1 Application of a Fuzzy MCDM Model to the Evaluation and Selection of Smart Phones Advisor:Prof....

1

Application of a Fuzzy MCDM Modelto the Evaluation and Selection of

Smart Phones

Advisor:Prof. Chu, Ta-Chung

Student: Chen, Chih-kai

2

INTRODUCTION

LITERATURE REVIEW

FUZZY SET THEORY

MODEL ESTABLISHMENT

NUMERICAL EXAMPLE

CONCLUSIONS

3

Background and Motivation

Why We Apply FMCDM

Research Objectives

Research Framework

INTRODUCTION

4

According to Gartner, International Research and Consulting:

1. In the third quarter of 2010, worldwide mobile phone sales volume of 417 million represented the growth of 35% over the same period in 2009.

2. In addition, smart phone sales volume is more substantial in the third quarter of 2009 which grew 96%.

3. The proportion of mobile phone sales volume has slightly increased to 19.3%.

5

The prices of the mobile phones continue to lower and the market tends to be saturated, the manufacturers cannot get as high gross margin as they did in the past.

Therefore, manufacturers have begun to research and produce smart phones.

The stereotype about the smart phones:

1. These devices are just for the businessmen.

2. It’s expensive.

3. Interfaces are too incomprehensible.

6

Why We Apply FMCDMEvaluating smart phone many criteria (or factors) need to be

considered.

Different decision-makers also have different thoughts about the weight of each criterion.

evaluating smart phone

Quantitative Qualitative

Hardware PriceBrand Awareness

of Smart Phone

Operating SystemsNumber of

ApplicationsSecurity

7

Research Objectives The objectives of this study are listed as follows:

1. Smart phone selection related to literature is investigated.

2. Criteria for selecting smart phones are analyzed.

3. A fuzzy MCDM approach is established.

4. A total relative area for ranking fuzzy numbers is suggested.

5. A numerical example is used to demonstrate the computational process of the proposed model.

88

Chapter 1Introduction

Chapter 2Literature Review

Chapter 3Fuzzy Set Theory

Chapter 4Model Establishment

Chapter 5Numerical Example

Chapter 6Conclusion

9

LITERATURE REVIEWDefinition of Mobile Phone What is a System of Smart PhoneThe overview of the system development of the

smart phone Criteria AssessmentRelated literatures on smart phonesRelated Works on FMCDMFuzzy Number RankingCriteria Assessment

10

Definition of Mobile PhoneFeature Phones Smart Phones

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Feature Phones Feature phone has its own mobile phone manufacturer’s opera

ting system (OS), which has a basic audio and video call beside the additional features (e.g. taking photos, sending text message, listening to music, etc.) but it is not allowed to install or remove software (e.g. remove preset program or install GPS software).

However, if the phone supports JAVA and BREW, it is able to install applications.

The software being developed through the two systems is not user-friendly.

12

Smart PhonesSu (2009) Smart phone is a phone equipped with function of

phone and PDA.

Yang (2009) Smart phones are developed because industrial technology progresses and consumers require integrating multiple requirements into one device.

Hsu (2004) In addition to the original function of voice communications, a mobile phone should also be equipped with an open operating system, and sufficient processing power, allowing users to choose application freely and expand multiple or limitless functions.

13

According to the above-mentioned definitions of smart phones, this study defined “Smart Phone” as follows:

1. Opening source operating system platform.

2. Strong support on the third party's applications, and is allowed to install or remove software freely.

3. A strong hardware performance and faster processing power.

14

What is a System of Smart PhoneAlter (2002)

Operating System (OS) “The system that controls the execution of all other programs, communication with peripheral devices and use of memory and resources.”

Malykhina (2007) OS “the heart of the smart phone, it determines a phone's features, performance, security, and application installation.”

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The overview development of the smart phone

According to Gartner's statistics, the leading market of smart phone operating system:

Smart Phone Operating Systems

3Q10 sales

3Q10 Market

share

3Q09 sales

3Q09 Market

share Symbian 29,480.1 36.6% 18,314.8 44.6%

Android 20,500.0 25.5% 1,424.5 3.5%

iOS 13,484.4 16.7% 7,040.4 17.1%

Research In Motion 11,908.3 14.8% 8,522.7 20.7%

Microsoft Windows Mobile 22,247.9 2.8% 3,259.9 7.9%

Linux 1,697.1 2.1% 1,918.5 4.7%

Other OS 1,214.8 1.5% 612.5 1.5%

Total 80,532.6 100.0% 41,093.3 100.0%

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Related literatures on smart phonesYang (2009), “ The Development Trend Analysis of

Smart Phone Industry”, he designs an expert question-naire and interviews with 7 experts.

operating interface entertainment platform mobile business specification of software and hardware.

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Lin and Ye (2009), “ Operating System Battle in the Ecosystem of Smart phone Industry”, they adopt concept of Food Web to explain ecosystem of various smart phone OS.

They found out: device maker third-party application developer

are two key sources.

Information Week (2007), “Survey of Emphasized Functions of Smart phones towards 325 Experts in Related Field”.

The results demonstrated that security and easy integration with PC obtain the first and second place respectively.

Gizmodo (2008), “Smart phone OS Comparison Chart”.17 functions are listed in a table and the presence and absence of each function are compared.

