European Journal of Operational Research 233 (2014) 159–170

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

A scenario-based stochastic model for supplier selection in globalcontext with multiple buyers, currency fluctuation uncertainties,and price discounts

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.08.020

⇑ Corresponding author. Address: ESC Rennes School of Business, 2 Rue Robertd’Arbrissel, 35000 Rennes, France. Tel.: +33 (0) 2 99 54 63 63; fax: +33 (0) 2 99 3308 24.

E-mail address: hammami.ramzi@gmail.com (R. Hammami).

Ramzi Hammami a,⇑, Cecilia Temponi b, Yannick Frein c

a ESC Rennes School of Business, Rennes, Franceb McCoy College of Business, Texas State University, San Marcos, TX 78666, USAc LABORATOIRE G-SCOP, Grenoble-INP/UJF, CNRS, Grenoble, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 July 2013Accepted 14 August 2013Available online 30 August 2013

Keywords:Supplier selectionCurrency fluctuation uncertaintyMultiple buyersPrice discountsGlobal purchasing

Suppliers network in the global context under price discounts and uncertain fluctuations of currencyexchange rates have become critical in today’s world economy. We study the problem of suppliers’ selec-tion in the presence of uncertain fluctuations of currency exchange rates and price discounts. We specif-ically consider a buyer with multiple sites sourcing a product from heterogeneous suppliers and addressboth the supplier selection and purchased quantity decision. Suppliers are located worldwide and pricingis offered in suppliers’ local currencies. Exchange rates from the local currencies of suppliers to the stan-dard currency of the buyer are subject to uncertain fluctuations overtime. In addition, suppliers offer dis-counts as a function of the total quantity bought by the different customer’ sites over the time horizonirrespective of the quantity purchased by each site.

We first provide a literature review on the overlapping items of suppliers’ selection and risk due to cur-rency. Then, we model the problem using the mixed integer scenario-based stochastic programmingmethod. The objective is to minimize the total system expected cost (purchased price + inventorycost + transportation cost + supplier management cost). Finally, we conduct numerical studies to showthe value of the proposed model and we discuss some relevant managerial insights into the theoryand practice of supply chain management research.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The selection of suppliers is a strategic process that determinesthe long viability of a company, especially when purchasing costsconstitute a significant portion of the operating costs. The suppli-ers’ selection plays a key role in achieving the objectives of aneffective supply chain (Ng, 2008). The suppliers’ selection is a com-plex decision making problem, which should include both quanti-tative, and qualitative criteria, as well as global factors to accounteffectively for suppliers’ performance (Amid, Ghodsypour, &O’Brien, 2009; Aissaoui, Haouari, & Hassini, 2007; Sawik, 2010;Xu & Nozick, 2009). Some researchers agreed that a combinationof factors should fit not only the technical requirements, but alsothe company’s strategy. Some of these factors might include: finan-cial metrics, consistency of the supplier, relationship, flexibility,technological capability, customer service, reliability and price

(Bode, Wagner, Petersen, & Ellram, 2011; Choi & Hartley, 1996;Qu, Huang, Chen, & Chen, 2009). The combination of factors tobetter select suppliers is not trivial, we refer the reader to Dickson(1966), Weber, Current, and Benton (1991), Bhutta and Huq (2002),Demirtas and Ustum (2008), Chen (2011) for more details aboutthe suppliers’ selection criteria. The decisions related to the suppli-ers’ selection problem in the relevant literature are, mainly, whichsuppliers to select and how much to order from each supplier ineach period and over the planning horizon.

With the globalization of industrial activities and the expansionof offshoring there has been a steady increase in the global pur-chasing. Most companies have several sites located worldwideand are concerned with purchasing raw materials and componentsfor these sites from a global network of suppliers. The sourcingeconomic development and the need for competitive advantagehave increased the trend of sourcing products across the globalmarketplace. The global purchasing environment has unintendedconsequences; it exposes organizations to considerable sourcingrisks and imposes new challenges that must be considered in thesupplier selection process (Aloini, Dulmin, & Mininno, 2011;Hopkins, 2010). Some researchers (Aloini et al., 2011; Christopher,

160 R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170

Mena, Khan, & Yurt, 2011; Sawik, 2010) outlined and classified thesignificant risks associated with global outsourcing.

One of these significant risks includes the currency exchangerates. Fluctuations in the exchange rates impact the sourcing froma global network of suppliers using different currencies. Differentand fluctuating currencies expose the buyer to the currency risk.When different currencies are involved, the buyer should copewith the uncertain fluctuations of the exchange rates. Recently,the executive vice president procurement of Airbus argued thattheir procurement strategy is highly influenced by the €/$ fluctua-tions. Airbus decided to reduce $ exposure by increasing $ spendvolume. It seems necessary that the suppliers’ selection modelsevolve to take into account the currency fluctuation uncertaintyin order to be in line with the new challenges faced by purchasingmanagers in global environments. The literature seems to be ratherscanty on analytical approaches that integrate the currency fluctu-ation uncertainty in the suppliers’ selection process; which is ourunderlying tenet in this research paper. We model the currencyfluctuations through a scenario-based approach. The motivationbehind this modeling approach will be presented in detail inSections 3 and 4.

If one considers the exchange rate fluctuations between thesupplier currency and the currency of reference then the purchas-ing price (expressed in the currency of reference) would depend onthe order placement time. In this case, the buyer may have interestin ordering large quantities when the exchange rate is attractive.Therefore, the orders amounts and placement times decisionsshould be considered. The orders are likely to impact the total pur-chasing cost and, consequently, the suppliers to be selected. Thisalso raises related inventory issues at the buyer’s site. The inven-tory level incurred at the buyer’s site and, consequently, the inven-tory cost are highly correlated to the amount and placement timeof orders. In our model, we propose that the inventory costs withcurrency risk be taken into account in the assessment of suppliers’selection decisions in the global context.

Often complicating the supplier selection problem for the buyeris the presence of price discounts offered by suppliers (Xia & Wu,2007). Crama, Pascual, and Torres (2004) indicated that traditionaldiscount models involve one of two types of discounts: quantitydiscounts – based on the quantity of each item ordered – or busi-ness volume discounts – based on the total dollar value of all itemsordered. In the suppliers’ selection literature that incorporates dis-counts, the discounts are commonly calculated based on the natureof business done with each buyer. The discounts are not calculatedbased on multiple buyers’ sites. It should be noted that in manyglobal companies there is a central purchasing department thatis concerned with the negotiations with suppliers and the provi-sions to the different subsidiaries. In our research, this is the com-monly found situation in the automotive sector; companies havegenerally many offshore sites that consume the same type of inputitems. Hence, an increasing number of suppliers are finding moremeaningful to offer discounts as a function of the total quantityof each item bought by the different customer’ sites over a giventime horizon, irrespective of the quantity purchased by each site.It seems to be a limited supplier selection models and methodsaddressing suppliers’ discounts from this perspective. Our researchwork capitalizes on this void in the literature and proposes a modelwhere suppliers are assumed to offer discounts based on the totalquantity supplied to the different buyer’s sites.

