Maximizing value and Minimizing base on Fuzzy TOPSIS model

1. Maximizing value and Minimizing base on Fuzzy TOPSIS model Advisor: Prof. Ta Chung ChuStudent : Pham Hoang Chien (Rhett) 林師賢(Shih Hsien Lin)

2. 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)

3. 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)

4. The algorithm • Assume: Committee of k decision makers (i.e. Dt, t=1k) Responsible for selection m alternative (i.e. Ai,i=1m) Under n criteria (Cj, j=1n) A classic fuzzy multi-criteria decision making problem can be expressed in matrix format as follows:

5. Before normalization Quantitative criteria Qualitativecriteria

6. After normalization Quantitative criteria Qualitative criteria

7. 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

8. Chen’smethod 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

9. Weight matrix After we calculate , we get result

10. Weighted normalized decision matrix Weighted normalized decision matrix is obtained by multiplying normalized matrix with the weights of the criteria

11. Weighted normalized decision matrix

12. Weighted normalized decision matrix (con’t) • The final fuzzy evaluation values can be developed via arithmetic operation of fuzzy numbers as

13. Weighted normalized decision matrix (con’t) • Let assume we have (4.10) (4.11)

14. Weighted normalized decision matrix (con’t) • For convenience, we make some assumptions:

15. Weighted normalized decision matrix (con’t) Equations 4.10 and 4.11 can be expressed as:

16. Defuzzification Our 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

17. Chen’s maximizing set and minimizing set approach Definition 1 The maximizing set M is a fuzzy subset with as (4.12) The minimizing set N is a fuzzy subset with as (4.13) where

18. Chen’s maximizing set and minimizing set approach Definition 2 The right utility value of each is defined (4.14) The left utility value of each is defined (4.15)

19. Chen’s maximizing set and minimizing set approach Definition 3 The total utility value of each is defined as (4.16) The total utility is used to rank fuzzy number. The larger the , the larger the

20. Final Ranking Values In our model we modify Chen’s maximizing and minimizing set approach Definition 1 The maximizing set M is a fuzzy subset with as The minimizing set N is a fuzzy subset with as

21. Reference • Fuzzy performance evaluation in Turkish Banking Sector using Analytic Hierarchy Process and TOPSIS • An interval arithmetic based fuzzy TOPSIS model

22. Thank you for your attention