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Many works investigated smart phone operating system as the research object, however, these works cannot offer consumers a method to evaluate and select smart phones.

Most importantly, when consumers choose to buy smart phones, they will usually consider both hardware and operating system.

This thesis proposes a fuzzy multiple criteria decision making approach to comprehensively consider criteria in hardware and operating system in order to help consumers evaluate and select smart phones.

19

Related literature on fuzzy MCDMIn 1970, Bellman and Zadeh introduced fuzzy set theory to

multi-criteria decision making, which involved fuzzy decision analysis concepts and models for solving the problem of uncertainty in decision-making.

Since then, the fuzzy multiple criteria decision making has resulted in many researches.

Fuzzy numbers can be used to better describe suitability of alternatives versus qualitative criteria under fuzzy multiple criteria decision making environment.

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Fuzzy MCDM is mainly divided into two categories:

1. fuzzy multiple-objective decision-making:

It is mainly used in the "planning; design aspects ".

2. fuzzy multiple criteria (attributes) decision-making:It is mainly used in the "assessment; selection aspects".

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Chang et al. (2009) applied the fuzzy multi-criteria decision making method in enterprise organization to establish the key influential factors for the success of knowledge management.

Chou (2007) used fuzzy multiple criteria decision making method to resolve the selection problem of transshipment container port in marine transportation industry.

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Fuzzy Number RankingIn fuzzy multiple criteria decision making, the final evaluation

values are usually still fuzzy numbers.

A ranking method is needed to transform these final fuzzy evaluation values into crisp values for decision making.

At present, there are many defuzzification methods which have been investigated for ranking fuzzy numbers.

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Some methods are briefly introduced as follows:

Liou and Wang (1992) introduced a total integral value generated by the left and right integral values of a fuzzy number for ranking fuzzy numbers;

Chen and Hwang (1992) proposed ranking fuzzy numbers by preference relations, the average of fuzzy number and degree, fuzzy rating, and linguistic terms;

Abbasbandy and Hajjari (2009) presented a new approach for ranking trapezoidal fuzzy numbers based on left and right spreads at some α-levels of trapezoidal fuzzy numbers;

Farhadinia (2009) proposed a new approach to rank fuzzy numbers based on the concept of lexicographical ordering in order to provide decision makers algorithm in a simple and efficient way.

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In this research, we suggest total relative area to rank the final fuzzy numbers, and it is developed based on the concept of Chen’s (1985) maximizing set and minimizing set which is one of the most frequently used methods for the problems under fuzzy MCDM environment.

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Evaluation criteria Nature Literature source

Qualitative

brand awareness of smart phone

C1 Benefit This Study

brand awareness of OS

C2Benefit J.D. Power(2010)

Designs C3Benefit J.D. Power(2010)

Security C4Benefit Jakajima (2008)

Operability C5Benefit Jakajima (2008)

Entertainment C6Benefit Su (2009)

Execution efficiency C7Benefit Gizmodo (2009)

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Quantitative

Screen size C8 Benefit This Study

Camera resolution C9 Benefit This Study

Number of applications

C10Benefit Lin and Ye (2009)

Price of applications C11 Cost Distimo (2010)

Price C12 Cost This Study

Weight C13

Cost This Study

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Fuzzy SetsFuzzy Numbersα-cutArithmetic Operations on Fuzzy NumbersLinguistic Values

29

The fuzzy set A can be expressed as:

(3.1)

where U is the universe of discourse, x is an element in U, A is a fuzzy set in U, is the membership function of A at x. The larger , the stronger the grade of membership for x in A.

{( , ( )) | }AA x f x x U

Af x

Af x

30

A real fuzzy number A is described as any fuzzy subset of the real line R with membership function which possesses the following properties (Dubois and Prade, 1978):

(a) is a continuous mapping from R to [0,1];(b) (c) is strictly increasing on [a ,b];(d)(e) is strictly decreasing on [c ,d];(f)

where, A can be denoted as .

Af

Af

; ] ,( ,0)( axxf A

Af

; , ,1)( cbxxf A

Af

; ) ,[ ,0)( dxxf A

dcba , , ,

31

The membership function of the fuzzy number A can also be expressed as:

(3.2)

where and are left and right membership functions of A,respectively.

, ,0

,),(

,1

,),(

)(

otherwise

dxcxf

cxb

bxaxf

xfR

A

LA

A

Af

( )L

Af x ( )R

Af x

32

The α-cuts of fuzzy number A can be defined as:

(3.3)

where is a non-empty bounded closed interval contained in R and can be denoted by , where and are its lower and upper bounds, respectively.

| , 0, 1AA x f x

A[ , ]l uA A A

lAuA

33

Given fuzzy numbers A and B, , the α-cuts of A and B are and respectively. By interval arithmetic, some main operations of A and B can be expressed as follows (Kaufmann and Gupta, 1991): (3.4)

(3.5)

(3.6)

(3.7)

(3.8)

, A B R[ , ]l uA A A [ , ],l uB B B

] , [ uull BABABA

] , [ luul BABABA

] , [ uull BABABA

] , [ )(

l

u

u

l

B

A

B

ABA

RrrArArA ul , ,

34

According to Zadeh (1975), the concept of linguistic variable is very useful in dealing with situations which are complex to be reasonably described by conventional quantitative expressions.

