A Technique for Supply Chain Network Design under Uncertainty

Proceedigs of the 15th IFAC Symposium on Information Control Problems in Manufacturing Proceedigs of the 15th IFAC Symposium on Proceedigs of theOttawa, 15th IFAC Symposium on May 11-13, 2015. Canada Available online at www.sciencedirect.com Information Control Problems in Manufacturing Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada

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IFAC-PapersOnLine 48-3 (2015) 634–639 A Technique for Supply Chain Network Design under Uncertainty A Technique for Supply Network Design Uncertainty using Cross-Efficiency Fuzzy Data Envelopment A Technique for Supply Chain Chain Network Design under underAnalysis Uncertainty using Cross-Efficiency Fuzzy Data Envelopment Analysis using Cross-Efficiency Fuzzy Data Envelopment Analysis Mariagrazia Dotoli*, Nicola Epicoco**, Marco Falagario***

Mariagrazia Dotoli*, Nicola Epicoco**, Marco Falagario*** Mariagrazia Dotoli*, Nicola Epicoco**, Marco Falagario*** * Department of Electrical and Information Engineering, Politecnico di Bari, Bari, Italy 080 5963667; e-mail: Engineering, [email protected]). * Department(Tel:+39 of Electrical and Information Politecnico di Bari, Bari, Italy * Department of Electrical and Information Engineering, Politecnico di Bari, Bari, Italy ** Department of Electrical and Information Engineering, Politecnico di Bari, Bari, Italy (Tel:+39 080 5963667; e-mail: [email protected]). (Tel:+39 080 5963667; e-mail: [email protected]). [email protected]). ** Department of Electrical(e-mail: and Information Engineering, Politecnico di Bari, Bari, Italy ** Department of Electrical and Information Engineering, Politecnico di Bari, Bari, Italy *** Department of Mathematics, Mechanics and Management Engineering, Politecnico di Bari, Bari, Italy (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). *** Department of Mathematics, Mechanics and Management Engineering, Politecnico di Bari, Bari, Italy *** Department of Mathematics, Mechanics and Management Engineering, Politecnico di Bari, Bari, Italy (e-mail: [email protected]). (e-mail: [email protected]). Abstract: The paper focuses on Supply Chain Network Design (SCND) under uncertainty. We propose a SCND method extending anon approach proposed by some of the authors for supplier ranking.a Abstract: The paper focuses Supply originally Chain Network Design (SCND) under uncertainty. We propose Abstract: The paper focuses on Supply Chain Network Design (SCND) under uncertainty. We propose a The novel method integrates cross-efficiency Envelopment (DEA)for andsupplier fuzzy setranking. theory SCND method extending an the approach originallyData proposed by someAnalysis of the authors SCND method extending an approach originally proposed by some of the authors for supplier ranking. to manage the SCND problem nondeterministic input and output(DEA) data. After ranking all the The novel method integrates theconsidering cross-efficiency Data Envelopment Analysis and fuzzy set theory The novel method integrates the cross-efficiency Data Envelopment Analysis (DEA) and fuzzy set theory actors belonging to each SCN stage, a linear integer programming model is stated and solved to manage the SCND problem considering nondeterministic input and output data. After rankingforalleach the to manage the SCND problem considering nondeterministic input and output data. After ranking all the pair ofbelonging subsequent maximize overall SCN efficiency, respecting the available actors to SC eachstages SCN to stage, a linearthe integer programming modelwhile is stated and solved for each actors belonging to each SCN stage, a linear integer programming model is stated and solved for each capacity at each node satisfying customers’ A case study is presented to show the pair of subsequent SCand stages to maximize the demand. overall SCN efficiency, while respecting the technique available pair of subsequent SC stages to maximize the overall SCN efficiency, while respecting the available effectiveness. capacity at each node and satisfying customers’ demand. A case study is presented to show the technique capacity at each node and satisfying customers’ demand. A case study is presented to show the technique effectiveness. Keywords: supply chain network design, uncertainty, data envelopment analysis, fuzzy © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd.cross-efficiency, All rights reserved. effectiveness. logic. Keywords: supply chain network design, uncertainty, data envelopment analysis, cross-efficiency, fuzzy Keywords: supply chain network design, uncertainty, data envelopment analysis, cross-efficiency, fuzzy logic. logic. enhanced discriminative power among suppliers with respect 1. INTRODUCTION to traditional DEA (Charnes al., 1978). In with fact, respect crossenhanced