Abstract
Fuzzy Inf. Eng. (2011) 4: 411-431 DOI 10.1007/s12543-011-0095-4 ORIGINAL ARTICLE Integrated Fuzzy AHP and Fuzzy VIKOR Model for Supplier Selection in an Agile and Modular Virtual Enterprise Peyman Mohammady · Amin Amid Received: 30 January 2011/ Revised: 25 September 2011/ Accepted: 20 November 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this ever-changing world, organizations need to outsource parts of their processes for having agile response to market’s needs and varying demands. Because of temporal nature of virtual enterprises (VE’s), the situation of outsourcing process in this kind of organizations is a vital situation. The main idea of this paper aims to present a decision-making framework for speciﬁc area that is appropriated for com- plex states. Its contribution is developing a fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method and combining it with fuzzy analytic hier- archy process (AHP). This extension suitable for decision-making situations which is faced with mixture appraisement that simultaneously regarded to both “group utility” or majority and “individual regret” of the opponent. The Integrated and developed model suits to inconsistent conditions that we face to collection of criteria and sub- criteria that should satisfy some of them collectively and simultaneously and in other attainment of some individual criteria is desirable. This framework then extended to a case study with varied criteria for outsourcing process. Keywords Series system · Fuzzy VIKOR · Supplier selection · Virtual enterprise 1. Introduction Among organizations, VE’s have higher complexity because of dynamic and tempo- ral nature. The different deﬁnitions of VE have been used in researches. Camarinha- Matos et al [1] introduce it as a temporal alliance that organizations is based on it, share their competencies and resources. Gazendam [2] deﬁne it as multi-actor sys- tems that composed of human resources and virtual factors. There are few differences Peyman Mohammady () Department of Management and Accounting, Shahid Beheshti University, Tehran, Iran email: Pa.mohammady@yahoo.com Amin Amid Department of Management and Accounting, Shahid Beheshti University, Tehran, Iran 412 Peyman Mohammady · Amin Amid (2011) between VE’s and other form of organizations. Putnik et al [3] summarize them to three fundamental features: dynamics of network reconﬁguration, virtuality and ex- ternal entities. VE’s have shorter life cycle than others. As illustrated in Fig. 1, the life cycle of VEs depends on business opportunities or need to adoption with chang- ing environment. In current hyper competition era, some philosophies and concepts are inevitable for VEs’ as temporal alliance through it. Agility, modularity, and interoperation are of such concepts. For agility concept, different deﬁnitions have been presented. Al- though common aspect of them is emphasized on ﬂexibility and speed as primary properties of agility [4] and [5]. An important characteristic that has been considered in dominant researches is proactive and speedy response to change [6-8]. Fig. 1 The life cycle of VE, Kim et al At changing environment, the modularity concept is a critical philosophy for mass- customization production. This concept is related to agile production as some re- searchers such Anderson showed with higher levels of modularity, higher levels of ﬂexibility and compatibility are attainable [10]. However this concept is less studied in the services sector. The vital role of modularity is related to postponement strategy and combination of both lean and agile philosophies. Note that modular products and services don’t depend on modular processes only but modular supply chain has important impact on it. Lau et al approved relation between supply chain integration processes with modular products [11]. This ﬁnding is consistent with ﬁne’s research [12]. Corvello and Migliarese point out if processes haven’t modularity characteris- tics, it is impossible to ﬁnd suppliers with competencies for doing production phases [13]. Baldwin and Clarck describe modularity concept as complex product and pro- cess composed of series of sub-systems that each of them design independently and act with together as whole [14]. This deﬁnition is consistent with towill’s deﬁnition of agility and referees to efforts aggregation for component’s seamless of supply chain due to perceive themselves as whole [15]. This means joint and linkage of agility, interoperability and modularity concepts in VE’s is establishment of seamless suppli- ers in supply chain of VE. Thus as cited in [16], selection’s process of supplier(s) has Fuzzy Inf. Eng. (2011) 4: 411-431 413 a vital situation in outsourcing process. This importance may be because of supply chain and production strategies’ affecting from this selection. In outsourcing process of agile and modular virtual enterprise, we face with two kinds of main criteria. First category includes sub-criteria from “agility” criteria and next is related to sub-criteria from “modularity” criteria for supplier’s selection. Ac- cording to Qumer and Henderson’s 4-DAT, we can extract and deﬁne sub-criteria of agility level. 