A1=(0,0,0.2)=Unimportant A2=(0.1,0.3,0.5)= Between Unimportant and Important A3=(0.3,0.5,0.7)=Important A4=(0.5,0.7,0.9)=Very important A5=(0.8,1,1)=Absolutely important

Figure 3.1. Linguistic Values and Fuzzy Numbers for Degree of Importance

35

Average Importance WeightsAggregate Ratings of Alternatives versus

Qualitative CriteriaNormalize Values of Alternatives versus Quantitative

CriteriaAggregate the Ratings and WeightsRank Fuzzy Numbers

3636

decision makers, candidate of smart phones, selected criteria, In model development process, criteria are categorized

into three groups:Benefit qualitative criteria: Benefit quantitative criteria: Cost quantitative criteria:

ltDt ,...,2,1, tDmiAi ,...,2,1, iA

jC njC j ,...,2,1,

gjC j ,...,1,

hgjC j ,...,1, nhjC j ,...,1,

37

Average Importance WeightsAssume , , ,jt jt jt jt jtw a b c w R ( ) , 1,..., ,j n 1,..., ,t l

1 21

...j j j jlw w w wl

( ) (4.1)

where, 1

1,

l

j jtt

a al

1

1,

l

j jtt

b bl

1

1.

l

j jtt

c cl

jtw represents the weight assigned by each decision maker for each criterion.

jw represents the average importance weight of each criterion.

38

Aggregate Ratings of Alternatives versus Qualitative Criteria

Assume , , 1,..., , 1,..., , 1,...,ijt ijt ijt ijtx d e f i m j g t l ( ) ,

1 21

( ... )ij ij ij ijlx x x xl

(4.2)

where 1

1,

l

ij ijtt

d dl

1

1,

l

ij ijtt

e el

1

1 l

ij ijtt

f fl

ijtx denotes ratings assigned by each decision maker for each alternative versus each

qualitative criterion.

ijx denotes averaged ratings of each alternative versus each qualitative criterion.

3939

is the value of an alternative versus a benefit quantitative criterion and cost quantitative criterion .

denotes the normalized value of

(4.3)

For calculation convenience, assume

),,( ijijijij qpoy ,,...,2,1, miAi ,,...,1, hgjj

nhjj ,...,1,

ijr ijy

* * *( , , ) ,ij ij ij

ijij ij ij

o p qr

q q q * max ,ij ijq q j B

( , , ) ,ij ij ij ijr a b c . ,...,1 ngj

* * *( , , ) ,ij ij ij

ijij ij ij

o o or

q p o * min ,ij ijo o j C

40

The membership function of the final fuzzy evaluation value, , 1,...,iT i n of each

alternative can be developed as follows:

1 1 1

,g h n

i j ij j ij j ijj j g j h

T w x w x w x

(4.4)

The membership functions are developed as:

[( ) , ( ) ] ,j j j j j j jw b a a b c c (4.5)

[(e ) , (e ) ] .ij ij ij ij ij ij ijx d d f f (4.6)

41

From Eqs. (4.5) and (4.6), we can develop Eqs. (4.7) and (4.8) as follows:

2

2

[( )( ) ( ( ) ( )) ,

( )( ) ( ( ) ( )) ] .

j ij j j ij ij ij j j j ij ij j ij

j j ij ij ij j j j ij ij j ij

w x b a e d d b a a e d a d

b c e f f b c c e f c f

(4.7)

2

1 1 1 1

n2

j 1 1 1

[ ( )( ) ( ( ) ( )) ,

( )( ) ( ( ) ( ))) ] .

n n n n

j ij j j ij ij ij j j j ij ij j ijj j j j

n n

j j ij ij ij j j j ij ij j ijj j



(4.8)

42

When applying Eq. (4.8) to Eq.(4.4), three equations are developed:

2

1 1 1 1

g2

j 1 1 1

[ ( )( ) ( ( ) ( )) ,

( )( ) ( ( ) ( )) ] .

g g g g

j ij j j ij ij ij j j j ij ij j ijj j j j

g g

j j ij ij ij j j j ij ij j ijj j



(4.9)

2

1 1 1 1

h2

j 1 1 1

[ ( )( ) ( ( ) ( )) ,

( )( ) ( ( ) ( )) ] .

h h h h

j ij j j ij ij ij j j j ij ij j ijj g j g j g j g

h h

j j ij ij ij j j j ij ij j ijg j g j g



(4.10) 2

1 1 1 1

n2

j 1 1 1

[ ( )( ) ( ( ) ( )) ,

( )( ) ( ( ) ( )) ] .

n n n n

j ij j j ij ij ij j j j ij ij j ijj h j h j h j h

n n

j j ij ij ij j j j ij ij j ijh j h j h



(4.11)

43

To simplify equation, assume:

11

21

31

11

2

( )( )

( )( )

( )( )

( ) ( )

( ) ( )

g

i j j ij ijj

h

i j j ij ijj g

n

i j j ij ijj h

g

i ij j j j ij ijj

i ij j j j ij ijj g

A b a e d

A b a e d

A b a e d

B d b a a e d

B d b a a e d

1

31

( ) ( )

h

n

i ij j j j ij ijj h

B d b a a e d

11

21

31

11

2

( )( )

