1 A Maximizing Set and Minimizing Set Based Fuzzy MCDM Approach for the Evaluation and Selection of the Distribution Centers Advisor:Prof. Chu, Ta-Chung Student: Chen, Chun Chi

2 Outline Introduction Fuzzy set theory Model development Numerical example Conclusion

3 Introduction Properly selecting a location for establishing a distribution center is very important for an enterprise to effectively control channels, upgrade operation performance, service level and sufficiently allocate resources, and so on. Selecting an improper location of a distribution center may cause losses for an enterprise. Therefore, an enterprise will always conduct evaluation and selection study of possible locations before determining distribution center.

Introduction (cont.) Evaluating a DC location, many conflicting criteria must be considered: 1. objective – these criteria can be evaluated quantitatively, e.g. investment cost. 2. subjective – these criteria have qualitative definitions, e.g. expansion possibility, closeness to demand market, etc. Perez et al. pointed out “Location problems concern a wide set of fields where it is usually assumed that exact data are known, but in real applications is full of linguistic vagueness. 4

Introduction (cont.) Fuzzy set theory, initially proposed by Zadeh, and it can effectively resolve the uncertainties in an ill-defined multiple criteria decision making environment. Some recent applications on locations evaluation and selection can be found, but despite the merits, most of the above papers can not present membership functions for the final fuzzy evaluation values and defuzzification formulae from the membership functions. To resolve the these limitations, this work suggests a maximizing set and minimizing set based fuzzy MCDM approach. 5

6 Introduction (cont.) Purposes of this paper: Develop a fuzzy MCDM model for the evaluation and selection of the location of a distribution center. Apply maximizing set and minimizing set to the proposed model in order to develop formulae for ranking procedure. Conduct a numerical example to demonstrate the feasibility of the proposed model.

Fuzzy set theory 2.1 Fuzzy sets 2.2 Fuzzy numbers 2.3 α-cuts 2.4 Arithmetic operations on fuzzy numbers 2.5 Linguistic values 7

8 2.1 Fuzzy set The fuzzy set A can be expressed as: (2.1) where U is the universe of discourse, x is an element in U, A is a fuzzy set in U, is the membership function of A at x. The larger, the stronger the grade of membership for x in A.

9 2.2 Fuzzy numbers A real fuzzy number A is described as any fuzzy subset of the real line R with membership function which possesses the following properties (Dubois and Prade, 1978): –(a) is a continuous mapping from R to [0,1]; –(b) –(c) is strictly increasing on [a,b]; –(d) –(e) is strictly decreasing on [c,d]; –(f) where, A can be denoted as.

Fuzzy numbers (Cont.) The membership function of the fuzzy number A can also be expressed as: (2.2) where and are left and right membership functions of A, respectively.

α-cut The α-cuts of fuzzy number A can be defined as: (2.3) where is a non-empty bounded closed interval contained in R and can be denoted by, where and are its lower and upper bounds, respectively.

Arithmetic operations on fuzzy numbers Given fuzzy numbers A and B,, the α-cuts of A and B are and respectively. By interval arithmetic, some main operations of A and B can be expressed as follows (Kaufmann and Gupta, 1991): – (2.4) – (2.5) – (2.6) – (2.7) – (2.8)

Linguistic variable According to Zadeh (1975), the concept of linguistic variable is very useful in dealing with situations which are complex to be reasonably described by conventional quantitative expressions. A 1 =(0,0,0.25)=Unimportant A 2 =(0,0.25,0.5)=Less important A 3 =(0.25,0.50,0.75)=important A 4 =(0.50,0.75,1.00)=More important A 5 =(0.75,1.00,1.00)=Very important Figure 2-1. Linguistic values and triangular fuzzy numbers

14 Model development 3.1 Aggregate ratings of alternatives versus qualitative criteria 3.2 Normalize values of alternatives versus quantitative criteria 3.3 Average importance weights 3.4 Develop membership functions 3.5 Rank fuzzy numbers

15 Model development decision makers candidate locations of distribution centers selected criteria, In model development process, criteria are categorized into three groups: –Benefit qualitative criteria: –Benefit quantitative criteria: –Cost quantitative criteria:

Aggregate ratings of alternatives versus qualitative criteria Assume (3.1) –where – Ratings assigned by each decision maker for each alternative versus each qualitative criterion. – Averaged ratings of each alternative versus each qualitative criterion.

