A hierarchical model for optimal supplier selection in multiple sourcing contexts

Transcription

1 A hierarchical model or optimal supplier selection in multiple sourcing contexts Mariagrazia Dotoli, Marco Falagario To cite this version: Mariagrazia Dotoli, Marco Falagario. A hierarchical model or optimal supplier selection in multiple sourcing contexts. International Journal o Production Research, Taylor Francis, 0, <0.00/000.0.>. <hal-00> HAL Id: hal-00 Submitted on Jul 0 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 International Journal o Production Research A hierarchical model or optimal supplier selection in multiple sourcing contexts Journal: International Journal o Production Research Manuscript ID: TPRS-0-IJPR-000.R Manuscript Type: Original Manuscript Date Submitted by the Author: -Feb-0 Complete List o Authors: Dotoli, Mariagrazia; Politecnico di Bari, DEE Falagario, Marco; Politecnico di Bari, DIMEG Keywords: SUPPLY CHAIN DESIGN, PURCHASING, DATA ENVELOPMENT ANALYSIS, TOPSIS, LINEAR PROGRAMMING Keywords (user): Supplier evaluation and eiciency, Optimal supplier selection

3 Page o International Journal o Production Research A Hierarchical Model or Optimal Supplier Selection in Multiple Sourcing Contexts M. DOTOLI and M. FALAGARIO * Politecnico di Bari, Dipartimento di Elettrotecnica ed Elettronica, Via Re David 00, 0 Bari, Italy Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Via Japigia, 0 Bari, Italy *Corresponding author. m.alagario@poliba.it Abstract. The paper addresses a crucial objective o the strategic purchasing unction in supply chains, i.e., optimal supplier selection. We present a hierarchical extension o the Data Envelopment Analysis (DEA), the most widespread method or supplier rating in the literature, or application in a multiple sourcing strategy context. The proposed hierarchical technique is based on three levels. First, a modiied DEA approach is used to evaluate the eiciency o each supplier according to some criteria proposed by the buyer. Second, the well known Technique or Order Preerence by Similarities to Ideal Solution (TOPSIS) is applied to rank the maximally eicient suppliers given by the previous step. Third and inally, a linear programming problem is stated and solved to ind the quantities to order rom each maximally eicient supplier in the multiple sourcing context. The presented approach is able to straightorwardly discern between eicient and ineicient partners, avoid the conusion between eicient and eective suppliers, and split the supply in a multiple sourcing context. The hierarchical model is applied to the supply o a C class component to show its robustness and eectiveness, while comparing it with the DEA and TOPSIS approaches. Keywords. Supply chain, Supplier Evaluation, Supplier Eiciency, Optimal Supplier Selection, Data Envelopment Analysis, Technique or Order Preerence by Similarities to Ideal Solution, Linear Programming.. Introduction and motivation A Supply Chain (SC) is a business network interconnecting independent manuacturing and logistics companies that perorm critical unctions in the order ulilment process (Dotoli et al., 00). The extended enterprise strategic and tactical planning reers to the SC network design, a complex decision process (see Biswas and Narahari, 00, or a discussion on the topic). In the new global business environment purchasing is becoming one o the most signiicant and strategic decision areas o the physical SC design, because external suppliers now exert a major inluence on a company s success or ailure (Karpak et al., 00). In the purchasing management domain, the

4 International Journal o Production Research Page o supplier selection process has started to receive considerable attention in the business management literature, especially with reerence to the private sector (de Boer et al., 00). Indeed, incorrect decisions about supplier selections may lead to serious proit losses up to the exit rom a product s market, as shown by numerous examples o problems suered by the SC o leading world enterprises (Piramuthu, 00). This paper ocuses on optimal supplier evaluation and selection. Supplier rating systems identiy the candidate partners that are best equipped to meet the SC expected level o perormance and check them periodically. Typically, supplier selection is a multi-objective decision problem, including conlicting objectives such as, besides the obvious goal o (low) price, quality, quantity, delivery, perormance, capacity, communication, service, geographical location, etc. (Degraeve et al., 000). The problem is urther complicated by the possibility to consider either a single sourcing or a multiple sourcing solution. The wide literature connected to supplier evaluation and selection is synonymous o the importance o such a choice. However, a thorough enumeration and discussion o the many techniques or supplier rating available in the related literature is beyond the scope o this paper: the interested reader is reerred to the comprehensive reviews by de Boer et al. (00) and Ho et al. (00) or an in-depth discussion on this topic. Broadly, the numerous multi-criteria decision making approaches suggested in the literature to solve the supplier evaluation and selection problem may be classiied into individual approaches and integrated ones. The most widespread individual methods are (Ho et al., 00): the Data Envelopment Analysis (DEA), mathematical programming, the Analytic Hierarchy Process (AHP), case-based reasoning, uzzy decision making, genetic algorithms, the Analytic Network Process (ANP), the Simple Multi-Attribute Rating Technique (SMART) and many more. The so-called integrated approaches join together dierent techniques (e.g., integrated AHP, DEA, and artiicial neural networks, integrated AHP and goal programming, etc.). Individual approaches are more popular than integrated ones, with the most widespread individual technique being DEA, due to its robustness (Ho et al. 00) and its ability to be implemented also considering qualitative criteria: as an example, Talluri et al. (00) extend the classical DEA technique considering risk evaluation. Two examples o DEA integrated approach are proposed by Sevkli et al. (00), presenting the DEAHP method and by Wu and Blackhurst (00), proposing an augmented DEA approach. However, DEA presents the drawback that its eicient alternatives are by deinition equally optimal and no dierence can be singled out with respect to their eectiveness. Moreover, the method cannot be straightorwardly applied to multiple sourcing supplier selection.

