Mitigating Supply Disruption: The Interplay between Responsive Pricing and Information Sharing under Dual Sourcing

Abstract

Facing supply disruptions that often occur in business, firms can increase redundancy through supplier diversification and manage demand-side problems through responsive pricing and demand information sharing. We consider a Stackelberg–Nash game consisting of two competing suppliers with heterogeneity in terms of reliability and production cost and a manufacturer, and study the manufacturer’s strategy choice problem. The manufacturer has two strategies, namely responsive pricing and information sharing. The interplay between responsive pricing and information sharing is analyzed by defining the value of responsive pricing. The results of the study show that responsive pricing always benefits the manufacturer. Responsive pricing increases the manufacturer’s incentive to use dual sourcing, whereas the manufacturer with committed pricing will only choose single sourcing. Under responsive pricing, the demand potential and the unreliable supplier’s disruption probability similarly affect the manufacturer’s sourcing decision. As the demand potential or the unreliable supplier’s disruption probability increases, the manufacturer will tend to prefer dual sourcing. When the reliable supplier’s production cost is moderate, the manufacturer with responsive pricing will choose to share demand information, while the opposite is true for the manufacturer with committed pricing. At this point, responsive pricing and information sharing will complement each other.

1. Introduction

Sustainable development is a common global goal in the 21st century [1], and it inherently includes sustainable supply process management and supply chain network management [2,3]. In recent years, various internal and external emergencies have occurred frequently, such as geopolitical threats, natural disasters, equipment failures, etc. [4,5,6]. These emergencies have resulted in significant losses of human lives and financial losses while also disrupting the normal operation of the supply chain. For instance, Ericsson lost 400 million euros in 2000 due to a fire in a supplier’s semiconductor factory, and Apple lost many customer orders due to a shortage of a DRAM chip supply after the 1999 Taiwan earthquake [7]. The catastrophic supply chain disruptions caused by the 2011 earthquake in Japan led to widespread component shortages in the global automotive, semiconductor, and electronics industries [8]. These examples illustrate that supply disruptions pose a major threat to supply chain operations and business efficiency, and they have become a non-negligible problem in the field of supply chain management.

To mitigate supply disruption risk, firms can implement several effective strategies, such as multi-sourcing, responsive pricing, and demand information sharing.

Procurement, as a core component of a firm’s production and operation process, directly affects the efficiency of its overall production and operation [9]. Sourcing from multiple suppliers to increase redundancy is a strategy that is widely used in practice to mitigate supply disruption risk. Firms that rely solely on a single supplier could suffer greatly in the event of a disruption in the supply, but those that have a backup supplier can better prepare for unpredictable supply disruptions [10]. In order to mitigate potential supply disruptions, firms may consider spreading their orders among competing suppliers [11]. In addition to the direct benefit of spreading risks, an indirect benefit of multi-sourcing is inducing multiple suppliers to compete in order to lower wholesale prices [12]. Wu and Choi’s empirical study of firms, such as Coach, suggests that two key motivations for buyers to seek alternative suppliers are reducing supply disruption risk and inducing supplier competition [13].

Aside from managing the supply side, firms can also manage the demand side to reduce supply disruption risk through two effective strategies: responsive pricing and demand information sharing. After resolving supply uncertainty, responsive pricing aligns supply and demand by adjusting product prices to regulate market demand. Responsive pricing is widely used in several industries. In agriculture, farmers can set food prices after resolving production uncertainty, thus optimizing the use of the available supply. Consequently, many farmers prefer to sell their produce in the spot market rather than entering into forward contracts with fixed prices [14]. Similarly, Dell addressed a shortage of memory cards by adjusting prices to shift customer demand to low-memory PCs [15]. Furthermore, the role of information in the decision-making process has received extensive academic attention [16]. With the development of information technology, the sharing of demand information between upstream and downstream has become a crucial method for enhancing supply chain transparency. Usually, suppliers do not have sufficient knowledge of market demand information because they are far away from the consumer market. Downstream firms, which are closer to the consumer market, can help suppliers make better operational decisions by sharing market demand information, thereby facilitating supply chain coordination.

Implementing a single strategy to mitigate supply disruption risk is often insufficient. Instead, firms typically need to adopt multiple strategies simultaneously in order to effectively manage supply chain risks. This raises several important questions: (1) What is the optimal sourcing strategy when facing multiple heterogeneous upstream suppliers? (2) How do the choices of pricing and information-sharing strategies influence the sourcing strategy? (3) How do the three strategies interplay to reduce supply disruption risk? Our analysis addresses these questions and identifies the corresponding conditions.

To solve the above problems, we consider a scenario in which a manufacturer purchases from a low-cost unreliable supplier and a high-cost reliable supplier, and there is a horizontal Nash game between the two suppliers. In general, more reliable suppliers have more expensive costs or prices, which is common in model settings and practice [15,17,18]. The manufacturer has two pricing strategies, i.e., responsive pricing and committed pricing, and they decide whether or not to share market demand information with the two suppliers. This creates four strategic combination scenarios. We obtain the equilibrium decisions and expected profits of supply chain members in each scenario. Based on the equilibrium decisions and expected profits, we determine the optimal pricing and information-sharing strategies for the manufacturer under different conditions and examine the interplay between these strategies.

We describe a real case of Dell to understand the real-world context of our study, which exhibits all of the key modeling features. Both Intel and Advanced Micro Devices (AMDs) serve as important chip suppliers to Dell. In low-end chips, for example, Intel sells at a higher price and has a larger market share (meaning more capacity) compared to AMDs [19]. Dell used a responsive pricing strategy to respond to supply disruptions by modulating product prices to shift market demand [15]. At the same time, Dell engages in full information sharing with suppliers. Suppliers can view the sales of their own products in Dell’s IT system to better manage inventory [20].

Although there is considerable research on the problem of responsive pricing and information sharing under supply diversification, to the best of our knowledge, no one has studied the interplay between responsive pricing and information sharing. We address this problem in the context of dual sourcing, as shown by the example of Dell and other enterprises, considering this interplay is quite necessary and fills a gap in the relevant literature. Our conclusion shows that responsive pricing and information sharing can complement each other under certain conditions, which can guide firms to effectively adopt multiple strategies to manage supply chain risk.

The remainder of this article is organized as follows. Section 2 reviews the related literature. Section 3 describes the model, including the notation and assumptions used in this study. Section 4 presents the specific model and equilibrium decisions. Section 5 examines the manufacturer’s optimal strategic choice and numerically analyzes the interplay between responsive pricing and information-sharing strategies. Finally, Section 6 summarizes the main conclusions of the analysis and suggests directions for future research. For convenience, the proof of lemmas, theorems and propositions are provided in the Supplementary Materials.

2. Literature Review

Snyder et al. [21] argued that under output stochastic risk, the quantity produced by a supplier’s delivery or production process is a random variable that depends on the number of orders, and supply disruption can be regarded as a special case of output stochastic risk fixing, i.e., the output is a Bernoulli random variable. To effectively manage supply disruption risk, Shekarian and Parast [22] demonstrated that supply chain flexibility is the most effective strategy for mitigating supply disruption risk, followed by collaboration, redundancy, and agility. Chen et al. [23] found that supplier cooperation can effectively manage supply disruption risk, with information being central to demand risk reduction. In a collaborative relationship, customers have a greater incentive to share reliable demand information with manufacturers to ensure that manufacturers’ forecasts align with customer orders. Additionally, Golmohammadia and Hassini [24] suggest that to enhance decision-making efficiency, firms should devise strategies for addressing both supply side and demand-side risks. Their review emphasizes two approaches—diversified supply and responsive pricing—to managing these risks. Consequently, our literature review focuses on three areas: multi-sourcing, responsive pricing, and information sharing considering supply disruptions.

