Backup Sourcing Decisions for Coping with Supply Disruptions under Long-Term Horizons

This paper studies a buyer’s inventory control problem under a long-term horizon. The buyer has one major supplier that is prone to 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/or has yield uncertainty, whereas in dual sourcing the backup supplier has an incentive to offer output flexibility during disrupted periods.Thebuyer’s expected cost functions and the optimal base-stock levels using each sourcingmethod under long-termhorizon are obtained, respectively.The effects of three risk parameters, disruption probability, contingent capacity or uncertainty, and backup flexibility, are examined using comparative studies and numerical computations. Four sourcing methods, namely, single sourcing with contingent supply, dual sourcing, and single sourcing from either of the two suppliers, are also compared. These findings can be used as a valuable guideline for companies to select an appropriate sourcing strategy under supply disruption risks.


Introduction
With widespread applications of outsourcing, supply chains are becoming increasingly dependent upon suppliers, and supply interruption can obstruct the normal operations of the entire supply chain.The situation with Toyota in Japan serves as a perfect example.After the disasters hit Japan in March of 2011, Toyota and their part suppliers struggled to resume operations, which paralyzed the downstream supply chain; for example, General Motors was reported to be the first US auto maker to close a factory because of a short supply of a Japan-made part.Overall, the global auto industry is suffering losses in the production of hundreds of thousands of vehicles.Additionally, companies around the world are feeling the impacts of Japan's disasters as various supplies fall, because Japan accounts for roughly one-fifth of the world's supply of silicon wafers used to make semiconductors, is home to a large number of manufacturers for a key material in liquid-crystal-display panels, and supplies about 90% of the world's need of a chemical used in making circuit boards for telephone handsets (Supply Chain Asia [1]).
As the world is becoming increasingly volatile and uncertain, one important strategy and tactic for building supply chain resilience is to establish backup supplies, in the form of contingency supplies or dual sourcing, even if they cost more (Supply Chain Asia [1]).While the appropriate level of resilience depends heavily on the time horizon being considered (Swaminathan and Tomlin [2]), supply chain managers must not only consider all possible risks around, but also plan for sufficiently long term when making inventory and supply decisions.Most research to date, however, has studied the optimal solutions for finite-horizon situations under supply disruption risks.These solutions are not always optimal for infinite-horizon problems and could even lead to either incorrect financial investment or wrong partners.Therefore, it is important to examine decisions under longterm horizons so as to reduce the impact of supply risk.Unlike most of the existing research efforts on supply risk management that look for optimal decision parameters and strategies in finite horizon, especially in single period, this work aims to study the optimal backup sourcing methods and inventory management decisions from the viewpoint of minimizing the buyer's long-term cost.Note we use the term, buyer, in the paper to generally refer to any firm procuring physical items from outside vendors.
We study the situation in which a buyer has two sources for the same product: a main source and a backup supply source, where the former is prone to supply disruptions during which it provides no supply of the critical component and the latter offers higher wholesale price.The buyer needs to decide whether to use the latter as a contingent source or a regular one, that is, single sourcing with contingent supply or dual sourcing.Under single sourcing with contingent supply, since the backup supplier replenishes inventory at unplanned moments when the main supplier is disrupted, its production may be insufficient to meet the buyer's order (Chen et al. [3]).Consequently, the buyer may suffer from the contingent supplier's yield uncertainty and/or limited capacity during disrupted periods.For example, when Japan earthquake occurred in March 2011, a plasma display panel maker in Anhui province of China was facing supply disruptions from its main supplier located in Japan.The company had to resort to a small domestic supplier for contingent supply of raw materials.As the contingent supplier's invested capacity was limited and its yield rate was uncertain, the panel maker suffered a great loss during that period.The dual sourcing method may avoid contingent supplier's problems but encounter new issues; because the buyer allocates a proportion of order to the more expensive supplier, its total purchasing costs during nondisrupted periods can be higher; on the other hand, the buyer would obtain more stable delivery from the backup supplier during disrupted periods.
Under each sourcing method, the buyer needs to control its inventory level to minimize the expected understock and overstock costs, as well as backup purchasing cost.He/she would estimate how much should be invested in supply chain resilience and prepare the stocks based on the likelihood of disruptions in next and more periods.It is reasonable to assume that the backup supplier provides no larger quantity than the cycle demand each time and that the buyer accepts the entire delivery from the backup supplier.We will analyze the impacts of the backup supply parameters on the sourcing method selection and inventory management decisions.
The paper is organized as follows.The related literature is reviewed briefly in the next section.Section 3 describes the problems and lists the assumptions used in our model.In Sections 4 and 5, the inventory decisions under single sourcing with contingent supply and dual sourcing are examined and analyzed separately.The comparisons between the two sourcing methods are explored in Section 6. Section 7 summarizes our work, discusses the model limitations, and suggests future research directions.

