maximizing value and minimizing base on fuzzy topsis model

Maximizing value and Minimizing base on Fuzzy TOPSIS model

Advisor: Prof. Ta Chung ChuStudent : Pham Hoang Chien (Rhett)

林師賢 (Shih Hsien Lin)

Introduction

• Among many famous MCDM methods, Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) is a practical and useful technique for ranking and selection of a number of possible alternatives through measuring Euclidean distances.

• TOPSIS was first developed by Hwang and Yoon (1981)

Introduction (con’t)

• TOPSIS bases upon the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) the solution that maximizes the benefit criteria and minimizes the cost criteria; and the farthest from the negative ideal solution (NIS)

The algorithm• Assume:

Committee of k decision makers (i.e. Dt, t=1k)

Responsible for selection m alternative (i.e. Ai, i=1m)

Under n criteria (Cj, j=1n)

A classic fuzzy multi-criteria decision making problem

can be expressed in matrix format as follows:

( , , )ijt ijt ijt ijtx a b c

is rating versus by x A C Dtijt i j1 , 1 ,i m j g t t k

11 1 1( 1) 1 1( 1) 1

1 ( 1) ( 1)

1 ( 1) (

1

1)

t gt g h h n

i t igt i g ihi

m

i h in

m t mgt m g mh m h mn

x x r r r r

x x r r r r

x x r r r

A

A

A r

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

Qualitative criteria Quantitative criteria

Before normalization

is the normalize x rij ij

Suppose ( , , ) 1 , ( 1)r e f g i m j g hij ij ij ij

After normalization

11 1 1( 1) 1 1( 1) 1

1 ( 1) ( 1)

1 ( 1) (

1

1)

t gt g h h n

i t igt i g ihi

m

i h in

m t mgt m g mh m h mn

x x x x x x

x x x x x x

x x x x x

A

A

A x

1 1 1g g h h nC C C C C C 1 ( 1) ( 1)t gt g t ht h t ntw w w w w w

Qualitative criteria Quantitative criteria

Chen’s method

• Both B and C are further normalized by the Chen method into comparable scales respectively. This method preserves the property in which the ranges of normalized triangular fuzzy number belong to [0,1]. The normalization of the averaged ratings, as follows

ijr

Chen’s method

objective criteria can be classified to benefit (B) and cost (C). Benefit criterion has the characteristics: the larger the better. The cost criterion has the characteristics: the smaller the better

, , , , ( 1)max max max

ij ij ijij

ij ij ij

e f gx j B j g h

g g g

where is the normalized value of , or x r j B j Cij ij

min min min, , , , ( 1)ij ij ij

ijij ij ij

e e ex j C j h n

e f g

Weight matrix

1

1 k

jt ijtt

w xk

1 11 1

1 1k k

t i t gt igtt t

w x w xk k

( 1) ( 1)1 1

1 1k k

g t i g ht iht t

w x w xk k

( 1) ( 1)1 1

1 1k k

h t i h nt int t

w x w xk k

After we calculate , we get result

Weighted normalized decision matrix

Weighted normalized decision matrix is obtained by multiplying

normalized matrix with the weights of the criteria

1

1( )

k

ij jt ijst

v w xk

( , , ), , 1, 2..., ; 1, 2,..., .jt jt jt jt jtw o p q w R j n t k

when 1

when 1

ijs ijt

ijs ij

x x j g

x x j g n

Weighted normalized decision matrix

,jt jt jt ijt jt jt jtw p o o p q q

,ijs ijs ijs ijs ijs ijs ijsx b a a b c c

2

2

[ ] ,

[ ]

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jt

ijt jt

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jt

b a p o a p o o b a a ox w

b c p q c p q q b c c q

1

1( )

k

ij jt ijst

v w xk

Weighted normalized decision matrix (con’t)

• The final fuzzy evaluation values can be developed via arithmetic operation of fuzzy numbers as