discriminative poweretamong suppliers enhanced discriminative power among suppliers with respect 1. INTRODUCTION DEADEA steps(Charnes from a self-evaluation Supply Chains (SCs) are a set of independent business efficiency to traditional et al., 1978).toIna comparative fact, cross1. INTRODUCTION to traditional DEA (Charnes et al., 1978). In fact, crossevaluation DEA of each SCfrom actor’s efficiency (Dotoli al., 2014). entities (e.g., suppliers, distributors and efficiency steps a self-evaluation to a et comparative Supply Chains (SCs) are manufacturers, a set of independent business DEA steps from a self-evaluation to a comparative Supply Chains (SCs) are a set of independent business efficiency As a result,ofthe original method allows discriminating retailers) (e.g., working together manufacturers, to acquire raw materials, convert each SC actor’s efficiency (Dotoli et al., among 2014). entities suppliers, distributors and evaluation of each SC actor’s efficiency (Dotoli et al., 2014). entities (e.g., suppliers, manufacturers, distributors and evaluation a set of suppliers while method trading-off between the precision of them intoworking final products, such raw goods to retailers and As a result, the original allows discriminating among retailers) togetherdeliver to acquire materials, convert a result, the original method allows discriminating among retailers) working together to acquire raw materials, convert As and the while uncertainty of thebetween problem.theInprecision the present supplyinto them to products, final customers (Costantino et retailers al., 2012a). set of suppliers trading-off of them final deliver such goods to and aoutcomes set of suppliers while trading-off between the precision of them into final products, deliver such goods to retailers and apaper, theand novel a recursive the Hence, optimal design typically refers to taking advantage the approach uncertaintyapplies of the in problem. In theway present supply them toSC final customers (Costantino et al., 2012a). outcomes and the uncertainty of the problem. In the present supply them to final customers (Costantino et al., 2012a). outcomes cross-efficiency DEA method the efficiency from aoptimal correctSCSC activity coordination with advantage the final paper, the novelfuzzy approach applies to inevaluate a recursive way the Hence, design typically refers to taking the novel approach applies in a recursive way the Hence, optimal SC design typically refers to taking advantage paper, of all the actors belonging to each stage of SCN. objective of satisfying customers’ needs. In with recenttheyears fuzzy DEA method to evaluate the the efficiency from a correct SC activity coordination finala cross-efficiency fuzzy DEA method to evaluate the efficiency from a correct SC activity coordination with the final cross-efficiency the belonging techniqueto solves a linear large number of criteriacustomers’ are adopted jointly with price of all the actors each stage of the integer SCN. objective of satisfying needs. In recent years ina Subsequently, all the actors belonging to each stage of the SCN. objective of satisfying customers’ needs. In recent years a of programming optimization of order select of SCcriteria stakeholders (Liang,jointly 2011). with Indeed, a SC the techniqueproblem solves for a each linear pair integer large to number are adopted price in Subsequently, the technique solves a linear integer large number of criteria are adopted jointly with price in Subsequently, subsequent SC optimization stages to choose amongfor the each actors pair of each not only to SC minimize the overall costs, but should be programming problem of order to has select stakeholders (Liang, 2011). Indeed,also a SC optimization problem for each pair of order to select SC stakeholders (Liang, 2011). Indeed, a SC programming stage the actual SC to stakeholders, withthe theactors objective of flexible environmental SC stages choose among of each not only and has responsive to minimizetothe overall costs,and but social shouldchanges also be subsequent SC stages to choose among the actors of each not only has to minimize the overall costs, but should also be subsequent maximizing the overall SCN efficiency while (Farahaniand et responsive al., 2014).toFurthermore, many of the changes change stage the actual SC stakeholders, with the respecting objective the of flexible environmental and social the actual SC stakeholders, with the objective of flexible and responsive to environmental and social changes stage constraints on capacity each nodewhile and respecting satisfying the drivers in etsupply chainFurthermore, management,many like ofglobalization, thethe overall SCNatefficiency (Farahani al., 2014). the change maximizing the overall SCN efficiency while respecting the (Farahani et al., 2014). Furthermore, many of the change