Fuzzy Logic Based Vendor Selection for the Public Procurement Sector: a Case Study
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1 Fuzzy Logic Based Vendor Selection for the Public Procurement Sector: a Case Study Nicola Costantino 1, Mariagrazia Dotoli, Marco Falagario 3, Maria Pia Fanti 4 1 Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, costantino@poliba.it Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, dotoli@de .poliba.it 3 Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, Via Japigia 18, 7016 Bari, Italy, m.falagario@poliba.it, PH FAX (corresponding author) 4 Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, fanti@de .poliba.it Keywords: Purchasing, Public Procurement, Supplier Evaluation, Vendor Rating, Fuzzy Logic, Linear Weighting Method, Fuzzy Technique for Order Preference by Similarity to Ideal Solution, Fuzzy Analytic Hierarchy Process. Abstract The paper addresses the strategic and relatively recent problem of supplier selection in the public procurement sector. After discussing the similarities and peculiarities in comparison with the procurement of services and goods in the private sector, we formalize the decision problem and apply and compare the well known linear weighting method and two prominent multiple criteria decision making approaches based on fuzzy logic, i.e. the fuzzy technique for order preference by similarity to ideal solution and the fuzzy analytic hierarchy process. We enlighten the three approaches by way of a real case study, whose results show the strengths and limitations of the proposed methods in the considered purchasing problem.
2 1. Introduction Nowadays the supplier or vendor selection process receives considerable attention in the business management literature, especially with reference to the private sector [7]. Indeed, incorrect decisions about supplier selections may lead to serious profit losses [15]. As a result, in the last decade many organizations have changed their focus from the classical purchasing concept to an effective tactical management of the procurement task, including identification of supplier selection criteria, supplier selection decisions, and monitoring of supplier performance [13]. However, the supplier selection process may turn out to be deeply complex, since it incorporates a great variety of incontrollable and unpredictable factors affecting decisions [5]. In particular, often the vendor choice is based on quantitative and qualitative judgments, e.g. the trust component affecting the buyer/supplier relationship or the importance of a certain component along the considered organization. Because of the inherent complexity of decision making in the procurement sector, the help of decision making experts is frequently needed for effective purchasing management [6]. Accordingly, with the aim of achieving benefits already realized by private companies, in most countries also the public sector has recently started using new methodologies for supplier selection and, more generally, for the procurement of services and goods [10, 14]. Despite the similarities between private and public (or governmental) purchasing, the latter type of procurement exhibits some peculiarities [14]. In particular, in most countries the public sector is covered by a number of public procurement regulations (e.g. in the EU a number of public procurement directives are effective), bringing legislative requirements into force. As a consequence, although governmental and private procurement share the same essential purpose of finding supply sources at the cheapest price and at acceptable quality, several dissimilarities arise between these two procurement systems. In particular, public procurement differs from the private one in the fact that prescribed procedures are to be followed and transparency is imperative [14]. In other words, it is crucial that public procurement follows strict and clear business models that optimize the specific service objectives and considers the impact on processes across the considered governmental organization, avoiding (or at least reducing) excessively subjective evaluations by the buyers. However, in most governmental areas selecting one of the numerous alternative vendors offering a bid for a transaction may be a very complex task, since usually the dimension of the vendor set is excessively large. Moreover, although such 1