Literature Review
The issue of supply disruptions has received a great deal of attention lately in the literature.We concentrate on the papers most directly relevant to our research and refer the reader to the survey paper of Snyder et al. [4] for a comprehensive review of recent literature on the subject.

Backup Supply under Supply Disruptions.
Perfectly reliable backup suppliers are considered by many researchers.For example, Yu et al. [5] evaluate the impacts of supply disruption risks on the choice between single and dual sourcing methods based on the assumption that the backup supplier is perfectly reliable.Hou et al. [6] focus on the backup contract between a buyer and their reliable backup supplier to mitigate supply disruptions.Some existing studies assume a spot market as a contingent supply that is totally reliable (e.g., Li et al. [7]), while in reality, the buyer often cannot receive the desirable amount due to high market uncertainty, and the accessible delivery thus may be limited or stochastic.
Different types of risks associated with a backup supplier or multiple suppliers are considered in some publications, including lead time uncertainty, random defaults, limited capacity, and asymmetrical information.Babich et al. [8] study the effects of disruption risk where one buyer deals with risky suppliers that not only compete against each other, but are subject to random defaults as well.Serel [9] examines the relationship between one buyer and one reliable manufacturer when there is competition from a second supplier that is prone to supply disruptions.Yang et al. [10] study a manufacturer's strategic use of a dual sourcing option when both suppliers have private, reliable information.In the work of Sajadieh and Eshghi [11], a dual sourcing model with constant demand and stochastic lead time is established.Sting and Huchzermeier [12] analyze how firms should contract with backup suppliers so that the latter would install responsive capacity to mitigate imminent mismatches of uncertain supply and demand.The recent work of Xu et al. [13] studies the contract between the buyer and an urgent supplier with private cost information.
In addition, there are a number of papers that focus on modeling supplier selection and order allocation decisions under supply risks.Berger et al. [14], Ruiz-Torres and Mahmoodi [15,16], and Sarkar and Mohapatra [17] have addressed the problem of the optimal size of supply base under the situation where every supplier is prone to supply disruptions.The research conducted by Burke et al. [18] relies on the classic newsvendor framework to determine the optimal number of suppliers with the consideration of each supplier's capacity and reliability.Dada et al. [19] consider a newsvendor served by multiple suppliers, each of which delivers an amount strictly less than the amount desired with certain probabilities.The work of Sawik [20] deals with the selection of supply portfolio in the presence of supply chain disruption risks, and each supplier is assumed to have limited capacity, unique price, and different level of quality for the purchased parts.
The above studies consider different backup supply types and aim to derive optimal solutions in single-period models.In our study, we consider single and dual sourcing with capacitated and/or uncertain backup supply under long-term planning horizon.The impacts of the supplier's capacity and uncertainty on the buyer's decisions under two sourcing methods are examined, respectively.