maximizing required demand. The method allowsand SCND under outsourcing, IT diffusion, product complexity, have constraints on the capacity at each node satisfying the drivers in supply chain management, like globalization, on the capacity at each node and satisfying the drivers in supply chain management, like globalization, constraints uncertaintydemand. thanks to the of fuzzy numbers. exacerbated theITuncertainty the outcomes of thehave SC required Theusemethod allows SCND under outsourcing, diffusion,about product complexity, outsourcing, IT diffusion, product complexity, have required demand. The method allows SCND under (Pfohl et al., the 2010). Such uncertainty means the possibility of uncertainty thanks to the use of fuzzy numbers. exacerbated uncertainty about the outcomes of the SC 2. to LITERATURE REVIEW the use of fuzzy numbers. exacerbated the uncertainty about the outcomes of the SC uncertainty thanks undesired during the SCmeans operation, with the (Pfohl et al.,outcomes 2010). Such uncertainty the possibility of 2. LITERATURE REVIEW (Pfohl et al., 2010). Such uncertainty means the possibility of resulting risks (Costantino Consequently, 2. literature LITERATURE REVIEW undesired outcomes during ettheal.,SC2011a). operation, with the 2.1 Related SCND undesired outcomes during operation, the Supply Chain Network Design toolsConsequently, havewith to deal resulting risks (Costantino etthe(SCND) al.,SC2011a). Related SCND literature resulting risks (Costantino et al., 2011a). Consequently, 2.1 Related SCND with uncertainty in orderDesign to help (SCND) stakeholders Supply Chain Network toolstaking have optimal to deal 2.1 A complete reviewliterature of SCND is presented by Matinrad et al. Supply Chain Network Design (SCND) tools have to deal decisions even in order case to of help deviation from taking the ideal SC (2013). with uncertainty stakeholders optimal The authors show most methods for SCND A complete review SCNDthat is presented by Matinrad et al. with uncertainty in order to help stakeholders taking optimal A complete review of of SCND is presented by Matinrad et al. operating conditions. decisions even in case of deviation from the ideal SC (2013). The mixedauthors integershow programming, typically for aiming at that most methods SCND decisions even in case of deviation from the ideal SC employ (2013). The authors show that most methods for SCND operating conditions. minimizing costs (in about 75% of the literature, see Melo et employ mixed integer programming, typically aiming This paperconditions. presents an original method for the optimal SCND employ mixed integer programming, typically aiming at operating at al. (2009)) or maximizing profit (in about 16% of minimizing costs (in about 75% of the literature, see Melo et with paper uncertain data.an The approach extends a cross-efficiency This presents original method for the optimal SCND costs (in about 75% of the literature, see Melo et This paper presents an original method for the optimal SCND minimizing contributions, see Melo et al. (2009)). Only few contributions al. (2009)) or maximizing profit (in about 16% of fuzzyuncertain Data Envelopment proposed with data. The Analysis approach (DEA) extendstechnique a cross-efficiency (2009)) or maximizing profit (in about 16% of with uncertain data. The approach extends a cross-efficiency al. exist on other SCND objectives such as: minimizing transport see Melo et al. (2009)). Only few contributions by some the authors for suppliers ranking (Dotoli et al., contributions, fuzzy DataofEnvelopment Analysis (DEA) technique proposed see Melo et time, al. (2009)). Only few contributions fuzzy Data Envelopment Analysis (DEA) technique proposed contributions, time, total tardiness, lead maximizing the service level, exist on other SCND objectives such as: minimizing transport 2014). previous technique integrates fuzzy logicet and by someThe of the authors for suppliers ranking (Dotoli al., on other SCND objectives such as: minimizing transport by some of the authors for suppliers ranking (Dotoli et al., exist and in a single word, the SC efficiency. In fact, several time, total tardiness, lead time, maximizing the service level, cross efficiency DEA approach, is characterized an 2014). The previous technique which integrates fuzzy logicbyand total tardiness, lead time, maximizing the service level, 2014). The previous technique integrates fuzzy logic and time, methods have been developed only to face the problem of and in a single word, the SC efficiency. In fact, several cross efficiency DEA approach, which is characterized by an and in a single word, the SC efficiency. In fact, several cross efficiency DEA approach, which is characterized by an methods have been developed only to face the problem of methods have been developed only to face the problem of

