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) There are 2 types of criteria: a. Qualitative (all of them are benefit), C j =1~g b. Quantitative For benefit: C j =g+1~h For cost: C j =h+1~n

2. Ratings of Each Alternative versus Criteria Qualitative criteria Quantitative criteria BenefitCost Let X ijt = (a ijt,b ijt,c ijt ), i= 1,…,m, j= 1,…,g, t=1,…,k, is the rating assigned to alternative A i by decision maker D t under criterion C j.

2. Ratings of Each Alternative versus Criteria X ij = (a ij,b ij,c ij ) is the averaged rating of alternative A i versus criterion C j assessed by the committee of decision makers. Then:(4.1) Where: j=1~g

2. Ratings of Each Alternative versus Criteria Qualitative (subjective) criteria are measured by linguistic values represented by fuzzy numbers. Linguistic ValueFuzzy 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 r ij =(e ij,f ij,g ij ) is the performance value of alternative A i versus criteria C j, j=g+1 ~ n. The normalization of the r ij is as follows: (4.2)

3.Normalization of the Averaged Ratings The fuzzy multi-criteria decision making decision can be concisely expressed in matrix format after normalization as follow: j = 1~n

3.Averaged Importance Weights Let j=1,…,n t=1,…,k be the weight of importance assigned by decision maker D t to criterion C j. Wj = (o j,p j,q j ) is the averaged weight of importance of criterion Cj assessed by the committee of k decision makers, then: (4.3) Where:

4.Averaged Importance Weights The degree of importance is quantified by linguistic terms represented by fuzzy numbers Linguistic valuesFuzzy 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 A i can be obtained by using the Simple Addictive Weighting (SAW) concept as follow: Here, P i is the final fuzzy evaluation values of each alternative A i. i=1,2,…,m,

4.Final Fuzzy evaluation Value The membership functions of the P i can be developed as follows: and

4.Final Fuzzy evaluation Value By applying Eq. (4.4) and (4.5), one obtains the  -cut of P i as follows: (4.6) There are now two quotations to solve, there are: (4.7) (4.8)

4.Final Fuzzy evaluation Value We assume: So, Eq. (4.7) and (4.8) can be expressed as:

5.Final Fuzzy evaluation Value The left membership function and the right membership function of the final fuzzy evaluation value P i can be produced as follows: (4.11) (4.12) Only when G i1 =0 and G i2 =0, Pi is triangular fuzzy number, those are: For convenience, Pi can be donated by: (4.13)

5.An Improved Fuzzy Preference Relation To define a preference relation of alternative A h over A k, we don’t directly compare the membership function of P h (-) P k. We use the membership function of P h (-) P k. to indicate the prefer ability of alternative A h over alternative A k, and then compare P h (-) P k. with zero. The difference P h (-) P k. here is the fuzzy difference between two fuzzy numbers. Using the fuzzy number, P h (-) P k., one can compare the difference between P h and P k. for all possibly occurring combinations of P h and P k.

5.An Improved Fuzzy Preference Relation The final fuzzy evaluation values P h and P k are triangular fuzzy numbers. The difference between P h and P k is also a triangular fuzzy number and can be calculated as: Let Z hk =P h -P k, h,k=1,2,…m, the  -cut of Z hk can be expressed as: Where

5.An Improved Fuzzy Preference Relation By applying Eq. (4.6) to (4.13) to obtain results as follows: (4.14) (4.15)

5.An Improved Fuzzy Preference Relation Because the formula is too complicated, then we make some assumptions as follows:

5.An Improved Fuzzy Preference Relation There 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 Z hk =P h -P k can be produced as follows: (4.18) (4.19)

5.An Improved Fuzzy Preference Relation Obviously, Z hk =P h -P k may not yield a triangular shape as well. Only when G hk1 =0 and G hk2 =0, is a triangular fuzzy number, that is: For convenience, Z hk can be denoted by: (4.20)