The global purchasing context, where buyer’ sites and suppliersmight be geographically dispersed, rises the impact of transporta-tion costs on the suppliers’ selection decision. Zelda and Zabinsky(2011) argue that the purchasing cost must include the costsassociated with the whole purchasing process and over thepurchased item’s entire life in addition to the purchasing price.Among these costs, transportation costs constitute a significant

bulk. Nevertheless, Aissaoui et al. (2007) indicate that very littleattention has been paid to transportation/logistics costs in theexisting supplier selection research. Our proposed model explicitlyconsiders the transportation costs in the global suppliers’ selectionprocess.

In this paper, we propose a mixed integer stochastic program-ming model for the supplier selection in the global context. Theuse of this mathematical programming/optimization approach isdriven by two main conditions. First, the model allows the flexibil-ity for selecting a variety of variables and facilitates the inclusion ofuncertainty into these variables. Second, it allows allocating ordersto each supplier with probabilistic conditions. Moreover, the se-lected technique takes into account the different constraints ofthe problem (Ding, Benyoucef, & Xie, 2003), which we have brieflyoutlined and will be discussed in detailed in Section 4. We considerthe typical case where a global company is concerned with pur-chasing a given product for its different sites from different poten-tial suppliers. We take into account the global supplier selectionaspects discussed above.

The remainder of this paper is organized as follows: Section 2presents a literature review on existing suppliers’ selection models,and on a variety of risks associated with the selection of suppliers,additional emphasis is provided on currency fluctuation risk anddiscounts. Section 3 is dedicated to the problem statement.Section 4 addresses the mathematical formulation for the problemunder study. In Section 5, the computational experiments andmanagerial insights are presented. Section 6 provides concludingremarks and future research directions.

2. Literature review

A key component in developing a reliable supply chain is theselection of suppliers. The sourcing of products from acrossthe world has become an increasing trend by companies aroundthe world looking for new sources of competitive advantage. Thesecorporate initiatives to have prompted unexpected and unintendedissues that have increased the complexity of supply chains. Someof these issues expose companies to different types of risks(Christopher & Lee, 2004) and traditional suppliers’ selectionmodels are evolving to incorporate such uncertainties. Our paperaddresses some of these uncertainties. We review the analyticalsuppliers’ selection literature from the perspective of assessinghow well existing models fit in the context of global suppliersselection. In particular, we identify how sourcing risks, pricediscounts and logistics costs have been accounted for in themathematical model-based suppliers selection approaches. Ourmain focus is on research published in the last decade.

2.1. Suppliers selection models

Many analytical techniques have been used to address thesuppliers’ selection problem. An innovative research presented byDe Boer, Labro, and Morlacchi (2001) reviewed the literature onsuppliers’ selection and classified it according to the supplier’sselection stage. De Boer et al.’s framework was revisited by Wuand Barnes (2011) in the context of agile supply chains. Their find-ings showed the need for the development of methods able toincorporate qualitative and quantitative goals. Wu’s and Barnes’classification clustered the analytical models by linear weighting,by mathematical programming, by fuzzy set theory and by analyt-ical hierarchical process. For more details on the different tech-niques and related references, please refer to Wu and Barnes(2011). Wadhwa and Ravindran (2007) classified the supplierselection to enhance outsourcing operations. These researchersbroadly clustered the related literature by the method used in

R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170 161

modeling the problem. The three clusters include: single or multi-objective mathematical programming methods, game theoreticmethods, and artificial intelligence applications to supplier selec-tion. For a detailed review of these three clusters, we kindly referthe reader to Wadhwa and Ravindran (2007).

2.2. Sourcing risks

Sourcing risks associated with supply chain management havebeen a topic of great interest by academicians and practitioners.Risk makes the supply chains vulnerable and consequently impactsthe organization’s performance. Global outsourcing has brought asignificant risk to organizations worldwide; many of these risksevolved due to the nature of worldwide transactions and opera-tions. Risk has been addressed from different perspectives.

Some researchers explored the qualitative aspects of risk anduncertainty and its associated mitigation strategies (Andreas &Wallenburg, 2012; Cavusgil & Deligonul, 2012; Choi & Linton,2011; Wilson, 2012). In the same line of research, Christopheret al. (2011) classified the different types of global sourcing risksas Supply risk, Environmental and sustainability risk, Process andcontrol risk, and Demand risk. Their work expanded on Christopherand Peck’s (2004) framework. As it is pertinent to our researchpaper, we are addressing a segment of the Environmental and sus-tainability risk. This includes fluctuations of interest rate, currencyfluctuations, quota restrictions, high levels of CO2, and so forth. Foran in-depth analysis of a variety of global risks, we refer the readerto Christopher et al.’s (2011) research.

Others researchers have addressed risk uncertainty or sourcingrisk through quantitative models. Xu and Nozick (2009) formu-lated a two-stage stochastic program to optimize suppliers selec-tion and hedge against the disruption caused by the loss ofproduction capability at the suppliers’ sites. Xu and Nozik consid-ered different disruption scenarios. One of the scenarios specifiedthe severity of the production capability lost at each suppliers’ sitein each time period. In this work, the researchers integrated thetransportation mode selection with the supplier selection. How-ever, Xu and Nozik’s model ignored price discounts and inventorycosts. Their model’s objective function was to minimize theexpected cost over the planning horizon.

Sawik (2011a) addressed the optimal selection of supply portfo-lio in a make-to-order environment in the presence of supply chaindisruption risks. Two types of disruption scenarios were consid-ered: scenarios with independent local disruptions of each supplierand scenarios with local and global disruptions that may result inall suppliers disruption simultaneously. Disruptions may arisefrom equipment break downs, local labor strike, bankruptcy, localnatural disasters, etc. The problem was formulated as a single-orbi-objective mixed integer program and considered a single buyer.Moreover, price discounts and logistics costs (transportation andinventory) were not included. The model minimized the total ex-pected cost, which includes the ordering cost, the purchasing price,the cost of defects and the cost of shortages. The minimization ofthe total expected worst-case cost is also considered in this work.Sawik (2011b) investigated the problem of multi-period supplierselection and order allocation in make-to-order environment whileconsidering supply chain disruption and delay risks. The mainselection criteria are: price, quality of purchased parts, and reliabil-ity of supplies. The author proposed a mixed-integer programmingapproach to incorporate risk that uses conditional value-at-risk viascenario analysis. Sawik (2013) addressed the issues of supplychain materials flows due to disruption risks. In his mixed integerprogramming approach he accounted for disruption risks such asrisk-neutral, risk-averse or mean-risk supply portfolio. Theseworks did not account for currency exchange risk.

Zelda and Zadinsky (2011) developed a scenario-based stochas-tic model and a chance-constrainted model for suppliers selection.Their model captured the risk associated with uncertain customerdemand and supplier capacity. Also, the model considered the pro-vision of multiple plants with multiple products. The total purchas-ing cost to minimize included purchasing price with businessvolume discounts, pipeline inventory costs and transportationcosts, coordination costs and penalty costs for exceeding thesupplier’s capacity. The problem did not account for the timedimension and inventory cost at the buyer’s sites.

