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Int. J. Production Economics 171 (2016) 8–21

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Int. J. Production Economics

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journal homepage: www.elsevier.com/locate/ijpe

Mitigating supply chain disruptions through the assessmentof trade-offs among risks, costs and investments in capabilities

S. Vahid Nooraie a,1, Mahour Mellat Parast b,n

a Department of Industrial and Systems Engineering, North Carolina A&T State University, 1601 E Market Street, Greensboro, NC 27411, United Statesb Technology Management North Carolina A&T State University, 1601 E Market Street, Greensboro, NC 27411, United States

a r t i c l e i n f o

Article history:Received 26 October 2014Accepted 20 October 2015Available online 29 October 2015

Keywords:Supply chain risk managementDisruptionsSupply riskDemand riskHeuristic

x.doi.org/10.1016/j.ijpe.2015.10.01873/& 2015 Elsevier B.V. All rights reserved.

esponding author. Tel.: þ336 285 3111.ail addresses: [email protected] (S. [email protected] (M.M. Parast).l.: þ336 285 3723.

a b s t r a c t

One of the central questions in supply chain design is how to properly invest in supply chain capabilitiesin order to be more responsive to supply chain disruptions. This new perspective in supply chain designrequires an understanding of the relationships among costs, supply chain risk drivers, and investments insupply chain capabilities. In this paper, we develop a multi-objective stochastic model for supply chaindesign under uncertainty and time-dependency. Sources of risk are modeled as a set of scenarios, and therisk of the system is determined. The objective is to examine the trade-offs among investments inimproving supply chain capabilities and reducing supply chain risks, and to minimize cost of supplychain disruptions. Due to the NP-hard nature of the problem, a heuristic algorithm based on a relaxationmethod is designed to determine an optimal or near-optimal solution. To examine the efficiency of theheuristic algorithm, a numerical example is provided. Our findings suggest that increasing supply chaincapabilities can be viewed as a mitigation strategy that enables a firm to reduce the total expected cost ofa supply chain subject to disruptions.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

The design of a supply chain that can be efficient whileresponsive to disruptions is a significantly complex and challen-ging task (Christopher and Peck, 2004; Ponomarov and Holcomb,2009; Pettit et al., 2010). Supply chain managers are striving toachieve the goal of fully integrated supply chains that are efficientand competitive, yet responsive to risks and disruptions. This is adaunting task due to the inherent risks in global supply chains,ranging from demand uncertainty to environmental turbulence(Chopra and Sodhi, 2004; Roh et al., 2014). While investment insupply chain capabilities increases the ability of the firm to bemore resilient and responsive to supply chain disruptions, it has itsown costs (Juttner, 2005; Chopra and Sodhi, 2014). Thus, organi-zations are faced with the evaluating the cost-benefit of invest-ments in supply chain capabilities to address supply chain risks.

Although a focus on the design of efficient supply chains hashelped organizations reduce their costs, it has increased their vul-nerability to disruptions (Wright, 2013). Previous studies show thatdue to economies of scale, firms would be able to minimize their fixedcost through minimizing investment in the number of facilities

hid Nooraie),

(Goetschalckx et al., 2013; Huang and Goetschalckx, 2014). Thus,addressing the overall effectiveness of a supply chain requires exam-ining the trade-off between investments in supply chain capabilitiesand the costs associated with disruptions. This requires a significantlydifferent approach to supply chain design, using a perspective thatincorporates the responsiveness and resiliency of a supply chain.

In recent years, academics and practitioners have focused onsupply chain risks and the impact of such risks on supply chaindesign decisions (Blackhurst et al., 2005; Craighead et al., 2007;Elkins et al., 2005; Hendricks and Singhal, 2003, 2005; Kleindorferand Saad, 2005; Rice and Caniato, 2003; Tang, 2006). A great deal ofwork has focused on evaluating different sources of risk and dis-ruption in supply chains, and how firms can develop mitigationstrategies to respond to disruptions. Nevertheless, there is a gap inthe literature on the trade-off between increased investment insupply chain capabilities and reduced supply chain risks. Chopraand Sodhi (2014) discuss the importance of development andimplementation of risk management plans that reduce risks withlimited impact on cost efficiency. While there is some anecdotalevidence on the benefits of implementing risk management plans,the cost-effectiveness of these programs has not been fully exam-ined. To address this gap in the literature, we aim to provide a moreholistic assessment of the trade-off between investment in supplychain capabilities and minimizing supply chain risk and cost.

The study makes two contributions to the literature in supplychain risk management. It develops a decision model for supply

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S. Vahid Nooraie, M.M. Parast / Int. J. Production Economics 171 (2016) 8–21 9

chain risk management with respect to the tradeoff between thecost associated with supply chain disruptions and the revenuegenerated as the result of investment in supply chain capability,where supply chain capability as investment in new facilities, pro-ducts sites, and distribution channels, which are usually regarded asimproving redundancy in the supply chain design. The existingstudies in supply chain design do not examine the impact of supplychain capability on mitigating supply chain disruptions. Previousstudies (e.g. Guille'n et al. (2005)) provides a decision model forsupply chain under uncertainty. However, whether firms would beable to mitigate supply chain disruptions through investment insupply chain capability remains unclear. Chopra and Sodhi (2014)argued that managers usually do not invest in supply chain cap-abilities because they view these investment as costs. We deter-mine whether decisions to improve supply chain capability throughinvestment in supply chain components such as facility, plant, anddistribution channels has a positive impact on mitigating supplychain disruptions and minimizing supply chain cost. Such anapproach to supply chain design has important managerial impli-cations since manager would be able to incorporate supply chainrisk decision into their supply chain design as part of their supplychain practices. Methodologically, we develop a heuristic algorithmto find the (near) optimal solution due to the NP-hard nature of themodel. This algorithm is new and novel, which is used for problemsthat have binary variables and optimum solution is not alwaysaccessible for large scale problems, which is an extension of themethod proposed by Narenji et al. (2011).

The remainder of this paper is organized as follows. In the nextsection, we discuss the importance of supply chain design as a riskmitigation strategy, and examine the scholarly work on supplychain risk management. Later, we introduce a multi-objectivesupply chain model that incorporates supply chain capabilityinvestment, supply chain risks and costs. Then we provide modelinterpretations and define our heuristic method based on arelaxation and decomposition method. Finally, we discuss thefindings of the study, its contribution to the theory and practice ofsupply chain risk management and directions for future research.

2. Supply chain design as a risk mitigation strategy

While supply chain design may involve many strategic, tactical andoperational decisions, most supply chain design decisions are concernedwith location decisions, i.e., where to locate facilities such as plants,processing units, warehouses, and retail stores to minimize the totalcost of transportation (Speier et al., 2011). With the emergence ofintegrated logistics, integrated manufacturing, and strategic procure-ment, supply chain design goals have expanded beyond their limitedfocus on cost, and have begun to focus on minimizing the total landedcost, including factors such as material acquisition, production, inven-tory, and logistics (Bowersox et al., 2006). Further developments insupply chain design have incorporated the idea of segmental customerservice requirements, which proposes that manufacturers need multi-ple supply chains to satisfy the individual service requirements of dif-ferent customer segments while also being efficient and cost-effective.This approach attempts to minimize total delivered cost while devel-oping innovative design strategies to satisfy delivery requirements interms of time and availability (Speier et al., 2011). We propose a newperspective on supply chain design where we incorporate supply chaincapabilities into the design of the supply chain, and the objective ofsatisfying customers is achieved through minimizing the risk andincreasing the responsiveness of the supply chain to disruptions.

2.1. Supply chain capability

Supply chain capability refers to the ability of an organization toidentify, use, and assimilate both internal and external resourcesand information to facilitate supply chain activities (Bharadwaj,2000; Wu et al., 2006). Previous research classifies supply chaincapabilities into efficiency-related and efficacy-related capabilities(Chen et al., 2009). Efficiency-related capabilities enable firms toreduce the cost of logistics and supply chain activities (Wu et al.,2006; Chen et al., 2009), while efficacy-related capabilities alloworganizations to maintain keep connections with supply chainparticipants as well as respond to consumer needs (Chen et al.,2009; Kim et al., 2006). Morash and Lynch (2002) view supply chaincapabilities as logistics-oriented capabilities and customer-service-oriented capabilities. In this research, we use both efficiency-relatedcapabilities and efficacy-related capabilities, because we minimizetransportation cost as well as responding to customer demands(Rajaguru and Matanda, 2013).

