A Grey-Based Approach to Suppliers Selection Problem

Transcription

1 A Grey-Based Approach to Suppliers Selection Problem Guo-Dong Li Graduate School of Science and Engineer Teikyo University Utsunomiya City, JAPAN Masatake Nagai Faculty of Engineering Kanagawa University Yokohama City, JAPAN Daisuke Yamaguchi Graduate School of Engineering Kanagawa University Yokohama City, JAPAN Abstract The suppliers selection is a multiple attribute decision making (MADM) problem Since the decision maker (DM)s like preferences on alternatives or on attributes of suppliers are often uncertain, and thus the selection of good suppliers becomes more difficult Grey theory is one of the methods that are used to study uncertainty, it is superior in mathematical analysis of systems with uncertain information In this paper, we proposed a new grey-based approach to deal with selection problem of suppliers The work procedure is shown as follows briefly: First, the weight and rating of attribute for all alternatives are described by linguistic variables that can be expressed in grey number Second, proposed grey possibility degree to determine the ranking order of all alternatives Finally, an example of selection problem of suppliers was used to illustrate the proposed approach Keywords: suppliers selection, multiple attribute decision making (MADM), grey number, grey possibility degree, grey weight and grey rating 1 Introduction With the globalization of economic market and the development of information technology, suppliers selection problem become one of the most important components in supply chain management [1]-[4] The suppliers selection is a multiple attribute decision making (MADM) problem The decision maker (DM)s always express their preferences on alternatives or on attributes of suppliers, which can be used to help ranking the suppliers or selecting the most desirable one The preference information on alternatives of suppliers and on attributes belongs to DMs subective udgments Generally, DMs udgment are often uncertain and can t be estimated by the exact numerical value And thus the selection problem of suppliers has many uncertainties and becomes more difficult Grey theory [5], [6] is one of the methods that are used to study uncertainty, it is superior in mathematical analysis of systems with uncertain information Up to present, fuzzybased approach has been proposed to deal with the suppliers selection problem under certainty [7] The advantage of grey theory over fuzzy theory [8], [9] is that grey theory considers the condition of the fuzziness That is, grey theory can flexibly deal with the fuzziness situation [10] In this paper, we proposed a new grey-based approach to deal with selection problem of suppliers under uncertainty environment The work procedure is shown as follows briefly: First, the weight and rating of attribute for all suppliers alternatives are described by linguis-

2 tic variables that can be expressed in grey number Then we can obtain the grey decision matrix of suppliers Second, proposed grey possibility degree to determine the ranking order of all alternatives of suppliers Finally, an example of selection problem of suppliers was used to illustrate the proposed approach This paper is organized as follows: Section 2 describes preliminaries of grey theory and grey number comparison Section 3 introduces proposed grey-based approach In Section 4 the proposed approach is applied to the suppliers selection problem Finally, conclusions are described in Section 5 2 Preliminaries 21 Grey Theory Grey theory [5], originally developed by Prof Deng in 1982, has become a very effective method of solving uncertainty problems under discrete data and incomplete information Grey theory has now been applied to various areas such as