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Research ArticleBackup Sourcing Decisions for Coping withSupply Disruptions under Long-Term Horizons

Jing Hou1 and Lijun Sun2

1School of Business, Hohai University, Nanjing, Jiangsu 211100, China2Institute of Systems Engineering, Faculty of Management and Economics, Dalian University of Technology,Dalian, Liaoning 116023, China

Correspondence should be addressed to Jing Hou; [email protected]

Received 27 April 2016; Accepted 24 August 2016

Academic Editor: Elmetwally Elabbasy

Copyright © 2016 J. Hou and L. Sun. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies a buyer’s inventory control problem under a long-term horizon. The buyer has one major supplier that is proneto disruption risks and one backup supplier with higher wholesale price. Two kinds of sourcingmethods are available for the buyer:single sourcingwith/without contingent supply and dual sourcing. In contingent sourcing, the backup supplier is capacitated and/orhas yield uncertainty, whereas in dual sourcing the backup supplier has an incentive to offer output flexibility during disruptedperiods.Thebuyer’s expected cost functions and the optimal base-stock levels using each sourcingmethod under long-termhorizonare obtained, respectively.The effects of three risk parameters, disruption probability, contingent capacity or uncertainty, and backupflexibility, are examined using comparative studies and numerical computations. Four sourcing methods, namely, single sourcingwith contingent supply, dual sourcing, and single sourcing from either of the two suppliers, are also compared. These findings canbe used as a valuable guideline for companies to select an appropriate sourcing strategy under supply disruption risks.

1. Introduction

With widespread applications of outsourcing, supply chainsare becoming increasingly dependent upon suppliers, andsupply interruption can obstruct the normal operations ofthe entire supply chain. The situation with Toyota in Japanserves as a perfect example. After the disasters hit Japan inMarch of 2011, Toyota and their part suppliers struggled toresume operations, which paralyzed the downstream supplychain; for example, General Motors was reported to be thefirstUS automaker to close a factory because of a short supplyof a Japan-made part. Overall, the global auto industry issuffering losses in the production of hundreds of thousandsof vehicles. Additionally, companies around the world arefeeling the impacts of Japan’s disasters as various supplies fall,because Japan accounts for roughly one-fifth of the world’ssupply of silicon wafers used to make semiconductors, ishome to a large number ofmanufacturers for a keymaterial inliquid-crystal-display panels, and supplies about 90% of theworld’s need of a chemical used in making circuit boards fortelephone handsets (Supply Chain Asia [1]).

As the world is becoming increasingly volatile and uncer-tain, one important strategy and tactic for building supplychain resilience is to establish backup supplies, in the formof contingency supplies or dual sourcing, even if they costmore (Supply Chain Asia [1]). While the appropriate levelof resilience depends heavily on the time horizon beingconsidered (Swaminathan and Tomlin [2]), supply chainmanagers must not only consider all possible risks around,but also plan for sufficiently long term when making inven-tory and supply decisions. Most research to date, however,has studied the optimal solutions for finite-horizon situationsunder supply disruption risks. These solutions are not alwaysoptimal for infinite-horizon problems and could even leadto either incorrect financial investment or wrong partners.Therefore, it is important to examine decisions under long-term horizons so as to reduce the impact of supply risk.Unlike most of the existing research efforts on supply riskmanagement that look for optimal decision parameters andstrategies in finite horizon, especially in single period, thiswork aims to study the optimal backup sourcing methodsand inventory management decisions from the viewpoint of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016, Article ID 6716058, 16 pageshttp://dx.doi.org/10.1155/2016/6716058

2 Discrete Dynamics in Nature and Society

minimizing the buyer’s long-term cost. Note we use the term,buyer, in the paper to generally refer to any firm procuringphysical items from outside vendors.

We study the situation in which a buyer has two sourcesfor the same product: a main source and a backup supplysource, where the former is prone to supply disruptionsduring which it provides no supply of the critical componentand the latter offers higher wholesale price. The buyer needsto decide whether to use the latter as a contingent source or aregular one, that is, single sourcing with contingent supply ordual sourcing. Under single sourcing with contingent supply,since the backup supplier replenishes inventory at unplannedmoments when themain supplier is disrupted, its productionmay be insufficient to meet the buyer’s order (Chen et al.[3]). Consequently, the buyer may suffer from the contingentsupplier’s yield uncertainty and/or limited capacity duringdisrupted periods. For example, when Japan earthquakeoccurred in March 2011, a plasma display panel maker inAnhui province of China was facing supply disruptions fromits main supplier located in Japan. The company had toresort to a small domestic supplier for contingent supply ofraw materials. As the contingent supplier’s invested capacitywas limited and its yield rate was uncertain, the panelmaker suffered a great loss during that period. The dualsourcing method may avoid contingent supplier’s problemsbut encounter new issues; because the buyer allocates aproportion of order to the more expensive supplier, its totalpurchasing costs during nondisrupted periods can be higher;on the other hand, the buyer would obtain more stabledelivery from the backup supplier during disrupted periods.

Under each sourcing method, the buyer needs to controlits inventory level to minimize the expected understock andoverstock costs, as well as backup purchasing cost. He/shewould estimate howmuch should be invested in supply chainresilience and prepare the stocks based on the likelihood ofdisruptions in next and more periods. It is reasonable toassume that the backup supplier provides no larger quantitythan the cycle demand each time and that the buyer acceptsthe entire delivery from the backup supplier. We will analyzethe impacts of the backup supply parameters on the sourcingmethod selection and inventory management decisions.

The paper is organized as follows. The related literatureis reviewed briefly in the next section. Section 3 describesthe problems and lists the assumptions used in our model.In Sections 4 and 5, the inventory decisions under singlesourcing with contingent supply and dual sourcing areexamined and analyzed separately.The comparisons betweenthe two sourcingmethods are explored in Section 6. Section 7summarizes our work, discusses the model limitations, andsuggests future research directions.

2. Literature Review

The issue of supply disruptions has received a great deal ofattention lately in the literature.We concentrate on the papersmost directly relevant to our research and refer the readerto the survey paper of Snyder et al. [4] for a comprehensivereview of recent literature on the subject.

2.1. Backup Supply under Supply Disruptions. Perfectly reli-able backup suppliers are considered by many researchers.For example, Yu et al. [5] evaluate the impacts of supply dis-ruption risks on the choice between single and dual sourcingmethods based on the assumption that the backup supplieris perfectly reliable. Hou et al. [6] focus on the backupcontract between a buyer and their reliable backup supplier tomitigate supply disruptions. Some existing studies assume aspot market as a contingent supply that is totally reliable (e.g.,Li et al. [7]), while in reality, the buyer often cannot receivethe desirable amount due to highmarket uncertainty, and theaccessible delivery thus may be limited or stochastic.

Different types of risks associated with a backup supplieror multiple suppliers are considered in some publications,including lead time uncertainty, random defaults, limitedcapacity, and asymmetrical information. Babich et al. [8]study the effects of disruption risk where one buyer dealswith risky suppliers that not only compete against eachother, but are subject to random defaults as well. Serel[9] examines the relationship between one buyer and onereliable manufacturer when there is competition from asecond supplier that is prone to supply disruptions. Yanget al. [10] study a manufacturer’s strategic use of a dualsourcing option when both suppliers have private, reliableinformation. In the work of Sajadieh and Eshghi [11], a dualsourcing model with constant demand and stochastic leadtime is established. Sting and Huchzermeier [12] analyzehow firms should contract with backup suppliers so that thelatter would install responsive capacity to mitigate imminentmismatches of uncertain supply and demand. The recentwork of Xu et al. [13] studies the contract between the buyerand an urgent supplier with private cost information.

In addition, there are a number of papers that focus onmodeling supplier selection and order allocation decisionsunder supply risks. Berger et al. [14], Ruiz-Torres and Mah-moodi [15, 16], and Sarkar andMohapatra [17] have addressedthe problemof the optimal size of supply base under the situa-tion where every supplier is prone to supply disruptions. Theresearch conducted by Burke et al. [18] relies on the classicnewsvendor framework to determine the optimal number ofsuppliers with the consideration of each supplier’s capacityand reliability. Dada et al. [19] consider a newsvendor servedby multiple suppliers, each of which delivers an amountstrictly less than the amount desired with certain probabili-ties.The work of Sawik [20] deals with the selection of supplyportfolio in the presence of supply chain disruption risks,and each supplier is assumed to have limited capacity, uniqueprice, and different level of quality for the purchased parts.

The above studies consider different backup supply typesand aim to derive optimal solutions in single-period models.In our study, we consider single and dual sourcing withcapacitated and/or uncertain backup supply under long-termplanning horizon. The impacts of the supplier’s capacity anduncertainty on the buyer’s decisions under two sourcingmethods are examined, respectively.

2.2. Supply Risk Management under Long-Term PlanningHorizon. There has been a group of works that considersmultiperiod or long-term system optimization under supply

Discrete Dynamics in Nature and Society 3

disruption risks. Parlar [21] and Parlar and Perry [22] con-sider stochastic inventory model with supply disruptions. Byusing the renewal reward theorem, the long-run average costand reordering policies are derived. These models focus onderiving optimal multiperiod ordering policies but do notconsider using a backup supplier.

