# Optimal Supply Diversification Strategy under Supply Disruption

Authors: Xiaomeng Luo; Jia Ren
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

When a disruption caused by human or environmental accident occurs in production systems, it may cause a shortage of the supply, and thus the buyers’ procurement behaviors will be influenced. This paper studies a supply chain comprised of a buyer and two types of suppliers: one is cheap but unreliable and the other is reliable but expensive. If there is a major disruption, the unreliable supplier may not be able to fully satisfy the buyer’s order, despite the fact that it exerts additional effort to rebuild capacity; at the same time, the reliable supplier cannot fulfill extra orders from the buyer due to capacity constraints. In this way, the buyer should strategically allocate its order between the two types of suppliers by offering different contracts at the very beginning, and then the unreliable supplier chooses its optimal restoration effort according to the contract if a disruption occurs. The model is built based on the real-life cases such as Walmart and Apple such that it is the buyer who determines the wholesale price of the unreliable supplier’s products. The results show the optimal contracts provided by the buyer under different circumstances, which aims to help managers design their contracts under disruption risks to maximize the company’s profit.


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## Body

## 1. Introduction

It is common to observe a variety of disruptions occurring in global supply chain, and thus firms are exposed to risks and uncertainties [1]. The disruption can be attributed to various factors such as natural disasters, financial bankruptcy, government policy, and human factors. When a disruption happens, firms would be quite passive if they did not prepare backup plans. For example, two deadly earthquakes that happened in Southern Japan halted Sony’s production of image sensors in 2016. This affected the procurement of its major customers, and, thus, for instance, Apple had to postpone the supply of its new iPhone while Samsung shifted to alternative suppliers at a much higher price although its production process is not affected (“Japan Quakes Close Sony Factory Making Key iPhone Part,” Fortune, 16 April, 2016, https://fortune.com/2016/04/16/japan-quakes-disrupt-sony-image-sensors-apple-iphones/, accessed on January 2020). More recently, a series of actions have been taken by Apple to mitigate the effects of possible disruptions: execs from Apple, together with representatives from Google, Microsoft, and Amazon, visited Korea to ensure consistent supplies of critical chips, which might be affected by the trade war between Japan and Korea; and, considering the disruption risk caused by the US-China trade war, Apple has planned to shift 15% to 30% production out of China to reduce its dependence on Chinese suppliers (“Apple Execs Rush to Samsung in Korea Fearing Disruption in iPhone Chip Supplies,” iGeeksBlog, 20 July, 2019, https://www.igeeksblog.com/apple-execs-rush-samsung-iphone-chip-supplies-135428/, accessed on January 2020).Although it is more efficient to procure from a single supplier aiming at developing long-term relationship, firms tend to diversify their sources to prevent supply disruption in practice. A classic case regarding two famous mobile communication producers shows the benefits brought by diversifying procurement sources. In March 2000, a Philips Semiconductor plant has been shut down due to a sudden fire caused by lightning in Albuquerque, New Mexico, for six weeks. This leads to a shortage of components for both Ericsson and Nokia, but the situations of the two companies are very different. Although Nokia has lost all of the supply from the Philips plant, it was able to source from alternative suppliers by temporarily increasing their production and thus suffered little financial loss at last. However, Ericsson did not have backup suppliers to shift supply, so it eventually lost at least 400 million dollars in potential revenue [2]. To adopt multisource strategy, some critical decisions need to be determined by firms at the very beginning, such as how many suppliers to choose and how many quantities to procure from each of them. The common practice is to source the majority of products from one or two major suppliers which provide favorable terms and the minority from other small suppliers, just as Samsung and Nokia do.The fact is if firms are sufficiently powerful, they can procure products at the prices they are pleased of. For example, more than 450 US companies competed to provide 750 products in the stores of Walmart nationwide in 2016, and only some of them won the chance to get into the stores. The unequal power in the supply and demand relationship between Walmart and its suppliers adds bargaining power to Walmart. Thus, it is Walmart’s decision to set the final wholesale price based on the cost structure of the product, which requires the suppliers to eliminate excess costs (“Amy Feldman, “What It Takes To Sell To Walmart: A Senior Buyer Tells All,” Forbes, 11 May,https://www.forbes.com/sites/amyfeldman/2017/05/11/what-it-takes-to-sell-to-walmart-a-senior-buyer-tells-all/#25ef71e5352b, accessed on January 2020). Other large companies such as Costco, Amazon, and Apple are similar to Walmart when bargaining with their suppliers on price. The suppliers can choose to accept the prices offered by those big names or they can leave. Most suppliers will choose to stay considering the large volume and reputation provided by big companies. Thus, the procurement decisions of those companies become how many suppliers to choose and which wholesale price and order quantity to provide to each of the suppliers.Based on the above background, this paper aims to study the optimal procurement strategy of a powerful firm (referred to as “buyer”). Basically, the buyer can source from two types of suppliers, reliable but expensive ones and unreliable ones which may suffer production disruptions. The buyer has to decide which type of suppliers to partner with and how many quantities to procure at what wholesale price. In all, there are three possible strategies: procuring from the reliable suppliers only, sourcing from the unreliable ones only, and ordering from both types of suppliers. The buyer chooses the best answer by making a tradeoff between the profit and the risk. We are interested in which strategy would be the optimal choice for the buyer when the risk of production disruption exists. The remainder of this paper is organized as follows: Section2 summarizes the relevant literature, Section 2 provides the model description, and optimal results are given in Section 4. Section 5 provides a numerical study. Conclusion is presented in Section 6.


## 2. Literature Review

The management of supply disruption has long been an interesting topic for both researchers and practitioners in the Operations Management field. We may refer to Vakharia and Yenipazarli [3] for an excellent review of relevant literature, Snyder et al. [4] for a summary for OR/MS models used in the study of supply chain disruption, and Ivanov et al. [5] for quantitative tools used in supply chain disruption recovery.To fight against supply disruption, which is modeled as an all-or-nothing yield model, supply diversification is a useful strategy [6, 7]. Other operational tools include multiple sourcing [6, 8], improvement in reliability and process (Liu et al. [9] and Wang et al. [10]), and cost-sharing mechanism [11]. Iyer et al. (2005) studied the supplier’s optimal contract structure and explored the impact of an alternate supplier in a supply chain consisting of a monopolist supplier and multiple buyers. By conducting a detailed study, Tomlin [12] considers firm characteristics such as risk tolerance and supplier characteristics such as percentage uptime, disruption length, capacity, and flexibility to study the supply disruption management problem. They provide various strategies adopted by buyers to manage supply disruption risk, including investment in inventory, cooperation with multiple suppliers, and preparation for backup orders to alternative suppliers. Hu et al. [13] study how a buyer can use incentive mechanisms to motivate a supplier’s investment in capacity restoration and compare this approach with the traditional approach of diversifying part of the order to an expensive but reliable supplier. Chen et al. [14] compare two incentive mechanisms, i.e., providing financial subsidy and adjusting the wholesale price(namely, direct and indirect contract, respectively) and study their effects on the supplier’s investment behavior.A lot of factors are involved in the study for supply disruption. Gupta et al. [15] consider competition in the buyer level and find that both supply disruption and procurement times will affect the buyers’ procurement decisions. Ray and Jenamani [16] analyze the uncertainty from both the supply side and the demand side, providing the optimal order allocation to hedge against supply disruption risk. Besides all-or-nothing supply, Tang et al. [17] involve partial disruption and find that subsidy is preferred in all-or-nothing setting, while both subsidy and order inflation are optimal in partial disruption setting. Tsai [18] proposes a dynamic sourcing strategy that helps managers determine the optimal number of suppliers required in different scenarios. Based on a multitier supply chain structure, Ang et al. [19] explore the interactions between the manufacturer and suppliers and show that the degree of overlap in the supply chain may impact the optimal strategies of both manufacturer and tier 1 supplier. A series of articles analyze the supply chain coordination under supply disruption risks [20–22]. Specifically, Sawik [23] provides the stochastic and deterministic approaches to coordinate the supply chain and finds that the stochastic approach offers a more diversified supply portfolio which helps deal with a variety of scenarios.Although managers know clearly the impact of supply chain disruptions, they actually have done little to prevent the incidents and corresponding effects [24]. Chopra and Sodhi [25] thus remind managers to notice the disruption risk when they pursue cost efficiency by providing feasible advices on real actions. Based on the real situation of the Chinese dairy market, He et al. [26] explore the optimal ordering strategy for a retailer who can source from a local supplier and an oversea supplier under supply disruption. With the help of real-option pricing methodology and actuarial techniques, they help managers to make decisions on procurement. Our model studies a supply chain composed of one buyer and two suppliers, and there are three options for buyer to choose: the first one is ordering from the reliable supplier only, the second one is ordering from the unreliable supplier only, and the third one is ordering from both of them. It is obvious that the first one is more expensive and the second one is more risky, and thus the intermediate choice not only avoids some risk but also reduces cost. The main features of our proposed model are the focus on supplier’s restoration effort after disruption and buyer’s order allocation strategy and pricing strategy.