3 a decision problem is often characterized by conflicting objectives and imprecise and qualitative information, as previously remarked decisions have to be based on transparency. For these reasons, it may be advisable to employ fuzzy multi-criteria optimization to enhance the classical vendor rating technique and rank bids related to a public procurement transaction using fuzzy logic. Indeed, fuzzy logic provides a natural framework to incorporate qualitative knowledge with quantitative information such as real data. Therefore, fuzzy multi-objective optimization [4] is particularly suitable for evaluating and ranking, on the basis of qualitative knowledge but a quantitative and unambiguous decision process, the available bids [1,, 5, 8]. With a view to contribute to the relatively new field of supplier selection for governmental purchasing [10, 17], the present paper applies to the vendor rating problem in the public procurement sector and compares the well known linear weighting method and two well known multiple criteria decision making approaches based on fuzzy logic, i.e. FTOPSIS and FAHP. More precisely, the former and simpler method (nowadays often used by European public agencies) ranks the suppliers assigning them an overall performance index that is a weighted sum of the performance indices characterizing the considered vendor. On the other hand, the first fuzzy multi-objective technique is based on the idea that the best alternative among a set of solutions of a given problem exhibits the shortest distance from the ideal solution and the farthest distance from the negative ideal solution in a Euclidean sense [0]. Moreover, the second fuzzy multi-objective decision technique considered in this work arranges all the elements involved in the decision problem (overall goal, criteria, alternatives) in a hierarchical structure and objectives are of varying degrees of importance: the ranking is achieved by assigning to each of the available supplies a power indicative of its importance and then raising each fuzzy value to the appropriate power [16, 0]. Naturally, all the considered methods are compatible with the current EU directives. We enlighten and compare the three approaches by way of a real case study, namely the contract regarding the renovation of a building facility of Politecnico of Bari, Italy, which is adapted to the simulation investigation that we carry on. For the sake of simplicity and with the perspective of extending the proposed approach, here we choose to address a multiple source single-item vendor rating problem considering only three dimensions: price (i.e. the offered bid), reduction in the execution time and free maintenance time. The case study results show the strengths and limitations of the proposed methods.
4 . Fuzzy Logic Based Vendor Selection for the Public Procurement Sector For any organization, the choice of the supplier is one of the most important responsibilities of the purchasing function, and the vendor rating process is perhaps the most critical step in such a selection problem. In the private sector, vendor rating systems identify top suppliers, i.e. the candidate partners that are best equipped to meet the customer s expected level of performance, and check them periodically [3]. Hence, vendor selection is a multi-objective decision problem, including many conflicting objectives such as, besides the obvious goal of (low) price, quality, quantity, delivery, performance, capacity, communication, service, geographical location etc. [11]. In the public procurement sector, usual regulations impose that all the potential suppliers satisfying the required characteristics may bid; public agencies have therefore to evaluate all the different bids received according to the offered prices or to a prefixed set of parameters (price, delivery time, quality, etc.). In order to define the vendor selection problem for the procurement of a new product or service, a set of bidding suppliers S=s 1,s,,s m and a set of conflicting criteria C=c 1,c,,c n is introduced. Moreover, the different vendors and their bids have to be ranked, taking into account several parameters connected to the supplier characteristics and the product/service features. Accordingly, we associate to each supplier s i S the following n-tuple: (d i1, d i,,d in ), where d ij represents the value of the performance index characterizing the i-th supplier with respect to the j-th criterion with j=1,,n. In addition, the input data are collected in a mxn decision matrix D M, where m is the number of available vendors and n is the number of criteria on the grounds of which the suppliers ranking is performed. Hence, the generic element d m ij of D M, with i=1,,m and j=1,,n, represents the j-th performance value of the i-th alternative supplier s i. The input data are completed by the criteria importance, i.e., each criterion c j with j=1,,n is associated to a weight w j, with n w j 1 j1. Typically, in a vendor selection problem of the public procurement sector the n considered criteria include price and other indices such as reduction in the planned work execution time, free maintenance period post delivery, certified supplier quality, etc. Accordingly, since normally the most important factor in a vendor selection problem is price, particularly in the case of public procurement, usually its corresponding weight is the highest one while the others are 3