In regard to long-termmitigation strategies under supplydisruptions, Tomlin [23] focuses on the value of mitigationand contingency strategies for managing supply chain dis-ruption risks under the situation where the reliable suppliermay ormay not possess volume flexibility. Tomlin and Snyder[24] characterize the optimal threat-dependent inventorylevels in both infinite- and finite-horizon settings.The backupsupplier in the sourcing mitigation is assumed to be totallyreliable.The work of Serel [25] studies a multiperiod capacityreservation contract between a manufacturer and a long-term supplier when the random amount of supply availablefrom the spot market is independently and identically dis-tributed in each period. Schmitt et al. [26] develop a closed-form, approximate solution for a buyer who faces stochasticdemand or supply yield. Schmitt and Snyder [27] comparethe optimal inventory control solutions in single-period andinfinite-horizon models.Their work assumes that the backupsupplier is totally reliable.

This paper differs from the existing studies in the fol-lowing two major ways. First, unlike most of the existingstudies that investigate the optimal inventory system solu-tions under supply disruption risks with reliable or stochasticbackup supply, we distinguish between the characteristics ofbackup supply under contingent sourcing and dual sourcing;specifically, under single sourcing with contingent supply, thebackup supplier may have insufficient capacity and/or yielduncertainty, while under dual sourcing, the backup supplier’syield is dependent on both the buyer’s order allocation andoutput flexibility. Second, both inventory control policies andbackup sourcingmethod selection under long-term planninghorizon are examined in our research. An in-depth analysisis also conducted to examine the impacts of the supply risksand the cost parameters.We show (1) how the buyer’s optimalbase-stock level and the corresponding expected cost aredifferent under the two sourcing methods; (2) when singlesourcing will outperform dual sourcing and vice versa; and(3) how the backup supply parameters affect the decisions ina long-term planning model.

3. Problem Description and Assumptions

We consider a buyer that has two supply options for a criticalcomponent: one is cheaper but subject to randomdisruptions(Supplier 1) and the other is perfectly reliable and responsive,but more expensive (Supplier 2). The demand at the buyer iscertain and denoted as 𝑑. This assumption helps us focus onthe impact of supply risks. Besides, it is reasonable to supposestable demand for a lot of products, such as grocery and basicapparel (Lee [28]). The buyer operates by a periodic-review,base-stock policy. Unsatisfied demands are backordered witha unit penalty cost of 𝑝. The lead time is negligible, and theunit overstock cost is ℎ. Under such a situation, the buyerneeds to choose the sourcing method and determine the

optimal base-stock level 𝑠 by minimizing its expected long-term cost consisting of overstock and understock cost, as wellas backup purchasing cost.

Two sourcing methods are available for the buyer: singlesourcing with contingent backup supply and dual sourcing.Specifically, (1) under single sourcingwith contingent backupsupply, the buyer places an order to their major supplierreferred to as Supplier 1 at the beginning of each cycle. Whena disruption breaks down the main supplier’s capacity, thebackup supplier (Supplier 2) is available for meeting thebuyer’s demandbutmayhave limited capacity and/or randomyield; and (2) under dual sourcing, the buyer uses Supplier 2as a second regular source. Previous studies (e.g., Yan and Liu[29], Chen et al. [3]) have found that splitting orders amongmultiple suppliers is usually an effective approach to mitigatesupply disruptions. Let 𝜃𝑖 (𝑖 = 1, 2) denote the proportionof demand allocated to supplier 𝑖 where 𝜃1 + 𝜃2 = 1. In thiscase, as a regular supplier in a long-term relationship with thebuyer, Supplier 2 has the incentive to increase its output withextra capacity related to its order proportion 𝜃2 in the eventof Supplier 1 failure. Thus, the buyer has to determine boththe order allocation and the optimal base-stock level 𝑠. Notethat there is a possibility that the buyer single-sources fromSupplier 2 which is completely reliable; in this case the buyer’soptimal base-stock level can easily be deduced to be equal todemand 𝑑. Therefore, in what follows, we will first analyzethe buyer’s expected costs and optimal decisions under singlesourcing with contingent backup supply and dual sourcing,respectively, and then compare these two sourcing methods.

To formulate the supply disruption process, we followthe assumptions made in Schmitt et al. [26] and Schmittand Snyder [27]. Specifically, an infinite-state, discrete-timeMarkov chain is used to represent the disruption that lastsfor a number of periods, with 𝜋𝑖 representing the steady-stateprobability that the major supply has been disrupted for 𝑖consecutive periods, 𝑖 ∈ [0,∞). In addition, denote 𝛼 as thetransition probability from the normal to the disrupted stateand 𝛽 as the recovery probability from the disrupted to thenormal state. It is easy to see that the smaller the value of 𝛽 is,the longer the disruption would last. Thus, the parameter set(𝛼, 𝛽) represents the disruption probability and the averageduration, respectively, and can be used to characterize eachdisruption. The notation 𝜋𝑖 is used to denote the probabilityof being in state 𝑖 where the state represents the number ofconsecutive disrupted periods. In addition,when a disruptionoccurs, since the buyer cannot assess the disruption duration,it is logical to assume that the buyer will start the backupsupply from the first cycle of the disruption. A complete list ofsymbols used in the paper is provided inThe List of SymbolsUsed in the Paper.

4. Single Sourcing withContingent Backup Supply

During normal situations, the inventory position of the buyeris maintained at a level of 𝑠, while during disruptions thebuyer can order only from the backup supplier (Supplier 2),whose capacity is limited and/or yield is uncertain.


Before the single sourcingwith contingent supplymethodis analyzed, we first give the following results under singlesourcing without contingent supply, as presented by Tomlin[23] and Schmitt et al. [26].

Theorem1. For a buyer with deterministic demand and supplydisruptions,

(a) the expected cost per period is

𝐶1 (𝑠) = ∞∑𝑖=0

𝜋𝑖 [ℎ (𝑠 − (𝑖 + 1) 𝑑)+ + 𝑝 ((𝑖 + 1) 𝑑 − 𝑠)+] (1)

and is a convex function;(b) 𝑠∗1 = 𝑗∗𝑑, where 𝑗∗ (>1) is the smallest integer such that

∑𝑗∗−1𝑖=0 𝜋𝑖 ≥ 𝑝/(𝑝 + ℎ).4.1. Analysis of Contingent Backup Supply with Capacitatedand Uncertain Yield. Under single sourcing with capacitatedand uncertain backup supply, the backup capacity is limitedto 𝑦 (𝑦 ≤ 𝑑) in each period. That is, to satisfy the demandand maintain the inventory level of 𝑠, the maximum orderquantity the buyer can place to the contingent supplier in eachcycle during disruption periods is 𝑦. Moreover, the backupyield has an additive random quantity 𝑤, which is generallydistributed with a normal distribution, 𝑁(𝑢𝑤, 𝜎𝑤), and isindependent of the order quantity and may be positive ornegative (Schmitt and Snyder [27]). Thus the actual backupdelivery is 𝑦 + 𝑤 (we assume 𝐹(−𝑦) ≈ 0).

Under such assumption, the only difference betweenthe buyer’s expected cost with and without (as shown inTheorem 1) contingent supply is the added backuppurchasingcost from the contingent supplier and larger inventory levelper period; therefore, the buyer’s cost can be written bymodifying (1) as follows:

𝐶𝑐𝑢 = 𝜋0 [ℎ (𝑠 − 𝑑)+ + 𝑝 (𝑑 − 𝑠)+] + (𝑦 + 𝑢𝑤) 𝑐2 ∞∑𝑖=1

𝜋𝑖𝑖+ ∞∑𝑖=1

𝜋𝑖 [ℎ∫∞(𝑖+1)𝑑−𝑠−𝑖𝑦

(𝑠 + 𝑖𝑦 + 𝑤𝑖 − (𝑖 + 1) 𝑑)⋅ 𝑓𝑖 (𝑤𝑖) 𝑑𝑤𝑖+ 𝑝∫(𝑖+1)𝑑−𝑠−𝑖𝑦−∞

((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑤𝑖) 𝑓𝑖 (𝑤𝑖) 𝑑𝑤𝑖] ,

(2)

where 𝑤𝑖 is the total random quantity of 𝑤 for 𝑖 periods ofdisruption and follows a normal distribution with the densityfunction of 𝑓𝑖(𝑤𝑖), mean of 𝑖𝑢𝑤, and standard deviation of√𝑖𝜎𝑤. The first term is the inventory holding cost and thepenalty cost during normal periods.The second is the backupsupply purchasing cost. The last is the holding and penaltycost when disruption occurs. Note that the purchasing costfrom the major supplier would not affect the buyer’s decisionon inventory level 𝑠 under single sourcing; we thus do notconsider it in the cost function.