5 Page o International Journal o Production Research The motivation o this paper is to propose a novel ormulation o the most widespread method or supplier selection in the literature, namely DEA, with the aim o overcoming some o its recalled drawbacks: in particular, assuming a given number o pre-existing suppliers, we wish to evaluate and rank them, so as to choose only the most eicient ones to cooperate with and determine the product quantities to order rom such suppliers. To this aim, we propose a novel three-level hierarchical method that overcomes the drawbacks o the single DEA method. The presented hierarchical method or supplier evaluation and selection is structured as ollows. First, the method uses a modiied version, called DEA-P, o the well-known DEA method, to make a irst distinction o suppliers into two categories: eicient and ineicient ones. More precisely, the DEA-P approach, presented by the authors in Dotoli et al. (00), personalizes DEA to evaluate the weights o input and output criteria in terms o percentages. In the second step, we apply the so-called Technique or Order Preerence by Similarities to Ideal Solution (TOPSIS) (Hwang and Yoon, ) to rank the maximally eicient suppliers determined by DEA. Although in many cases methods such as AHP, TOPSIS and Linear Weighting (LW) lead to similar results (Costantino et al., 00), here we adopt TOPSIS because it groups the optimal and worst perormances o the alternatives, respectively, showing them in the so-called Best Ideal Solution (BIS) and Worst Ideal Solution (WIS), providing a clear idea o how close (ar) an optimal solution is rom the BIS (WIS). Third, ater ranking the maximally eicient suppliers by TOPSIS, a linear programming problem is solved to calculate the quantities o product to require rom each such supplier in the multiple sourcing context. Summing up, the paper provides a novel decision support tool or supplier business intelligence, i.e., or ranking suppliers and providing the buyer with a simple and lexible instrument or determining the quantities to order rom each maximally eicient supplier in a multiple sourcing strategy context. The remainder o the paper is organized as ollows: Section presents a review o the related literature on DEA ocusing particularly on the multiple sourcing context; Section, ater recalling the DEA traditional approach, presents the hierarchical method based on DEA, TOPSIS and linear programming; Section reports a numerical example to demonstrate the application o the method and its eectiveness; inally, Section presents the conclusions and urther research suggestions.. Related Literature on DEA in a Multiple Sourcing Context In a recent contribution, Ho et al. (00) provide a thorough overview o the literature addressing the supplier selection problem in the last decade. The so-called individual approaches are a little

6 International Journal o Production Research Page o more popular than the integrated techniques, with the most accepted individual approach being DEA, apparently due to its robustness and straightorwardness. DEA, irst proposed by Charnes et al. (), is a linear programming-based technique or determining the eiciency o dierent decision making units. Currently, in the related literature there exist numerous approaches that, starting rom the traditional DEA technique, are personalized by the authors to better it their own case study (see or instance the recalled paper by Talluri et al., 00 and the work by Saen, 00). As regards the application o DEA to supplier selection, the strong point o this technique is its straightorwardness in practical implementation: the DEA approach does not require the decision maker to pre-deine the criteria weights but these are endogenously determined. A urther strength o the approach lies in the distinction between input and output perormance measures. The input perormance is given by the amount o resource used by the supplier to carry out the supply process (or instance, the purchasing price), while output parameters express how good the service is (examples are the quality o the purchased product or the timeliness o deliveries). Although DEA is by ar the most widespread technique or supplier selection, some remarks are in order. First, while in the past this method was typically used only on quantitative data, nowadays it is implemented also considering qualitative criteria. Second, although price is generally the chosen criterion against which to rank suppliers (particularly using DEA), additional criteria are nowadays being used, the most popular one being quality. Third, DEA presents the drawback o making a conusion between eicient suppliers and eective suppliers (Ho et al. 00). As a matter o act, DEA simply classiies suppliers into two categories (eicient and ineicient ones), while other approaches (e.g., AHP, TOPSIS, and LW) simply rank suppliers, disregarding their eiciency level. In the private sector, the buyer addresses the supplier selection problem choosing between a single or multiple sourcing approach. On the one hand, single sourcing is deined as the ulillment o all corporate requirements or a particular product by a selected supplier. On the other hand, multiple sourcing is the splitting o an order among multiple sellers. Obviously, each solution presents advantages and drawbacks (Costantino and Pellegrino, 00). The problems connected to single sourcing are related to the disruption risk, while those connected to the multiple sourcing strategy can involve higher managing costs due to dealing with more than one contract/supplier. The most important advantage presented by the ormer approach is the cooperation between buyer and seller in improving quality, due to their long term relationship; the latter type o sourcing, on the contrary, presents as an advantage the increasing competition among partners that oten lead to improvements in the supply quality.