First, we review the literature on multi-sourcing. Babich et al. [12] study a problem of a retailer purchasing from multiple suppliers with default correlations. In their model, the market demand is exogenous, but our study sets the price sensitive demand. Dada [11] et al. considered a newsvendor’s procurement problem served by multiple fully reliable or unreliable suppliers and found that cost usually outweighs reliability when choosing a supplier and that supplier reliability affects the order size. Our study leads to a similar conclusion as Dada et al. Hu et al. [17] found the optimal multi-sourcing strategy for a retailer when some but not all of the suppliers are exposed to the risk of complete supply disruptions. Goldschmidt et al. [25] studied supplier diversification through a behavioral lens, and their study’s conclusions suggested that firms should design mechanisms to eliminate supplier diversification bias in order to enhance supplier diversification strategies. Yan et al. [26] built a supply chain model consisting of two unreliable suppliers and one retailer, and illustrated the effect of disruption time, disruption probability and fill rate on the optimal decisions and expected profit. Shan et al. [27] studied a problem of the retailer using sales effort to cope with the suppliers’ disruption risk under dual sourcing. While market demand is regulated through sales effort in their study, our study regulates it through responsive pricing. Lv and Yin [28] consider a problem of two competing firms sourcing from a reliable supplier and an unreliable supplier. The result shows that when consumers are socially conscious, sourcing diversification may be detrimental to firms. Pan et al. [29] consider an assembly system for a final product composed of multiple components and analyze the benefits of backup suppliers in reducing the adverse effects of supply uncertainty when purchasing parts from the primary suppliers.

The next stream of the literature review concerns responsive pricing. Tang and Yin [30] showed that retailers always earn higher expected profits with responsive pricing strategies. Feng et al. [31] studied the profit growth obtained by using dynamic pricing relative to static pricing. The results of the study showed that significant gains can be obtained from dynamic pricing for both the supply constraint effect and the supply uncertainty effect. Li et al. [32] studied the sourcing and pricing decisions of relevant suppliers and firms with price-dependent demand and, finally, gave an algorithm for solving the optimal diversified sourcing problem of a firm. Li et al. [33] considered the purchasing problem of a firm sourcing from a reliable supplier and an unreliable supplier in two price-setting scenarios. By comparing the firm’s optimal diversification decisions in the two pricing scenarios, the interplay between the supply diversification strategy and the responsive pricing strategy in mitigating supply uncertainty was examined. When the lost revenue effect dominates the lost goodwill effect, the two strategies are complementary; otherwise, they are substitutes. Geng et al. [34] investigated the interplay between responsive pricing strategies and delayed payment strategies for retailers, and by setting up a Stackelberg game and solving the equilibrium for each of the four scenarios in which the retailer used different combinations of strategies, they determined that delayed payment and responsive pricing are strategically complementary conditions for retailers. The research in this study is partly inspired by Shan et al. [19], who considered a problem in which a retailer with responsive pricing orders from competing strategic suppliers set wholesale prices in a Nash game. The results showed that an increase in supplier reliability may be detrimental to the supplier’s interests, as suppliers compete with each other to sell products to retailers with responsive pricing.

The final stream of the literature is on supply chain information sharing when considering supply disruption risk. The literature review by Dominguez et al. [35] showed that information transparency is an effective means to improve supply chain resilience. Yang et al. carved out a model in which a retailer is faced with holding private information about supply disruptions, and they investigated how the retailer’s risk management strategy varies and whether the corresponding strategy is more valuable in the presence of such asymmetric information [36]. Immediately after, Yang et al. continued to apply mechanism design theory to determine how buyers optimally deploy their dual-sourcing options in the presence of asymmetric information about supply risk [37]. Wu et al. studied the multi-sourcing and vertical information-sharing problem in a supply chain, where the supply chain consists of two retailers competing in a Cournot competition and Bertrand competition with N suppliers, and upstream and downstream members of the supply chain form a Stackelberg game where supply is disrupted with a certain probability [38]. Yoon et al. explored the three-tier supply chain problem with disruption risk in which the retailer devises mechanisms to induce the primary supplier to share risky information from the secondary supplier [39]. Li et al. found that the retailer may have incentives to share information with suppliers, and such information sharing can improve supply reliability [40].

In summary, multi-sourcing has received extensive attention in the literature as an important measure for mitigating supply disruption risk on the supply side. Much of the literature considers responsive pricing or information sharing for downstream firms when faced with multiple suppliers, thus jointly considering the two-sided supply disruption risk management problem on both the supply and demand sides. However, the interplay of responsive pricing and information sharing in mitigating supply disruption risk, which are important tools for firms’ supply chain management in practice, has been neglected in the existing literature. We contribute to the research in these areas by considering the problem of selecting a pricing strategy and information-sharing strategy for a manufacturer when faced with two suppliers.

3. Model Descriptions, Notations, and Assumptions

We consider a supply chain consisting of two suppliers and a manufacturer, where both suppliers are leaders and the manufacturer is a follower in a Stackelberg game. Supplier 1 is an unreliable supplier with lower production cost and supply reliability $\theta$, where $0<\theta <1$. Without the loss of generality, we focus on the impact of Supplier 2’s production cost by assuming that Supplier 1’s production cost is 0. This assumption is frequently used in the literature [41,42]. Supplier 2 is a reliable supplier, i.e., is able to deliver all orders placed by the manufacturer, but has a more expensive production cost $c_r>0$. The two suppliers produce homogeneous products, and set their wholesale prices $w_{1i}$ and $w_{2i}$ simultaneously in a Nash game. The manufacturer orders quantities $Q_{1i}$ and $Q_{2i}$ from Supplier 1 and Supplier 2, respectively, based on the wholesale prices. Due to the disruption risk of Supplier 1, the actual delivered quantities are $S\left(Q_{1i},Q_{2i}\right)$, where $S\left(Q_{1i},Q_{2i}\right)=S_1\left(Q_{1i}\right)+Q_{2i}$.

The market demand is determined and price-sensitive. The demand function is of the form $D\left(p\right)=a_i-b{p}_i$, where $a_i>0$ represents the market demand potential and $p_i$ represents the selling price. The market demand potential $a_i$ is a random variable that is of the high type ($a_i=a_h$) with a probability of 1/2 or of the low type ($a_i=a_l$) with a probability of 1/2, and $a_h>a_l$. The manufacturer can observe the actual demand information $a_i$, while the uninformed supplier has only the prior distribution of the demand information [40,43,44]. To avoid trivial situations, we assume that demand fluctuation $a_h/a_l$ will not be too large, i.e., $1<a_h/a_l<2/(2-\theta )$, and $a_l>b{c}_r$. At the end of the selling season, the remaining product has no salvage value.

The sequence of events in the responsive pricing strategy is as follows:

In stage 1, the manufacturer chooses the pricing strategy.

In stage 2, the manufacturer needs to decide whether to share the demand infor-mation with suppliers before observing it, and acts on the decision after observing the demand information.

In stage 3, Supplier 1 and Supplier 2 set their wholesale prices simultaneously in a Nash game.

