119867119865119878

3

(119860 119861) le 1(2) 119860 = 119861 hArr 120588

1198631198671198651198783

(119860 119861) = 1(3) 120588119863119867119865119878

3

(119860 119861) = 120588119863119867119865119878

3

(119861 119860)



3

(119860 119861) le 1

Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

for 119894 = 1 2 119899

=(Δ120574120582

minΔ120574120582

max) + 1

(Δ120574120582119894Δ120574120582max) + 1

+(Δ120578120582

minΔ120578120582

max) + 1

(Δ120578120582119894Δ120578120582max) + 1

le 2

(19)

We obtain

120588DHFS3(119860 119861)

=1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

)

le1

2119899sdot 2119899

= 1

(20)






119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend



(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)





3(typhoid) 119876

4(stomach problem)



2(headache)

1198783(cough) 119878


5(chest pain) Xu


119894119895) and (120578

119894119895)


119894














Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2

(119860 119861) is defined by

120588DHFS2(119860 119861)

=119862DHFS

1(119860 119861)

max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

= (

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(10)


119863119867119865119878119894


(1) 0 le 120588119863119867119865119878

119894

(119860 119861) le 1

(2) 119860 = 119861 rArr 120588119863119867119865119878

119894

(119860 119861) = 1

(3) 120588119863119867119865119878

119894

(119860 119861) = 120588119863119867119865119878

119894

(119861 119860) 119894 = 1 2


(119860 119861) and 0 le

120588DHFS2


1

(119860 119861) le 1 120588DHFS2

(119860 119861) le 1

119862DHFS1(119860 119861)

=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894))

=

119899

sum119894=1

(

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

119897ℎ(119894)

+

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)

119897119892(119894)

)

= (

119897ℎ(1)

sum119895=1

120574119860120590(119895)

(1199091)

radic119897ℎ(1)

sdot120574119861120590(119895)

(1199091)

radic119897ℎ(1)

+

119897ℎ(2)

sum119895=1

120574119860120590(119895)

(1199092)

radic119897ℎ(2)

sdot120574119861120590(119895)

(1199092)

radic119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

120574119860120590(119895)

(119909119899)

radic119897ℎ(119899)

sdot120574119861120590(119895)

(119909119899)

radic119897ℎ(119899)

)

+(

119897119892(1)

sum119896=1

120578119860120590(119896)

(1199091)

radic119897119892(1)

sdot120578119861120590(119896)

(1199091)

radic119897119892(1)

+

119897119892(2)

sum119896=1

120578119860120590(119896)

(1199092)

radic119897119892(2)

sdot120578119861120590(119896)

(1199092)

radic119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

120578119860120590(119896)

(119909119899)

radic119897119892(119899)

sdot120578119861120590(119896)

(119909119899)

radic119897119892(119899)

)

(11)


(11990911199101+ 11990921199102+ sdot sdot sdot + 119909

119899119910119899)2

le (1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

(12)

where (1199091 1199092 119909

119899) isin 119877119899

(1199101 1199102 119910

119899) isin 119877119899 we obtain

119862DHFS1(119860 119861)

2

le (

119897ℎ(1)

sum119895=1

1205742

119860120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119860120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119860120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119860120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119860120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119860120590(119896)

(119909119899)

119897119892(119899)

)

sdot (

119897ℎ(1)

sum119895=1

1205742

119861120590(119895)

(1199091)

119897ℎ(1)

+

119897ℎ(2)

sum119895=1

1205742

119861120590(119895)

(1199092)

119897ℎ(2)

+ sdot sdot sdot +

119897ℎ(119899)

sum119895=1

1205742

119861120590(119895)

(119909119899)

119897ℎ(119899)

+

119897119892(1)

sum119896=1

1205782

119861120590(119896)

(1199091)

119897119892(1)

+

119897119892(2)

sum119896=1

1205782

119861120590(119896)

(1199092)

119897119892(2)

+ sdot sdot sdot +

119897119892(119899)

sum119896=1

1205782

119861120590(119896)

(119909119899)

119897119892(119899)

)


=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

= 119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861)

(13)

Therefore

119862DHFS1(119860 119861) le (119862DHFS

1

(119860 119860))12

sdot (119862DHFS1(119861 119861))

12

(14)