17 3.2Normalize values of alternatives versus quantitative criteria is the value of an alternative versus a benefit quantitative criteria or cost quantitative criteria. denotes the normalized value of (3.2) For calculation convenience, assume

Average importance weights Assume (3.3) where represents the weight assigned by each decision maker for each criterion. represents the average importance weight of each criterion.

Develop membership functions The membership function of final fuzzy evaluation value, of each candidate distribution center can be developed as follows: The membership functions are developed as: (3.4) (3.5) (3.6)

Develop membership functions (cont.) From Eqs.(3.5) and (3.6),we can develop Eqs.(3.7) and (3.8)as follows: (3.7) (3.8)

Develop membership functions (cont.) When applying Eq.(3.8)to Eq.(3.4), three equations are developed: (3.9) (3.10) (3.11)

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Develop membership functions (cont.) By applying the above equations, Eqs.(3.9)-(3.11) can be arranged as Eqs.(3.12)-(3.14)as follows: (3.12) (3.13) (3.14)

Develop membership functions (cont.) Applying Eqs.(3.12)-(3.14) to Eq.(3.4) to produce Eq.(3.15): (3.15) The left and right membership function of can be obtained as shown in Eq. (3.16) and Eq. (3.17) as follows: (3.16) If (3.17) If

Rank fuzzy numbers In this research, Chen’s maximizing set and minimizing set (1985) is applied to rank all the final fuzzy evaluation values. Definition 1. The maximizing set M is defined as: (3.18) The minimizing set N is defined as: (3.19) where usually k is set to 1.

Rank fuzzy numbers (cont.) Definition 2. The right utility of is defined as: (3.20) The left utility of is defined as: (3.21) The total utility of is defined as: (3.22)

Rank fuzzy numbers (cont.) Figure 3-1. Maximizing and minimizing set

Rank fuzzy numbers (cont.) Applying Eqs.(3.16)~ (3.22), the total utility of fuzzy number can be obtained as: (3.23)

Rank fuzzy numbers (cont.) is developed as follows: Assume: (3.24) (3.25)

Rank fuzzy numbers (cont.) is developed as follows: Assume: (3.26) (3.27)

31 NUMERICAL EXAMPLE 4.1 Ratings of alternatives versus qualitative criteria 4.2 Normalization of quantitative criteria 4.3 Averaged weights of criteria 4.4 Development of membership function 4.5 Defuzzification

32 NUMERICAL EXAMPLE Assume that a logistics company is looking for a suitable city to set up a new distribution center. Suppose three decision makers, D 1, D 2 and D 3 of this company is responsible for the evaluation of three distribution center candidates, A 1, A 2 and A 3.

33 Criteria Figure 4-1. Selected criteria

34 Unimportant(0.00,0.00,0.25) Less important(0.00,0.25,0.50) Important(0.25,0.50,0.75) More important(0.50,0.75,1.00) Very important(0.75,1.00,1.00) Very low (VL) /Very difficult (VD) /Very far (VF)(0.00,015,0.30) Low (L)/Difficult (D)/Far (F)(0.15,0.30,0.50) Medium (M)(0.30,0.50,0.70) High (H)/Easy (E)/Close (C)(0.50,0.70,0.85) Very high (VH)/Very easy (VE)/Very close (VC)(0.70,0.85,1.00) Table 4-1 Linguistic values and fuzzy numbers for importance weights Table 4-2 Linguistic values and fuzzy numbers for ratings