In stage 4, the manufacturer decides the order quantities from two suppliers.

In stage 5, after resolving supply uncertainty, based on the actual delivered quantities, the manufacturer decides the unit selling price of the product.

The sequence of events in the committed pricing strategy is consistent with the responsive pricing strategy in stage 1–3, but in stage 4, the manufacturer needs to decide order quantities and the unit selling price simultaneously. The sequence of events is shown in Figure 1.

It is assumed that the suppliers’ production cost and reliability are common knowledge; see Shan et al. for a detailed discussion [19]. We call the case with the use of a responsive pricing strategy R and that with the use of a committed pricing strategy C. This is case S if the manufacturer shares demand information, and case N otherwise, resulting in four strategy combinations: RS, RN, CS, and CN.

Table 1. The main notations used in the model.

Decision Variables
$Q_{1i}$	$\mathrm{Quantity} \mathrm{purchased} \mathrm{from} \mathrm{Supplier} 1,i\in \left\{h,l\right\}$
$Q_{2i}$	$\mathrm{Quantity} \mathrm{purchased} \mathrm{from} \mathrm{Supplier} 2,i\in \left\{h,l\right\}$
$p_i$	$\mathrm{Sales} \mathrm{price},i\in \left\{h,l\right\}$
$w_{1i}$	$\mathrm{Wholesale} \mathrm{prices} \mathrm{from} \mathrm{Supplier} 1,i\in \left\{h,l\right\}$
$w_{2i}$	$\mathrm{Wholesale} \mathrm{prices} \mathrm{from} \mathrm{Supplier} 2,i\in \left\{h,l\right\}$
Parameters
$a_h$	High-type demand potential
$a_l$	Low-type demand potential
$b$	Price sensitivity factor
$c_r$	Unit production costs for Supplier 2
$\theta$	Reliability of Supplier 1
$S\left(Q_{1i},Q_{2i}\right)$	Total actual deliveries by suppliers

explains the main notations used in the model.

4. Equilibrium Analysis

4.1. Equilibrium Decision Making with the Responsive Pricing Strategy

This section is solved using backward induction when the manufacturer uses a responsive pricing strategy. Therefore, we first solve for the manufacturer’s optimal selling price $p_i$, then solve for the manufacturer’s optimal purchase quantities $Q_{1i}$ and $Q_{2i}$, and finally solve for the equilibrium wholesale prices $w_{1i}$ and $w_{2i}$ of the two suppliers.

4.1.1. Optimal Selling Price Decision

With the responsive pricing strategy, the manufacturer’s dual-sourcing problem can be formulated as follows:

$$\underset{Q_{1i},Q_{2i}⩾0}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_M=E\left[\pi \left(Q_{1i},Q_{2i}\right)-w_{1i}{S}_1\left(Q_{1i}\right)-w_{2i}{Q}_{2i}\right]\right\}$$

(1)

where $\pi \left(Q_{1i},Q_{2i}\right)=\underset{p_i\ge 0}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{p_i\mathrm{m}\mathrm{i}\mathrm{n}\left\{D\left(p_i\right),S\left(Q_{1i},Q_{2i}\right)\right\}\right\}$. The manufacturer needs to address the problem of $\pi \left(Q_{1i},Q_{2i}\right)$ in Stage 5. In the following, $D\left(p_i\right)$, $S\left(Q_{1i},Q_{2i}\right)$, and $S_1\left(Q_{1i}\right)$ will be abbreviated as $D$, $S$, and $S_1$.

Lemma 1.

The optimal selling price with the responsive pricing strategy is

$${\tilde{p}}_i=\left\{\begin{array}{cc}\frac{a_i}{2b},\hfill & S>A_i,\\ \frac{a_i-S}{b}0⩽& S⩽A_i.\end{array}\right.$$

where $A_i=a_i/2$.

We define $A_i$ as the adequate supply quantity; Lemma 1 is consistent with the findings of Li et al. and Shan et al. [19,32]. This suggests that when the actual delivery quantity is smaller than the adequate supply quantity, the manufacturer sets the selling price so that the demand equals the actual delivery quantity in order to avoid shortages. The smaller the actual delivery quantity, the larger the selling price set by the manufacturer to reduce the market demand to match the supply and demand. The optimal selling price is a fixed value when the actual delivery quantity $S$ is greater than the sufficient supply quantity $A_i$. In addition, the setting of the selling price is also related to the demand potential $a_i$, where a higher demand potential ($a_i=a_h$) motivates the manufacturer to set a higher selling price and vice versa.

4.1.2. Optimal Sourcing Quantity Decisions

Based on the optimal selling price obtained in the previous subsection, the manufacturer’s problem in Stage 4 can be formulated as follows:

$$\underset{Q_{1i},Q_{2i}⩾0}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_M=E\left[{\tilde{p}}_i\mathrm{m}\mathrm{i}\mathrm{n}\left\{D,S\right\}-w_{1i}{S}_1-w_{2i}{Q}_{2i}\right]\right\}$$

(2)

Lemma 2.

The optimal purchase quantity with the responsive pricing strategy is

$$\left\{\begin{array}{cc}{\tilde{Q}}_{1i}=\frac{b{w}_{2i}-b{w}_{1i}}{2(1-\theta )},{\tilde{Q}}_{2i}=\frac{a_i-\theta a_i+b\theta w_{1i}-b{w}_{2i}}{2(1-\theta )},& w_{1i}<w_{2i}<\frac{a_i-\theta a_i+b\theta w_{1i}}{b},\\ {\tilde{Q}}_{1i}=0,{\tilde{Q}}_{2i}=\frac{a_i-b{w}_{2i}}{2},\hfill & \hfill w_{2i}\le w_{1i},\\ {\tilde{Q}}_{1i}=\frac{a_i-b{w}_{1i}}{2},{\tilde{Q}}_{2i}=0,\hfill & \hfill w_{2i}\ge \frac{a_i-\theta a_i+b\theta w_{1i}}{b}.\end{array}\right.$$

Lemma 2 states that there are three scenarios when a manufacturer using responsive pricing is faced with two suppliers. When Supplier 1’s wholesale price $w_{1i}$ is greater than Supplier 2’s wholesale price $w_{2i}$, Supplier 1 will not receive the order, and there is no reason for the manufacturer to choose to purchase from the less reliable supplier at the higher price. Similarly, when $w_{2i}$ is greater than $(a_i-\theta a_i+b\theta w_{1i})/b$, the benefits of reliability are canceled out by the excessively high wholesale price, and the manufacturer will not spend so much to purchase from Supplier 2. Only when $w_{2i}$ is moderate does the manufacturer resort to dual sourcing, sending orders to both suppliers at the same time. Thus, the purchasing structure of the manufacturer is determined by the wholesale prices of the two suppliers.

4.1.3. Equilibrium Wholesale Price Decisions

If the manufacturer shares information, the supplier problem can be formulated as follows:

$$\underset{w_{1i}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_S\left(w_{1i}|w_{2i}\right)=w_{1i}\theta {\tilde{Q}}_{1i}\left(w_{1i},w_{2i}\right)\right\}$$

(3)

The problem of Supplier 2 can be formulated as follows:

$$\underset{w_{2i}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_S\left(w_{2i}|w_{1i}\right)=\left(w_{2i}-c_r\right){\tilde{Q}}_{2i}\left(w_{1i},w_{2i}\right)\right\}$$

(4)

Theorem 1.