5 lower and correspond to criteria that are less crucial..1. The Linear Weighting Method The first method considered in this paper to rank the pool of supplier candidates is the Linear Weighting (LW) technique, one of the best-known and most widely used decision making techniques both in private and in public procurement [1, 17]. The LW technique may be formalized by the following steps. Step 1. Constructing the normalized decision matrix. Determine each element n ij of the mxn normalized decision matrix N as follows: d j d max ij d j d max jmin nij dij d jmin d j d max jmin if the j-th criterion values have to be minimized if the j-th criterion values have to be maximized for i=1,,m and j=1,,n, (1) with d j min dij and d j max dij min i1,..., m max i1,..., m, respectively representing the minimum and maximum values of the performance indices for the m suppliers and for the j-th criterion with j=1,,n. Note that the first (second) expression in (1) associates a normalized performance index that rewards low (high) values, associating a performance index of 1 (0) to suppliers that exhibit a performance value equal to that exhibit a performance value equal to d j min and, on the contrary, a performance index of 0 (1) to suppliers d j max. Step. Constructing the weighted normalized decision matrix. Determine the mxn weighted normalized decision matrix W, where w ij =n ij w j, for i=1,,m and j=1,,n. Step 3. Calculating the overall performance index of the alternatives. Determine the overall performance index PI i_lw of each alternative supplier s i S as follows: 4
6 n PIi _ LW wij j1, for i=1, m. () Step 4. Ranking the alternatives. Obviously, the best supplier is the one showing the highest index PI i_lw obtained by (). Hence, the ranked vector of alternatives is =[ 1 m ] T, where i for i=1,,m is the generic i-th supplier and exhibits a performance index PI i_lw PI i+1_lw : hence, 1 is the best supplier and m is the worst vendor... The Fuzzy Multi-Objective Techniques The fuzzy multi-objective techniques considered in this work as an alternative to the widespread LW method are the following well-known techniques [0]: 1) the Fuzzy Technique for Order Preference by Similarity to Ideal Solution (FTOPSIS) and ) the Fuzzy Analytic Hierarchy Process (FAHP). Just as the LW method, both the considered fuzzy multi-objective techniques require as input data the mxn decision matrix D M and the weights w j with j=1,,n and n w j 1 j1 measuring the criteria importance. Moreover, a fuzzification process associates to each value d m ij of D M a fuzzy value d ij, with 0 d ij 1, defining the mxn fuzzified decision matrix D, that depends on the vendors satisfaction degree with respect to the criteria. More precisely, if the j-th performance index has to be minimized (maximized), for each d ij with i=1 m, representing the fuzzified index for the i-th supplier, the membership function used to fuzzify this performance index is chosen sigmoidal and decreasing (increasing) [18]. Accordingly, the generic element of the fuzzified decision matrix is obtained as follows: 5
7 1 if dij d jmin dij d j d min j d min jmax 1 if d j d min ij d j d max j min d ' ij d j d max ij d j d min j max if dij d j d max j d max j min 0 dij d jmax if the j-th criterion values have to be minimized and 0 if dij d jmin dij d j d min j d min jmax if d j d min ij d j d max j min d ' ij d j d max ij d j d min j max 1 if dij d j d max j d max j min 1 dij d jmax if the j-th criterion values have to be maximized., for i=1, m, i=1,,n. (3a), for i=1, m, i=1,,n. (3b)..1. The FTOPSIS Approach The Fuzzy Technique for Order Preference by Similarity to Ideal Solution (FTOPSIS) [0] is a fuzzy multi-objective decision technique based on simple geometric concepts: the best alternative exhibits the shortest distance from the Best Ideal Solution (BIS) and the farthest distance from the Worst Ideal Solution (WIS) in a geometrical (i.e., Euclidean) sense. The FTOPSIS technique consists of the following steps. Step 1. Determining the fuzzified decision matrix. Determine the mxn fuzzified decision matrix D by (3a)-(3b). Step. Constructing the normalized fuzzified decision matrix. Determine each element n ij of the mxn normalized fuzzified decision matrix N as follows: 6