forecasting, system control, decision making and computer graphics Here, we give some basic definitions of grey system, grey set and grey number in grey theory Definition 1 A grey system [10] is defined as a system containing uncertain information presented by grey number and grey variables The concept of a grey system is shown in Fig1 Definition 2 Let X be the universal set Then a grey set G of X is defined by its two Input Grey variables Grey system Unknown information Known information Grey number Output Grey variables Figure 1: The concept of a grey system mappings µ G (x) and µ G (x) { µg (x) : x [0, 1] µ G (x) : x [0, 1] (1) µ G (x) µ G (x), x X, X = R, µ G (x) and µ G (x) are the upper and lower membership functions in G respectively When µ G (x) = µ G (x), the grey set G becomes a fuzzy set It shows that grey theory considers the condition of the fuzziness and can flexibly deal with the fuzziness situation Definition 3 A grey number is one of which the exact value is unknown, while the upper and/or the lower limits can be estimated Generally grey number is written as G, ( G = G µ µ) Definition 4 If only the lower limit of G can be possibly estimated and G is defined as lower limit grey number G = [G, ) (2) Definition 5 If only the upper limit of G can be possibly estimated and G is defined as lower limit grey number G = (, G] (3) Definition 6 If the lower and upper limits of G can be estimated and G is defined as interval grey number G = [G, G] (4) Definition 7 The basic operation laws of grey numbers G 1 [G 1, G 1 ] and G 2 = [G 2, G 2 ] can be expressed as follows: G 1 + G 2 = [G 1 + G 2, G 1 + G 2 ] (5) G 1 G 2 = [G 1 G 2, G 1 + G 2 ] (6) G 1 G 2 = [G 1, G 1 ] [G 2, G 2 ] = [Min(G 1 G 2, G 1 G 2, G 1 G 2, G 1 G 2 ), Max(G 1 G 2, G 1 G 2, G 1 G 2, G 1 G 2 )] (7) G 1 G 2 = [G 1, G 1 ] [ 1 1, ] (8) G 2 G 2 Definition 8 The lenth of grey number G is defined as L( G) = [G G] (9)

3 22 Comparison of Grey Number We proposed grey possibility degree to compare the ranking of grey numbers Definition 9 For two grey numbers G 1 [G 1, G 1 ] and G 2 = [G 2, G 2 ], the possibility degree of G 1 G 2 can be expressed as follows [11] : P { G 1 G 2 } = Max(0, L Max(0, G 1 G 2 )) L (10) where L = L( G 1 ) + L( G 2 ) For the position relationship between G 1 and G 2, there exist four possible cases on the real number axis The relationship between G 1 and G 2 are determined as follows: 1 If G 1 = G 2 and G 1 = G 2, we say that G 1 is equal to G 2, denoted as G 1 = G 2 Then P { G 1 G 2 }=05 2 If G 2 > G 1, we say that G 2 is larger than G 1, denoted as G 2 > G 1 Then P { G 1 G 2 }=1 3 If G 2 < G 1, we say that G 2 is smaller than G 1, denoted as G 2 < G 1 Then P { G 1 G 2 }=0 4 If there is an intercrossing part in them, when P { G 1 G 2 } >05, we say that G 2 is larger than G 1, denoted as G 2 > G 1 When P { G 1 G 2 } <0,5, we say that G 2 is samller than G 1, denoted as G 2 < G 1 3 Proposed Approach A new approach based on grey possibility degree is proposed to make the order preference of suppliers This method is very suitable for solving the group decision-making problem under uncertainty environment Assume that S = {S 1, S 2,, S m } is a discrete set of m possible suppliers alternatives Q = Table 1: The scale of attribute weights w Scale w Very low (VL) [00,01] Low (VL) [01,03] Medium low (ML) [03,04] Medium (M) [04,05] Medium high (MH) [05,06] High (H) [06,09] Very high (VH) [09,10] Table 2: The scale of attribute ratings G Scale G Very poor (VP) [0,1] Poor (P) [1,3] Medium poor (MP) [3,4] Fair (F) [4,5] Medium good (MG) [5,6] Good (G) [6,9] Very good (VG) [9,10] {Q 1, Q 2,, Q n } is a set of n attributes of suppliers The attributes are additively independent w = { w 1, w 2,, w n } is the vector of attribute weights In this paper, the attribute