4-DAT propose framework to evaluate agility of software’s development methods [17]. For virtual enterprise with IS/IT context, this framework is useful and can be used for construction of ﬁrst level. With attention to this, outsourcing process has two aspects; ﬁrst is extraction and determination of processes that should out- source and next is supplier’s evaluation and selection for each or collection of these processes; we need to select some of commensurable criteria inevitably for context of virtual enterprise supply chain. From other hand such criteria should be compatible with modularity’s preservation goal. With analysis of criteria based on cited aspects, the suitable sub-criteria are ﬂexibility, speed, leanness, learning and responsiveness. However modularity has multidisciplinary but schilling categorizes this concept and cites several properties of concept at different domain [18]. Among of these proper- ties, some characteristics are suitable for virtual enterprise with software development context. It includes re-combinability, expandability, decomposability and module as homologue. These properties absolutely are compatible with VE’s life cycle. We eliminate last property because its concept is goal of outsourcing process in this con- text. As discussed above, the main goal of evaluation is seamless preservation due to guarantee all of supply chain components and so Virtual enterprise remains agile and modular. 2. Multi-criteria Decision Making (MCDM) Multi-criteria optimization (MCO) is considered as the process of determining the best feasible solution according to established criteria representing different effects. However, these criteria usually conﬂict with each other and in practical problems are often characterized by several non-commensurable or competing criteria ever there is no solution satisfying all criteria simultaneously. Multi-criteria decision-making (MCDM) at new changing and turbulent environment should deal with decision- making situations with uncertainty and non-crisp conditions. Thus several techniques and methods have been introduced and discussed for deal- ing with imprecise, uncertain, and complex decision-making problems; researchers have proposed different MCDM approaches, such as the technique for order prefer- ence by similarity to ideal solution (TOPSIS). Further studies have extended MCDM in a fuzzy environment and proposed var- ied fuzzy multi-criteria decision-making (FMCDM) methods or other advanced tech- niques. These effective proposed techniques connected decision making with fuzzy set applications, to solve the problem for the optimal selection such as the VIKOR, fuzzy TOPSIS and fuzzy AHP. Each of these methods has its context and need to perceive along it. First we will describe fuzzy VIKOR and fuzzy AHP summarily below. Next we develop fuzzy VIKOR for speciﬁc and multi-purpose situations and combine it with 414 Peyman Mohammady · Amin Amid (2011) pairwise comparisons as a core of it. 2.1. Fuzzy VIKOR The VIKOR method was developed for multi-criteria optimization of complex sys- tems. This method was once developed by opricovic in 1998 to solve MCDM prob- lems with conﬂicting and non-commensurable criteria [19]. This method focuses on ranking and selecting from a set of alternatives in the pres- ence of conﬂicting criteria. It determines the compromise ranking-list, the compro- mise solution, and the weight stability intervals for preference stability of the compro- mise solution obtained with the initial (given) weights. It introduces an aggregating function representing the distance from the ideal solution. This ranking index is an aggregation of all criteria, the relative importance of the criteria, and a balance between total and individual satisfaction [20]. The main distinctive characteristic of fuzzy VIKOR is authority of decision-making based on group utility or individual regret of the opponent. According to VIKOR’s algorithm, we can select alternatives with considering two viewpoints. Upon on these states, it may requires us to select alternatives with higher scores based on all of criteria or ranked alternatives considering distinctive and higher distance from other alternatives in one or several criteria. It is suitable