With the responsive pricing strategy, there exists a unique Nash equilibrium of the wholesale price $\left({\tilde{w}}_{1i}^{RS},{\tilde{w}}_{2i}^{RS}\right)$ when the manufacturer shares demand information.

(1) 

When $0<c_r\le {\overline{C}}_{1i}^{RS}$, ${\tilde{w}}_{1i}^{RS}=\frac{a_i-\theta a_i+b{c}_r}{b(4-\theta )}$, ${\tilde{w}}_{2i}^{RS}=\frac{2(a_i-\theta a_i+b{c}_r)}{b(4-\theta )}$;

(2) 

When ${\overline{C}}_{1i}^{RS}<c_r\le {\widehat{C}}_{1i}^{RS}$, ${\tilde{w}}_{1i}^{RS}=\frac{b{c}_r-a_i+\theta a_i}{b\theta }$, ${\tilde{w}}_{2i}^{RS}=c_r$;

(3) 

When $c_r>{\widehat{C}}_{1i}^{RS}$, ${\tilde{w}}_{1i}^{RS}=\frac{a_i}{2b}$, ${\tilde{w}}_{2i}^{RS}=c_r$.

where ${\overline{C}}_{1i}^{RS}=\frac{2a_i(1-\theta )}{b(2-\theta )}$, ${\widehat{C}}_{1i}^{RS}=\frac{(2-\theta )a_i}{2b}$.

Theorem 1 shows that the manufacturer will always send orders to unreliable Supplier 1, and there is no single-sourcing situation where it only sources from reliable Supplier 2. With the equilibrium decision, the wholesale price of Supplier 2 is always greater than the wholesale price of Supplier 1, and the manufacturer will purchase the product from the cheaper supplier even if it has to bear some disruption risk. When the equilibrium supply structure has a single sourcing (Theorem 1 (2) and (3)), Supplier 2 sets its wholesale price equal to the production cost in order to avoid negative profits. The result of Theorem 1 confirms the conclusion of Dada et al. [11] that “cost is prioritized over reliability in supplier selection”. Figure 2 illustrates the result of Theorem 1.

The manufacturer’s purchasing decisions are influenced by the production costs of Supplier 2. Drawing on the analytical process of Shan et al. [19], by treating unreliable Supplier 1 as a market incumbent and reliable Supplier 2 as a potential market entrant, we refer to ${\widehat{C}}_{1i}^{RS}$ as the threat threshold of Supplier 1 and ${\overline{C}}_{1i}^{RS}$ as Supplier 1’s entry barrier for Supplier 2. The relationship between $c_r$ and the threat threshold and entry barriers determines the equilibrium.

When $c_r$ is greater than the threat threshold of Supplier 1, the manufacturer’s equilibrium decision is to purchase only from Supplier 1, and the existence of Supplier 2 does not threaten Supplier 1, which acts as a monopolist and charges the manufacturer a monopoly price of $a_i/2b$ to maximize its profits. The threat threshold can be viewed as the expectation of the monopoly prices $a_i/2b$ and $a_i/b$, which represent the maximum marginal costs that the manufacturer is willing to pay in the two scenarios of no disruptions and disruptions, respectively, for Supplier 1. That is, the manufacturer pays the monopoly price of $a_i/2b$ to Supplier 1 in order to purchase the product if Supplier 1 does not experience disruptions, and the manufacturer is willing to pay the wholesale price per unit $a_i/b$ in order to earn positive profits if Supplier 1 experiences disruptions. This suggests that a potential market entrant will only pose a threat to the market incumbent if Supplier 2’s production costs are lower than the manufacturer’s expected willingness to pay per unit of supply.

When $c_r$ is below the threat threshold but above Supplier 1’s barrier to entry for Supplier 2, even though the manufacturer’s equilibrium supply structure remains single- sourcing, the presence of Supplier 2 threatens Supplier 1’s profitability, and Supplier 1 chooses to lower its equilibrium wholesale price in response to the threat of a potential market entrant, preventing the manufacturer from sourcing from Supplier 2. As $c_r$ decreases further below the barrier to entry, Supplier 1 allows Supplier 2 to enter the market and charge a higher equilibrium wholesale price, and dual sourcing becomes the equilibrium.

Information sharing gives suppliers the ability to adjust their decision-making levels based on actual demand information. When the production cost $c_r$ of reliable Supplier 2 is small or large, Supplier 1 sets higher wholesale prices at a high demand potential and lower wholesale prices at a low demand potential. While the relationship between ${\tilde{w}}_{1i}^{RS}$ and $c_r$ shows an opposite trend when $c_r$ is moderate, Corollary 1 provides ideas to explain this phenomenon.

Corollary 1.

With the RS strategy, $\frac{\partial {\overline{C}}_{1i}^{RS}}{\partial \theta }<0$, $\frac{\partial {\widehat{C}}_{1i}^{RS}}{\partial \theta }<0$, $\frac{\partial {\overline{C}}_{1i}^{RS}}{\partial a_i}>0$ and $\frac{\partial {\widehat{C}}_{1i}^{RS}}{\partial a_i}>0$ are satisfied.

The magnitude of the barriers to entry and threat thresholds represent, to some extent, the manufacturer’s incentives for dual sourcing, and Corollary 1 describes the tendency of the two thresholds to vary with Supplier 1’s reliability $\theta$ and market demand potential $a_i$. As the barriers to entry increase, the manufacturer’s willingness for dual sourcing increases as he chooses to dual source in a wider range. As the threat threshold increases, the probability that Supplier 1 is threatened increases, and Supplier 2 enters the market and competes with greater incentives.

Increased (decreased) reliability of Supplier 1 reduces (rises) the incentive for the manufacturer to engage in dual sourcing. Intuitively, as $\theta$ increases (decreases), Supplier 1 is less (more) prone to disruptions, the benefits of sourcing from the reliable supplier using more expensive prices are decreased (increased), and the need for dual sourcing is decreased (increased). This finding demonstrates the benefit of addressing supply disruption risk through supplier diversification. The empirical results of Wang et al. show that a diversified supplier can build supply chain resilience under the COVID-19 shock [45]. In contrast to the effect of $\theta$, an increase in market demand potential can increase a manufacturer’s incentive to engage in dual sourcing. As the market demand potential increases, any possible disruptions result in larger sales losses, making it necessary to purchase from Supplier 2 to ensure a reliable supply. In addition, an increase in $a_i$ incentivizes Supplier 2 to enter the market with greater willingness, and Supplier 1 must take more aggressive defensive measures to block Supplier 2’s actions in order to maintain a monopoly position in the market, which explains the decrease in ${\tilde{w}}_{1i}^{RS}$ with the increase in $a_i$ in Theorem 1 (2). Next, we solve for the equilibrium decisions of the supply chain members with the RN strategy.

If the manufacturer does not share information, Supplier 1’s problem can be formulated as follows:

$$\underset{w_1}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_S\left(w_1|w_2\right)=w_1\theta [\frac{1}{2}{\tilde{Q}}_{1h}\left(w_{1,}{w}_2\right)+\frac{1}{2}{\tilde{Q}}_{1l}\left(w_{1,}{w}_2\right)]\right\}$$

(5)

Supplier 2’s problem can be formulated as follows:

$$\underset{w_2}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_S\left(w_2|w_1\right)=\left(w_2-c_r\right)[\frac{1}{2}{\tilde{Q}}_{2h}\left(w_{1,}{w}_2\right)+\frac{1}{2}{\tilde{Q}}_{2l}\left(w_{1,}{w}_2\right)]\right\}$$

(6)

Theorem 2.