Supply Risk Management under Long-Term Planning
Horizon.There has been a group of works that considers multiperiod or long-term system optimization under supply disruption risks.Parlar [21] and Parlar and Perry [22] consider stochastic inventory model with supply disruptions.By using the renewal reward theorem, the long-run average cost and reordering policies are derived.These models focus on deriving optimal multiperiod ordering policies but do not consider using a backup supplier.
In regard to long-term mitigation strategies under supply disruptions, Tomlin [23] focuses on the value of mitigation and contingency strategies for managing supply chain disruption risks under the situation where the reliable supplier may or may not possess volume flexibility.Tomlin and Snyder [24] characterize the optimal threat-dependent inventory levels in both infinite-and finite-horizon settings.The backup supplier in the sourcing mitigation is assumed to be totally reliable.The work of Serel [25] studies a multiperiod capacity reservation contract between a manufacturer and a longterm supplier when the random amount of supply available from the spot market is independently and identically distributed in each period.Schmitt et al. [26] develop a closedform, approximate solution for a buyer who faces stochastic demand or supply yield.Schmitt and Snyder [27] compare the optimal inventory control solutions in single-period and infinite-horizon models.Their work assumes that the backup supplier is totally reliable.
This paper differs from the existing studies in the following two major ways.First, unlike most of the existing studies that investigate the optimal inventory system solutions under supply disruption risks with reliable or stochastic backup supply, we distinguish between the characteristics of backup supply under contingent sourcing and dual sourcing; specifically, under single sourcing with contingent supply, the backup supplier may have insufficient capacity and/or yield uncertainty, while under dual sourcing, the backup supplier's yield is dependent on both the buyer's order allocation and output flexibility.Second, both inventory control policies and backup sourcing method selection under long-term planning horizon are examined in our research.An in-depth analysis is also conducted to examine the impacts of the supply risks and the cost parameters.We show (1) how the buyer's optimal base-stock level and the corresponding expected cost are different under the two sourcing methods; (2) when single sourcing will outperform dual sourcing and vice versa; and (3) how the backup supply parameters affect the decisions in a long-term planning model.

Problem Description and Assumptions
We consider a buyer that has two supply options for a critical component: one is cheaper but subject to random disruptions (Supplier 1) and the other is perfectly reliable and responsive, but more expensive (Supplier 2).The demand at the buyer is certain and denoted as .This assumption helps us focus on the impact of supply risks.Besides, it is reasonable to suppose stable demand for a lot of products, such as grocery and basic apparel (Lee [28]).The buyer operates by a periodic-review, base-stock policy.Unsatisfied demands are backordered with a unit penalty cost of .The lead time is negligible, and the unit overstock cost is ℎ.Under such a situation, the buyer needs to choose the sourcing method and determine the optimal base-stock level  by minimizing its expected longterm cost consisting of overstock and understock cost, as well as backup purchasing cost.
Two sourcing methods are available for the buyer: single sourcing with contingent backup supply and dual sourcing.Specifically, (1) under single sourcing with contingent backup supply, the buyer places an order to their major supplier referred to as Supplier 1 at the beginning of each cycle.When a disruption breaks down the main supplier's capacity, the backup supplier (Supplier 2) is available for meeting the buyer's demand but may have limited capacity and/or random yield; and (2) under dual sourcing, the buyer uses Supplier 2 as a second regular source.Previous studies (e.g., Yan and Liu [29], Chen et al. [3]) have found that splitting orders among multiple suppliers is usually an effective approach to mitigate supply disruptions.Let   ( = 1, 2) denote the proportion of demand allocated to supplier  where  1 +  2 = 1.In this case, as a regular supplier in a long-term relationship with the buyer, Supplier 2 has the incentive to increase its output with extra capacity related to its order proportion  2 in the event of Supplier 1 failure.Thus, the buyer has to determine both the order allocation and the optimal base-stock level .Note that there is a possibility that the buyer single-sources from Supplier 2 which is completely reliable; in this case the buyer's optimal base-stock level can easily be deduced to be equal to demand .Therefore, in what follows, we will first analyze the buyer's expected costs and optimal decisions under single sourcing with contingent backup supply and dual sourcing, respectively, and then compare these two sourcing methods.
To formulate the supply disruption process, we follow the assumptions made in Schmitt et al. [26] and Schmitt and Snyder [27].Specifically, an infinite-state, discrete-time Markov chain is used to represent the disruption that lasts for a number of periods, with   representing the steady-state probability that the major supply has been disrupted for  consecutive periods,  ∈ [0, ∞).In addition, denote  as the transition probability from the normal to the disrupted state and  as the recovery probability from the disrupted to the normal state.It is easy to see that the smaller the value of  is, the longer the disruption would last.Thus, the parameter set (, ) represents the disruption probability and the average duration, respectively, and can be used to characterize each disruption.The notation   is used to denote the probability of being in state  where the state represents the number of consecutive disrupted periods.In addition, when a disruption occurs, since the buyer cannot assess the disruption duration, it is logical to assume that the buyer will start the backup supply from the first cycle of the disruption.A complete list of symbols used in the paper is provided in The List of Symbols Used in the Paper.