3. Literature review

Supply chain risk management (SCRM) is defined as thedevelopment and implementation of strategies to manage bothday-to-day and exceptional risks along a supply chain, with theobjective of reducing vulnerability and ensuring business con-tinuity (Zsidisin et al., 2005; Wieland and Wallenburg, 2012).Sources of risk include (but are not limited to) supply disruptions,demand fluctuations, environmental uncertainty and turbulence,equipment breakdown, procurement failures, and forecast inac-curacies (Harland et al., 2003; Zsidisin, 2003; Chopra and Sodhi,2004; Spekman and Davis, 2004).

In order to minimize the impact of disruptions on supply chainperformance, several attempts have been made to model andoptimize supply chain design, mostly utilizing a deterministicapproach to supply chain modeling and analysis (Timpe andKallrath, 2000; Gjerdrum et al., 2001; Azaron et al., 2008). How-ever, most real supply chain design problems are characterized bymultiple sources of risks and uncertainties inherent in the designof such systems. Thus, in order to obtain a more realistic assess-ment of supply chain risks and their impact on supply chain per-formance, the model parameters such as cost coefficients, supplies,and demand should be implemented in a stochastic model.

Few research studies have used two-stage stochastic models toexamine the comprehensive design of supply chain networks. Mir-Hassani et al. (2000) considered a two-stage model for multi-periodcapacity planning of supply chain networks. The authors applied aBenders decomposition to solve the resulting stochastic integer pro-gram. Santoso et al. (2005) unified a sampling strategy with anaccelerated Benders decomposition to solve supply chain design pro-blems assuming continuous distributions for the non-deterministicparameters. They designed a computational model involving two realsupply chain networks to highlight the significance of the stochasticmodel as well as the efficiency of the proposed solution strategy. Gohet al. (2007) developed a probabilistic model of the multi-stage globalsupply chain network problem, assuming supply, demand, exchange,and disruption as the deterministic parameters. Azaron et al. (2008)developed a multi-objective stochastic programming approach forsupply chain design under uncertainty.

It should be noted that assessment of the optimal supply chainconfiguration is a real challenge, because many factors andobjectives must be assumed when designing the network underuncertainty. However, the robustness of such a decision to non-deterministic parameters is not considered in the above cases. Toovercome the above limitations, this paper considers the mini-mization of the expected total cost, and the financial risk in a

Process Risk

Environmental Risk

Demand Risk

Supply Risk

Control Risk

Fig. 1. Scope of supply chain risk management.

S. Vahid Nooraie, M.M. Parast / Int. J. Production Economics 171 (2016) 8–2110

multi-objective model to design a robust supply chain network.Such an approach in using expected value to reduce model com-plexity is suggested in previous studies (e.g., Li et al. (2013), Aielloet al. (2015), Nooraie and Parast (2015)). We follow the recentdevelopments in multi-objective optimization through utilizingthe concept of solution robustness; we assume stability of anoptimal solution, considering errors in the objective functionparameters (Cromvik et al., 2011).

3.1. Methodological approaches

Huang and Goetschalckx (2014) and Goetschalckx et al. (2013)proposed supply chain systems when there are a large number ofdiscrete configurations. Sources of risks are modeled as a series ofscenarios. The risk of the supply chain is formulated as the stan-dard deviation of the revenues of the different scenarios. Anoptimizing algorithm that efficiently determines all Pareto-optimal figures of a supply chain is determined. The resultsshow that a general risk mitigation strategy for supply chains is toenhance the total capacity of the supply chain by either buildingmore facilities or by raising the capacity of individual facilities.

Manymethods and approaches have been applied to solve problemsconnected to supply chain design. Thesemethods include mathematicalmodeling, heuristics and artificial intelligence. In recent years, alter-native methods have been applied; one of the most widely used ismeta-heuristics. Although such methods are not guaranteed to providean optimal solution, they make a helpful compromise between themeasure of computation time spent and the quality of the approxi-mated solution area. Silva et al. (2005) offered a supply chain design as alogistic process that comprises order arrival, components request,components arrival, components assignment and order delivery. Thecase is to define the sequence in which orders should be performed.The ensuring scheduling problem is solved applying Genetic Algorithms(GA) and Ant Colony Optimization (ACO). Altiparmak et al. (2006)offered a method, based on GA, for designing a four-echelon supplychain (suppliers, plants, warehouses and customers).

Several heuristic techniques such as Simulated Annealing (SA) andGA have been used on a number of problems. First, ACO was used ondecision problems including a single objective. Later efforts inserted asecond objective or multiple objectives (McMullen, 2001; Doerner et al.,2004, 2006, 2008; Stummer and Sun, 2005). McMullen (2001) devel-oped a multi-objective production sequencing case where the objec-tives are the number of set-ups and the stability of materials used. Asingle combined pheromone matrix is applied in this technique.Moncayo-Martіnez and Zhang (2011) proposed an algorithm based onPareto Ant Colony Optimization as an effective meta-heuristic techniquefor solving multi-objective supply chain design problems. This techni-que is efficient but rather complex; when the dimension of the problemincreases, the complexity increases dramatically.

Guille´n et al. (2005) developed a supply chain model using amulti-objective stochastic Mixed-Integer Programming (MIP)approach in which uncertainty is examined with demand forecasting.Themodel is solved by branch and bound techniques. Objectives are tomaximize profit over the time period, maximize demand satisfactionand minimize financial risk. While mathematical models have theadvantage of providing the optimal solution, the level of complexity ofan NP-hard problem requires too much computation for mathematicalcalculation to be a realistic approach. Thus, the development ofalternative methodological approaches is needed (Dabia et al., 2013).

In this paper, we develop an efficient heuristic method wherebranch and bound methods cannot be used to determine theoptimal solution. Our heuristic method does not guarantee anoptimal solution, but it provides a near-optimal solution where thedeviation between near-optimal and optimal is negligible.

3.2. Sources of risk in the supply chain

The scope of our model includes supply, process, demand andcontrol under risk. Fig. 1 presents the conceptual model of our supplychain risk scope, where all main elements of supply chain design areunder risk (Christopher and Peck, 2004). In our model, we define risk asprobabilistic scenarios that have a direct effect on the value of ourmodel parameters.

The availability of more facilities provides more capability toovercome disruptions and risks; however cost of facility establish-ment and deployment should be considered (Chopra and Sodhi,2004; Talluri et al., 2013; Chopra and Sodhi, 2014). This perspectiveprovides a more balanced approach toward supply chain risk man-agement in which the benefits of establishing more facilities (cap-abilities) should be examined in view of the associated risks andcosts within the supply chain. In other words, the design of supplychain systems should examine the associated costs of trade-offsamong capabilities, risks and vulnerabilities (Pettit et al., 2010, 2013).

In this research, we examine the effectiveness of a supply chaindesign system through investment in new facilities such aswarehouses, distribution centers and manufacturers to meet cus-tomer needs in the face of disruptions. All of our parameters areassumed time-dependent, because the time factor has a con-siderable effect on the value of the model parameters. We willdevelop an efficient heuristic method to identify all feasiblesolutions based on a decomposition method to break MIP into sub-linear models, and to compute the relaxed binary variables. Toevaluate probabilistic scenarios, we calculate the expected valuefor each parameter. This heuristic method is based on the devel-opment of previous heuristic methods for situations when themodel is NP-hard and complex (Narenji et al., 2011).

4. Problem description

In the strategic design of supply chains, there are possiblyseveral parameters whose measures cannot be determined accu-rately, and their values are considered to be stochastic. In ourmodel, the probability of each scenario is determined as a discretevalue and we use the expected value approach to evaluate it.Moreover, each scenario is time-dependent and dynamicallyaffects all parameters. In other words, the probability of eachscenario is not fixed, and it is changed during the time horizon(Huang and Goetschalckx, 2014).