With the responsive pricing strategy, there exists a unique Nash equilibrium of the wholesale price $\left({\tilde{w}}_1^{RN},{\tilde{w}}_2^{RN}\right)$when the manufacturer does not share demand information.

(1) 

When $0<c_r\le {\overline{C}}_1^{RN}$, ${\tilde{w}}_1^{RN}=\frac{(1-\theta )(a_h+a_l)+2b{c}_r}{2b(4-\theta )}$, ${\tilde{w}}_2^{RN}=\frac{(1-\theta )(a_h+a_l)+2b{c}_r}{b(4-\theta )}$;

(2) 

When ${\overline{C}}_1^{RN}<c_r\le {\widehat{C}}_1^{RN}$, ${\tilde{w}}_1^{RN}=\frac{(\theta -1)\left(a_h+a_l\right)+2b{c}_r}{2b\theta }$, ${\tilde{w}}_2^{RN}=c_r$;

(3) 

When $c_r>{\widehat{C}}_1^{RN}$, ${\tilde{w}}_1^{RN}=\frac{a_h+a_l}{4b}$, ${\tilde{w}}_2^{RN}=c_r$;

where ${\overline{C}}_1^{RN}=\frac{\left(a_h+a_l\right)(1-\theta )}{b(2-\theta )}$, ${\widehat{C}}_1^{RN}=\frac{(2-\theta )\left(a_h+a_l\right)}{4b}$.

Theorem 2 shows that, consistently with the case of the RS strategy, there exists a unique Nash equilibrium of the wholesale price for suppliers in the RN strategy, and the equilibrium wholesale prices have a similar structure. Since suppliers only have the prior distributions of the market demand potential, their equilibrium wholesale prices, entry barriers ${\overline{C}}_1^{RN}$, and threat thresholds ${\widehat{C}}_1^{RN}$ are equal to the expectations of the equilibrium wholesale prices, entry barriers ${\overline{C}}_{1i}^{RS}$, and threat thresholds ${\widehat{C}}_{1i}^{RS}$ in the RS strategy. With the RS strategy, suppliers and the manufacturer have symmetric information, and the equilibrium wholesale price and the equilibrium purchasing decision are set in the same way. For example, when $c_r>{\overline{C}}_{1i}^{RS}$, the manufacturer always chooses to purchase from Supplier 1 regardless of the type of demand potential.

However, with the RN strategy, the equilibrium purchasing decision may not match the setting of the equilibrium wholesale price due to the asymmetric information between the two parties, and the manufacturer may still purchase from Supplier 2 despite the setting of the monopoly wholesale price by Supplier 1. Corollary 2 gives the equilibrium purchasing quantity of the manufacturer.

Corollary 2.

With the RN strategy, given the supplier Nash equilibrium wholesale price, the manufacturer’s equilibrium purchasing decision is the following:

(1) 

When $0<c_r<\frac{(1-\theta )\left[(6-\theta )a_l-(2-\theta )a_h\right]}{2b(2-\theta )}$, ${\tilde{Q}}_{1h}^{RN}={\tilde{Q}}_{1l}^{RN}=\frac{(1-\theta )(a_h+a_l)+2b{c}_r}{4(1-\theta )(4-\theta )}$, ${\tilde{Q}}_{2h}^{RN}=\frac{(6-\theta )a_h-(2-\theta )a_l}{4(4-\theta )}-\frac{b(2-\theta )c_r}{2(4-\theta )(1-\theta )}$, ${\tilde{Q}}_{2l}^{RN}=\frac{(6-\theta )a_l-(2-\theta )a_h}{4(4-\theta )}-\frac{b(2-\theta )c_r}{2(4-\theta )(1-\theta )}$;

(2) 

When $\frac{(1-\theta )\left[(6-\theta )a_l-(2-\theta )a_h\right]}{2b(2-\theta )}\le c_r\le {\overline{C}}_1^{RN}$, ${\tilde{Q}}_{1h}^{RN}=\frac{(1-\theta )(a_h+a_l)+2b{c}_r}{4(1-\theta )(4-\theta )}$, ${\tilde{Q}}_{2h}^{RN}=\frac{(6-\theta )a_h-(2-\theta )a_l}{4(4-\theta )}-\frac{b(2-\theta )c_r}{2(4-\theta )(1-\theta )}$, ${\tilde{Q}}_{1l}^{RN}=\frac{(7-\theta )a_l-(1-\theta )a_h-2b{c}_r}{4(4-\theta )}$, ${\tilde{Q}}_{2l}^{RN}=0$;

(3) 

When ${\overline{C}}_1^{RN}<c_r\le {\widehat{C}}_1^{RN}$, ${\tilde{Q}}_{1h}^{RN}=\frac{a_h+a_l-2b{c}_r}{4\theta }$, ${\tilde{Q}}_{2h}^{RN}=\frac{a_h-a_l}{4}$, ${\tilde{Q}}_{1l}^{RN}=\frac{(1-\theta )a_h+(1+\theta )a_l-2b{c}_r}{4\theta }$, ${\tilde{Q}}_{2l}^{RN}=0$;

(4) 

When ${\widehat{C}}_1^{RN}<c_r<\frac{4a_h-3\theta a_h+\theta a_l}{4b}$, ${\tilde{Q}}_{1h}^{RN}=\frac{4b{c}_r-a_h-a_l}{8(1-\theta )}$, ${\tilde{Q}}_{2h}^{RN}=\frac{(4-3\theta )a_h+\theta a_l-4b{c}_r}{8\left(1-\theta \right)}$, ${\tilde{Q}}_{1l}^{RN}=\frac{3a_l-a_h}{8}$, ${\tilde{Q}}_{2l}^{RN}=0$;

(5) 

When $\frac{4a_h-3\theta a_h+\theta a_l}{4b}\le c_r<\frac{a_l}{b}$, ${\tilde{Q}}_{1h}^{RN}=\frac{3a_h-a_l}{8}$, ${\tilde{Q}}_{2h}^{RN}=0$, ${\tilde{Q}}_{1l}^{RN}=\frac{3a_l-a_h}{8}$, ${\tilde{Q}}_{2l}^{RN}=0$.

Figure 3 illustrates the results of Corollary 2. In region (a), where the reliable supplier has sufficiently low production costs and Supplier 2 sets a low wholesale price, the manufacturer chooses dual sourcing regardless of the type of market demand potential. In region (b), the equilibrium decision of the manufacturer takes on a different structure for different demand potentials, i.e., dual sourcing for high demand potentials and single sourcing for low demand potentials, as the demand potential increases. In region (c), the higher lead causes the manufacturer to abstain from sending orders to Supplier 2, even at higher demand potentials. Consistently with the results in the information-sharing case, a manufacturer will never choose Supplier 2 for single sourcing.

4.2. Equilibrium Decision Making with the Committed Pricing Strategy

With the committed pricing strategy, the optimal selling price and purchase quantity decisions of the manufacturer are first solved for in Stage 4, and then the optimal wholesale price decisions of the two suppliers are solved for in Stage 3.