Single Sourcing with Contingent Backup Supply
During normal situations, the inventory position of the buyer is maintained at a level of , while during disruptions the buyer can order only from the backup supplier (Supplier 2), whose capacity is limited and/or yield is uncertain.
Before the single sourcing with contingent supply method is analyzed, we first give the following results under single sourcing without contingent supply, as presented by Tomlin [23] and Schmitt et al. [26].
Theorem 1.For a buyer with deterministic demand and supply disruptions, (a) the expected cost per period is and is a convex function;

Analysis of Contingent Backup Supply with Capacitated and Uncertain
Yield.Under single sourcing with capacitated and uncertain backup supply, the backup capacity is limited to  ( ≤ ) in each period.That is, to satisfy the demand and maintain the inventory level of , the maximum order quantity the buyer can place to the contingent supplier in each cycle during disruption periods is .Moreover, the backup yield has an additive random quantity , which is generally distributed with a normal distribution, (  ,   ), and is independent of the order quantity and may be positive or negative (Schmitt and Snyder [27]).Thus the actual backup delivery is  +  (we assume (−) ≈ 0).Under such assumption, the only difference between the buyer's expected cost with and without (as shown in Theorem 1) contingent supply is the added backup purchasing cost from the contingent supplier and larger inventory level per period; therefore, the buyer's cost can be written by modifying (1) as follows: where   is the total random quantity of  for  periods of disruption and follows a normal distribution with the density function of   (  ), mean of   , and standard deviation of √   .The first term is the inventory holding cost and the penalty cost during normal periods.The second is the backup supply purchasing cost.The last is the holding and penalty cost when disruption occurs.Note that the purchasing cost from the major supplier would not affect the buyer's decision on inventory level  under single sourcing; we thus do not consider it in the cost function.
Using the standard normal loss function, (), we can rewrite the cost function as The next proposition provides the buyer's optimal base-stock level in a long-term horizon.
Proposition 2. Under single sourcing with a backup supplier that has capacitated yield and additive uncertainty, the buyer's optimal base-stock level under long-term horizon,  *  , should satisfy Proof.See Appendix A.
The final optimal value of   can be obtained by comparing   () and The formula in Proposition 2 suggests that the buyer would decrease their base-stock level  *  if the backup supply capacity  increases.Since no close-form solution of  *  can be obtained, we will rely on numerical examples to examine its properties in later part of this section.To examine the impacts of the backup capacity limitation and yield uncertainty, respectively, we consider two special cases below.

Two Special Cases
Case 1 (contingent supply with capacitated yield).Consider a backup supplier that can provide only a finite quantity  ( ≤ ) each period when disruption occurs; to satisfy the demand and maintain the inventory level, the buyer orders  units from the backup supplier in each cycle.Based on the assumptions above, the buyer's expected cost function is given by the following expression: The buyer's optimal decision is presented in Proposition 3.