The main idea is to develop a model to maximize total revenueand minimize total cost related to opening new facilities: ware-houses, distribution centers and manufacturers. Our objective is tofind an optimal point as a trade-off between revenue and cost, tounderstand how the establishment of new facilities (i.e. capabilities)affects disruptions and risk while meeting customer demands.Overall, we need to design a supply chain that incorporates risk ofdisruptions, and to determine whether more investment in cap-abilities and facilities enhances supply chain responsiveness todisruptions. Thus, the goal is to understand how much investmentin new facilities is justified to mitigate supply chain risks, and todetermine the trade-off between costs and investment in cap-abilities in a supply chain that is exposed to risks.

Supplier (S) Manufacturer (F) Warehouse (W) Distributer (DC) Customers (C)

S F W DC C

Supply chain channels


In the next section, we define the mathematical formulationsfor the multi-objective supply chain problem. Then, we define thenotation and decision variables, and present the mathematicalmodel for a multi-objective stochastic supply chain.

CDUWUF=TN = S U C U T

Fig. 2. Graphical model of the facilities in a supply chain design.
4.1. Notation and model variables
Sets and Indices

S: set of supplier facilitiesC: set of customer facilitiesW: set of warehouse facilitiesDC: set of distribution centersF: set of manufacturing factoriesT: set of transformation facilities: T ¼ F [ DC [ W (union of F,DC, and W is the definition of transformation facilities)N: set of all facilities: N¼ S [ C [ T (N is the union of allfacilities)A: set of channels: A¼{(i, j) | i, jA N} (channels come from anycombination of F, DC, W, S, C and T)SC: set of scenariosK: set of productsTP: set of time periods (total of different time periods forms theset of time periods, for instance, seasonal periods and yearlyperiods)

Parameters

copenksit: the cost for establishing facility i to produce product k,

i.e., its initial investment costunder scenario s in period tPopenk

sit: the revenue from facility i from product k, under sce-nario s in period tps: the probability of scenario s,

Psps ¼ 1

ctransksijt: the unit cost to transport product k from location i tolocation j under scenario s in period tcapkslt: the capacity for product k of facility l under scenario s inperiod tdemk

slt: the forecast demand for the product k of customer lunder scenario s in period tsupkslt: the supplier capacity for product k of supplier l underscenario s in period tsrksct: the unit sales revenue for product k sold to customer cunder scenario s in period t

Variables

yi: binary variable equal to 1 if facility i is opened and equal to0 otherwisexksijt: the flow variable for the product k from location i to loca-tion j under scenario s in period texp z1: the expected value of objective function z1, the firstobjective function is the maximization of total revenue, whereps is the probability of each parameterexp z2: the expected value of objective function z2, the secondobjective function is the minimization of total cost, where ps isthe probability of each parameter

Expected value of Z1¼Psðps � z1Þ

Expected value of Z2¼Psðps � z2Þ

Note that the model can be expanded to handle multiple per-iods that correspond to seasons in the planning horizon.

4.2. Mathematical model for multi-objective stochastic supply chaindisruption

Following the procedure suggested by Nooraie and Parast (2015),the trade-off between investment in supply chain capability andcosts is formulated as the expected values of Z1 and Z2.

Max Z1 :Xs

psXcAC

Xk

Xt

srksctXi;cð ÞAA

Xt

xksict

1A

0@

þXs

psXiAT

Xt

Popensit

1A� yi8Y ; 8sϵSC

0@ ð1Þ

Min Z2 :Xs

psXi;jð ÞAA

Xt

ctransksijt � xksijt

1A

0@

þXs

psXiAT

Xt

Copensit

1A� yi 8Y ; 8sϵSC

0@ ð2Þ

s.t.Xi;lð ÞϵA

Xt

xksilt�Xl;jð ÞϵA

Xt

xksljt ¼ 0 8kAK; 8 lϵT ; 8sASC; 8 tATP ð3Þ

Xi;lð ÞϵA

Xt

xksiltrcapkslt � yl 8kAK; 8 lϵT ; 8sASC; 8 tATP ð4Þ

Xl;ið ÞϵA

Xt

xkslitrsupkslt 8kAK ; 8 lϵS; 8sASC; 8 tATP ð5Þ

Pi;lð ÞϵA

Ptxksiltrdemk

slt 8kAK ; 8 lϵC; 8sASC; 8 tATPyiϵ 0;1f g 8 iAT

ð6ÞThe objective function (1) is the expected value of the total

revenue of opening facilities and revenue of selling products aswell. The objective function (2) is the expected value of the totalcost of opening facilities and transportation cost of products aswell. Constraint (3) is a flow constraint that ensures that the inputquantity for each product is equal to the output quantity in eachfacility; it means that the arrival products to each manufacturer(F), warehouse (W), and distributer (DC) are equal to the productsthat leave these facilities. Constraint (4) ensures that materialquantities do not exceed the given capacity. Constraint (5) enforcesthat a supplier does not provide more of a product than its capa-city for that product. Constraint (6) ensures that the quantity ofthe finished products delivered to the customer does not exceedthe demand for the customer.

4.3. Model descriptions

Our model consists of the main elements of a supply chainsystem: suppliers, manufacturers, warehouses, distribution cen-ters and customers (Fig. 2). We can choose several numbers foreach element, as each element has a one-to-many relation withthe next elements in the supply chain; therefore, we have differentchannels within the supply chain. The summation of the manu-facturers, warehouses, and distribution centers is equal to T.Moreover, T and the summation of suppliers and customers is

1. . . . n

Sup F W DC J

n n n n n

= = = Equal materials flow

Fig. 3. Constraint (3).

1 . . . . n

Sup F W DC

n n n n

Capacity limit from manufacturer, warehouse, distributer


1

.

.

.

.

n

Sup

n

Supplier capacity limit


Customer demand

DC

n



equal to N, suggesting that the summation of all elements in oursupply chain equals N. These definitions are useful when we needto formulate and develop our mathematical model.

Constraint (3) includes all elements where k is a product, s is ascenario, and t is a time period. In this constraint, suppliers andcustomers are connected together via T, which is the summationof manufacturers, distribution centers and warehouses; i comesfrom a supplier and passing through F, W, and DC reaches thecustomer. Based on different values of these parameters, we havedifferent channels. The arrival products to a manufacturer (F),warehouse (W), or distributer (DC) are equal to the products thatleave these facilities. Fig. 3 shows balanced flows in the supplychain (from supplier to the manufacturer, to the warehouse, to thedistribution center, and to the customer). Arrows show that eachelement has a relationship with all next-stage facilities. Eachsupplier has a relationship with all manufacturers. The smallarrows on the upper side show that the quantity of materialsmoving between facilities is unchanged.P

i;lð ÞϵA

Ptxksilt�

Pl;jð ÞϵA

Ptxksljt ¼ 0 8kAK ; 8 lϵT ; 8sASC; 8 tATP

Constraint (4) includes all elements except the customer, wherek is a product, s is a scenario, and t is a time period. Fig. 4 showsthat there is a capacity limit in each element of T that affects eachprevious facility, where T is a combination of the manufacturer, thewarehouse and the distribution center. Based on the capacity limitfor each combination of T elements, there is a limit for product kfrom location i under scenario s in period t. This constraint ensuresthat material quantities do not exceed the given capacity. Thereare M�N relations between elements; upper arrows from right toleft show that the capacity of each element affects previous ele-ments in the chain. For example, the capacity limit of a manu-facturer affects supplier output.Xi;lð ÞϵA

Xt

xksiltrcapkslt � yl 8kAK ; 8 lϵT ; 8sASC; 8 tATP

Constraint (5) includes all elements where k is a product, s is ascenario, and t is a time period. Fig. 5 shows that product k fromlocation i under scenario s in period t should be produced, con-sidering the supplier limit provided by supplier l to produce pro-duct k under scenario s in period t. This constraint enforces that asupplier does not provide more of a product than its capacity forthat product. The upper arrow shows that the capacity limit of a

supplier affects the next element in the chain, which is a manu-facturer.Xl;ið ÞϵA

Xt

xkslitrsupkslt 8kAK ; 8 lϵS; 8sASC; 8 tATP ð7Þ

Constraint (6) includes all elements where k is a product, s is ascenario, and t is a time period; i comes from a supplier withdifferent channels and is connected to the customer. Fig. 6 showsthat the quantity of the finished products delivered to the custo-mer does not exceed the demand for the customer. The upperarrow shows that customer demand affects warehouses.Xi;lð ÞϵA

Xt

xksiltrdemkslt 8kAK ; 8 lϵC; 8sASC; 8 tATP

4.4. Numerical example

A numerical example shows how the model determines theoptimal solution. After obtaining the solution, we use our pro-posed heuristic algorithm to determine how the algorithm deter-mines the solution, and how much deviation exists between theoptimal value and the heuristic solution. This numerical example

Table 1Solution for the flow variable for product k from location i under scenario s in period t.