Hammami, Frein, and Hadj-Alouane (2012) developed an opti-mization model for supplier selection. The focus of the modelwas on the low-cost suppliers’ issues. Such issues included theuncertainties of delivery lead times from suppliers’ sites to buyer’ssites, and the impact of the safety stock at the buyer’s sites. Theresearchers minimized the total cost. This included the purchasingprice, the transportation cost, the inventory cost and the suppliermanagement cost. The price discounts were ignored in their model.

Ravindran, Bilsel, Wadhwa, and Yang (2010) developed a riskadjusted supplier selection problem as a multicriteria optimiza-tion. The models accounted for value-at-risk type risk of disruptiondue to natural events and miss-to-target type of risk of quality. Themodels offered four different alternatives via the value path ap-proach, which seems a valuable tool to compare objectives amongsuppliers. Their model did not address the currency risk or dis-counts. Also Bilsel and Ravindran (2011) developed a supplierselection approach incorporating uncertainties due to demand forproducts, capacities at suppliers as well as transportation andother variable costs. The disruptions considered are due to exoge-nous events but did not include currency risk. Bilsel and Ravin-dran’s model provided proactive mitigation strategies for thetype of uncertainties addressed.

2.3. Discounts

The price discounts offered by suppliers are included in manysupplier selection models in the literature. Different types ofdiscounts are considered in the literature. Crama et al. (2004) pro-posed a procurement decision model where each supplier offersquantity discounts based on the total quantity of items purchasedby the company. This model considered local discounts to whicheach plant is entitled independently of the company’s consolidatedpurchases and group discounts to which each plant is entitled as aconsequence of the company’s cumulative purchases. The problemwas formulated for both single plant and multi-plant situations asa non-linear mixed 0–1 programming model. The objective func-tion minimized the total purchasing price.

Murthy, Soni, and Ghosh (2004) addressed the vendor selectionproblem for make-to-order items in the presence of volume-baseddiscounts for bundles. The researchers considered bundling dis-counts being offered by suppliers when suppliers can gain savingsin setup costs associated with producing a family of items. Theproblem is formulated for a single buyer as a mixed-integerprogramming model where the objective is to minimize the fixedand variable sourcing costs.

Xia and Wu (2007) developed a multi-criteria suppliers selec-tion model for a single buyer where suppliers offer price discountson total business volume for different products. The researchersproposed a multi-objective mixed integer programming modelwhere the price and the number of defective items were mini-mized while the qualitative weight of suppliers and the numberof items delivered on time were maximized.

A fuzzy multi-objective model to deal with the suppliers selec-tion problem for a single buyer was proposed by Amid et al. (2009).The problem was formulated in such a way as to simultaneouslyconsidered the imprecision of information and the order quantities

162 R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170

assigned to each supplier based on price breaks. Suppliers offeredprice breaks as a function of the order quantity. The problemincluded three objective functions: minimizing the net cost, mini-mizing the net rejected items, and minimizing the net latedeliveries.

Sawik (2010) developed single and multi-objective mixed-inte-ger programming models where the selection of suppliers is basedon price and quality of purchased parts and reliability of on timedelivery. Two types of discounts were considered: quantity andbusiness volume discounts. Sawik postulated the situation with asingle buyer.

One can note that these previous research have made greatcontributions to the supplier selection literature, but most of themignored sourcing uncertainties and did not address the transporta-tion and inventory issues.

2.4. Overall comments

Previous research shows limited supplier selection models con-sidering the impacts of currency fluctuation uncertainty on sourc-ing decisions. In previous research, sourcing risks often refer to theuncertainties on the buyer’s demand and supplier’s capability, toquality of materials flow or quality of products, to natural eventsas disruptive to the supply chain and other; please refer toChristopher et al. (2011) for a complete review of the types of risksand their categorization. A supply chain disruption as a Bernoullirandom variable was introduced in the model developed by Babich(2005). He also used the theory of financial options, which are be-yond the scope of this paper. In Babich’s (2005) model, a firm hasthe option to address disruptions by deferring ordering decisionsuntil uncertainty is unfolded and the supplier has the option to de-fer pricing decisions. This model provides interesting options tomitigate uncertainty; however, most industries do not have theluxury to adopt such approach. The product or finish good needsto be in a timely manner in the market to allow the firm to staycompetitive. Thus, it is very likely, that the currency fluctuationrisk addressed in our approach has not been fully explored yet.

Moreover, relatively few papers have developed effective mod-els for supplier selection problems addressing simultaneously thesourcing uncertainties and the price discounts offered by suppliers.One can also note that supplier selection methods under price dis-counts often ignore the inventory and transportation costs that arecorrelated to the sourcing decisions. As underlined by Aissaouiet al. (2007), very little attention has been paid to logistics costsin the existing suppliers selection research. Furthermore, previousresearch has focused on the single buyer situation, but it is unclearhow the developed models can be generalized to consider the sit-uation where the buyer has multiple sites.

In this research paper, we develop a supplier selection modelthat integrates the exchange rate fluctuation uncertainties withprice discounts while explicitly considering transportation andinventory costs in a context of multiple buyer’s sites. Our researchmakes contributions: First by addressing complexities in the sup-plier selection models that has not been presented in the literature.Second, by using a well-known methodology – mixed integer sto-chastic programming – to capture a new paradigm in the area ofsupply chain management. Third, our model present opportunitiesfor further research on the subject of supplier selection with theintegration of other disciplines such as social sciences, manage-ment and so forth.

3. Problem statement

We consider the typical global purchasing case where a globalcompany is concerned by purchasing a product for its different

sites from different potential suppliers over a given planning hori-zon. The set of potential suppliers could be obtained after a prese-lection process in which several qualitative criteria might beconsidered. The sets of buyers’ sites and potential suppliers arerespectively denoted by J and I.

The supplier selection decisions are typically undertaken overthe long-term with a time horizon ranging from one to three years.At the beginning of the planning horizon, the agreements must beestablished with suppliers. The purchasing managers should deter-mine the suppliers to be selected and the total amount to be allo-cated to each selected supplier over the planning horizon. Wedenote by Qi the total quantity to be sourced from supplier i overthe planning horizon. Based on the quantity Qi, the supplier offersprice discounts. The discount schedule of a given supplier i is char-acterized by Ni discount intervals and a set of threshold quantitiesAn

i . The quantity Ani denotes the lower limit on the total purchased

quantity from supplier i over the planning horizon to obtain a unitprice that corresponds to the discount interval n. If An

i 6 Qi < Anþ1i

then the unit price offered by supplier i on all purchased units andfor all buyer’s sites is Un

i . Clearly, the larger n is, the smaller Uni be-

comes. For each supplier i, the quantity A1i of the first discount

interval is equal to the null value.The currency of payment may be different from one supplier to

another. In order to assess the total purchasing cost, all prices mustbe expressed in a unique standard currency. As typically assumed,the currency of the parent company is taken as reference. The cur-rency exchange rates are usually subject to fluctuations over time;these fluctuations are very difficult to predict on the long term. Thedifficulty economists have had in finding an empirically successfulexchange rate theory is well documented (e.g., Taylor, 1995).Nonetheless, a company has to forecast what the exchange ratewould be in the future to make decisions. In order to predict cur-rency fluctuations on the long term, financial analysts typicallyelaborate different exchange rate scenarios on a quarterly basis.Some currency fluctuations scenarios over a time horizon rangingfrom one to three years are published by financial organisms andcan then be used in our modeling approach. Consequently, we for-mulate the problem under study as a scenario-based stochasticmodel to address currency fluctuations.

