Supplier selection and evaluation using generalized fuzzy multi criteria decision making approach

2016 Eighth International Conference on Knowledge and Systems Engineering (KSE) Supplier Selection and Evaluation Using Generalized Fuzzy Multi-Criteria Decision Making Approach Luu Huu Van Vincent F Yu Department of Industrial Management National Taiwan University of Science and Technology Taipei 10607, Taiwan Email: vanluuhuu82@gmail.com Department of Industrial Management National Taiwan University of Science and Technology Taipei 10607, Taiwan Email: vincent@mail.ntust.edu.tw Shuo-Yan Chou Department of Industrial Management National Taiwan University of Science and Technology Taipei 10607, Taiwan Email: sychou@mail.ntust.edu.tw Luu Quoc Dat Department of Development Economics University of Economics and Business, Vietnam National University Hanoi, Vietnam Email: datlq@vnu.edu.vn The supplier selection process mainly involves evaluation of different alternative suppliers based on different qualitative and quantitative criteria This process is essentially considered as a multiple criteria decision making (MCDM) problem which is affected by different tangible and intangible criteria including price, quality, technology, flexibility, delivery, etc MCDM methods incorporate with fuzzy set theory has been used popular to solve uncertainty in the supplier selection decision process and to it provide a language appropriate to dispose imprecise criteria [1] Fuzzy MCDM approach allows decision makers to evaluate alternatives suing linguistic terms such as high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides and algorithm to aggregate the assessments of alternatives And the FMCDM approach offers a flexible practical and effective way of group decision making Abstract—Supplier selection and evaluation plays an importance role for companies to gain competitive advantage and achieve the objectives of the whole supply chain To select the appropriate suppliers, many qualitative and quantitative criteria are needed consider in the decision process Therefore, supplier selection and evaluation can be seem as a multi-criteria decision making (MCDM) problem in vague environment However, most existing fuzzy MCDM approaches have been developed using normal fuzzy numbers or converting generalized fuzzy numbers into normal fuzzy numbers through normalization process This leads to a restriction in the application of the fuzzy MCDM approaches In this study, a generalized fuzzy MCDM approach is proposed to select and evaluate suppliers In the proposed approach, the ratings of alternatives and important weights of criteria are expressed in linguistic terms using generalized fuzzy numbers Then, the membership functions of the final fuzzy evaluation value are developed To make procedure easier and more practical, the weighted ratings are defuzzified into crisp values by employing the maximizing and minimizing set ranking approach to determine the ranking order of alternatives Finally, a numerical example is presented to illustrate the applicability and efficiency of the proposed method Keywords—Generalized fuzzy supplier selection numbers, Although numerous fuzzy MCDM approaches and applications have been investigated in literature [1, 5, 6, 9, 11], most of these approaches have been developed using normal fuzzy numbers or converting generalized fuzzy numbers into normal fuzzy numbers