weights and ratings of suppliers are considered as linguistic variables [12] Here these linguistic variables can be expressed in grey numbers by 1-7 scale shown in Table 1 The attribute ratings G can be also expressed in grey numbers by 1-7 scale shown in Table 2 The procedures are summarized as follows: Step 1: Form a committee of decision-maker and identify attribute weights of suppliers Assume that a decision group has K persons, then the attribute weight of attribute Q can be calculated as w = 1 K [ w1 + w w K ] (11) where w K ( = 1, 2,, n) is the attribute weight of Kth DM and can be described by

4 Select an ideal supplier Q 1 : Product quality Q 2 : Service quality Q 3 : Delivery time Q 4 : Price S 1 : Supplier 1 S 2 : Supplier 2 S 6 : Supplier 6 Table 3: Attribute weights for six suppliers Q D 1 D 2 D 3 D 4 w Q 1 VH H H H [0675, 0925] Q 2 H VH VH H [0750, 0950] Q 3 MH H H MH [0550, 0750] Q 4 M M MH MH [0450, 0550] Figure 2: The selection structure of suppliers grey number w K = [w K, wk ] Step 2: Using linguistic variables for the ratings to make attribute rating value Then the rating value can be calculated as G i = 1 K [ G1 i + G 2 i + + G K i ] (12) where G K i (i = 1, 2,, m; = 1, 2,, n) is the attribute rating value of Kth DM and can be described by grey number G K i = [G K i, G K i ] Step 3: Establishment of grey decision matrix G 11 G 12 G 1n G 21 G 22 G 2n D = (13) G m1 G m2 G mn where G i are linguistic variables based on grey number Step 4: Normalization grey decision matrix G 11 G 12 G 1n D G 21 G 22 G 2n = (14) G m1 G m2 G mn where For benefit attribute, G i [ G G i i = G Max, G ] i G Max is expressed as (15) G Max = Max 1 i m {G i } For cost attribute, G i is expressed as G i = [G Min G i, GMin ] G i (16) G Min = Min 1 i m {G i } The normalization method mentioned above is to preserve the property that the ranges of normalized grey number belong to [0, 1] Step 5: Establishment of weighted normalized grey decision matrix Considering the different important of each attribute, the weighted normalized grey decision matrix can be established as V 11 V 12 V 1n D V 21 V 22 V 2n = (17) V m1 V m2 V mn where V i = G i w Step 6: Making the ideal alternative as referential alternative For m possible suppliers alternatives set S = {S 1, S 2,, S m }, the ideal referential supplier alternative S Max = { G Max 1, G Max 2,, G Max n } can be obtained by S Max = {[Max 1 i m V i1,max 1 i m V i1 ], [Max 1 i m V i2, Max 1 i m V i2 ],, [Max 1 i m V in, Max 1 i m V in ]}(18) Step 7: Calculation grey possibility degree between compared suppliers alternatives set S =

5 {S 1, S 2,, S m } and ideal referential supplier alternative S Max P {S i S Max } = 1 n n =1 P { V i G Max } (19) Step 8: Ranking order of suppliers alternatives When P {S i S Max } is smaller, the ranking order of S i is better Otherwise, the ranking order is worse According to above procedures, we can determine the ranking order of all suppliers alternatives and select the best one form among a set of feasible suppliers 4 Application and Analysis There are six suppliers S i (i = 1, 2,, 6) are as selected alternatives against four attributes Q ( = 1, 2,, 4) The four attributes are product quality, service quality, delivery time and price respectively [13] Q 1, Q 2 and Q 3 are benefit attributes, the larger values are better Q 4 is cost attributes, the smaller values are better The selection structure is shown in Fig 2 The calculation procedures are shown as follows: Step 1: Making the attribute weights of attributes Q 1, Q 2, Q 3 and Q 4 A committee of four DMs, D 1, D 2, D 3 and D 4 has been formed to express their preferences and to select the most best suppliers According to Eq (11), the evaluation values of attribute weights from four MDs can be obtained and the