Using the standard normal loss function, 𝐺(𝑥), we canrewrite the cost function as

𝐶𝑐𝑢 (𝑠) = ∞∑𝑖=1

𝜋𝑖 [𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤)+ (𝑝 + ℎ)√𝑖𝜎𝑤𝐺((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 )]+ 𝜋0 [ℎ (𝑠 − 𝑑)+ + 𝑝 (𝑑 − 𝑠)+] + (𝑦 + 𝑢𝑤) 𝑐2 ∞∑

𝑖=1

𝜋𝑖𝑖.(3)

The next proposition provides the buyer’s optimal base-stocklevel in a long-term horizon.

Proposition 2. Under single sourcing with a backup supplierthat has capacitated yield and additive uncertainty, the buyer’soptimal base-stock level under long-term horizon, 𝑠∗𝑐𝑢, shouldsatisfy

∞∑𝑖=1

𝜋𝑖 (Φ((𝑖 + 1) 𝑑 − 𝑠∗𝑐𝑢 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 )) = ℎℎ + 𝑝,or 𝑠∗𝑐𝑢 = 𝑑.

(4)

Proof. See Appendix A.

The final optimal value of 𝑠𝑐𝑢 can be obtained by compar-ing 𝐶𝑐𝑢(𝑑) and 𝐶𝑐𝑢(𝑠∗𝑐𝑢) with 𝑠∗𝑐𝑢 that satisfies ∑∞𝑖=1 𝜋𝑖(Φ(((𝑖 +1)𝑑 − 𝑠∗𝑐𝑢 − 𝑖𝑦 − 𝑖𝑢𝑤)/√𝑖𝜎𝑤)) = ℎ/(ℎ + 𝑝).

The formula in Proposition 2 suggests that the buyerwould decrease their base-stock level 𝑠∗𝑐𝑢 if the backup supplycapacity 𝑦 increases. Since no close-form solution of 𝑠∗𝑐𝑢can be obtained, we will rely on numerical examples toexamine its properties in later part of this section. To examinethe impacts of the backup capacity limitation and yielduncertainty, respectively, we consider two special cases below.

4.2. Two Special Cases

Case 1 (contingent supply with capacitated yield). Considera backup supplier that can provide only a finite quantity 𝑦(𝑦 ≤ 𝑑) each period when disruption occurs; to satisfy thedemand and maintain the inventory level, the buyer orders𝑦 units from the backup supplier in each cycle. Based onthe assumptions above, the buyer’s expected cost function isgiven by the following expression:

𝐶𝑐 (𝑠) = ∞∑𝑖=0

𝜋𝑖 [ℎ (𝑠 + 𝑖𝑦 − (𝑖 + 1) 𝑑)++ 𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦)+ + 𝑐2𝑦𝑖] .

(5)

The buyer’s optimal decision is presented in Proposition 3.

Proposition 3. With a capacitated backup supply, the buyer’soptimal base-stock level under the long-term horizon, 𝑠∗𝑐 ,should satisfy the following condition:

𝑠∗𝑐 = 𝑗∗𝑑 − (𝑗∗ − 1) 𝑦, (6)


where 𝑗∗ (>1) is the smallest integer such that ∑𝑗∗−1𝑖=0 𝜋𝑖 ≥𝑝/(𝑝 + ℎ).Proof. See Appendix B.

It is clearly shown by Proposition 3 that (1) either higherunderstock cost or lower overstock cost implies higher base-stock level and (2) the higher the capacity the backup supplierhas, the less the buyer stock should be to deal with supplydisruption risks. Note that one special case exists when 𝑦 = 0,the buyer single-sources from S1, and in this case, 𝑠∗1 = 𝑗∗𝑑.Case 2 (contingent backup supply with uncertain yield). Ineach period, the buyer observes the current inventory level(IL) and places an order of 𝑠 − IL. The backup supplier’sdelivery then brings the buyer’s IL to a value of 𝑠+V during dis-rupted period, where V indicates the yield uncertainty whichis independent of the order quantity. We use the notations𝑢V, 𝜎V, 𝑚(V), and 𝑀(V) to denote the mean, deviation, pdf,and cdf of V, respectively, and suppose 𝑀(−𝑠) ≈ 0. Thebuyer’s expected cost in a long-term planning situation canbe calculated below and the buyer’s optimal base-stock levelis presented in Proposition 4.

𝐶𝑢 (𝑠) = ∞∑𝑖=1

𝜋𝑖 [ℎ∫∞𝑑−𝑠(𝑠 + V − 𝑑)𝑚 (V) 𝑑V

+ 𝑝∫𝑑−𝑠−∞

(𝑑 − 𝑠 − V)𝑚 (V) 𝑑V + 𝑐2 (𝑑 + 𝑢V)]+ 𝜋0 [ℎ (𝑠 − 𝑑)+ + 𝑝 (𝑑 − 𝑠)+] .

(7)

Proposition 4. With an uncertain backup supply, the buyer’soptimal base-stock level under the long-term horizon, 𝑠∗𝑢 ,should satisfy the following condition:

𝑠∗𝑢 = 𝑑 −𝑀−1 ((𝛼 + 𝛽) ℎ𝛼 (ℎ + 𝑝)) , if(𝛼 + 𝛽) ℎ𝛼 (ℎ + 𝑝) < 0.5

or 𝑠∗𝑢 = 𝑑.(8)

Proof. See Appendix C.

Given the value of 𝜎V, as𝑀(𝑑 − 𝑠∗𝑢) = (𝛼 + 𝛽)ℎ/𝛼(ℎ + 𝑝),we see that 𝑑 − 𝑠∗𝑢 increases with 𝑢V; that is, 𝑠∗𝑢 decreases with𝑢V. In the following analysis, we will further examine howthe various input parameters impact the buyer’s decision andexpected cost.

4.3. Numerical Analysis of Single Sourcing with ContingentSupply. Wenowprovide somenumerical examples to investi-gate the properties of single sourcing with contingent supply.The proposed base values of the input parameters are listedbelow:

𝑑: 100;𝑝: 18;ℎ: 2;𝛼: 0.1;

s∗

s∗1

s∗cu

s∗c

s∗u

0

50

100

150

200

250

300

350

400

450

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

Figure 1: The buyer’s optimal decision 𝑠∗ under single sourcingversus 𝛼.

𝛽: 0.5;𝑐1: 8;𝑐2: 11;𝑦: 50;(𝑢𝑤, 𝜎𝑤): (−15, 5);(𝑢V, 𝜎V): (0, 5).

4.3.1. The Impact of Supply Disruptions under Single Sourcing.We first calculate the optimal base-stock level 𝑠∗ by changing𝛼 from 0.01 to 0.9 with an increment of 0.1 (𝛽 = 0.5). Figure 1shows how the buyer’s optimal decisions are under singlesourcing with (𝑠∗𝑐𝑢, 𝑠∗𝑐 , 𝑠∗𝑢) and without the backup supply(𝑠∗1 ). It is seen that the buyer’s base-stock level under singlesourcing increases with the disruption probability and the useof a contingent backup supplier decreases the buyer’s stocklevel. In addition, the line of 𝑠∗𝑢 is flatter than the other two,meaning that the impact of disruption probability on 𝑠∗ ishighly affected by the type of backup supplier, which is largewhen the backup supply is capacitated (𝑠∗𝑐𝑢 or 𝑠∗𝑐 ). In addition,the optimal value of 𝑠∗𝑐𝑢 under both capacitated and uncertainbackup yield is larger than 𝑠∗𝑐 (under capacitated yield) and 𝑠∗𝑢(under uncertain yield).This implies that either larger backupuncertainty or smaller backup capacity would increase thebuyer’s inventory level.

While the use of a contingent backup supplier wouldalways lower the buyer’s inventory level, the impacts of supplydisruption on its expected costs vary for different types ofbackup supplier. Figure 2 plots the buyer’s expected profitsunder single sourcing with (𝐶𝑐𝑢, 𝐶𝑐, 𝐶𝑢) and without (𝐶1)backup supplier.

According to Figure 2, the buyer’s expected cost 𝐶increases with 𝛼. In particular, when the disruption riskis small enough (𝛼 ≤ 0.1 in this case), the buyer alwaysbenefits from using a contingent backup supplier, because𝐶𝑐𝑢, 𝐶𝑐, 𝐶𝑢 < 𝐶1. On the other hand, when the disruptionrisk becomes larger (0.1 < 𝛼 ≤ 0.6 in this case), the buyeronly benefits from using a contingent backup supplier with


Cc

Cu

C1

0

200

400

600

800

1000

1200

Cos

t

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

Ccu

Figure 2: The buyer’s expected cost 𝐶 under single sourcing versus𝛼.

no capacity limitation (𝐶𝑢 < 𝐶1 < 𝐶𝑐𝑢, 𝐶𝑐). Moreover, ifthe disruption risk is large enough (𝛼 ≥ 0.6 in this case), thebuyer would single-source from Supplier 1 and stock more tomitigate disruptions.

4.3.2. The Impact of Contingent Backup Supply under SingleSourcing. To learn the impacts of backup capacity and uncer-tainty, which are also influenced by the disruption probability,we illustrate the buyer’s decision and expected cost undersingle sourcing with capacitated or uncertain backup supplyin Figures 3 and 4, respectively.