through normalization process [7] This leads to a restriction in the application of the fuzzy MCDM approaches Additionally, [8] pointed out that the normalization process is a serious disadvantage ranking method, I INTRODUCTION In this study, a generalized fuzzy MCDM approach is proposed to select and evaluate suppliers In the proposed approach, the ratings of alternatives and importance weights of criteria are expressed in linguistic terms using generalized fuzzy numbers Then, the membership functions of the final fuzzy evaluation value are developed To make the procedure easier and more practical, the weighted ratings are defuzzified into crisp values by employing the maximizing and minimizing set ranking approach to determine the ranking order of alternatives Finally, a numerical example is presented to illustrate the applicability and efficiency of the proposed method Supplier selection process is one of the most important components of production and logistics management for many companies Selection of the appropriate supplier can significantly lessen purchasing costs, consequently enhance the enterprise competitiveness in the market, and increase end user satisfaction [5] The best supplier selection can also create a great contributes to the quality of goods for company and to achieve the objectives to overall the supply chain performance of organizations [4] However, selection of wrong supplier could be enough to upset the company’s financial and operation position 978-1-4673-8929-7/16/$31.00 ©2016 IEEE 31 The rest of this paper is organized as follows Section briefly reviews the basic definitions and arithmetic operations of generalized fuzzy number Section develops the fuzzy MCDM using generalized fuzzy numbers Section presents a numerical example to demonstrate the feasibility of the proposed mode Finally, the conclusion and discussion are presented in section d = Max( a × b , a × b , a × b , a × b ) f A is strictly decreasing on [c, d ]; f A ( x) = 0, for all x ∈ ( d , ∞ ] , Where a, b, c and d are real numbers Unless elsewhere specified, it is assumed that A is convex and bounded (i.e (f) (A B Some arithmetic operators between the generalized fuzzy numbers A and B are defined as follows: ( A⊗ r) α A B (1) where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values 2) Generalized trapezoidal fuzzy numbers subtraction: A(−) B = (a1 , a2 , a3 , a4 ; wA )(−)(b1 , b2 , b3 , b4 ; wB ) (2) where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values 1 4 4 b = Min( a × b , a × b , a × b , a × b ), 2 3 3 c = Max( a × b , a × b , a × b , a × b ), 2 3 3 1 3 4 A B 3 A (4) B ] [ ] (6) (7) B ) = êơ Al Bu , Au Bl ẳ (8) = êơ Al r , Au ⋅ r º¼ , r ∈ R + (9) In fuzzy set theory, conversion scales are employed to convert linguistic values into fuzzy numbers Determining the number of conversion scales is generally intuitive and subjective, in this study a five-point scale has been used to convert linguistic values into triangular fuzzy numbers (TFNs), as introduced (1) in TABLE and TABLE 3) Generalized trapezoidal fuzzy numbers multiplication: Where a = Min( a × b , a × b , a × b , a × b ), E Linguistic Values and Fuzzy Numbers There are decision situations in which the information can not be assessed precisely in a quantitative form but may be in a qualitative one, and thus, the use of a linguistic approach is necessary The concept of linguistic variable is to provide