1

1( )

k

ij jt ijtt

v w xk

2

1 1 1

2

1 1 1

1 1 1[ ] ,

1 1 1[ ]

n n n

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jtj j j

n n n

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jtj j j

b a p o a p o o b a a ok k k

b c p q c p q q b c c qk k k

Weighted normalized decision matrix (con’t)

• Let assume we have

2

1 1 1

1 1 1[ ] 0

n n n

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jtj j j

b a p o a p o o b a a o xk k k

2

1 1 1

1 1 1[ ] 0

n n n

ijs ijs jt jt ijs jt jt jt ijs ijs ijs jtj j j

b c p q c p q q b c c q xk k k

(4.10)

(4.11)

Weighted normalized decision matrix (con’t)

• For convenience, we make some assumptions:

11

1[ ]

n

i ijs jt jt jt ijs ijsj

J a p o o b ak

2

1

1 n

i ijs ijs jt jtj

I b c p qk

21

1[ ]

n

i ijs jt jt jt ijs ijsj

J c p q q b ck

1

1 n

i ijs jtj

Z c qk

11

1 n

i ijs ijs jt jtj

I b a p ok

1

1 n

i ijs jtj

Q a ok

Weighted normalized decision matrix (con’t)

Equations 4.10 and 4.11 can be expressed as:2

1 1 0i i iI J Q x 2

2 2 0i i iI J Z x

The left and right membership function and

of can then be produced as

Lvij

v

f xRf x vijij

1 1 2 2( , , , , , , , ) where 1i i i i i i i iv Q Y Z I J I J i m

1

2 21 1 1 14 ( ) / 2 ,

ij

Lv i i i i i i if x J J I x Q I Q x Y

1

2 22 2 2 24 ( ) / 2 ,

ij

Rv i i i i i i if x J J I x Z I Y x Z

DefuzzificationOur model uses Chen’s maximizing set and minimizing set approach to defuzzify the final fuzzy number

Suppose there are n fuzzy numbers

with in R

Figure 1.1. Maximizing Set and Minimizing Set

, , , 1i i i iA a b c i n ( )

iAf x

Chen’s maximizing set and minimizing set approach

Definition 1The maximizing set M is a fuzzy subset with as

min max 1inf , sup , , | ( ) 0i

ni i i Ax S x S S U S S x f x

Nf

min

min maxmax min( ) ,

0,

i

i

k

R

M R

x x

f x x x xx x

otherwise

where

Mf

max

min maxmin max( ) ,

0,

i

i

k

L

N L

x x

f x x x xx x

otherwise

The minimizing set N is a fuzzy subset with as

(4.12)

(4.13)

Chen’s maximizing set and minimizing set approach

Definition 2The right utility value of each is defined

iA( ) sup( ( ) ( )), 1

iM i M AU A f x f x i n

The left utility value of each is defined

iA

( ) sup( ( ) ( )), 1iN i N AU A f x f x i n (4.15)

(4.14)

Chen’s maximizing set and minimizing set approach

Definition 3The total utility value of each is defined as

( )T iU A

1( ) ( ( ) 1 ( )), 1

2T i M i N iU A U A U A i n

The total utility is used to rank fuzzy number. The larger the , the larger the

iA

iA( )T iU A

(4.16)

Final Ranking ValuesIn our model we modify Chen’s maximizing and minimizing set approach

Definition 1The maximizing set M is a fuzzy subset with as

Nf

min

min axax min( ) ,

0,

i

i

knew

R

new newnew newM R mm

x x

f x x x xx x

otherwise

Mf

ax

min axmin ax( ) ,

0,

i

i

knew

L m

new newnew newN L mm

x x

f x x x xx x

otherwise

The minimizing set N is a fuzzy subset with as

min ax 1where inf , sup , , | ( ) 0i

new new nm i i i Ax S x S S U S S x f x

ax max min min axand max{ , }, new new newm mx x x x x

Reference

• Fuzzy performance evaluation in Turkish Banking Sector using Analytic Hierarchy Process and TOPSIS

• An interval arithmetic based fuzzy TOPSIS model

Thank you for your attention