8 n' ij d ' ij m ' d ij i1, i=1,,m, j=1,,n. (4) Step 3. Constructing the weighted normalized decision matrix. Determine the mxn weighted normalized decision matrix W, where w ij =n ij w j, for i=1,,m and j=1,,n. Step 4. Determining the best and worst ideal solutions. Determine the BIS as the ideal solution with performance indices given by the row vector G [ G1... G n ], where G j =max(n 1j,,n mj ) with j=1,,n. Moreover, determine the WIS as the ideal solution associated to performance indices of the row vector H [ H1... H n ], where H j =min(n 1j,,n mj ) with j=1,,n. Step 5. Calculating the separation measure. Calculate the separation distance S G,i from the BIS of each alternative supplier s i with i=1,,m as follows: n S G, i ( n' ij G j ) j1. (5) Moreover, determine the separation distance S H,i of s i with i=1,,m from the WIS as follows: n S H, i ( n' ij H j ) j1. (6) Step 6. Calculating the relative closeness of alternatives to the ideal solution. Determine the overall performance index PI i_ftopsis measuring the closeness to the WIS of each alternative s i with i=1,,m as follows: SH, i PIi _ FTOPSIS SG, i SH, i. (7) Step 7. Ranking the alternatives. In the same way as for the LW method, the suppliers are ranked according to the index PI i_ftopsis. Obviously, the best supplier is the one showing the highest index PI i_ftopsis obtained by (7). Hence, the ranked vector of alternatives is =[ 1 m ] T, 7
9 where i for i=1,,m is the generic i-th supplier and exhibits a performance index PI i_ftopsis PI i+1_ FTOPSIS : hence, 1 is the best supplier and m is the worst vendor.... The FAHP Approach The Fuzzy Analytic Hierarchy Process (FAHP) [16, 0] is the second fuzzy multi-objective decision technique considered in this work. All the elements involved in the decision problem (overall goal, criteria, alternatives) are arranged in a hierarchical structure and objectives are of varying degrees of importance. The ranking is achieved by assigning to each of the available suppliers a power indicative of its importance and then raising each fuzzy value to the appropriate power. Such powers are obtained by determining the eigenvector of the maximum eigenvalue of the so-called comparison matrix. The technique consists of the following steps. Step 1. Structuring the decision problem as a hierarchy. Select the first level of the hierarchical structure as the overall goal Supplier Efficiency. Define the second level, that is composed by the n considered criteria contributing to the goal. Moreover, determine the third level as the m alternative supplier configurations to be ranked in terms of the criteria defined in the second level. Step. Determining the fuzzified decision matrix and. Determine the mxn fuzzified decision matrix D by (3a)-(3b). Step 3. Constructing the pairwise comparison matrix C M. Compare the n criteria that define the second level of the hierarchical structure with each other and construct the nxn pairwise comparison matrix C M by Saaty s original AHP scale in Table 1. More precisely, determine each element c m ij of C M with i,j=1,,n, representing the relative importance of the i-th criterion compared to the j-th one, by evaluating the difference 100 w i -w j of the respective performance indices weights and associating it an integer value from 1 to 9 according to Table 1. Step 4. Determining the eigenvector associated to the maximum eigenvalue of the comparison matrix. Calculate the eigenvalues set { 1,,, R } of C M, where R is the matrix rank. Let max 8
10 be the maximum eigenvalue of C M, then determine its eigenvector v max. Compute the priority vector: P vmax n [ p1... p ] T n, (8) where each element p j with j=1,,n of P represents the importance degree of the j-th performance index associated to the j-th column of D : the greater p j, the more important the j-th performance index. Step 5. Raising alternatives to the criteria power. Determine the alternative values associated to each j-th performance index as follows: CRITj [ d ' 1 j... d ' mj ] (9) for each j=1,,n. Determine the following vectors: p j p j p j C j =[c 1j c mj ]= CRITj d 1' j... d ' mj for each j=1,,n. (10) Step 6. Determining the decision model. For each alternative s i with i=1,,m, determine: PIi _ FAHP min ci1,..., cin so that PI i_fahp provides information about the satisfaction of alternative s i with respect to the performance indices and their importance degree. (11) Step 7. Ranking the alternatives. In the same way as for the other techniques, the suppliers are ranked according to the index PI i_fahp. Obviously, the best supplier is the one showing the highest index PI i_fahp obtained by (11). Hence, the ranked vector of alternatives is =[ 1 m ] T, where i for i=1,,m is the generic i-th supplier and exhibits a performance index PI i_fahp PI i+1_ FAHP : hence, 1 is the best supplier and m is the worst vendor. Pairwise differences AHP scale Table 1: Saaty s original AHP scale. 9