Copyright © 2015, 2015 IFAC 666 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2015 responsibility IFAC 666Control. Peer review©under of International Federation of Automatic Copyright © 2015 IFAC 666 10.1016/j.ifacol.2015.06.153

INCOM 2015 Mariagrazia Dotoli et al. / IFAC-PapersOnLine 48-3 (2015) 634–639 May 11-13, 2015. Ottawa, Canada

maximizing the SC efficiency at a single stage level, evaluating the efficiencies of the stage candidates, typically at the suppliers’ level.

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distribution networks employing digraph modelling and fuzzy mixed integer linear programming. More complex fuzzy possibilistic models for SCND can be stated and solved using heuristics approaches. For instance, Xu et al. (2008) employ heuristics to achieve cost and service target under uncertain demand and objective function’s coefficients.

Some recent techniques for multi-stage SCND state that to evaluate a network’s performance, it is possible to subdivide all the SC into sub-stages, and evaluate every sub-stage’s efficiency in order to assure an optimal performance of the whole network (Troutt et al., 2004). Among multi-stage approaches for SC efficiency maximization, some recent methods employ the DEA technique in a SCN setting. In particular, Liang et al. (2006), use several DEA-based approaches to estimate the SC performance. In addition, Kao (2009a) builds a relational network DEA model to measure the efficiency of the system and those of its sub-processes. In another work by Kao (2009b) a model for efficiency measurement for parallel production systems is presented. In addition, Yang et al. (2011) propose a DEA-based performance evaluation approach for the overall SC under the control of a unique decision maker. Moreover, Amirteimoori et al. (2012) present a DEA-based production planning problem involving all the SC individual units, each contributing in part to the total production.

To the best of the authors’ knowledge, only few models simultaneously consider the overall efficiency maximization in the SCND problem and the treatment of uncertainty. In the context of SCND models integrating DEA and fuzzy logic to maximize the SC efficiency under uncertainty, we recall only the recent approaches by Lozano et al. (2014), proposing a network fuzzy DEA to handle fuzzy data characterizing the SCND, and by Kao (2014), who presents a network DEA technique for fuzzy observations proposing to measure the system and process efficiencies via two-level mathematical programming: the membership grade and the α-cut. Once the two-level problem is solved, the approach is transformed into a conventional one level program. 2.2 Contribution and positioning within the related literature We present a method to solve the SCND problem while maximizing the overall chain efficiency under uncertainty. In particular, the stage candidates’ selection and ranking problem is stated and solved in a cross-efficiency fuzzy DEA setting. Subsequently, the SCND is solved for each pair of subsequent SC stages with a fuzzy linear programming optimization with the aim of satisfying consumers’ estimated demand while respecting the constraints on the candidates’ capacities. Results are defuzzified using the distributions centre of area.

