Multi-Criteria Decision Making Method using Intuitionistic fuzzy sets

Deepa Joshi

Ph.D Mathematics

G. B. Pant University of Agriculture & Technology

Pantnagar

1

Intuitionistic Fuzzy sets

An intuitionistic fuzzy set(IFS) A on a universe X is defined as an object of the following form

A={(x, μA(x), νA(x))| x X}where

0 ≤ μA(x) + νA(x) ≤ 1

is called intuitionistic fuzzy set (IFS) and functionsμA : X→ [0, 1] and νA : X → [0, 1] represent thedegree of membership and the degree of non-membership respectively.

is called degree of hesitation.

2

xxxAAA

1

Multi-Criteria Decision Making (MCDM)

Multi-Criteria Decision Making (MCDM) means the process of determining the best feasible

solution according to the given criteria.

3

Approaches For MCDM

ANP (Analytic network process)

AHP (The Analytical Hierarchy Process)

SIR (superiority and inferiority ranking method)

SMART (The Simple Multi Attribute Rating Technique )

SCORE FUNCTION

TOPSIS (Technique for Order Preference by Similarity to the Ideal Solution)

4

Score function definition

Let be an intuitionistic fuzzy value for

The score function(S) of is given by

and

5

),(ijijijx1

ijij

xij

2

13)(

ijij

ijxS

]1,1[)(xijS

Score function

If is the hesitation degree of a decision maker

then the value of the Score function is given by

Where

= criteria , j=1,2……..n

6

)().()()( ccccS jjjj

]1,1[)(cS j

c j

Example using Score function method

Objective

- To select best air-condition system

Criteria

- Economical, Function, Operative

with weight vector W=(0.3,0.3,0.4)

Alternatives

- A, B and C

7

Applying Score function method to example

Step1- We provide intuitionistic values for each criteria and construct the intutionistic group multi-criteria decision matrix as follows

A

D = B

C

8

)6.0,3.0()9.0,1.0()6.0,3.0(

)1.0,7.0()5.0,5.0()5.0,5.0(

)2.0,8.0()1.0,7.0()2.0,8.0(

Applying Score function method to example

Step2-Using intuitionistic fuzzy arithmetic averaging operator to aggregate all over all the criteria.

,I, j, k=1,2,3

= criteria ,j=1,2,3

n = no. of criteria

S = score function

9

xk

ij

)(

)(1

1

)()(

cSxxj

n

j

k

ij

k

i n

c j

Applying Score function method to example

Putting the values from decision matrix we get

=(0.310697, 0.00058)

=(0.2351, 0.00142)

=(0.04914, 0.00062)

10

x)1(

1

x)2(

1

x)3(

1

Applying Score function method to example

Step3-Using intuitionistic weighted arithmetic averaging operator to aggregate all

, I, j, k=1,2,3

Where W= weight of each criteria

11

xk

i

)(

n

k

k

ii xwx k1

)(

Applying Score function method to example

Putting the values from decision matrix in previous formula we get

=(0.09321,0.000174)

=(0.07053, 0.000426)

=(0.01966, 0.000248)

12

x1

x2

x3

Applying Score function method to example

Step4-Using Score function formula

to get Score functions

& each alternative A, B & C.

13

2

13)(

vx

ijij

ijS

)(),(21 xx SS

)(3xS

Applying Score function method to example

= -0.36037

= -0.39441

=-0.47063

14

)(1xS

)(2xS

)(3xS

Applying Score function method to example

Step5- Rank all the alternatives A, B,C and select the best one in accordance with the values of Score function .

Now,

Therefore

Hence A > B > C A is best.

15

)(&)(),(321 xxx SSS

)()()(321 xxx SSS

xxx 321

REFERENCES

Atanassov K., “Intuitionistic fuzzy sets .Fuzzy Sets and System”,110(1986) 87-96

Atanassov K., “ More on intuitionistic fuzzy Sets,Fuzzy Sets and Systems”,33(1989) 37-46

Bustine H. and Burillo P., “Vauge sets are intuitionistic fuzzy sets,Fuzzy sets and systems”,79(1996) 403-405

Xu Z.S., “Intuitionistic preference relations and their applications in group decision making.Information Sciences”,177(2007) 2263-2379

Zadeh L.A., “Fuzzy Sets.Information and control”,8(1965) 338-353

16