7 Page o International Journal o Production Research In this paper the application o DEA in the multiple sourcing context is analyzed, i.e., we assume that the buyer has already chosen the set o partners to provide a certain component by a multiple sourcing. As it is common in the literature, the spare parts in inventory are divided into three classes (A, B, and C) according to their money usage (corresponding respectively to 0%, %, and % o money usage and 0%, 0% and 0% o the items, see Krajewski and Ritzman, 00). Typically, or A and B components a strategic partnership between buyer and seller is created. Hence, all suppliers can roughly satisy the buyer s requirements o demand, quality, delivery, etc., so that the buyer only needs to make one decision, i.e., which supplier is the best (Xia and Wu, 00). On the contrary, C components are such that an increasing competition among several suppliers can allow the buyer to obtain a better price. In other words, the buyer needs to purchase some part o the demand rom a supplier and the other rom other ones, e.g., to compensate or shortage o capacities, or low quality o some suppliers. Summing up, in general any supplier selection technique has to consider the sourcing strategy chosen by the buyer, so that it results in being generally inluenced by three undamental decisions that the buyer has to take (Burke and Vakharia, 00): the criteria or identiying potential suppliers, the criteria or choosing the appropriate or eicient suppliers (a subset o the potential ones), and the optimal quantity o goods to order to each selected supplier. DEA alone, although widespread or its simplicity, cannot answer to all the above questions. Some authors in the related literature proposed several contribution on the subject. In particular, Weber et al. (000) developed an optimization approach that uses multi-objective programming to select suppliers and then evaluate their eiciency on multiple criteria using DEA. However, the approach does not consider the limitations on suppliers capacity nor answer to the question o which quantities to order rom which supplier. Later, Talluri and Baker (00) presented a comprehensive model or eectively designing the entire SC by considering not only the eiciency and capacity o participating candidates, but also location and transportation issues. However, the technique addresses too many strategic problems simultaneously, resulting in a complex procedure. Later on, Talluri and Narasimhan (00) proposed a ramework or strategic sourcing using a combination o DEA models or eectively discriminating supplier perormance. However, the method requires numerous data and utilizes statistical techniques that make the overall approach quite complex. Further, Ha and Krishnan (00) used DEA and neural networks to measure supplier perormance eiciency more accurately. Similarly, Wu (00) proposed a hybrid model using DEA, decision trees and neural networks to assess supplier perormance. However, both these techniques are

8 International Journal o Production Research Page o hindered by the use o advanced sot computing methods that will not be straightorward or most users and make the process quite cumbersome. In conclusion, all the cited works either result in complex techniques, oten requiring numerous parameters that are not straightorward to determine or the buyer, or do not explicitly consider the optimal supplier selection problem in a multiple sourcing context. On the contrary, the presented approach uses three dierent techniques, i.e., DEA, TOPSIS, and linear programming, to obtain a simple and yet eective method in order to measure supplier perormance eiciency (DEA), determine a ranking (TOPSIS) and calculate the quantities to order (linear programming). Since these techniques can complement each other, using the three o them is helpul to eectively discriminate among suppliers in a multiple sourcing strategy, while avoiding the typical pitalls associated with traditional DEA models, and straightorwardly answering the buyers questions: which suppliers are the best, which suppliers should be selected, and how much should be purchased rom each selected supplier.. Methodology Optimal supplier selection may be deined as a multi-objective decision problem, including conlicting objectives such as, besides the obvious goal o (low) price, quality, quantity, delivery, perormance, capacity, communication, service, geographical location etc. (Degraeve et al., 000). Generally, a supplier selection problem is deined by a set o bidding suppliers S = { s s s } and a set o conlicting criteria C { c c c },,..., F =,,..., n, according to which suppliers have to be ranked. We assume that the criterion set is partitioned into C CI CO {,,..., } O H+ H+ H+ K =, with C { c, c,..., c } I = and C = c c c respectively representing the input and output criteria sets, with H+K=n being the criteria number. The input criteria may be deined as the supplier attributes considered beore the supply takes place (e.g., price, geographical distance o the supplier, ICT integration, etc.), while the output criteria characterize the supplier ater the goods arrive at the irm (e.g., quality, reliability, lead time, etc.). Although qualitative criteria may be addressed, to avoid conusion on subjective judgments only quantitative criteria are considered in this paper. Figure shows the presented hierarchical integrated approach to determine eicient suppliers and the optimal product quantities in the multiple sourcing context. H

9 Page o International Journal o Production Research The DEA Method Take in Figure. Figure. The hierarchical model or supplier selection. The ounding method o the supplier selection approach in Fig. is DEA (Charnes et al., ). DEA is a linear programming-based technique or determining the eiciency o dierent Decision Making Units (DMUs). Eiciency is a unction o some input values (which are the resources used by the DMUs) and output values (expressing the results o the DMUs activities). Eicient DMUs are the vertices o a Pareto ace: based on this, other DMUs eiciencies are calculated. In the traditional DEA approach or supplier selection, the eiciency o the generic supplier s is deined as ollows: E = K uk k= H vh h= y x k h with =,,F, () where y k (x h ) is the k-th (h-th) output (input) perormance value or the -th supplier and u k (v h ) is its weighting coeicient. The eiciency o each alternative is obtained by determining the set o coeicients u k with k=,,k and v h with h=,,h which maximizes such a value, taking into account that or each supplier s S it holds by deinition E. Hence, the measure o supplier eiciency is obtained solving the ollowing optimization problem or each considered supplier: max E with =,,F, () subject to (s.t.): K k= H h= u v k h y x kg hg with g=,,f, () u, v 0 or k=,,k; h=,,h. () k h The optimization problem ()-()-() can be linearized by maximizing the outputs and keeping ixed input values, obtaining the so-called output-oriented method (Wang and Chin, 00). Hence, the problem is modiied into a linear programming problem as ollows: S