( )( )

( )( )

( ) ( )

( ) ( )

g

i j j ij ijj

h

i j j ij ijj g

n

i j j ij ijj h

g

i ij j j j ij ijj

i ij j j j ij ij

C b c e f

C b c e f

C b c e f

D f b c c e f

D f b c c e f

1

31

( ) ( )

h

j g

n

i ij j j j ij ijj h

D f b c c e f

11

21

31

11

21

31

1

g

i j ijj

h

i j ijj g

n

i j ijj h

g

i j ijj

h

i j ijj g

n

i j ijj h

i j ijj

O a d

O a d

O a d

P b e

P b e

P b e

Q c f

1

21

31

g

h

i j ijj g

n

i j ijj h

Q c f

Q c f

44

By applying the above equations, Eqs. (4.9)-(4.11) can be arranged as Eqs. (4.12)-(4.14) as follows:

2 21 1 1 1 1 1

1

[ , ] .g

j ij i i i i i ij

w x A B O C D Q

(4.12)

2 22 2 2 2 2 2

1

[ , ] .h

j ij i i i i i ij g

w x A B O C D Q

(4.13)

2 23 3 3 3 3 3

1

[ , ] .n

j ij i i i i i ij h

w x A B O C D Q

(4.14)

45

Applying Eqs.(4.12)-(4.14) to Eq.(4.4) to produce Eq.(4.15):

Assume:

21 2 3 1 2 3 1 2 3

21 2 3 1 2 3 1 2 3

[( ) ( ) ( ) ,

( ) ( ) ( )] .

i i i i i i i i i i

i i i i i i i i i

T A A C B B D O O Q

C C A D D B Q Q O

(4.15)

1 1 2 3

1 1 2 3

2 1 2 3

2 1 2 3

1 2 3

1 2 3

1 2 3

i i i i

i i i i

i i i i

i i i i

i i i i

i i i i

i i i i

I A A C

J B B D

I C C A

J D D B

Q O O Q

Y P P P

Z Q Q O

46

By applying the above equations, Eqs. (4.15) can be arranged as Eqs. (4.16) and (4.17) as follows:

The left and right membership function of Ti can be obtained as shown in Eq. (4.18) and (4.19) as follows:

21 1 0i i iI J Q x (4.16)

22 2 0i i iI J Z x (4.17)

12 2

1 1 1

1

4, ;

2i

i i i iLi iT

i

J J I x Qa x Q x Y

If

(4.18)

12 2

2 2 2

2

4,

2i

i i i iRi iT

i

J J I x Za x Y x Z

If

(4.19)

47

Rank Fuzzy Numbers

In this research, we applied total relative area to rank the final fuzzy numbers, and it is developed based on the concept of Chen’s (1985) maximizing set and minimizing set.

Table 4.1 Chen’s maximizing set and minimizing set ranking outcomes

Example A1 A2 A3 A4 Ranking

1 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (8, 9, 10) A1=A2

2 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (6, 7, 8) A1<A2

3 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (10, 11, 12) A1>A2

48

Using Chen’s maximizing set and minimizing set on the given sets of examples, we can see three rankings for the same alternatives.

To resolve the inconsistency problem above, modification was made to Chen’s ranking method.

49

Definition 1

The maximizing set M is a fuzzy subset with fM as

(4.20)

The minimizing set N is a fuzzy subset with fN as

(4.22)

Where usually k is set to 1.

1

1

minmin max

max min

( ) , ,

0,otherwise .

i

i

R kR

M

x xx x x

x x xf

1

1

maxmin max

min max

( ) , ,

0,otherwise ,

i

i

L kL

N

x xx x x

x x xf

min max 1inf , sup , , { ( ) 0} ,i

ni i i A

x xx S x S S S S x f x

50

1

1

( )i

i ijRi

ZR

R Vx

S x x dxf

1 1

1

( )i

i i ijLi

YL

L i L Vx

S x Y x x dxf 2

2

2 2

max

min max

( )Li

i

i iiji

xLR

L L iVY

x xS x x dx x Y

x xf

2

2( )

Ri

i iji

xL

R VQ

S x x dxf

1 1 2 2

1( ) ( ( ) ( ) ( ) ( )) , 1 ~ .

4 i i i iT i R L R LS A S x S x S x S x i n

Figure 4.1. Total Relative Area

Table 4.4 Total relative area ranking outcomes


1 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (8, 9, 10) A1>A2

2 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (6, 7, 8) A1>A2

3 (3, 5, 7) (4, 5, 51/8) (2, 3, 5) (10, 11, 12) A1>A2

51

By applying the new method, we can see that the rankings have changed and are now consistent.

Compare the Chen’s (2010) maximizing area and minimizing area and the total relative area method by changing values of A2.

Using Chen’s maximizing area and minimizing area on the given sets of examples, we can see three rankings for the same alternatives as shown in Table 4.6.

Table 4.6 Maximizing area and minimizing area ranking outcomes

The Chen’s (2010) method have the inconsistency problem.52


1 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (8, 9, 10) A1=A2

2 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (6, 7, 8) A1>A2

3 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (10, 11, 12) A1<A2

By applying the total relative area method, we can see that the rankings have changed and are now consistent as shown in Table 4.8.