## 3. Model Description

We consider a supply chain that consists of a buyer and two suppliers: one is reliable supplier and the other one is unreliable supplier. For simplicity, we assume that the market demandd and retail price p are fixed and given. We further assume that the original capacity of either supplier is enough to cover the market demand. Specifically, the reliable supplier can fully satisfy the buyer’s order, and the unreliable supplier also can if there is no disruption. Note that once the order has been offered to the reliable supplier, no replenishment order can be placed later if the unreliable supplier suffers disruption. To better model the practice in the real world, we assume that the wholesale price of the reliable supplier’s product is deterministic, which is determined by the market, while the wholesale price of the unreliable supplier’s product is decided by the buyer since the buyer has power towards the unreliable supplier. Thus, the buyer has to decide an order allocation between the two suppliers and the wholesale price from the unreliable supplier to maximize its own profit.For the unreliable supplier, we usequ,wu to denote the contract offered by the buyer. Following Anupindi and Akella [6] and Gurnani et al. [10], we assume that the risk of disruption is dichotomous (all or nothing), which means that the unreliable supplier may lose all of its capacity if there is an unexpected disruption. There is no disruption with probability β and disruption occurs with probability 1−β. When disruption happens, the unreliable supplier has to decide whether and how much costly effort to exert to rebuild capacity. The amount of restored capacity is proportional to the supplier’s effort level e; that is, with effort e, the restored capacity is αe, where α is a deterministic factor and we further assume that α=1. Restoration effort incurs a cost determined by a convex and increasing function CRe, representing the diminishing return of effort on capacity recovery [27]. The production cost of either supplier is normalized to 0, and the unit production cost of the unreliable supplier after disruption is assumed to be cd.For the reliable supplier, we useqr,wr to represent the long-term contract provided by the buyer, where wr is deterministic and is exogenously given. We assume that the reliable supplier has unlimited capacity of satisfying the order but cannot afford any more orders if the unreliable supplier is not able to meet the order. To ignore the trivial case, we assume wr>wu.The buyer can source from two suppliers to reduce the risk of the disruption and save cost through two contracts: one is a long-term contractqr,wr for the reliable supplier and the other one is qu,wu for the unreliable supplier. The buyer’s problem is whether to source from two suppliers and how many to order from either. Note that the buyer does not offer any extra incentive to the unreliable supplier for restoring capacity if a disruption occurs. This is because the buyer can still receive a part of the order from the reliable supplier even if the unreliable supplier suffers disruption and does not exert any restoration effort. If there is a disruption, the sequence of the events is shown in Figure 1.Figure 1
Sequence of the events.


## 4. Equilibrium Analysis

Our goal is to find the optimal restoration effort exerted by the unreliable supplier,e∗, the optimal order quantities, qu∗ and qr∗, and the optimal wholesale price offered by the buyer to the unreliable supplier, wu∗.


### 4.1. Optimal Strategy of Unreliable Supplier

If a disruption occurs, the unreliable supplier exerts restoration efforte based on the contract qu,wu. For analytical tractability, we assume a quadratic restoration cost function CRe=ke2/2, where e is the effort that unreliable supplier exerts and k is a coefficient related to the rebuild cost. Similar cost functions have been widely used in the operation and marketing literature to model the diminishing impact of effort [28–30]. With the restoration effort e, we can formulate the unreliable supplier’s profit function if there is a disruption for given values of qu as follows:(1)Πse=−ke22+wu−cdminqu,e.If the wholesale pricewu is less than the postdisruption unit production cost (i.e., wu≤cd), the unreliable supplier will not exert effort to restore capacity. Note that our assumption about the restoration cost CRe=ke2/2 implies that the marginal effort cost increases with the amount of effort. The unreliable supplier’s optimal effort is summarized as follows.Proposition 1.
Ifwu≤cd, the unreliable supplier will not exert any effort to restore capacity for order fulfillment; otherwise, the supplier’s optimal restoration effort level e∗qu is increasing in qu and(2)e∗qu=minwu−cdk,qu.Obviously, if the unreliable supplier decides to exert effort to rebuild capacity after disruption, the amount of restored capacity will not be more than the order quantity (i.e.,e∗≤qu). As shown in Figure 2, if qu falls in 0≤qu≤wu−cd/k, the supplier’s restoration effort level e increases with qu , which means that the supplier can fully meet the order even if there is a disruption without any additional incentive from the buyer. While qu is falls in qu>wu−cd/k, the supplier’s restoration effort level e is fixed and is independent with qu, and thus the order is not fully satisfied.Figure 2
The relationship between restoration effort and order quantity.


### 4.2. Optimal Procurement Strategy

In all, the buyer can choose among three strategies. The first strategy is ordering from the reliable supplier only (strategy R), the second one is ordering from the unreliable supplier only (strategy U), and the last one is ordering from both of the suppliers, also referred to as supply diversification strategy (strategy D).


#### 4.2.1. Ordering Only from Reliable Supplier

The buyer orders only from the reliable supplier with fixed wholesale pricewr. We use qr to denote the order quantity under this strategy. The profit function of the buyer is as follows:(3)ΠbRqr=p−wrminqr,d.Under the wholesale contract, it is easy to obtainqr∗=d.


#### 4.2.2. Ordering Only from Unreliable Supplier

The buyer orders only from the unreliable supplier through a wholesale contractqu,wu. If the disruption occurs with the possibility of 1−β, the unreliable supplier will lose all the capacity and decide whether to exert restoration effort e to rebuild capacity. We assume that the unreliable supplier will get the restoration capacity e through the restoration effort e. From Proposition 1, we know that the restoration capacity depends on the values of qu and wu, and the profit function of the buyer is formulated as follows:(4)ΠbUqu,wu=p−wuβminqu,d+1−βmine∗,d.Taking (2) into (4), we have ΠbUqu,wu=p−wuβminqu,d+1−βminwu−cd/k,qu,d. We consider two cases: when qu<d, we have ΠbUqu,wu=p−wuβqu+1−βminwu−cd/k,qu, which increases with qu; when qu≥d, we have ΠbUqu,wu=p−wuβd+1−βminwu−cd/k,d, which is unrelated to qu. We summarize the above results in Lemma 1.Lemma 1.
If the seller orders from the unreliable supplier only, thenqu∗=d.From Lemma1, we rewrite equation (4) as follows:(5)ΠbUwu=p−wuβd+1−βminwu−cdk,d,where 0≤wu≤p.To solve the optimal wholesale price provided to the unreliable supplier, we conduct our analysis based on two casescd+kd≤p and cd+kd>p as follows.Whencd+kd≤p, we consider two subcases wu−cd/k≥d and wu−cd/k<d as follows:(1)
Ifwu−cd/k≥d, then the profit function is(6)ΠbUwu=p−wud.SinceΠbUwu decreases with wu, we obtain the local maximizer that w^u=kd+cd and ΠbUw^u=p−kd−cdd.(2)
Ifwu−cd/k<d, we have cd≤wu<kd+cd; then, the profit function is(7)ΠbUwu=−1−βwu2k+1−βp+cdk−βdwu−1−βpcdk+βpd.Since the axis of symmetry of equation (7) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Whenp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbUw^u=βdp−cd.(b)
Whencd<p+cd/2−βkd/21−β<cd+kd, we have w^u=p+cd/2−βkd/21−β and ΠbUw^u=p−cd/2+βkd/21−ββd/2+1−βp−cd/2k.(c)
Whenp+cd/2−βkd/21−β≥cd+kd, we have w^u=cd+kd and ΠbUw^u=dp−kd−cd.Thus, we conclude the equilibrium outcomes whencd+kd≤p as follows:(1)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≤0 or cd<p+cd/2−βkd/21−β<cd+kd and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kd, then w^u=cd+kd and ΠbU=p−cd−kdb.(2)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≥0, then w^u=cd and ΠbU=βbp−cd.(3)
Ifcd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β, then w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.Whencd+kb>p, we know that wu−cd/k<b,i.e.,cd<wu<p; thus, the profit function is as follows:(8)ΠbU=maxwu−1−βwu2k+1−βp+cdk−βbwu−1−βpcdk+βpb.Since the axis of symmetry of Function (8) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Ifp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbU=βbp−cd.(b)
Ifcd<p+cd/2−βkd/21−β<p, we have w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(c)
Ifp+cd/2−βkd/21−β≥p, we have w^u=p and ΠbU=0.According to the above analysis, we summarize our equilibrium results in Proposition2.Proposition 2.
If the buyer only procures from the unreliable supplier, the equilibrium decision and profit of the buyer are summarized as follows:(i)
Ifcd+kb≤p, (1) p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≤0 or (2) cd<p+cd/2−βkd/21−β<cd+kb and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kb, then wu∗=cd+kb and ΠbU=p−cd−kbb.(ii)
If (1)cd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β or (2) cd+kb>p and cd<p+cd/2−βkd/21−β<p, then wu∗=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(iii)
If (1)cd+kb≤p, p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≥0 or (2) cd+kb>p and p+cd/2−βkd/21−β≤cd, then wu∗=cd and ΠbU=βbp−cd.(iv)
Ifcd+kb>p and p+cd/2−βkd/21−β≥p, then wu∗=p and ΠbU=0.


