Supplier Quality Evaluation Using a Fuzzy Multi Criteria Decision Making Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 199

3.2.1 Fuzzy TOPSIS

The fuzzy TOPSIS approach involves fuzzy assessments of criteria and alternatives
in TOPSIS (Hwang and Yoon 1981 ). The TOPSIS approach chooses alternative
that is closest to the positive ideal solution and farthest from the negative ideal
solution. A positive ideal solution is composed of the best performance values for
each criterion whereas the negative ideal solution consists of the worst performance
values. The various steps of Fuzzy TOPSIS are presented as follows:

Step 1: Assignment of ratings to the criteria and the alternatives.

Let us assume there are j possible candidates (in our case suppliers) called A ¼

f A 1 ; A 2 ; ...; A j g which are to be evaluated against n criteria, C ¼ f C 1 ; C 2 ; ...; C n g .

The criteria weights are denoted by wiði i ð

; ; ...; Þ

i ¼ 1 2 m . The performance ratings of

decision maker D k ð

¼ ; ; ...; Þ

for each alternative A j ð

¼ ; ; ...; Þ

k 1 2 K j 1 2 n with

are denoted by R k ¼ ~ respect to criteria Ciði i ð i ¼ 1 ; 2 ; ...; m Þ x ijk ð i ¼ 1 ; 2 ; ...; m ; j ¼

; ; ...; ; ¼ ; ; ...; Þ

l R k ð

Þ

1 2 n k 1 2 K with membership function x .

Step 2: Compute aggregate fuzzy ratings for the criteria and the alternatives.

If the fuzzy ratings of decision makers are described by triangular fuzzy number
R k ¼ ð

then the aggregated fuzzy rating is given by R a k ; b k ; c k Þ; k ¼ 1 ; 2 ; ...; K ; ¼

ð a ; b ; c Þ; k ¼ 1 ; 2 ; ...; K where;

K X

K

1

a ¼ min

k

f a k g; b ¼ b k ; c ¼ max

k

f c k g

k ¼ 1

If the fuzzy rating and importance weight of the kth decision maker are ~ x ijk ¼

ð a ijk ; b ijk ; c ijk Þ and w ijk ¼ ð ~ w jk1 ; w jk2 ; w jk3 Þ; i ¼ 1 ; 2 ; ...; m ; j ¼ 1 ; 2 ; ... n respec-

~

tively, then the aggregated fuzzy ratings ( x ij ) of alternatives with respect to each

criteria are given by ~ x ij ¼ ð a ij ; b ij ; c ij Þ where

K X

K

1

a ij ¼ min

k

f a ijk g; b ij ¼ b ijk ; c ij ¼ max

k

f c ijk g ð 2 Þ

k¼1

The aggregated fuzzy weights ( w ij ) of each criterion are calculated as w j ¼

ð w j1 ; w j2 ; w j3 Þ where

K X

K

1

w j1 ¼ min

k

f w jk1 g; w j2 ¼ w jk2 ; w j3 ¼ max

k

f c jk3 g ð 3 Þ

k¼1

Step 3: Compute the fuzzy decision matrix.