Ratings of alternatives versus qualitative criteria Ratings of distribution center candidates versus qualitative criteria are given by decision makers as shown in Table 4-3. Through Eq. (3.1), averaged ratings of distribution center candidates versus qualitative criteria can be obtained as also displayed in Table 4-3. CandidatesCriteriaD1D1 D2D2 D3D3 Averaged Ratings A1A1 C1C1 VHH (0.63,0.80,0.95) C2C2 VEEM(0.50,0.68,0.85) C3C3 CVC (0.63,0.80,0.95) C4C4 MHH(0.43,0.63,0.80) A2A2 C1C1 VH H(0.63,0.80,0.95) C2C2 MME(0.37,0.57,0.75) C3C3 CCVC(0.57,0.75,0.90) C4C4 VH (0.70,0.85,1.00) A3A3 C1C1 LLH(0.27,0.43,0.62) C2C2 VEE (0.63,0.80,0.95) C3C3 MMC(0.37,0.57,0.75) C4C4 LMH(0.32,0.50,0.68)

Normalization of quantitative criteria Evaluation values under quantitative criteria are objective. Suppose values of distribution center candidates versus quantitative criteria are present as in Table 4-4. Table 4-4 Values of distribution center candidates versus quantitative criteria Criteria Distribution Center Candidates Units A1A1 A2A2 A3A3 C5C hectare C6C6 2510million

Normalization of quantitative criteria (Cont.) According to Eq. (3.2), values of alternatives under benefit and cost quantitative criteria can be normalized as shown in Table 4-5. Table 4-5 Normalization of quantitative criteria Criteria Distribution Center Candidates A1A1 A2A2 A3A3 C5C C6C

Averaged weights of criteria The linguistic values and its corresponding fuzzy numbers, shown in Table 4-1, are used by decision makers to evaluate the importance of each criterion as displayed in Table 4-6. The average weight of each criterion can be obtained using Eq. (3.3) and can also be shown in Table 4-6. Table 4-6 Averaged weight of each criterion D1D1 D2D2 D3D3 Averaged weights C1C1 MIVIIM(0.50,0.75,0.92) C2C2 IMMILI(0.25,0.50,0.75) C3C3 LI VI(0.25,0.53,0.67) C4C4 UIIMVI(0.33,0.50,0.67) C5C5 MIVIIM(0.50,0.75,0.92) C6C6 VI (0.75,1.00,1.00)

Apply Eqs. (3.4)-(3.15) and g=4, h=5, n=6 to the numerical example to produce the following values For each candidate as displayed in Table Table 4-7 Values for A1A1 A2A2 A3A3 A i A i A i B i B i B i C i C i C i A1A1 A2A2 A3A3 D i D i D i O i O i O i P i P i P i A1A1 A2A2 A3A3 Q i Q i Q i

the calculation values for are shown in table Table 4-8 Values for A1A1 A2A2 A3A3 A i1 +A i2 -C i B i1 +B i2 -D i O i1 +O i2 -Q i C i1 +C i2 -A i D i1 +D i2 -B i P i1 +P i2 -P i Q i1 +Q i2 -O i

Development of membership function Through Eqs. (3.17) - (3.18), the left,, and right,, membership functions of the final fuzzy evaluation value,, of each distribution center candidate can be obtained and displayed in Table 4-9. Table 4-9 Left and right membership functions of G i

Defuzzification By Eqs. (3.23)-(3.28) in Model development for defuzzification, the total utilities, and can be obtained and shown in Table Table 4-10 Total utilities and Then according to values in Table 4-10, candidate A 2 has the largest total utility, Therefore becomes the most suitable distribution center candidate. AlternativesT1T1 T2T2 T3T

43 CONCLUSIONS A fuzzy MCDM model is proposed for the evaluation and selection of the locations of distribution centers Chen’s maximizing set and minimizing set is applied to the model in order to develop ranking formulae. Ranking formulae are clearly developed for better executing the decision making A numerical example is conducted to demonstrate the computational procedure and the feasibility of the proposed model.