53


1 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (8, 9, 10) A1>A2

2 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (6, 7, 8) A1>A2

3 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (10, 11, 12) A1>A2

Table 4.8 Total relative area ranking outcomes

Using Liou and Wang method on the given sets of examples, we can see three rankings for the same alternatives as shown in Table 4.10.

According to Table 4.10, the rankings are consistent; therefore, the feasibility of this model can be demonstrated.

54


1 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (8, 9, 10) A1>A2

2 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (6, 7, 8) A1>A2

3 (3, 5, 7) (41/20, 5, 55/8) (2, 3, 5) (10, 11, 12) A1>A2

Table 4.10 The Liou and Wang method ranking outcomes

5555

is developed as follows:

(4.29)

1iRx

1

1

1min 2 2

2 2 2 2max min

= +4 2i

i

Ri i i R i i

x xJ J I x Z I

x x

1

12 2

max min 2 max min 2 min max min 2 max min 2 min

2

2 +4 =

2i

i i i i i

Ri

x x J x x I x x x J x x I x Zx

I

5656


(4.30)

1iLx

1

1

1max 2 2

1 1 1 1min max

= + +4 2i

i

Li i i L i i

x xJ J I x Q I

x x

1

12 2

min max 1 min max 1 max min max 1 min max 1 min

1

2 + +4=

2i

i i i i i

Li

x x J x x I x x x J x x I x Qx

I

5757


(4.33)

2iRx

2

2

1min 2 2

1 1 1 1max min

= + +4 2i

i

Ri i i R i i

x xJ J I x Q I

x x

2

12 2

max min 1 max min 1 min max min 1 max min 1 min

1

2 + +4=

2i

i i i i i

Ri

x x J x x I x x x J x x I x Qx

I

5858


(4.34)

2iLx

2

2

1max 2 2

2 2 2 2min max

= +4 2i

i

Li i i L i i

x xJ J I x Z I

x x

2

12 2

min max 2 min max 2 max min max 2 min max 2 min

2

2 +4

2i

i i i i i

Li

x x J x x I x x x J x x I x Zx

I

5959

By equations(4.29) ， the area, denoted as:

(4.31)

=equations(4.29)1iRx

1

1 1

332 22 2

2 222 22

14

2 1212i

i i

i i R iR i i R i

i ii

J Z x JS x J I x Z

I II

1iRS x

6060


(4.32)

=equations(4.30)

1

1 1 1

331 2 2 22

1 1 1 12 21 1 1

1 14 4

2 12 12i

i i i

i i LL i L i i i i i i L i

i i i

J Y xS x Y x J I Y Q J I x Q

I I I

1iLS x

1iLx

6161


(4.35)

=equations(4.33)2iRx

2iRS x

2

2 2

3 31 2 12

1 12 21 1 1

14

2 12 12i

i i

i R i iR i i R i

i i i

J x Q JS x J I x Q

I I I

6262


(4.36)

=equations(4.34)

2iLS x

2iLx

2

2 2

2 2

3 32 2 22 2

2 2 2 22 22 2 2

2max max

min max

1 1+4 +4

2 12 12i

i i

i i

i L iL i i L i i i i i

i i i

L L i i

J x YS x J I x Z J I Y Z

I I I

x x x Y Y x

x x

6363

By equations (4.31),(4.32),(4.35)and(4.36), the total area, denoted as:

(4.37)

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

( ) , 1 ~ .4

i i i iR L R LT i

S x S x S x S xS A i n

64

Averaged Weights of CriteriaRatings of Alternatives versus Qualitative CriteriaNormalization of Quantitative CriteriaDevelopment of Membership Function Defuzzification

65

Assume that a user is looking for a suitable new smart phone.

Suppose three professional persons, D1, D2 and D3, who have the knowledge background in smart phone.

The five candidate smart phone are A1 (Nokia), A2

(Apple), A3 (HTC), A4 (Sony Ericsson) and A5 (RIM) .

66

Quantitative

Benefit Cost

Screen size

Camera resolution

Number of applications

Price of applications

Price

Weight

Qualitative

Brand awareness of smart phone

Brand awareness of OS

Designs

Security

Operability

Entertainment Execution efficiencyQualitative

Figure 5.1. Evaluation Criteria

67

Table 5.1 Linguistic Values and Fuzzy Numbers for Ratings

Linguistic values Triangular fuzzy numbers

Very Unsatisfactory (0.0 ,0.1,0.3),

Unsatisfactory (0.1,0.3,0.5)

Ordinary (0.3,0.5,0.7)

Satisfactory (0.5,0.7,0.9)

Very Satisfactory (0.7,0.9,1.0)

Table 5.2 Linguistic Values and Fuzzy Numbers for Weights

Linguistic values Triangular fuzzy numbers

Unimportant (0.0 ,0.1,0.3),

Ordinary Important (0.1,0.3,0.5)

Important (0.3,0.5,0.7)

Very Important (0.5,0.7,0.9)

Absolutely Important (0.7,0.9,1.0)