We divide the planning horizon into periods k (k 2 {1. . .K})according to the time periods of the available currency fluctuationforecasts. For instance, if the exchange rates scenarios are elabo-rated on a quarterly basis then a period k would designate a quar-ter. We let X denote the set of all possible scenarios defined by ascenario tree. A particular instance of exchange rates fluctuationsscenarios is referred to by x. The exchange rate from the currencyof supplier i to the standard currency in period k under scenarioxis denoted by akx

i . A probability of occurrence, Px, is associatedwith every scenario x.

Under currency fluctuations, the purchasing price (expressed inthe currency of reference) depends on the order placement period.For a given scenario, the buyer may have interest in ordering largequantities when the exchange rate is attractive in order to get asmaller price (expressed in the currency of reference). We let qkx

ij

represents the quantity ordered by site j from supplier i in periodk under scenario x. This also raises the inventory issue since itemswould be hold in stock when the available quantity in a given periodis larger then the demand in this period. We denote by skx

j the inven-tory level in site j at the beginning of period k under scenario x.

The first-stage decisions are basically the suppliers to be se-lected and the total quantity to be ordered from each supplier.The second-stage decisions are associated with each exchangerates scenario. They represent the quantity ordered by each sitein each time period and the inventory level in the buyer’s sites.These second-stage decisions depend on the first-stage decisionsand affect the suppliers’ evaluation at this stage.

R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170 163

The objective is to minimize the total expected purchasing costover the planning horizon while satisfying the different problemconstraints. The total cost is the sum of the supplier managementcost, the purchasing price, the transportation cost, and the inven-tory cost. The supplier management cost is incurred once the sup-plier is selected and refers to the cost of doing business with thesupplier. It basically represents the cost of integrating and develop-ing suppliers. When low-cost suppliers are involved, the manage-ment cost may be substantial as experienced by many globalcompanies. Indeed, a hard work may be required to get a low-costsupplier in line with the strategy of the company. Quantity dis-counts are considered in the purchasing price. The transportationcost depends on the location of the supplier’s site and the buyer’ssite. Finally, the inventory cost in a buyer’s site is calculated foreach time period based on the average inventory level in thisperiod.

4. Problem formulation

We formulate the supplier selection problem as a mixed integerscenario-based stochastic programming model. The main decisionsthat must be made immediately (first-stage decisions) are the sup-pliers to select and the total quantity to purchase from each sup-plier over the planning horizon. There are different scenarios forfuture exchange rates fluctuations over the planning horizon. Asso-ciated with each scenario are the second-stage variables: the quan-tities ordered in each period and the incurred inventory levels inbuyers’ sites. In the model, future decisions depend on the first-stage decisions made today and affect their evaluation. In theobjective function, we minimize the expected total cost (in a statis-tical sense) over all scenarios included in the model.

4.1. Notations

First-stage variables:

� Yi: binary integer variables, Yi = 1 if supplier i is selected,0otherwise.� Qi: total amount ordered from supplier i over the planning

horizon.� an

i : binary integer variables, ani ¼ 1 if the quantity purchased

from supplier i falls on the discount interval n of its discountschedule, 0 otherwise.

Second-stage variables:

� qkxij : quantity ordered by site j from supplier i in period k under

scenario x.� skx

j : inventory level in site j at the beginning of period k underscenario x.

Cost factors:

� Uni : unit purchasing price from supplier i associated with inter-

val n of supplier i’s discount. It is expressed in the currency ofthe supplier.� TCij: unit transportation cost from supplier i to buyer’s site j.� ICk

j : unit inventory holding cost in buyer’s site j over period k.� MCi: management cost associated with supplier i.

Parameters:

� Dkj : demand of buyer’s site j in period k.

� Ani : lower limit on the business volume of supplier i that corre-

sponds to the discount interval n ðA1i ¼ 08i 2 IÞ.

� Lki : Capacity of supplier i in period k.

� akxi : exchange rate from the currency of supplier i to the stan-

dard currency in period k under scenario x.� Px: probability associated with scenario x.

4.2. Objective function

The objective function (1) minimizes the total expected cost Zexpressed in the currency of reference. This cost is the sum ofthe supplier management cost, the purchasing price, the transpor-tation cost, and the inventory cost. Unlike the purchasing price, allthe cost factors are assumed to be directly expressed in the cur-rency of reference. Indeed, the impact of the currency risk associ-ated with these costs on the supplier selection decisions can beneglected if one considers the price risk.

The management cost MCi is charged once the supplier is se-lected (Yi = 1). The total management cost for all selected suppliersisP

i2IMCiYi. The unit purchasing price from supplier i depends onthe considered interval of supplier i’s discount. It is given byPn¼N

n¼1 ani Un

i . Under scenario x, the purchasing price associated with

the quantity qkxij in the currency of supplier i is

Pn¼Nn¼1 an

i Uni

� �qkx

ij . It

becomes akxi

Pn¼Nn¼1 an

i Uni

� �qkx

ij in the currency of reference. Hence,

the total expected purchasing price is obtained byPj2J

Pi2I

Pk¼Kk¼1

Px2XPxakx

i

Pn¼Nn¼1 an

i Uni

� �qkx

ij .

The transportation cost of the quantity qkxij from supplier i to

buyer’s site j in period k under scenario x is TCijqkxij . Therefore,

the total expected transportation cost isP

j2J

Pi2I

Pk¼Kk¼1

Px2X

PxTCijqkxij . The average inventory level in buyer’s site j in period k

under scenario x isskx

jþsðkþ1Þx

j

2 . The expected inventory cost for j, k,

x is then given by PxICkj

skxjþsðkþ1Þx

j

2 . We assume that for every buyer’ssite j, there is no inventory at the beginning and at the end of theplanning horizon. The expected inventory cost for all sites over the

planning horizon isP

j2J

Pk¼Kk¼1

Px2XPxICk

j skxj .

Min Z ¼Xi2I

MCiYi þXj2J

Xi2I

Xk¼K

k¼1

Xx2X

Pxakxi

Xn¼N

n¼1

ani Un

i

!qkx

ij

þXj2J

Xi2I

Xk¼K

k¼1

Xx2X

PxTCijqkxij þ

Xj2J

Xk¼K

k¼1

Xx2X

PxICkj skx

j ð1Þ

The objective function is not linear. In order to linearize theobjective function we define new non-negative variables xkxn

ij such

as xkxnij ¼ an

i qkxij . Then, we replace the objective function (1) by the

new linear function (2) where xkxnij is used instead of an

i qkxij . We add

constraints (3)–(5) in order to guarantee that xkxnij ¼ an

i qkxij for all

i, j, k, w, n. The parameter W designates a sufficiently big number.Indeed, if an

i ¼ 0 then xkxnij ¼ 0 according to constraints (4). If an

i ¼ 1then the combination of constraints (3) and (5) ensures thatxkxn

ij ¼ qkxij . Thus, xkxn

ij ¼ ani qkx

ij in all cases.