results are shown in Table 3 Step 2: Making attribute rating values for six suppliers alternatives According to Eq (12), The results of attribute rating values are shown in Table 4 Step 3: Establishment of grey decision matrix According to Eq (13), we can obtain the grey decision matrix of suppliers Step 4: Table 4: Attribute rating values for suppliers Q S i D 1 D 2 D 3 D 4 G i Q 1 S 1 G MG G G [575, 825] S 2 MG G F MG [500, 650] S 3 F F MG G [475, 625] S 4 F MG MG F [450, 550] S 5 MG F F MG [450, 550] S 6 G MG MG MG [525, 675] Q 2 S 1 G G MG MG [550, 750] S 2 G MG MG G [550, 750] S 3 F F P F [325, 450] S 4 P MP MP P [200, 350] S 5 MP MP M MP [250, 375] S 6 MP P P MP [200, 350] Q 3 S 1 G MG MG G [550, 750] S 2 MG G G G [575, 825] S 3 G G F MG [525, 725] S 4 G MG MG G [550, 750] S 5 MG F F MG [450, 550] S 6 F F MG F [425, 525] Q 4 S 1 F G G G [550, 800] S 2 G G F MG [575, 825] S 3 VG VG G G [750, 950] S 4 G MG G G [575, 825] S 5 MG MG G MG [525, 675] S 6 G VG VG G [750, 950] Establishment of grey normalized decision table According to grey normalized decision matrix shown in Eq (14), the grey normalized decision table is shown in Table 5 Step 5: Establishment of grey weighted normalized decision table According to grey weighted normalized decision matrix shown in Eq (17), the grey weighted normalized decision table is shown in Table 6 Step 6: Making the ideal supplier S Max as referential alternative According to Eq (18), the the

6 Table 5: Grey normalized decision table S i Q 1 Q 2 Q 3 Q 4 S 1 [0697, 1000] [0733, 1000] [0667, 0909] [0656, 0955] S 2 [0606, 0788] [0733, 1000] [0697, 1000 [0724, 1000] S 3 [0576, 0758] [0433, 0600] [0636, 0879] [ ] S 4 [0545, 0667] [0267, 0467] [0667, 0909] [0636, 0913] S 5 [0545, 0667] [0333, 0500] [0545, 0545] [0778, 1000] S 6 [0636, 0818] [0267, 0467] [0515, 0636] [0553, 0700] Table 6: Grey weighted normalized decision table S i Q 1 Q 2 Q 3 Q 4 S 1 [0470, 0925] [0550, 0950] [0367, 0682] [0295, 0525] S 2 [0409, 0729] [0550, 0950] [0383, 0750] [0326, 0550] S 3 [0389, 0701] [0325, 0570] [0350, 0659] [0249, 0385] S 4 [0368, 0617] [0200, 0443] [0367, 0682] [0286, 0502] S 5 [0368, 0617] [0250, 0475] [0300, 0500] [0350, 0550] S 6 [0430, 0757] [0200, 0443] [0283, 0477] [0249, 0385] ideal supplier S Max is shown as follows: S Max = {[0470, 0925], [0550, 0950], [0383, 0750], [0350, 0550]} Step 7: Calculation grey possibility degree between compared alternatives of six suppliers S i (i = 1, 2,, 6) and ideal referential supplier alternative S Max According to Eq (19), the results of grey possibility degree are shown as follows: P (S 1 S Max )=0539, P (S 2 S Max )=0549, P (S 3 S Max )=0789, P (S 4 S Max )=0747, P (S 5 S Max )=0772, P (S 6 S Max )=0841 Step 8: Ranking order of six suppliers S i (i = 1, 2,, 6) According to Step 7, the result of ranking order is shown as follows: S 1 > S 2 > S 4 > S 5 > S 3 > S 6 We can say that the supplier S 1 is the best supplier in six suppliers S 1 should be as an important alternative for company The next important alternative is S 2 Because of the grey possibility degrees of S 1 and S 2 against ideal S Max are almost equal S 4, S 5 and S 3 are better suppliers and S 6 is the worse supplier The selection problem of suppliers is a MADM problem In conventional MADM methods, the ratings and the weights of the attribute must be known precisely [14]-[16] But, DMs udgment are often uncertain and cannot be estimated by the exact numerical value And thus the selection problem of suppliers has many uncertainties and becomes more difficult Thus, conventional MADM methods is limited We can change our attribute and look at the real world from a different point of view, system analysis can be treated from the viewpoint of the degree of information availability Grey theory is one of new mathematical fields that was born by the concept of grey set It is one of the methods that are used to study uncertainty of system In grey theory, according to the degree of information, if the system information is fully known, the system is called a