10 International Journal o Production Research Page o K max E = u y with =,,F, () s.t.: K k k k= u y v x 0 with g=,,f, () k kg h hg k= h= H h= H vh xhg = with g=,,f, () and (). Summing up, the eiciency o the analyzed suppliers can be ound solving problem ()-()-()-() or each -th supplier with =,,F. Obviously, the -th supplier is maximally eicient i it holds E =. Thereore, suppliers can be rated based on their eiciency value E.. The First Level o the Hierarchical Supplier Selection: The DEA-Percentage (DEA-P) Method to Single out Maximally Eicient Suppliers The irst level o the supplier selection approach in Fig. employs the DEA-Percentage (DEA-P) method, a novel ormulation o the DEA technique proposed by the authors in Dotoli et al. (00) to single out the potential suppliers. DEA-P allows the buyer to compare the dierent supplier evaluation criteria by assigning a percentage index, which expresses the importance o each criterion. Hence, the approach can ind a useul application to choose one or more suppliers or a product or service on the basis o past perormance evaluations or o oered quotations. DEA-P evaluates the eiciency o a supplier by measuring the dierence between weighted input values and weighted output values. Such an index o the usage o input resources is minimized and imposed to be non-negative or each analyzed supplier. A normalization constraint or weighting coeicients allows us to obtain a non-zero optimum or each supplier. The solution o the resulting linear programming problem provides the purchasing manager with a set o values or the criteria weights that can be compared with each other. These values express the percentages o the relevance o each criterion (a eature o the DEA-P technique that is instead not directly apparent in the DEA traditional technique). Hence, deining the eiciency as in (), in DEA-P a Perormance Evaluation Index (PEI) or each supplier s H S is deined as the dierence between weighted input and output perormance values: PEI = v x u y h h k k h= k= K with =,,F. ()

11 Page o International Journal o Production Research Obviously, using the dierence as a novel measure o comparison, the -th supplier is maximally eicient (i.e., E = ) i PEI = 0. Indeed, this means that all the resources used by the supplier in the supply process are returned as an output. Hence, in this case, the supplier with =,,F is determined by minimizing PEI while imposing that such a value is nonnegative: PEI index or each -th min PEI () s.t.: H v x u y 0 with g=,,f, (0) h hg k kg h= k= H K h h= k= K v + uk =, () and (). The constraint (0) is imposed to avoid negative PEI values. In addition, constraint () is added to avoid the trivial solution in which all the coeicients u k or k=,,k and v h or h=,,h are equal to zero. Moreover, such a constraint allows to compare the dierent criteria weights with each other orcing the coeicients to be percentage contributions o input and output perormance values. In this way, the minimization admits non-zero solutions only. The problem ()-(0)-()-() is a linear programming problem to be solved F times; consequently, and remarkably, there is no need or linearization. Moreover, in the presented approach the input and output values are introduced in an already normalized ashion. In this way, input and output coeicients are not aected by dierent orders o magnitude among input and output values, but are only the expression o the contribution o every attribute to the overall eectiveness o the alternative. In Dotoli et al. (00), the authors demonstrate that DEA-P leads to the same eiciency results as the traditional DEA approach, but with dierent criteria weight values. Moreover, the DEA-P model exhibits the same computational complexity o the traditional DEA. Indeed, the two methods are both linear programming problems, sharing the same number o variables u k with k=,,k and v h with h=,,h and the same number o constraints equal to F+. The main advantage o DEA-P with respect to DEA is that it provides the weights o the input and output criteria directly in percentages. In this way, it is possible to overcome the problem o aggressive values or these attributes, ocusing only on the contribution o each input and output criterion to the overall eiciency. In order to limit the oten large number o eicient suppliers, we add to the DEA-P optimization problem some constraints related to the importance o one or more criteria, as ollows:

12 International Journal o Production Research Page 0 o u v k h α with k=,...,k, () k β, with h=,...,h, () where h α k and β h are coeicients assigned by the buyer or each k-th (h-th) output (input) criterion, with k=,,k (h=,,h) such that α, β [0;] and k h α + βh. k k= h= Constraints () and () are related to the buyer judgments on criteria and enable him to assign the range in which a criterion weight can vary depending on how strategic it is in the supplier evaluation. In this way, it is possible to determine a subset o eicient suppliers satisying the buyer requirements. Hence, constraints ()-() play a role similar to that o assurance regions (Thompson et al., 0) in traditional DEA. These allow to rank the dierent weights o the DEA input and output criteria and can be useul when dealing with components o dierent typologies (e.g., A, B or C class component) in the multiple sourcing context. The advantage o DEA-P with respect to the assurance regions approach is that by the ormer technique the buyer can indicate in an absolute way how much a given criteria weighs on the overall set o criteria weights. Instead, the assurance regions approach only allows to compare in a pairwise manner the relative importance o input and/or output criteria. Summing up, the irst level o the hierarchical supplier selection uses the DEA-P procedure to provide as a solution to problem ()-(0)-()-()-()-() a subset o the suppliers set S collecting the m maximally eicient suppliers (i.e., the partners showing the maximum eiciency level) with the corresponding perormance values against the H+K=n criteria.. The Second Level o the Hierarchical Supplier Selection: The TOPSIS Method to Rank the Maximally Eicient Suppliers The second level o the supplier selection approach in Fig. employs TOPSIS to rank the potential suppliers obtained by the irst level and select the most eicient ones according to the given criteria. TOPSIS, developed by Hwang and Yoon (), is a multi-objective decision technique or ranking a set o alternatives according to a set o conlicting criteria o various degrees o importance. This decision technique is based on simple geometric concepts: the best alternative exhibits the shortest distance rom the BIS and the arthest distance rom the WIS in a Euclidean sense. Deining the decision matrix D o dimensions mxn where m is the number alternatives (the maximally eicient suppliers), with the generic element d ij with i=,,m and j=,,n the corresponding perormance values against the j-th criterion, the TOPSIS technique consists o the ollowing six steps. K H 0

13 Page o International Journal o Production Research Constructing the normalized decision matrix. The generic element n' ij = d ' ij m ' d ij i= n ' ij o the mxn normalized decision matrix N' is determined as ollows:, i=,,m, j=,,n (with n=h+k). (). Constructing the weighted normalized decision matrix. Assigning a vector W = [ w... w n ] with n w j = j= as the vector o criteria weights, the mxn weighted normalized decision matrix W is determined, where w' = n' w, or i=,,m and j=,,n.. Determining the best and worst ideal solutions. ij ij j The BIS is determined as the ideal solution with perormance indices given by the row vector G, where G = opt( n',..., n' ) with j=,,n is obtained applying unction opt deined = [ G... G n ] j j mj by the buyer to the normalized decision matrix j-th column. Hence, each element o G is the optimum perormance value o all suppliers with respect to the j-th criterion: as an example, opt may be the minimum unction i a perormance index - such as price - is to be minimized. As a result, vector G is composed by the best perormance values o the suppliers against all the criteria. Similarly, the WIS is determined as the ideal solution associated to perormance indices o the row vector P = [ P... P n ], where Pj = wor( n' j,..., n' mj ) with j=,,n and wor is a unction providing the worst perormance value or the j-th criterion (e.g., wor may be the maximum unction i a perormance index - such as price - is to be minimized). As a result, vector P collects the worst perormance values o the suppliers against all criteria.. Calculating the separation measure. The separation distance S G, i rom the BIS o each supplier s i with i=,,m is calculated as ollows: n S G, i = ( n' ij G j ) j=. () Moreover, the separation distance S H, i o s i with i=,,m rom the WIS is as ollows: n S H, i = ( n' ij H j ) j=. ()

14 International Journal o Production Research Page o Hence, the above two values respectively measure the relative distance o each alternative rom the BIS and WIS ideal solutions.. Calculating the relative closeness o alternatives to the ideal solution. The overall perormance index PI i_topsis, measuring the closeness to the BIS o each maximally eicient supplier s i is thus determined as ollows: SH, i PIi _ TOPSIS = SG, i + SH, i or each i=,,m. (). Ranking the alternatives. Finally, the maximally eicient suppliers are ranked according to the values o the PI i_topsis index with i=,,m. Obviously, the best supplier is the one showing the highest value obtained by (). Hence, the PI i_topsis index can be considered as a buyer satisaction priority to be maximized. Summing up, the second level o the hierarchical supplier selection uses the TOPSIS approach to rank only the m maximally eicient suppliers in set S against the n conlicting criteria.. The Third Level o the Hierarchical Supplier Selection: The Linear Programming Method to Split the Supply among the Maximally Eicient Suppliers The third and inal level o the technique employs linear programming to split the supply among the maximally eicient suppliers, as ranked by the second and higher level, and determine the quantities to order rom them. Linear programming is an optimization mathematical process: an objective unction states what is being maximised, e.g., proit, or minimized, e.g., cost or scrap. To determine the quantities to require rom the eicient suppliers, we deine the Supply Evaluation Index (SEI) as ollows: m SEI = qi PIi _ TOPSIS i=, () that is an overall index measuring the eiciency o the supply by the m maximally eicient suppliers obtained by the irst-level DEA optimization as ranked by the second-level TOPSIS optimization. In particular, variables q i with i=,.., m are the unknown quantities o product (expressed as percentage o the supply with values ranging rom 0 to ) to request rom each optimal supplier. Note that, or the sake o simplicity, we assume that the supply is ulilled by a multiple sourcing procedure including all the maximally eicient suppliers. Obviously, the present third level