a means of approximating characterization of ill-defined phenomena in a system or model Linguistic values are those values represented in words or sentences in natural or artificial languages, where each linguistic value can be modeled by a fuzzy set A ( + ) B = ( a1 , a , a3 , a ; w A )( + )(b1 , b2 , b3 , b4 ; wB ) = (a1 − b4 , a2 − b3 , a3 − b2 , a4 − b1 ;min( wA , wB )), = êơ Al Bl , Au Bu ẳ α 1) Generalized trapezoidal fuzzy numbers addition: = ( a1 + b1 , a + b2 , a3 + b3 , a + b4 ; min( w A , wB )), B α (A B ) = êơ Al Bu , Au Blα º¼ and ≤ w ≤ [ ( A ⊗ B) α - Cuts of Fuzzy Numbers where a , a , a , a , b , b , b and b4 are real values, ≤ w ≤ 1 cuts of A and B are Aα = Alα , Auα and B α = Blα , Buα , respectively By the interval arithmetic, some main operations of A and B can be expressed as follows: α (5) ( A ⊕ B ) = êơ Al + Bl , Au + Buα º¼ Let A and B are two generalized trapezoidal fuzzy numbers, i.e., A = (a , a , a , a ; w ) and B = (b , b , b , b ; w ), 3 Given fuzzy numbers A and B, where A, B ∈ R + , the Į- B Arithmetic Operations [2] presented arithmetical operations between generalized trapezoidal fuzzy numbers based on the extension principle 2 D Arithmetic Operations on Fuzzy Numbers −∞ < a, d < ∞) The α - cuts of fuzzy number A can be defined as (Kaufmann and Gupta, 1991): α Aα ={ xf A ( x )≥α }, α ∈[0,1] , where A is a non-empty bounded closed interval contained in R and can be denoted by Aα =[ Alα , Auα ] where Alα and Auα are its lower and upper bounds of the closed interval respectively f A is strictly increasing on [ a, b]; A (d) f A ( x) = ϖ, for all x ∈ [b, c ] ; 4 = (a / b , a / b , a / b , a / b ;min(w , w )), (b) f A ( x) = 0, for all x ∈ ( −∞ , a ] ; 3 C real numbers Then, the division of A and B is as follows: A(/)B = (a , a , a , a ; w )(/)(b , b , b , b ; w ) interval [0,ϖ ], ≤ ϖ ≤ 1; 4) Generalized trapezoidal fuzzy numbers division: The inverse of the fuzzy number B is 1/ B = (1/ b ,1/ b ,1/ b ,1/ b ; w ) where b , b , b and b are non-zero positive numbers or all non-zero negative real numbers Let a , a , a , a , b , b , b and b be non-zero positive f A is a continuous mapping from R to the closed (e) A Trapezoidal Fuzzy Numbers A generalized fuzzy number A = (a, b, c, d; ϖ ) is described as any fuzzy subset of the real line R with membership function f A that can be generally defined as [2]: (c) It is obvious that if a , a , a , a , b , b , b and b are all positive real numbers, then: A(x) B = (a1 × b1 , a2 × b2 , a3 × b3 , a4 × b4 ; min( wA , wB )) (3) II PRELIMINARIES (a) 32 TABLE THE LINGUISTIC TERMS AND RELATED FUZZY NUMBERS OF EVALUATION RATINGS C Aggregate the Weighted Ratings The membership function Ti i = 1,…, m, j = 1,…, n is the final fuzzy evaluation value of each alternative: Ratings Linguistic variables TFNs Very Low (VL) (0.0, 0.1, 0.2) Low (L) ( 0.1, 0.3, 0.5) Fair (F) (0.3, 0.5, 0.7) Good (G) (0.6, 0.7, 1.0) Very Good (VG) (0.8, 0.9, 1.0) n T i = ¦ w j ⊗ x ij (12) j =1 α - Cuts is used to develop the membership function: n α α α T i = ¦ w j ⊗ x ij TABLE THE LINGUISTIC TERMS AND RELATED FUZZY NUMBERS OF CRITERIA WEIGHTS Thus, the membership function is developed as follows: Ratings Linguistic variables TFNs Unimportant (UI) (0.0, 0.1, 0.3) Less Important (LI) ( 0.2, 0.3, 0.4) Important (IM) (0.3, 0.5, 0.7) More Important (MI) (0.7, 0.8, 0.9) Very Imprortant (VI) (0.8, 0.9, 1.0) α w j = [( u j − t j ) α ⁄ϖ w + t j , ( u j − v j ) α ⁄ϖ w + v j ] ( ­°(u j − t j )(n ij − mij )α 2⁄ϖ wϖ x + α ⊗ = w j x ij ® °¯[(u j − t j )m ij ⁄ϖ w + (n ij − mij )t j ⁄ϖ x ]α + mijt j ; ẵ ắ +[(u j v j ) p ij ⁄ϖ w + (n ij − p ij )v j ⁄ϖ x ]α + p ij v j ° ¿ (u j − v j )(n ij − p ij )α 2⁄ϖ wϖ x ­n ( u j − t j )( nij − mij ) α2⁄ϖwϖx °°¦ α α j =1 n ¦wj ⊗ xij = ® n j =1 °+¦[( u j − t j ) mij⁄ϖw + ( nij − mij ) t j⁄ϖx ]α + ¦mijt j; °¯ j=1 j =1 be the importance n weights assigned by decision maker Dt to criterion C The j averaged weight w j = (t j , u j , v j ,ϖ jt ) of criterion C assessed by committee of k decision makers can be evaluated as: j ẵ ắ n +Ưêơ( u j − v j ) pij⁄ϖw + ( nij pij ) v jx ẳ + Ư pijv j ° °¿ j =1 j =1 n ¦( u − v ) ( n − p ) α ⁄ϖ ϖ j w j = (1/ k ) ⊗ ( w j1 ⊕ w j ⊕ ⊕ w jk ) (10) w ij x n Aij1 = ¦ ( u j − t j )( n ij − mij )⁄ϖ wϖ x , j =1 j = 1, 2, , h, n B ij1 = ¦ [( u j − t j ) m ij ⁄ϖ w + ( n ij − m ij ) t j ⁄ϖ x ] , t = 1, 2, , k , be the suitability rating assigned to alternative Ai by decision maker Dt under criterion C j The averaged rating j =1 n ( ) C ij1 = ¦ ( u j − v j ) n ij − p ij ⁄ϖ w ϖ x , xij = (mij , nij , pij , ϖij ), of alternative Ai versus criteria C assessed by the committee of k decision makers can be expressed as follow j ( xij1 ⊕ xij ⊕ ⊕ xijk ) k ij Suppose that: B Agrregate Rating of Alteratives Versus Criteria xij = j j =1 n k k k = t jt , u j = ¦ u jt , v j = ¦ v jt where t j k ¦ k t =1 k t =1 t =1 ϖ j = min{ϖ ,ϖ , ,ϖ j } Let xijt = ( mijt , nijt , pijt , ϖ ijt ), i = 1, 2, , m , ) x ij = êơ ( n ij − m ij ) α ⁄ϖ x + m ij , n ij − p ij α ⁄ϖ x + p ij º¼ III MODEL ESTABLISHMENT A Aggregate the Importance Weights Let W jt = (t jt , u jt , v jt , ϖ jt ), Wjt ∈ R + (13) j =1 j =1 n ( ) n Dij1 = Ư êơ( u j v j ) pij⁄ϖw + nij − pij v j⁄ϖx º¼, O ij1 = ¦ mijt j, j =1 (11) n Qij1 = ¦ pij v j , P ij1 = where mij = mijt , nij = nijt , pij = pijt k k k j =1 j =1 n ¦n u , ij j j =1 Then we have: ϖij = min{ϖij1 , ϖij , , ϖijk } n ¦w j =1 33 α j ⊗ xαij = [ Ai1α + B i1α + O i1, C i1α + D i1α + Q i1] We have two simplified equation as follows: ⇔ 4C i2 x 2R i1 + 4C i êơ D i ( x max − x ) − 2C i x º¼ x R i1 Aiα + Biα + Oi x = + êơ D i ( x max − x ) − 2C i x º¼ C iα + D iα + Q i − x = = ( x max − x ) êơ D i2 + 4C i ( x R i1 − Q i ) º¼ From the above two equations, we have: α= 1/ − B i ± [ Bi + Ai ( x − Oi )] Ai α= 1/ − D i ± [ Di + 4Ci ( x − Qi )] 2Ci ⇔ 4C i2 x 2R i1 + 4C i êơ D i ( x max x ) − 2C i x º¼ x R i1 + D i2( x max − x ) − D iC i x ( x max − x ) + 4C i2 x 2min 2 = D i2( x max − x ) + 4C i x R i1( x max − x ) − 4C iQ i( x max − x ) , Oi ≤ x ≤ Pi , Pi ≤ x ≤ Qi ⇔ 4C i2 x 2Ri1 + 4C i x Ri1 ª Di ( x max − x ) − 2C i x − ( x max − x ) º ¬ ¼ 2 + 4C i2 x − 4DiC i x ( x max − x ) + 4C iQi( x max − x ) = Since α ∈ [0,1], then the left and right membership functions f i L (x) and f TRi (x) of Ti can be produced as: α = f Ti ( x) = L α = f T i ( x) = R 1/ −Bi + [ Bi + Ai ( x − Oi )] Ai , Oi ≤ x ≤ Pi ⇔xRi1 ={2Cx i +(xmax − xmin)(−Di + xmax − xmin) − (xmax −xmin)[(−Di + xmax −xmin)2 +4Ci(xmin −Qi)]1/2} /2Ci (14) 1/2 −Di −[Di + 4Ci (x − Qi )] 2Ci , Pi ≤ x ≤ Qi (15) Ti = (Oi , Pi , Qi , Ai , Bi , Ci , Di ) , i = 1, , m D Defuzzification The conversion from a fuzzy set to a crisp number is called defuzzification Numerous ranking methods have been investigated to rank the fuzzy numbers in literature This study employs the ranking method proposed by [3] to defuzzify all the final fuzzy evaluation