4.2.1. Optimal Selling Price and Sourcing Quantity Decisions

With the committed pricing strategy, the manufacturer’s problem at Stage 4 can be formulated as follows:

$$\underset{Q_{1i},Q_{2i},p_i⩾0}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\pi }_M=E\left[p_i\mathrm{m}\mathrm{i}\mathrm{n}\left\{D\left(p_i\right),S\left(Q_{1i},Q_{2i}\right)\right\}-w_{1i}{S}_1\left(Q_{1i}\right)-w_{2i}{Q}_{2i}\right]\right\}$$

(7)

Lemma 3.

The optimal purchase quantity with the committed pricing strategy is the following:

(1) 

When $0<w_{2i}\le \frac{a_i}{b}\left(1-\sqrt{\theta }\right)$, ${\tilde{Q}}_{1i}=0$, ${\tilde{Q}}_{2i}=\frac{a_i-b{w}_{2i}}{2}$;

(2) 

When $\frac{a_i}{b}\left(1-\sqrt{\theta }\right)<w_{2i}<\frac{a_i}{b}$, $\left\{\begin{array}{c}{\tilde{Q}}_{1i}=\frac{a_i-b{w}_{1i}}{2},{\tilde{Q}}_{2i}=0,{ w}_{1i}\le {\widehat{w}}_{1i},\\ {\tilde{Q}}_{1i}=0,{\tilde{Q}}_{2i}=\frac{a_i-b{w}_{2i}}{2},{ w}_{1i}>{\widehat{w}}_{1i}.\end{array}\right.$

where ${\widehat{w}}_{1i}=\frac{\sqrt{\theta }{a}_i-a_i+b{w}_{2i}}{b\sqrt{\theta }}$.

Lemma 3 exhibits a different sourcing decision structure from that for responsive pricing. The manufacturer has only a form of single sourcing in the committed pricing strategy, and the decision of the supplier from which to purchase depends on the magnitude of the wholesale prices of the two suppliers. When the Supplier 2’s wholesale price $w_{2i}$ is small, the manufacturer purchases products only from Supplier 2. As $w_{2i}$ increases, the impact of Supplier 1’s wholesale price $w_{1i}$ starts to come to the fore. Only when the $w_{1i}$ is low, the manufacturer’s single sourcing is to Supplier 1, and if $w_{1i}$ is too large, the single sourcing is still with Supplier 2. Due to the inflexibility of committed pricing, the manufacturer bases his purchasing decision on the known $w_{1i}$ and $w_{2i}$, and either purchases from Supplier 1 or from Supplier 2, thus losing the possibility of dual sourcing. Thus, responsive pricing increases the incentive for the manufacturer to adopt dual sourcing. This finding may explain Dell’s dual-sourcing behavior from the perspective of pricing strategy.

4.2.2. Equilibrium Wholesale Price Decisions

The suppliers’ problem with committed pricing is consistent with that of responsive pricing. Based on the result of Lemma 3, we first solve for the Nash equilibrium of the wholesale price of the supplier in the CS strategy.

Theorem 3.

With the committed pricing strategy, there exists a unique Nash equilibrium of the wholesale price $\left({\tilde{w}}_{1i}^{CS},{\tilde{w}}_{2i}^{CS}\right)$ when the manufacturer shares demand information.

(1) 

When $0<c_r\le \frac{a_i}{b}\left(1-\sqrt{\theta }\right)$, ${\tilde{w}}_{1i}^{CS}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{0,\frac{(2\sqrt{\theta }-1)a_i+b{c}_r}{2b\sqrt{\theta }}\right\}$, ${\tilde{w}}_{2i}^{CS}=\frac{a_i+b{c}_r}{2b}$;

(2) 

When $\frac{a_i}{b}\left(1-\sqrt{\theta }\right)<c_r<\frac{a_i}{b}$, ${\tilde{w}}_{1i}^{CS}=\frac{a_i}{2b}$, ${\tilde{w}}_{2i}^{CS}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{c_r,\frac{(2-\sqrt{\theta })a_i}{2b}\right\}$.

Consistently with the case of the RS strategy, there is still a unique Nash equilibrium of the wholesale price for suppliers in the CS strategy, and the wholesale price decision depends on the production cost of the reliable supplier $c_r$. The supplier market behaves as a single-supplier monopoly scenario, and we note that the monopoly wholesale price $a_i/2b$ of Supplier 2 is greater than the monopoly wholesale price $\left(a_i+b{c}_r\right)/2b$ of Supplier 1. The trade-off between wholesale price and reliability determines the manufacturer’s equilibrium purchasing decision. When $c_r$ is small, the benefits of a more reliable supply outweigh the disadvantages resulting from a more expensive wholesale price, and the manufacturer chooses to engage in single sourcing from Supplier 2 to maximize profits. When $c_r$ is large, the overly expensive monopoly wholesale price of ${\tilde{w}}_{2i}^{CS}$ makes it unaffordable for the manufacturer, at which point the cost advantage prevails in the trade-off, and the manufacturer chooses to engage in single sourcing from Supplier 1, even if it has to bear some disruption risk. Theorem 3 extends the findings of Dada [11] and others, which showed that the conclusion that “cost is prioritized over reliability when choosing a supplier” does not always hold in our model setup. There is no situation with committed pricing where there will not be single sourcing from Supplier 2, and reliability is clearly prioritized over cost when $c_r$ is small. Next, Theorem 4 gives the unique Nash equilibrium of the wholesale price of the suppliers $\left({\tilde{w}}_{1i}^{CN},{\tilde{w}}_{2i}^{CN}\right)$ with the CN strategy.

Theorem 4.

With the committed pricing strategy, there exists a unique Nash equilibrium of the wholesale price $\left({\tilde{w}}_{1i}^{CN},{\tilde{w}}_{2i}^{CN}\right)$ when the manufacturer does not share demand information.

(1) 

When $0<c_r\le \frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)$, ${\tilde{w}}_1^{CN}=max\left\{0,\frac{(2\sqrt{\theta }-1)\left(a_h+a_l\right)+2b{c}_r}{4b\sqrt{\theta }}\right\}$, ${\tilde{w}}_2^{CN}=\frac{a_h+a_l+2b{c}_r}{4b}$;

(2) 

When $\frac{a_h+a_l}{2b}\left(1-\sqrt{\theta }\right)<c_r<\frac{a_l}{b}$, ${\tilde{w}}_1^{CN}=\frac{a_h+a_l}{4b}$, ${\tilde{w}}_2^{CN}=max\left\{c_r,\frac{(2-\sqrt{\theta })\left(a_h+a_l\right)}{4b}\right\}$.

Similarly, since the suppliers in the CN strategy only have the prior distributions of the demand information, the Nash equilibrium of the wholesale price and threshold are then equal to the expectation of the Nash equilibrium of the wholesale price and threshold with shared information. Since the information asymmetry upstream and downstream of the supply chain leads to a mismatch between the wholesale price decision and the purchasing decision (similar to Corollary 2), we give the equilibrium purchasing decision of the manufacturer with the CN strategy in Corollary 3.

Corollary 3.