11 3. The Case Study The case study, useful for enlightening and comparing the presented approaches, consists in the contract regarding the renovation of a building facility of Politecnico of Bari, Italy, opportunely modified and integrated. In particular, the number of bidding suppliers is m=45, each offering a bid characterized by n=3 parameters. More precisely, according to one of the options offered by the European legislation regulating public procurement, the following n=3 criteria are selected: 1) price (c 1 =p, with the corresponding performance value d i1 =p i expressed in for the i-the supplier and i=1,,m); ) reduction in the planned work execution time (c =r, with d i =r i in days and i=1,,m); 3) free maintenance period post delivery (c 3 =m, with d i3 =m i in days and i=1,,m). Accordingly, to each supplier s i S we associate the following triplet collecting the elements of the i-th row of the decision matrix: (d i1,d i,d i3 )=(p i,r i,m i ). Note that these data are obtained integrating the economic data, i.e. the actually offered prices, with simulated data about reductions in the planned work time and free maintenance period. The latter data are obtained by a probabilistic simulation, with a probability distribution designed in accordance with the evaluation of the competent buyer. Moreover, we remark that the first criterion values (prices) have to be minimized, while the other two conflicting objectives correspond to two criteria whose values (reduction in delivery time and free maintenance period) have to be maximized. In addition, we select the weights w j for j=1,,3 for price, reduction in execution time and free maintenance time, respectively. More precisely, to show the effectiveness of the techniques we evaluate two alternative rankings as follows: case a) w 1 =0.70, w =0.10, w 3 =0.; case b) w 1 =w =w 3 = In other words, the first investigation corresponds to the usual public procurement situation in which the most important factor of the vendor selection problem is price, while the other criteria are less crucial. The second case corresponds to the less frequent situation in which all criteria have the same importance. The three methods for vendor rating presented in section are alternatively applied to the case study, and a comparison between their results is proposed, in order to focus on the differences and the similarities, as well as the advantages and limitations of the techniques. In particular, the different vendors and their bids are evaluated and ranked implementing the described techniques in the Matlab framework [19, 1] The vendor labels s i with i=1,..,m, their bids p i and their normalized values P i =n i1 obtained by () are collected in the first three columns of Table. Note that prices range from bids of about
12 k to about 13 k. In particular, according to price only, supplier s 4, highlighted in bold in the table, is the best supplier, while the worst one is s 9, also highlighted in bold. Moreover, the suppliers offered reduction in the planned work execution time r i for i=1,,m and their normalized values employed R i =n i obtained by () are collected in the subsequent two columns of Table. We remark that reductions in execution time range from bids of 10 days for the best suppliers according to this criterion (s 9, s 13, s 17, s 3, s 8, s 34, s 43 ), highlighted in bold in the table, to 0 days for the worst suppliers (s, s 5, s 15, s 0, s 7 ), also highlighted in bold. In addition, the vendors offered maintenance time m i for i=1,,m and their normalized values M i =n i3 obtained by () are collected in the subsequent two columns of Table. We observe that maintenance times range from bids of 360 days for the best suppliers according to this measure (s 7, s 14, s 18, s 4, s 7, s 30, s 35, s 40, s 45 ), highlighted in bold in the table, to 0 days for the worst suppliers (s 4, s 6, s 8, s 19, s 3, s 37, s 4 ), also highlighted in bold. The last six columns of Table show the results obtained applying the three decision making techniques to the case study in case a) (columns eight to ten) and in case b) (columns eleven to thirteen), reporting the