Proposition 3.
With a capacitated backup supply, the buyer's optimal base-stock level under the long-term horizon,  *  , should satisfy the following condition: where  * (>1) is the smallest integer such that ∑ It is clearly shown by Proposition 3 that (1) either higher understock cost or lower overstock cost implies higher basestock level and (2) the higher the capacity the backup supplier has, the less the buyer stock should be to deal with supply disruption risks.Note that one special case exists when  = 0, the buyer single-sources from S1, and in this case,  * 1 =  * .
Case 2 (contingent backup supply with uncertain yield).In each period, the buyer observes the current inventory level (IL) and places an order of  − IL.The backup supplier's delivery then brings the buyer's IL to a value of +V during disrupted period, where V indicates the yield uncertainty which is independent of the order quantity.We use the notations  V ,  V , (V), and (V) to denote the mean, deviation, pdf, and cdf of V, respectively, and suppose (−) ≈ 0. The buyer's expected cost in a long-term planning situation can be calculated below and the buyer's optimal base-stock level is presented in Proposition 4.
Proposition 4. With an uncertain backup supply, the buyer's optimal base-stock level under the long-term horizon,  *  , should satisfy the following condition: Proof.See Appendix C.
Given the value of  V , as ( −  *  ) = ( + )ℎ/(ℎ + ), we see that  −  *  increases with  V ; that is,  *  decreases with  V .In the following analysis, we will further examine how the various input parameters impact the buyer's decision and expected cost.( V ,  V ): (0, 5).

The Impact of Supply Disruptions under Single Sourcing.
We first calculate the optimal base-stock level  * by changing  from 0.01 to 0.9 with an increment of 0.1 ( = 0.5). Figure 1 shows how the buyer's optimal decisions are under single sourcing with ( *  ,  *  ,  *  ) and without the backup supply ( * 1 ).It is seen that the buyer's base-stock level under single sourcing increases with the disruption probability and the use of a contingent backup supplier decreases the buyer's stock level.In addition, the line of  *  is flatter than the other two, meaning that the impact of disruption probability on  * is highly affected by the type of backup supplier, which is large when the backup supply is capacitated ( *  or  *  ).In addition, the optimal value of  *  under both capacitated and uncertain backup yield is larger than  *  (under capacitated yield) and  *  (under uncertain yield).This implies that either larger backup uncertainty or smaller backup capacity would increase the buyer's inventory level.
While the use of a contingent backup supplier would always lower the buyer's inventory level, the impacts of supply disruption on its expected costs vary for different types of backup supplier.Figure 2 plots the buyer's expected profits under single sourcing with (  ,   ,   ) and without ( 1 ) backup supplier.
According to Figure 2, the buyer's expected cost  increases with .In particular, when the disruption risk is small enough ( ≤ 0.1 in this case), the buyer always benefits from using a contingent backup supplier, because   ,   ,   <  1 .On the other hand, when the disruption risk becomes larger (0.1 <  ≤ 0.6 in this case), the buyer only benefits from using a contingent backup supplier with no capacity limitation (  <  1 <   ,   ).Moreover, if the disruption risk is large enough ( ≥ 0.6 in this case), the buyer would single-source from Supplier 1 and stock more to mitigate disruptions.

The Impact of Contingent Backup Supply under Single
Sourcing.To learn the impacts of backup capacity and uncertainty, which are also influenced by the disruption probability, we illustrate the buyer's decision and expected cost under single sourcing with capacitated or uncertain backup supply in Figures 3 and 4, respectively.
As shown in Figure 3, although larger backup capacity  would decrease the buyer's inventory level, only when the disruption probability is small ( ≤ 0.1 in this case), the buyer would benefit from larger backup capacity.Additionally, Figure 4 implies that the buyer always prefers lower backup uncertainty in order to reduce both inventory level and expected cost.

Dual Sourcing
Under dual sourcing, the buyer maintains the inventory level at  and receives  units of components in each nondisrupted period, with  1  units from Supplier 1 and  2  units from Supplier 2 ( 1 +  2 = 1).In the event of Supplier 1 failure, Supplier 2, as a regular supplier, has an incentive to increase its output to   2 (0 ≤  ≤ 1), where  indicates its output flexibility with  values close to 0 indicating high output flexibility and close to 1 suggesting low output flexibility (as in Ruiz-Torres and Mahmoodi [16]).It is worth mentioning that, compared with single sourcing with contingent backup supply discussed in Section 4, a regular supplier is likely more reliable or capable than a contingent one, which indicates that Supplier 2 output in disrupted periods under dual sourcing,   2 , is larger than its delivery  +  when it is used as a contingent source.In what follows, we will consider   2 ≤  +  as a possible scenario when contingent sourcing outperforms dual sourcing.

