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Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers

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 )

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Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers

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  1. Chapter 4MODEL ESTABLISMENTThe Preference Degree of Two Fuzzy Numbers Advisor: Prof. Ta-Chung Chu Graduate: Elianti (李水妮) M977z240

  2. 1. Introduction • Assume: 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: • Qualitative (all of them are benefit), Cj=1~g • Quantitative For benefit: Cj=g+1~h For cost: Cj=h+1~n

  3. 2. Ratings of Each Alternative versus Criteria Qualitative criteria Quantitative criteria BenefitCost 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.

  4. 2. Ratings of Each Alternative versus Criteria Xij= (aij,bij,cij) is the averaged rating of alternative Aiversus criterion Cjassessed by the committee of decision makers. Then: (4.1) Where: j=1~g

  5. 2. Ratings of Each Alternative versus Criteria Qualitative (subjective) criteria are measured by linguistic values represented by fuzzy numbers.

  6. 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 rij=(eij,fij,gij) is the performance value of alternative Aiversus criteria Cj, j=g+1 ~ n. • The normalization of the rijis as follows: (4.2)

  7. 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

  8. 3. Averaged Importance Weights 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: (4.3) Where:

  9. 4. Averaged Importance Weights • The degree of importance is quantified by linguistic terms represented by fuzzy numbers

  10. 4. Final Fuzzy evaluation Value • The final fuzzy evaluation value of each alternative Aican be obtained by using the Simple Addictive Weighting (SAW) concept as follow: Here, Piis the final fuzzy evaluation values of each alternative Ai. i=1,2,…,m,

  11. 4. Final Fuzzy evaluation Value The membership functions of the Pican be developed as follows: and

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

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

  14. 5. Final Fuzzy evaluation Value The left membership function and the right membership function of the final fuzzy evaluation value Pi can be produced as follows: (4.11) (4.12) Only when Gi1 =0 and Gi2=0, Pi is triangular fuzzy number, those are: For convenience, Pi can be donated by: (4.13)

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

  16. 5. An Improved Fuzzy Preference Relation • The final fuzzy evaluation values Ph and Pkare triangular fuzzy numbers. The difference between Ph and Pkis alsoa triangular fuzzy number and can be calculated as: Let Zhk=Ph-Pk, h,k=1,2,…m, the -cut of Zhk can be expressed as: Where

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

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

  19. 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 Zhk=Ph-Pkcan be produced as follows: (4.18) (4.19)

  20. 5. An Improved Fuzzy Preference Relation • Obviously, Zhk=Ph-Pk may not yield a triangular shape as well. Only when Ghk1=0 and Ghk2=0, is a triangular fuzzy number, that is: • For convenience, Zhkcan be denoted by: (4.20)

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