As shown in Figure 3, although larger backup capacity𝑦 would decrease the buyer’s inventory level, only when thedisruption probability is small (𝛼 ≤ 0.1 in this case), the buyerwould benefit from larger backup capacity. Additionally,Figure 4 implies that the buyer always prefers lower backupuncertainty in order to reduce both inventory level andexpected cost.

5. Dual Sourcing

Under dual sourcing, the buyer maintains the inventory levelat 𝑠 and receives 𝑑 units of components in each nondisruptedperiod, with 𝜃1𝑑 units from Supplier 1 and 𝜃2𝑑 units fromSupplier 2 (𝜃1 + 𝜃2 = 1). In the event of Supplier 1 failure,Supplier 2, as a regular supplier, has an incentive to increaseits output to 𝑑𝜃𝑘2 (0 ≤ 𝑘 ≤ 1), where 𝑘 indicates its outputflexibility with 𝑘 values close to 0 indicating high outputflexibility and close to 1 suggesting low output flexibility (asin Ruiz-Torres and Mahmoodi [16]). It is worth mentioningthat, compared with single sourcing with contingent backupsupply discussed in Section 4, a regular supplier is likelymorereliable or capable than a contingent one, which indicates thatSupplier 2 output in disrupted periods under dual sourcing,𝑑𝜃𝑘2 , is larger than its delivery 𝑦 + 𝑤 when it is used as acontingent source. In what follows, we will consider 𝑑𝜃𝑘2 ≤𝑦 + 𝑤 as a possible scenario when contingent sourcingoutperforms dual sourcing.

5.1. Analysis of Dual Sourcing. The buyer’s long-termexpected cost under dual sourcing can be calculated as

𝐶𝑑 (𝑠, 𝜃1, 𝜃2) = ∞∑𝑖=1

𝜋𝑖 [ℎ (𝑠 + 𝑖𝑑𝜃𝑘2 − (𝑖 + 1) 𝑑)+

+ 𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑑𝜃𝑘2)+ + 𝑑𝑐2𝜃𝑘2] + 𝜋0 [ℎ (𝑠 − 𝑑)+ Δ𝑐𝜃2𝑑] .

(9)

Note thatΔ𝑐𝜃2𝑑 = (𝑐2−𝑐1)𝜃2𝑑 = (𝑐1𝜃1𝑑+𝑐2𝜃2𝑑)−𝑐1𝑑 denotesthe backup purchasing cost in nondisrupted period, whichis essentially the additional cost incurred by sourcing fromSupplier 2.

In this situation, the buyer needs to allocate their totalorder optimally and set the corresponding base-stock levelby minimizing the expected total cost. We first derive theoptimal base-stock level 𝑠∗𝑑 under dual sourcing with a givenorder allocation (𝜃1, 𝜃2) in the following proposition.

Proposition 5. Under dual souring, given backup outputflexibility k, the relationship between the buyer’s optimal base-stock level 𝑠∗𝑑 and the order allocation parameters (𝜃∗1 , 𝜃∗2 )satisfies

𝑠∗𝑑 = 𝑗∗𝑑 (1 − 𝜃∗𝑘2 ) + 𝑑𝜃∗𝑘2 , (10)

where 𝑗∗ (>1) is the smallest integer such that ∑𝑗∗−1𝑖=0 𝜋𝑖 ≥𝑝/(𝑝 + ℎ).Proof. See Appendix D.

It can be seen that if the buyerwould lower their inventorylevel, he/she should ordermore from Supplier 2 and less fromSupplier 1 with disruption risks. In contrast, if the buyer reliesmore on Supplier 1, he/she would stock more to mitigate thesupply risks. With the allocation set, (𝜃1, 𝜃2), the larger theoutput flexibility Supplier 2 has, the lower the inventory levelthe buyer should maintain.

Before the optimal order allocation under dual sourcing isobtained, we first compare the dual sourcing method and thesingle sourcing method with Supplier 1 as the major source.We write the buyer’s expected cost by using (9):

𝐶𝑑 = (𝑑 − 𝑑𝜃𝑘2)

⋅ [[ℎ𝑗∗−1∑𝑖=0

𝜋𝑖 (𝑗∗ − 𝑖 − 1) + 𝑝 ∞∑𝑖=𝑗∗𝜋𝑖 (𝑖 + 1 − 𝑗∗)]]

+ ∞∑𝑖=1

𝜋𝑖𝑑𝑐2𝜃𝑘2 + 𝜋0Δ𝑐𝜃2𝑑,

(11)

then the following result can be arrived for 0 < 𝑘 < 1.


y = 20y = 30y = 40y = 50

s∗

0

50

100

150

200

250

300

350

400

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

(a) 𝑠∗ versus 𝑦

y = 20y = 30y = 40y = 50

0

200

400

600

800

1000

1200

Cos

t

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

(b) Cost versus 𝑦

Figure 3: The impact of backup capacity under single sourcing with contingent supplier.

s∗

𝜎� = 25

𝜎� = 20

𝜎� = 15

𝜎� = 10

𝜎� = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

95

100

105

110

115

120

125

130

(a) 𝑠∗ versus 𝜎V

𝜎� = 25

𝜎� = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

0

100

200

300

400

500

600

700

800

900C

ost

(b) cost versus 𝜎V

Figure 4: The impact of backup uncertainty under single sourcing with contingent supplier.

Proposition 6. For 0 < 𝑘 < 1, if the backup supplier’s unitwholesale price (𝑐2) satisfies the relationship𝑐2 ≥ 𝑐𝐿2= ℎ∑𝑗

∗−1𝑖=0 𝜋𝑖 (𝑗∗ − 𝑖 − 1) + 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖 (𝑖 + 1 − 𝑗∗)1 − 𝜋0

(12)

single sourcing from Supplier 1 (the one with disruption risksand lower wholesale price) then outperforms the dual sourcingmethod.

Proof. Given 𝑠∗𝑑 = 𝑗∗𝑑(1 − 𝜃𝑘2) + 𝑑𝜃𝑘2 we have 𝑑𝐶𝑑/𝑑𝜃2 =𝑑𝑘𝜃𝑘−12 [𝑐2∑∞𝑖=1 𝜋𝑖−ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗−𝑖−1)−𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖+1−𝑗∗)]+𝜋0𝑑(𝑐2 −𝑐1). If 𝑐2∑∞𝑖=1 𝜋𝑖 −ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ −𝑖−1)−𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖+1 − 𝑗∗) ≥ 0, then 𝐶𝑑 is a convex and increasing function of𝜃2, and the optimal value 𝜃∗2 = 0; that is, the buyer wouldsingle-source from Supplier 1. The proof is complete.

Next, we compare dual sourcing and single sourcing withSupplier 2 and arrive at the following result for decision-making.

Proposition 7. For 0 < 𝑘 < 1, if the backup supplier’s outputflexibility (or wholesale price) meets the requirement𝑘 ≥ 𝑘𝐿= 𝜋0Δ𝑐ℎ∑𝑗∗−1𝑖=0 𝜋𝑖 (𝑗∗ − 𝑖 − 1) + 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖 (𝑖 + 1 − 𝑗∗) − 𝑐2∑∞𝑖=1 𝜋𝑖

(13)

or 𝑐2 < 𝑐𝑆2= 𝑘 (ℎ∑

𝑗∗−1𝑖=0 𝜋𝑖 (𝑗∗ − 𝑖 − 1) + 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖 (𝑖 + 1 − 𝑗∗)) + 𝜋0𝑐1𝜋0 + 𝑘 (1 − 𝜋0)

(14)

using Supplier 2 (the one with larger wholesale price butno disruptions) as the only source is better than using bothsuppliers.


Proof. Given 𝑑𝐶𝑑/𝑑𝜃2 = 𝑑𝑘𝜃𝑘−12 [𝑐2∑∞𝑖=1 𝜋𝑖 − ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ −𝑖−1)−𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖+1−𝑗∗)]+𝜋0𝑑(𝑐2−𝑐1), if (𝑑𝐶𝑑/𝑑𝜃2)|𝜃2=1 =𝑑𝑘[𝑐2∑∞𝑖=1 𝜋𝑖 − ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) − 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 −𝑗∗)] + 𝜋0𝑑(𝑐2 − 𝑐1) < 0, that is, 𝑘 ≥ 𝜋0Δ𝑐/(ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ −𝑖 − 1) + 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗) − 𝑐2∑∞𝑖=1 𝜋𝑖), then 𝑐2∑∞𝑖=1 𝜋𝑖 −ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) − 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗) < 0 and 𝑑𝐶𝑑/𝑑𝜃2increases with 𝜃2. Therefore, 𝑑𝐶𝑑/𝑑𝜃2 ≤ 0 for all 𝜃2 ∈ [0, 1],and the optimal value 𝜃∗2 = 1; that is, the buyer would single-source from Supplier 2. The proof is complete.

The above results indicate that the backup supplier’wholesale price has a critical impact on the buyer’s optimaldecisions. Specifically, if the backup supplier’s wholesale priceis too high, the buyer would not use it even if itmay have greatoutput flexibility and no risks. But if the backup supplier’soutput flexibility is small enough or its wholesale price issmall enough, the buyer would prefer to allocate the totalorder to it.Therefore, we could summarize themost profitablesourcing method and its corresponding decisions in thefollowing proposition.