MinXi2I

MCiYi þXj2J

Xi2I

Xk¼K

k¼1

Xx2X

Pxakxi

Xn¼N

n¼1

Uni xkxn

ij

!

þXj2J

Xi2I

Xk¼K

k¼1

Xx2X

PxTCijqkxij þ

Xj2J

Xk¼K

k¼1

Xx2X

PxICkj skx

j ð2Þ

xkxnij 6 qkx

ij i 2 I; j 2 J; 1 6 k 6 K; x 2 X; 1 6 n 6 Ni ð3Þ

xkxnij 6 Wan

i i 2 I; j 2 J; 1 6 k 6 K; x 2 X; 1 6 n 6 Ni ð4Þ

164 R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170

xkxnij P qkx

ij þWðani � 1Þ i 2 I; j 2 J; 1 6 k 6 K; x 2 X; 1 6 n 6 Ni

ð5Þ

4.3. Constraints

The second-stage decisions may vary from one scenario to an-other but must be in line with the first-stage decisions. Accordingto constraints (6), the total quantity purchased from supplier i un-

der any given scenario xP

j2J

Pk¼Kk¼1 qkx

ij

� �must be equal to the total

quantity allocated to supplier i (Qi).If supplier i is selected (Yi = 1) then the quantity ordered by all

buyer’s sites from supplier i in period k under any given scenarioxðP

j2Jqkxij Þ must not be larger than supplier i’s capacity in this

same period ðLki Þ. Otherwise (i.e. supplier i is not selected and

Yi = 0), the quantity ordered from supplier i must take the null va-lue for every scenario x. This is guaranteed by constraints (7).

Q i ¼Xj2J

Xk¼K

k¼1

qkxij i 2 I; x 2 X ð6Þ

Xj2J

qkxij 6 Lk

i Yi i 2 I; 1 6 k 6 K; x 2 X ð7Þ

In order to obtain the price associated with the discount inter-val n from supplier i the total purchased quantity Qi must satisfyAn

i 6 Q i < Anþ1i . Recall that for the first discount interval (n = 1),

the parameter A1i ¼ 0; 8i 2 I. Only one discount interval n can be

selected for each supplier i. Since we minimize the purchasing costin the objective function, the model will try to get the most profit-able discount interval. Therefore, discount constraints can be for-mulated as follows:

ani An

i 6 Q i i 2 I; 1 6 n 6 Ni ð8ÞXn¼Ni

n¼1

ani ¼ 1 i 2 I ð9Þ

According to constraints (10), in buyer’s site j in period k (k < K)under scenario x, the sum of the stock at the beginning of period k

ðskxj Þ and the total quantity received in period k

Pi2Iq

kxij

� �is equal

to the sum of the stock at the beginning of period k + 1 sðkþ1Þxj

� �and the demand of period k (Dk

j ). The inventory levels at the begin-ning of the planning horizon are null as given in constraints ( 11).No inventories are kept at the end of the last period (k = K). In thislast period, the inventory constraints can be formulated as given inconstraints (12).

skxj þ

Xi2I

qkxij ¼ sðkþ1Þx

j þ Dkj j 2 J; 1 6 k 6 K � 1; x 2 X ð10Þ

s1xj ¼ 0 j 2 J; x 2 X ð11Þ

sKxj þ

Xi2I

qKxij ¼ DK

j j 2 J; x 2 X ð12Þ

Finally, we include the constraints on the domain of variables.

Yi 2 0;1f g i 2 I ð13Þan

i 2 0;1f g i 2 I; 1 6 n 6 Ni ð14Þ

Q i 2 IRþ i 2 I ð15Þ

qkxij 2 IRþ i 2 I; j 2 J; 1 6 k 6 K; x 2 X ð16Þ

skxj 2 IRþ j 2 J; 1 6 k 6 K; x 2 X ð17Þ

xkxnij 2 IRþ i 2 I; j 2 J; 1 6 k 6 K; x 2 X; 1 6 n 6 Ni ð18Þ

4.4. Case of purchasing contracts in buyer’s currency

In our model, the purchasing contracts are assumed to be ex-pressed in the currency imposed by the supplier which is generallythe currency of the supplier’s country. This is a valid assumption inmany situations especially for North American and European sup-pliers. However, in some cases, the purchasing contracts may beexpressed in the buyer’s currency. The model can be easily modi-fied to take this situation into account. Indeed, we replace the pur-chasing price Un

i expressed in the supplier currency by the newprice defined as follows:

� Unij: unit purchasing price from supplier i associated with inter-

val n of supplier i’s discount and expressed in the currency ofbuyer j.

Hence, the purchasing price of the quantity qkxij purchased by

site j from supplier i in period k under scenario w in the currency

of buyer j isPn¼N

n¼1 ani Un

ij

� �qkx

ij . It becomes akxj

Pn¼Nn¼1 an

i Unij

� �qkx

ij in

the currency of reference where akxj denotes the exchange rate

from the currency of buyer site j to the standard currency in periodk under scenario x. Consequently, when the purchasing constractwith given supplier i is expressed in the buyer’s site we must

replace in the objective function akxi

Pn¼Nn¼1 an

i Uni

� �qkx

ij by

akxj

Pn¼Nn¼1 an

i Unij

� �qkx

ij .

In this case, the suppliers profits are subject to currency fluctu-ation uncertainty and suppliers may see their profits going down.We can also consider alternative situations where the currency riskis shared between the buyer and the supplier. For instance, we canextend the model to consider the case where the purchasing pricechanges if the currency fluctuation exceeds a given level (purchas-ing price increases if the fluctuation is profitable to the buyer andvice versa). This aspect can be integrated in our model but willcomplicate the solving approach as it will introduce other non-lin-earities in the objective function.

5. Experiments

In order to conduct experiments, we solved the deterministicmixed integer programming equivalent of the proposed stochasticprogram by using the commercial optimization software Cplex.This is an efficient solving approach given the reasonable size ofthe obtained mixed-integer program. Our experiments are per-formed based on a realistic case study that we describe in the nextsection. The experiments are guided by two main goals:

� Evaluate the cost saving obtained by applying the proposedmodel to the suppliers’ selection problem,� Derive managerial insights that may be interesting for both aca-

demicians and practitioners.

5.1. Case study description

The computational experiments conducted with the proposedmathematical model are inspired from the case study presentedin Xu and Nozick (2009). The structure of this case study and theparameters used are based on a realistic problem faced by a USautomotive manufacturer. It concerns the provisioning of twoassembly plants with a single common part. One plant is locatedin Detroit, Michigan, and the second is located in Russelsheim,Germany. The demand at each plant is characterized by an annualvolume of 1,084,500 and 723,000 units in Detroit and Russelsheim,respectively. The planning horizon is assumed to be one year and

Table 2Unit transportation costs (USD).

