Advisor: Prof. Ta-Chung ChuGraduate: Elianti (李水妮 )

M977z240

1. IntroductionAssume:k decision makers (i.e. Dt,t=1~k)

m alternative (i.e. Ai,i=1~m)

n criteria (Cj,j=1~n)

There are 2 types of criteria:a.Qualitative (all of them are benefit), Cj=1~gb.Quantitative

For benefit: Cj=g+1~h

For cost: Cj=h+1~n

2. Ratings of Each Alternative versus CriteriaQualitative criteria Quantitative criteria

Benefit Cost

Let Xijt= (aijt,bijt,cijt), i= 1,…,m, j= 1,…,g, t=1,…,k, is the rating assigned to alternative Ai by decision maker Dt under criterion Cj.

1 1 1g g h h nC C C C C C

2. Ratings of Each Alternative versus CriteriaXij= (aij,bij,cij) is the averaged rating of alternative Ai versus

criterion Cj assessed by the committee of decision makers.

Then: (4.1)

Where:

j=1~g

xxxx ijtijijij k ...

121

k

tijtij bb k 1,

1

k

tijtij cc k 1

1

k

tijtij aa k 1

1

2. Ratings of Each Alternative versus CriteriaQualitative (subjective) criteria are measured by linguistic values

represented by fuzzy numbers.

Linguistic Value Fuzzy Numbers

Very Poor (VP) (0, 0.1, 0.3)

Poor (P) (0.1, 0.3, 0.5)

Medium (M) (0.3, 0.5, 0.7)

Good (G) (0.5, 0.7, 0.9)

Very Good (VG) (0.7, 0.9, 1.0)

3. Normalization of the Averaged Ratings

Values under quantitative criteria may have different units and then must be normalized into a comparable scale for calculation rationale. Herein, the normalization is completed by the approach from (Chu, 2009), which preserves by property where the ranges of normalized triangular fuzzy numbers belong to [0,1].

Let’s suppose rij=(eij,fij,gij) is the performance value of alternative Ai versus criteria Cj, j=g+1 ~ n.

The normalization of the rij is as follows:

(4.2)

Bjd

e jg ij

d

e jf ij

d

e jeij

jd

eijg j

d

f ijg j

d

g ijg jijijij cbaxij

,*

,*

,*

C,*

*

,*

*

,*

*,,

3. Normalization of the Averaged RatingsThe fuzzy multi-criteria decision making decision can be concisely

expressed in matrix format after normalization as follow:

mnmjm

iniji

nj

xxx

xxx

xxx

D

1

1

1111

j = 1~n

3. Averaged Importance Weights

Let j=1,…,n t=1,…,k be the weight of importance assigned by decision maker Dt to criterion Cj.

Wj = (oj,pj,qj) is the averaged weight of importance of criterion Cj assessed by the committee of k decision makers, then:

(4.3)

Where:

k

tjtj

ppk 1

,

1

k

tjtj oo k 1,

1

k

tjtj

qqk 1

1

,,,, \Rwqpow jtjt jtjtjt

wwwkw jtjjj...............

211

4. Averaged Importance WeightsThe degree of importance is quantified by linguistic terms

represented by fuzzy numbers

Linguistic values Fuzzy numbers

Ver y Low (VL) (0, 0.1, 0.3)

Low (L) (0.1, 0.3, 0.7)

Medium (M) (0.3, 0.5, 0.7)

High (H) (0.5, 0.7, 0.9)

Very High (VH) (0.7, 0.9, 1.0)

4. Final Fuzzy evaluation Value

The final fuzzy evaluation value of each alternative Ai can be obtained by using the Simple Addictive Weighting (SAW) concept as follow:

Here, Pi is the final fuzzy evaluation values of each alternative Ai.

n

jjiji wxP

1, i=1,2,…,m,

4. Final Fuzzy evaluation ValueThe membership functions of the Pi can be developed as

follows:

and

qqpoopw jjjjjjj

,

ccbaabx ijijijijijijij

,

,

2

oaaboopaopabwx jijijijjjjijjjijijjij

qccbqqpcqpcb jijijijjjjijjjijij

2

4. Final Fuzzy evaluation Value

,11

2

11

n

jjij

n

jijijjjjijjj

n

jijij

n

jjiji oaaboopaopabwxP

,111

2

n

jjij

n

jijijjjjij

n

jjjijij qccbqqpcqpcb

n

jjij

n

jijijjjjijjj

n

jijij xoaaboopaopab

11

2

10

0111

2

xqccbqqpcqpcbn

jjij

n

jijijjjjij

n

jjjijij

4. Final Fuzzy evaluation ValueWe assume:

opabG jj

n

jijiji

11

n

jijijjjjiji aboopaH

11

qpcbG jj

n

jijiji

12

n

jijijjjjiji cbqqpcH

12

n

jjiji oaV

1

,

1

n

jjiji pbY

n

jjiji qcZ

1

01

2

1 xVHG iii

02

2

2 xZHG iii

So, Eq. (4.7) and (4.8) can be expressed as:

5. Final Fuzzy evaluation ValueThe left membership function and the right membership function of the final

fuzzy evaluation value Pi can be produced as follows:

(4.11)

(4.12)

Only when Gi1 =0 and Gi2 =0, Pi is triangular fuzzy number, those are:

For convenience, Pi can be donated by:

(4.13)

,

2

4)(1

2/11

211

G

VxGHHxfi

iiiiLPi

,,YxV ii

,

2

4)(

2

2/12

222

G

ZxGHHxfi

iiiiRPi

,,ZxY ii

,,)(1

YxVH

Vxxf iii

iLPi

,,)(

2ZxY

H

Zxxf iii

iRPi

.,....,1,2,2;1,1;,, miH iGiH iGiZ iY iV iPi

5. An Improved Fuzzy Preference RelationTo define a preference relation of alternative Ah over

Ak, we don’t directly compare the membership function of Ph (-) Pk. We use the membership function of Ph (-) Pk. to indicate the prefer ability of alternative Ah over alternative Ak, and then compare Ph (-) Pk.with zero.

The difference Ph (-) Pk. here is the fuzzy difference between two fuzzy numbers. Using the fuzzy number, Ph (-) Pk. , one can compare the difference between Ph and Pk. for all possibly occurring combinations of Ph and Pk.

5. An Improved Fuzzy Preference RelationThe final fuzzy evaluation values Ph and Pk are triangular

fuzzy numbers. The difference between Ph and Pk is also a triangular fuzzy number and can be calculated as:

 Let Zhk=Ph-Pk, h,k=1,2,…m, the -cut of Zhk can be expressed as:

Where

,PPZ khhk

PPP huhlh ,

PPP kuklk ,

PPPPPP klhukuhlkh

,

5. An Improved Fuzzy Preference Relation

\1

2

1 1

n

jhjhjjjjhj

n

j

n

jjjkjkjjjhjhjhkl aboopaqpcbopabZ

n

j

n

jjkjjhj

n

jkjkjjjjkj qcoacbqqpc

1 11

,

\1

2

1 1

n

jhjhjjjjhj

n

j

n

jjjkjkjjjhjhjhku cbqqpcopabqpcbZ

n

j

n

jjkjjhj

n

jkjkjjjjkj oaqcaboopa

1 11

,

ZZZ hkuhklhk ,

5. An Improved Fuzzy Preference RelationBecause the formula is too complicated, then we make some

assumptions as follows:

n

j

n

jkhjjkjkjjjhjhjhk GGqpcbopabG

1 1211,

,2

11

11 HHcbqqpcaboopaH k

n

jhkjkjjjjkj

n

jhjhjjjjhjhk

,12

1 12 GGopabqpcbG kh

n

j

n

jjjkjkjjjhjhjhk

,12

112 HHaboopacbqqpcH khkjkj

n

jjjjkj

n

jhjhjjjjhjhk

n

jjhjh oaV

1

, ,

1

n

jjhjh pbY

n

jjhjh qcZ

1

,

n

jjkjk oaV

1

, ,

1

n

jjkjk pbY

n

jjkjk qcZ

1

.

5. An Improved Fuzzy Preference RelationThere are two equations to solve:

(4.16)

(4.17)

Using Eq. (4.16) and (4.17), the left and right membership functions of the difference Zhk=Ph-Pk can be produced as follows:

(4.18)

(4.19)

0,1

2

1 xZVHG khhkhk

0.2

2

2 xVZHG khhkhk

,,

2

4)(1

2/11

211

YYxZVG

ZVxGHHxf khkhhk

khhkhkhkLZ hk

,,

2

4)(2

2/12

222

VZxYYG

VZxGHHxf khkhhk

khhkhkhkRZ hk

5. An Improved Fuzzy Preference RelationObviously, Zhk=Ph-Pk may not yield a triangular shape as well.

Only when Ghk1=0 and Ghk2=0, is a triangular fuzzy number, that is:

For convenience, Zhk can be denoted by:

(4.20)

,,)(

1YYxZV

H

ZVxxf khkhhk

khLPi

,,)(2

ZZxYYH

VZxxf khkhhk

khRPi

.,....,2,1,,2,2;1,1;,,,,, mkhH hkGhkH hkGhkZ kY kV kZ hY hV hZ hk