So 0 le 120588DHFS1


(1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

le (max ((11990921+ 1199092


2

119899) (1199102

1+ 1199102


2

119899)))2

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

le (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)+

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

2

(15)

Then

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

le max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

(16)





1

(119860 119861) ge 120588119863119867119865119878

2

(119860 119861)



3

(119860 119861) is defined by

120588DHFS3(119860 119861)

= (1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(17)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(18)


3



3


(1) 0 le 120588119863119867119865119878

3

(119860 119861) le 1(2) 119860 = 119861 hArr 120588

1198631198671198651198783

(119860 119861) = 1(3) 120588119863119867119865119878

3

(119860 119861) = 120588119863119867119865119878

3

(119861 119860)



3

(119860 119861) le 1

Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

for 119894 = 1 2 119899

=(Δ120574120582

minΔ120574120582

max) + 1

(Δ120574120582119894Δ120574120582max) + 1

+(Δ120578120582

minΔ120578120582

max) + 1

(Δ120578120582119894Δ120578120582max) + 1

le 2

(19)

We obtain

120588DHFS3(119860 119861)

=1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

)

le1

2119899sdot 2119899

= 1

(20)






119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend



(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)





3(typhoid) 119876

4(stomach problem)



2(headache)

1198783(cough) 119878


5(chest pain) Xu


119894119895) and (120578

119894119895)


119894














Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

= 119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861)

(13)

Therefore

119862DHFS1(119860 119861) le (119862DHFS

1

(119860 119860))12

sdot (119862DHFS1(119861 119861))

12

(14)

So 0 le 120588DHFS1


(1199092

1+ 1199092


2

119899) sdot (1199102

1+ 1199102


2

119899)

le (max ((11990921+ 1199092


2

119899) (1199102

1+ 1199102


2

119899)))2

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

le (max

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)+

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

2

(15)

Then

(119862DHFS1(119860 119860) sdot 119862DHFS

1(119861 119861))

12

le max 119862DHFS1(119860 119860) 119862DHFS

1(119861 119861)

(16)





1

(119860 119861) ge 120588119863119867119865119878

2

(119860 119861)



3

(119860 119861) is defined by

120588DHFS3(119860 119861)

= (1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(17)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(18)


3



3


(1) 0 le 120588119863119867119865119878

3

(119860 119861) le 1(2) 119860 = 119861 hArr 120588

1198631198671198651198783

(119860 119861) = 1(3) 120588119863119867119865119878

3

(119860 119861) = 120588119863119867119865119878

3

(119861 119860)



3

(119860 119861) le 1

Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

for 119894 = 1 2 119899

=(Δ120574120582

minΔ120574120582

max) + 1

(Δ120574120582119894Δ120574120582max) + 1

+(Δ120578120582

minΔ120578120582

max) + 1

(Δ120578120582119894Δ120578120582max) + 1

le 2

(19)

We obtain

120588DHFS3(119860 119861)

=1

2119899

119899

sum119894=1

(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

)

le1

2119899sdot 2119899

= 1

(20)






119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend



(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)





3(typhoid) 119876

4(stomach problem)



2(headache)

1198783(cough) 119878


5(chest pain) Xu


119894119895) and (120578

119894119895)


119894














Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894isin 119883 is 119908

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum119899

119894=1119908119894= 1 then we extend



(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times((

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894) +

1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

sdot

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894)))

12

)

minus1

(21)


(119860 119861)

= (

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

120574119860120590(119895)

(119909119894) 120574119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

120578119860120590(119896)

(119909119894) 120578119861120590(119896)

(119909119894)))

times (max

119908119894

119899

sum119894=1

(1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119860120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119860120590(119896)

(119909119894))

119899

sum119894=1

119908119894(

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

1205742

119861120590(119895)

(119909119894)

+1

119897119892(119894)

119897119892(119894)

sum119896=1

1205782

119861120590(119896)

(119909119894))

)

minus1

(22)


(119860 119861)

= (1

2

119899

sum119894=1

119908119894(Δ120574120582

min + Δ120574120582

max

Δ120574120582119894+ Δ120574120582max

+Δ120578120582

min + Δ120578120582

max

Δ120578120582119894+ Δ120578120582max

))

1120582

(23)

where

Δ120574120582

119894=

1

119897ℎ(119894)