for problems and situations that we face with criteria including of several sub-criteria and we want to have appropriate decisions with ﬂexibility of group utility or individual regret of the opponent. In real conditions, it may be some criteria attainable with attaining of maximum of sub-criteria and some criteria have speciﬁc construction attainable by attaining each of sub-criteria independently and in absence of other sub-criteria. Main advantage of fuzzy VIKOR rather than other is such distinction. This method applied a numerical weight to percentage for shift between two cited philosophy which refereed to envi- ronment and context of organization and its appraisement. Note that one of two states can be attained and in some context we face with all of these states, fuzzy VIKOR method don’t appropriate for them. We developed this method for covering of such problems-inconsistent situations that some criteria and their sub-criteria should si- multaneously are attainable and some criteria can be attained based on attaining of one or several their sub-criteria. The main contribution of this paper is relevant to this domain and then combination of it to fuzzy AHP methods. We describe some of necessary deﬁnitions and details of developed fuzzy VIKOR brieﬂy below: 2.1.1. Primary Deﬁnitions We describe some of fuzzy sets and fuzzy mathematics’ deﬁnitions brieﬂy, then pro- pose mathematical steps of proposed methodology below: Deﬁnition 1 A fuzzy set A in a universe of discourse X is characterized by a mem- bership function f (x) which associates with each element x in X, a real number in the interval [0, 1]. The function value f (x) is termed as grade of membership of x in A. The current study employs triangular form of fuzzy numbers deﬁned by a triple Fuzzy Inf. Eng. (2011) 4: 411-431 415 (l, m, u). The membership function f (x) deﬁned as shown in Equation (1): 0, x < l, x− l , l ≤ x ≤ m, f (x) = (1) A ⎪ m− l ⎪ u− x ⎩ , m ≤ x ≤ u. u− m The diagram of such membership function illustrated in Fig. 2. Fig. 2 Membership function graph Assume A(l , m , u ) and B(l , m , u ) are two triangular fuzzy numbers. Opera- 1 1 1 2 2 2 tional laws of these two triangular fuzzy numbers shown as follow: A⊕ B = (l + l , m + m , u + u ), (2) 1 2 1 2 1 2 A B = (l − u , m − m , u − l ), (3) 1 2 1 2 1 2 A⊗ B = (l × l , m × m , u × u ), (4) 1 2 1 2 1 2 A B = (l /u , m /m , u /l ), (5) 1 2 1 2 1 2 for l , l , u , u , m , m > 0. 1 2 1 2 1 2 Deﬁnition 2 A linguistic variable is a variable values of which are linguistic terms. The application of such concept returns to complex and uncertain situations which are ill-deﬁned too. For example, the following fuzzy numbers deﬁned by Chen and Huang (1992). Very Low(VL) = (0.00 , 0.00 , 0.25); Low(L) = (0.00 , 0.25 , 0.50); Medium(M) = (0.25 , 0.50 , 0.75); High(H) = (0.50 , 0.75 , 1.00); Very High(VH) = (0.75 , 1.00 , 1.00). 2.1.2. Developed Fuzzy VIKOR Method Step 1: Determining of hierarchy’s dimensions Suppose following variables: m: number of alternatives; n: number of decision-makers(experts team); k : number of collective’s sub-criteria of j th criteria; l : number of individual’s sub-criteria of j th criteria. j 416 Peyman Mohammady · Amin Amid (2011) Due to simplify calculations and representation of equations and operations, we assume that models have one main-criteria of collective criteria and one main-criteria of individual criteria( j = j = 1). Extension of operations to each main-criteria is simple. So we repeated Equation (6) to Equation (7) for each main criteria. We establish CC and IC matrixes. First matrix should be established for collective main criteria and next for individual main criteria. Previously, the concepts of both matrixes have been described: SC SC ··· SC 1 2 j ⎡ ⎤ SC ⎢ w w ··· w ⎥ 1 ⎢ 11 12 1 j ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ SC ⎢ w w ··· w ⎥ 2 21 22 2 j CC = ⎢ ⎥ (6) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ . . . . ⎥ ⎢ ⎥ . ⎢ . . . . ⎥ ⎢ ⎥ . ⎢ . . . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ SC w w ··· w j l1 l2 j j SC SC ··· SC 1 2 j ⎡ ⎤ SC ⎢ w w ··· w ⎥ 1 11 12 1 j ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ SC ⎢ w w ··· w ⎥ IC = 2 ⎢ 21 22 2 j ⎥ (7) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ . . . . ⎥ ⎢ ⎥ . ⎢ . . . . ⎥ ⎢ ⎥ . . . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ SC w w ··· w j k1 k2 j j 1 1 1 −1 where w = (w ) = ( , , ). ij ij u m l ij ij ij Weight of pairwise comparisons of criteria are mean of weight which experts- decision-makers assigned them to each comparisons by linguistic variables and should be transformed to fuzzy triangular numbers, w = [ w ], (8) ij ijr r=1 where r = 1, 2,···, n. For collective sub-criteria; j = j = 1, 2,···, k, and for individual sub-criteria j = j = 1, 2,···, l. Here w is weight of sub criteria i against sub criteria j. First, these weights ij assigned by experts (decision-makers) to each comparisons using linguistic variables and then those should be transformed to fuzzy triangular numbers or other form of fuzzy numbers. In continuance, pairwise comparisons should be calculated separately for each sub-criteria based on fuzzy AHP. In this stage, opinions about comparisons need to be aggregated and averaged and then SC calculated for all of them (sub-criteria) in class of main criteria (see Equation (9)). The degree of possibility of SC (l , m , u ) ≥ SC (l , m , u ) deﬁned 1 1 1 1 2 2 2 2 as indicated in Equation (10). This equation depicts possibility’s degree of largeness of SC rather than SC . These calculation is applied for sub-criteria of individual- 1 2 main criteria-IC matrix(es), second section of previous appraisement, k k l l k k 1 1 1 SC (l , m , u ) = ( l , m , u )× ( , , ), (9) 1 1 1 1 1i 1i 1i u m u 1i 1i 1i i=1 i=1 i=1 i=1 i=1 i=1 Fuzzy Inf. Eng. (2011) 4: 411-431 417 0, l ≥ u , ⎪ 2 1 u − l 1 2 V(SC ≥ SC ) = , otherwise, (10) 1 2 (u − l )+ (m − m ) ⎪ 1 2 2 1 1, m ≥ m . 1 2 In order to compare degree of largeness of criteria or sub-criteria more than all of other criteria or sub-criteria for example for SC s calculation, we need to calculate both V(SC ≥ SC ) and V(SC ≥ SC ) synchronously as depicted in Equation (11). 1 j j 1 V(SC ≥ SC ) = V(SC ≥ SC ,···, SC ) = minV(SC ≥ SC ), (11) 1 j 1 2 j 1 2 where j 1 and for collective sub-criteria j = j = 1, 2,··· , k, and for individual sub-criteria j = j = 1, 2,··· , l. Next matrixes of sub-criteria’ weights as shown in Equation (12) and Equation (13) should be normalized. Note, normalization of refereed matrixes in this step are related to sub criteria of decision-making structure and should be calculated using fuzzy AHP, W = [V(SC ≥ SC ), V(SC ≥ SC ),···, V(SC ≥ SC )] , (12) 1 j 2 j k j where for collective sub-criteria j = j = 1, 2,··· , k, and for individual sub-criteria = 1, 2,··· , l j = j W = . (13) These ﬁnal weights are used in second construct of model- developed fuzzy VIKOR method as shown in Equation (17) and Equation (18). Step 2: Determining of alternatives’ comparisons 1) Separately on each sub criteria, we establish matrixes of alternatives. The men- tioned matrix should be established according to Equation (6) or Equation (7) with rows and columns which consisted of alternatives instead of sub criteria. Then, Equations (8)-(11) for alternatives’ comparisons calculated. We prepare matrixes for applying developed and extended fuzzy VIKOR method. This revised methods described brieﬂy below. Note normalization of mentioned matrixes should be per- formed by using fuzzy VIKOR. For describing of developed fuzzy VIKOR in this step, suppose X is ij A A ··· A 1 2 m ⎡ ⎤ A ⎢ (1, 1, 1) x ··· x ⎥ 1 12 1k ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A ⎢ x (1, 1, 1)··· x ⎥ 2 ⎢ 21 2k ⎥ D = (14) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . . . ⎢ . ⎥ . . . . . ⎢ ⎥ ⎢ . ⎥ . . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ A x x ··· (1, 1, 1) m m1 m2 where i = 1, 2,··· , m for collective sub-criteria j = j = 1, 2,··· , k ; for individual sub-criteria j = j = 1, 2,··· , l. Here, x is rating of alternative i against alternative ij j respect to each criteria. These variables are same to w in Equation (7), so we sum- ij marize some matrixes and calculations below. 418 Peyman Mohammady · Amin Amid (2011) 2) Determine the fuzzy best value (FBV) and fuzzy worst value (FWV) for collec- tive and individual main-criteria, separately: f = max x , (15) ij f = min x , (16) ij where i = 1, 2,··· , m for collective sub-criteria j = j = 1, 2,··· , k, for individual sub-criteria j = j = 1, 2,··· , l. The points of combination of fuzzy AHP and fuzzy VIKOR in proposed method occur at two aspects. First aspect is that, the weights of sub criteria as shown in Equa- tions (6)-(14) should be calculated using fuzzy AHP and next be used as weights of Equation (17) and Equation (18) in fuzzy VIKOR method. Second and most important aspect belonged to here Equation (15) and Equation (16); our criteria for recognition of minimum and maximum of x . For each of com- ij parisons in alternatives’ matrixes, we calculate Equations (9)-(11) separately on sub criteria. Then we determine minimum and maximum of x with considering of S and SC . ij i i On the other hand, the probability’s degree of largeness (PDL) of each of sub criteria against others is most important criteria for recognition of minimum and maximum of x . For example minimum of x respect to alternative which has smallest PDL ij