Detroit Russelsheim

Cleveland 0.180 3.344Tokyo 4.400 7.388Shanghai 4.930 6.974Madrid 3.316 1.312

Table 3Discount schedule for the different suppliers.

n Ani Quantity Discount (%)

1 0 0 to under 300,000 02 300,000 300,000 to under 500,000 13 500,000 500,000 and over 3

Table 4Exchange rates baseline forecasts.

Quarter 1 Quarter 2 Quarter 3 Quarter 4

USD/EUR 0.752 0.763 0.781 0.787USD/JPY 75 76 73 71USD/CNY 6.30 6.30 6.20 6.10

R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170 165

each period k is assumed to be a quarter. The demand of the prod-uct is assumed to be evenly distributed over the periods. Therefore,the quarterly volume of demand is 271,125 and 180,750 in Detroitand Russelsheim, respectively. The US dollar is taken as thecurrency of reference.

The potential suppliers are listed in Table 1. Since there is onlyone supplier in each city we reference the supplier by thecityname. Table 1 gives the unit price offered by each supplier inits local currency and in USD (the considered exchange rates aregiven in Table 4). We also give the supplier management cost(in USD) that represents, in this case study, the cost to engage witha supplier. Shanghai is the lowest cost supplier but with the high-est management cost. The maximum amount that can be obtainedfrom any supplier in a quarter is also given in Table 1. Note that thelow-cost supplier Shanghai has the lowest capacity.

Parts are shipped under a multi-modal transportation formatthat might include truck, rail, or ship. For instance, to supplyDetroit from Shanghai, we use ships from Shanghai to Los Angelesand then trains from Los Angeles to Detroit. From Shanghai toRusselsheim, we use ships to enter Europe via Rotterdam and thentrucks from Rotterdam to Russelsheim. The unit transportationcosts from suppliers’ sites to buyers’ sites are given in Table 2. Theywere calculated based on the distance travelled by each transpor-tation mode and based on standard costs. The per-unit quarterlyholding cost is assumed to be $2.56 and $2.78 in Detroit andRusselsheim, respectively.

We assume that suppliers offer quantity discount schedule withthree intervals as shown in Table 3. These discount rates are usedto calculate the unit purchasing prices Un

i . For example, the unitprice offered by the supplier in Cleveland becomes $21.34 if thetotal ordered quantity over the planning horizon is larger than500,000 units.

The baseline forecasts of currency exchange rates are takenfrom the financial market forecasts that have been published bythe Royal Bank of Canada (2012). These forecasts are given inTable 4. They predict the exchange rate fluctuations of EUR, JPY,and CNY compared to USD on a quarterly basis over one year.The baseline forecasts will be used to build the different scenariosof exchange rates fluctuations as will be detailed later. Note that ascenario in the model is obtained by specifying the exchange ratesof EUR, JPY, and CNY in each quarter over the planning horizon.

5.2. Relevance of taking the exchange rate into account

The purpose here is to show the relevance of taking theexchange rates into account when undertaking supplier selectiondecisions in the global context. It is important to note that we donot deal here with the fluctuations of exchange rates or the uncer-tainties of these fluctuations. Indeed, we only evaluate the impactof considering constant and known exchange rates on the supplierselection decisions.

For every currency, we consider a unique constant value ofexchange rate over the planning horizon (fluctuations anduncertainties are ignored). Consequently, there is only one scenarioof currency exchange rates in the model. For the CNY and JPYcurrencies, we take the average value of the exchange rates

Table 1Suppliers data.

Supplier Unit baseprice

Unit base pricein USD

Managementcost (USD)

Quarterlycapacity

Cleveland 22 USD 22 37,500 247,600Tokyo 1575 JPY 21 37,500 247,600Shanghai 107.1 CNY 17 150,000 139,200Madrid 18.9 EUR 25 37,500 247,600

presented on Table 4. Hence, for instance, the exchange rateUSD/CNY is assumed to be 6.225 in each quarter. Then we solvethe model for different values of the exchange rate $/€. Figs. 1and 2, present the total cost and the quantity allocated to theEuropean supplier (Madrid) as a function of the exchange rate $/€.

One can first note that the purchasing cost and the model solu-tion are very sensitive to the value of the exchange rate. The in-crease of the exchange rate $/€ leads to significantly increasingthe total quantity Q purchased from the European supplier in Ma-drid. For instance, if the exchange rate goes up from 0.75 to 0.8then the quantity purchased from Madrid increases by 92%. If theexchange rate $/€ is smaller than 0.65 then the supplier in Madridis not selected by the model. Our results validate the relevance ofincluding currency exchange rates in the evaluation of supplierselection decisions.

5.3. Impact of over-time fluctuating exchange rates

In real-world situations, the currency exchange rates are notconstant as assumed in the previous experiment but fluctuate overtime. Here, we let the exchange rates fluctuate over time but westill consider only one scenario of exchange rates (uncertainty isnot considered). For the CNY and JPY currencies, we take the ex-change rates given in Table 4. We generate different forecasts of$/€ exchange rates over the planning horizon using the followingmethod. Indeed, we first specify a rate of fluctuation; for instance,5%. Then, we randomly generate exchange rates while ensuringthat, in each quarter, the $/€ exchange rate has the following lowerand upper limits: 0.771⁄(1 � 5%) and 0.771⁄(1 + 5%). Note that0.771 is the average value of $/€ exchange rates over theplanning horizon for the baseline forecast given in Table 4 (Indeed,0.771 = (0.752 + 0.763 + 0.781 + 0.787)⁄1/4). We also ensure that,for each generated forecast, the average value of $/€ exchange ratesover the planning horizon is equal to 0.771 as for the baselineforecast.

Twenty forecasts of $/€ exchange rates over the planning hori-zon are randomly generated according to the above conditionsfor each rate of fluctuation. Thus, we obtain twenty instances ofthe model for each fluctuation rate. We solve the differentinstances of the model and obtain the optimal costs and decisions.

Fig. 1. Total cost vs. the exchange rate $/€.

Fig. 2. Quantity from Madrid vs. the exchange rate $/€.

Fig. 3. Total cost vs. rate of fluctuation $/€.

Fig. 4. Quantity from Madrid vs. rate of fluctuation $/€.

166 R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170

We compute the average cost and quantity allocated to supplierMadrid for each fluctuation rate. Figs. 3 and 4 show the average to-tal cost and the average quantity purchased from Madrid as a func-tion of the fluctuation rate.

These experiments show that a company can capitalize on thecurrency fluctuation if this fluctuation can be predicted andknown. According to Fig. 3, the higher the percentage of fluctuationis, the smaller the total purchasing cost becomes. The model deter-mines the placement times of orders that allows taking profit dur-ing the periods when the $/€ exchange rate increases. Results fromFig. 3 explain that higher exchange rates fluctuations generatelarger opportunities for optimizing the orders placement times.From Fig. 4, it is also observed that the increasing rate of fluctua-tion entices an increase in the quantity of products purchased fromMadrid. According to our model, to capitalize on the periods whenthe value of the exchange rate $/€ increases, companies shouldpurchase more products from Madrid and enhance their gains.