119897ℎ(119894)

sum119895=1

100381610038161003816100381610038161003816120574119860120590(119895)

(119909119894) minus 120574119861120590(119895)

(119909119894)100381610038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120574120582

119894 Δ120574

120582

max = max119894

Δ120574120582

119894

Δ120578120582

119894=

1

119897119892(119894)

119897119892(119894)

sum119896=1

10038161003816100381610038161003816120578119860120590(119896)

(119909119894) minus 120578119861120590(119896)

(119909119894)10038161003816100381610038161003816

120582

Δ120574120582

min = min119894

Δ120578120582

119894 Δ120574

120582

max = max119894

Δ120578120582

119894

120582 gt 0

(24)





3(typhoid) 119876

4(stomach problem)



2(headache)

1198783(cough) 119878


5(chest pain) Xu


119894119895) and (120578

119894119895)


119894














Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Viral fever (06 04 03 )(02 00)

(07 05 03 02)(03 01)

(05 03)(05 04 02)

(05 04 03 02 01)(0503)

(05 04 02 01)(05 04 03)

Malaria (09 08 07)(01 00)

(05 03 02 01)(04 03)

(02 01)(07 06 05)

(06 05 03 02 01)(03 02)

(04 03 02 01)(06 05 04)

Typhoid (06 03 01)(03 02)

(09 08 07 06)(01 00)

(05 03)(05 04 03)

(05 04 03 02 01)(05 04)

(06 04 03 02)(04 03 02)


(04 03 02 01)(04 03)

(04 03)(06 05 04)

(09 08 07 06 05)(01 00)

(05 04 02 01)(05 04 03)

Chest problem (03 02 01)(07 06)

(05 03 02 01)(05 03)

(03 02)(06 04 03)

(07 06 05 03 02)(02 01)

(08 07 06 05)(02 01 00)



Al (09 07 05) (01 00) (04 03 02 01)(05 04) (04 03) (05 04 02) (06 05 04 02

01) (0302) (04 03 02 01) (05 04 03)

Bob (05 04 02) (05 03) (05 04 03 01)(04 03) (02 01) (07 06 05) (09 08 06 05

04) (01 00) (05 04 03 02) (05 04 03)

Joe (09 07 06) (01 00) (07 04 03 01)(02 01) (03 02) (05 04 03) (06 04 03 02

01) (04 03) (06 03 02 01) (04 03 02)

Ted (08 07 05) (02 01) (06 05 04 02)(04 03) (05 03) (05 04 03) (06 04 03 02

01) (04 03) (05 04 02 01) (05 04 03)



Chestproblem

Al 09257 09620 07957 08680 07110Bob 08380 08791 08041 09922 09035Joe 09427 09521 08757 09329 07026Ted 09718 09472 08890 08790 07644



(120588119894119895)119898times119898


119894 119860119895)




(1) 0 le 120588119894119895le 1 for all 119894 119895 = 1 2 119898

(2) 120588119894119895= 1 if and only if 119860

119894= 119860119895

(3) 120588119894119895= 120588119895119894 for all 119894 119895 = 1 2 119898



119894119895)119898times119898



119894119896 120588119896119895 for

all 119894 119895 = 1 2 119898






2119896+1

derived from 1198622119896+1

= 1198622119896

∘ 1198622119896




max119896

min 120588119894119896 120588119896119895 le 120588

119894119895 forall119894 119895 = 1 2 119898 (25)





2

rarr 1198624



119896

= 1198622119896+1

and 1198622119896




120582= (120582

120588119894119895)119898times119898


120582

120588119894119895=

0 if 120588119894119895lt 120582

1 if 120588119894119895ge 120582

119894 119895 = 1 2 119898 (26)








Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Step 1 Let 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)



119862 where

1198622

= 119862 ∘ 119862 = (120588119894119895)119898times119898

120588119894119895= max119896

min 120588119894119896 120588119896119895

119894 119895 = 1 2 119898

(27)


119896

119862 rarr 1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898





119894and119860

119895are


119895(119895 = 1 2 119898)



119901 when 120588(119909 119909

1) ge 120582 120588(119909

1 1199092) ge 120582

120588(1199092 1199093) ge 120582 120588(119909

119901 119910) ge 120582 then 119909 119909

1 1199092 119909

119901and 119910



1 1199092 1199093 119909119901 when (119909 119909

1) isin 119877 (119909

1 1199092) isin

119877 (119909119896 119909119896+1




Step 1 Let 119860 = 1198601 1198602 119860


1199091 1199092 119909



(120588119894119895)119898times119898

where 120588119894119895= 120588(119860

119894 119860119895)