ij and so on. 3) We determine normalized values of pervious matrixes according to Equation (17) and Equation (18) for each of alternatives. Note that Values of Equation (17) and Equation(18) should be calculated upon to collective and individual main-criteria separately. As mentioned, we extract weights of sub-criteria according to Equation (13), where S is A with respect to all criteria calculated by the sum of the distance for the FBV, i i and R is A with respect to the j th criteria calculated by the maximum distance of i i FBV. In Equation (19) to Equation (23), we separate equation dependent on j -collective criteria and j -Individual criteria, korl f − x ij S = w ( ), (17) i j ∗ − f − f j=1 j j f − x ij R = max[w ( )], (18) i j ∗ − f − f j j where i = 1, 2,··· , m for collective sub-criteria j = j = 1, 2,··· , k, and for individ- ual sub-criteria j = j = 1, 2,··· , l, S = min S , (19) S = max S , (20) i Fuzzy Inf. Eng. (2011) 4: 411-431 419 R = min R , (21) R = max R . (22) Here S is the minimum value of S , which is the maximum majority rule or max- imum group utility, and R is the minimum value of R , which is the minimum in- dividual regret of the opponent. Thus, the index Q is obtained and is based on the consideration of both the group utility and individual regret of the opponent. 4) De- termine values of Q , sort them in increasing order and rank them from smaller to larger, ∗ ∗ S − S R − R i i i i Q = v [ ]+ (1− v )[ ], (23) − ∗ − ∗ S − S R − R i i i i ∗ ∗ S − S R − R i i i i Q = v [ ]+ (1− v )[ ], (24) − ∗ − ∗ S − S R − R i i i i Q = ω× Q + (1−ω)× Q . (25) i i i Step 3: Analyzing of calculated results In the last step, depended on values of acceptable advantage and acceptable stabil- ity of decision (C and C ), we determine ﬁnal decision as an optimal solution for 1 2 evaluation. Assume A is the ﬁrst optimal solution and A is the second, 1 2 [C ]: Q(A )− Q(A ) ≥ DQ, (26) 1 2 1 DQ = (If m ≤ 4, then DQ = 0.25), (27) m− 1 Q(a ) = S (a ) or/and R(a ), (28) [C ]: Q(A )− Q(A ) < DQ. (29) 2 m 1 If C and C are both accepted, then solution is an optimal one, otherwise if [C ] is not 1 2 1 accepted and Equation (29) is not accepted, then A and A are the same compromise 1 m solution. However, A does not have a comparative advantage, so the compromise solutions are the same. If [C ] is not accepted, the stability in decision-making is deﬁcient, although A 2 1 has a comparative advantage. Hence, compromise solutions of A and A are the 1 m same. 2.2. Fuzzy AHP 420 Peyman Mohammady · Amin Amid (2011) The AHP introduced by Saaty [21], directs how to determine the priority of a set of alternatives and the relative importance of attributes in a multi-criteria decision- making problems. Through AHP, the importance of several attributes is obtained from a process of paired comparison, in which the relevance of the attributes’ class or drivers’ cate- gories of intangible assets are matched two-on-two in a hierarchic structure. However as cited by Yang and Chen’s research [22], the pure AHP model has some shortcomings. They pointed out that the AHP method is mainly used in nearly crisp-information decision applications; the AHP method creates and deals with a very unbalanced scale of judgment; the AHP method does not take into account the uncertainty associated with the mapping of human judgment to a number by natural language; the ranking of the AHP method is rather imprecise; and the subjective judgment by perception, evaluation, improvement and selection based on preference of decision-makers have great inﬂuence on the AHP results. To overcome these problems, several researchers integrate fuzzy theory with AHP to improve the uncertainty and some researchers integrated fuzzy AHP with other fuzzy methods such as fuzzy TOPSIS [23-25]. The main advantage of AHP and fuzzy AHP is hidden in this note that pairwise comparisons led to more convenient, realistic and logical appraisement of alternatives rather than other methods and techniques. This advantage of AHP and fuzzy AHP can led to more usability of them as core of model’s evaluation. In along of it other methods can be used as core of ranking operation, so fuzzy AHP plays a complementary role in model and is base of comparisons. 3. The Integrated Methodology Due to enrichment and improvement of pairwise comparisons in fuzzy AHP, we inte- grated the developed fuzzy VIKOR method after normalization in fuzzy AHP. Advan- tage of such integration is capability of dual-disciplinary of fuzzy VIKOR to fuzzy AHP. In proposed method, fuzzy VIKOR has been developed for situations that we face with both group utility and individual regret of the opponent simultaneously. At this development, fuzzy VIKOR appropriated at two calculations levels and one decision-making level. Cited method is able to apply to any MCDM with inconsistent judgment’s conditions. At ﬁrst level, we calculate scores for evaluation’s criteria that should be attainable all of them and then do calculations which related to sub-criteria that attainment of them individually can estimated main criteria. Fig. 3 illustrated a schematic step-by-step diagram of proposed methodology. 