Therefore, the real problem faced by purchasing managers isnot the fluctuation of currency exchange rates but rather theuncertainty of this fluctuation. Uncertainty will be addressed inthe next section.

5.4. Value of the stochastic programming

The proposed model adopts a scenario-based stochastic pro-gramming approach to model the uncertainties of fluctuating ex-change rates in the suppliers selection problem. It is important toevaluate the relevance of such an approach. A simpler approach

would be to consider the expected value problem, where the deci-sion maker replaces all random variables by their expected valuesand solves a deterministic model. We can distinguish between twocases: (1) the case where the expected values of exchange rates aretime-dependent (fluctuation is considered) and, (2) the case wherethe expected values are constant (fluctuation is not considered).

This first deterministic model is called deterministic modelwith fluctuations (DMF) since exchange rates are time-dependentbut uncertainty is ignored. The second deterministic model iscalled deterministic model with constant exchange rates (DMC).We evaluate hereafter the value of the stochastic solution com-pared to the deterministic solutions in the above cases. It is impor-tant to note that most existing supplier selection models in theliterature implicitly assume that exchange rates are constant(DMC) or fluctuating but known (DMF).

5.4.1. Value of the stochastic solutionIn order to measure the importance of considering the stochas-

tic version of a model, the Value of the Stochastic Solution (VSS)has always been used in the literature. It measures the expectedgain from solving a stochastic model rather than its deterministiccounterpart (where all random variables are replaced by theirmeans) (Birge, 1982).

In our case, VSS = Z(optimal solution of the counterpart deter-ministic model) � Z(optimal stochastic solution). It measures theexpected decrease in cost from solving the stochastic version ofthe suppliers selection model rather than the simpler deterministicone. Recall that Z is the objective function of the proposedstochastic model.

R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170 167

In order to conduct our experiments, we consider 3 forecasts ofexchange rates fluctuations from each foreign currency to the stan-dard currency over the planning horizon: the first one correspondsto the baseline forecast (Table 4) and the others are randomly gen-erated with a maximum fluctuation rate of 20% (compared to theexchange rates of the baseline forecast). Since we have 3 foreigncurrencies, the total number of scenarios in the stochastic modelis 33 = 27.

As stated earlier, two deterministic models can correspond toour stochastic model: DMF and DMC. For DMF, there is only onescenario where the exchange rate from a given currency to thestandard currency in a given quarter is the average value of the ex-change rates of the different forecasts of this currency in this quar-ter weighted by the probability of each forecast. For instance, if theforecasts of exchange rates from EUR to USD are those given in thefirst three rows of Table 5, then the exchange rates USD/EUR forDMF will be those of row 4. For instance, in the first quarter,(0.752⁄0.4) + (0.793⁄0.3) + (0.700⁄0.3) = 0.749.

For DMC, the exchange rate from a given currency to the stan-dard currency is constant. It is computed as the average value ofthe exchange rates of DMF. In the example of Table 5, the exchangerate USD/EUR for DMC is 0.759 in each quarter. Indeed,(0.749 + 0.799 + 0.762 + 0.724)/4 = 0.759.

We consider different classes of instances according to theprobabilities associated with the different forecasts. For instance,for the first class of instances, the baseline forecast of the exchangerates from a given foreign currency to the standard currency has aprobability of 0.5 and each randomly generated forecast has aprobability of 0.25. The different scenarios of the stochastic model(27 scenarios) are obtained by making all the possible combina-tions of the forecasts of the different currencies. The probabilityof each resulting scenario is calculated accordingly. Twenty in-stances are generated in each class. For each instance, we solvethe stochastic model and the two deterministic models DMF andDMC. We then compute the VSS with regards to DMF and DMC. Gi-ven the space restrictions, we show in Table 6 the results for onlythree instances from each class.

We thought the uncertainty of exchange rate fluctuations isimportant for the supplier selection problem. Observing Table 6,we conclude that, in fact, it is. Indeed, the gains resulting fromthe use of stochastic solutions instead of the deterministic solution,either with or without fluctuations, can be very large. For instance,the gain with regards to DMC is above $1,000,000 in many cases.We tested 80 instances (20 instances ⁄ 4 classes). For all these in-stances (not only those shown in Table 6), the average gains were182,566 and 811,802 with regards to DMF and DMC, respectively.This clearly proves the importance of capturing the uncertainties ofcurrency exchange rates fluctuations in the supplier selectionproblem.

As may be expected, the gain with comparison to DMC is largerthan the gain compared to DMF. Many suppliers selection modelsin the literature consider the exchange rates to be constant. Thesemodels are clearly not suited for the global context and can lead toundertake inadequate decisions and to significantly increase thepurchasing cost. The use a deterministic suppliers selection modelwhere the fluctuations of exchange rates are considered but not

Table 5Example of calculation of deterministic scenarios.

Q1 Q2 Q3 Q4 Prob.

$/€ forecast 1 0.752 0.763 0.781 0.787 0.4$/€ forecast 2 0.793 0.887 0.867 0.693 0.3$/€ forecast 3 0.700 0.760 0.631 0.671 0.3$/€ forecast for DMF 0.749 0.799 0.762 0.724 –$/€ forecast DMC 0.759 0.759 0.759 0.759 –

the uncertainty (like the DMF) is a trade-off between capturingthe currency risk and considering a simple optimization model.

5.4.2. Stochastic and deterministic solutionsIt is also interesting to compare between the proposed stochas-

tic model, DMF, and DMC in terms of purchasing decisions. Fromthe instances of Table 6, we selected one representative instancefor each class. The instances 1, 4, 7 and 11 of Table 6 were selected.In the following figures, we present the optimal solution obtainedwith each model and for each instance.

As shown in the figures below, the purchasing decisions are dif-ferent from one model to another. Regarding instances, 1, 4, and11, the same suppliers are selected by the different models. How-ever, the quantity purchased from each supplier (Qi) is differentfrom one case to another. In these cases, the solution of DMF is clo-ser to the stochastic solution than the DMC solution. This is in linewith the results of the previous section. Regarding instance 7, it isimportant to note that the stochastic model selects 3 suppliers(Cleveland, Shanghai, and Madrid) while DMF selects four suppli-ers. Hence, ignoring the uncertainty of exchange rate fluctuationsin supplier selection models may lead to select a supplier thatshould not be selected if the currency uncertainty is considered.This proves once again the interest of our proposed model (seeFigs. 5–8).

5.5. Impact of discounts

Unlike many works in the literature, we calculate the discountsas a function of the total quantity bought by the different cus-tomer’ sites over the time horizon, irrespective of the quantity pur-chased by each site. In practice, we observed that many multi-sitescompanies are still negotiating locally with their suppliers. It isthen interesting to evaluate the gain that these companies canachieve if they negotiate globally (for the requirements of the dif-ferent sites) with suppliers that offer quantity discounts.