I j X11ij1 X1

1ij2 X12ij1 X1

2ij2 X21ij1 X2

1ij2 X22ij1 X2

2ij2

1 2 o1 24 0 0 0 0 0 0 0 01 3 o1 34 0 0 0 0 0 0 0 01 4 o1 44 0 0 0 0 0 0 0 01 5 o1 54 0 0 0 0 0 0 0 01 6 o1 64 0 0 0 0 0 0 0 01 7 o1 74 0 0 0 0 0 0 0 01 8 o1 84 0 0 0 0 0 0 0 01 9 o1 94 0 0 0 0 0 0 0 93,7751 10 o1 104 0 65,700 0 0 0 0 0 02 1 o2 14 0 0 0 0 0 0 0 02 3 o2 34 0 0 0 0 0 0 0 02 4 o2 44 0 0 0 0 0 0 0 33002 5 o2 54 0 0 0 0 0 0 0 02 6 o2 64 0 0 0 0 0 0 0 02 7 o2 74 0 0 0 0 0 0 0 02 8 o2 84 0 0 0 0 0 0 0 02 9 o2 94 0 0 0 0 0 0 825 02 10 o2 104 57,150 0 65722.5 0 93,500 0 105,875 03 1 o3 14 0 0 0 0 0 0 0 03 2 o3 24 0 0 0 0 0 0 0 03 4 o3 44 0 0 0 0 0 0 0 03 5 o3 54 0 0 0 0 0 0 0 03 6 o3 64 0 0 0 0 0 0 0 03 7 o3 74 0 0 0 0 0 0 0 03 8 o3 84 0 0 0 0 0 0 0 03 9 o3 94 0 0 0 0 0 0 0 03 10 o3 104 0 0 0 0 0 0 0 04 1 o4 14 0 0 0 0 0 0 0 04 2 o4 24 0 0 0 0 0 0 0 04 3 o4 34 0 0 0 0 0 0 0 04 5 o4 54 0 0 0 0 0 0 0 04 6 o4 64 0 0 0 0 0 0 0 33004 7 o4 74 0 0 0 0 0 0 0 04 8 o4 84 0 0 0 0 0 0 0 04 9 o4 94 67,500 0 0 0 2750 85,250 0 04 10 o4 104 0 1800 0 0 0 0 0 05 1 o5 14 0 0 0 0 0 0 0 05 2 o5 24 0 0 0 0 0 0 0 05 3 o5 34 0 0 0 0 0 0 0 05 4 o5 44 0 0 0 0 0 0 0 05 6 o5 64 0 0 0 0 0 0 0 05 7 o5 74 0 0 0 0 0 0 0 05 8 o5 84 0 0 0 0 0 0 0 05 9 o5 94 0 0 0 0 0 0 0 05 10 o5 104 0 0 0 0 0 0 0 06 1 o6 14 0 0 0 0 0 0 0 06 2 o6 24 0 0 0 0 0 0 0 06 3 o6 34 0 0 0 0 0 0 0 06 4 o6 44 0 0 0 0 0 0 0 06 5 o6 54 0 0 0 0 0 0 0 06 7 o6 74 0 0 0 0 0 0 0 06 8 o6 84 0 0 0 0 0 0 0 06 9 o6 94 0 0 0 64,350 0 0 0 06 10 o6 104 0 0 0 0 0 0 0 33007 1 o7 14 0 0 0 0 0 0 0 07 2 o7 24 0 0 0 0 0 0 0 07 3 o7 34 0 0 0 0 0 0 0 07 4 o7 44 0 0 0 0 0 0 0 07 5 o7 54 0 0 0 0 0 0 0 07 6 o7 64 0 0 0 0 0 0 0 07 8 o7 84 0 0 0 0 0 0 0 07 9 o7 94 0 0 0 0 0 0 0 07 10 o7 104 0 0 0 0 0 0 0 08 1 o8 14 0 0 0 0 0 0 0 08 2 o8 24 0 0 0 0 0 0 0 08 3 o8 34 0 0 0 0 0 0 0 08 4 o8 44 0 0 0 0 0 0 0 08 5 o8 54 0 0 0 0 0 0 0 08 6 o8 64 0 0 0 64,350 0 0 0 08 7 o8 74 0 0 0 0 0 0 0 08 9 o8 94 0 0 0 0 88,000 0 99,000 08 10 o8 104 0 0 3577.5 3150 0 0 0 93,5009 1 o9 14 0 0 0 0 0 0 0 09 2 o9 24 0 0 0 0 0 0 0 09 3 o9 34 0 0 0 0 0 0 0 0


Table 1 (continued )

9 4 o9 44 67,500 0 0 0 2750 85,250 0 09 5 o9 54 0 0 0 0 0 0 0 09 6 o9 64 0 0 0 0 0 0 0 09 7 o9 74 0 0 0 0 0 0 0 09 8 o9 84 0 0 3577.5 0 88,000 0 0 93,5009 10 o9 104 5850 0 0 71,100 2750 88,000 0 0

10 1 o10 14 0 0 0 0 0 0 0 010 2 o10 24 0 0 0 0 0 0 0 010 3 o10 34 0 0 0 0 0 0 0 010 4 o10 44 0 1800 0 0 0 0 0 010 5 o10 54 0 0 0 0 0 0 0 010 6 o10 64 0 0 0 0 0 0 0 010 7 o10 74 0 0 0 0 0 0 0 010 8 o10 84 0 0 0 67,500 0 0 99,000 010 9 o10 94 0 58,500 74,250 0 0 0 0 0

For j = 2…T define Yi’s=1 then calculate Z and count the number of satisfied constraints

Zj < = upper bound?

Yes No

j = j +1

Stop when j = T + 1

Fig. 7. Minimization algorithm.

For j = 2…T define Yi’s=1 then calculate Z and count the number of satisfied constraints

Zj < = upper bound?

Yes No

j = j +1

Stop when j = T + 1

Fig. 8. Maximization algorithm.


consists of two products, two periods, two scenarios, two suppli-ers, two customers, two warehouses, two distribution centers, andtwo manufacturers (the data is provided in Appendix A). Toeliminate the probabilistic aspects of scenarios, we calculate theexpected value for each parameter. To further reduce the com-plexity of the model, we define only one objective function basedon minimization of cost minus maximization of revenue. Table 1shows the cost of each component of facilities under each scenarioin each period. Facilities include suppliers, customers, warehouses,distribution centers and manufacturers. Table 1 is the solution ofthe model on decision variable xksijt , the flow variable of eachproduct under each scenario in each period, that moves betweenfacilities i and j, where i j come from all facilities.

According to the solution for this numerical example by Cplexsoftware, we see that all binary variables y1, y2, y3, y4, y5 and y6take on the value of 1. This suggests that the optimal policy isachieved when all manufacturers, warehouses, and distributioncenters are open. In this case, the objective function Z¼�$18,955,628. The negative symbol shows that the total cost is lessthan total revenue. Gross revenue is the absolute value of thisnegative amount (Gross revenue¼$18,955,628).

Dabia et al. (2013) showed that a multi-objective time-depen-dent optimization problem is an NP- hard problem. The supplychain risk management system presented in this study is a multi-objective, time-dependent, and multi-item capacitated model.Thus, it has a much higher level of complexity due to the numberof model parameters. To find the solution for such a complexproblem, we develop a heuristic algorithm to determine the near-optimum or optimum value with minimum deviation from theoptimal solution (Figs. 7 and 8).