3.1. Identiﬁcation of Necessary Criteria and Structuring of Evaluation Hierar- chy In this step, with considering of business opportunities and determining of objective and strategies according to those opportunities, appropriated criteria and their sub- criteria have been selected and then decision hierarchy should be structured based on them. At this stage, it should be established two kind of mentioned criteria. First those can be attainable with attainment of respective collection of their sub-criteria and the other can be attainable with attainment of one or several sub-criteria. We referee to Fuzzy Inf. Eng. (2011) 4: 411-431 421 ﬁrst category as “collective criteria” and next “individual criteria”. Also according to this step, it’s inevitable, identiﬁcation of potential partners and determining of appro- priated and proposed alternatives. Tasks of this step should be adapted with different aspects of VE’s life cycle. Fig. 3 Schematic diagram of proposed methodology 3.2. Determining of Weight of Hierarchy’s Dimensions In this step, we need to deﬁne fuzzy number and determine linguistic variables for comparisons. A linguistic variable is a variable values which are linguistic terms [26]. Application of such concept return to complex and uncertain situations which are ill-deﬁned too. For example, Chen and Huang deﬁne the (0.00, 0.00, 0.25) as fuzzy numbers for very low (VL) linguistic variable and so (0.00, 0.25, 0.50) for low (L) linguistic variable and so on [27]. These terms used as important compar- isons of each criteria rather than others. Then for each main criteria, matrixes of sub-criteria for pairwise-comparisons separately have been established. First matrix of sub-criteria which has been established belonged to collective criteria and next to individual criteria. Such as it mentioned above, with performing or attaining of each of second categorized sub-criteria independently, respective main criteria can be at- tainable. Then we should complete pairwise-comparisons separately by respective sub-criteria using of fuzzy AHP technique. Also we should calculate weight of sub- criteria at pairwise approach. For all kind of main criteria, comparisons’ matrixes with looking to Equations (6)-(8) should be determined and normalized. 3.3. Determining of Comparisons between Alternatives In this step, linguistic variables for alternative evaluation should be determined. Then pairwise comparisons of alternatives should be established and calculated separately 422 Peyman Mohammady · Amin Amid (2011) on each sub-criteria. Due to considering different decision-makers opinions, aggre- gation of votes needs to be run. Next according to section of developed fuzzy VIKOR from proposed methodology, we determine FBV and FWV separately on collective and individual main criteria. Against previous calculations that belonged to weight of decision-making structure, the normalization of matrixes in this step performed by using fuzzy VIKOR methods. Then matrixes should be normalized and ranked-list extracted. Note that, v in Equation (23) means the weight of the strategy of the maximum group utility. At v > 0.5, the decision tends toward the maximum majority rule; and if v < 0.5, the decision tends toward the individual regret of the opponent. We use v > 0.8 because of the fact that we face with sub-criteria of collective main-criteria and in calculation of v , we assign v < 0.3 to nature of its main-criteria. We referee to authority of strategic plan and objective where organization want to emphasize on individual main-criteria or collective main-criteria in its construct of appraisement. w < 0.5 apply ﬁrst strategy and w > 0.5 emphasize on second strategy. 3.4. Analyzing of Results and Final Decision-making Finally, considering decision-making status and typical evaluation of alternatives, we calculate ﬁnal results, examine it according to Equations (26)-(29) and recognize an optimal solution or solutions. Dependent on acceptable advantage and acceptable stability of decision C and C ; it may has unique optimal solution or not. The ﬁnal 1 2 result depends on experts’ appraisements, typical main criteria, sub-criteria and or- ganization attention to evaluation process as collective process or individual process. At the end we report status of results and ﬁnal decision. 