The recalled models for multi-stage SCND are all deterministic. Both Matinrad et al. (2013) and Melo et al. (2009) observe that surprisingly few literature contributions consider in the SCND the uncertainty, which is conversely a very important issue in real world. Roughly speaking, the uncertainty analysis in the SC can be addressed according to four main approaches (Mirzapour-Al-e-Hashem et al., 2011): 1) stochastic programming; 2) stochastic dynamic programming; 3) robust optimization; 4) fuzzy programming. Despite their wide use, stochastic tools cannot always face uncertainty in the SCND. In fact, probability distributions are derived from values that a variable assumed in the past, but does not indispensably have to occur in the future. When no historical observations are available the probabilistic approach is not appropriate. In such cases, uncertain parameters can be estimated considering the possibility that such parameters have for a given value, and fuzzy set theory (Zimmermann, 2001) is particularly suitable. In fact, fuzzy logic provides a natural framework to incorporate qualitative judgments with quantitative information. In the context of SCND methods using fuzzy programming, Torabi and Hassini (2008) propose a possibilistic SCND approach under uncertainty in demand, capacity and lead time with cost and total value of purchasing as objectives: constraints’ defuzzification is made by a weighting average of characteristic values of uncertain triangular parameter at a given alpha-cut. Another approach is proposed by Peidro et al. (2009): the objective is minimizing costs of a SCN under uncertain values of demand, supply and process and they use ranking functions in order to compare fuzzy numbers. A similar solution, but with a bi-objective optimization of total costs and total delivery lateness, is addressed by Pishvaee and Torabi (2010). Another method using fuzzy theory is presented by Costantino et al. (2012b): they employ fuzzy integer linear programming to design the optimally efficient and sustainable SCN. Moreover, Costantino et al. (2011b) present a similar approach for a strategic design of

With respect to the recalled approaches by Lozano et al. (2014) and by Kao (2014), the novel technique allows deploying the cross-efficiency DEA discriminative power between SC stakeholders, while being able to deal with data uncertainty thanks to the fuzzy approach. Uncertain data are estimated through a triangular fuzzy distribution and the fuzzy efficiency of each SC candidate is assessed in a crossefficiency evaluation setting. To discern among efficient actors in the same stage, coefficients resulting from maximizing the efficiency of each candidate are used to estimate the efficiency of all stakeholders and the cross efficiency of each actor is the mean value of its efficiency measures varying weights, i.e., stepping from a self to a comparative evaluation. 3. THE PROPOSED TECHNIQUE We characterize the generic SCN by n stages {S1,S2,…,Sn} connected by material flows: a generic couple of partners belonging to two subsequent stages can be connected by links representing the physical transportation of material between two partners. We assume that material only flows through different subsequent stages. We wish to determine the actual SCN from the whole set of candidates while maximizing the overall SC efficiency and taking into account uncertain data characterizing stakeholders. In particular, actors in each stage are ranked according to their efficiency - as calculated with the fuzzy-cross efficiency DEA approach in (Dotoli et al., 2014). Hence, we determine the SCN structure by 667


636

maximizing its overall efficiency in a stage-by-stage setting while providing the required demand and given the actors’ supply/production capacities.

START

Set k=1

Assume each stage Sk with k∈{1,…,n} includes mk stakeholders. Let us consider two consecutive k-th and (k+1)th stages, respectively formed by mk and mk+1 actors. Suppose that between the i-th actor in Sk with i∈{1,…, mk} and the j-th actor in the subsequent stage Sk+1 with j∈{1,…, mk+1} an amount of product Qji≤min{Ci; Cj} is traded, where Ci and Cj are the maximum supply/production capacities of stakeholders (i,j)∈Sk, due to production or warehouse limitations. Let us call Ri and Rj. the fuzzy cross efficiencies associated to the actors. The overall efficiency of each connection may be defined as Rji=Ri·Rj.

Choose input and output criteria

Import k-th stage’s input and output data

1ºSTEP

Calculate the efficiencies and rank the actors on k-th stage (cross fuzzy DEA)

Set k=k+1

k=n ?