To conduct our experiments, we consider only one exchangerate fluctuation forecast for each currency (baseline forecast). Wethen have only one scenario. We compare the cost obtained bythe proposed model (multi-site discount) to the cost given by amodified version of the model where discounts are calculatedbased on the quantity purchased by each site (single site discount).In this modified version, the decision variables an

i and Qi have beenreplaced by an

ij and Qij, respectively. The objective function and con-straints have also been modified accordingly.

� anij: binary integer variables, an

ij ¼ 1 if the quantity purchasedfrom supplier i by site j falls on the discount interval n of thediscount schedule of supplier i,0 otherwise.� Qij: total amount ordered from supplier i by site j over the plan-

ning horizon.

We report in Table 7 the percentage of gain resulting from amulti-site discount compared to a single site discount for differentvalues of the discount parameter An

i (we multiplied the value of Ani

given in Table 3 by 0.5, 1, 1.5, 2, and 3). For the five tested in-stances, the average gain is 0.73 %. This constitutes a significantgain for global companies for which the annual purchasing costis large such as in the automotive sector.

5.6. Impact of transportation costs

Transportation costs are very important in the global purchas-ing context. In this section, we focus on the impact of the transpor-tation cost on the model solution. We consider only one currencyexchange rate scenario (baseline forecasts). We vary the transpor-tation costs by multiplying the transportation costs of our case

Table 6Value of stochastic solution.

Class of instances Optimal cost with stochastic model VSS regarding DMF VSS regarding DMC

0.5-0.25-0.25 40926100 286000 78850041319300 20000 115450041564600 41500 41500

0.4-0.3-0.3 40789100 48500 45350041308500 0 71900040878200 0 996000

0.3-0.35-0.35 40990100 310500 153400040495300 57000 62700041122300 314000 1729500

0.2-0.4-0.4 40978700 391000 103550040745600 66500 150400041758600 0 654500

Fig. 5. Model solution (1): purchased quantities.

Fig. 6. Model solution (4): purchased quantities.

Fig. 7. Model solution (7): purchased quantities.

Fig. 8. Model solution (11): purchased quantities.

Table 7Gains with multi-sites discounts.

Ani �0.5 �1 �1.5 �2 �3

Gain (%) 0 0.68 1.91 0.23 0.84

168 R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170

study by 0.5, 1, 1.5, etc. Fig. 9 shows how the purchasing decisionsvary with the increasing values of transportation costs.

The model solution is still unchanged when increasing the valueof the transportation cost from 0.5⁄baseline value to 3⁄baseline va-lue. Then, for transportation costs 4 times larger than the baselinevalues the quantity purchased from Shanghai decreases while thequantity allocated to Cleveland goes up. The quantity purchasedfrom Shanghai takes the null value if the transportation costs aremultiplied by 5. Thus, only a significant increase of transportationcosts can lead the model to opt for local suppliers instead of distantlow-cost suppliers. Clearly, this result cannot be generalized sinceit depends on the values of the other costs.

5.7. Computational complexity

The difficulty with the stochastic programming model is its sizeand the corresponding computational difficulty required to obtain

Fig. 9. Model solution vs. transportation cost.

Table 8Computational complexity.

No.scenarios

No.variables

No.constraints

Average computational time(s)

2 293 652 53 429 970 64 565 1288 128 1109 2560 229 1245 2878 2716 2197 5104 5727 3693 8602 12681 11037 25774 1008

R. Hammami et al. / European Journal of Operational Research 233 (2014) 159–170 169

a solution. In order to evaluate the solvability of the proposed mod-el with the commercial optimization software Cplex, we tested dif-ferent classes of instances while varying the number of scenariosand their corresponding probabilities. For each class of instances,we randomly generated 20 scenarios. Experiments are conductedon an Intel Core duo 2.10 GHz computer with 2.00 Go memory.We used Cplex 11.0 coupled with C++. We give in the following ta-ble the average computational time (in seconds) required to obtainthe optimal solution for each class of instances.

The results of Table 8 show that it is possible to solve the modelwith Cplex in an acceptable amount of time even for relativelylarge instances for such a kind of problem. For instance, we wereable to solve the model for our realistic case study in less than17 mn with up to 81scenarios. However, as expected, the largerthe size of the instance is the larger the computational time be-comes. The use of a commercial optimization software is likelynot a viable option for solving very large instances of our model.In this case, a specially tailored method needs to be developed. Thisis beyond the scope of the present paper.

6. Conclusion

We developed a mixed integer scenario-based stochastic pro-gramming model for the supplier selection problem in the globalcontext. The buyer has multiple sites and sources a product froman international network of heterogenous suppliers. Suppliers offerprices in their local currencies. Exchange rates from the local cur-rencies of suppliers to the standard currency of the buyer are sub-ject to uncertain fluctuations over time. In addition, suppliers offerdiscounts as a function of the total quantity bought by the differentcustomer’ sites over the time horizon irrespective of the quantitypurchased by each site. The first-stage model decisions are whichsuppliers to select and which total quantity to order form eachsupplier over the planning horizon. The objective is to minimizethe total expected cost which is given by the sum of the purchasing

price, the inventory cost, the transportation cost, and the suppliermanagement cost. In the archive literature, the suppliers selectionproblem has not been approached from our perspective.

We conducted a number of computational experiments on arealistic and documented case study. We experimentally sup-ported the inclusion of exchange rates in the suppliers selectionproblem and showed that our model solution is sensitive to the va-lue of exchange rates. We then evaluated the impact of exchangerate fluctuations on the system cost and model decisions. Weshowed how the customer can capitalize on theses currencies fluc-tuations to decrease the total purchasing cost. Therefore, the realproblem faced by purchasing managers is the uncertainty of cur-rencies fluctuations and not the fluctuations themselves.

Our proposed model can assist managers to undertake purchas-ing decisions under currency fluctuations uncertainties. We thenevaluated the value of the stochastic solution obtained by our sce-nario-based stochastic programming model compared to its deter-ministic counterpart when (1) there are fluctuations of exchangerates but uncertainty is ignored and (2) exchange rates are con-stant and uncertainty is ignored. Our results indicated that the va-lue of the stochastic solution can be very large (over $ 1,000,000 insome cases) especially in comparison to the second deterministicmodel (DMC). The relevance of our model is supported.

In addition, we showed that ignoring the uncertainty of ex-change rate fluctuations in supplier selection models may lead toselect a supplier that should not be selected if the currency uncer-tainty is considered. Our model validated the proposition regardingthe necessity of suppliers selection model that accommodate theuncertainty of exchange rate fluctuations in the global context.

We also evaluated the gain that a global company can achieve ifit negotiates globally (for the requirements of the different sites ofthe company) with suppliers that offer quantity discounts ratherthan locally (each sites negotiates alone). We finally tested the im-pact of the increase of transportation costs on the suppliers selec-tion decisions. We found out that only a large increase of thesecosts can lead to replace the low-cost supplier by a local supplier.

In our future research, it would be important to developefficient methods to solve the large size instances of the proposedmodel for which the computational times with commercialsoftwares may be very large. It would be interesting to work onan efficient heuristic approach that explores and uses the specificcharacteristics of the model.

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