4. Case Study Alpha system is a virtual enterprise with information, virtual-training and automation services. Some of its services include: - Consultation about system analysis and design; - Web-based training program; - Consultation about business process reengineering on web-based services; - Web-based organizational project handling and management; - Appraisement of web-based services’ quality; - Consultation about ERP selection and implementation; - BPMS designing and planning in sector of public services. According to revised strategies, objective and goals by adopting it to new opportu- nities, enterprise decides to outsource section of web-based training in supporting of respective consultation. Top manager and change management believe outsourcing cited process led to cost optimization and focusing on main tasks and critical opera- tions. Board of directors speciﬁes ﬁve members of organizational experts as team of decision-makers for evaluation of alternatives. This process including several of sub-process and main aim of enterprise is per- forming of process as agile and speedy changeable services. From managers perspec- tive agility factor and modularity have vital position in evaluation of alternatives. Fuzzy Inf. Eng. (2011) 4: 411-431 423 Second factor explained because of enterprise need to be compatible with environ- mental opportunities and have agile dissolution and reconﬁguration. Fig. 4 Depicts criteria and sub-criteria that are vital for alpha system’s objectives and strategies. Ac- cording to it, we establish hierarchy dimensions and determine weight of criteria. Experts vote to equality of dual main criteria thus there is no need to matrix’s establishment for pairwise comparisons of both mentioned main criteria-agility and modularity. First considering experts’ knowledge, we deﬁne linguistic variables for evaluation process. Then pairwise comparisons of sub-criteria according to Equation (6) and and W for sub-criteria Equation (8) performed. According to them we calculate W in ﬁrst and second main criteria and determine normalization vector. The ﬁnal results which showed in Table 5 are aligned with second step of proposed methodology. Then we acquire experts’ votes about comparisons of alternatives base on sub-criteria that grouped in each main criteria then aggregate and calculate average of them. Fig. 4 The decision hierarchy of partner’s selection of VE for alpha-system In the next step, we determine matrix of pairwise comparisons of alternatives for each sub-criteria and perform related calculations (Table 6-16). According to Table 15, in ﬂexibility sub-criteria, S and S are larger than other. With referee to Table 1 4 14, it has been seem that selection of S is a better decision. These results should be for each sub-criteria. Because of nature of alpha-system enterprise, we need to have one matrix for agility and one matrix for modularity. In the continuance of calculation we complete some remained cells of matrixes CC and IC, determine the FBV and FWV. Then according to Equation (17) and Equation (18), we normalize mentioned ma- trixes based on typical main criteria-individual and collective criteria, separately. Note, the weights of sub-criteria calculated in previous pairwise comparisons using fuzzy AHP for main criteria and sub criteria, however expert cited same importance for both main criteria. The detailed calculations have been described below: 424 Peyman Mohammady · Amin Amid (2011) Table1: Linguistic variables for comparisons. Table 2: The construct of decision model. Table 3: Average of aggregated fuzzy numbers extracted from expert’s opinions about collective sub-criteria. S = (0.24, 0.36, 0.54); S = (0.17, 0.26, 0.39); S = (0.07, 0.13, 0.21); 1 2 3 S = (0.07, 0.12, 0.20); S = (0.08, 0.13, 0.24). 4 5 After determining of S C , we need to calculate PDL of each sub criteria against S i each other’s as shown below: V(SC >= SC ) = 1, i = 2, 3, 4, 5; 1 i V(SC >= SC ) = 0.5999, V(SC >= SC ) = 1, V(SC >= SC ) = 1, 2 1 2 3 2 4 V(SC >= SC ) = 1; 2 5 V(SC >= SC ) = 0, V(SC >= SC ) = 0.2, V(SC >= SC ) = 1, 3 1 3 2 3 4 V(SC >= SC ) = 0.99; 3 5 V(SC >= SC ) = 0, V(SC >= SC ) = 0.18, V(SC >= SC ) = 0.94, 4 1 4 2 4 3 V(SC >= SC ) = 0.94; 4 5 V(SC >= SC ) = 0.009, V(SC >= SC ) = 0.34, V(SC >= SC ) = 1, 5 1 5 2 5 3 V(SC >= SC ) = 1. 5 4 The weigh vector from refereed calculation will be W = (1, 0.59, 0, 0, 0.009). Therefore the normalized weight is W = (0.62, 0.37, 0, 0, 0.01). These calculations also should be performed for modularity’s criteria. This criteria considering of ex- perts opinions has three sub-criteria. Fuzzy Inf. Eng. (2011) 4: 411-431 425 Table 4: The Aggregated fuzzy numbers extracted from expert’s opinions about individual sub-criteria. S = (0.18, 0.32, 0.53); S = (0.30, 0.45, 0.71); S = (0.15, 0.22, 0.38) 1 2 3 After determining of S C , these result can be calculated: S i V(SC >= SC ) = 0.64, V(SC >= SC ) = 1; 1 2 1 3 V(SC >= SC ) = 1, i = 1, 3; 2 i V(SC >= SC ) = 0.67, V(SC >= SC ) = 0.27. 