68

Table 5.3 The Candidate Smart Phones

Manufacrer Nokia Apple HTC Sony

Ericsson RIM

Smart

Phone

Type E7 iPhone 4 7 Mozart XPERIA

Arc

BlackBerry

Bold 9780

OS Symbian^3 iOS 4 Windows

Phone 7 Android 2.3

BlackBerry

OS 6.0

Screen

size 4.0 inches 3.5 inches 3.7 inches 4.2 inches 2.44 inches

camera

resolution

8 million

pixels

5 million

pixels

8 million

pixels

8.1 million

pixels

5 million

pixels

screen

resolution

640*360

pixels

640*960

pixels

480*800

pixels

480*854

pixels

480*360

pixels

Price NT. 20400 NT.22200 NT.11200 NT.18600 NT.15000

Weight 176 g 137 g 130 g 117 g 122 g

69

Assume that the importance weights given by decision makers to each criterion are shown in Table 5.4.

Table 5.4 Linguistic Weights of Criteria

DM

Criteria 1D 2D 3D

1C AI VI I

2C OI VI I

3C I VI AI

4C AI I UI

5C AI AI VI

6C AI I AI

7C UI OI VI

8C VI UI I

9C UI I OI

10C OI I VI

11C OI I UI

12C I VI AI

13C I VI OI

70

Table 5.5 Fuzzy Weights of Criteria

DM

Criteria 1D 2D 3D

1C (0.7,0.9,1.0) (0.5,0.7,0.9) (0.3,0.5,0.7)

2C (0.1,0.3,0.5) (0.5,0.7,0.9) (0.3,0.5,0.7)

3C (0.3,0.5,0.7) (0.5,0.7,0.9) (0.7,0.9,1.0)

4C (0.7,0.9,1.0) (0.3,0.5,0.7) (0.0,0.1,0.3)

5C (0.7,0.9,1.0) (0.7,0.9,1.0) (0.5,0.7,0.9)

6C (0.7,0.9,1.0) (0.3,0.5,0.7) (0.7,0.9,1.0)

7C (0.0,0.1,0.3) (0.1,0.3,0.5) (0.5,0.7,0.9)

8C (0.5,0.7,0.9) (0.0,0.1,0.3) (0.3,0.5,0.7)

9C (0.0,0.1,0.3) (0.3,0.5,0.7) (0.1,0.3,0.5)

10C (0.1,0.3,0.5) (0.3,0.5,0.7) (0.5,0.7,0.9)

11C (0.1,0.3,0.5) (0.3,0.5,0.7) (0.0,0.1,0.3)

12C (0.3,0.5,0.7) (0.5,0.7,0.9) (0.7,0.9,1.0)

13C (0.3,0.5,0.7) (0.5,0.7,0.9) (0.1,0.3,0.5)

According to Table 5.2, the corresponding triangular fuzzy numbers are shown in Table 5.5.

71

Table 5.6 Averaged Weights of Criteria

Criteria Averaged Weights

1C 0.5 0.7 0.86

2C 0.3 0.5 0.7

3C 0.5 0.7 0.86

4C 0.33 0.5 0.66

5C 0.63 0.83 0.96

6C 0.56 0.76 0.9

7C 0.2 0.36 0.56

8C 0.26 0.43 0.63

9C 0.13 0.3 0.5

10C 0.3 0.5 0.7

11C 0.13 0.3 0.5

12C 0.5 0.7 0.86

13C 0.3 0.5 0.7

By Eq. (4.1), the averaged weight of each criterion can be obtained as shown in Table 5.6.

72

Ratings of Alternatives versus Qualitative Criteria

According to the step in Section 4.2, we obtain the ratings of alternatives versus qualitative criteria as shown in Table 5.7.

Criteria Smart Phone

Candidates

Decision Makers

1D 2D 3D

1C

1A VS S O

2A S VS S

3A VS O VS

4A O O O

5A S O O

2C

1A S O O

2A VS O VS

3A S O S

4A VS S O

5A VS S VS

3C

1A U O U

2A VS VS O

3A U VS O

4A S O U

5A S S S

Table 5.7 Ratings of Smart Phone Candidates versus Qualitative Criteria

4C

1A S VS O

2A S O VS

3A O S VS

4A U S O

5A O O VS

5C

1A O O VS

2A VS S VS

3A VS VS VS

4A O S O

5A O O O

6C

1A S VS O

2A S S VS

3A VS S S

4A S S O

5A O S O

7C

1A U O VU

2A O O O

3A U O O

4A U VS O

5A U O O

73

By Eq. (4.2), the averaged fuzzy evaluation values of candi-dates versus qualitative criteria can be obtained as shown in Table 5.8.