We remark that we consider a single product. Moreover, we shall not consider the choice of transportation mode since we are focusing on a unique objective, i.e., satisfying consumers’ demand with the maximum overall efficiency. In addition, we assume that stakeholders’ capacities are known parameters.

NO

YES Import data on actors capacities and required demand

Set k=1

3.1 Step 1: Actors ranking at each SC stage 2ºSTEP

Figure 1 shows the proposed model flow chart, organized in two main steps: 1) actors ranking at each SC stage; 2) SCND design. The first step is based on the fuzzy cross-efficiency DEA problem in Dotoli et al. (2014). In this way we are able to select which is the most efficient candidate at each stage of the SCN. In particular, candidates are assessed and ranked against a set of conflicting objectives C = CI ∪ CO formed

Set k=k+1

NO

/ y gf = ( y gfp , y gfm , y gfo ) with g=1,..,G. These values refer to a pessimistic, a modal, and an optimistic estimate of each input/output data. The procedure aims at maximizing the efficiency of an actor, defined as the ratio between the weighted sum of outputs and the weighted sum of inputs. This is possible by determining some input/output coefficients vh with h=1,…,H / ug with g=1,…,G and imposing that the efficiencies of other suppliers do not overcome the unitary value (Dotoli et al., 2014):

(1)

g =1

subject to: G

g =1 H

∑v h =1

H

g ⋅ y gf ' − ∑ vh ⋅ xhf ' ≤ 0 with f ' = 1, 2,..., mk

h

The described procedure to rank the candidate actors in each stage has to be repeated for each stage for k=1,…,n-1. With regard to customers’ evaluation, we rank consumers according to their orders’ urgency. Hence, we associate each f-th consumer in Sn with f=1,…,mn with an efficiency E f = 1/ DT f , where DTf is the consumer’s required delivery time. In such a way the presented model, while taking into

(2)

h =1

⋅ xhf = 1

u g , vh ≥ 0 for g =1, 2, ..., G and h=1, 2, ..., H

Design the optimized SCN

The fuzzy efficiency E f of the generic f-th candidate in Sk is determined by using only one set of weights vh / ug. This set is actually selected in order to find a trade-off among different objectives regarding the shape of the efficiency possibility function: the maximization of the modal value, the minimization of the distance of the pessimistic value from the modal value and the maximization of the distance of the optimistic value from the modal one. Consequently, a Positive Ideal Solution or PIS (Negative Ideal Solution or NIS) is defined as the solution ideally satisfying (missing) all the three objectives. The resulting fuzzy efficiencies p m o E f = ( E f , E f , E f ) of each f-th candidate in Sk is defuzzified by using the Centre of Area (COA) of the distribution. The resulting crisp efficiencies E f allow ranking the stakeholders of the k-th SC stage. For the sake of brevity we do not report here the mathematical formulation. The reader is referred to Dotoli et al. (2014) for all details.

Consider a particular stage Sk. Each f-th candidate in Sk with f=1,…,mk is characterized by some uncertain input/output indicator defined by a triple xhf = ( xhfp , xhfm , xhfo ) with h=1,..,H

∑u

YES

Fig. 1. Flow chart representation of the proposed method.

while CO = {cH +1 , cH + 2 ,....., cH + G } is the output one.

G

k=n ?

END

by two subsets: CI = {c1 , c2 ,....., cH } is the input criteria set,

max E f = max ∑ u g ⋅ y gf with f = 1, 2,..., mk

Solve the optimization problem for k-th and (k+1)-th stages (fuzzy linear method)

(3) (4)

668


account each SCN stakeholder’s efficienccy, also assigns a priority in satisfying the required demand.

637

Table 1. Data of caandidate suppliers.

