Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers. Advisor: Prof. Ta-Chung Chu Graduate: Elianti ( 李水妮 ) M977z240. 1. Introduction. Assume: k decision makers (i.e. D t ,t=1~ k ) m alternative (i.e. A i ,i=1~ m ) n criteria ( C j ,j=1~ n )
Chapter 4MODEL ESTABLISMENTThe Preference Degree of Two Fuzzy Numbers
Advisor: Prof. Ta-Chung Chu
Graduate: Elianti (李水妮)
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:
For benefit: Cj=g+1~h
For cost: Cj=h+1~n
Qualitative criteria Quantitative criteria
Let Xijt= (aijt,bijt,cijt), i= 1,…,m, j= 1,…,g, t=1,…,k, is the rating assigned to alternative Aiby decision maker Dt under criterion Cj.
Xij= (aij,bij,cij) is the averaged rating of alternative Aiversus criterion Cjassessed by the committee of decision makers.
Qualitative (subjective) criteria are measured by linguistic values represented by fuzzy numbers.
The fuzzy multi-criteria decision making decision can be concisely expressed in matrix format after normalization as follow:
j = 1~n
Let j=1,…,n t=1,…,k be the weight of importance assigned by decision maker Dtto criterion Cj.
Wj = (oj,pj,qj) is the averaged weight of importance of criterion Cj assessed by the committee of k decision makers, then:
Here, Piis the final fuzzy evaluation values of each alternative Ai.
The membership functions of the Pican be developed as follows:
There are now two quotations to solve, there are:
So, Eq. (4.7) and (4.8) can be expressed as:
The left membership function and the right membership function of the final fuzzy evaluation value Pi can be produced as follows:
Only when Gi1 =0 and Gi2=0, Pi is triangular fuzzy number, those are:
For convenience, Pi can be donated by:
Let Zhk=Ph-Pk, h,k=1,2,…m, the -cut of Zhk can be expressed as:
Because the formula is too complicated, then we make some assumptions as follows:
Using Eq. (4.16) and (4.17), the left and right membership functions of the difference Zhk=Ph-Pkcan be produced as follows: