# Retailer’s Optimal Procurement Strategy under Supply Disruption and Stochastic Demand: Backup Sourcing or Simultaneous Sourcing

Authors: Feng Lin; Jinzhao Shi; Peng Wu; Xingxuan Zhuo
Journal: Complexity
 (2020)

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## Abstract

Practically, supply disruption may lead production process to entirely halt (completely disrupted) or the output to differ in the order size (partially disrupted), which makes it more difficult for the retailer to satisfy stochastic market demand. Under the circumstance, the retailer is likely to procure products from two suppliers to effectively alleviate the demand-supply mismatches. Thus, under supply disruption and stochastic demand, this paper develops both backup sourcing and simultaneous sourcing (SS) strategies to analyze the retailer’s performance, where backup sourcing includes wholesale price priority (WPP) and supply reliability priority (SRP). Specifically, (1) under WPP, when the selling price is relatively lower (higher), the retailer is suggested to activate the reliable backup supplier after the realization of supply disruption (demand uncertainty). (2) Under SRP, two scenarios including minor disruption and major disruption can be identified, where the retailer’s order quantity from the reliable (unreliable) supplier under minor disruption scenario is more (less) than that under major. (3) Finally, this paper systematically compares the retailer’s preferences among WPP, SRP, and SS via theoretical results and numerical examples. That is, when the unreliable supplier is more likely to work normally or shortage cost (selling price) is relatively lower, the retailer prefers SPR regarding the unreliable supplier as backup sourcing due to its lower wholesale price and acceptable supply disruption. Otherwise, the retailer is inclined to WPP regarding the reliable supplier as backup sourcing for ensuring all market demand to be satisfied. In addition, unless the emergency prices of two suppliers are extremely higher, backup sourcing strategies could perform better than simultaneous sourcing strategy.


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## Body

## 1. Introduction

In practice, the upstream supply process is highly vulnerable to supply disruption, ranging from major breakdowns because of natural or man-made disasters (e.g., earthquakes and floods) to minor interruptions due to material contamination and labour problems [1, 2]. As a result, the supply process may entirely halt (completely shut down) or the output differs in the order size (partially disrupted), leading to supply shortage, loss of goodwill, and the collapse of supply chain, which has already been viewed as one of the major challenges to supply chains success and caused massive economic consequences [3, 4]. For example, the volcanic eruption in Iceland in March 2010 caused the air cargo transportation to be completely paralyzed. Therefore, many companies with production plants in Europe (such as BMW, Nissan, and Audi) had to suspend production and shut down the factory due to the supply interruption of key parts [5]. Further, as Silbermayr and Minner [6] noted, according to a survey of 519 respondents from 71 countries across 15 industry sectors in 2015, 75% of the respondents experienced at least one supply chain disruption. Further, from the downstream side, the supply chain is also confronted with demand risk due to the variation of market demand, which further complicates the matches between supply and demand [7]. Under the circumstance, the retailer is subject to the combined negative effects of supply disruption and stochastic demand, which may significantly lower operational performances.To mitigate negative effects of stock-out risk, it is gradually acknowledged that the retailer could adopt backup sourcing strategy for hedging against supply disruption and stochastic demand, which has been widely adopted in aircraft, pharmaceutical, and electronic industries [8]. That is, two suppliers provide products with similar quality but differ in supply uncertainty and selling price, where one supplier is viewed as major source and the other is taken as backup sourcing which would be initiated after observing the major supplier’s supply [1]. This procurement strategy is able to prevent the retailer from potential supply disruption and simultaneously control operating costs, which enhances the flexibility and stability of the supply chain [9]. Taking an example of Japan tsunami on 2011 March, Toyota North America only operated its 30% capacity due to the lack of part supply from Japan, while General Motors successfully solved the supply shortages of 118 parts by implementing backup plans [10]. Generally, under backup sourcing strategy, the retailer could procure products from major sourcing under demand uncertainty, and the backup sourcing can be postponed after market demand or supply disruption is realized. Thus, under supply disruption and stochastic demand, there are two types of backup sourcing strategies based on the retailer’s order sequences [1, 3].One of the typical forms is wholesale price priority (WPP) strategy, where the unreliable supplier with lower wholesale price is regarded as the major sourcing, and the reliable supplier can be treated as the backup supplier. Further, the reliable backup supplier can be activated after the supply disruption occurs or both supply and demand risks are realized, which can be labeled as WPP-SR and WPP-DR strategy, respectively. Another typical form is supply reliability priority (SRP) strategy, where the reliable supplier with higher wholesale price is viewed as major sourcing, while the unreliable supplier with lower wholesale price is viewed as backup supplier.To sum up, the backup sourcing strategy can be viewed as emergency procurement requiring higher emergency cost, which mitigates the high loss due to the occurrence of supply disruption [5]. Although WPP and SRP have found their wide applications to prevent the retailer from stock risk, it is urgent for the retailer to identify the most appropriate backup sourcing strategy via trading off supply disruption and stochastic demand as well as wholesale prices.In addition to backup sourcing, simultaneous sourcing (SS) strategy from two suppliers is another effective policy for the retailer to hedge against supply disruption. That is, when market demand is stochastic, the retailer procures products from both the unreliable supplier and the reliable supplier at the same time. Under this policy, the retailer could take the advantage of lower wholesale price from two suppliers and avoid expensive emergency costs, while it is responsible for demand risk. To summarize, under supply disruption and stochastic demand, the following three urgent issues are required to be further analyzed:(1)
Under WPP, what are the differences between activating backup reliable supplier after supply disruption and for demand uncertainty?(2)
Under SRP, how does supply disruption of the unreliable supplier impact the retailer’s procurement decisions from major and backup suppliers?(3)
When involving both backup sourcing (including WPP and SRP) and simultaneous sourcing strategies, what is the retailer’s optimal sourcing strategy under supply disruption and stochastic demand?To solve the previous issues, in this paper, the retailer is assumed to procure products from two suppliers for hedging against supply disruption and stochastic demand. And different backup sourcing and simultaneous sourcing strategies are involved to address the retailer’s optimal procurement policy. And the major contributions of this paper can be summarized as follows:(1)
Firstly, two forms of WPP strategies are further classified on the basis of the reliable supplier to be initiated after the occurrence of supply disruption (i.e., WPP-SR) or the realization of both supply and demand uncertainties (i.e., WPP-DR). Specifically, WPP-SR ensures the retailer’s order quantity from unreliable supplier to be always met, while WPP-DR guarantees the market demand to be entirely satisfied with more expensive emergency cost. Further, positive relationships between selling price (shortage cost) and the retailer’s order quantity from the reliable supplier can be identified under WPP-SR, which do not exist under WPP-DR. In addition, when the procurement or emergency price of the reliable supplier is relatively higher, the retailer’s order quantity from the reliable supplier under WPP-SR is less than that of WPP-DR.(2)
Secondly, under SRP, it is surprising that the retailer’s optimal procurement solutions from two suppliers consist of two scenarios including minor disruption and major disruption. Specifically, (i) under minor disruption scenario, the unreliable supplier is more likely to supply normally or offer a higher value of emergency price, implying that the retailer would assume a higher-order quantity from the reliable supplier when demand is uncertain. And after the realization of the demand, the retailer accurately places order quantity from the unreliable supplier based on the difference between the supply of the reliable sourcing and the demand. (ii) Under major disruption scenario, on the contrary, the retailer would decrease the order quantity from the reliable supplier (compared with that under minor disruption scenario) and set the order quantity from the unreliable supplier to exceed the gap between the reliable supplier’s products and the realized demand.(3)
Finally, under supply disruption, this paper systematically compares the retailer’s preferences towards backup sourcing and simultaneous sourcing strategies via theoretical results and numerical examples. (i) Since WPP-DR could ensure market demand to be entirely satisfied by expensive emergency replenishment from the reliable supplier, the retailer would regard it as the optimal backup sourcing when selling price or shortage cost is relatively higher. On the contrary, the retailer would adopt SRP for directly improving the supply reliability and avoiding expensive emergency replenishment from the reliable supplier. (ii) When the unreliable supplier is less likely to work normally, the retailer would prefer WPP-DR and regard the reliable supplier as backup sourcing for ensuring all market demand to be satisfied. Otherwise, the retailer would prefer SPR regarding the reliable supplier as backup sourcing due to its lower wholesale price and its acceptable supply disruption. (iii) In addition, unless the emergency prices of two suppliers are extremely higher, backup sourcing strategies could perform better than simultaneous sourcing strategies.The remainder of this paper is organized as follows. Section2 presents a series of literature reviews concerning supply disruption, backup sourcing, and simultaneous sourcing strategies. And Section 3 depicts key assumptions and relevant notations which are used to present the models. Then, Sections 4–6 take views of WPP, SRP, and SS to investigate the retailer’s optimal procurement policies, respectively. Furthermore, in Section 7, several numerical examples are provided to explicitly derive more managerial insights. Finally, derived results are summarized in Section 8.


## 2. Literature Review

In view of negative impacts of supply disruption towards operational practices, a series of researches have been conducted to optimize operational performance of firms. Schmitt et al. [11] integrated discrete-event uncertainty (disruptions) and continuous sources of uncertainty to address their effects on optimal solutions for newsvendor models. Under the deterministic demand setting, Tang et al. [12] assumed the buyer to offer incentives (including investment subsidy and inflated order quantity) to the supplier under supply disruption for enhancing process reliability. Under supply and demand uncertainty, Ray and Jenamani [13] applied newsvendor approach to formulate two models for optimal order allocation of a risk-neutral or risk-averse decision-maker. Inderfurth and Kiesmuller [14] established a periodic-review inventory model for a single item, which involves two different types of uncertain supply, including binomial and stochastically proportional yield. Nevertheless, besides the all-or-nothing setting (i.e., binomial) concerning supply disruption, several scholars have pointed out that supply disruption may also lead to partial disruption, where the manufacturer’s final output could be a fraction between complete shutdown and working normally. Based on this assumption, Hou et al. [1] applied capacity reservation and minimum order quantity to mitigate the disruption risk, while Zeng and Xia [10] adopted a decision-tree approach to consider disruption situations at all levels.To hedge against supply disruption, a series of researches have incorporated wholesale price priority (WPP), the most widely adopted backup sourcing strategy, to improve supply reliability under the deterministic demand setting. Yin and Wang [15] proposed three common options including advance purchase, reservation, and contingency purchase for firms to adopt when cooperating with the backup supplier to hedge against the disruption risk. Via a sequential optimization process, Hou et al. [16] applied a buy-back contract to coordinate a buyer with a backup supplier, when the main supplier experiences supply disruption. In a continuous-review inventory problem, Qi [17] used the order-up-to policy to explore the retailer’s operational solutions when the primary supplier is subject to random disruptions.Further, in the context of WPP, several studies have also analyzed the retailer’s procurement policy under supply disruption and stochastic demand. Giri and Bardhan [5] built a two-echelon supply chain under uncertain demand and supply disruption, where the penalty contract mechanism is applied to achieve supply chain coordination. Zhang et al. [4] adopted a procurement strategy with backup sourcing under stochastic supply and demand, where the game-theoretic framework is involved to explore optimal decisions of the players. Nevertheless, it can be concluded that these papers simply assume backup supplier to be initiated after supply disruption is realized, which do not distinguish WPP-SR from WPP-DR. Additionally, in their studies, partial disruption is excluded from supply disruption, which certainly merits additional attention.In addition to backup sourcing, simultaneous sourcing from two suppliers is another effective policy to hedge against supply disruption. Under supply disruption, Silbermayr and Minner [6] compared risk reduction via dual sourcing with learning effects when negotiating with a single supplier via a long-term contract. Under disruption risk, Kumar, Basu, and Avittathur [9] analyzed the retailer’s optimal pricing and dual-sourcing policies for competing with another retailer supplied by a reliable supply chain. In a two-stage supply chain with a nonstationary and price-sensitive deterministic demand, Yu et al. [18] quantified impacts of supply disruption towards single and dual sourcing. Nevertheless, existing researches are still lack of addressing simultaneous sourcing policy under partial supply disruption and demand uncertainty.To our best knowledge, Li [3] is the closest to our study, which systematically addresses the retailer’s procurement policy from two unreliable suppliers under deterministic demand. Specifically, Li [3] incorporated both the all-or-nothing supply disruption and the proportional random yield to analyze how supply risks influence the retailer’s order decisions. Further, under deterministic demand, Li [3] compared different decision sequences including backup sourcing and simultaneous sourcing strategies, suggesting the retailer to regard the relatively expensive supplier as a backup source via computational experiments. Unlike Li [3], this paper desires to incorporate both stochastic demand and supply disruption to vary decision sequences to generate more procurement strategies. It indicates that the retailer could place orders from two suppliers before or after demand uncertainty (or supply disruption), resulting in WPP-SR, WPP-DR, SRP, and SS. And compared with Li [3], due to the joint effects of stochastic demand and partial disruption, this paper identified that the retailer shows different order incentives and procurement preferences towards backup sourcing and simultaneous sourcing strategies.To sum up, in order to explicitly distinguish this paper from previous studies, the following table is adopted to summarize key characteristics of the above-mentioned papers under supply disruption (Table1).Table 1
Literature reviews of backup (simultaneous) sourcing models under supply disruption.
AuthorsSupply disruptionDemandBackup sourcingSimultaneous sourcingWPPSRPHou et al. [1]PartialDet.✓Li [3]BinomialDet.WPP-SR✓✓Giri and Bardhan [5]BinomialSto.WPP-DRSilbermayr and Minner [6]BinomialDet.✓Kumar et al. [9]BinomialDet.✓Zeng and Xia [10]PartialSto.✓Tang et al. [12]BinomialDet.✓Yin and Wang [15]BinomialDet.WPP-SR✓Hou et al. [16]BinomialDet.WPP-SRQi [17]BinomialDet.WPP-SRZhang and Wang [19]BinomialSto.WPP-DRYu et al. [18]BinomialDet.WPP-SR✓Present paperPartialSto.WPP-DR, WPP-SR✓✓Note. For convenience, let terms Det. and Sto. denote deterministic and stochastic, respectively.


## 3. Model Description

The following notations and assumptions are adopted to develop new proposed models throughout the paper.


### 3.1. Notations

i: i=1, 2 represent supplier 1 and supplier 2, respectivelyj: j = WS, WD, SR, and SS denote WPP-SR, WPP-DR, SRP, and SS, respectivelyp: the unit selling price of the retailerwi: the unit procurement price of supplier i when demand is stochasticw˜i: the unit emergency price of supplier i when demand is deterministiccs: the unit shortage cost of the retailerco: the unit holding cost of the retailerQ: the retailer’s order quantity from supplier 1 (decision variable)R: the retailer’s order quantity from supplier 2 (decision variable)y: the disruption magnitude (0≤y≤1), indicating the fraction from supplier 2m: the probability of supplier 2 to completely break down; that is,y=0n: the probability of supplier 2 to work normally; that is, y=1Πj: the retailer’s expected profit under j strategy


### 3.2. Assumptions

(1)
In a single period, this paper focuses on the retailer’s procurement problem where two suppliers offer similar products. Supplier 1 is perfectly reliable with a higher wholesale price, while supplier 2 is subject to supply disruption with a lower wholesale price.(2)
At the beginning of the period (i.e., stage 1), the market demandD is assumed to be stochastic, where the cumulative and probability distribution functions of D are FD and fD, respectively. And market demand FD is differentiable, strictly increasing, while its complementary CDF F¯D=1−FD. Further, at the end of the period (i.e., stage 2), the market demand D can be realized, where the emergency replenishment can be proceeded via backup sourcing.(3)
The retailer’s order quantities from suppliers 1 and 2 areQ and R, respectively. At stage 1 when market demand is stochastic, wholesale prices of products of suppliers 1 and 2 are w1 and w2, respectively, where p>w1>w2. And at stage 2 after demand is revealed, wholesale prices of suppliers 1 and 2 would increase to w˜1 and w˜2, respectively, where w˜1>w˜2.(4)
Supplier 1 can fully fulfill the retailer’s order quantityQ, while supplier 2 is subject to supply disruption, leading final output to be yR. By referring to Hou et al. [1] and Zeng and Xia [10], y is a fraction between 0 and 1 representing the magnitude of a disruption, which can be(1)gy=m,y=0,1−m−n,0<y<1,n,y=1.From (1), the supplier 2’s probability of entire shutdown (y=0) and entire reliable (y=1) is m and n, respectively. And the probability of being partially disrupted is 1−m−n, which is continuous for 0<y<1. In addition, the retailer should pay off the purchase cost based on the received quantity as soon as receiving products from supplier 1 or supplier 2.(5)
In this paper, the following two backup sourcing strategies are involved.(i)
Wholesale price priority (WPP): the retailer procuresR units from the unreliable supplier 2 with w2 when demand is stochastic. Further, the reliable backup supplier would be ordered by the retailer with w1 after the supply disruption occurs or with w˜1 after both supply and demand risks are realized, which can be labeled as WPP-SR and WPP-DR strategy, respectively.(ii)
Supply reliability priority (SRP): the retailer procuresQ units from the reliable supplier 1 with w1 when market demand is uncertain. And the retailer conducts emergency procurement from the unreliable supplier 2 with w˜2 when market demand is realized.(iii)
Simultaneous sourcing (SS): when demand is stochastic, the retailer simultaneously procuresQ and R units from the reliable supplier 1 and the unreliable supplier 2, respectively.


## 3.1. Notations

i: i=1, 2 represent supplier 1 and supplier 2, respectivelyj: j = WS, WD, SR, and SS denote WPP-SR, WPP-DR, SRP, and SS, respectivelyp: the unit selling price of the retailerwi: the unit procurement price of supplier i when demand is stochasticw˜i: the unit emergency price of supplier i when demand is deterministiccs: the unit shortage cost of the retailerco: the unit holding cost of the retailerQ: the retailer’s order quantity from supplier 1 (decision variable)R: the retailer’s order quantity from supplier 2 (decision variable)y: the disruption magnitude (0≤y≤1), indicating the fraction from supplier 2m: the probability of supplier 2 to completely break down; that is,y=0n: the probability of supplier 2 to work normally; that is, y=1Πj: the retailer’s expected profit under j strategy


## 3.2. Assumptions

(1)
In a single period, this paper focuses on the retailer’s procurement problem where two suppliers offer similar products. Supplier 1 is perfectly reliable with a higher wholesale price, while supplier 2 is subject to supply disruption with a lower wholesale price.(2)
At the beginning of the period (i.e., stage 1), the market demandD is assumed to be stochastic, where the cumulative and probability distribution functions of D are FD and fD, respectively. And market demand FD is differentiable, strictly increasing, while its complementary CDF F¯D=1−FD. Further, at the end of the period (i.e., stage 2), the market demand D can be realized, where the emergency replenishment can be proceeded via backup sourcing.(3)
The retailer’s order quantities from suppliers 1 and 2 areQ and R, respectively. At stage 1 when market demand is stochastic, wholesale prices of products of suppliers 1 and 2 are w1 and w2, respectively, where p>w1>w2. And at stage 2 after demand is revealed, wholesale prices of suppliers 1 and 2 would increase to w˜1 and w˜2, respectively, where w˜1>w˜2.(4)
Supplier 1 can fully fulfill the retailer’s order quantityQ, while supplier 2 is subject to supply disruption, leading final output to be yR. By referring to Hou et al. [1] and Zeng and Xia [10], y is a fraction between 0 and 1 representing the magnitude of a disruption, which can be(1)gy=m,y=0,1−m−n,0<y<1,n,y=1.From (1), the supplier 2’s probability of entire shutdown (y=0) and entire reliable (y=1) is m and n, respectively. And the probability of being partially disrupted is 1−m−n, which is continuous for 0<y<1. In addition, the retailer should pay off the purchase cost based on the received quantity as soon as receiving products from supplier 1 or supplier 2.(5)
In this paper, the following two backup sourcing strategies are involved.(i)
Wholesale price priority (WPP): the retailer procuresR units from the unreliable supplier 2 with w2 when demand is stochastic. Further, the reliable backup supplier would be ordered by the retailer with w1 after the supply disruption occurs or with w˜1 after both supply and demand risks are realized, which can be labeled as WPP-SR and WPP-DR strategy, respectively.(ii)
Supply reliability priority (SRP): the retailer procuresQ units from the reliable supplier 1 with w1 when market demand is uncertain. And the retailer conducts emergency procurement from the unreliable supplier 2 with w˜2 when market demand is realized.(iii)
Simultaneous sourcing (SS): when demand is stochastic, the retailer simultaneously procuresQ and R units from the reliable supplier 1 and the unreliable supplier 2, respectively.


## 4. Retailer’s Optimal Procurement Policy under WPP

As previously mentioned, there are two types of WPP on the basis of the emergency order from supplier 1 to be conducted after supply disruption is realized (WPP-SR) or both supply and demand are realized (WPP-DR).


### 4.1. Optimal Procurement Policy under WPP-SR

Under WPP-SR (see Figure1(a)), the retailer should firstly order R units with w2 from the unreliable supplier 2 to satisfy the stochastic demand. And after supply disruption is revealed, the retailer receives yR from the unreliable supplier 2 and then further procures Q units (i.e., R−yR) with w1 from supplier 1 to ensure that the amount of available products equals R units. Therefore, considering the supply disruption of supplier 2, the retailer’s expected profit function can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted, we haveyR=0 and the retailer would procure R (i.e., Q=R−yR) units from supplier 1 with the wholesale price w1. Under the circumstance, Π11WS=EpminD,R−csD−R+−coR−D+−w1R.(2)
When the supplier’s supply process is entirely normal, we haveyR=R and there is no need for the retailer to further order products from supplier 1. Thus, Π12WS=EpminD,R−csD−R+−coR−D+−w2R.(3)
When the supplier’s supply process is partially disrupted, that is,R>yR>0, the retailer would procure the rest of the products from supplier 1. Thus, we have Π13WS=EpminD,R−csD−R+−coR−D+−w11−yR−w2yR.Figure 1
Retailer’s order sequences under different backup and simultaneous sourcing strategies.
(a)(b)(c)(d)Hence, the retailer’s expected profit function under WPP-SR can be depicted as follows:(2)Π1WSR=mΠ11WS+nΠ12WS+1−m−n∫01Π13WSdy.According to (2), the following theorem can be summarized.Theorem 1.
Under WPP-SR, the retailer’s optimal order quantityRWS from the unreliable supplier 2 can be uniquely obtained by the following expression: F¯RWS=Δ1, where Δ1=1+m−nw1+1+n−mw2+2co/2p+cs+co.Proof.
The details can be found at AppendixA. From Theorem 1, it can be found that WPP-SR ensures that the retailer’s order quantity RWS from the unreliable supplier 2 can be always met by emergency replenishment from the reliable supplier 1 (i.e., 1−yRWS) and enables the retailer to avoid higher emergency procurement price w˜1 of supplier 1 after both supply and demand uncertainty are realized. Furthermore, it is apparent that RWS is positively related to p and cs, while it is negatively related to w1 and w2. That is, under WPP-SR, the retailer would increase (decrease) the order quantity from the unreliable supplier 2 when confronted with higher value of selling profit and shortage cost (procurement prices). In addition, since w1>w2, the retailer would place a higher-order quantity from the unreliable supplier 2 when supplier 2 is more (less) likely to work normally (complete shutdown).


### 4.2. Optimal Procurement Policy under WPP-DR

Under WPP-DR strategy, the decision-making process of the retailer’s order allocation scheme consists of two stages (see Figure1(b)). (1) In the first stage, when the market demand is stochastic, the retailer firstly orders R units of the product from the unreliable supplier 2 with the lower price w2. (2) In the second stage, the retailer has received the final output yR, and the market demand can be realized. Then, the retailer will decide whether to place an emergency order Q from supplier 1 with w˜1. Thus, under WPP-DR strategy, the backward induction method can be used to obtain the retailer’s optimal procurement policy.


#### 4.2.1. Procurement Policy at the Second Stage

In the second stage, the market demandD and the supplier 2’s final output yR are revealed to the retailer. Therefore, the retailer should order Q units from supplier 1 on the basis of the difference between supply and demand in the first stage, that is, D−yR. In summary, the following lemma can be obtained.Lemma 1.
Under the second stage of WPP-DR, the retailer’s optimal order quantity from supplier 1QWD=D−yR+.From Lemma1, since the supplier 1’s supply process is reliable, the market demand can be always satisfied under WPP-DR with higher emergency cost. Specifically, if D<yR, the retailer would not adopt emergency order; otherwise, the supplier 1 could fully deliver D−yR units to the retailer.


#### 4.2.2. Procurement Policy at the First Stage

In the first stage, market demandD is stochastic and the retailer should order R units from the unreliable supplier 2. Therefore, considering the supply disruption of supplier 2, the retailer’s expected profit function can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted, we haveyR=0 and the retailer would procure D−R (i.e., Q=R−yR) units from supplier 1 with the wholesale price w˜1. Under the circumstance, Π11WD=Ep−w˜1D.(2)
When the supplier 2 supply process is entirely normal, we haveyR=R. Then, if R>D, the retailer should pay off additional holding cost; otherwise, the retailer should further order D−R units from supplier 1. Thus, based on the stochastic demand D, Π12WD=∫0RpD−coR−DfDdD+∫R+∞pD−w˜1D−RfDdD−w2R.(3)
When the supplier 2 supply process is partially disrupted, that is,R>yR>0, then, the retailer should either pay off holding cost for the surplus products D−yR or procure the shortages from supplier 1. Thus, we have Π13WD=∫0yRpD−coyR−DfDdD+∫yR+∞pD−w˜1D−yRfDdD−w2yR.Hence, the retailer’s expected profit function under WPP-DR can be depicted as follows:(3)Π1WDR=mΠ11WD+nΠ12WD+1−m−n∫01Π13WDdy.Based on (3), the following theoretical results can be summarized.Theorem 2.
Under WPP-DR, the retailer’s optimal order quantityRWD from the unreliable supplier 2 can be uniquely obtained by the following expression: nF¯RWD+1−n−m∫01yF¯yRWDdy=1+n−mco+w2/2co+w˜1.Proof.
The details can be found at AppendixB. From Theorem 2, WPP-DR ensures that the market demand can be always satisfied by the reliable backup sourcing with expensive emergency cost from supplier 1 with w˜1. Under the circumstance, the retailer’s order quantity RWD is irrelevant to p and cs, which is totally different from the results derived from WPP-SR. Further, the retailer would increase order quantity from the unreliable supplier 2 when confronted with higher (lower) values of w˜1w2. And based on Theorems 1 and 2, the following theoretical results can be obtained.Corollary 1.
LetΔ2=co+w2/co+w˜1 and Δ3=1+n−mw2−1−n−mw˜1+2nco/2nco+w˜1, the following hold true:(1)
IfΔ1≥Δ2,thenRWS≤RWD,(2)
IfΔ1<Δ3,thenRWS>RWD.Proof.
The details can be found at AppendixC. From Corollary 1, the differences concerning the retailer’s order quantity from sourcing reliable supplier 1 between WPP-SR and WPP-DR can be explicitly identified. Specifically, when p or cs is relatively lower or w1w˜1 is relatively higher, Δ1 is more likely to be greater than Δ2, implying that the retailer’s order quantity RWS from the unreliable supplier 2 under WPP-SR is less than that of WPP-DR. Otherwise, when p or cs increases to some degree or w1 (w˜1) is relatively lower, Δ1 may be less than Δ3, indicating that the retailer would order less quantities from the unreliable supplier 2 under WPP-DR.


## 4.1. Optimal Procurement Policy under WPP-SR

Under WPP-SR (see Figure1(a)), the retailer should firstly order R units with w2 from the unreliable supplier 2 to satisfy the stochastic demand. And after supply disruption is revealed, the retailer receives yR from the unreliable supplier 2 and then further procures Q units (i.e., R−yR) with w1 from supplier 1 to ensure that the amount of available products equals R units. Therefore, considering the supply disruption of supplier 2, the retailer’s expected profit function can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted, we haveyR=0 and the retailer would procure R (i.e., Q=R−yR) units from supplier 1 with the wholesale price w1. Under the circumstance, Π11WS=EpminD,R−csD−R+−coR−D+−w1R.(2)
When the supplier’s supply process is entirely normal, we haveyR=R and there is no need for the retailer to further order products from supplier 1. Thus, Π12WS=EpminD,R−csD−R+−coR−D+−w2R.(3)
When the supplier’s supply process is partially disrupted, that is,R>yR>0, the retailer would procure the rest of the products from supplier 1. Thus, we have Π13WS=EpminD,R−csD−R+−coR−D+−w11−yR−w2yR.Figure 1
Retailer’s order sequences under different backup and simultaneous sourcing strategies.
(a)(b)(c)(d)Hence, the retailer’s expected profit function under WPP-SR can be depicted as follows:(2)Π1WSR=mΠ11WS+nΠ12WS+1−m−n∫01Π13WSdy.According to (2), the following theorem can be summarized.Theorem 1.
Under WPP-SR, the retailer’s optimal order quantityRWS from the unreliable supplier 2 can be uniquely obtained by the following expression: F¯RWS=Δ1, where Δ1=1+m−nw1+1+n−mw2+2co/2p+cs+co.Proof.
The details can be found at AppendixA. From Theorem 1, it can be found that WPP-SR ensures that the retailer’s order quantity RWS from the unreliable supplier 2 can be always met by emergency replenishment from the reliable supplier 1 (i.e., 1−yRWS) and enables the retailer to avoid higher emergency procurement price w˜1 of supplier 1 after both supply and demand uncertainty are realized. Furthermore, it is apparent that RWS is positively related to p and cs, while it is negatively related to w1 and w2. That is, under WPP-SR, the retailer would increase (decrease) the order quantity from the unreliable supplier 2 when confronted with higher value of selling profit and shortage cost (procurement prices). In addition, since w1>w2, the retailer would place a higher-order quantity from the unreliable supplier 2 when supplier 2 is more (less) likely to work normally (complete shutdown).


## 4.2. Optimal Procurement Policy under WPP-DR

Under WPP-DR strategy, the decision-making process of the retailer’s order allocation scheme consists of two stages (see Figure1(b)). (1) In the first stage, when the market demand is stochastic, the retailer firstly orders R units of the product from the unreliable supplier 2 with the lower price w2. (2) In the second stage, the retailer has received the final output yR, and the market demand can be realized. Then, the retailer will decide whether to place an emergency order Q from supplier 1 with w˜1. Thus, under WPP-DR strategy, the backward induction method can be used to obtain the retailer’s optimal procurement policy.


### 4.2.1. Procurement Policy at the Second Stage

In the second stage, the market demandD and the supplier 2’s final output yR are revealed to the retailer. Therefore, the retailer should order Q units from supplier 1 on the basis of the difference between supply and demand in the first stage, that is, D−yR. In summary, the following lemma can be obtained.Lemma 1.
Under the second stage of WPP-DR, the retailer’s optimal order quantity from supplier 1QWD=D−yR+.From Lemma1, since the supplier 1’s supply process is reliable, the market demand can be always satisfied under WPP-DR with higher emergency cost. Specifically, if D<yR, the retailer would not adopt emergency order; otherwise, the supplier 1 could fully deliver D−yR units to the retailer.


### 4.2.2. Procurement Policy at the First Stage

In the first stage, market demandD is stochastic and the retailer should order R units from the unreliable supplier 2. Therefore, considering the supply disruption of supplier 2, the retailer’s expected profit function can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted, we haveyR=0 and the retailer would procure D−R (i.e., Q=R−yR) units from supplier 1 with the wholesale price w˜1. Under the circumstance, Π11WD=Ep−w˜1D.(2)
When the supplier 2 supply process is entirely normal, we haveyR=R. Then, if R>D, the retailer should pay off additional holding cost; otherwise, the retailer should further order D−R units from supplier 1. Thus, based on the stochastic demand D, Π12WD=∫0RpD−coR−DfDdD+∫R+∞pD−w˜1D−RfDdD−w2R.(3)
When the supplier 2 supply process is partially disrupted, that is,R>yR>0, then, the retailer should either pay off holding cost for the surplus products D−yR or procure the shortages from supplier 1. Thus, we have Π13WD=∫0yRpD−coyR−DfDdD+∫yR+∞pD−w˜1D−yRfDdD−w2yR.Hence, the retailer’s expected profit function under WPP-DR can be depicted as follows:(3)Π1WDR=mΠ11WD+nΠ12WD+1−m−n∫01Π13WDdy.Based on (3), the following theoretical results can be summarized.Theorem 2.
Under WPP-DR, the retailer’s optimal order quantityRWD from the unreliable supplier 2 can be uniquely obtained by the following expression: nF¯RWD+1−n−m∫01yF¯yRWDdy=1+n−mco+w2/2co+w˜1.Proof.
The details can be found at AppendixB. From Theorem 2, WPP-DR ensures that the market demand can be always satisfied by the reliable backup sourcing with expensive emergency cost from supplier 1 with w˜1. Under the circumstance, the retailer’s order quantity RWD is irrelevant to p and cs, which is totally different from the results derived from WPP-SR. Further, the retailer would increase order quantity from the unreliable supplier 2 when confronted with higher (lower) values of w˜1w2. And based on Theorems 1 and 2, the following theoretical results can be obtained.Corollary 1.
LetΔ2=co+w2/co+w˜1 and Δ3=1+n−mw2−1−n−mw˜1+2nco/2nco+w˜1, the following hold true:(1)
IfΔ1≥Δ2,thenRWS≤RWD,(2)
IfΔ1<Δ3,thenRWS>RWD.Proof.
The details can be found at AppendixC. From Corollary 1, the differences concerning the retailer’s order quantity from sourcing reliable supplier 1 between WPP-SR and WPP-DR can be explicitly identified. Specifically, when p or cs is relatively lower or w1w˜1 is relatively higher, Δ1 is more likely to be greater than Δ2, implying that the retailer’s order quantity RWS from the unreliable supplier 2 under WPP-SR is less than that of WPP-DR. Otherwise, when p or cs increases to some degree or w1 (w˜1) is relatively lower, Δ1 may be less than Δ3, indicating that the retailer would order less quantities from the unreliable supplier 2 under WPP-DR.


## 4.2.1. Procurement Policy at the Second Stage

In the second stage, the market demandD and the supplier 2’s final output yR are revealed to the retailer. Therefore, the retailer should order Q units from supplier 1 on the basis of the difference between supply and demand in the first stage, that is, D−yR. In summary, the following lemma can be obtained.Lemma 1.
Under the second stage of WPP-DR, the retailer’s optimal order quantity from supplier 1QWD=D−yR+.From Lemma1, since the supplier 1’s supply process is reliable, the market demand can be always satisfied under WPP-DR with higher emergency cost. Specifically, if D<yR, the retailer would not adopt emergency order; otherwise, the supplier 1 could fully deliver D−yR units to the retailer.


## 4.2.2. Procurement Policy at the First Stage

In the first stage, market demandD is stochastic and the retailer should order R units from the unreliable supplier 2. Therefore, considering the supply disruption of supplier 2, the retailer’s expected profit function can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted, we haveyR=0 and the retailer would procure D−R (i.e., Q=R−yR) units from supplier 1 with the wholesale price w˜1. Under the circumstance, Π11WD=Ep−w˜1D.(2)
When the supplier 2 supply process is entirely normal, we haveyR=R. Then, if R>D, the retailer should pay off additional holding cost; otherwise, the retailer should further order D−R units from supplier 1. Thus, based on the stochastic demand D, Π12WD=∫0RpD−coR−DfDdD+∫R+∞pD−w˜1D−RfDdD−w2R.(3)
When the supplier 2 supply process is partially disrupted, that is,R>yR>0, then, the retailer should either pay off holding cost for the surplus products D−yR or procure the shortages from supplier 1. Thus, we have Π13WD=∫0yRpD−coyR−DfDdD+∫yR+∞pD−w˜1D−yRfDdD−w2yR.Hence, the retailer’s expected profit function under WPP-DR can be depicted as follows:(3)Π1WDR=mΠ11WD+nΠ12WD+1−m−n∫01Π13WDdy.Based on (3), the following theoretical results can be summarized.Theorem 2.
Under WPP-DR, the retailer’s optimal order quantityRWD from the unreliable supplier 2 can be uniquely obtained by the following expression: nF¯RWD+1−n−m∫01yF¯yRWDdy=1+n−mco+w2/2co+w˜1.Proof.
The details can be found at AppendixB. From Theorem 2, WPP-DR ensures that the market demand can be always satisfied by the reliable backup sourcing with expensive emergency cost from supplier 1 with w˜1. Under the circumstance, the retailer’s order quantity RWD is irrelevant to p and cs, which is totally different from the results derived from WPP-SR. Further, the retailer would increase order quantity from the unreliable supplier 2 when confronted with higher (lower) values of w˜1w2. And based on Theorems 1 and 2, the following theoretical results can be obtained.Corollary 1.
LetΔ2=co+w2/co+w˜1 and Δ3=1+n−mw2−1−n−mw˜1+2nco/2nco+w˜1, the following hold true:(1)
IfΔ1≥Δ2,thenRWS≤RWD,(2)
IfΔ1<Δ3,thenRWS>RWD.Proof.
The details can be found at AppendixC. From Corollary 1, the differences concerning the retailer’s order quantity from sourcing reliable supplier 1 between WPP-SR and WPP-DR can be explicitly identified. Specifically, when p or cs is relatively lower or w1w˜1 is relatively higher, Δ1 is more likely to be greater than Δ2, implying that the retailer’s order quantity RWS from the unreliable supplier 2 under WPP-SR is less than that of WPP-DR. Otherwise, when p or cs increases to some degree or w1 (w˜1) is relatively lower, Δ1 may be less than Δ3, indicating that the retailer would order less quantities from the unreliable supplier 2 under WPP-DR.


## 5. Retailer’s Optimal Procurement Policy under SRP

In SRP strategy, the retailer’s procurement decision process can be divided into two stages (see Figure1(c)). Specifically, (1) in the first stage (at the beginning of the selling period), the retailer should order Q units from the reliable supplier 1 at a higher price w1 to meet the stochastic market demand. (2) In the second stage (at the end of selling period), the market demand can be revealed by the retailer. And based on the gap between the realized demand and the supplied quantity Q by supplier 1, the retailer can urgently order R units from unreliable supplier 2 with a lower price w˜2. To sum up, the retailer optimal procurement policy under SRP can be solved by the backward induction method, which can be formulated as follows.


### 5.1. Optimal Procurement Policy at the Second Stage

In the second stage, the retailer is aware of both market demandD and supplier 1’s delivery quantity Q. Therefore, according to the differences between supply and demand in the first stage (i.e., D−Q), there are two situations to analyze:(1)
IfD≤Q, the supplied quantity of supplier 1 can effectively meet the market demand, indicating that the retailer will not order any product from the unreliable supplier 2 at this stage; that is, RSR=0.(2)
IfD>Q, the retailer should urgently purchase R units from the unreliable supplier 2. According to the probability density function gy, three situations can be obtained concerning the actual output yR.(i)
When supplier 2 is completely disrupted with the probability ofm (yR=0), the retailer’s profit equals Π21SR=pQ−csD−Q+.(ii)
When supplier 2 fully supplies products with probabilityn (i.e., yR=R), the retailer’s profit equals Π22SR=pminD,Q+R−csD−Q−R+−coQ+R−D+−w˜2R.(iii)
When supplier 2’s product supply is partially disrupted (i.e.,R>yR>0), the retailer’s profit equals Π23SR=pminD,Q+yR−csD−Q−yR+−coQ+yR−D+−w˜2yR.To sum up, in stage 2, since the market demand is deterministic, in the context ofD>Q, the retailer’s expected profit function at the second stage can be presented as follows:(4)Π2SRR=mΠ21SR+nΠ22SR+1−m−n∫01Π23SRdy.Furthermore, according to the relationship betweenR and D−Q (in the second stage, the market demand D is deterministic), Π2SRR can be rewritten as follows:(5)Π2SRR=Π2SR1R,R≥D−Q>0,Π2SR2R,R≤D−Q,where(6)Π2SR1R=mpQ−csD−Q+npD−coQ+R−D−w˜2R+∫0D−Q/RpQ+yR−csD−Q−yR−w˜2yR1−m−ndy+∫D−Q/R1pD−coQ+yR−D−w˜2yR1−m−ndy,(7)Π2SR2R=mQp+cs−csD+npQ+R−csD−Q−R−w˜2R+∫01pQ+yR−csD−Q−yR−w˜2yR1−m−ndy.In addition, sinceΠ2SR1D−Q=Π2SR2D−Q,Π2SRR is continuous in R. Combining cases of D≤Q and D>Q, the following lemma can be obtained via analyzing the properties of (5).Lemma 2.
Under the second stage of SRP and letΔ4=p+cs+co1−m−n/w˜2+co1−m+n, the following hold true:(1)
WhenΔ4≤1, the retailer’s optimal order quantity from supplier 2 RSR=D−Q+(2)
WhenΔ4>1, the retailer’s optimal order quantity from supplier 2 RSR=Δ4D−Q+Proof.
The details can be found at AppendixD. From Lemma 2, when D>Q, the retailer’s emergency procurement quantity from unreliable supplier 2 would be definitely not less than D−Q considering the supply disruption. And under SRP, two scenarios can be identified concerning the retailer’s order quantity from the unreliable supplier 2. That is, (i) when the emergency (shortage) cost is relatively higher (lower) or the supplier 2 is more likely to supply normally (i.e., higher values of n), Δ4 is more likely to be less than 1 (denoted as minor disruption scenario), implying that the retailer’s emergency order quantity from the unreliable supplier 2 remains at D−Q+. (ii) On the contrary, when supplier 2 is more possible to be entirely or partially disrupted, Δ4 would increase to exceed 1 (denoted as major disruption scenario), indicating that the retailer will set the order quantity from the supplier 2 equal to Δ4D−Q+, aiming to avoid the shortage and minimize the operating cost.


### 5.2. Optimal Procurement Policy at the First Stage

In the first stage, the market demandD is stochastic, and the retailer should order Q units from the reliable supplier 1. Considering the impact of Δ4 on the retailer’s order quantity from unreliable supplier 2 (from Lemma 2), the following two cases are considered:(1)
WhenΔ4≤1, then RSR=D−Q+. Based on the relationship between D and Q mentioned in Lemma 2, we have the following two cases:(i)
WhenD≤Q, we have RSR=0 and the retailer’s profit equals Π11SRQ=pD−coQ−D−w1Q.(ii)
WhenD>Q, we have RSR=D−Q, indicating that the retailer’s profit equals Π12SRQ=mQp+cs−csD+nDp−w˜2+w˜2Q+∫01Qp+cs1−y+w˜2y+p−w˜2y−cs1−yD1−m−ndy−w1Q.(2)
WhenΔ4>1, then RSR=Δ4D−Q+. Similarly,(i)
WhenD≤Q, we have RSR=0 and Π11SRQ=pD−coQ−D−w1Q(ii)
WhenD>Q, we have RSR=Δ4D−Q, indicating that the retailer’s profit equals Π12SRQ=mQp+cs−csD+npD−coΔ4−1+w2Δ4D−Q+∫01/Δ4pQ−cs1−yΔ4+w2yΔ4D−Q1−m−ndy+∫1/Δ41pD−coyΔ4−1+w2yΔ4D−Q1−m−ndy−w1Q.Under the circumstance, the market demandD is stochastic, indicating that the retailer’s profit function including both Δ4≤1 and Δ4>1 can be presented by the following maximization problem:(8)Π1SRQ=∫0QΠ11SRQfxdx+∫Q+∞Π12SRQfxdx.Thus, under different conditions ofΔ4, the following theorem can be obtained.Theorem 3.
Under the first stage of SRP, letΔ5=p+cs1−n+m+w˜21+n−m and Δ6=p+cs+com+p+cs+co1−m−nw˜2+co1−m+n; the following hold true: (1)
WhenΔ4≤1, then QSR=F−11−2w1+co/Δ5+2co(2)
WhenΔ4>1, then QSR=F−11−w1+co/Δ6Proof.
The details can be found at AppendixE. From Theorem 3, under both scenarios including minor and major disruption (corresponding to Δ4≤1 and Δ4>1, resp.), it is apparent that QSR is negatively associated with w1, while it is positively related to p, cs, and w˜2. That is, the retailer would procure more quantities from the reliable supplier 1 to improve supply reliability if shortage cost plays a more important role or the selling price can be set higher values. Further, the comparison concerning QSR under different scenarios can be presented in the following.Corollary 2.
In SRP, the retailer’s order quantity from the reliable supplier 1 under minor disruption scenario (i.e.,Δ4≤1) is not less than that under major disruption scenario (i.e., Δ4>1).Proof.
The details can be found at AppendixF. From Corollary 2, it is apparent that, under minor disruption scenario, the retailer would increase order quantity from the reliable supplier 1 compared with that of major disruption scenario. And combined with Lemma 2, it can be found that, under SRP, the retailer could assume different procurement policies according to the possibility of supply disruption. That is, (i) under minor disruption scenario, the retailer would assume a higher-order quantity from supplier 1 when demand is uncertain and accurately place order quantity from the unreliable supplier 2 based on D−Q+ after demand is realized. (ii) On the contrary, under major disruption scenario, the retailer would decrease the order quantity from supplier 1 (compared with that under minor disruption scenario) and set the order quantity from the supplier 2 equal to Δ4D−Q+.


## 5.1. Optimal Procurement Policy at the Second Stage

In the second stage, the retailer is aware of both market demandD and supplier 1’s delivery quantity Q. Therefore, according to the differences between supply and demand in the first stage (i.e., D−Q), there are two situations to analyze:(1)
IfD≤Q, the supplied quantity of supplier 1 can effectively meet the market demand, indicating that the retailer will not order any product from the unreliable supplier 2 at this stage; that is, RSR=0.(2)
IfD>Q, the retailer should urgently purchase R units from the unreliable supplier 2. According to the probability density function gy, three situations can be obtained concerning the actual output yR.(i)
When supplier 2 is completely disrupted with the probability ofm (yR=0), the retailer’s profit equals Π21SR=pQ−csD−Q+.(ii)
When supplier 2 fully supplies products with probabilityn (i.e., yR=R), the retailer’s profit equals Π22SR=pminD,Q+R−csD−Q−R+−coQ+R−D+−w˜2R.(iii)
When supplier 2’s product supply is partially disrupted (i.e.,R>yR>0), the retailer’s profit equals Π23SR=pminD,Q+yR−csD−Q−yR+−coQ+yR−D+−w˜2yR.To sum up, in stage 2, since the market demand is deterministic, in the context ofD>Q, the retailer’s expected profit function at the second stage can be presented as follows:(4)Π2SRR=mΠ21SR+nΠ22SR+1−m−n∫01Π23SRdy.Furthermore, according to the relationship betweenR and D−Q (in the second stage, the market demand D is deterministic), Π2SRR can be rewritten as follows:(5)Π2SRR=Π2SR1R,R≥D−Q>0,Π2SR2R,R≤D−Q,where(6)Π2SR1R=mpQ−csD−Q+npD−coQ+R−D−w˜2R+∫0D−Q/RpQ+yR−csD−Q−yR−w˜2yR1−m−ndy+∫D−Q/R1pD−coQ+yR−D−w˜2yR1−m−ndy,(7)Π2SR2R=mQp+cs−csD+npQ+R−csD−Q−R−w˜2R+∫01pQ+yR−csD−Q−yR−w˜2yR1−m−ndy.In addition, sinceΠ2SR1D−Q=Π2SR2D−Q,Π2SRR is continuous in R. Combining cases of D≤Q and D>Q, the following lemma can be obtained via analyzing the properties of (5).Lemma 2.
Under the second stage of SRP and letΔ4=p+cs+co1−m−n/w˜2+co1−m+n, the following hold true:(1)
WhenΔ4≤1, the retailer’s optimal order quantity from supplier 2 RSR=D−Q+(2)
WhenΔ4>1, the retailer’s optimal order quantity from supplier 2 RSR=Δ4D−Q+Proof.
The details can be found at AppendixD. From Lemma 2, when D>Q, the retailer’s emergency procurement quantity from unreliable supplier 2 would be definitely not less than D−Q considering the supply disruption. And under SRP, two scenarios can be identified concerning the retailer’s order quantity from the unreliable supplier 2. That is, (i) when the emergency (shortage) cost is relatively higher (lower) or the supplier 2 is more likely to supply normally (i.e., higher values of n), Δ4 is more likely to be less than 1 (denoted as minor disruption scenario), implying that the retailer’s emergency order quantity from the unreliable supplier 2 remains at D−Q+. (ii) On the contrary, when supplier 2 is more possible to be entirely or partially disrupted, Δ4 would increase to exceed 1 (denoted as major disruption scenario), indicating that the retailer will set the order quantity from the supplier 2 equal to Δ4D−Q+, aiming to avoid the shortage and minimize the operating cost.


## 5.2. Optimal Procurement Policy at the First Stage

In the first stage, the market demandD is stochastic, and the retailer should order Q units from the reliable supplier 1. Considering the impact of Δ4 on the retailer’s order quantity from unreliable supplier 2 (from Lemma 2), the following two cases are considered:(1)
WhenΔ4≤1, then RSR=D−Q+. Based on the relationship between D and Q mentioned in Lemma 2, we have the following two cases:(i)
WhenD≤Q, we have RSR=0 and the retailer’s profit equals Π11SRQ=pD−coQ−D−w1Q.(ii)
WhenD>Q, we have RSR=D−Q, indicating that the retailer’s profit equals Π12SRQ=mQp+cs−csD+nDp−w˜2+w˜2Q+∫01Qp+cs1−y+w˜2y+p−w˜2y−cs1−yD1−m−ndy−w1Q.(2)
WhenΔ4>1, then RSR=Δ4D−Q+. Similarly,(i)
WhenD≤Q, we have RSR=0 and Π11SRQ=pD−coQ−D−w1Q(ii)
WhenD>Q, we have RSR=Δ4D−Q, indicating that the retailer’s profit equals Π12SRQ=mQp+cs−csD+npD−coΔ4−1+w2Δ4D−Q+∫01/Δ4pQ−cs1−yΔ4+w2yΔ4D−Q1−m−ndy+∫1/Δ41pD−coyΔ4−1+w2yΔ4D−Q1−m−ndy−w1Q.Under the circumstance, the market demandD is stochastic, indicating that the retailer’s profit function including both Δ4≤1 and Δ4>1 can be presented by the following maximization problem:(8)Π1SRQ=∫0QΠ11SRQfxdx+∫Q+∞Π12SRQfxdx.Thus, under different conditions ofΔ4, the following theorem can be obtained.Theorem 3.
Under the first stage of SRP, letΔ5=p+cs1−n+m+w˜21+n−m and Δ6=p+cs+com+p+cs+co1−m−nw˜2+co1−m+n; the following hold true: (1)
WhenΔ4≤1, then QSR=F−11−2w1+co/Δ5+2co(2)
WhenΔ4>1, then QSR=F−11−w1+co/Δ6Proof.
The details can be found at AppendixE. From Theorem 3, under both scenarios including minor and major disruption (corresponding to Δ4≤1 and Δ4>1, resp.), it is apparent that QSR is negatively associated with w1, while it is positively related to p, cs, and w˜2. That is, the retailer would procure more quantities from the reliable supplier 1 to improve supply reliability if shortage cost plays a more important role or the selling price can be set higher values. Further, the comparison concerning QSR under different scenarios can be presented in the following.Corollary 2.
In SRP, the retailer’s order quantity from the reliable supplier 1 under minor disruption scenario (i.e.,Δ4≤1) is not less than that under major disruption scenario (i.e., Δ4>1).Proof.
The details can be found at AppendixF. From Corollary 2, it is apparent that, under minor disruption scenario, the retailer would increase order quantity from the reliable supplier 1 compared with that of major disruption scenario. And combined with Lemma 2, it can be found that, under SRP, the retailer could assume different procurement policies according to the possibility of supply disruption. That is, (i) under minor disruption scenario, the retailer would assume a higher-order quantity from supplier 1 when demand is uncertain and accurately place order quantity from the unreliable supplier 2 based on D−Q+ after demand is realized. (ii) On the contrary, under major disruption scenario, the retailer would decrease the order quantity from supplier 1 (compared with that under minor disruption scenario) and set the order quantity from the supplier 2 equal to Δ4D−Q+.


## 6. Retailer’s Optimal Procurement Policy under SS

Under SS, the retailer simultaneously procures products from two suppliers when market demand is stochastic, aiming to take the advantage of lower wholesale prices (see Figure1(d)). And considering supply disruption of supplier 2, the retailer’s expected profit can be presented as follows:(1)
When the supplier 2 supply process is entirely disrupted (i.e.,yR=0), the retailer could only apply Q units offered by supplier 1 to satisfy the market demand. And we have Π11SSQ,R=EpminD,Q−csD−Q+−coQ−D+−w1Q.(2)
When the supplier’s supply process is entirely normal (i.e.,yR=R), the market demand can be met by the retailer’s total received products Q+R units. And we have Π12SSQ,R=EpminD,Q+R−csD−Q−R+−coQ+R−D+−w1Q−w2R.(3)
When the supplier’s supply process is partially disrupted; (i.e.,R>yR>0), the retailer’s total available products are Q+yR units. And we have Π13SSQ,R=EpminD,Q+yR−csD−Q−yR+−coQ+yR−D+−w1Q−w2yR.Thus, the retailer’s expected profit function under SS can be depicted as follows:(9)Π1SSQ,R=mΠ11SS+nΠ12SS+1−m−n∫01Π13SSdy.By analyzing the property of (9), the following theorem can be obtained.Theorem 4.
Under SS, whenn>m1−m−n and p+cs−w1/p+cs−w2>1−m+n/1−m−n, the retailer’s optimal order quantity from supplier 1 and supplier 2 can be obtained by the following expressions: ∫01FQSS+yRSSdy+nFQSS+RSS+mFQSS=p+cs−w1/p+cs+co and ∫01yFQSS+yRSSdy+nFQSS+RSS/1−m−n=1−m+np+cs−w2/2p+cs+co1−m−n.Proof.
The details can be found at AppendixG. From Theorem 4, under SS, it can be identified that both QSS and RSS are decreasing with the procurement prices of two supplies, including w1 and w2. Furthermore, according to Theorems 1–4, it is possible to partially compare the retailer’s order quantity from two suppliers under different sourcing strategies, which can be formulated as follows.Corollary 3.
Whenn>m1−m−n and p+cs−w1/p+cs−w2>1−m+n/1−m−n, the following hold true.(1)
Regarding WPP-SR, ifw2≥1−m−np+cs+w11+m−n/1−n1+n−m, then RWS≥RSS(2)
With respect to WPP-DR, ifw˜1≥p+cs−co, then RSS≤RWD(3)
With respect to SRP andm+n2−m+n>0(i)
ifΔ4≤1 and p+cs−w2/p+cs+co≤1−2w1+co/Δ5+2co, then QSS≤QSR(ii)
ifΔ4>1 and p+cs−w2/p+cs+co≤1−w1+co/Δ6, then QSS≤QSRProof.
The details can be found at AppendixH.
From Corollary3, under mild assumptions, a series of results can be obtained to present differences of the retailer’s procurement solutions under different sourcing strategies. Specifically, (1) with respect to WPP, provided that the selling price or cs is relatively lower, the retailer’s order quantity from the unreliable supplier 2 under WPP (including WS and WD) is more than that of SS. In addition, if the wholesale price (emergency cost) of the reliable supplier 1 is relatively lower (higher), supplier 1 still plays a more important role under WPP than under SS. (2) Further, with the increase of the procurement (emergency cost) of the unreliable supplier 2, the retailer’s order quantity from the reliable supplier 1 under SRP is likely to be greater than that under SS.


## 7. Numerical Examples and Discussion

Example 1.
In this section, the following numerical examples are provided to emphasize the retailer’s preferences towards backup sourcing and simultaneous sourcing strategies in an explicit manner. Specifically, referring to Giri and Bardhan [5] and Zeng and Xia [10], let the market demand D follow a normal distribution with μ=150 and σ=5. Other parameters are set as follows: p=27,cs=0.5,co=1, w1=7, w2=5, w˜1=20, w˜2=15, m=0.2, and n=0.4. As a result, the form of gy can be presented as follows:(10)gy=0.2,y=0,0.4,0<y<1,0.4,y=1.
In this numerical example, by changing the values ofp, cs, and μ, the retailer’s expected profit under different backup sourcing and simultaneous sourcing can be depicted in Figure 2.
From Figure2, under most of circumstances, the retailer achieves better performance under backup sourcing strategy than under simultaneous sourcing. It implies that when market demand is relatively stochastic, it is unwise for the retailer to simultaneously procure products from both suppliers. And more conclusions can be further obtained as follows:(1)
It is apparent that, under any sourcing strategy, the retailer can earn more expected profit when products can be sold at higher selling pricep (see Figure 2(a)). Further, when the selling price is relatively lower, the retailer could earn less profit margin, leading the retailer to adopt SRP for directly improving the supply reliability and avoiding expensive emergency replenishment from the reliable supplier. And as the selling price of products increases, the retailer is less sensitive to supply disruption and becomes more inclined to order quantities for achieving better performance. As a result, WPP-DR would be regarded as the optimal backup sourcing by the retailer for lower wholesale prices from the unreliable supplier.(2)
Along with the increase of unit shortage costcs, Π1SR and Π1WS and Π1SS decrease, while Π1WD remains the same (see Figure 2(b)). That is, since WPP-DR could ensure market demand to be entirely satisfied with expensive emergency replenishment from the reliable supplier, WPP-DR would be the priority strategy for the retailer when cs is relatively higher. Nevertheless, other sourcing strategies are still under the effects of shortage cost. Furthermore, when cs is insignificant, the retailer is better off under SRP to take the advantage of both supply reliability and lower wholesale prices of the reliable supplier.(3)
Higher values ofμ lead to the increases of Π1SR and Π1WD, while resulting in the slight decreases of Π1SR and Π1SS (see Figure 2(c)). In other words, under SRP and WPP-DR, since one of the suppliers can be activated after the realization of market demand, the retailer could gain more profit by increasing the emergence quantity from the backup sourcing. On the contrary, the retailer under WPP-SR and SS would pay off more shortage costs in the same situation.Figure 2
The retailer’s expected profit under different values ofp, cs, and μ.
(a)(b)(c)Example 2.
In this example, via changing values ofm and n, this part aims to further verify the effects of supply disruption on the retailer’s preferences towards different sourcing strategies. And two types of data concerning wholesale prices are further proposed including (i) w1≤w˜2 (i.e., w1=11, w˜2=16) and (ii) w1>w˜2 (i.e., w1=16, w˜2=11), while other parameters remain the same in Example 1. Additionally, numerical results for w1≤w˜2 are presented in Figures 3(a) and 3(c), while others for w1>w˜2 are shown in Figures 3(b) and 3(d).
According to Figure3, it can be found that the retailer’s expected profit under backup sourcing strategies can be improved to some degree when the supply process of the unreliable supplier 2 grows to be more reliable, i.e., the decrease of m or the increase of n. Further, the retailer could earn more profit under SS when either m or n increase, indicating that the retailer can always benefit as long as the possibility of partial disruption reduces. In addition, in this example, the retailer is better off under backup sourcing than simultaneous sourcing. And the variation of m and n would impact the retailer’s preference towards different backup sourcing strategies:(1)
From Figures3(a) and 3(b), the retailer is under the combined effects of supply disruption (i.e., m) and the relationship of wholesale prices (i.e., between w1 and w˜2). Specifically, if the unreliable supplier 2 is less likely to entirely disrupt (i.e., m is relatively lower), the retailer would pay off more emergency cost to purchase products from the unreliable supplier 2, leading SRP to be the priority strategy. And along with the increase of m, the retailer is less convinced to receive products from the unreliable supplier 2 when demand is realized. In this case, the retailer would choose WPP-DR to regard the reliable supplier 1 as the backup supplier.(2)
From Figures3(c) and 3(d), when the unreliable supplier 2 is less likely to work normally (i.e., n is relatively lower), the retailer is better off under WPP-DR. That is, the retailer would regard the unreliable supplier 2 as major sourcing for purchasing products with the lowest wholesale price and take the reliable supplier 2 as backup sourcing for ensuring all market demand to be satisfied. Further, when the unreliable supplier 2 becomes more reliable, the retailer would prefer SPR, which regards supplier 1 as backup sourcing due to the lower wholesale price and its acceptable supply disruption. And this finding is irrelevant to the relationships between w1 and w˜2.Figure 3
The retailer’s expected profit under different values ofm and n.
(a)(b)(c)(d)Example 3.
In this example, we aim to change values of wholesale prices to demonstrate the differences among different backup sourcing strategy and simultaneous sourcing policy. Apparently, along with the decreases ofw˜1 and w˜2, all of three backup sourcing strategies would achieve better performance, while simultaneous sourcing strategy remains the same. Thus, in this example, we aim to change values of w1 and w2 to quantify their impacts. For simplicity, let p = 28, μ=100, and other parameters keep the same as those in Example 1.
From Figure4, along with the increases of w1 and w2, the retailer’s expected profit under both backup sourcing and simultaneous sourcing would be lowered. And since WPP-DP and SRP are irrelevant to w1 and w2, respectively, it can be found that the retailer is better off under WPP-DP or SRP when w1 or w2 is relatively higher. In other words, compared with simultaneous sourcing strategy, the retailer is able to choose the specific backup sourcing strategies on the basis of wholesale price and the possibility of supply disruption. Additionally, unless the emergency prices are extremely higher, backup sourcing could perform better than simultaneous sourcing strategy.Figure 4
The retailer’s expected profit under different values ofw1 and w2.
(a)(b)


## 8. Conclusions

Subject to the combined negative effects of supply disruption and stochastic demand, the retailer would suffer from supply shortage and loss of goodwill, which may significantly lower operational performances. To mitigate negative effects of supply uncertainty, this paper proposes different backup sourcing and simultaneous strategies, where backup sourcing includes wholesale price priority (WPP) and supply reliability priority (SRP). Although these different types of sourcing strategies have found their wide applications to prevent the retailer from stock risk, it is urgent for the retailer to identify the most appropriate sourcing strategy via trading off supply disruption and stochastic demand as well as wholesale prices:(1)
Under supply disruption and stochastic demand, two types of WPP strategies including WPP-SR and WPP-DR are available for the retailer to regard the unreliable supplier with lower wholesale price as the major sourcing. Specifically, (i) under WPP-SR, the reliable supplier is initiated after the occurrence of supply disruption for ensuring the retailer’s order quantity from unreliable supplier to be always met. (ii) Under WPP-DR, the retailer would replenish from the reliable supplier after the realization of both supply and demand uncertainties, which guarantees the market demand to be entirely satisfied with more expensive emergency price. Besides, it is apparent that, along with the increases of selling price and shortage cost, the retailer definitely orders more quantity from the reliable supplier under WPP-SR, while it remains the same under WPP-DR. In addition, when the procurement or emergency price of the reliable supplier is relatively higher, the retailer’s order quantity from the reliable supplier under WPP-SR is less than that of WPP-DR.(2)
Further, under SRP, there are two scenarios, including minor disruption and major disruption, when addressing the retailer’s optimal order quantity from two suppliers. That is, (i) under minor disruption scenario, the unreliable supplier 2 is more likely to supply normally or offer a higher value of emergency price. Thus, the retailer would assume a higher-order quantity from supplier 1 when demand is uncertain and accurately place order quantity from the unreliable supplier 2 based onD−Q+ after demand is realized. (ii) Under major disruption scenario, the unreliable supplier 2 is more possible to be entirely or partially disrupted; the retailer would decrease the order quantity from supplier 1 (compared with that under minor disruption scenario) and set the order quantity from the supplier 2 equal to Δ4D−Q+. In addition, under both minor and major disruption scenarios, the retailer would procure more quantities from the reliable supplier to improve supply reliability if shortage cost or selling price plays a more important role.(3)
Unlike existing studies under supply disruption, this paper systematically compares the retailer’s preferences towards backup sourcing and simultaneous sourcing strategies via theoretical results and numerical examples. (i) Since WPP-DR could ensure market demand to be entirely satisfied by expensive emergency replenishment from supplier 1, the retailer would regard it as the optimal backup sourcing when selling price or shortage cost is relatively higher. On the contrary, the retailer would adopt SRP for directly improving the supply reliability and avoiding expensive emergency replenishment from the reliable supplier. (ii) When the unreliable supplier 2 is less likely to work normally, the retailer would regard WPP-DR the reliable supplier 2 as backup sourcing for ensuring all market demand to be satisfied. Otherwise, the retailer would prefer SPR, which regards supplier 2 as backup sourcing due to the lower wholesale price and its acceptable supply disruption. (iii) In addition, unless the emergency prices are extremely higher, backup sourcing could perform better than simultaneous sourcing.



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*Source: 9612648-2020-08-28.xml*