3.2 Step 2: Supply chain network design The second step of the proposed approach consists in designing the SCN in order to satisfy the cuustomers’ demand, given the candidate actors’ production capacities. c To this aim, a linear fuzzy programming method is i applied, in order to keep the uncertainty in the model until the final step. The objective of this step is to maximize the quantities that the c provide to the generic k-th stage’s most efficient actor can subsequent (k+1)-th stage’s most efficcient one, while satisfying the latter demand. In case of incapability to satisfy all the demand of the actor on stage (kk+1) by the most efficient stakeholder in the previous k-th sttage, the remaining quantity will be provided by the second most m efficient actor on stage k and so on until the fulfilment of the t demand.

Table 2. Efficiencies an nd ranking of suppliers.

Hence, working in pairs of stages, i..e., starting from consumer/retailer trade and working backw ward, the problem is formulated as follows: max ∑ i =k1 ∑ j =k +11 Q ji ⋅R ji ∀( k , k + 1) with k = 1,..., n − 1 m

m

4. THE CAS SE STUDY The proposed method is applied to a case study SC with n=5 stages, comprising m1=8 supppliers, m2=3 manufacturers, m3=2 distributors, m4=2 retaailers, m5=3 consumers. The optimization problem is implemented and solved in the MATLAB environment, using the t GLPK optimizer tool.

(5)

subject to:

∑ ∑

mk i =1

Q ji =C j ∀j = 1,..., mk +1

mk +1 j =1

Q ji ≤ Ci ∀i = 1,..., mk

(6)

The first step of the method is employed to firstly determine among the eight candidate suppliers s which is the most efficient one. Seven performannce indices are considered for the supplier evaluation, dividded into five inputs and two outputs (Table 1). The former are: the total costs for the supply expressed in euro [€]; the t energy required [MJ]; the CO2 emission [kgCE]; the ordeer fulfilment lead time [days]; the geographical distance betweeen the buyer and its suppliers [Km]. On the other hand, the output o indices are: the supplied components quality, i.e., thee percentage ratio between working units and total suppliedd units; the delivery reliability, defined as the percentage ratio between filled-on time orders and total orders. All indicatoors are triangular fuzzy sets, defined over a suitable range, except for the distance that is obviously crisp. Table 1 lists row r by row the input data and the output data of each supplierr, providing the triples of most pessimistic, most possible and most m optimistic estimates.

(7)

m, representing the where Qji is the variable of the problem quantity of products that an actor can provide to a stakeholder in next stage. Thus, (5) is the objectivve function to be maximized. Constraint (6) is related to cuustomer’s demand, which has to be fully satisfied. The equatioon explains that the sum of all the quantities arriving from the t previous stage actors must be equal to its demand. Connstraint (7) assures that, once completed the most efficient actoor’s capacity, if the demanding actor j has not yet fulfilledd its demand, the remaining quantity will be provided by the second most efficient in the ranking. Moreover, once first actor’s demand is fulfilled, and the following actor deemand has to be satisfied, if the most efficient provider acttor has finished its stock, automatically to the remaining quantity will be provided by the second most efficient one.

As shown in Table 1, the closeest supplier to the buying firm is supplier S4. Thanks to the low west transportation costs, S4 is able to offer a low price compaared to the others. However, its energy consumption is quite more m significant than that of other suppliers and it also exxhibits considerably high lead times (except than with respecct to S2). Referring to quality, suppliers S1 and S5 provide the highest values, while the best reliability is provided by candidates S5 and S8. It is also noticeable that supplier S7 em mits high CO2 quantities. By applying the first step of the prroposed procedure we estimate that the most efficient supplier is S4, which is favored by the geographical closeness to thee buying firm, followed by supplier S5 who is the next one closer to the buying firm, and then all the others. Table 2 repoorts the obtained results, which takes about 10 seconds on a PC with an Intel Core 2 Duo 3.40 GHz processor and 8.00 GB G of RAM.