#### 4.2.3. Supply Diversification Strategy

In this part, we discuss the optimal strategy of the buyer when ordering from both suppliers. We assume thatqu+qr=b and the buyer’s profit is(9)ΠbDwu,qu=βpb−wuqu−wrqr+1−βpe∗+qr−wue∗−wrqr,where e∗=minwu−cd/k,qu and qr=b−qu. We further consider two cases qu≤wu−cd/k and qu>wu−cd/k as follows.Whenqu≤wu−cd/k, we have(10)ΠbDwu,qu=pb−quwu−b−quwr=p−wrb+wr−wuqu.SinceΠbDwu,qu is decreasing in wu, we have wu=kqu+cd. Taking it back into equation (10), we have(11)ΠbDqu=−kqu2+wr−cdqu+p−wrb.Since the axis of symmetry of the function isqu=wr−cd/2k, for 0≤qu≤b, we have the following two subcases 0≤wr−cd/2k≤b and wr−cd/2k>b:(1)
If0≤wr−cd/2k≤b, then qu^=wr−cd/2k, w^u=wr+cd/2, and ΠbDqu^,w^u=wr−cd2/4k+p−wrb.(2)
Ifwr−cd/2k>b, then qu^=b, w^u=kb+cd and ΠbDqu^,w^u=p−cdb−kb2.Whenqu>wr−cd/2k, we have(12)ΠbDwu,qu=wr−βwu−1−βpqu+p−wrb+1−βk−w2+p+cdw−pcd.We can easily obtain thatcd≤wu≤kb+cd; then we consider three subcases wr−1−βp/β<cd, cd≤wr−1−βp/β<kb+cd, and wr−1−βp/β≥kb+cd as follows:(1)
Ifwr−1−βp/β<cd, we have the coefficient of qu in (13) which is wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu; then qu^=wu−cd/k. Then, we can rewrite equation (12) as follows:(13)ΠbDwu=−wu2k+wr+cdwuk−wrcdk+p−wrb.Since the axis of symmetry of equation (13) is wu=wr+cd/2, for cd≤wu≤kb+cd, then we have the following two scenarios:(a)
Ifwr+cd/2≤kb+cd, we have w^u=wr+cd/2, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.(b)
Ifwr+cd/2>kb+cd, we have w^u=kb+cd, qu^=b and ΠbDw^u,qu^=p−cdb−kb2.(2)
Ifcd≤wr−1−βp/β<kb+cd, we have the following two scenarios:(a)
Ifcd≤wu≤wr−1−βp/β, then wr−βwu−1−βp≥0, so ΠbDwu,qu is increasing in qu, for 0≤qu≤b; we obtain qu^=b. Taking it into equation (12), we have(14)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Since the axis of symmetry of equation (14) is wu=p+cd/2−βkd/21−β, we thus conduct the following discussion:Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kw^u=p+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥wr−1−βp/β, then w^u=wr−1−βp/β, qu^=b, and ΠbDw^u,qu^=pwr+cd−1−βp2+wrcd.(b)
Ifwr−1−βp/β<wu≤kb+cd, then wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu, for wu−cd/k≤qu≤b; we obtain qu^=wu−cd/k. Taking it into (13), we haveSince the axis of symmetry of (16) is wu=wr+cd/2k, we thus conduct the following discussion:Ifwr+cd/2k<wr−1−βp/β, then w^u=wr−1−βp/β, qu^=p−cd/k−p−wr/βk, and ΠbDw^u,qu^=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb.Ifwr−1−βp/β≤wr+cd/2k<kb+cd, then w^u=wr+cd/2k, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.Ifwr+cd/2k≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.(3)
Ifwr−1−βp/β≥kb+cd, we have wr−βwu−1−βp>0, so qu^=b. Taking it into equation (12), we have(15)ΠbDwu=−1kwu2−wr+cdwu+wrcd+p−wrb.Since the axis of symmetry of equation (16) is w^u=p+cd/2−βkd/21−β, for cd≤wu≤kb+cd, we thus conduct the following discussion:(16)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<kb+cd, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.According to the above analysis, we have the following discussions to obtain our equilibrium results when the buyer adopts supply diversification strategy.Case 1.
Ifwr−1−βp/β<cd and wr+cd/2k≤kd+cd, the analysis is shown in Figure 3.
As we can see, for0≤qu≤b and cd≤wu≤kb+cd, the feasible region is a rectangle with an oblique line cut into two parts, where the solution becomes better along the direction of the arrow. So under this circumstance, we obtain the global optimal solutions wu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 3
Case1: wr−1−βp/β<cd and wr+cd/2≤kd+cd.Case 2.
Ifwr−1−βp/β<cd≤kd+cd<wr+cd/2, we have our analysis as shown in Figure 4.
It is easy to obtain the global optimal solution which iswu∗=kb+cd and qu∗=b, which is the intersection of the lines qu=b and qu=wu−cd/k, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 4
Case2: wr−1−βp/β<cd≤kd+cd<wr+cd/2.Case 3.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd, the analysis is shown in Figure 5.
We can see that the feasible region is cut into three parts with a local optimal solution in each part. We can easily find the global optimumswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 5
Case3: p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd.Case 4.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd, the analysis is shown in Figure 6.
As we can see, the feasible region is divided into three parts which is the same as in Figure5, but the difference is that there are two similar local optimums and we cannot intuitively obtain the global optimum. Thus, we need to compare the profit of the two points cd,b and wr+cd/2,wr−cd/2k, and the optimal profit is maxΠbDcd,b,ΠbDwr+cd/2,wr−cd/2k.Figure 6
Case4: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd.Case 5.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2, the analysis is shown in Figure 7.
As we can see, there are two local optimums. Sincep+cd/2−βkd/21−β<cd, we compare the profits of the two pointscd,b and kb,b and find that ΠbDcd,b≥ΠbDkb,b, so the global optimum is wu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 7
Case5: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2.Case 6.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd, the analysis is shown in Figure 8.
Similarly, there are three local optimums, but we can easily obtain that the global optimal solution iswu∗=p+cd/2−βkd/21−β and qu∗=b.Figure 8
Case6: cd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd.Case 7.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β<wr+cd/2≤kb+cd, the analysis is shown in Figure 9.
We can obtain that the local optimums arep+cd/2−βkb/21−β,b and wr+cd/2,wr−cd/2k. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDwr+cd/2,wr−cd/2k.Figure 9
Case7: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<wr+cd/2≤kb+cd.Case 8.
Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2, the analysis is shown in Figure 10.
We can obtain the two local optimumsp+cd/2−βkd/21−β,b and kb+cd,b. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDkb+cd,b.Figure 10
Case8: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2.Case 9.
Ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd, the analysis is shown in Figure 11.
We can see thatwr+cd/2,wr−cd/2k is the worst solution, so the best solution is either wr−1−βp/β,b or wr−1−βp/β,p+cd/2−βkb/21−β. The profit of the points is ΠbDwr−1−βp/β,b=pwr+cd−1−βp2+wrcd and ΠbDwr−1−βp/β,p+cd/2−βkd/21−β=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb, and the optimal profit is the bigger one.Figure 11
Case9: cd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd.Case 10.
Ifcd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β, the analysis is shown in Figure 12.
We can easily obtain the optimal solutionswu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 12
Case10: cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β.Case 11.
Ifcd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, the analysis is shown in Figure 13.
It is easy to find that the optimal solution iswu∗=b+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 13
Case11: cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd.Case 12.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 14.
As we can see, the global optimum iswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 14
Case12: p+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β.Case 13.
Ifp+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 15.
At this circumstance, the global optimum is the same as the one above.Figure 15
Case13: p+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 14.
Ifcd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 16.
We can see that the global optimum iswu∗=p+cd/2−βkd/21−β and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Figure 16
Case14: cd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β.Case 15.
Ifcd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 17.
The solution is the same as the one above.Figure 17
Case15: cd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 16.
Ifcd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 18.
As we can see, the global optimum iswu∗=kb+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 18
Case16: cd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd.Case 17.
Ifwr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 19.
The solution is the same with the one above.
According to our analysis, we come to the conclusions which are summarized in Proposition 3.Figure 19
Case17: wr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd.Proposition 3.
In all, the optimal strategy of the buyer might be divided into six different scenarios as follows:(i)
wu∗=wr+cd/2 and qu∗=wr−cd/2k if(1)wr−1−βp/β<cd and wr+cd/2≤kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ∏bDwr+cd/2,wr−cd/2k≥∏bDcd,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, ∏bDwr+cd/2,wr−cd/2k≥∏bDp+cd/2−βkb/21−β,b, or (4) cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkb/21−β≥wr−1−βp/β.(ii)
wu∗=kb+cd and qu∗=b if(1)wr−1−βp/β<cd<kb+cd<wr+cd/2 or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b≥ΠbDp+cd/2−βkd/21−β,b or (3) cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, or (4) wr−1−βp/β≥kb+cd and p+cd/2−βkd/21−β≥kb+cd.(iii)
wu∗=cd and qu∗=b if(1)p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ΠbDwr+cd/2,wr−cd/2k<ΠbDcd,b or (3) p+cd/2−βkd/21−β<cd<wr−1−βp/β<kb+cd<wr+cd/2, or (4) p+cd/2−βkd/21−β<cd<kb+cd<wr−1−βp/β.(iv)
wu∗=p+cd/2−βkd/21−β and qu∗=b if(1)cd≤p+cd/2−βkd/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, and ΠbDwr+cd/2,wr−cd/2k<ΠbDp+cd/2−βkd/21−β,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b<ΠbDp+cd/2−βkd/21−β,b, or (4) cd≤p+cd/2−βkd/21−β<kb+cd≤wr−1−βp/β.(v)
wu∗=wr−1−βp/β and qu∗=b ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b≥ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.(vi)
wu∗=wr−1−βp/β and qu∗=p−cd/k−p−wr/βk ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b<ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.Furthermore, from Proposition3, we can obtain the optimal restoration effort of the unreliable supplier as shown in Proposition 4.Proposition 4.
Given the optimal contract provided by the buyerwu∗,qu∗, the optimal restoration effort exerted by the unreliable supplier is e∗wu∗,qu∗=minwu∗−cd/k,qu∗.


## 4.1. Optimal Strategy of Unreliable Supplier

If a disruption occurs, the unreliable supplier exerts restoration efforte based on the contract qu,wu. For analytical tractability, we assume a quadratic restoration cost function CRe=ke2/2, where e is the effort that unreliable supplier exerts and k is a coefficient related to the rebuild cost. Similar cost functions have been widely used in the operation and marketing literature to model the diminishing impact of effort [28–30]. With the restoration effort e, we can formulate the unreliable supplier’s profit function if there is a disruption for given values of qu as follows:(1)Πse=−ke22+wu−cdminqu,e.If the wholesale pricewu is less than the postdisruption unit production cost (i.e., wu≤cd), the unreliable supplier will not exert effort to restore capacity. Note that our assumption about the restoration cost CRe=ke2/2 implies that the marginal effort cost increases with the amount of effort. The unreliable supplier’s optimal effort is summarized as follows.Proposition 1.
Ifwu≤cd, the unreliable supplier will not exert any effort to restore capacity for order fulfillment; otherwise, the supplier’s optimal restoration effort level e∗qu is increasing in qu and(2)e∗qu=minwu−cdk,qu.Obviously, if the unreliable supplier decides to exert effort to rebuild capacity after disruption, the amount of restored capacity will not be more than the order quantity (i.e.,e∗≤qu). As shown in Figure 2, if qu falls in 0≤qu≤wu−cd/k, the supplier’s restoration effort level e increases with qu , which means that the supplier can fully meet the order even if there is a disruption without any additional incentive from the buyer. While qu is falls in qu>wu−cd/k, the supplier’s restoration effort level e is fixed and is independent with qu, and thus the order is not fully satisfied.Figure 2
The relationship between restoration effort and order quantity.


## 4.2. Optimal Procurement Strategy

In all, the buyer can choose among three strategies. The first strategy is ordering from the reliable supplier only (strategy R), the second one is ordering from the unreliable supplier only (strategy U), and the last one is ordering from both of the suppliers, also referred to as supply diversification strategy (strategy D).


### 4.2.1. Ordering Only from Reliable Supplier

The buyer orders only from the reliable supplier with fixed wholesale pricewr. We use qr to denote the order quantity under this strategy. The profit function of the buyer is as follows:(3)ΠbRqr=p−wrminqr,d.Under the wholesale contract, it is easy to obtainqr∗=d.


### 4.2.2. Ordering Only from Unreliable Supplier

The buyer orders only from the unreliable supplier through a wholesale contractqu,wu. If the disruption occurs with the possibility of 1−β, the unreliable supplier will lose all the capacity and decide whether to exert restoration effort e to rebuild capacity. We assume that the unreliable supplier will get the restoration capacity e through the restoration effort e. From Proposition 1, we know that the restoration capacity depends on the values of qu and wu, and the profit function of the buyer is formulated as follows:(4)ΠbUqu,wu=p−wuβminqu,d+1−βmine∗,d.Taking (2) into (4), we have ΠbUqu,wu=p−wuβminqu,d+1−βminwu−cd/k,qu,d. We consider two cases: when qu<d, we have ΠbUqu,wu=p−wuβqu+1−βminwu−cd/k,qu, which increases with qu; when qu≥d, we have ΠbUqu,wu=p−wuβd+1−βminwu−cd/k,d, which is unrelated to qu. We summarize the above results in Lemma 1.Lemma 1.
If the seller orders from the unreliable supplier only, thenqu∗=d.From Lemma1, we rewrite equation (4) as follows:(5)ΠbUwu=p−wuβd+1−βminwu−cdk,d,where 0≤wu≤p.To solve the optimal wholesale price provided to the unreliable supplier, we conduct our analysis based on two casescd+kd≤p and cd+kd>p as follows.Whencd+kd≤p, we consider two subcases wu−cd/k≥d and wu−cd/k<d as follows:(1)
Ifwu−cd/k≥d, then the profit function is(6)ΠbUwu=p−wud.SinceΠbUwu decreases with wu, we obtain the local maximizer that w^u=kd+cd and ΠbUw^u=p−kd−cdd.(2)
Ifwu−cd/k<d, we have cd≤wu<kd+cd; then, the profit function is(7)ΠbUwu=−1−βwu2k+1−βp+cdk−βdwu−1−βpcdk+βpd.Since the axis of symmetry of equation (7) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Whenp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbUw^u=βdp−cd.(b)
Whencd<p+cd/2−βkd/21−β<cd+kd, we have w^u=p+cd/2−βkd/21−β and ΠbUw^u=p−cd/2+βkd/21−ββd/2+1−βp−cd/2k.(c)
Whenp+cd/2−βkd/21−β≥cd+kd, we have w^u=cd+kd and ΠbUw^u=dp−kd−cd.Thus, we conclude the equilibrium outcomes whencd+kd≤p as follows:(1)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≤0 or cd<p+cd/2−βkd/21−β<cd+kd and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kd, then w^u=cd+kd and ΠbU=p−cd−kdb.(2)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≥0, then w^u=cd and ΠbU=βbp−cd.(3)
Ifcd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β, then w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.Whencd+kb>p, we know that wu−cd/k<b,i.e.,cd<wu<p; thus, the profit function is as follows:(8)ΠbU=maxwu−1−βwu2k+1−βp+cdk−βbwu−1−βpcdk+βpb.Since the axis of symmetry of Function (8) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Ifp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbU=βbp−cd.(b)
Ifcd<p+cd/2−βkd/21−β<p, we have w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(c)
Ifp+cd/2−βkd/21−β≥p, we have w^u=p and ΠbU=0.According to the above analysis, we summarize our equilibrium results in Proposition2.Proposition 2.
If the buyer only procures from the unreliable supplier, the equilibrium decision and profit of the buyer are summarized as follows:(i)
Ifcd+kb≤p, (1) p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≤0 or (2) cd<p+cd/2−βkd/21−β<cd+kb and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kb, then wu∗=cd+kb and ΠbU=p−cd−kbb.(ii)
If (1)cd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β or (2) cd+kb>p and cd<p+cd/2−βkd/21−β<p, then wu∗=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(iii)
If (1)cd+kb≤p, p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≥0 or (2) cd+kb>p and p+cd/2−βkd/21−β≤cd, then wu∗=cd and ΠbU=βbp−cd.(iv)
Ifcd+kb>p and p+cd/2−βkd/21−β≥p, then wu∗=p and ΠbU=0.


### 4.2.3. Supply Diversification Strategy

In this part, we discuss the optimal strategy of the buyer when ordering from both suppliers. We assume thatqu+qr=b and the buyer’s profit is(9)ΠbDwu,qu=βpb−wuqu−wrqr+1−βpe∗+qr−wue∗−wrqr,where e∗=minwu−cd/k,qu and qr=b−qu. We further consider two cases qu≤wu−cd/k and qu>wu−cd/k as follows.Whenqu≤wu−cd/k, we have(10)ΠbDwu,qu=pb−quwu−b−quwr=p−wrb+wr−wuqu.SinceΠbDwu,qu is decreasing in wu, we have wu=kqu+cd. Taking it back into equation (10), we have(11)ΠbDqu=−kqu2+wr−cdqu+p−wrb.Since the axis of symmetry of the function isqu=wr−cd/2k, for 0≤qu≤b, we have the following two subcases 0≤wr−cd/2k≤b and wr−cd/2k>b:(1)
If0≤wr−cd/2k≤b, then qu^=wr−cd/2k, w^u=wr+cd/2, and ΠbDqu^,w^u=wr−cd2/4k+p−wrb.(2)
Ifwr−cd/2k>b, then qu^=b, w^u=kb+cd and ΠbDqu^,w^u=p−cdb−kb2.Whenqu>wr−cd/2k, we have(12)ΠbDwu,qu=wr−βwu−1−βpqu+p−wrb+1−βk−w2+p+cdw−pcd.We can easily obtain thatcd≤wu≤kb+cd; then we consider three subcases wr−1−βp/β<cd, cd≤wr−1−βp/β<kb+cd, and wr−1−βp/β≥kb+cd as follows:(1)
Ifwr−1−βp/β<cd, we have the coefficient of qu in (13) which is wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu; then qu^=wu−cd/k. Then, we can rewrite equation (12) as follows:(13)ΠbDwu=−wu2k+wr+cdwuk−wrcdk+p−wrb.Since the axis of symmetry of equation (13) is wu=wr+cd/2, for cd≤wu≤kb+cd, then we have the following two scenarios:(a)
Ifwr+cd/2≤kb+cd, we have w^u=wr+cd/2, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.(b)
Ifwr+cd/2>kb+cd, we have w^u=kb+cd, qu^=b and ΠbDw^u,qu^=p−cdb−kb2.(2)
Ifcd≤wr−1−βp/β<kb+cd, we have the following two scenarios:(a)
Ifcd≤wu≤wr−1−βp/β, then wr−βwu−1−βp≥0, so ΠbDwu,qu is increasing in qu, for 0≤qu≤b; we obtain qu^=b. Taking it into equation (12), we have(14)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Since the axis of symmetry of equation (14) is wu=p+cd/2−βkd/21−β, we thus conduct the following discussion:Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kw^u=p+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥wr−1−βp/β, then w^u=wr−1−βp/β, qu^=b, and ΠbDw^u,qu^=pwr+cd−1−βp2+wrcd.(b)
Ifwr−1−βp/β<wu≤kb+cd, then wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu, for wu−cd/k≤qu≤b; we obtain qu^=wu−cd/k. Taking it into (13), we haveSince the axis of symmetry of (16) is wu=wr+cd/2k, we thus conduct the following discussion:Ifwr+cd/2k<wr−1−βp/β, then w^u=wr−1−βp/β, qu^=p−cd/k−p−wr/βk, and ΠbDw^u,qu^=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb.Ifwr−1−βp/β≤wr+cd/2k<kb+cd, then w^u=wr+cd/2k, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.Ifwr+cd/2k≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.(3)
Ifwr−1−βp/β≥kb+cd, we have wr−βwu−1−βp>0, so qu^=b. Taking it into equation (12), we have(15)ΠbDwu=−1kwu2−wr+cdwu+wrcd+p−wrb.Since the axis of symmetry of equation (16) is w^u=p+cd/2−βkd/21−β, for cd≤wu≤kb+cd, we thus conduct the following discussion:(16)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<kb+cd, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.According to the above analysis, we have the following discussions to obtain our equilibrium results when the buyer adopts supply diversification strategy.Case 1.
Ifwr−1−βp/β<cd and wr+cd/2k≤kd+cd, the analysis is shown in Figure 3.
As we can see, for0≤qu≤b and cd≤wu≤kb+cd, the feasible region is a rectangle with an oblique line cut into two parts, where the solution becomes better along the direction of the arrow. So under this circumstance, we obtain the global optimal solutions wu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 3
Case1: wr−1−βp/β<cd and wr+cd/2≤kd+cd.Case 2.
Ifwr−1−βp/β<cd≤kd+cd<wr+cd/2, we have our analysis as shown in Figure 4.
It is easy to obtain the global optimal solution which iswu∗=kb+cd and qu∗=b, which is the intersection of the lines qu=b and qu=wu−cd/k, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 4
Case2: wr−1−βp/β<cd≤kd+cd<wr+cd/2.Case 3.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd, the analysis is shown in Figure 5.
We can see that the feasible region is cut into three parts with a local optimal solution in each part. We can easily find the global optimumswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 5
Case3: p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd.Case 4.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd, the analysis is shown in Figure 6.
As we can see, the feasible region is divided into three parts which is the same as in Figure5, but the difference is that there are two similar local optimums and we cannot intuitively obtain the global optimum. Thus, we need to compare the profit of the two points cd,b and wr+cd/2,wr−cd/2k, and the optimal profit is maxΠbDcd,b,ΠbDwr+cd/2,wr−cd/2k.Figure 6
Case4: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd.Case 5.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2, the analysis is shown in Figure 7.
As we can see, there are two local optimums. Sincep+cd/2−βkd/21−β<cd, we compare the profits of the two pointscd,b and kb,b and find that ΠbDcd,b≥ΠbDkb,b, so the global optimum is wu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 7
Case5: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2.Case 6.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd, the analysis is shown in Figure 8.
Similarly, there are three local optimums, but we can easily obtain that the global optimal solution iswu∗=p+cd/2−βkd/21−β and qu∗=b.Figure 8
Case6: cd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd.Case 7.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β<wr+cd/2≤kb+cd, the analysis is shown in Figure 9.
We can obtain that the local optimums arep+cd/2−βkb/21−β,b and wr+cd/2,wr−cd/2k. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDwr+cd/2,wr−cd/2k.Figure 9
Case7: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<wr+cd/2≤kb+cd.Case 8.
Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2, the analysis is shown in Figure 10.
We can obtain the two local optimumsp+cd/2−βkd/21−β,b and kb+cd,b. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDkb+cd,b.Figure 10
Case8: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2.Case 9.
Ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd, the analysis is shown in Figure 11.
We can see thatwr+cd/2,wr−cd/2k is the worst solution, so the best solution is either wr−1−βp/β,b or wr−1−βp/β,p+cd/2−βkb/21−β. The profit of the points is ΠbDwr−1−βp/β,b=pwr+cd−1−βp2+wrcd and ΠbDwr−1−βp/β,p+cd/2−βkd/21−β=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb, and the optimal profit is the bigger one.Figure 11
Case9: cd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd.Case 10.
Ifcd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β, the analysis is shown in Figure 12.
We can easily obtain the optimal solutionswu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 12
Case10: cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β.Case 11.
Ifcd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, the analysis is shown in Figure 13.
It is easy to find that the optimal solution iswu∗=b+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 13
Case11: cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd.Case 12.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 14.
As we can see, the global optimum iswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 14
Case12: p+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β.Case 13.
Ifp+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 15.
At this circumstance, the global optimum is the same as the one above.Figure 15
Case13: p+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 14.
Ifcd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 16.
We can see that the global optimum iswu∗=p+cd/2−βkd/21−β and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Figure 16
Case14: cd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β.Case 15.
Ifcd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 17.
The solution is the same as the one above.Figure 17
Case15: cd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 16.
Ifcd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 18.
As we can see, the global optimum iswu∗=kb+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 18
Case16: cd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd.Case 17.
Ifwr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 19.
The solution is the same with the one above.
According to our analysis, we come to the conclusions which are summarized in Proposition 3.Figure 19
Case17: wr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd.Proposition 3.
In all, the optimal strategy of the buyer might be divided into six different scenarios as follows:(i)
wu∗=wr+cd/2 and qu∗=wr−cd/2k if(1)wr−1−βp/β<cd and wr+cd/2≤kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ∏bDwr+cd/2,wr−cd/2k≥∏bDcd,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, ∏bDwr+cd/2,wr−cd/2k≥∏bDp+cd/2−βkb/21−β,b, or (4) cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkb/21−β≥wr−1−βp/β.(ii)
wu∗=kb+cd and qu∗=b if(1)wr−1−βp/β<cd<kb+cd<wr+cd/2 or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b≥ΠbDp+cd/2−βkd/21−β,b or (3) cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, or (4) wr−1−βp/β≥kb+cd and p+cd/2−βkd/21−β≥kb+cd.(iii)
wu∗=cd and qu∗=b if(1)p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ΠbDwr+cd/2,wr−cd/2k<ΠbDcd,b or (3) p+cd/2−βkd/21−β<cd<wr−1−βp/β<kb+cd<wr+cd/2, or (4) p+cd/2−βkd/21−β<cd<kb+cd<wr−1−βp/β.(iv)
wu∗=p+cd/2−βkd/21−β and qu∗=b if(1)cd≤p+cd/2−βkd/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, and ΠbDwr+cd/2,wr−cd/2k<ΠbDp+cd/2−βkd/21−β,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b<ΠbDp+cd/2−βkd/21−β,b, or (4) cd≤p+cd/2−βkd/21−β<kb+cd≤wr−1−βp/β.(v)
wu∗=wr−1−βp/β and qu∗=b ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b≥ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.(vi)
wu∗=wr−1−βp/β and qu∗=p−cd/k−p−wr/βk ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b<ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.Furthermore, from Proposition3, we can obtain the optimal restoration effort of the unreliable supplier as shown in Proposition 4.Proposition 4.
Given the optimal contract provided by the buyerwu∗,qu∗, the optimal restoration effort exerted by the unreliable supplier is e∗wu∗,qu∗=minwu∗−cd/k,qu∗.


## 4.2.1. Ordering Only from Reliable Supplier

The buyer orders only from the reliable supplier with fixed wholesale pricewr. We use qr to denote the order quantity under this strategy. The profit function of the buyer is as follows:(3)ΠbRqr=p−wrminqr,d.Under the wholesale contract, it is easy to obtainqr∗=d.