5. The heuristic algorithm

We examined the number of constraints, and we learned thatthe redundancy of each parameter significantly affects total con-straints. For instance, if the number of each parameter is equal to 2(parameters include number of products, scenarios, periods, sup-pliers, manufacturers, warehouses, distribution centers and

customers), we should define more than 500 constraints. Narenjiet al. (2011) reported that when the number of parametersincreases in MIP, a branch and bound algorithm is not capable ofproviding the optimal solution. They proposed a heuristic algo-rithm based on relaxation and decomposition methods. In thispaper, we use a relaxation method for binary variables to developour heuristic method. Based on maximization or minimization, weneed to define two types of heuristic methods for our objectivefunctions. Simply, our heuristic algorithm activates binary vari-ables (yi¼1) each time, and gives us a primary idea to performideal combinations for the binary variables. We choose the bestcombination of activated variables; then, we solve different LPproblems. We look for the best LP problems based on a greaternumber of satisfied constraints.

5.1. Satisfied constraints method for minimization

This method determines the (near) optimal solution based onsatisfying the greatest number of constraints. This approach isused when the goal is to minimize the objective function (Nooraieand Parast, 2015). We need to follow all the following steps todetermine an optimal or near-optimal solution.

Step 1. Calculate Z when all binary variables are one (Yi¼1where i ε T ), then set the upper bound Zupper to Z.Step 2. For i¼1….T (for T¼F U DC U W, combination of manu-facturers, distribution centers and warehouses), set Yi¼1, set allother Y¼0, and calculate Zi. Then for Z¼Zi………ZT allocate agroup number from 1….TStep 3. Sort groups from minimum to maximum. We define j asa combinations number of Yiwhere j¼2…T. For example: if j¼3,it means that we should set Yi¼1 for the first three sortedgroups, then calculate Z.Step 4. Follow this algorithm:Step 5. Compare all Z values from Steps 3 and 4 and report anoptimal or near-optimal solution based on the largest number ofsatisfied constraints.

Table 2Satisfied constraints heuristic approach for minimization.

Yi Cost Revenue Profit¼Cost-Revenue Profit¼ |Cost-Revenue| Upper-bound

Min 6 $2,052,416 $17,723,155 �$15,670,739 $15,670,739 $18,955,6285 $2,113,465 $17,803,695 �$15,690,230 $15,690,230 $18,955,6284 $2,027,103 $17,841,795 �$15,814,692 $15,814,692 $18,955,6283 $2,402,493 $18,266,139 �$15,863,646 $15,863,646 $18,955,6282 $1,872,173 $17,747,859 �$15,875,686 $15,875,686 $18,955,628

Max 1 $2,022,575 $18,487,339 �$16,464,764 $16,464,764 $18,955,62865 $2,812,988 $18,940,071 �$16,127,083 $16,127,083 $18,955,628654 $3,487,199 $20,195,087 �$16,707,888 $16,707,888 $18,955,6286543 $4,468,351 $21,874,447 �$17,406,096 $17,406,096 $18,955,62865432 $5,033,388 $23,035,527 �$18,002,139 $18,002,139 $18,955,628654321 $5,980,459 $24,936,087 �$18,955,628 $18,955,628 $18,955,628

$0

$20,00,000

$40,00,000

$60,00,000

$80,00,000

$1,00,00,000

$1,20,00,000

$1,40,00,000

$1,60,00,000

$1,80,00,000

$2,00,00,000

|Cost-Profit|LowerboundMiddle boundUpperbound

Fig. 9. Minimization algorithm (numerical example).


5.2. Satisfied constraints method for maximization

This approach is used when the objective function is definedbased on maximization. We need to follow the following steps foroptimal or near-optimal solution.

Step 1. Calculate Zupper when all binary variables are one (Yi¼1where i ε T ), then set upper bound based on Zupper (Upperbound¼Zupper).Step 2. For i¼1….T (T is the combination of warehouse, manu-facturer and distribution center) Yi¼1 while other Yis¼0 andcalculate Z then for Z¼Z i………ZT allocate group number from1….TStep 3. Sort groups from Max to MinWe define j as a combinations number of Yi where j¼2…TStep 4. Follow this algorithm:

Step 5. Compare all Z values from Steps 3 and 4 and reportoptimum/near optimum based most satisfied constraints (SeeAppendix B).

Now, we consider a situation when we are dealing with a large-scale problem. In this situation, we are not able to determine theoptimal point due to the complexity of the model. According to theheuristic algorithm, we have the following data for our numerical case.

According to Table 2, the most constraints satisfied belong to thelast row, where y1, y2, y3, y4, y5, and y6 are set to 1, and the optimalsolution is $18,955,628. In this case, we have no deviation between theoptimal solution and the solution from the heuristic algorithm.

To assess the efficiency of the heuristic algorithm, we generateddata on each parameter, and ran our heuristic for 70 times; wecompared the heuristic solutions with the optimum values. Ourfindings suggests that the mean value of deviation between opti-mum and heuristic solutions is less than 5%. Thus, we concludethat our heuristic algorithm is robust enough.

6. Discussion

In this study, to solve an NP-hard problem related to supplychain design under risks and disruptions, we designed a heuristicalgorithm based on a relaxation method and the number of satisfiedconstraints. In the process of decision-making to choose the bestsolution, different strategies affect our alternative solutions (Fig. 9).The decision model can assist a manager to manage the trade-offamong investment, risk, and costs within a supply chain. Amongsingle Yi's (decision variables), the best choice is Y1, where theobjective function is $16,464,764 and deviation from the optimalsolution is 13%. In the case of using a balanced emphasis on eachobjective function (50% weight is given to each objective function),the closest choice (where cost and revenue have almost equalweights) is the combination of Y6, Y5, Y4 and Y3, where theobjective function is equal to $17,406,096 and the deviation is 10%.

In our model, we are interested in opening facilities as much aspossible, because more active facilities provide us more revenueand the capability to cope with more cost and risk within thesupply chain. This is well in line with the notion of enhancingredundancy in supply chain systems in order to mitigate thenegative impact of disruptions (Christopher and Peck, 2004; Sheffi,2005; Sheffi and Rice, 2005; Zsidisin and Wagner, 2010). There-fore, our alternative solution is based on y6, y5, y4, y3 and y2binary variables, and the objective function is $17,406,096, wheredeviation from the optimal solution is only 5%.

Our results show that we would be able to use the heuristicmethod to estimate the optimal solution when we have a large-scale NP-hard problem. We should mention that due to the NP-hardnature of the problem, the optimal solution would not be deter-mined easily. The relaxation method is very efficient, since devia-tion of the heuristic solution from the optimal solution is negligible.Based on this strategy, for example, if our objective function isminimization, we need to sort binary variables from minimum tomaximum; then, we need to set the objective function based onvarious combinations of binary variables. This method (sortingbinary variables) provides near-optimal or optimal solutions.

According to our findings, if we activate all facilities and expandcapabilities, the result of the objective function shows a trade-offbetween total revenue and total cost. Opening all facilities providesmore capability to meet customer demands (efficacy-related cap-ability) as well as minimizing transportation costs (efficiency relatedcapability). According to the probabilistic nature of scenarios, eachscenario directly affects all parameters because we use the expectedvalue technique, which decreases the real value of each parameter.Furthermore, such probabilistic risks deviate from both total revenueand total cost. This suggests that the results always are an approx-imation of real amounts, and deviations are determined based onrisk probabilities. If the probability of a scenario is high, our heuristicalgorithm solutions have less deviation from the actual value.


6.1. Theoretical contributions

In this study, we developed a supply chain risk management modelwhere the tradeoffs among risks, supply chain costs and investment insupply chain capabilities were examined. We added a time parameterto the basic model to investigate the effect of time-dependency on allparameters, where risks of disruption were incorporated into thesupply chain design. We also realize that a time parameter imposesmore complexity on the model. Moreover, we used expected value toeliminate the stochastic nature of scenarios. We investigated thetrade-off between risk, cost and revenue, wheremore facilities providemore capability to meet customer needs, and decrease the negativeeffect of disruptions. Our finding shows that among those invest-ments, any combination of facilities (manufacturer, distribution centerand warehouse) that can satisfy customer demand decreases the riskof lost sales and lost revenue. It is important that we have to beconcerned with the total cost. It is very clear that we need to increasethe total revenue through increasing total capabilities, while the totalcost should be decreased as much as possible.