Analysis of Dual Sourcing. The buyer's long-term expected cost under dual sourcing can be calculated as
Note that Δ 2  = ( 2 −  1 ) 2  = ( 1  1 + 2  2 ) −  1  denotes the backup purchasing cost in nondisrupted period, which is essentially the additional cost incurred by sourcing from Supplier 2.
In this situation, the buyer needs to allocate their total order optimally and set the corresponding base-stock level by minimizing the expected total cost.We first derive the optimal base-stock level  *  under dual sourcing with a given order allocation ( 1 ,  2 ) in the following proposition.

Proof. See Appendix D.
It can be seen that if the buyer would lower their inventory level, he/she should order more from Supplier 2 and less from Supplier 1 with disruption risks.In contrast, if the buyer relies more on Supplier 1, he/she would stock more to mitigate the supply risks.With the allocation set, ( 1 ,  2 ), the larger the output flexibility Supplier 2 has, the lower the inventory level the buyer should maintain.
Before the optimal order allocation under dual sourcing is obtained, we first compare the dual sourcing method and the single sourcing method with Supplier 1 as the major source.We write the buyer's expected cost by using (9): then the following result can be arrived for 0 <  < 1.  Proposition 6.For 0 <  < 1, if the backup supplier's unit wholesale price ( 2 ) satisfies the relationship single sourcing from Supplier 1 (the one with disruption risks and lower wholesale price) then outperforms the dual sourcing method.

Proof. Given 𝑠
then   is a convex and increasing function of  2 , and the optimal value  * 2 = 0; that is, the buyer would single-source from Supplier 1.The proof is complete.
Next, we compare dual sourcing and single sourcing with Supplier 2 and arrive at the following result for decisionmaking.

Proposition 7. For 0 < 𝑘 < 1, if the backup supplier's output flexibility (or wholesale price) meets the requirement
using Supplier 2 (the one with larger wholesale price but no disruptions) as the only source is better than using both suppliers.
The above results indicate that the backup supplier' wholesale price has a critical impact on the buyer's optimal decisions.Specifically, if the backup supplier's wholesale price is too high, the buyer would not use it even if it may have great output flexibility and no risks.But if the backup supplier's output flexibility is small enough or its wholesale price is small enough, the buyer would prefer to allocate the total order to it.Therefore, we could summarize the most profitable sourcing method and its corresponding decisions in the following proposition.
Proof.As Proposition 8 showed, under dual sourcing, the buyer's optimal order allocation probability satisfies  * 2 = −1 ln   > 0, then there must be  =  0 within (0, 1) that satisfies 1 − 1/ 0 + ln(  / 0 ) = 0, and when  <  0 ,  * 2 increases with  and when  ≥  0 ,  * 2 decreases with ; else if   < 1, ln   < 0, then 1−1/+ln(  /) < 0, and  * 2 / > 0, which means  * 2 increases with .The proof is complete.These results indicate that the buyer should not always decrease their order allocation to Supplier 2 if its output flexibility increases and that Supplier 2 should not always lower its output flexibility to obtain more orders.Both of them should evaluate the sign of   − 1 before making decisions.
Moreover, when  = 0, that is, Supplier 2 would satisfy the buyer's total demand during disrupted periods, the buyer would decrease the order proportion  2 to the smallest value that Supplier 2 could accept, denoted as  2 =  2 for illustration purpose, and  * = .
When  = 1, that is, Supplier 2 would not expand its output during disrupted periods, then This implies that if Supplier 2 has no output flexibility, the buyer would single-source from either Supplier 1 or Supplier 2, depending on whether its wholesale price is smaller or larger than   2 (1 −  0 ) +  0  1 .

Numerical Analysis of Dual
Sourcing.Similar to Section 4.3, the impacts of supply disruptions on the buyer's optimal decisions and expected cost are examined through numerical analysis, and the value of Supplier 2 output flexibility is also investigated with the basic parameters set in Section 4.3.