As the optimization yet includes a fuzzy formulation, by reasoning with the same logic adopted in step 1, a PIS and NIS are defined as the solutions ideally satiisfying/missing the desired objectives: the maximization of thee median value, the minimization of the distance of the pessimiistic value from the median, and the maximization of the distance of the optimistic value from the median one. Thhe obtained results are defuzzified by using the Centre of Area A (COA) of the distribution. Thus the results of the optimizzation problem will be the quantities defined above, that will heelp us to design the optimal SC, which is the most efficient. Summing up, the model optimizes the SCN S configuration providing the exchange of the maxim mum quantities of products between the most efficient actorrs and considering urgencies in deliveries (by the consumers’ efficiency). e 669


638

Table 3. Data of candidate manufacturers.

458 CS4=900

442 322

CS5=1200

CM3=1500

1200

1500

228

QM3’=1500

29

1272

CS6=450

CD1=4800

128 CS2=400

CM1=1000

720

Table 4. Data of candidate distributors.

100

2150

CR2=3500 QR2’=2850

QD1’=2179

1350 CC2=2000

QM1’=100

650

CS8=900

CC1=1500

2821

2000

1951 600 CD2=4200

900 CS3=1000

750

2049

30 100

CS1=750

650

CM2=4000

QD2’=3421

CR1=3400 QR1’=2750

CC3=2100

QM2’=4000

CS7=650

Table 5. Data of candidate retailers.

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Suppliers

Manufacturers

Distributors

Retailers

Consumers

Fig. 2.The optimal SCN with traded quantities. Hence, the second step of the SCND method described in Section 3.2 is applied. The fuzzy efficiency values obtained in the previous step are the inputs for this new step, together with the consumers’ required demands (Table 9) and the candidates’ production capacities (Table 10). The resulting mathematical programming problem is solved in about 25 seconds.

Table 6. Efficiencies and ranking of manufacturers.

Table 7. Efficiencies and ranking of distributors.

Figure 2 depicts the optimal SCN highlighting the links activated between the nodes of the SC, the traded quantities (reported on the corresponding arrows), and the total quantities arriving at each node Qi’. By analyzing the resulting SCN in Fig. 2, we remark that supplier S2 is the only stakeholder not involved in the chain, although it is ranked as the fourth most efficient supplier. This is due to the fact that the other suppliers exhibit similar efficiencies (see Table 2) and a much bigger supply capacity, so that, when optimizing the whole SCN, it is preferable to consider all the other suppliers, except for the case when the total demand saturates the capacity of all other suppliers. All the other actors are involved in the SCND with the quantities required to ensure the maximum overall efficiency.

Table 8. Efficiencies and ranking of retailers.

Table 9. Demand and efficiencies of customers. Customer C1 C2 C3

Demand [nr] 1,500 2,000 2,100

DT [days] 120 110 130

Efficiency 0.008333 0.009091 0.007692

5. CONCLUSIONS AND FURTHER RESEARCH

Table 10. Supply/Production capacities of SC stakeholders.

We present a technique for Supply Chain Network Design (SCND) under uncertainty. The approach comprises two steps: the first one ranks all the actors of each SCN stage by recursively applying an integrated fuzzy cross-efficiency Data Envelopment Analysis that allows dealing with uncertain data; the second one solves a linear integer programming model for each pair of subsequent SCN stages to maximize the overall SCN efficiency, while respecting the available capacity at each node and satisfying the customers’ demand and the priority required in the delivery. A case study is presented to show the technique effectiveness. Future research will address modelling uncertainty in consumers’ demand and in stakeholders’ supply/production capacities, e.g., to represent the possibility that an actor simultaneously belongs to multiple SC. Another possible improvement will consider the cost for trading, so as to limit the exchange of small quantities where trade is not economically convenient.

The first step is repeated for all actors belonging to the same stage. Data characterizing the candidate manufacturers, distributors and retailers, can be found respectively on Table 3-5. The obtained results are reported in Tables 6-8. We omit the description of these results for the sake of brevity. Finally, Table 9 reports the customers’ delivery times and corresponding efficiency values, as well as the required demand.

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