## 4.2.2. Ordering Only from Unreliable Supplier

The buyer orders only from the unreliable supplier through a wholesale contractqu,wu. If the disruption occurs with the possibility of 1−β, the unreliable supplier will lose all the capacity and decide whether to exert restoration effort e to rebuild capacity. We assume that the unreliable supplier will get the restoration capacity e through the restoration effort e. From Proposition 1, we know that the restoration capacity depends on the values of qu and wu, and the profit function of the buyer is formulated as follows:(4)ΠbUqu,wu=p−wuβminqu,d+1−βmine∗,d.Taking (2) into (4), we have ΠbUqu,wu=p−wuβminqu,d+1−βminwu−cd/k,qu,d. We consider two cases: when qu<d, we have ΠbUqu,wu=p−wuβqu+1−βminwu−cd/k,qu, which increases with qu; when qu≥d, we have ΠbUqu,wu=p−wuβd+1−βminwu−cd/k,d, which is unrelated to qu. We summarize the above results in Lemma 1.Lemma 1.
If the seller orders from the unreliable supplier only, thenqu∗=d.From Lemma1, we rewrite equation (4) as follows:(5)ΠbUwu=p−wuβd+1−βminwu−cdk,d,where 0≤wu≤p.To solve the optimal wholesale price provided to the unreliable supplier, we conduct our analysis based on two casescd+kd≤p and cd+kd>p as follows.Whencd+kd≤p, we consider two subcases wu−cd/k≥d and wu−cd/k<d as follows:(1)
Ifwu−cd/k≥d, then the profit function is(6)ΠbUwu=p−wud.SinceΠbUwu decreases with wu, we obtain the local maximizer that w^u=kd+cd and ΠbUw^u=p−kd−cdd.(2)
Ifwu−cd/k<d, we have cd≤wu<kd+cd; then, the profit function is(7)ΠbUwu=−1−βwu2k+1−βp+cdk−βdwu−1−βpcdk+βpd.Since the axis of symmetry of equation (7) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Whenp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbUw^u=βdp−cd.(b)
Whencd<p+cd/2−βkd/21−β<cd+kd, we have w^u=p+cd/2−βkd/21−β and ΠbUw^u=p−cd/2+βkd/21−ββd/2+1−βp−cd/2k.(c)
Whenp+cd/2−βkd/21−β≥cd+kd, we have w^u=cd+kd and ΠbUw^u=dp−kd−cd.Thus, we conclude the equilibrium outcomes whencd+kd≤p as follows:(1)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≤0 or cd<p+cd/2−βkd/21−β<cd+kd and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kd, then w^u=cd+kd and ΠbU=p−cd−kdb.(2)
Ifcd+kd≤p, p+cd/2−βkd/21−β≤cd, and kd−1−βp−cd≥0, then w^u=cd and ΠbU=βbp−cd.(3)
Ifcd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β, then w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.Whencd+kb>p, we know that wu−cd/k<b,i.e.,cd<wu<p; thus, the profit function is as follows:(8)ΠbU=maxwu−1−βwu2k+1−βp+cdk−βbwu−1−βpcdk+βpb.Since the axis of symmetry of Function (8) is p+cd/2−βkd/21−β, we have the following scenarios:(a)
Ifp+cd/2−βkd/21−β≤cd, we have the local maximizer w^u=cd and ΠbU=βbp−cd.(b)
Ifcd<p+cd/2−βkd/21−β<p, we have w^u=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(c)
Ifp+cd/2−βkd/21−β≥p, we have w^u=p and ΠbU=0.According to the above analysis, we summarize our equilibrium results in Proposition2.Proposition 2.
If the buyer only procures from the unreliable supplier, the equilibrium decision and profit of the buyer are summarized as follows:(i)
Ifcd+kb≤p, (1) p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≤0 or (2) cd<p+cd/2−βkd/21−β<cd+kb and ΠbUcd+kb>ΠbUp+cd/2−βkd/21−β or (3) p+cd/2−βkd/21−β≥cd+kb, then wu∗=cd+kb and ΠbU=p−cd−kbb.(ii)
If (1)cd+kb≤p, cd<p+cd/2−βkd/21−β<cd+kb, and ΠbUcd+kb≤ΠbUp+cd/2−βkd/21−β or (2) cd+kb>p and cd<p+cd/2−βkd/21−β<p, then wu∗=p+cd/2−βkd/21−β and ΠbU=p−cd/2+βkd/21−ββb+1−β/kp−cd/2−βkd/21−β.(iii)
If (1)cd+kb≤p, p+cd/2−βkd/21−β≤cd, and kb−1−βp−cd≥0 or (2) cd+kb>p and p+cd/2−βkd/21−β≤cd, then wu∗=cd and ΠbU=βbp−cd.(iv)
Ifcd+kb>p and p+cd/2−βkd/21−β≥p, then wu∗=p and ΠbU=0.