We designed a heuristic method to reduce the complexity of themodel due to the NP-hard nature of a multi-objective time-dependent optimization problem (Dabia et al., 2013). Our model canbe applied when we need to meet customer needs in a riskyenvironment where the capability of each facility decreases theimpact of risks that are defined as probabilistic scenarios. In addi-tion, our heuristic method is efficient, and can be applied wherethere are several objective functions with opposite directions.

6.2. Managerial implications

Our model includes two main goals. The first one is based oninvestment in new facilities that provide financial benefits throughincreasing revenue; the second goal is to minimize the total cost ofthe supply chain under risks. Our mathematical model is a type ofdecision-making tool that a manager may use to examine decisionsregarding investment in developing new capabilities in a supplychain vis-à-vis the cost associated with development of these cap-abilities to mitigate disruptions and risks in the supply chain.

By evaluating the sources of risk and their frequency, a managerwould be able to evaluate how to design the proper supply chain tomitigate the effects of those risks. For example, assume a manger hasintended to invest in a food supply chain in Florida, where tsunami,storm and other natural disasters pose significant risks to the supplychain. If the probability of tsunami in Florida is estimated to bearound 80–90%, that high level of risk could have a significant impacton customer demand. In this situation, the supply chain managershould estimate the future demands with a lower probability (e.g.,45–55%); thus, we expect a much higher deviation of the expectedvalue for demand from the actual demand, due to the higher level ofenvironmental risk, as presented in the numerical example. Alter-natively, when the risk of tsunami or other natural disasters are onthe lower side (e.g. 10–20%), the future demand would be morestable, and the supply chain manager would be able to predict futuredemand with more certainty, as the expected value for demand ismuch closer to the actual demand. The parameters in our modelassist managers to evaluate the cost of facilities, revenue of facilities,transportation cost, production capacity, customer demands, suppliercapacity and product revenue. The numerical results of the mathe-matical model give a clear vision to the manager to decide onopening manufacturers, warehouses and distribution centers in orderto be more responsive to supply chain disruptions.

6.3. Limitations and future research

In this research, the risk was measured as the expected value ofparameters. Various other risk measures have been proposed in

the literature and in stochastic optimization (Goetschalckx et al.,2013). Examples are downside risk, conditional value at risk andupper partial mean of the scenario profits. Investigation of thoserisk measures, their relationships, and their impact on the supplychain configuration is a fertile area of future research. Differentrisk measures may also require the development of differentoptimization and heuristic algorithms.

Another interesting area of research is the impact of othercapabilities such as agility and responsiveness in our model. Inaddition, to prevent generating lower probabilities for risks, futureresearch should consider efficient solution techniques for high riskscenarios. Alternatively, we can assume that our model is com-pletely deterministic and there is no probabilistic risk in ourproblem. In this case, we will have an accurate solution when wealready have designed preventive policies to mitigate risk situa-tions or improve our supply chain design through investment inother capabilities using new techniques such as flexibility.

7. Conclusion

The strategic design of a supply chain system is very important tothe long-term profitability and survival of firms. One of the key ques-tions in supply chain design is how to determine the trade-off betweenthe capability of the supply chain (investment) and vulnerability tosupply chain disruptions under a variety of uncertain future conditions(its risk). Firms would be able to decrease the negative impact of risksthrough investment in more capabilities; however these investmentsincrease costs as well. Therefore, in supply chain design, we need toexamine investment in capabilities and their associated costs.

While obtaining an optimal solution for the model is a chal-lenge due to the NP-hard nature of the problem, one efficient wayto make an informed decision on the trade-off involved indesigning a supply chain from a risk management perspective is touse the largest number of satisfied constraints heuristic approachand the corresponding solution algorithms.

Acknowledgment

This research is based upon work supported by the NationalScience Foundation (NSF) under Grant number 123887 (ResearchInitiation Award: Understanding Risks and Disruptions in SupplyChains and their Effect on Firm and Supply Chain Performance).

Appendix A. Solution Results

Table A2 shows the expected values of the entries in Table A1.(We use the expected value method to make deterministic values.)

Opening each facility in each period under each scenario foreach product generates revenue that is shown in Table A3. Theexpected values of these numeric entries are shown in Table A4.

Table A5 shows transportation between facilities (supplier,manufacturer, warehouse, distributer and customer facilities) foreach product, scenario, and time period separately (These data arebased on the expected values).

Table A6 shows that our model is capacitated, because eachfacility (manufacturer, warehouse and distributor) has a capacityconstraint based on different periods, scenarios and products.Table A7 shows the expected values for these entries.

Table A8 shows the customer demand that comes from demandforecasting, where it varies by scenario, product, time period, andcustomer. Table A9 shows the expected values of customer demand.

Table A1Cost of establishing facility i under scenario s in period t.

Copensit Scenario s 45% 55% 45% 55%facility Period t 1 2

I S1 $ 100,000 $ 125,000 $ 300,000 $ 450,000I S2 $ 150,000 $ 148,000 $ 250,000 $ 300,000I C1 $ 250,000 $ 143,000 $ 115,000 $ 155,000I C2 $ 300,000 $ 147,000 $ 450,000 $ 477,500I W1 $ 450,000 $ 160,000 $ 376,000 $ 461,000I W2 $ 500,000 $ 117,000 $ 394,000 $ 484,000I DC1 $ 550,000 $ 112,000 $ 410,000 $ 502,500I DC2 $ 600,000 $ 82,000 $ 430,000 $ 525,000I F1 $ 700,000 $ 390,000 $ 465,000 $ 565,000I F2 $ 750,000 $ 555,000 $ 480,000 $ 585,000

Table A2Cost of establishing facility i under scenario s in period t.

E(Copensit) Scenario s Expected V Expected V Expected V Expected Vfacility Period t 1 2

I S1 $45,000 $68,750 $135,000 $247,500I S2 $67,500 $81,400 $112,500 $165,000I C1 $112,500 $78,650 $51,750 $85,250I C2 $135,000 $80,850 $202,500 $262,625I W1 $202,500 $88,000 $169,200 $253,550I W2 $225,000 $64,350 $177,300 $266,200I DC1 $247,500 $61,600 $184,500 $276,375I DC2 $270,000 $45,100 $193,500 $288,750I F1 $315,000 $214,500 $209,250 $310,750I F2 $337,500 $305,250 $216,000 $321,750

Table A3Revenue from establishing facility i under scenario s in period t.

Popensit Scenario s 45% 55% 45% 55%facility Period t 1 2

I S1 $ 160,000 $ 200,000 $ 480,000 $ 720,000I S2 $ 240,000 $ 236,800 $ 400,000 $ 480,000I C1 $ 400,000 $ 228,800 $ 184,000 $ 248,000I C2 $ 480,000 $ 235,200 $ 720,000 $ 764,000I W1 $ 720,000 $ 256,000 $ 601,600 $ 737,600I W2 $ 800,000 $ 187,200 $ 630,400 $ 774,400I DC1 $ 800,000 $ 179,200 $ 656,000 $ 804,000I DC2 $ 960,000 $ 131,200 $ 688,000 $ 840,000I F1 $ 1,120,000 $ 624,000 $ 744,000 $ 904,000I F2 $ 1,200,000 $ 888,000 $ 768,000 $ 936,000

Table A4Revenue from establishing facility i under scenario s in period t.

E (Popensit) Scenario s Expected V Expected V Expected V Expected Vfacility Period t 1 2

I S1 $72,000 $110,000 $216,000 $396,000I S2 $108,000 $130,240 $180,000 $264,000I C1 $180,000 $125,840 $82,800 $136,400I C2 $216,000 $129,360 $324,000 $420,200I W1 $324,000 $140,800 $270,720 $405,680I W2 $360,000 $102,960 $283,680 $425,920I DC1 $360,000 $98,560 $295,200 $442,200I DC2 $432,000 $72,160 $309,600 $462,000I F1 $504,000 $343,200 $334,800 $497,200I F2 $540,000 $488,400 $345,600 $514,800


Table A5Unit transportation cost for product k from location i to j under scenario s in period t.