Proposition 8. In making the optimal decisions of base-stocklevel 𝑠∗𝑑 and order allocation proportion (𝜃1, 𝜃2) for 0 < 𝑘 < 1under dual sourcing, there are two critical values for Supplier2 wholesale prices, (𝑐𝑆2 , 𝑐𝐿2 ), given by (10) and (13), respectively,which guide the decisions in the following way:

(i) if 𝑐𝑆2 < 𝑐2 < 𝑐𝐿2 , then choose dual sourcing with

𝜃∗2 = 𝑘−1√𝑘𝐿𝑘 ,𝜃∗1 = 1 − 𝜃∗2 ;𝑠∗𝑑 = 𝑗∗𝑑 − 𝑑 (𝑗∗ − 1) (𝑘𝐿𝑘 )

𝑘/(𝑘−1) ;(15)

(ii) if 𝑐2 > 𝑐𝐿2 , then choose Supplier 1 as the sole source with𝑠∗𝑑 = 𝑗∗𝑑;(iii) if 𝑐2 < 𝑐𝑆2 (or 𝑘 ≥ 𝑘𝐿), then choose Supplier 2 as the sole

source with 𝑠∗𝑑 = 𝑑,where 𝑗∗ (>1) is the smallest integer such that ∑𝑗∗−1𝑖=0 𝜋𝑖 ≥𝑝/(𝑝 + ℎ).Proof. Using the results in Propositions 6 and 7 yields theconclusions in (ii) and (iii) directly. Besides, under dualsourcing, as

𝑑2𝐶𝑑𝑑𝜃22 = 𝑑𝑘 (𝑘 − 1) 𝜃𝑘−22 [[𝑐2 ∞∑𝑖=1

𝜋𝑖 − ℎ𝑗∗−1∑𝑖=0

𝜋𝑖 (𝑗∗ − 𝑖 − 1)

− 𝑝 ∞∑𝑖=𝑗∗𝜋𝑖 (𝑖 + 1 − 𝑗∗)]]

,𝑐𝑆2 < 𝑐2 < 𝑐𝐿2 ,

(16)

𝐶𝑑 is a concave function of 𝜃2. The optimal decision shouldsatisfy 𝑑𝐶𝑑/𝑑𝜃2 = 0; that is, 𝜃∗2 = 𝑘−1√𝑘𝐿/𝑘, 𝑠∗𝑑 = 𝑗∗𝑑 − 𝑑(𝑗∗ −1)(𝑘𝐿/𝑘)𝑘/(𝑘−1). The proof is complete.

Proposition 9. Under dual sourcing (0 < 𝑘 < 1), thebuyer’s optimal order allocation to backup Supplier S2, 𝜃∗2 , is anincreasing function of 𝑘 if 𝑘𝐿 < 1; or if 𝑘𝐿 > 1, 𝜃∗2 first increasesuntil 𝑘 = 𝑘0, with 𝑘0 satisfying 1 − 1/𝑘0 + ln(𝑘𝐿/𝑘0) = 0, andthen decreases.

Proof. As Proposition 8 showed, under dual sourcing, thebuyer’s optimal order allocation probability satisfies 𝜃∗2 =𝑘−1√𝑘𝐿/𝑘, where 𝑘𝐿 = 𝜋0Δ𝑐/(ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) +𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗) − 𝑐2∑∞𝑖=1 𝜋𝑖). Then we have 𝑑𝜃∗2 /𝑑𝑘 =𝑘−1√𝑘𝐿/𝑘 = 𝑘−1√𝑘𝐿/𝑘(1/(1 − 𝑘))(1/𝑘 + (1/(𝑘 − 1)) ln(𝑘𝐿/𝑘)) =− 𝑘−1√𝑘𝐿/𝑘(1/𝑘2(1 − 𝑘)2)(1 − 1/𝑘 + ln(𝑘𝐿/𝑘)); therefore thesign of 𝑑𝜃∗2 /𝑑𝑘 is opposite to 1 − 1/𝑘 + ln(𝑘𝐿/𝑘). As 𝑑(1 −1/𝑘 + ln(𝑘𝐿/𝑘))/𝑑𝑘 = (1/𝑘)(1/𝑘 − 1) > 0, lim𝑘→0+(1 − 1/𝑘 +ln(𝑘𝐿/𝑘)) < 0, and lim𝑘→1(1 − 1/𝑘 + ln(𝑘𝐿/𝑘)) = ln 𝑘𝐿, if𝑘𝐿 > 1, ln 𝑘𝐿 > 0, then there must be 𝑘 = 𝑘0 within (0, 1)that satisfies 1 − 1/𝑘0 + ln(𝑘𝐿/𝑘0) = 0, and when 𝑘 < 𝑘0, 𝜃∗2increases with 𝑘 and when 𝑘 ≥ 𝑘0, 𝜃∗2 decreases with 𝑘; else if𝑘𝐿 < 1, ln 𝑘𝐿 < 0, then 1−1/𝑘+ln(𝑘𝐿/𝑘) < 0, and 𝑑𝜃∗2 /𝑑𝑘 > 0,which means 𝜃∗2 increases with 𝑘. The proof is complete.

These results indicate that the buyer should not alwaysdecrease their order allocation to Supplier 2 if its outputflexibility increases and that Supplier 2 should not alwayslower its output flexibility to obtain more orders. Both ofthem should evaluate the sign of 𝑘𝐿 − 1 before makingdecisions.

Moreover, when 𝑘 = 0, that is, Supplier 2 would satisfythe buyer’s total demand during disrupted periods, the buyerwould decrease the order proportion 𝜃2 to the smallestvalue that Supplier 2 could accept, denoted as 𝜃2 = 𝜃2 forillustration purpose, and 𝑠∗ = 𝑑.

When 𝑘 = 1, that is, Supplier 2 would not expandits output during disrupted periods, then (𝑑𝐶𝑑/𝑑𝜃2)|𝑘=1 =𝑑(∑∞𝑖=1 𝜋𝑖𝑐2 − ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) − 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗)) +𝜋0(𝑐2 − 𝑐1)𝑑. If 𝑐2 > 𝑐𝐿2 (1 − 𝜋0) + 𝜋0𝑐1 = ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 −1)+𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 +1−𝑗∗)+𝜋0𝑐1,𝐶𝑑|𝑘=1 increases with 𝜃2; thus𝜃∗2 = 0; else 𝜃∗2 = 1. This implies that if Supplier 2 has nooutput flexibility, the buyer would single-source from eitherSupplier 1 or Supplier 2, depending on whether its wholesaleprice is smaller or larger than 𝑐𝐿2 (1 − 𝜋0) + 𝜋0𝑐1.5.2. Numerical Analysis of Dual Sourcing. Similar to Sec-tion 4.3, the impacts of supply disruptions on the buyer’soptimal decisions and expected cost are examined throughnumerical analysis, and the value of Supplier 2 outputflexibility is also investigated with the basic parameters set inSection 4.3.

5.2.1. The Impact of Supply Disruptions under Dual Sourcing.Under dual sourcing, the buyer’s decisions are order allo-cation proportion (𝜃1, 𝜃2) and base-stock level 𝑠. Intuitively,from the buyer’s perspective, higher disruption probability


s∗

c2 = 13c2 = 12

c2 = 11

c2 = 10

c2 = 9

0

50

100

150

200

250

300

350

400

450

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

(a) 𝑠∗ versus 𝛼

c2 = 13

c2 = 12

c2 = 11

c2 = 10

c2 = 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

0

0.2

0.4

0.6

0.8

1

1.2

𝜃2

(b) 𝜃2 versus 𝛼

Figure 5: The buyer’s optimal decisions 𝑠 and 𝜃2 under dual sourcing versus 𝛼.

k = 0.9

k = 0.8

k = 0.7

k = 0.6

k = 0.1

𝜃2

00.10.20.30.40.50.60.70.80.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

(a) 𝜃2 versus 𝛼

𝜃2

𝛼 = 0.4

𝛼 = 0.5

𝛼 = 0.6

𝛼 = 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.2 0.3 0.4 0.5 0.6 0.7 0.80.1k

(b) 𝜃2 versus 𝑘

Figure 6: The optimal value of 𝜃2 under dual sourcing versus (𝛼, 𝑘).

may lead the buyer to allocate a larger order proportion 𝜃2;on the other hand, the larger the order proportion to Supplier2 is, the higher the purchasing cost would be incurred tothe buyer. Thus, we will examine the impacts of 𝛼 on 𝑠 and𝜃2, respectively, under different values of wholesale price 𝑐2(𝑘 = 0.7).

It is indicated by Figure 5 that the increase of supplydisruption risks raises the buyer’s inventory level under dualsourcing, but the buyer would not always allocate more toSupplier 2. Specifically, when 𝛼 is small (𝛼 ≤ 0.2 in this case),the buyer allocates more to Supplier 2 as disruption risksincrease; but when 𝛼 is large, the buyer decreases Supplier 2order proportion 𝜃2. This result implies that if the disruptionrisk is large enough, the buyer prefers to order less fromSupplier 2 and stock more to avoid large purchasing cost. Inaddition, the buyer’s expected cost increases with disruptionprobability according to our data results.