With the CN strategy, given the Nash equilibrium of the wholesale price from the suppliers, the manufacturer’s equilibrium purchasing decision is the following:

(1) 

When $0<c_r\le \frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)$:

For $0<\theta <\frac{1}{4}$, $1<\frac{a_h}{a_l}<min\left\{3-4\sqrt{\theta },\frac{2}{2-\theta }\right\}$, $0<c_r\le \frac{(3-4\sqrt{\theta })a_l-a_h}{2b}$, ${\tilde{Q}}_{1h}^{CN}={\tilde{Q}}_{1l}^{CN}=0$, ${\tilde{Q}}_{2h}^{CN}=\frac{3a_h-a_l-2b{c}_r}{8}$, ${\tilde{Q}}_{2l}^{CN}=\frac{3a_l-a_h-2b{c}_r}{8}$;

For $0<\theta <\frac{1}{4}$, $1<\frac{a_h}{a_l}<min\left\{3-4\sqrt{\theta },\frac{2}{2-\theta }\right\}$, $\frac{(3-4\sqrt{\theta })a_l-a_h}{2b}<c_r\le \frac{(1-2\sqrt{\theta })(a_h+a_l)}{2b}$ or $3-4\sqrt{\theta }<\frac{a_h}{a_l}<\frac{2}{2-\theta }$, $0<c_r\le \frac{(1-2\sqrt{\theta })(a_h+a_l)}{2b}$, ${\tilde{Q}}_{1h}^{CN}=0$, ${\tilde{Q}}_{1l}^{CN}=\frac{a_l}{2}$, ${\tilde{Q}}_{2h}^{CN}=\frac{3a_h-a_l-2b{c}_r}{8}$, ${\tilde{Q}}_{2l}^{CN}=0$;

For $0<\theta <\frac{1}{4}$, $c_r>\frac{(1-2\sqrt{\theta })(a_h+a_l)}{2b}$ or $\frac{1}{4}<\theta <1$, ${\tilde{Q}}_{1h}^{CN}=0$, ${\tilde{Q}}_{1l}^{CN}=\frac{a_h+a_l-2\sqrt{\theta }\left(a_h-a_l\right)-2b{c}_r}{8\sqrt{\theta }}$, ${\tilde{Q}}_{2h}^{CN}=\frac{3a_h-a_l-2b{c}_r}{8}$, ${\tilde{Q}}_{2l}^{CN}=0$.

(2) 

When $\frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)<c_r<\frac{a_l}{b}$:

For $\frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)<c_r\le \frac{(2-\sqrt{\theta })\left(a_h+a_l\right)}{4b}$, ${\tilde{Q}}_{1h}^{CN}=0$, ${\tilde{Q}}_{1l}^{CN}=\frac{3a_l-a_h}{8}$, ${\tilde{Q}}_{2h}^{CN}=\frac{(2+\sqrt{\theta })a_h-(2-\sqrt{\theta })a_l}{8}$, ${\tilde{Q}}_{2l}^{CN}=0$;

For $c_r>\frac{(2-\sqrt{\theta })\left(a_h+a_l\right)}{4b}$, ${\tilde{Q}}_{1h}^{CN}=0$, ${\tilde{Q}}_{1l}^{CN}=\frac{3a_l-a_h}{8}$, ${\tilde{Q}}_{2h}^{CN}=\frac{a_h-b{c}_r}{2}$, ${\tilde{Q}}_{2l}^{CN}=0$.

The manufacturer’s choice of who to buy from is influenced by the type of demand potential. Regardless of the reliable supplier’s production cost $c_r$ and how it varies, the manufacturer always engages in single sourcing from Supplier 2 when the demand potential is high. When the demand potential is high, the uncertainty in Supplier 1’s supply can be very damaging to the manufacturer, and any disruptions can lead to a large loss of sales, so the manufacturer chooses to purchase from the reliable supplier even if it costs more to purchase. When the demand potential is low, on the other hand, a fully reliable but expensive supply of the product is not necessary for the manufacturer, who chooses to engage in single sourcing from Supplier 1 in some cases. Specifically, the manufacturer chooses to purchase products from Supplier 2 whether the demand potential is high or low only when $c_r$ is small ($0<c_r\le \left((3-4\sqrt{\theta })a_l-a_h\right)/2b$) and when there is a greater probability of disruption ($0<\theta <1/4$) with Supplier 1. Once $c_r$ is large ($c_r>\left((3-4\sqrt{\theta })a_l-a_h\right)/2b$) or Supplier 1 is more reliable ($\theta >1/4$), the manufacturer has no incentive to engage in single sourcing from Supplier 2 under low demand potential.

5. Comparison and Discussion

In this section, we determine the optimal strategic choice for a manufacturer over a range of parameters by comparing the manufacturer’s expected profits in four strategy combination scenarios.

5.1. Demand-Information-Sharing Strategy Options

This subsection describes the manufacturer’s optimal information-sharing strategy choice in two pricing scenarios. To avoid trivial cases, we focus only on the manufacturer’s information-sharing strategy choice when the condition ${\overline{C}}_{1h}^{RS}<{\widehat{C}}_{1l}^{RS}$ (when $1<a_h/a_l<\mathrm{m}\mathrm{i}\mathrm{n}\left\{\left(4-4\theta +{\theta }^2\right)/\left(4-4\theta \right),2/\left(2-\theta \right)\right\}$) is satisfied, and the result is given by Proposition 1.

Proposition 1.

With the responsive pricing strategy, the manufacturer chooses to share demand information when $c_r^{\left(1\right)}<c_r\le c_r^{\left(2\right)}$ and chooses not to share demand information when $0<c_r\le c_r^{\left(1\right)}$ or $c_r>c_r^{\left(2\right)}$. Here,

$$c_r^{\left(1\right)}=\frac{(4-5\theta +{\theta }^2)}{2b}\sqrt{\frac{(32-40\theta -4{\theta }^2+10{\theta }^3-{\theta }^4){\left(a_h-a_l\right)}^2+16{\theta }^2{a}_h^2}{{\left(16+12{\theta }^2-28\theta -{\theta }^3\right)}^2}}+\frac{32a_h-56\theta a_h+24{\theta }^2{a}_h}{2b(16+12{\theta }^2-28\theta -{\theta }^3)},$$

$$c_r^{\left(2\right)}=\frac{4a_h-\sqrt{2}\theta \sqrt{3{\left(a_h-a_l\right)}^2+2a_h^2}}{4b}.$$

Intuitively, if supply reliability is exogenous and wholesale prices are endogenous, demand information sharing would deprive the manufacturer of an information advantage and increase the bargaining power of suppliers to the detriment of the manufacturer. Proposition 1 points out that this intuition does not always hold. With a responsive pricing strategy, the manufacturer refuses to share demand information only if it is large or small. When moderate, information sharing increases the manufacturer’s expected profit. Figure 4 presents the results of Proposition 1.

This phenomenon occurs because when $c_r$ is small, the manufacturer has a higher probability of purchasing from two suppliers at the same time, and information sharing induces suppliers to set wholesale prices according to the actual demand potential, which can harm the manufacturer’s expected profit. When $c_r$ is greater than the threshold $c_r^{\left(1\right)}$ (recall Theorem 1 and Corollary 1 of the analysis), Supplier 2 is a potential market entrant, forcing Supplier 1 to reduce its wholesale prices to maintain a monopoly in the market. With a higher demand potential, information sharing is exacerbated by Supplier 1 in defense, and the manufacturer is able to lower the wholesale price of purchases from Supplier 1 to obtain a greater sales revenue. This time, the information sharing works in favor of the manufacturer. When $c_r$ further increases to exceed the threshold $c_r^{\left(2\right)}$, sourcing from reliable suppliers is no longer profitable. Supplier 1 is in an almost absolute monopoly, and information sharing gives Supplier 1 decision-making flexibility, so it has both the first-mover advantage and an information advantage, which undermine the manufacturer’s expected profits.