The Impact of Supply Disruptions under Dual Sourcing.
Under dual sourcing, the buyer's decisions are order allocation proportion ( 1 ,  2 ) and base-stock level .Intuitively, from the buyer's perspective, higher disruption probability   may lead the buyer to allocate a larger order proportion  2 ; on the other hand, the larger the order proportion to Supplier 2 is, the higher the purchasing cost would be incurred to the buyer.Thus, we will examine the impacts of  on  and  2 , respectively, under different values of wholesale price  2 ( = 0.7).
It is indicated by Figure 5 that the increase of supply disruption risks raises the buyer's inventory level under dual sourcing, but the buyer would not always allocate more to Supplier 2. Specifically, when  is small ( ≤ 0.2 in this case), the buyer allocates more to Supplier 2 as disruption risks increase; but when  is large, the buyer decreases Supplier 2 order proportion  2 .This result implies that if the disruption risk is large enough, the buyer prefers to order less from Supplier 2 and stock more to avoid large purchasing cost.In addition, the buyer's expected cost increases with disruption probability according to our data results.
It is also seen that the backup Supplier S2 wholesale price  2 plays an important role; in particular, as  2 increases, the buyer tends to order less from S2 and stock more with a result of larger expected cost.Furthermore, the impact of  2 decreases as the value of  2 increases.

The Impact of Output Flexibility under Dual Sourcing.
From the results in Propositions 3 and 5, we can deduce that  *  ≤  *  holds if  ≤ log  * 2 (/); that is, if Supplier 2 output flexibility is large enough, dual sourcing would reduce its inventory level compared to single sourcing with capacitated backup supply.For more implications concerning the impact of output flexibility, we will rely on numerical results shown in Figures 6-7.
Figure 6 provides the following interesting results: (1) larger output flexibility would bring down the order allocation proportion  2 to S2 under low disruption risks ( < 0.4 in this case; except at  = 0.4 when  2 is too small), indicating that backup Supplier S2 would decrease output flexibility to obtain a larger order; but from the buyer's perspective, he/she prefers to order more from S1 if S2 is very flexible; and (2) when the disruption probability is large ( ≥ 0.4 in this case), the curve of  2 presents a concave function of  (as shown in Figure 6(b)); that is,  2 first increases with  and then decreases.This behavior suggests an optimal output flexibility  * for S2 to maximize its order allocation proportion.Furthermore, the value of  * tends to decrease with the disruption probability, implying that the larger the disruption risk, the higher the output flexibility S2 should have to maximize order allocation and the smaller the order the buyer should allocate to S2.
Figure 7 presents that higher backup flexibility would help decrease the buyer's expected cost, but its impact is negligible when supply disruption risk is very small ( ≤ 0.1 in this case) or very large ( ≥ 0.7 in this case).

Comparative Studies of Single and Dual Sourcing
In this section, we will rely on numerical examples to find out the most appropriate method among the four sourcing options: single sourcing from S1 (parameters with subscript 1), single sourcing from S2 (subscript 2), dual sourcing (subscript ), and single sourcing from S1 with S2 as a capacitated and uncertain contingent supply (subscript ).
First, based on the analysis in Section 5.1, the following results can be given with the consideration of the decision situations of dual sourcing and single sourcing.Proposition 10.In selecting the optimum sourcing method, the borderline between dual sourcing or single sourcing from S1 is not affected by S2 output flexibility, while the lower the output flexibility is, the more possible the buyer would select single sourcing from S2 over dual sourcing.
Proof.As Proposition 8 stated, there are two critical values of Supplier 2 wholesale prices, (  2 ,   2 ), and  2 <   2 suggests the possibility of selecting single sourcing from S2 over dual sourcing.As the value of   2 = (ℎ∑ ) is an increasing function of , we can conclude that the possibility of  2 <   2 increases with  with a given  2 .The proof is complete.
The optimal base-stock levels under four sourcing methods are plotted in Figure 8, with the base-stock level of selected sourcing methods shown in dotted line.We consider four cases: low or high output flexibility ( = 0.2 or 0.7) and low or high backup purchasing cost ( 2 = 9 or 12).
Two observations can be made from Figure 8 By comparing the buyer's expected costs under the four sourcing methods with disruption parameter  changing from 0.01 to 0.9 and backup output flexibility  from 0.1 to 0.9, we can pinpoint the optimal sourcing method under different values of wholesale price  2 as shown in Figure 9.As seen from Figure 9, if the difference between  1 (=8) and  2 is small (Δ = 1), the buyer should single-source from S2 except when their output flexibility is very large (dual sourcing) or the supply disruption probability is very small (single sourcing with contingent backup supply).As S2 wholesale price becomes larger, the range of single sourcing from S2 shrinks towards the upper-left corner until it disappears, and the 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 singlesource from S2 decreases, especially under large disruption risk or output flexibility, and the likelihood of single sourcing from S1 increases especially under large disruption risk.
In addition, single sourcing with contingent supply always outperforms the other methods at very low disruption risk ( < 0.1) except when output flexibility is very high ( ≤ 0.2), with its range slightly expanding with Δ.On the other hand, dual sourcing method always performs the best under small disruption risk and large backup output flexibility.