120582= (120582

120588119894119895)119898times119898

If 120588119894119895= 1




1 2 119898)


2

rarr

1198624



= 1198622119896+1


= (120582

120588119894119895)119898times119898



120582= (120582

120588119894119895)119898times119898


119895(119895 =





isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











isin 119860exist1199091 1199092 1199093 119909119901 if 120588(119860

119894 1199091) ge 120582 120588(119909

1 1199092) ge 120582 120588(119909

2 1199093) ge

120582 120588(119909119901 119860119895) ge 120582 then 119860

119894and 119860



matrix 1198622119896


1198622119896 (119860119894 119909119895) ge 120582 Consider

1205881198622 (119860119894 119860119895)

= max119860119902isin119860

min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119894= 119860119895

max min 120588119862(119860119894 119860119902) 120588119862(119860119902 119860119895)

120588119862(119860119894 119860119895)

ge 120588119862(119860119894 119860119895)

(28)

So 1198622


supe

1198622119896minus1

supe 1198622119896minus2

supe 1198622


1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896 (119860119894 119860119902) 1205881198622119896 (119860119902 119860119895)

119894 119895 119902 = 1 2 119898

= max119860119902isin119860119860

119902= 119860119909119901

maxmin 1205881198622119896 (119860119894 119860119901) 1205881198622119896 (119860119901 119860119895)

min 1205881198622119896 (119860119894 119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 119860119909119901minus1

)

1205881198622119896 (119860119909119901minus1

119860119909119901

) 1205881198622119896 (119860119909119901

119860119895)

ge min 1205881198622119896 (119860119894 1198601199091

) 1205881198622119896 (1198601199091

1198601199092

)

1205881198622119896 (119860119894 119860119909119901minus1

) 1205881198622119896 (119860119909119901minus1

119860119909119901

)

1205881198622119896 (119860119909119901

119860119895)

ge min 120588119862(119860119894 1198601199091

) 120588119862(1198601199091

1198601199092

)

120588119862(119860119894 119860119909119901minus1

)

120588119862(119860119909119901minus1

119860119909119901

) 120588119862(119860119909119901

119860119895)

ge 120582

(29)


119895are




1205881198622119896 (119860119894 119909119895) ge 120582 then 119860

119894


Let

1205881198622119896 (119860119894 119860119895)

= max119860119902isin119860

min 1205881198622119896minus1 (119860

119894 119860119902) 1205881198622119896minus1 (119860

119902 119860119895)

(30)

Thenexist1199091 1205881198622119896 (119860119894 119860119895) = min120588

1198622119896minus1 (119860119894 1198601199091

) 1205881198622119896minus1 (1198601199091

119860119895) ge 120582 120588

1198622119896minus1 (119860119894 1198601199091

) ge 120582 1205881198622119896minus1 (1198601199091

119860119895) ge 120582

So 119860119894and 119860


2119896minus1



) ge 120582 1205881198622119896minus2 (1198601199092

1198601199091

) ge 120582 exist1199093 1205881198622119896minus2 (1198601199091

1198601199093

) ge 120582 1205881198622119896minus2 (1198601199093

119860119895) ge 120582

119860119894and119860


119896minus2


So exist1199091 1199092 1199093 119909

2119896 120588119862(119860119894 1198601199091

) ge 120582 120588119862(1198601199091

1198601199092

) ge

120582 120588119862(1198601199092

1198601199093

) ge 120582 120588119862(1198601199092119896 119860119895) ge 120582

119860119894and 119860



119894and 119860

119895are of


We assume 119860 = 1198601 1198602 119860



119862 rarr

1198622

rarr 1198624



= 1198622119896+1


120582= (120582

120588119894119895)119898times119898


3

+ 1198961198982


2


2


120582= (120582

120588119894119895)119898times119898








(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(05 04 03) (04 02) (06 05) (03 02 01) (06 04 03) (04 02 01) (06) (04)1198602