## 4.2.3. Supply Diversification Strategy

In this part, we discuss the optimal strategy of the buyer when ordering from both suppliers. We assume thatqu+qr=b and the buyer’s profit is(9)ΠbDwu,qu=βpb−wuqu−wrqr+1−βpe∗+qr−wue∗−wrqr,where e∗=minwu−cd/k,qu and qr=b−qu. We further consider two cases qu≤wu−cd/k and qu>wu−cd/k as follows.Whenqu≤wu−cd/k, we have(10)ΠbDwu,qu=pb−quwu−b−quwr=p−wrb+wr−wuqu.SinceΠbDwu,qu is decreasing in wu, we have wu=kqu+cd. Taking it back into equation (10), we have(11)ΠbDqu=−kqu2+wr−cdqu+p−wrb.Since the axis of symmetry of the function isqu=wr−cd/2k, for 0≤qu≤b, we have the following two subcases 0≤wr−cd/2k≤b and wr−cd/2k>b:(1)
If0≤wr−cd/2k≤b, then qu^=wr−cd/2k, w^u=wr+cd/2, and ΠbDqu^,w^u=wr−cd2/4k+p−wrb.(2)
Ifwr−cd/2k>b, then qu^=b, w^u=kb+cd and ΠbDqu^,w^u=p−cdb−kb2.Whenqu>wr−cd/2k, we have(12)ΠbDwu,qu=wr−βwu−1−βpqu+p−wrb+1−βk−w2+p+cdw−pcd.We can easily obtain thatcd≤wu≤kb+cd; then we consider three subcases wr−1−βp/β<cd, cd≤wr−1−βp/β<kb+cd, and wr−1−βp/β≥kb+cd as follows:(1)
Ifwr−1−βp/β<cd, we have the coefficient of qu in (13) which is wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu; then qu^=wu−cd/k. Then, we can rewrite equation (12) as follows:(13)ΠbDwu=−wu2k+wr+cdwuk−wrcdk+p−wrb.Since the axis of symmetry of equation (13) is wu=wr+cd/2, for cd≤wu≤kb+cd, then we have the following two scenarios:(a)
Ifwr+cd/2≤kb+cd, we have w^u=wr+cd/2, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.(b)
Ifwr+cd/2>kb+cd, we have w^u=kb+cd, qu^=b and ΠbDw^u,qu^=p−cdb−kb2.(2)
Ifcd≤wr−1−βp/β<kb+cd, we have the following two scenarios:(a)
Ifcd≤wu≤wr−1−βp/β, then wr−βwu−1−βp≥0, so ΠbDwu,qu is increasing in qu, for 0≤qu≤b; we obtain qu^=b. Taking it into equation (12), we have(14)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Since the axis of symmetry of equation (14) is wu=p+cd/2−βkd/21−β, we thus conduct the following discussion:Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kw^u=p+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥wr−1−βp/β, then w^u=wr−1−βp/β, qu^=b, and ΠbDw^u,qu^=pwr+cd−1−βp2+wrcd.(b)
Ifwr−1−βp/β<wu≤kb+cd, then wr−βwu−1−βp<0, so ΠbDwu,qu is decreasing in qu, for wu−cd/k≤qu≤b; we obtain qu^=wu−cd/k. Taking it into (13), we haveSince the axis of symmetry of (16) is wu=wr+cd/2k, we thus conduct the following discussion:Ifwr+cd/2k<wr−1−βp/β, then w^u=wr−1−βp/β, qu^=p−cd/k−p−wr/βk, and ΠbDw^u,qu^=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb.Ifwr−1−βp/β≤wr+cd/2k<kb+cd, then w^u=wr+cd/2k, qu^=wr−cd/2k, and ΠbDw^u,qu^=wr−cd2/4k+p−wrb.Ifwr+cd/2k≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.(3)
Ifwr−1−βp/β≥kb+cd, we have wr−βwu−1−βp>0, so qu^=b. Taking it into equation (12), we have(15)ΠbDwu=−1kwu2−wr+cdwu+wrcd+p−wrb.Since the axis of symmetry of equation (16) is w^u=p+cd/2−βkd/21−β, for cd≤wu≤kb+cd, we thus conduct the following discussion:(16)ΠbDwu=−1−βkwu2−p+cd−βkb1−βwu+pcd+βpb.Ifp+cd/2−βkd/21−β<cd, then w^u=cd, qu^=b, and ΠbDw^u,qu^=βp−cdb.Ifcd≤p+cd/2−βkd/21−β<kb+cd, then w^u=p+cd/2−βkd/21−β, qu^=b, and ΠbDw^u,qu^=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Ifp+cd/2−βkd/21−β≥kb+cd, then w^u=kb+cd, qu^=b, and ΠbDw^u,qu^=p−cdb−kb2.According to the above analysis, we have the following discussions to obtain our equilibrium results when the buyer adopts supply diversification strategy.Case 1.
Ifwr−1−βp/β<cd and wr+cd/2k≤kd+cd, the analysis is shown in Figure 3.
As we can see, for0≤qu≤b and cd≤wu≤kb+cd, the feasible region is a rectangle with an oblique line cut into two parts, where the solution becomes better along the direction of the arrow. So under this circumstance, we obtain the global optimal solutions wu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 3
Case1: wr−1−βp/β<cd and wr+cd/2≤kd+cd.Case 2.
Ifwr−1−βp/β<cd≤kd+cd<wr+cd/2, we have our analysis as shown in Figure 4.
It is easy to obtain the global optimal solution which iswu∗=kb+cd and qu∗=b, which is the intersection of the lines qu=b and qu=wu−cd/k, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 4
Case2: wr−1−βp/β<cd≤kd+cd<wr+cd/2.Case 3.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd, the analysis is shown in Figure 5.
We can see that the feasible region is cut into three parts with a local optimal solution in each part. We can easily find the global optimumswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 5
Case3: p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd.Case 4.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd, the analysis is shown in Figure 6.
As we can see, the feasible region is divided into three parts which is the same as in Figure5, but the difference is that there are two similar local optimums and we cannot intuitively obtain the global optimum. Thus, we need to compare the profit of the two points cd,b and wr+cd/2,wr−cd/2k, and the optimal profit is maxΠbDcd,b,ΠbDwr+cd/2,wr−cd/2k.Figure 6
Case4: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd.Case 5.
Ifp+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2, the analysis is shown in Figure 7.
As we can see, there are two local optimums. Sincep+cd/2−βkd/21−β<cd, we compare the profits of the two pointscd,b and kb,b and find that ΠbDcd,b≥ΠbDkb,b, so the global optimum is wu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 7
Case5: p+cd/2−βkd/21−β<cd≤wr−1−βp/β<kb+cd<wr+cd/2.Case 6.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd, the analysis is shown in Figure 8.
Similarly, there are three local optimums, but we can easily obtain that the global optimal solution iswu∗=p+cd/2−βkd/21−β and qu∗=b.Figure 8
Case6: cd≤p+cd/2−βkb/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd.Case 7.
Ifcd≤p+cd/2−βkb/21−β<wr−1−βp/β<wr+cd/2≤kb+cd, the analysis is shown in Figure 9.
We can obtain that the local optimums arep+cd/2−βkb/21−β,b and wr+cd/2,wr−cd/2k. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDwr+cd/2,wr−cd/2k.Figure 9
Case7: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<wr+cd/2≤kb+cd.Case 8.
Ifcd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2, the analysis is shown in Figure 10.
We can obtain the two local optimumsp+cd/2−βkd/21−β,b and kb+cd,b. The optimal profit is maxΠbDp+cd/2−βkd/21−β,b,ΠbDkb+cd,b.Figure 10
Case8: cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2.Case 9.
Ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd, the analysis is shown in Figure 11.
We can see thatwr+cd/2,wr−cd/2k is the worst solution, so the best solution is either wr−1−βp/β,b or wr−1−βp/β,p+cd/2−βkb/21−β. The profit of the points is ΠbDwr−1−βp/β,b=pwr+cd−1−βp2+wrcd and ΠbDwr−1−βp/β,p+cd/2−βkd/21−β=−1−βp−wr2/kβ2+1−βp2+wrcd−pwr−pcd/kβ+p−wrb, and the optimal profit is the bigger one.Figure 11
Case9: cd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β and wr−1−βp/β<kb+cd.Case 10.
Ifcd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β, the analysis is shown in Figure 12.
We can easily obtain the optimal solutionswu∗=wr+cd/2 and qu∗=wr−cd/2k, and the optimal profit is ΠbDwu∗,qu∗=wr−cd2/4k+p−wrb.Figure 12
Case10: cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkd/21−β≥wr−1−βp/β.Case 11.
Ifcd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, the analysis is shown in Figure 13.
It is easy to find that the optimal solution iswu∗=b+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 13
Case11: cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd.Case 12.
Ifp+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 14.
As we can see, the global optimum iswu∗=cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=βp−cdb.Figure 14
Case12: p+cd/2−βkd/21−β<cd<wr+cd/2<kb+cd<wr−1−βp/β.Case 13.
Ifp+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 15.
At this circumstance, the global optimum is the same as the one above.Figure 15
Case13: p+cd/2−βkd/21−β<cd<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 14.
Ifcd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β, the analysis is shown in Figure 16.
We can see that the global optimum iswu∗=p+cd/2−βkd/21−β and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=1−β/kp+cd/2−βkd/21−β2−1−βpcd/k+βpb.Figure 16
Case14: cd<p+cd/2−βkd/21−β<kb+cd and wr+cd/2<kb+cd<wr−1−βp/β.Case 15.
Ifcd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 17.
The solution is the same as the one above.Figure 17
Case15: cd<p+cd/2−βkd/21−β<kb+cd<wr+cd/2 and wr−1−βp/β≥kb+cd.Case 16.
Ifcd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 18.
As we can see, the global optimum iswu∗=kb+cd and qu∗=b, and the optimal profit is ΠbDwu∗,qu∗=p−cdb−kb2.Figure 18
Case16: cd<wr+cd/2<kb+cd≤p+cd/2−βkd/21−β and wr−1−βp/β≥kb+cd.Case 17.
Ifwr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd, the analysis is shown in Figure 19.
The solution is the same with the one above.
According to our analysis, we come to the conclusions which are summarized in Proposition 3.Figure 19
Case17: wr+cd/2≥kb+cd, p+cd/2−βkd/21−β≥kb+cd, and wr−1−βp/β≥kb+cd.Proposition 3.
In all, the optimal strategy of the buyer might be divided into six different scenarios as follows:(i)
wu∗=wr+cd/2 and qu∗=wr−cd/2k if(1)wr−1−βp/β<cd and wr+cd/2≤kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ∏bDwr+cd/2,wr−cd/2k≥∏bDcd,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, ∏bDwr+cd/2,wr−cd/2k≥∏bDp+cd/2−βkb/21−β,b, or (4) cd≤wr−1−βp/β≤wr+cd/2<kb+cd and p+cd/2−βkb/21−β≥wr−1−βp/β.(ii)
wu∗=kb+cd and qu∗=b if(1)wr−1−βp/β<cd<kb+cd<wr+cd/2 or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b≥ΠbDp+cd/2−βkd/21−β,b or (3) cd≤wr−1−βp/β≤p+cd/2−βkd/21−β and wr+cd/2≥kb+cd, or (4) wr−1−βp/β≥kb+cd and p+cd/2−βkd/21−β≥kb+cd.(iii)
wu∗=cd and qu∗=b if(1)p+cd/2−βkd/21−β<cd<wr+cd/2<wr−1−βp/β<kb+cd or (2) p+cd/2−βkd/21−β<cd≤wr−1−βp/β<wr+cd/2<kb+cd and ΠbDwr+cd/2,wr−cd/2k<ΠbDcd,b or (3) p+cd/2−βkd/21−β<cd<wr−1−βp/β<kb+cd<wr+cd/2, or (4) p+cd/2−βkd/21−β<cd<kb+cd<wr−1−βp/β.(iv)
wu∗=p+cd/2−βkd/21−β and qu∗=b if(1)cd≤p+cd/2−βkd/21−β<wr−1−βp/β and wr+cd/2≤wr−1−βp/β<kb+cd or (2) cd≤p+cd/2−βkd/21−β<wr−1−βp/β, wr−1−βp/β<wr+cd/2<kb+cd, and ΠbDwr+cd/2,wr−cd/2k<ΠbDp+cd/2−βkd/21−β,b or (3) cd≤p+cd/2−βkd/21−β<wr−1−βp/β<kb+cd≤wr+cd/2 and ΠbDkb+cd,b<ΠbDp+cd/2−βkd/21−β,b, or (4) cd≤p+cd/2−βkd/21−β<kb+cd≤wr−1−βp/β.(v)
wu∗=wr−1−βp/β and qu∗=b ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b≥ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.(vi)
wu∗=wr−1−βp/β and qu∗=p−cd/k−p−wr/βk ifcd<wr+cd/2<wr−1−βp/β≤p+cd/2−βkd/21−β, wr−1−βp/β<kb+cd, and ΠbDwr−1−βp/β,b<ΠbDwr−1−βp/β,p−cd/k−p−wr/βk.Furthermore, from Proposition3, we can obtain the optimal restoration effort of the unreliable supplier as shown in Proposition 4.Proposition 4.
Given the optimal contract provided by the buyerwu∗,qu∗, the optimal restoration effort exerted by the unreliable supplier is e∗wu∗,qu∗=minwu∗−cd/k,qu∗.