values This ranking method in one of the most commonly used approaches of ranking fuzzy numbers in fuzzy decision making E Ranking Obtain Using [3] ranking method, the total utility value of each A is applied to defuzzify all the final fuzzy evaluation values i (21) x L i1 = {2 Ai x max + ( x max − x )( B i + x max − x ) − ( x max − x )[( B i + x max − x ) ẵ / ắ Ai + Ai ( x max − O i )]1/ ¿ (22) x Li = {2 Ai x + ( x max − x )(− B i + x max − x ) − )2 ½ ( x max − x ) [(− B i + x max − x 1/2 ¾ / Ai +4 Ai( x − O i )] ¿ (23) i1 i − D i − [ Di2 + 4Ci ( xRi1 − Qi )]1/ 2C i i1 − (16) + − D i − [ Di2 + 4Ci ( xRi − Qi )]1/ 2C i i2 UG = x Ri = {2C i x max + ( x max − x )( D i + x max − x ) − ª º 2ẵ ( x max x ) ô(+D4 i + x max −−xQmin ) » °¾ / 2C i i) ẳ C i ( x max Then, the total utility value of Ti , with index of optimism α = 0.5 is defined as: ­° − D i − [ Di2 + 4Ci ( xR − Qi ]1/ 0.5 i = ( ) uT ® 2C i °¯ T as follows: UG = (20) Similarly, we have: For convenience, Ti can be expressed as: UM = (17) i − 1/ − B i + [ B + Ai ( xLi1 − Oi )] Ai i1 (18) − D i − [ Di2 + 4Ci ( xR − Qi )]1/ i2 2C i − B i + [ Bi2 + Ai ( xL − Oi )]1/ i2 Ai − B i + [ Bi2 + Ai ( xL − Oi )]1/ i1 Ai ½° + 2¾ / °¿ (24) The greater the u T0.5 ( i ) , the bigger fuzzy number A and the higher its ranking order i UM = − B i + [ Bi2 + Ai ( xLi − Oi )]1/ 2 Ai i2 x R i1− x where: x max − x = (19) D i êơ D i2 + 4C i ( x R i1 − Q i ) º¼ IV NUMERICAL EXAMPLE This section applies the proposed approach to solve the supplier selection and evaluation problem to demonstrate the feasibility and applicability of the proposed approach 2C i Assume that a company desires to select the suitable material supplier for the company’s producing strategy After preliminary screening, three suppliers A1, A2 and A3 are chosen for further evaluation A committee of three decision makers, ⇔ 2C i x R i1 − 2C i x + D i ( x max − x ) = − ( x max x ) êơ D i2 + 4C i ( x R i1 − Q i ) º¼ 1/ 34 D1, D2 and D3, and, has been formed to conduct the assessment and to select the most suitable supplier using nine criteria: product/service quality (C1), customer satisfaction (C2), organization control (C3), technological capability (C4), relationship closeness (C5), complaints (C6), product/service warranty period (C7), punctuality of delivery (C8) , unit price (C9) The computational procedure is summarized as the following: A Step 1: Agrregate the Ratings of Alternatives Versus Criteria TABLE presents the suitability ratings of suppliers versus the nine criteria Using Equation (11) the aggregated suitability ratings are obtained in the last column of TABLE C1 C2 C3 C4 C5 C6 C7 C8 C9 Supplier A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 Decision Makers D1 D2 D3 G F G G F G VG G F F G VG VG G G F F G G VG G G G VG G VG F G G F G F VG VG F G F G G VG VG F F F G G G G F G G G G F VG G G F G G G F G G F VG G G F G F VG VG G G G VG G VG G G rij D2 D3 C1 AI AI VI (0.733, 0.867, 0.967; 0.9) C2 AI VI VI (0.667, 0.833, 0.933; 0.9) (0.400, 0.600, 0.767; 0.8) C4 I VI I (0.400, 0.600, 0.767; 0.8) C5 VI VI AI (0.667, 0.833, 0.933; 0.9) C6 AI AI VI (0.733, 0.867, 0.967; 0.9) C7 I I VI (0.400, 0.600, 0.767; 0.8) C8 I VI VI (0.500, 0.700, 0.833; 0.8) C9 VI I I (0.400, 0.600, 0.767; 0.8) Supplier uM1 uM2 uG1 uG2 uT0.5 ( Ai ) Ranking A1 A2 A3 0,635 0,593 0,630 0,343 0,269 0,335 0,569 0,605 0,574 0,288 0,351 0,296 0,530 0,476 0,524 Supplier selection and