It is also seen that the backup Supplier S2 wholesale price𝑐2 plays an important role; in particular, as 𝑐2 increases, the

buyer tends to order less from S2 and stock more with aresult of larger expected cost. Furthermore, the impact of 𝑐2decreases as the value of 𝑐2 increases.5.2.2. The Impact of Output Flexibility under Dual Sourcing.From the results in Propositions 3 and 5, we can deduce that𝑠∗𝑑 ≤ 𝑠∗𝑐 holds if 𝑘 ≤ log𝜃∗

2

(𝑦/𝑑); that is, if Supplier 2 outputflexibility is large enough, dual sourcing would reduce itsinventory level compared to single sourcing with capacitatedbackup supply. For more implications concerning the impactof output flexibility, we will rely on numerical results shownin Figures 6-7.

Figure 6 provides the following interesting results: (1)larger output flexibility would bring down the order alloca-tion proportion 𝜃2 to S2 under low disruption risks (𝛼 < 0.4in this case; except at 𝛼 = 0.4when 𝜃2 is too small), indicatingthat backup Supplier S2 would decrease output flexibilityto obtain a larger order; but from the buyer’s perspective,he/she prefers to order more from S1 if S2 is very flexible;


k = 0.9

k = 0.1

0

100

200

300

400

500

600

700

Cos

t

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼

Figure 7: The buyer’s expected cost 𝐶 under dual sourcing versus(𝛼, 𝑘).

and (2) when the disruption probability is large (𝛼 ≥ 0.4in this case), the curve of 𝜃2 presents a concave functionof 𝑘 (as shown in Figure 6(b)); that is, 𝜃2 first increaseswith 𝑘 and then decreases. This behavior suggests an optimaloutput flexibility 𝑘∗ for S2 to maximize its order allocationproportion. Furthermore, the value of 𝑘∗ tends to decreasewith the disruption probability, implying that the larger thedisruption risk, the higher the output flexibility S2 shouldhave to maximize order allocation and the smaller the orderthe buyer should allocate to S2.

Figure 7 presents that higher backupflexibilitywould helpdecrease the buyer’s expected cost, but its impact is negligiblewhen supply disruption risk is very small (𝛼 ≤ 0.1 in thiscase) or very large (𝛼 ≥ 0.7 in this case).

6. Comparative Studies ofSingle and Dual Sourcing

In this section, we will rely on numerical examples to findout the most appropriate method among the four sourcingoptions: single sourcing from S1 (parameters with subscript1), single sourcing from S2 (subscript 2), dual sourcing(subscript 𝑑), and single sourcing from S1 with S2 as acapacitated and uncertain contingent supply (subscript 𝑐𝑢).

First, based on the analysis in Section 5.1, the followingresults can be given with the consideration of the decisionsituations of dual sourcing and single sourcing.

Proposition 10. In selecting the optimum sourcing method,the borderline between dual sourcing or single sourcing fromS1 is not affected by S2 output flexibility, while the lower theoutput flexibility is, the more possible the buyer would selectsingle sourcing from S2 over dual sourcing.

Proof. As Proposition 8 stated, there are two critical valuesof Supplier 2 wholesale prices, (𝑐𝑆2 , 𝑐𝐿2 ), and 𝑐2 < 𝑐𝑆2 suggeststhe possibility of selecting single sourcing from S2 over dualsourcing. As the value of 𝑐𝐿2 = (ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) +𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗))/(1 − 𝜋0) does depend on 𝑘 and 𝑐𝑆2 =

(𝑘(ℎ∑𝑗∗−1𝑖=0 𝜋𝑖(𝑗∗ − 𝑖 − 1) + 𝑝∑∞𝑖=𝑗∗ 𝜋𝑖(𝑖 + 1 − 𝑗∗)) + 𝜋0𝑐1)/(𝜋0 +𝑘(1−𝜋0)) is an increasing function of 𝑘, we can conclude thatthe possibility of 𝑐2 < 𝑐𝑆2 increases with 𝑘 with a given 𝑐2. Theproof is complete.

The optimal base-stock levels under four sourcing meth-ods are plotted in Figure 8, with the base-stock level ofselected sourcingmethods shown in dotted line.We considerfour cases: low or high output flexibility (𝑘 = 0.2 or 0.7) andlow or high backup purchasing cost (𝑐2 = 9 or 12).

Two observations can bemade fromFigure 8: (1) 𝑠∗𝑑 , 𝑠∗𝑐𝑢 ∈[𝑠∗1 , 𝑠∗2 ] holds, and the base-stock level under dual sourcing(𝑠∗𝑑) can be higher or lower than that under single sourcingwith contingent backup supply (𝑠∗𝑐𝑢); (2) the buyer does notalways select the sourcing method with the lowest base-stocklevel and sometimes selects the one with the highest level(𝑐2 = 12, 𝛼 ≥ 0.6). Note that the basic value of backupcapacity 𝑦 under single sourcing with contingent supply isset to be 50 in this example, and for the situations in which𝑦 ≥ 𝑑𝜃𝑘2 , single sourcing with contingent supply is supposedto outperform dual sourcing due to smaller purchasing cost.

By comparing the buyer’s expected costs under the foursourcing methods with disruption parameter 𝛼 changingfrom0.01 to 0.9 and backup output flexibility 𝑘 from0.1 to 0.9,we can pinpoint the optimal sourcingmethod under differentvalues of wholesale price 𝑐2 as shown in Figure 9. As seenfrom Figure 9, if the difference between 𝑐1 (=8) and 𝑐2 is small(Δ𝑐 = 1), the buyer should single-source from S2 exceptwhen their output flexibility is very large (dual sourcing)or the supply disruption probability is very small (singlesourcing with contingent backup supply). As S2 wholesaleprice becomes larger, the range of single sourcing from S2shrinks towards the upper-left corner until it disappears, andthe range of single sourcing from S1 extends to the right.This indicates that if the difference between the two suppliers’wholesale price becomes larger, the possibility to single-source from S2 decreases, especially under large disruptionrisk or output flexibility, and the likelihood of single sourcingfrom S1 increases especially under large disruption risk.

In addition, single sourcing with contingent supplyalways outperforms the other methods at very low disruptionrisk (𝛼 < 0.1) except when output flexibility is very high (𝑘 ≤0.2), with its range slightly expanding with Δ𝑐. On the otherhand, dual sourcing method always performs the best undersmall disruption risk and large backup output flexibility.

7. Concluding Remarks

In this paper, we have studied a buyer’s decisions when theirmajor supplier encounters supply disruption and backupsuppliers are available for selection.The buyer’s optimal base-stock level is obtained under single sourcing with contingentsupply and dual sourcing, respectively, through minimizingthe buyer’s long-term cost, which includes overstock cost,penalty cost for unsatisfied demand, and backup purchas-ing cost. The impacts of three risk parameters, disruptionprobability, contingent capacity or uncertainty, and backupflexibility, are analyzed.


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s∗

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(k = 0.2, c2 = 9) (k = 0.2, c2 = 12)

(k = 0.7, c2 = 9) (k = 0.7, c2 = 12)

0.90.80.70.60.50.40.30.20.10.01 0.90.80.70.60.50.40.30.20.10.01

0.90.80.70.60.50.40.30.20.10.01 0.90.80.70.60.50.40.30.20.10.01

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Figure 8: The buyer’s optimal base-stock levels under the four sourcing methods.

Specifically, under single sourcingwith contingent supply,we show (1) how backup capacity and uncertainty affect thebuyer’s decision and expected cost, respectively and (2) howthe impacts of supply disruption on the buyer’s expectedcosts differ for different types of contingent supplier. Table 1summarizes our findings about the single sourcing withcontingent supply.

Under dual sourcing, we identify the buyer’s optimalbase-stock level, the optimal order allocations, and also thecritical values of wholesale price or output flexibility todetermine whether the buyer should select dual sourcing orsingle sourcing. Our major findings about the impacts ofsupply disruptions and backup output flexibility are reportedin Table 2.

Moreover, four sourcing methods, namely, single sourc-ingwith contingent supply, dual sourcing, and single sourcingfrom either of the two suppliers, are compared. In choosingthe optimal sourcing method, we find that (1) the lowerthe output flexibility is, the more possible the buyer wouldselect single sourcing from S2 over dual sourcing; (2) singlesourcing with contingent supply always outperforms theothermethods at very low disruption risk except when outputflexibility is very high; and dual sourcing method is always

selected under small disruption risk and large backup outputflexibility; and (3) if the difference between the two suppliers’wholesale prices is small enough, the buyer usually single-sources from S2; but when their difference becomes larger,single sourcing with S2 would be replaced by dual sourcingfirst and then single sourcing with S1 under large disruptionrisks.