I J CtransX11ij1 CtransX1

1ij2 CtransX12ij1 CtransX1



2ij2

1 2 o1 24 $1.46 $1.52 $1.38 $1.80 $1.28 $1.92 $1.94 $1.201 3 o1 34 $8.21 $4.10 $8.67 $6.38 $7.41 $3.65 $9.35 $9.581 4 o1 44 $7.79 $8.87 $9.84 $8.87 $5.41 $3.35 $1.95 $3.571 5 o1 54 $2.80 $3.61 $2.69 $1.13 $5.01 $2.37 $5.22 $2.261 6 o1 64 $8.20 $6.00 $1.10 $9.40 $7.20 $0.80 $0.40 $5.101 7 o1 74 $3.88 $8.21 $6.38 $8.44 $7.30 $10.49 $4.22 $9.461 8 o1 84 $6.02 $5.04 $2.59 $5.88 $5.95 $1.89 $5.25 $6.581 9 o1 94 $4.73 $7.06 $0.51 $3.64 $5.68 $6.92 $4.88 $0.511 10 o1 104 $2.61 $1.67 $5.64 $7.73 $8.04 $7.93 $2.51 $10.022 1 o2 14 $0.02 $1.82 $0.06 $1.08 $0.16 $1.32 $1.40 $1.802 3 o2 34 $10.14 $5.60 $5.87 $3.87 $5.34 $7.34 $8.27 $12.012 4 o2 44 $5.89 $1.75 $4.76 $3.63 $5.39 $0.63 $2.51 $0.382 5 o2 54 $3.85 $0.15 $1.70 $2.00 $4.40 $1.95 $4.80 $3.252 6 o2 64 $3.77 $9.89 $1.63 $7.04 $9.38 $2.65 $5.81 $4.082 7 o2 74 $8.06 $10.93 $3.98 $5.96 $1.66 $7.62 $5.74 $3.312 8 o2 84 $5.04 $3.33 $1.80 $2.25 $7.11 $4.86 $0.36 $8.192 9 o2 94 $3.69 $3.78 $6.82 $4.89 $6.18 $4.43 $2.30 $5.352 10 o2 104 $0.12 $9.15 $0.12 $4.33 $0.12 $5.07 $1.98 $6.803 1 o3 14 $2.05 $8.89 $2.85 $3.99 $5.24 $10.95 $11.06 $2.743 2 o3 24 $4.54 $5.87 $5.20 $7.87 $2.40 $8.14 $10.54 $1.733 4 o3 44 $0.32 $0.25 $2.52 $0.90 $2.24 $2.34 $3.21 $2.993 5 o3 54 $7.63 $1.45 $1.18 $3.29 $8.81 $4.74 $0.13 $4.743 6 o3 64 $9.13 $7.95 $8.08 $3.65 $12.39 $12.52 $10.17 $10.693 7 o3 74 $13.22 $4.03 $12.90 $9.35 $2.42 $11.45 $9.67 $15.163 8 o3 84 $0.65 $0.65 $3.80 $3.75 $0.85 $3.75 $0.80 $3.253 9 o3 94 $1.69 $0.54 $1.86 $2.19 $1.81 $3.92 $2.27 $0.993 10 o3 104 $3.04 $5.23 $3.77 $0.91 $3.22 $4.14 $0.85 $5.054 1 o4 14 $10.60 $1.08 $0.11 $10.28 $1.08 $6.17 $4.33 $3.464 2 o4 24 $0.00 $11.28 $5.39 $7.89 $9.02 $5.39 $12.40 $8.774 3 o4 34 $3.53 $1.59 $1.15 $1.73 $2.38 $2.31 $0.69 $2.164 5 o4 54 $8.50 $7.18 $10.93 $0.77 $4.75 $7.29 $3.53 $10.384 6 o4 64 $3.05 $1.87 $5.02 $5.81 $0.10 $3.55 $6.40 $0.394 7 o4 74 $2.60 $2.34 $2.34 $3.64 $6.50 $3.77 $9.36 $4.034 8 o4 84 $1.45 $0.25 $4.68 $3.29 $4.62 $1.71 $3.86 $3.604 9 o4 94 $0.58 $2.82 $0.09 $3.22 $2.19 $0.72 $1.97 $1.974 10 o4 104 $7.33 $1.63 $6.88 $1.63 $7.33 $7.61 $7.52 $6.165 1 o5 14 $2.10 $2.58 $2.26 $0.32 $1.67 $2.21 $5.33 $1.025 2 o5 24 $0.80 $0.90 $3.65 $2.85 $1.45 $4.75 $4.30 $2.355 3 o5 34 $9.86 $4.21 $2.10 $10.39 $10.00 $8.68 $5.66 $1.585 4 o5 44 $10.49 $9.17 $9.94 $10.82 $5.08 $5.52 $4.09 $0.115 6 o5 64 $4.47 $1.67 $5.33 $5.12 $3.39 $5.01 $0.27 $5.335 7 o5 74 $3.95 $5.60 $2.49 $2.92 $5.78 $2.37 $5.66 $0.675 8 o5 84 $7.21 $2.78 $1.96 $1.03 $0.21 $6.07 $2.27 $3.715 9 o5 94 $5.60 $3.04 $5.60 $8.25 $4.65 $7.02 $7.78 $3.985 10 o5 104 $3.75 $10.96 $7.79 $8.94 $13.56 $0.00 $13.12 $13.706 1 o6 14 $2.90 $5.50 $8.00 $2.90 $8.60 $5.30 $8.00 $7.806 2 o6 24 $0.71 $3.26 $2.35 $8.77 $7.14 $6.83 $9.89 $8.466 3 o6 34 $9.65 $5.22 $2.61 $7.95 $2.22 $6.00 $4.56 $10.69

6 4 o6 44 $8.27 $0.98 $7.29 $2.07 $2.56 $1.97 $4.24 $1.776 5 o6 54 $4.79 $1.99 $1.45 $4.58 $0.11 $0.05 $3.23 $3.076 7 o6 74 $1.49 $2.94 $0.95 $1.58 $2.09 $0.13 $1.87 $0.006 8 o6 84 $2.56 $7.57 $8.54 $4.27 $8.54 $0.61 $6.35 $3.546 9 o6 94 $5.53 $2.34 $4.04 $0.21 $2.44 $1.28 $3.61 $7.766 10 o6 104 $1.15 $7.05 $8.20 $5.25 $11.32 $0.33 $7.38 $2.307 1 o7 14 $10.15 $5.24 $6.27 $10.49 $5.93 $1.14 $6.61 $3.657 2 o7 24 $10.60 $2.54 $8.62 $6.07 $0.33 $3.98 $5.19 $4.207 3 o7 34 $15.64 $1.45 $14.67 $3.22 $13.38 $2.90 $0.16 $3.227 4 o7 44 $0.00 $1.56 $11.83 $7.54 $10.14 $7.15 $6.63 $12.487 5 o7 54 $1.28 $1.09 $3.22 $2.92 $2.68 $5.84 $2.19 $0.437 6 o7 64 $1.17 $2.47 $1.96 $1.96 $0.41 $1.01 $0.16 $1.147 8 o7 84 $12.49 $11.00 $4.31 $13.83 $2.23 $2.53 $1.64 $13.387 9 o7 94 $7.67 $9.82 $0.94 $5.11 $12.51 $12.24 $7.94 $3.507 10 o7 104 $14.14 $15.86 $6.11 $6.11 $11.65 $6.69 $17.96 $12.618 1 o8 14 $2.73 $3.15 $2.03 $5.04 $1.47 $6.16 $4.62 $1.898 2 o8 24 $4.32 $6.57 $8.55 $0.45 $4.32 $4.32 $1.26 $8.738 3 o8 34 $2.85 $3.20 $2.65 $3.70 $3.45 $3.55 $4.75 $3.408 4 o8 44 $0.25 $4.81 $4.87 $4.11 $2.40 $0.89 $5.31 $4.748 5 o8 54 $4.02 $3.60 $3.40 $9.68 $6.80 $0.21 $8.03 $3.608 6 o8 64 $0.00 $4.03 $1.83 $0.00 $8.79 $6.71 $5.62 $3.548 7 o8 74 $11.74 $1.64 $4.46 $9.37 $2.38 $12.49 $5.05 $8.188 9 o8 94 $0.98 $1.88 $1.84 $0.68 $0.32 $1.54 $1.76 $1.668 10 o8 104 $2.38 $2.72 $2.88 $0.81 $2.29 $1.15 $3.52 $2.429 1 o9 14 $1.60 $1.02 $3.71 $2.55 $1.60 $6.92 $2.40 $1.24