evaluation problem is a fuzzy MCDM problem that is affected by several qualitative and quantitative criteria In order to solve the supplier selection problem, this paper has proposed and extension of fuzzy MCDM In the proposed approach, the ratings of alternatives and relative importance weights of criteria for suppliers are expressed in linguistic values, which are represented by generalized fuzzy numbers The membership function of each weighted rating of each supplier for each criterion I then developed To avoid complicated calculations of fuzzy numbers, these weighted ratings are defuzzified into crisp values by using the new maximizing set and minimizing set ranking approach to determine the ranking order of alternatives A numerical example was given to illustrate the applicability of the proposed approach The results indicate that the proposed fuzzy MCDM approach is practical and useful The proposed approach can also be applied to other management problems under similar settings such as lecturer’s performance evaluation, project selection, hospital service quality evaluation, logistics center location selection, etc REFERENCES [1] wij D1 I V CONCLUSION TABLE THE IMPORTANCE WEIGHTS OF THE CRITERIA AND THE AGGREGATED WEIGHTS Criteria VI TABLE THE LEFT, RIGHT AND TOTAL UTILITIES OF EACH SUPPLIER (0,667, 0,833, 0,900; 0,9) (0,500, 0,700, 0,800; 0,8) (0,500, 0,700, 0,800; 0,8) (0,500, 0,700, 0,767; 0,8) (0,400, 0,600, 0,733; 0,8) (0,667, 0,833, 0,900; 0,9) (0,733, 0,867, 0,933; 0,9) (0,400, 0,600, 0,700; 0,8) (0,500, 0,700, 0,800; 0,8) (0,400, 0,600, 0,733; 0,8) (0,500, 0,700, 0,767; 0,8) (0,733, 0,867, 0,933; 0,9) (0,733, 0,867, 0,933; 0,9) (0,667, 0,833, 0,900; 0,9) (0,400, 0,600, 0,700; 0,8) (0,400, 0,600, 0,733; 0,8) (0,300, 0,500, 0,633; 0,8) (0,667, 0,833, 0,900; 0,9) (0,667, 0,833, 0,900; 0,9) (0,667, 0,833, 0,900; 0,9) (0,600, 0,800, 0,867; 0,9) (0,500, 0,700, 0,800; 0,8) (0,667, 0,833, 0,900; 0,9) (0,667, 0,833, 0,900; 0,9) (0,667, 0,833, 0,900; 0,9) (0,667, 0,833, 0,900; 0,9) (0,400, 0,600, 0,733; 0,8) B Step 2: Aggregate the Importance weights TABLE displays the importance weights of nine criteria from the three decision makers Using Equation (10) the aggregated weights of criteria from the decision makers as shown in the last column of TABLE Decision Makers I C Step 3: Aggregate the weighted ratings and defuzzification Using Equation (12) to (24), the left, right and total utilities with Į = 1/2 can be obtained as shown in TABLE It can be seen that from TABLE the ranking order of the three supplier is A1 > A3 > A2 Thus, the most suitable suppliers is A1, which has the largest total utility TABLE SUITABILITY RATINGS OF ALTERNATIVES VERUS CRITERIA Criteria C3 [2] [3] 35 J Chai, J.N.K Liu, and E.W.T Ngai, “Application 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[9] E.E Karsak and M Dursun, “An integrated fuzzy MCDM approach for supplier evaluation and selection, ” Com And Ind Eng Vol 82, pp 8293, 2015 [10] L.A Zadeh, Fuzzy sets,” Info and Cont., vol... between Fuzzy AHP and Fuzzy TOPSIS methods to supplier selection, ” Appl Soft Com., vol 21, pp 194-209, 2014 [7] A Kaur and A Kumar, “A new apporach for solving fuzzy transportation problems using generalized. .. defuzzify all the final fuzzy evaluation values This ranking method in one of the most commonly used approaches of ranking fuzzy numbers in fuzzy decision making E Ranking Obtain Using [3] ranking