overall performance index of each supplier PI i_lw, PI i_ftopsis and PI i_fahp for i=1,,m obtained under the considered methodologies. The overall best and worst suppliers (according to every decision technique) are highlighted in bold. Moreover, for the sake of simplicity Table 3 collects the vendors ranked as top five by the three selected techniques as well as the suppliers ranked as worst five. In particular, in case a) the LW, FTOPSIS and FAHP methods provide the same results and choose vendor s 4 as the best solution. Indeed, this supplier is characterized by one of the lowest values of price; moreover, among the low price solutions, it exhibits acceptable values of the less important criteria reduction in execution time and free maintenance time. 11
13 r i [days] m i [days] Supplier Case a) Case b) p s i [ ] P i R i M i i PI i_lw PI i_ftopsis PI i_fahp PI i_lw PI i_ftopsis PI i_fahp s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s Table : The vendor rating results for the case study for cases a) and b). 1
14 Position Case a) Case b) LW FTOPSIS FAHP LW FTOPSIS FAHP 1 s 4 s 4 s 4 s 34 s 3 s 3 s 34 s 35 s 35 s 3 s 34 s 34 3 s 35 s 3 s 1 s 4 s 4 s 1 4 s 6 s 34 s 3 s 35 s 35 s 4 5 s 3 s 1 s 34 s 1 s 1 s s 45 s 9 s 43 s 3 s 3 s 0 4 s 8 s 45 s 45 s 5 s 10 s 9 43 s 3 s 9 s 3 s 15 s 36 s s 9 s 3 s 9 s 0 s 9 s s 37 s 37 s 37 s 37 s 37 s 43 Table 3: Best and worst alternatives for the case study for cases a) and b). As regards case b), the FTOPSIS and FAHP methods both select supplier s 3 as the best solution, while the LW technique ranks this alternative at the second position. Indeed, such a vendor exhibits the highest value of reduction in execution time; moreover, among the high reduction in execution time solutions, it features a very low price and a very high value of free maintenance time. Similar remarks may be made for supplier s 34, which is selected as the best supplier in case b) by the LW method and is ranked at the second position by the other two techniques in the same case. Moreover, it is interesting to remark that in case a) the worst supplier is s 37 according to all the methods, since it is characterized by one of the highest prices (price being by far the most important criterion in this case), by an average performance index value for the second criterion and finally by the worst value with respect to the third one. Similar comments hold for the last ranking position in case b), which is still assigned to vendor s 37 when the LW and FTOPSIS methods are used, while the FAHP technique selects s 43 as the worst supplier. Furthermore, note that supplier s 4, offering the best price in the transaction, is never ranked among the first five positions by any technique neither in case a) nor in case b), since its reduction in execution time index is only average and its free maintenance time is the worst possible one, i.e. zero. We may additionally comment the obtained results in several respects. First of all, note that the outcomes of the three alternative methods are quite similar in both the considered cases with only minor differences arising in the obtained rankings. Indeed, we remark that the LW technique is compensatory in essence, since it simply determines a crisp or non fuzzy weighted sum of the normalized indices for each supplier, and a good performance of an alternative with respect to a particular criterion may easily balance another second-rate performance index of the 13
15 same supplier, which in many cases is not realistic [1, 17]. On the other hand, the two fuzzy methods are non-compensatory in nature [1, 17] partly due to the considered non linear rules in the vendors rating but mostly because of the non linear transformation of the input data resulting from their fuzzification by way of the sigmoidal membership functions, a process that tends to further reward suppliers exhibiting the best performance indices with respect to all criteria and penalize the worst ones. Hence, the differences between the former method and the two fuzzy techniques are more visible in case b), in which all criteria are assigned the same weights, rather than in case a), where price weighs so much that the taken decisions are close to those obtained in a single-objective decision problem and the three methods lead to very similar results. Additional simulations, in real cases performed using the customary LW technique, will allow to better evaluate the capacity of the fuzzy techniques to provide the better proxy of the synthetic global evaluation of a skilled buyer. Finally, it is interesting to remark that the use of one of the proposed fuzzy