## 5. Numerical Study

Since it is hard to obtain the closed forms of our decision variables, we conduct the following numerical analysis to generate some managerial insights. When the buyer orders only from the reliable supplier, the case would be easy to understand. In this way, we conduct numerical analyses for the case when the buyer orders only from the unreliable supplier and the case when the buyer adopts supply diversification strategy. Among all parameters, the probability without disruptionβ and the unit production cost after disruption cd would be the two most important factors that affect the buyer’s ordering strategy. Thus, we first assign the following values to other parameters: d=1,k=1/2,p=1,wr=1/2.


### 5.1. Ordering Only from Unreliable Supplier

When ordering only from the unreliable supplier, the buyer has to offer a contractqu,wu to the unreliable supplier. Through equilibrium analysis, it is easy to obtain qu∗=d. Here, we conduct a numerical study on wu∗ to show its property based on Proposition 2. The results are shown in Figure 20.Figure 20
Optimal wholesale pricewu∗ for unreliable supplier in the case of ordering only from the unreliable supplier.
(a)(b)(c)From Figure20, we find that the wholesale price for the unreliable supplier (wu∗) increases with the unit production cost after disruption (cd) and decreases with the probability without disruption (β). The interesting observation is that, if β is high enough, wu∗ would always be equal to cd (see Figure 20(c)); otherwise, there exists a threshold of cd where wu∗ would be higher when β is relatively low below this threshold (see Figures 20(a) and 20(b)). This reflects the fact that when the disruption is easy to occur, the buyer would provide a more favorable contract with a higher wholesale price to encourage the unreliable supplier to exert effort on restoring capacity.


### 5.2. Supply Diversification Strategy

In this part, we conduct a numerical study when the buyer adopts supply diversification strategy. From equilibrium analysis, we know that the buyer would always order the quantity that equals the deterministic demand from both retailers. Thus, the order quantity from the reliable supplierqr would be equal to d−qu. Here, we concentrate on the contract qu,wu provided to the unreliable supplier. The results are shown in Figures 21 and 22.Figure 21
Optimal wholesale pricewu∗ for unreliable supplier in the case of supply diversification strategy.
(a)(b)(c)Figure 22
Optimal order quantityqu∗ for unreliable supplier in the case of supply diversification strategy.
(a)(b)(c)From Figure21, we find that the wholesale price for the unreliable supplier (wu∗) is a nondecreasing function of the unit production cost after disruption (cd), while the effect of the probability without disruption (β) on the wholesale price for the unreliable supplier (wu∗) is not consistent. Specifically, if β is relatively low, wu∗ increases with cd at a constant rate (see Figure 21(a)). If β is intermediate, there would be a sudden increase in wu∗ when cd reaches a certain threshold, but the increase rate would be lower since then (see Figure 21(b)). If β is high enough, cd has no effect on wu∗ when cd is relatively low and would have a larger effect on wu∗ when cd is intermediate than when cd is high enough (see Figure 21(c)).This shows that if the disruption is easy to happen, the buyer would trade off between the wholesale price from the reliable supplier (wr) and the unit production cost after disruption (cd) to offer a good wholesale price to the unreliable supplier, encouraging it to restore capacity when the disruption happens. However, with the increase of the unit production cost after disruption (cd), the buyer would rely more on the reliable supplier than on the unreliable one by reducing the order quantity from the unreliable supplier (qu∗) (see Figure 22(a)). If the probability without disruption is neither too low nor too high, the buyer would tend to encourage the unreliable supplier to guarantee supply and source all the demand from the unreliable one when cd is below a threshold but would only offer a wu∗ that just equals cd when cd is very high (see Figure 22(b)). If the disruption seldom happens, sourcing from the unreliable supplier would be the best choice for the buyer especially when cd is relatively low. Under this circumstance, the buyer would choose a wholesale price that jointly considered the retail price (p), the probability without disruption (β), and the wholesale price from the reliable supplier (wr). However, when cd is intermediate, the buyer would offer a contract that sources all the demand from the unreliable supplier with a wholesale price that equals the unit production cost after disruption.


## 5.1. Ordering Only from Unreliable Supplier

When ordering only from the unreliable supplier, the buyer has to offer a contractqu,wu to the unreliable supplier. Through equilibrium analysis, it is easy to obtain qu∗=d. Here, we conduct a numerical study on wu∗ to show its property based on Proposition 2. The results are shown in Figure 20.Figure 20
Optimal wholesale pricewu∗ for unreliable supplier in the case of ordering only from the unreliable supplier.
(a)(b)(c)From Figure20, we find that the wholesale price for the unreliable supplier (wu∗) increases with the unit production cost after disruption (cd) and decreases with the probability without disruption (β). The interesting observation is that, if β is high enough, wu∗ would always be equal to cd (see Figure 20(c)); otherwise, there exists a threshold of cd where wu∗ would be higher when β is relatively low below this threshold (see Figures 20(a) and 20(b)). This reflects the fact that when the disruption is easy to occur, the buyer would provide a more favorable contract with a higher wholesale price to encourage the unreliable supplier to exert effort on restoring capacity.


## 5.2. Supply Diversification Strategy

In this part, we conduct a numerical study when the buyer adopts supply diversification strategy. From equilibrium analysis, we know that the buyer would always order the quantity that equals the deterministic demand from both retailers. Thus, the order quantity from the reliable supplierqr would be equal to d−qu. Here, we concentrate on the contract qu,wu provided to the unreliable supplier. The results are shown in Figures 21 and 22.Figure 21
Optimal wholesale pricewu∗ for unreliable supplier in the case of supply diversification strategy.
(a)(b)(c)Figure 22
Optimal order quantityqu∗ for unreliable supplier in the case of supply diversification strategy.
(a)(b)(c)From Figure21, we find that the wholesale price for the unreliable supplier (wu∗) is a nondecreasing function of the unit production cost after disruption (cd), while the effect of the probability without disruption (β) on the wholesale price for the unreliable supplier (wu∗) is not consistent. Specifically, if β is relatively low, wu∗ increases with cd at a constant rate (see Figure 21(a)). If β is intermediate, there would be a sudden increase in wu∗ when cd reaches a certain threshold, but the increase rate would be lower since then (see Figure 21(b)). If β is high enough, cd has no effect on wu∗ when cd is relatively low and would have a larger effect on wu∗ when cd is intermediate than when cd is high enough (see Figure 21(c)).This shows that if the disruption is easy to happen, the buyer would trade off between the wholesale price from the reliable supplier (wr) and the unit production cost after disruption (cd) to offer a good wholesale price to the unreliable supplier, encouraging it to restore capacity when the disruption happens. However, with the increase of the unit production cost after disruption (cd), the buyer would rely more on the reliable supplier than on the unreliable one by reducing the order quantity from the unreliable supplier (qu∗) (see Figure 22(a)). If the probability without disruption is neither too low nor too high, the buyer would tend to encourage the unreliable supplier to guarantee supply and source all the demand from the unreliable one when cd is below a threshold but would only offer a wu∗ that just equals cd when cd is very high (see Figure 22(b)). If the disruption seldom happens, sourcing from the unreliable supplier would be the best choice for the buyer especially when cd is relatively low. Under this circumstance, the buyer would choose a wholesale price that jointly considered the retail price (p), the probability without disruption (β), and the wholesale price from the reliable supplier (wr). However, when cd is intermediate, the buyer would offer a contract that sources all the demand from the unreliable supplier with a wholesale price that equals the unit production cost after disruption.


## 6. Conclusion

Various human or environmental factors such as natural disasters and political disputes can cause production disruption, which negatively affects the buyer’s procurement process and thus causes financial loss. This paper studies a noncooperative model that consists of two suppliers and one buyer. Since the buyer is a large-scale company with widespread reputation like Walmart and Apple, it chooses between the two types of suppliers: one is cheap but unreliable and the other is expensive but reliable and it offers contracts to them. Jointly considering the contract provided by the buyer and the cost of restoring capacity, the unreliable supplier may make efforts to rebuilt its capacity when a disruption occurs. The optimal procurement strategy is supply diversification strategy that is ordering from both suppliers with different proportions. This paper provides the contract offered by the buyer with optimal wholesale price and order quantity to the unreliable supplier in different scenarios. The main contributions to the literature are twofold. On the one hand, the proposed model characterizes the real situation of big retailers who have the power towards pricing when wholesale products from small suppliers and the contract offered by the buyer can affect the unreliable supplier’s optimal strategy on its restoration effort when a disruption occurs. On the other hand, the results generate the optimal contract containing the wholesale price and order quantity provided by the buyer to the unreliable supplier under different circumstances, which can be used by managers to design the procurement contract facing different types of suppliers and different environment. There are some possible ways to extend this research. First, the proposed model assumes that the market demand is deterministic, which might be extended to stochastic demand. Second, the restoration cost of the unreliable supplier is assumed to be deterministic, which is proportional to the restoration effort, and it is worthwhile to define the cost as other distribution functions of the restoration effort. At last, by assuming that the wholesale price for the reliable supplier is exogenously given and the total order quantity equals the demand, the proposed model mainly designs the contract for the unreliable supplier; it might be interesting to codetermine the contracts offered to both suppliers by relaxing the above assumptions.



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*Source: 8686371-2020-08-13.xml*