Next, Proposition 2 gives the manufacturer’s optimal information-sharing strategy choice with committed pricing. Again, in order to avoid trivial cases and, thus, highlight the main pattern of the manufacturer’s information-sharing strategy choice with committed pricing, we focus only on the case where Supplier 1 is more reliable (${\theta }^{\left(1\right)}<\theta <1$, where ${\theta }^{\left(1\right)}$ is the second root of $5648{\theta }^3-6816{\theta }^2+2816\theta -256=0$,${\theta }^{\left(1\right)}\approx 0.6$).

Proposition 2.

With the committed pricing strategy, the manufacturer chooses to share demand information when $0<c_r\le \frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)$ or $c_r>c_r^{\left(3\right)}$ and chooses not to share demand information when $\frac{\left(a_h+a_l\right)}{2b}\left(1-\sqrt{\theta }\right)<c_r\le c_r^{\left(3\right)}$.

Here, $c_r^{\left(3\right)}=\frac{4a_h-\sqrt{\theta (3a_h^2+6a_h{a}_l-5a_l^2)}}{4b}$.

With committed pricing, the manufacturer does not share demand information only when it is moderate, and shares demand information when it is large or small. The results of Corollary 3 show that if Supplier 1 is more reliable (${\theta }^{\left(1\right)}<\theta <1$), the equilibrium purchasing decision of the manufacturer who does not share information is unaffected by $c_r$, i.e., it purchases from Supplier 2 at a lower price at higher demand potentials, and it purchases from Supplier 1 at a monopoly price at lower demand potentials. When $c_r$ is moderate, the manufacturer who shares information can only purchase from Supplier 1 or Supplier 2 at the monopoly price, and, thus, the absence of information sharing at this point misleads Supplier 2’s wholesale price decision and reduces the manufacturer’s purchasing cost, thus favoring the manufacturer. When $c_r$ is small, purchasing from a reliable supplier is clearly profitable, and information sharing achieves supply chain coordination so that the manufacturer tends to purchase from Supplier 2 at any demand potential, thus increasing the supplier’s profits. Similarly, if $c_r$ is large, information sharing induces the manufacturer to purchase from Supplier 1 and abandon Supplier 2 for any demand potential, reducing the manufacturer’s purchasing cost compared with that in the no-information-sharing case and, therefore, increasing the manufacturer’s expected profit. Interestingly, the result of Proposition 2 is diametrically opposed to that of Proposition 1. When $c_r$ is moderate, the manufacturer using responsive pricing chooses to share demand information to increase supplier competition, but the manufacturer using committed pricing chooses not to share demand information at this point.

5.2. Value Analysis of Responsive Pricing

Intuitively, responsive pricing allows the manufacturer to set the selling price after resolving supply uncertainty in order to regulate demand for the purpose of supply–demand matching, which always favors the manufacturer. Therefore, this subsection explores the problem of the manufacturer’s pricing strategy choice by comparing the changes in the manufacturer’s expected profit margins before and after adopting responsive pricing with the same information-sharing strategy. It also defines the value of responsive pricing and analyzes the interplay of two supply and demand matching strategies, responsive pricing and information sharing, in mitigating the risk of supply disruption. We define the value of responsive pricing as the manufacturer’s expected profit, which is enhanced by using responsive pricing strategies. Specifically, the value of responsive pricing with information sharing and the value of responsive pricing with no information sharing are given by the following equation:

$$Vo{P}_S={\tilde{\mathsf{\Pi }}}_M^{RS}-{\tilde{\mathsf{\Pi }}}_M^{CS}, Vo{P}_N={\tilde{\mathsf{\Pi }}}_M^{RN}-{\tilde{\mathsf{\Pi }}}_M^{CN}.$$

(8)

In order to explore the interplay between responsive pricing and information sharing, it is necessary to compare the value of responsive pricing with information sharing and without information sharing. If $Vo{P}_S>Vo{P}_N>0$, which suggests that responsive pricing is more valuable with information sharing, and the manufacturer is more motivated to adopt responsive pricing strategies with information sharing, then we claim that the responsive pricing and information-sharing strategies complement each other; otherwise, the two are substitutes for each other.

Due to the mathematical complexity, this subsection uses numerical methods to analyze the interplay between the two strategies. We set the parameters $a_h=1.1$, $a_l=1$, and $b=1$ because $1<a_h/a_l<2/\left(2-\theta \right)$, and we let $\theta \in \left\{0.2,0.4,0.6,0.8\right\}$ to explore the effect of Supplier 1’s reliability on the interplay between the two strategies. Figure 5 illustrates the results of the numerical analysis.

The results of the numerical analysis show that responsive pricing gives the manufacturer the flexibility to set the selling price after the supply uncertainty is resolved, and it helps the manufacturer to regulate the market demand to increase its expected profit, so responsive pricing is always in favor of the manufacturer.

However, the joint adoption of responsive pricing and information sharing does not always achieve complementarity. Due to the cumbersome results involved, we only explore the general trend of this interplay and ignore the results within the trivial interval. In general, responsive pricing and information sharing complement each other only when the production cost $c_r$ of the reliable supplier is moderate, and this mutual complementary effect is further strengthened by an increase in the reliability $\theta$ of Supplier 1. Propositions 2 and 3 give reasons for this phenomenon. With a moderate value of $c_r$, the manufacturer using responsive pricing chooses to share demand information to increase competition, while the manufacturer using committed pricing chooses to retain demand information to mislead suppliers’ wholesale pricing decisions when responsive pricing and information sharing complement each other and together mitigate the effects of supply disruption risk. The opposite is true when $c_r$ is small or large, where either responsive pricing or information sharing is preferred for the manufacturer, and there is no need to employ both supply disruption risk mitigation strategies.

6. Conclusions

This study investigates the problem of a manufacturer’s choice of responsive pricing and demand-information-sharing strategies when facing supply disruption risk. By building a Stackelberg game model, the equilibrium decisions of supply chain members in four strategy combination scenarios are obtained, the conditions under which one strategy dominates are identified, and the interplay between responsive pricing and demand information sharing is explored. The main managerial findings of this study are as follows. First, there always exists a unique Nash equilibrium of the wholesale price for suppliers, and the production cost of Supplier 2 is a key factor influencing the purchasing decision. In order to obtain orders in the competition, the reliable supplier should minimize their production costs. Second, responsive pricing always favors the manufacturer and induces the manufacturer to engage in dual sourcing when the reliable supplier’s production cost is low and to purchase only from Supplier 1 when it is high. Third, whether information sharing benefits the manufacturer depends on the reliable supplier’s production costs and choice of pricing strategy. Finally, responsive pricing and information sharing complement each other only when the production costs of the reliable supplier are moderate. The research described here can help guide firms to comprehensively consider sourcing strategies, pricing strategies, and information-sharing strategies to mitigate supply disruption risk and, thus, achieve sustainable supply chain development.

We focus on responsive pricing and information sharing when sourcing from a reliable supplier and an unreliable supplier. However, it is possible to continue to extend our model for more in-depth research. For example, expand the number and structure of upstream suppliers, consider the existence of N competitive suppliers with supply correlation, and study the impact of changing suppliers on the manufacturer’s choices of pricing and information sharing. In addition, we assume that the manufacturer either shares demand information with all of their suppliers or none of them, and it would be valuable to consider partial sharing to explore the impact of supplier production costs and reliability differences on information sharing.