multi-objective techniques is undoubtedly advisable when some of the considered conflicting criteria refer to qualitative performance indices. These may be related to the requested goods or services, e.g. in the case of product standardization level, quality, strategic importance and availability, or to the supplier, such as for instance risk level, vendor dimension and reliability, partnership level with the buyer, experience and geographical location [9]. These aspects of the suppliers selection problem in the public procurement sector will be the subject of future research. Conclusions The paper contributes to the relatively new field of purchasing in the public sector focusing on the crucial issue of vendor assessment. More precisely, we apply to vendor rating in single-item multiple sourcing exchanges and compare the well known linear weighting method and two fuzzy logic based decision making approaches. A real case study illustrates the techniques effectiveness with their advantages and limitations. Future research may extend the methodology considering additional qualitative factors, that may be related either to the requested product/service or to the particular supplier, as well as the more complex case of multi items exchange. 14
16 References [1] Albino V., Garavelli A.C., Gorgoglione M., Fuzzy logic in vendor rating: a comparison between a fuzzy logic system and a neural network, Fuzzy Economic Review, Vol. 3, pp. 5-48, [] Amid A., Ghodsypour S.H., O Brien C., Fuzzy multiobjective linear model for supplier selection in a supply chain, International Journal of Production Economics, Vol. 104, pp , 006. [3] Baily P., Farmer D., Jessop D., Jones D., Purchasing, Principles and Management, Prentice Hall, London, UK, 9th ed., 005. [4] Bellmann R., Zadeh L.A., Decision making in a fuzzy environment, Management Science, Vol. 17, no. 4, pp , [5] Bevilacqua M., Ciarapica F.E., Giacchetta G., A fuzzy-qfd approach to supplier selection, Journal of Purchasing and Supply Management, Vol. 1, pp. 14-7, 006. [6] Biswas S., Narahari Y., Object oriented modelling and decision support for supply chains, European Journal of Operational Research, Vol. 153, no. 3, pp , 004. [7] Chen C.-C., Yeh T.-M., Yang C.-C., Customer-focused rating system of supplier quality performance, Journal of Manufacturing Technology Management, Vol. 15, no. 7, pp , 004. [8] Chen C.T., Lin C.T., Huang S.F., A fuzzy approach for supplier evaluation and selection in supply chain management, International Journal of Production Economics, Vol. 10, pp , 006. [9] Costantino N., Dotoli M., Falagario M., Fanti M. P., A fuzzy model for vendor rating with risk assessment, Proceedings of the 006 International Workshop on Logistics & Transportation, Hammamet, Tunisia, April 30 May 006, pp [10] de Boer L., Labro E., Morlacchi P., A review of methods supporting supplier selection, European Journal of Purchasing & Supply Management, Vol. 7, pp , 001. [11] Degraeve Z., Labro E., Roodhooft F., An evaluation of vendor selection models from a total cost of ownership perspective, European Journal of Operational Research, Vol. 15, pp , 000. [1] Dulmin R., Mininno V., Linear weighting per la vendor evaluation: alcune osservazioni sul metodo, RCS- Economia & Management Rivista on Line, 15
17 DulminMininno.jhtml, 004 (in Italian). [13] Karpak B., Kumcu E., Kasuganti R. R., Purchasing materials in the supply chain: managing a multi-objective task, European Journal of Purchasing and Supply Chain Management, Vol. 7, pp , 001. [14] Panayiotou N. A., Gayialis S. P., Tatsiopoulos I.P., An e-procurement system for governmental purchasing, International Journal of Production Economics, Vol. 90, pp , 004. [15] Piramuthu S., Knowledge-based framework for automated dynamic supply chain configuration, European Journal of Operational Research, Vol. 165, pp , 005. [16] Saaty T.L., How to make a decision: the Analytic hierarchy process, European Journal Operational Research, Vol. 48, pp. 9-6, [17] Sonmez M., A review and critique of supplier selection process and practices, Business School Occasional Papers Series, Paper 006:1, Loughborough University, ISBN , [18] The MathWorks Inc., Fuzzy Logic Toolbox for Use with MATLAB User s Guide Version, Natick, MA, 004. [19] The MathWorks Inc., MATLAB User s Guide Version 7, Natick, MA, 004. [0] Triantaphyllou E., Lin C., Development and evaluation of five fuzzy multiattribute decision-making methods, International Journal of Approximate Reasoning, Vol. 14, pp , [1] Venkataraman P., Applied Optimization with MATLAB Programming, Wiley Interscience,
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