15 Page o International Journal o Production Research optimization may also be straightorwardly modiied considering only a number µ m o maximally eicient suppliers assigned by the buyer. Hence, the ollowing problem is stated to ulill the supply in an optimal way: Max( SEI ) () s.t.: m qi = i= (0) qi γ i with 0 γi and i,..., m q δ with i i m δi i= = (). () In particular, δ i with i=,,m is a parameter measuring the minimum percentage quantity o the overall supply (eventually equal to zero) that the buyer decides to buy rom the i-th eicient supplier independently rom its ranking to keep the long-term partnership. In addition, γ i is the given production capacity (expressed in percentage values o the whole supply) o the i-th maximally eicient supplier with i=,,m. Hence, constraint (0) guarantees that the whole requested quantity is supplied (note that it is possible to increase the unitary term in the right member o the equation in order to consider saety stocks), constraints () take into account the quantities that each supplier is able to deliver, and inally constraints () model the buyer will o requiring products rom each eicient supplier independently rom their ranking. Summing up, the third level o the hierarchical optimal supplier selection uses linear programming to calculate the quantities to require rom the set o eicient suppliers as obtained by the DEA-P irst-level method and ranked through the TOPSIS approach in the second-level optimization.. Case study To show the eectiveness o the three step approach or optimal supplier evaluation and selection, we consider a simulated case study requiring the supply o C class components under multiple sourcing and assuming that the number o existing suppliers equals F=0. The presented method is compared with DEA and TOPSIS in order to underline dierences and similarities.

16 International Journal o Production Research Page o The Case Study Data Supplier eiciency is estimated using H= input criteria and K= output ones. In particular, the input criteria are price, ICT integration cost, and geographical distance. They are detailed as ollows. Price. This attribute measures the price oered by each supplier or the required product. Obviously, it is a very signiicant criterion, but nowadays its importance is becoming less and less central. ICT integration cost. This criterion addresses the costs necessary or improving the intercommunication ability between buyer and supplier. A high perormance value means that the two partners are not well integrated with each other rom a managerial point o view. Geographical distance. This criterion expresses the geographical distance o the buyer rom the supplier. Obviously, the nearer the supplier, the lower the transportation costs; moreover, the lexibility in delivery generally grows in case o a closer partner. The K output criteria considered or supplier eiciency evaluation are delivery on time, quality, and lead time. They are detailed as ollows. Delivery on time. This criterion can be related to numerous correlated indices, e.g., the appropriateness o the delivery date, the compliance with the due date, the degree o closeness, delivery, and location, and many more. In this paper a crisp or nonuzzy measure o the reliability level is deined as the ratio between the amount o dispatched orders and the overall number o orders. This is a very important issue, because delays in deliveries can cause disruptions in the production chain. Quality index. This criterion is strictly related to the number o accepted products among the veriied ones: indeed, a high number o deects means high costs o repairs or maintenance. A crisp perormance value is obtained or each supplier as the ratio between the percentage o accepted items and that o accepted lots. Lead time. This criterion is strictly related to the manuacturing capability o the supplier and its lexibility. The lead time is deined as the span o time between the placing o an order and the receipt o goods. Obviously, the shorter it is, the better the supplier in term o lexibility, production capability, and internal organization. The attributes or perormance values o each supplier against each criterion are determined in a normalized ashion, by dividing each attribute by the highest perormance value. In this way, normalization can be made only on coeicients and not on weights associated to input and output values. The considered input attributes o the -th supplier with =,,F are as ollows:

17 Page o International Journal o Production Research normalized product price normalized ICT integration cost cict attribute; normalized geographical distance p p =, where pmax pmax d cict = max ( p ) is the maximum oered price; =,,..., F =, where cictmax d cict max =, where dmax dmax =,,...,F = max ( cict ) is the maximum =,,...,F = max ( d ) is the maximum distance. In addition, the output attributes are as ollows or the -th supplier with =,,F: normalized delivery on time such an index and DT DT NDO DT =, where DTmax DT max =,,...,F = max ( DT ) is the maximum value o =, with NDO and NO representing the amount o dispatched orders NO by such a supplier and the overall number o orders, respectively; normalized quality index QI and QI pc lot QI =, where QImax QImax =,,...,F = max ( QI ) is the maximum quality index = a, a, pc v, lot, with pc a, (pc v, ) being the number o accepted (veriied) items and lot a, v, (lot v, ) being the amount o accepted (veriied) lots o the -th supplier; normalized lead time index LTI time. max LT =, where LTmax = max ( LT ) is the maximum lead LT =,,..., F. Application o the Proposed Hierarchical Method to the Case Study The normalized input perormance values o each supplier are collected in Table (columns,, and ). Note that normalized prices (column ) range rom a value o.000 or supplier (oering the highest price) to 0. or supplier 0 (the lowest price); second, ICT integration costs (column ) range rom.000 or supplier (the highest cost) to 0.0 or supplier ; third, geographical distances (column ) vary rom.000 or supplier (the arthest one rom the buyer irm) to 0.0 or supplier. Moreover, in columns,, and o Table the output indices are reported. In particular, the data in column show that the most reliable supplier is supplier, while the less reliable one is supplier 0; based on the quality attribute only (column ), the best supplier is supplier, while the worst one is supplier ; inally, column shows that the quickest partner in providing the required component is supplier 0, while the slowest one is supplier.