Table 5.8 Averages Fuzzy Evaluation Values of Candidates versus Qualitative

Criteria

Criteria

Alternative 1C 2C 3C 4C 5C 6C 7C

1A

0.5 0.36 0.16 0.5 0.43 0.5 0.13

0.7 0.56 0.36 0.7 0.63 0.7 0.3

0.86 0.76 0.56 0.86 0.8 0.86 0.5

2A

0.56 0.56 0.56 0.36 0.63 0.56 0.3

0.76 0.76 0.76 0.56 0.83 0.76 0.5

0.93 0.9 0.9 0.76 0.96 0.93 0.7

3A

0.56 0.43 0.36 0.43 0.7 0.56 0.3

0.76 0.63 0.56 0.63 0.9 0.76 0.5

0.9 0.83 0.73 0.83 1 0.93 0.7

4A

0.3 0.5 0.3 0.56 0.36 0.43 0.36

0.5 0.7 0.5 0.76 0.56 0.63 0.56

0.7 0.86 0.7 0.93 0.76 0.83 0.73

5A

0.36 0.63 0.5 0.43 0.3 0.36 0.23

0.56 0.83 0.7 0.63 0.5 0.56 0.43

0.76 0.96 0.9 0.8 0.7 0.76 0.63

74

Normalization of Quantitative Criteria

The suitability values of candidates versus quantitative criteria can be shown in Table 5.9.

* * *( , , ) ,ij ij ij

ijij ij ij

o p qr

q q q * max ,ij ijq q j B

* * *( , , ) ,ij ij ij

ijij ij ij

o o or

q p o * min ,ij ijo o j C

Criteria Candidates

1A 2A 3A 4A 5A

8C (3.9, 4.0, 4.1) (3.2, 3.5, 3.7) (3.5, 3.7, 4.0) (4.1, 4.2, 4.5) (2.25, 2.44, 2.64)

9C (730, 800, 870) (445, 500, 555) (720, 800, 880) (729, 810, 891) (450, 500, 550)

10C (6.0,6.5,6.7) (17.0,17.5,18.0) (5.5,6.0,6.5) (4.4,4.8,5.0) (4,4.2,4.5)

11C (6.4,6.6,6.8) (3.2,3.4,3.6) (3.0,3.2,3.4) (2.8,3.0,3.2) (7.8,8.0,8.2)

12C (19000,20400,20900) (21800,22200,23000) (10200,11200,12000) (18000,18600,19000) (14500,15000,15500)

13C (173, 176, 179) (134, 137, 140) (125, 130, 135) (115, 117, 119) (120, 122, 125)

75

By Eq. (4.3), the normalized values of candidates versus quantitative criteria can be obtained as shown in Table 5.10.

Table 5.10 Normalization of Quantitative Criteria

Criteria Smart Phone Candidates Normalization of Quantitative Criteria

8C

1A 0.8666 0.8888 0.9111

2A 0.7111 0.7777 0.8222

3A 0.7777 0.8222 0.8888

4A 0.9111 0.9333 1

5A 0.5 0.5422 0.5866

9C

1A 0.8193 0.8978 0.9764

2A 0.4994 0.5611 0.6228

3A 0.8080 0.8978 0.9876

4A 0.8181 0.9090 1

5A 0.5050 0.5611 0.6172

10C

1A 0.3333 0.3611 0.3888

2A 0.9444 0.9722 1

3A 0.3055 0.3333 0.3611

4A 0.2444 0.2666 0.2777

5A 0.2222 0.2333 0.25

11C

1A 0.4375 0.4242 0.4117

2A 0.875 0.8235 0.7777

3A 0.9333 0.875 0.8235

4A 1 0.9333 0.875

5A 0.3589 0.35 0.3414

12C

1A 0.5368 0.5 0.4880

2A 0.4678 0.4594 0.4434

3A 1 0.9107 0.85

4A 0.5666 0.5483 0.5368

5A 0.7034 0.68 0.6580

13C

1A 0.6647 0.6534 0.6424

2A 0.8582 0.8394 0.8214

3A 0.92 0.8846 0.8518

4A 1 0.9829 0.9663

5A 0.9583 0.9426 0.92

76

By Eqs. (4.4)-(4.19), the membership function, Ti, of the final fuzzy evaluation value of each smart phone candidate can be developed through α-cut and interval arithmetic operations.

77

Table 5.11 Values for 1 2 3 1 2 3 1 2 3 1 2 3, , , , , , , , , , ,i i i i i i i i i i i iA A A B B B C C C D D D

1 2 3 1 2 3 1 2 3, , , , , , , , ,i i i i i i i i iO O O P P P Q Q Q

1A 2A 3A 4A 5A 1iA 0.261 0.266 0.266 0.266 0.266

2iA 0.022 0.026 0.027 0.023 0.018

3iA -0.01 -0.014 -0.034 -0.018 -0.009

1iB 1.098 1.297 1.255 1.142 1.151

2iB 0.372 0.424 0.357 0.361 0.234

3iB 0.294 0.395 0.488 0.459 0.377

1iC 0.213 0.190 0.198 0.214 0.214

2iC 0.113 0.026 0.036 0.033 0.023

3iC 0.100 -0.015 -0.028 -0.017 -0.010

1iD -1.852 -1.885 -1.868 -1.964 -1.964

2iD -0.749 -0.509 -0.484 -0.473 -0.313

3iD -0.655 -0.392 -0.475 -0.521 -0.367

1iO 1.194 1.641 1.565 1.190 1.212

2iO 0.440 0.539 0.406 0.425 0.267

3iO 0.454 0.545 0.767 0.641 0.593

1iP 2.554 3.205 3.087 2.598 2.630

2iP 0.788 0.930 0.732 0.762 0.479

3iP 0.767 0.956 1.306 1.121 0.985

1iQ 4.193 4.901 4.757 4.348 4.380

2iQ 1.191 1.361 1.110 1.157 0.724

3iQ 1.095 1.396 1.877 1.634 1.389

11

21

31

11

2

( )( )