3 1 3 2 The weigh vector from previous calculation will be W = (0.64, 1, 0.27). There- fore the normalized weight is W = (0.34, 0.52, 0.14). The ﬁnal weight according to Equation (13) Table 5: The normalized weights of main-criteria and sub-criteria according to fuzzy AHP techniques. Table 6: Average of aggregated fuzzy numbers of pairwise comparisons of alternatives in ﬂexibility sub-criteria. Table 7: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in speed sub-criteria. 426 Peyman Mohammady · Amin Amid (2011) Table 8: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in Leanness sub-criteria. Table 9: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in learning sub-criteria. Table 10: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in responsiveness sub-criteria. Table 11: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in expandability sub-criteria. Table 12: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in decomposability sub-criteria. Fuzzy Inf. Eng. (2011) 4: 411-431 427 Table 13: Average of aggregated fuzzy numbers about pairwise comparisons of alternatives in recombinability sub-criteria. For Table 6, results will be: −1 S = (8.7, 10.7, 12.7)∗ (27.72, 33.50, 39.77) = (0.22, 0.32, 0.46), −1 S = (7.92, 9.48, 11.08)∗ (27.72, 33.50, 39.77) = (0.2, 0.28, 0.4), −1 S = (4.07, 5.20, 6.40)∗ (27.72, 33.50, 39.77) = (0.10, 0.16, 0.23), −1 S = (4.58, 5.34, 6.28)∗ (27.72, 33.50, 39.77) = (0.12, 0.16, 0.23), −1 = (2.44, 2.77, 3.31)∗ (27.72, 33.50, 39.77) = (0.06, 0.08, 0.12). Table 14: Calculation of possibility’s degree of relative largeness of S for Table 6. Table 15: Calculation of possibility’s degree of largeness of S mainly, for Table 6. Table 16: Average of aggregated opinions extract from S based on Equation (11). i 428 Peyman Mohammady · Amin Amid (2011) Table 17: FBV and FWV per each criteria. Table 18: Calculation of S and R for whole of sub-criteria. j j According to Equations (19)-(25) and considering of enterprise’s viewpoint, we assume v = 0.95 and v = 0.1 to have large appropriated effects on each stage in- cluding calculation of both collective and individual sub-criteria. w = 0.4, this means organization decides to emphasize on individual sub-criteria in its appraisement’s structure. According to Table 20, A and A are best solutions. With attention of C and C , 2 1 1 2 C is acceptable but C or acceptable stability of decision isn’t acceptable. However 1 2 acceptable advantage of selection of A against A In result A and A are the same. 2 1 1 2 Fuzzy Inf. Eng. (2011) 4: 411-431 429 ∗ − ∗ Table 19: Calculation of S , S , R and R for each of main-criteria. j j j Table 20: Final calculation and ranked list of alternatives. 5. Conclusion In this paper, we combine Fuzzy AHP method with developed Fuzzy VIKOR. The comparative advantage of proposed model from one viewpoint related to simplify- ing comparisons of alternatives with pairwise comparisons, and from other viewpoint related to authority of model to satisfy group utility and individual regret of the oppo- nent simultaneously in context of VEs. Then the model is applied to one VE which is willing to outsource their process for guarantee of agility and synergic responsive- ness to changing demand at whole of enterprise’s supply chain. Proposed model can be applied to the extended domain of appraisement and judgments’ activities. The main goal, which is satisﬁed by this model, is capability of multi-criteria decision- making in complex and heterogeneous context; situations which have incongruent nature affecting appraisement structure and decision-making process. 430 Peyman Mohammady · Amin Amid (2011) Acknowledgments The authors would like to thank to the supply chain manager and the other partici- pants of Alpha System for their collaborative participation which have led to an im- provement in both the quality and clarity of the paper. Peyman Mohammady wants to acknowledge the Dr. Masoud Alimohammady for his efforts on coordinating of assessment process. References 1. Camarinha-Matos L M, Afsarmanesh H, Garita C, Lima C (1998) Towards an architecture for virtual enterprises. Journal of Intelligent Manufacturing 9(2): 189-199 2. Gazendam H W M (2001) Semiotics, virtual organizations, and Information systems. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Dec 1, 2011
Keywords: Series system; Fuzzy VIKOR; Supplier selection; Virtual enterprise