We conclude by highlighting three research directions notconsidered in this paper. First, the backup supplier’s profitand decision-making can be included in the models and itwould be interesting to explore a Stackelberg game structure.Particularly, the backup supplier’s decisions of wholesaleprice and capacities can be analyzed tomaximize its expectedprofit. Second, the lead time of the backup suppliers can be animportant factor for the buyer to consider when selecting abackup source.That is, how to select between single sourcingand dual sourcing when the lead times of the suppliers aredifferent could be a worthwhile research problem. Finally,we can also examine the feasibility of using multiple backupsuppliers in the event of supply disruption and figure out theoptimal numbers of backup suppliers in long-term horizondecision-making. We hope that these directions for futureresearchwill further refine the insights provided in this paper.


k

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kDual sourcing Single sourcing from

Supplier 1

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼 (c2 = 10)

0.01 0.1 0.2 0.3 0.4 0.5𝛼 (c2 = 9)

0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01𝛼 (c2 = 11)

0.01 0.1 0.2 0.3 0.4𝛼 (c2 = 13)

0.5 0.6 0.7 0.8 0.9

0.01 0.1 0.2 0.3𝛼 (c2 = 12)0.4 0.5 0.6 0.7 0.8 0.9

Figure 9: The buyer’s optimal sourcing methods versus (𝛼, 𝑘).


Table 1: Major results of single sourcing with contingent backup supply.

Properties Results

The value of contingentsupply

Under small disruption risks, the buyer wouldbenefit from the contingent supplier, especially

the one with larger capacity.

Under large disruption risks, the buyer would stockmore instead of using a contingent supplier.

The impact of disruptionrisks Larger disruption probability increases both the buyer’s optimal base-stock level and expected cost.

The impact of backupcapacity Larger backup capacity increases the buyer’s optimal base-stock level.

The impact of backupuncertainty Larger backup uncertainty increases both the buyer’s optimal base-stock level and expected cost.

Table 2: Major results of dual sourcing.

Properties Results

The impact of disruptionrisks

Larger disruption probability increases both the buyer’s optimal base-stock level and expected cost.Under small disruption risks, the buyer

allocates more orders to the backup supplier asdisruption risks increase.

The backup supplier would decrease its outputflexibility to obtain larger order.

Under large disruption risks, the buyer prefers to orderless from the backup supplier and stock more inventory

as disruption probability increases.The backup supplier would increase its output flexibility.

The impact of backupoutput flexibility

Under small disruption risks, larger outputflexibility leads to smaller order proportion to

the backup supplier.

Under large disruption risks, the buyer’s optimal orderproportion to the backup supplier becomes a concave

function of the supplier’s output flexibility.The impact of output flexibility is very limited when supply disruption risk is very small or too large.

Appendix

A. Proof for Proposition 2

Since

𝐶𝑐𝑢 (𝑠)= ∞∑𝑖=1

𝜋𝑖 [𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤) + (𝑝 + ℎ)√𝑖𝜎𝑤𝐺((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 )] + 𝜋0 [ℎ (𝑠 − 𝑑)+ + 𝑝 (𝑑 − 𝑠)+] + 𝑐1𝑑𝜋0

+ (𝑦 + 𝑢𝑤) 𝑐2 ∞∑𝑖=1

𝜋𝑖𝑖

={{{{{{{{{{{

∞∑𝑖=1

𝜋𝑖 [𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤) + (𝑝 + ℎ)√𝑖𝜎𝑤𝐺((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 )] + 𝜋0ℎ (𝑠 − 𝑑) + 𝑐1𝑑𝜋0 + (𝑦 + 𝑢𝑤) 𝑐2 ∞∑𝑖=1

𝜋𝑖𝑖, for 𝑠 ≥ 𝑑,∞∑𝑖=1

𝜋𝑖 [𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤) + (𝑝 + ℎ)√𝑖𝜎𝑤𝐺((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 )] + 𝜋0𝑝 (𝑑 − 𝑠) + 𝑐1𝑑𝜋0 + (𝑦 + 𝑢𝑤) 𝑐2 ∞∑𝑖=1

𝜋𝑖𝑖, for 𝑠 < 𝑑,

𝑑𝐶𝑐𝑢𝑑𝑠 ={{{{{{{{{{{

−∞∑𝑖=1

𝜋𝑖 [𝑝 + (ℎ + 𝑝)(Φ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 ) − 1)] + 𝜋0ℎ, for 𝑠 ≥ 𝑑,−∞∑𝑖=1

𝜋𝑖 [𝑝 + (ℎ + 𝑝)(Φ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 ) − 1)] − 𝜋0𝑝, for 𝑠 < 𝑑,𝑑2𝐶𝑐𝑢𝑑𝑠2 = ∞∑

𝑖=1

𝜋𝑖 ℎ + 𝑝√𝑖𝜎𝑤 𝜙((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 ) ≥ 0,

(A.1)


𝐶𝑐𝑢 is a concave function of 𝑠. The optimal decision shouldsatisfy

∞∑𝑖=1

𝜋𝑖 (Φ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑦 − 𝑖𝑢𝑤√𝑖𝜎𝑤 ))

= {{{{{{{

ℎℎ + 𝑝, for 𝑠 ≥ 𝑑,ℎℎ + 𝑝 − 𝜋0 = 𝛼ℎ − 𝑝𝛽𝛼 (ℎ + 𝑝) , for 𝑠 < 𝑑.

(A.2)

As 𝛼 < 𝛽 and ℎ < 𝑝, (𝛼ℎ − 𝑝𝛽)/𝛼(ℎ + 𝑝) < 0; we can havethat when 𝑠 < 𝑑, 𝐶𝑐𝑢 decreases with 𝑠. Therefore, the optimalvalue of 𝑠∗ either equals 𝑑 or satisfies∑∞𝑖=1 𝜋𝑖(Φ(((𝑖 + 1)𝑑− 𝑠−𝑖𝑦 − 𝑖𝑢𝑤)/√𝑖𝜎𝑤)) = ℎ/(ℎ + 𝑝). The proof is complete.

B. Proof for Proposition 3

Suppose 𝑠∗ −𝑦 = 𝑗(𝑑−𝑦), where 𝑗 is an integer; then 𝑠∗ −𝑦−𝑖(𝑑 − 𝑦) = 𝑗(𝑑 − 𝑦) − 𝑖(𝑑 − 𝑦) = (𝑗 − 𝑖)(𝑑 − 𝑦). If 𝑑 − 𝑦 ≥ 0,when 𝑖 + 1 ≤ 𝑗, 𝑠∗ − 𝑦 − (𝑖 + 1)(𝑑 − 𝑦) ≥ 0, and vice versa.

Thus

𝐶𝑐 (𝑠) = ∞∑𝑖=0

𝜋𝑖 [ℎ (𝑠 − 𝑦 − (𝑖 + 1) (𝑑 − 𝑦))+

+ 𝑝 ((𝑖 + 1) (𝑑 − 𝑦) − 𝑠 + 𝑦)+] + 𝑐2𝑦∞∑𝑖=1

𝜋𝑖𝑖 =𝑗−1∑𝑖=0

𝜋𝑖⋅ ℎ (𝑠 − 𝑦 − (𝑖 + 1) (𝑑 − 𝑦)) + ∞∑

𝑖=𝑗

𝜋𝑖⋅ 𝑝 ((𝑖 + 1) (𝑑 − 𝑦) − 𝑠 + 𝑦) + 𝑐2𝑦∞∑

𝑖=1

𝜋𝑖𝑖,𝑑𝐶𝑐𝑑𝑠 = ℎ𝑗−1∑

𝑖=0

𝜋𝑖 − 𝑝∞∑𝑖=𝑗

𝜋𝑖.

(B.1)

If 𝑑𝐶𝑐/𝑑𝑠 = ℎ∑𝑗−1𝑖=0 𝜋𝑖 − 𝑝∑∞𝑖=𝑗 𝜋𝑖 ≥ 0, 𝐶𝑐 increases with𝑠. Therefore, 𝑠∗ takes the smallest value; that is, 𝑗 takes thesmallest integer so that 𝑑𝐶𝑐/𝑑𝑠 = ℎ∑𝑗−1𝑖=0 𝜋𝑖 − 𝑝∑∞𝑖=𝑗 𝜋𝑖 =ℎ∑𝑗−1𝑖=0 𝜋𝑖 − 𝑝 + 𝑝∑𝑗−1𝑖=0 𝜋𝑖−1 = (ℎ + 𝑝)∑𝑗−1𝑖=0 𝜋𝑖 − 𝑝 ≥ 0, whichis also ∑𝑗−1𝑖=0 𝜋𝑖 ≥ 𝑝/(𝑝 + ℎ). Else if 𝑑𝐶𝑐/𝑑𝑠 = ℎ∑𝑗−1𝑖=0 𝜋𝑖 −𝑝∑∞𝑖=𝑗 𝜋𝑖 ≤ 0, 𝑗 takes the largest integer so that ∑𝑗−1𝑖=0 𝜋𝑖 ≤𝑝/(𝑝 + ℎ). Because ∑𝑗−1𝑖=0 𝜋𝑖 increases with 𝑗, then the twovalues of 𝑗 are the same for 𝑑𝐶𝑐/𝑑𝑠 ≥ 0 and 𝑑𝐶𝑐/𝑑𝑠 ≤ 0.