( )( )

( )( )

( ) ( )

( ) ( )

g

i j j ij ijj

h

i j j ij ijj g

n

i j j ij ijj h

g

i ij j j j ij ijj

i ij j j j ij ijj g

A b a e d

A b a e d

A b a e d

B d b a a e d

B d b a a e d

1

31

( ) ( )

h

n

i ij j j j ij ijj h

B d b a a e d

11

21

31

11

2

( )( )

( )( )

( )( )

( ) ( )

( ) ( )

g

i j j ij ijj

h

i j j ij ijj g

n

i j j ij ijj h

g

i ij j j j ij ijj

i ij j j j ij ij

C b c e f

C b c e f

C b c e f

D f b c c e f

D f b c c e f

1

31

( ) ( )

h

j g

n

i ij j j j ij ijj h

D f b c c e f

11

21

31

11

21

31

1

g

i j ijj

h

i j ijj g

n

i j ijj h

g

i j ijj

h

i j ijj g

n

i j ijj h

i j ijj

O a d

O a d

O a d

P b e

P b e

P b e

Q c f

1

21

31

g

h

i j ijj g

n

i j ijj h

Q c f

Q c f

78

21 2 3 1 2 3 1 2 3

21 2 3 1 2 3 1 2 3

[( ) ( ) ( ) ,

( ) ( ) ( )] .

i i i i i i i i i i

i i i i i i i i i

T A A C B B D O O Q

C C A D D B Q Q O

(4.15)

Table 5.12 Values for 1 2 3 1 2 3 1 2 3, , ,i i i i i i i i iA A C B B D O O Q

1 2 3 1 2 3 1 2 3 1 2 3, , , ,i i i i i i i i i i i iC C A D D B P P P Q Q O

1A 2A 3A 4A 5A 1 2 3i i iA A C 0.19 0.31 0.32 0.31 0.28

1 2 3i i iB B D 2.23 2.12 2.10 2.04 1.60

1 2 3i i iO O Q 0.91 1.01 0.34 0.25 0.02

1 2 3i i iC C A 0.30 0.23 0.27 0.26 0.25

1 2 3i i iD D B -2.86 -2.74 -2.79 -2.84 -2.52

1 2 3i i iP P P 3.10 3.41 2.76 2.51 1.89

1 2 3i i iQ Q O 5.48 5.89 5.29 5.07 4.13

79

By Eq. (4.18)and(4.19), the left, , and right, , membership functions of the final fuzzy evaluation value

can be obtained as shown in Table 5.13.

( )i

LTf x ( )

i

RTf x

iT

81

Using the suggested total relative area method, we can obtain , , and using Eq.(4.31), (4.32), (4.35) and (4.36) respectively, and the total area value can be obtained by using Eq.(4.37), as shown in Table 5.15.

1iRS x 1iLS x 2iRS x 2iLS x

82

Table 5.15 1iRS x , 2iRS x , 1iLS x , 2iLS x and T iS A of Each Alternative

S(xRi1)

A1 A2 A3 A4 A5

34.235 48.762 38.451 41.085 30.423

S(xRi2)

A1 A2 A3 A4 A5

0.070 0.082 0.008 0.004 0.000

S(xLi1)

A1 A2 A3 A4 A5

0.184 0.210 0.135 0.116 0.058

S(xLi2)

A1 A2 A3 A4 A5

6.160 8.035 5.263 4.343 2.121

ST(Ai)

A1 A2 A3 A4 A5

10.162 14.272 10.964 11.387 8.150

RANK

A1 A2 A3 A4 A5

4 1 3 2 5

83

According to Table 5.15 the ranking order of the five candidate smart phones is A2 > A4 > A3 > A1 > A5 .

A2 has the largest area 14.272 ; therefore, A2 is the most suitable smart phone for decision makers under the evaluation procedure of the proposed model.

84

Conclusions

Suggestions for Future Research

85

1. The needed criteria for evaluating smart phones are carefully analyzed and selected through thoroughly comprehensive investigation on relevant literature survey.

2. This research uses a fuzzy MCDM to develop an evaluation and selection model for smart phones.

3. Ranking formulae are clearly developed for better executing the decision making.

4. A numerical example is used to demonstrate the feasibility of the proposed fuzzy MCDM model.

86

Suggestions for Future Research1. In this research, only one level in the criteria hierarchical

structure is considered. In future research, a multiple levels may be further investigated for better depicting the relationship in the criteria hierarchical structure for smart phone.

2. In the numerical example, ratings of alternatives and importance weights of criteria are subjectively assigned by decision makers. Some objective methods, such as survey, can be used to strengthen the effectiveness of the proposed model.

87

3. In fuzzy number ranking, this research suggests the total relative area based on Chen (1985) maximizing set and minimizing set for defuzzification. In future research, some other ranking approaches may be used for the proposed fuzzy MCDM model. However, the ranking results may be different.

4. Fuzzy numbers other than triangular can also be used for the proposed model, a comparison may be needed.

5. The linguistic values and their corresponding fuzzy numbers used in this research can be adjusted to fit different applications. A model may be needed to objectively produce these values.

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