Concluding Remarks
In this paper, we have studied a buyer's decisions when their major supplier encounters supply disruption and backup suppliers are available for selection.The buyer's optimal basestock level is obtained under single sourcing with contingent supply and dual sourcing, respectively, through minimizing the buyer's long-term cost, which includes overstock cost, penalty cost for unsatisfied demand, and backup purchasing cost.The impacts of three risk parameters, disruption probability, contingent capacity or uncertainty, and backup flexibility, are analyzed.Specifically, under single sourcing with contingent supply, we show (1) how backup capacity and uncertainty affect the buyer's decision and expected cost, respectively and (2) how the impacts of supply disruption on the buyer's expected costs differ for different types of contingent supplier.Table 1 summarizes our findings about the single sourcing with contingent supply.
Under dual sourcing, we identify the buyer's optimal base-stock level, the optimal order allocations, and also the critical values of wholesale price or output flexibility to determine whether the buyer should select dual sourcing or single sourcing.Our major findings about the impacts of supply disruptions and backup output flexibility are reported in Table 2.
Moreover, four sourcing methods, namely, single sourcing with contingent supply, dual sourcing, and single sourcing from either of the two suppliers, are compared.In choosing the optimal sourcing method, we find that (1) the lower the output flexibility is, the more possible the buyer would select single sourcing from S2 over dual sourcing; (2) single sourcing with contingent supply always outperforms the other methods at very low disruption risk except when output flexibility is very high; and dual sourcing method is always selected under small disruption risk and large backup output flexibility; and (3) if the difference between the two suppliers' wholesale prices is small enough, the buyer usually singlesources from S2; but when their difference becomes larger, single sourcing with S2 would be replaced by dual sourcing first and then single sourcing with S1 under large disruption risks.
We conclude by highlighting three research directions not considered in this paper.First, the backup supplier's profit and decision-making can be included in the models and it would be interesting to explore a Stackelberg game structure.Particularly, the backup supplier's decisions of wholesale price and capacities can be analyzed to maximize its expected profit.Second, the lead time of the backup suppliers can be an important factor for the buyer to consider when selecting a backup source.That is, how to select between single sourcing and dual sourcing when the lead times of the suppliers are different could be a worthwhile research problem.Finally, we can also examine the feasibility of using multiple backup suppliers in the event of supply disruption and figure out the optimal numbers of backup suppliers in long-term horizon decision-making.We hope that these directions for future research will further refine the insights provided in this paper.

Figure 2 :
Figure 2: The buyer's expected cost  under single sourcing versus .

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

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

Figure 5 :
Figure 5: The buyer's optimal decisions  and  2 under dual sourcing versus .

Figure 8 :
Figure 8: The buyer's optimal base-stock levels under the four sourcing methods.
(2)the buyer does not always select the sourcing method with the lowest base-stock level and sometimes selects the one with the highest level ( 2 = 12,  ≥ 0.6).Note that the basic value of backup capacity  under single sourcing with contingent supply is set to be 50 in this example, and for the situations in which  ≥   2 , single sourcing with contingent supply is supposed to outperform dual sourcing due to smaller purchasing cost.