Thus the proof is complete.

C. Proof for Proposition 4

Because

𝐶𝑢 (𝑠) = ∞∑𝑖=1

𝜋𝑖⋅ [ℎ∫∞𝑑−𝑠(𝑠 + V − 𝑑)𝑚 (V) 𝑑V + 𝑝∫𝑑−𝑠

−∞(𝑑 − 𝑠 − V)𝑚 (V) 𝑑V]

+ 𝜋0 [ℎ (𝑠 − 𝑑)+ + 𝑝 (𝑑 − 𝑠)+] + 𝑐2 (𝑑 + 𝑢V) ∞∑𝑖=1

𝜋𝑖,

𝑑𝐶𝑢𝑑𝑠 ={{{{{{{{{

∞∑𝑖=1

𝜋𝑖 [ℎ − (ℎ + 𝑝)∫𝑑−𝑠−∞

𝑚(V) 𝑑V] + 𝜋0ℎ for 𝑠 ≥ 𝑑∞∑𝑖=1

𝜋𝑖 [ℎ − (ℎ + 𝑝)∫𝑑−𝑠−∞

𝑚(V) 𝑑V] − 𝜋0𝑝 for 𝑠 < 𝑑

={{{{{{{{{

𝛼𝛼 + 𝛽 [ℎ − (ℎ + 𝑝)∫𝑑−𝑠

−∞𝑚(V) 𝑑V] + 𝛽𝛼 + 𝛽ℎ for 𝑠 ≥ 𝑑

𝛼𝛼 + 𝛽 [ℎ − (ℎ + 𝑝)∫𝑑−𝑠

−∞𝑚(V) 𝑑V] − 𝛽𝛼 + 𝛽𝑝 for 𝑠 < 𝑑

= {{{{{{{ℎ − 𝛼𝛼 + 𝛽 (ℎ + 𝑝)∫

𝑑−𝑠

−∞𝑚(V) 𝑑V for 𝑠 ≥ 𝑑

𝛼ℎ − 𝛽𝑝𝛼 + 𝛽 − 𝛼𝛼 + 𝛽 (ℎ + 𝑝)∫𝑑−𝑠

−∞𝑚(V) 𝑑V for 𝑠 < 𝑑,

𝑑2𝐶𝑢𝑑𝑠2 = ∞∑𝑖=1

𝜋𝑖 (ℎ + 𝑝)𝑚 (𝑑 − 𝑠) ≥ 0,

(C.1)

𝐶𝑢 is a concave function of 𝑠. And the optimal decision shouldsatisfy

∫𝑑−𝑠−∞

𝑚(V) 𝑑V

={{{{{{{{{

ℎℎ + 𝑝 11 − 𝜋0 =(𝛼 + 𝛽) ℎ𝛼 (ℎ + 𝑝) , for 𝑠 ≥ 𝑑,

ℎ − (ℎ + 𝑝) 𝜋0(ℎ + 𝑝) (1 − 𝜋0) =𝛼ℎ − 𝑝𝛽𝛼 (ℎ + 𝑝) , for 𝑠 < 𝑑.

(C.2)

As 𝛼 < 𝛽 and ℎ < 𝑝, (𝛼ℎ − 𝑝𝛽)/𝛼(ℎ + 𝑝) < 0, 𝐶𝑢 decreaseswith 𝑠 when 𝑠 < 𝑑.

Therefore, the optimal base-stock level should satisfy∫𝑑−𝑠∗−∞

𝑚(V)𝑑V = (𝛼 + 𝛽)ℎ/𝛼(ℎ + 𝑝) or 𝑠∗ = 𝑑. The proof iscomplete.

D. Proof for Proposition 5

Suppose 𝑠∗ − 𝑑𝜃𝑘2 = 𝑗(𝑑 − 𝑑𝜃𝑘2), where 𝑗 is an integer; thenwhen 𝑖 + 1 ≤ 𝑗, 𝑠∗ −𝑑𝜃𝑘2 − (𝑖 + 1)(𝑑 − 𝑑𝜃𝑘2) ≥ 0, and vice versa.Furthermore,

𝐶𝑑 (𝑠) = ∞∑𝑖=1

𝜋𝑖 [ℎ (𝑠 + 𝑖𝑑𝜃𝑘2 − (𝑖 + 1) 𝑑)+

+ 𝑝 ((𝑖 + 1) 𝑑 − 𝑠 − 𝑖𝑑𝜃𝑘2)+ + 𝑑𝑐2𝜃𝑘2]+ 𝜋0 [ℎ (𝑠 − 𝑑) + Δ𝑐𝜃2𝑑] =

𝑗−1∑𝑖=1

𝜋𝑖ℎ (𝑠 − 𝑑𝜃𝑘2


− (𝑖 + 1) (𝑑 − 𝑑𝜃𝑘2)) + ∞∑𝑖=𝑗

𝜋𝑖𝑝 ((𝑖 + 1) (𝑑 − 𝑑𝜃𝑘2)

− 𝑠 + 𝑑𝜃𝑘2) + ∞∑𝑖=1

𝜋𝑖𝑑𝑐2𝜃𝑘2 + 𝜋0 [ℎ (𝑠 − 𝑑) + Δ𝑐𝜃2𝑑] ,𝑑𝐶𝑑𝑑𝑠 = ℎ𝑗−1∑

𝑖=1


𝜋𝑖 + 𝜋0ℎ = ℎ𝑗−1∑𝑖=0


𝜋𝑖 = (ℎ

+ 𝑝) 𝑗−1∑𝑖=0

𝜋𝑖 − 𝑝(D.1)

increases with 𝑗 (which also means 𝑑𝐶𝑑/𝑑𝑠 increases with 𝑠;i.e., 𝑑2𝐶𝑑/𝑑𝑠2 > 0).

If 𝑑𝐶𝑑/𝑑𝑠 = (ℎ+𝑝)∑𝑗−1𝑖=0 𝜋𝑖−𝑝 ≥ 0, then𝐶𝑑 increases with𝑠, so 𝑠∗ takes the smallest value; that is, 𝑗 takes the smallestinteger so that 𝑑𝐶𝑑/𝑑𝑠 = (ℎ+𝑝)∑𝑗−1𝑖=0 𝜋𝑖−𝑝 ≥ 0, which is also∑𝑗−1𝑖=0 𝜋𝑖 ≥ 𝑝/(𝑝 + ℎ). Else if 𝑑𝐶𝑑/𝑑𝑠 = (ℎ + 𝑝)∑𝑗−1𝑖=0 𝜋𝑖 − 𝑝 < 0,𝑗 takes the largest integer so that∑𝑗−1𝑖=0 𝜋𝑖 < 𝑝/(𝑝+ℎ). Because∑𝑗−1𝑖=0 𝜋𝑖 increases with 𝑗, then the two values of 𝑗 are the samefor 𝑑𝐶𝑑/𝑑𝑠 ≥ 0 and 𝑑𝐶𝑑/𝑑𝑠 ≤ 0.

Thus the proof is complete.

The List of Symbols Used in the Paper

𝛼: The disruption probability or thetransition probability from the normal tothe disrupted state𝛽: The recovery probability from thedisrupted to the normal state𝜋𝑖: The steady-state probability that the majorsupply has been disrupted for 𝑖consecutive periods, 𝑖 ∈ [0,∞)𝑑: The average demand per cycle (unit)𝑝: Unit understock cost of the buyer ($/unit)(𝑝 > 𝑐2 + ℎ)ℎ: Unit overstock cost of the buyer ($/unit)𝑠: The buyer’s optimal base-stock level, adecision variable (unit)𝐿: The customer service-level𝐶: The buyer’s expected total cost per period

B(⋅): The standard normal distribution function𝐺(⋅): The standard normal loss function𝑦: Supplier 2 average backup capacity duringdisrupted period under single sourcing(𝑦 ≤ 𝑑)𝑤: Supplier 2 additive random quantity 𝑤under single sourcing with capacitatedand uncertain yield (it follows a normaldistribution𝑁(𝑢𝑤, 𝜎𝑤), with 𝐹(⋅) as itsdistribution function and 𝑓(⋅) as itsdensity function)

V: Supplier 2 yield uncertainty under singlesourcing with uncertain yield (it follows anormal distribution,𝑁(𝑢V, 𝜎V), with 𝐺(⋅) as itsdistribution function and 𝑔(⋅) as its densityfunction)𝜃𝑖: The proportion of demand allocated to supplier𝑖 under dual sourcing (𝑖 = 1, 2, and 𝜃1 + 𝜃2 = 1)𝑘: Supplier 2 output flexibility during disruptedperiod under dual sourcing (0 ≤ 𝑘 ≤ 1).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by National Natural ScienceFoundation of China (71201047, 71433003, and 71601069), theHumanities and Social Sciences Foundation of the Ministryof Education in China (12YJC630058), and the FundamentalResearch Funds for the Central Universities (2016B09314).

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