Table A5 (continued )

9 2 o9 24 $0.28 $2.77 $0.55 $0.00 $5.99 $5.81 $6.45 $1.949 3 o9 34 $4.08 $3.92 $2.56 $2.89 $3.96 $0.54 $2.80 $1.779 4 o9 44 $0.89 $4.29 $0.94 $0.76 $0.85 $0.49 $0.67 $3.139 5 o9 54 $7.40 $4.17 $1.42 $0.47 $7.68 $2.28 $6.55 $9.399 6 o9 64 $4.15 $9.99 $5.95 $5.00 $10.10 $0.96 $5.10 $8.199 7 o9 74 $2.83 $3.50 $6.19 $6.73 $3.63 $4.17 $8.74 $0.819 8 o9 84 $1.88 $0.68 $0.60 $1.92 $0.66 $0.62 $0.96 $0.349 10 o9 104 $0.52 $2.04 $5.42 $1.98 $0.41 $0.35 $2.45 $5.6610 1 o10 14 $1.36 $8.56 $3.65 $6.79 $2.30 $6.16 $8.35 $0.5210 2 o10 24 $10.51 $4.82 $5.32 $4.33 $9.28 $2.10 $11.26 $5.5710 3 o10 34 $0.79 $3.77 $5.35 $2.80 $2.86 $1.28 $1.40 $3.5310 4 o10 44 $5.43 $0.18 $1.63 $6.70 $8.42 $6.43 $8.69 $2.8110 5 o10 54 $11.11 $10.67 $10.38 $13.99 $4.47 $10.82 $2.02 $10.6710 6 o10 64 $15.09 $8.86 $5.41 $7.71 $9.35 $10.66 $5.74 $9.8410 7 o10 74 $4.39 $17.00 $14.90 $11.65 $3.82 $5.92 $7.64 $12.6110 8 o10 84 $1.44 $3.01 $3.10 $0.38 $3.39 $1.78 $0.17 $1.3610 9 o10 94 $1.63 $1.40 $0.17 $4.96 $3.91 $5.36 $4.26 $1.81

Table A6Capacity for product k of facility l under scenario s in period t.

Cap (k)slt ScenarioFacility

sPeriod t

Product 1 Product 2

45% 55% 45% 55% 45% 55% 45% 55%1 2 1 2

l F1 100,000 150,000 120,000 140,000 120,000 170,000 140,000 160,000l F2 120,000 160,000 130,000 150,000 140,000 180,000 150,000 170,000l W1 160,000 180,000 150,000 170,000 180,000 200,000 170,000 190,000l W2 180,000 190,000 160,000 180,000 200,000 210,000 180,000 200,000l DC1 200,000 200,000 170,000 190,000 220,000 220,000 190,000 210,000l DC2 220,000 210,000 180,000 200,000 240,000 230,000 200,000 220,000

Table A7Capacity for product k of facility l under scenario s in period t.

E(Cap (k)slt) ScenarioFacility

sPeriod t

Product 1 Product 2

Expd. V Expd. V Expd. V Expd. V Expd. V Expd. V Expd. V Expd. V1 2 1 2

l F1 45,000 82,500 54,000 77,000 54,000 93,500 63,000 88,000l F2 54,000 88,000 58,500 82,500 63,000 99,000 67,500 93,500l W1 72,000 99,000 67,500 93,500 81,000 110,000 76,500 104,500l W2 81,000 104,500 72,000 99,000 90,000 115,500 81,000 110,000l DC1 90,000 110,000 76,500 104,500 99,000 121,000 85,500 115,500l DC2 99,000 115,500 81,000 110,000 108,000 126,500 90,000 121,000

Table A8Demand forecast for the product k of customer l under scenario s in period t.

dem (k)slt ScenarioFacility

sPeriod t

Product 1 Product 2

45% 55% 45% 55% 45% 55% 45% 55%1 2 1 2

l C1 150,000 165,000 130,000 155,000 165,000 181,500 143,000 170,500l C2 140,000 175,000 150,000 160,000 154,000 192,500 165,000 176,000

Table A9Demand forecast for the product k of customer l under scenario s in period t (expected value).

E(dem (k)slt) ScenarioFacility

sPeriod t

Product 1 Product 2

Exped. V Exped. V Exped. V Exped. V Exped. V Exped. V Exped. V Exped. V1 2 1 2

l C1 67,500 90,750 58,500 85,250 74,250 99,825 64,350 93,775l C2 63,000 96,250 67,500 88,000 69,300 105,875 74,250 96,800


Table A10Supplier capacity for product k of supplier l under scenario s in period t.

Sup (k)sl ScenarioFacility

sPeriod t

Product 1 Product 2

45% 55% 45% 55% 45% 55% 45% 55%1 2 1 2

l S1 135,000 145,000 146,000 155,000 155,250 166,750 167,900 178,250l S2 127,000 170,000 140,000 165,000 146,050 195,500 161,000 189,750

Table A11Supplier capacity for product k of Supplier l under scenario s in period t.

E(Sup (k)sl) ScenarioFacility

SPeriod t

Product 1 Product 2


l S1 60,750 79,750 65,700 85,250 69,863 91,713 75,555 98,038l S2 57,150 93,500 63,000 90,750 65,723 107,525 72,450 104,363

Table A12Unit sales revenue for product k to customer c under scenario s in period t.

ScenarioSr (k)sc

SPeriod t

Product 1 Product 2

45% 55% 45% 55% 45% 55% 45% 55%1 2 1 2

C1 40 30 50 55 46 35 58 63C2 45 35 55 60 52 40 63 69

Table A13Unit sales revenue for product k to customer c under scenario s in period t.

ScenarioE(Sr (k)sc)

SPeriod t

Product 1 Product 2


C1 40 30 50 55 46 35 58 63C2 45 35 55 60 52 40 63 69


Table A10 shows the capacity limit for each supplier to produceeach product in each period and under each scenario. Table A11shows the expected values of the data on production capacity.

Table A12 shows the revenue of each product in each periodunder each scenario for each customer. Table A13 shows theexpected values of the entries in Table A12.

Appendix B. The heuristic algorithm

A) Minimization algorithm

This algorithm reports the optimal or near-optimal solutionsbased on the greatest number of satisfied constraints. We followthese steps for an optimal or near-optimal solution when theobjective function is minimization:1. Set all binary variables equal to 1, then calculate the objective

function. This result is an upper bound.2. For each individual binary variable, calculate the value of the

objective function.

For example, if we have 3 binary variables:Set Y1¼1, Y2¼0, Y3¼0, and calculate Z1.Set Y2¼1, Y1¼0, Y3¼0, and calculate Z2.Set Y3¼1, Y1¼0, Y2¼0, and calculate Z3.

3. Sort the objective function values in Step 2 from minimum tomaximum, then sort the related binary variables in thesame order.For example, if we have sorted objective function values fromminimum to maximum, and the result is Z3, Z1, Z2, then we seethat the sort of binary variables is Y3, Y1, Y2.

4. For all possible combinations of SORTED binary variables, cal-culate the objective function. Continuing our example:Set Y3¼1, Y1¼1, Y2¼0, and calculate Z3,1Set Y3¼1, Y2¼1, Y1¼0, and calculate Z3,2Set Y3¼1, Y1¼1, Y2¼1, and calculate Z3,1,2

5. Compare the objective function values from Step 2 and Step 4,and investigate which individual or combinations of binaryvariables meet all or most constraints.

6. Report optimum or near-optimum solutions based on theresults in Step 5.


7. Calculate the lower bound based on the first binary variable in Step 3.8. In our example, Y3¼1, so the lower bound is Z3.

B) Maximization algorithm

All steps are the same as in the minimization algorithm, exceptfor Steps 1, 3, 6:Step 1. Set all binary variables equal to 1, then calculate theobjective function. This result is a lower bound.Step 3. Sort the objective function values frommaximum tominimum.Step 6. Calculate the upper bound based on the first binaryvariable in Step 3.

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