7 white system, while the system information is unknown, it is called a black system A system with partial information known and partial information unknown is grey system The uncertain information can be analyzed by grey set consist of grey number, thus, it become possible for the analysis of uncertain system Grey theory over fuzzy theory is that grey theory considers the condition of the fuzziness That is, grey theory can flexibly deal with the fuzziness situation Grey theory expand the range of membership function of fuzzy theory to analyze uncertain problem When the upper membership is same to lower membership, the grey set become fuzzy set It is obvious that fuzzy set is a special type of grey set The selection problem of suppliers has many certainties, we view it as a grey process and can be resolved by grey theory As the same time, we introduced grey possibility degree to compare the ranking of grey numbers Through a verify example, we obtained the effectiveness of proposed approach 5 Conclusions In this paper, we proposed a new grey-based approach to deal with selection problem of suppliers under uncertainty environment An example of selection problem of suppliers was used to illustrate the proposed approach The experimental result shows that proposed approach is reliable and reasonable This proposed approach can help in more accurate selection problem, such as management and economic fields etc References [1] WD Hong, B Lyes and XL Xie A Simulation Optimization Methodology for Supplier Selection Problem International Journal of Computer Integrated Manufacturing, 18(2-3): , 2005 [2] NO Ndubisi, M Jantan, LC King and MS Ayub Supplier Selection and Management Strategies and Manufacturing Flexibility Journal of Enterprise Information Management, 18(3): , 2005 [3] FH Franklin Liu and HL Hai The Voting Analytic Hierarchy Process Method for Selecting Supplier International Journal of Production Economics, 97(3): , 2005 [4] R Lasch; CG Janker Supplier Selection and Controlling Using Multivariate Analysis International Journal of Physical Distribution and Logistics Management, 35(6): , 2005 [5] JL Deng Control Problems of Grey System System and Control Letters, 5: , 1982 [6] M Nagai and D Yamaguchi Grey Theory and Engineering Application Method, Kyoritsu publisher, 2004 [7] YX Wang Application of Fuzzy Decision Optimum Model in Selecting Supplier The Journal of Science Technology and Engineering, 5(15): , Aug 2005 [8] LA Zadeh Fuzzy sets Information and Control, 8: , 1965 [9] RE Bellman and LA Zadeh Decision- Making in a fuzzy environment Management Science, 17(4): , 1970 [10] J Xia Grey System Theory to Hydrology Huazhong Univ of Science and Technology Press, 2000 [11] JR Shi, SY Liu and WT Xiong A New Solution for Interval Number Linear Programming Journal of Systems Engineering Theory and Practice, 2: , Feb 2005 [12] JP Xu and M Sasaki Technique of Order Preference by Similarity for Multiple Attribute Decision Making Based on Grey Members Transof IEEJ, 124(10): , 2004 [13] SJ Wang and HA Hu Application of Rough Set on Supplier s Determination The Third Annual Conference on Uncertainty, , Aug 2005 [14] JS Dyer, PC Fishburn, RE Steuer, J Wallenius and S Zionts Multiple Criteria Decision Making Multi-attribute Utility Theory Management Science, 38(5): , 1992 [15] CL Hwang and K Yoon Multiple Attributes Decision Making Methods and Applications, 1981 [16] J Teghem, C Delhaye and PL Kunsch An Interactive Decision Support System for Multicriteria Decision Aid Math Computer Modeling, 12: , 1989