18 International Journal o Production Research Page o We apply the described irst-level optimization procedure deining the optimization problem ()- (0)-()-()-()-() with α k = 0 or k=,, and β h = 0 h=,, or each supplier, so that the results shown in Table are obtained. In particular, the table reports in columns,, and (,, ) the resulting input (output) weights obtained by the DEA-P optimization and in the second-last and last columns the resulting PEI and E indices, determining respectively the perormance evaluation index and eiciency o each -th supplier with =,,F. Obviously, the eiciency values reported in the last column o Table are determined by substituting in () the weights v, v, v, u, u, and u (reported in the previous columns o Table ) that are the solution o the linear programming problem ()-(0)-()-()-()-() or each supplier according to DEA-P. In the considered case study, or instance, the DEA-P procedure shows that, to obtain a maximum eiciency index equal to.000 or supplier, the normalized price weights or a percentage equal to.%, the normalized geographical distance or.%, the quality index or 0.%, the lead time index or.%, and the remaining criteria or a percentage equal to zero (see the results in Table ). We now consider a second case in which the buyer asks or a more severe requirement on the lead time: in particular, we solve problem ()-(0)-()-()-()-() assuming that β h = 0 h=,, and α = 0 or k=,, while the last constraint in () is speciied as ollows: k u () Hence, constraint () imposes that the weight o the lead time output criterion has to equal at least a percentage o 0%. Based on this urther constraint, the irst level optimization procedure now leads to the results shown in Table. The obtained results are analogous to the previous results reported in Table, but it is apparent that the number o eicient solutions decrease rom ive eicient suppliers (suppliers,,,, 0 in Table ) to only three eicient partners (suppliers,, and 0 in Table ). Such a reduction is expected, since the requirements o the buyer are in this case more severe and are not satisied by all the irst case eicient suppliers. As an example, in this second case, according to the data in Table, supplier is maximally eicient (i.e., E =) by weighting the price criterion or.%, ICT integration or 0.%, geographical distance or.0%, reliability or.%, quality or.%, and lead time or.% (that is obviously larger than 0%). On the contrary, it is worth underlining that supplier, which was maximally eicient in the previous case (see Table, showing E =), becomes totally ineicient (i.e., E =0) in the second case by applying constraint (). Indeed, Table shows that the lead time weight associated to this supplier equals zero (i.e., this supplier is the worst one with regard to this attribute only, which is coherent with the data in Table, reporting LT =0). Summing up, by the eiciency evaluation provided by DEA-P, the buyer has made the so called prequaliication o the existent partners.

19 Page o International Journal o Production Research Take in Table. THE DATA FOR THE DEA-P INPUT AND OUTPUT CRITERIA. Take in Table. THE RESULTS OF THE DEA-P FIRST LEVEL OPTIMIZATION (FIRST CASE). Take in Table. THE RESULTS OF THE DEA-P FIRST LEVEL OPTIMIZATION (SECOND CASE). Take in Table. THE RESULTS OF THE TOPSIS SECOND LEVEL OPTIMIZATION. Take in Table. THE DATA FOR THE LINEAR PROGRAMMING THIRD LEVEL OPTIMIZATION. Take in Table. THE RESULTS OF THE LINEAR PROGRAMMING THIRD LEVEL OPTIMIZATION. We now rank the m= maximally eicient suppliers (i.e., suppliers,,0) by determining their overall TOPSIS perormance index () against the n= conlicting criteria. In particular, the vector o input and output weights W = [ w... w ] has to be assigned based on the buyer judgements and obviously satisying constraints ()-(). In this case it holds β h = 0 h=,,, α k = 0 or k=, and α = 0. by (). Hence, we assign w = α = 0.. Considering that the supply regards a C class component, and supposing that price is twice as much important than quality and delivery on time, and inally that these last two attributes are twice as much important than ICT integration and geographical distance, the vector o weights W with n w j = j= is assigned as ollows: W = [ ]. () The results o the TOPSIS second level optimization are shown in Table. In particular, in columns to the perormance values o the maximally eicient suppliers s with =,,0 against the conlicting criteria are reported (according to Table ), while the third-last, second last, and last columns respectively show the novel supplier numbering index in the TOPSIS optimization, their overall perormance value, and their ranking. It is apparent that the best supplier is s 0, showing the

20 International Journal o Production Research Page o best lead time index (since this is the highest weight criterion) and the best oered price (the second highest weight criterion). Suppliers s and s ollow in the ranking. Having ranked the maximally eicient suppliers, we apply the described third-level optimization procedure, i.e., we solve the linear programming problem (), (), (0), (), and (). The assigned supplier capacities and minimum requested quantities are reported in Table, in columns and, respectively. Table shows the results o the linear programming problem solution, detailing the percentages according to which the supply is to be split. In particular, since s 0 is evaluated as the best supplier both in terms o eiciency (according to DEA) and eectiveness (according to TOPSIS), the hierarchical technique suggests that the buyer should buy his whole capacity, while the remaining part o the supply should be bought by the second and third eicient suppliers, avouring the second and more eective supplier.. Comparison with DEA and TOPSIS and Discussion In order to underline the eicacy o the presented approach, we apply the TOPSIS method alone to the case study, i.e., to the whole supplier set. For the sake o comparison, Table reports in column the DEA ranking o all the suppliers obtained by the eiciency evaluation (according to the results in the last column o Table ), in column the TOPSIS supplier rating, and in the last and ourth column the results obtained by the proposed three level procedure. In particular, we remark that, although the best two suppliers (s and s 0 ) are the same in the three approaches, the subsequent partners in the classiication are not the same. Indeed, under the presented technique only s is evaluated, since it is the only remaining eicient partner, while or example in TOPSIS suppliers s, s and s are valued better than s even though they are not eicient (as underlined by DEA). We point out some general remarks. On the one hand, DEA alone cannot lead to a detailed ranking (because it usually leads to equally maximally eicient suppliers). On the other hand, the TOPSIS approach leads to a more accurate classiication o suppliers but does not discriminate between eicient and ineicient partners. Conversely, the presented hierarchical model combines the DEA advantage o singling out ineicient suppliers with the TOPSIS ability o discerning among equally eicient partners, helping the buyer in the decision on how many suppliers to consider in the supply. In addition, the proposed model provides a simple procedure to determine the quantities to order rom the suppliers based on their characteristics and the buyer priorities. Summing up, the three-step integrated approach is a lexible and simple tool enabling the buyer to answer the main questions related to a supply in a multiple sourcing strategy context: which