(08 07 06) (02 01) (07 06) (03 02 01) (07 06 05) (03 02 01) (07) (02)1198603

(09 08 07) (01 00) (08 07) (02 01 00) (08 07 06) (02 01 00) (09) (01)1198604

(04 03 01) (06 05) (06 05) (04 02 01) (06 05 04) (03 02 01) (03) (06)1198605

(06 05 04) (03 02) (03 02) (06 05 04) (03 02 01) (06 05 04) (01) (08)1198606

(06 05 04) (04 02) (07 06) (03 02 01) (02 01 00) (07 02 01) (08) (01)1198607

(08 06 05) (02 01) (06 05) (03 02 01) (04 03 02) (05 04 03) (05) (04)1198608

(07 06 05) (02 00) (04 03) (06 05 04) (06 05 04) (04 03 02) (08) (02)1198609

(04 03 02) (06 05) (04 03) (06 05 04) (02 01 00) (08 06 05) (02) (06)11986010

(02 01 00) (07 06) (08 06) (02 01 00) (06 05 03) (04 02 01) (07) (03)





119862 =

((((((((

(

10000 09495 09010 09227 07571 08270 09542 09123 07778 09241

09495 10000 09853 08053 06436 08948 09457 09404 06226 08260

09010 09853 10000 07164 05146 08697 08904 09076 04867 07811

09227 08053 07164 10000 08174 07353 08490 07463 08484 09025

07571 06436 05146 08174 10000 06003 08240 06997 09316 05822

08270 08948 08697 07353 06003 10000 09048 08750 06948 08540

09542 09457 08904 08490 08240 09048 10000 09105 07993 08018

09123 09404 09076 07463 06997 08750 09105 10000 06826 07612

07778 06226 04867 08484 09316 06948 07993 06826 10000 07206

09241 08260 07811 09025 05822 08540 08018 07612 07206 10000

))))))))

)

(31)








1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

9 09542 lt 120582 le 09853 1198602 1198603 1198601 1198604 1198605 1198606 1198607 1198608 1198609 11986010

8 09495 lt 120582 le 09542 1198602 1198603 1198601 1198607 1198604 1198605 1198606 1198608 1198609 11986010

7 09404 lt 120582 le 09495 1198601 1198602 1198603 1198607 1198604 1198605 1198606 1198608 1198609 11986010

6 09316 lt 120582 le 09404 1198601 1198602 1198603 1198607 1198608 1198604 1198605 1198606 1198609 11986010

5 09241 lt 120582 le 09316 1198601 1198602 1198603 1198607 1198608 1198605 1198609 1198604 1198606 11986010

4 09227 lt 120582 le 09241 1198601 1198602 1198603 1198607 1198608 11986010 1198605 1198609 1198604 1198606

3 09123 lt 120582 le 09227 1198601 1198602 1198603 1198604 1198607 1198608 11986010 1198605 1198609 1198606

2 08484 lt 120582 le 09123 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010 1198605 1198609

1 0 lt 120582 le 08484 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010





10 1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198601 1198602 1198603 1198604 1198605

1198606 1198607 1198608 1198609 11986010

9 1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

1198602 1198603 1198601 1198604 1198605

1198606 1198607 1198608 1198609 11986010

8 1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198602 1198603 1198601 1198607 1198604 1198605

1198606 1198608 1198609 11986010

7 1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

1198601 1198602 1198603 1198607 1198604 1198605

1198606 1198608 1198609 11986010

6 1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608

1198604 1198605 1198606 1198609 11986010

1198601 1198602 1198603 1198607 1198608 1198604 1198605

1198606 1198609 11986010

5 1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

1198601 1198602 1198603 1198607 1198608

1198605 1198609 1198604 1198606 11986010

4 1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

1198601 1198602 1198603 1198607 1198608 11986010

1198605 1198609 1198604 1198606

3 1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

1198601 1198602 1198603 1198604 1198607 1198608 11986010

1198605 1198609 1198606

2 1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1198601 1198602 1198603 1198604 1198606 1198607 1198608 11986010

1198605 1198609

1 1198601 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010 119860

1 1198602 1198603 1198604 1198605 1198606 1198607 1198608 1198609 11986010

09495

09542

09854

09227

09241

09316

09404

08484

09123

120582 A2 A3 A1 A7 A8 A4 A6 A5 A9A10



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Correlation Measures of Dual Hesitant...

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