21 Page o International Journal o Production Research suppliers are the best, which suppliers should be selected, and how much should be purchased rom each selected supplier. Take in Table. THE COMPARISON BETWEEN DEA, TOPSIS AND THREE LEVEL OPTIMIZATION.. Conclusions The paper presents a novel three-step methodology or optimal supplier selection in a multiple sourcing context. The hierarchical model is based on the Data Envelopment Analysis (DEA) approach, the Technique or Order Preerence by Similarities to Ideal Solution (TOPSIS), and linear programming. To the best o the authors knowledge, no one in the related literature has ever joined these three approaches or supplier selection and evaluation in such a context. The proposed integrated technique takes the best o each method: irst, DEA is used to divide the suppliers into eicient and not eicient ones; second, TOPSIS is adopted to rank the eicient suppliers; and, inally, linear programming lets us calculate the quantities to order rom each supplier. In such a way, the hierarchical approach avoids the drawbacks o individual methods: it overcomes the well-known DEA inability to discriminate between eicient and eective suppliers, as well as the TOPSIS diiculty to discern between eicient and ineicient partners, and inally overcomes the inability o the two said methods to split the whole supply among suppliers, guaranteeing the will o the buyer to adopt a multiple sourcing strategy. A numerical case study is reported to show the eectiveness o the three step method or a C class component. The proposed methodology is compared to DEA and TOPSIS, showing its enhanced completeness and lexibility. Future perspectives are making the technique recursive (e.g., to increase the number o involved suppliers in one or more levels o the methods i the obtained solution is not satisactory or the buyer), identiying a real case study in order to urther veriy the approach lexibility and simplicity, and inally extending the approach considering uzzy techniques to take into account qualitative criteria.. Acknowledgements This work was supported by the TRASFORMA Reti di Laboratori network unded by Apulia Italian Region.

22 International Journal o Production Research Page 0 o Reerences Biswas S., Narahari Y., 00. Object oriented modelling and decision support or supply chains, European Journal o Operational Research, Vol., No., pp. 0-. Burke, G.J., Vakharia, A.J., 00. Supply chain management. In: The Internet Encyclopedia. Wiley, New York. Charnes A., Cooper W.W., Rhodes E.,. Measuring the eiciency o decision making units; European Journal o Operational Research,,. Costantino N., Dotoli M., Falagario M., Fanti M.P., 00. Fuzzy Logic Based Vendor Selection or the Public Procurement Sector: a Case Study; Proceedings o SIGEF 00, th Congress o the International Association or Fuzzy Set Management and Economy, Hammamet, Tunisia, 0 November December 00. Costantino N., Pellegrino R., 00. Choosing between single and multiple sourcing based on supplier deault risk: A real options approach; Journal o Purchasing & Supply Management, -0. de Boer L., Labro E., Morlacchi P., 00. A review o methods supporting supplier selection; European Journal o Purchasing and Supplying Management, Vol., pp. -. Degraeve Z., Labro E., Roodhoot F., 000. An evaluation o vendor selection models rom a total cost o ownership perspective; European J. o Operational Research, (), -. Dotoli M., Falagario M., Mangini A.M., Sciancalepore F., 00. An extension o the DEA model or supplier evaluation and selection, Proc. o the th IEEE International Conerence on Emerging Technologies and Factory Automation, Bilbao, Spain, September -. Dotoli M., Fanti M.P., Meloni C., Zhou M.C., 00. Design and optimization o integrated e-supply chain or agile and environmentally conscious manuacturing, IEEE Transactions on Systems Man and Cybernetics, part A, (), -. Ha S.H., Krishnan R., 00. A hybrid approach to supplier selection or the maintenance o a competitive supply chain, Expert Systems with Applications, Vol., No., pp. 0-. Ho W., Xu X., Dey P.K., 00. Multi-criteria decision making approaches or supplier evaluation and selection: A literature review, European Journal o Operational Research, 0, -. Hwang C.L., Yoon, K.,. Multiple Attribute Decision Making, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag: Berlin. Karpak B., Kumcu E., Kasuganti R.R., 00. Purchasing materials in the supply chain: managing a multi-objective task, European Journal o Purchasing and Supply Chain Management, Vol., No., pp