Supply Chain Performance with a Downside-Risk-Averse Retailer and Strategic Customers

Abstract

Predicting future promotion information on markdowns, customers can maximize their utilities by deciding when to buy. With this strategic behavior, this paper investigates a downside-risk-averse retailer’s integrated stock and pricing problem using a single case study method. Analyzing effects of the downside risk aversion and strategic customers is our purpose. By exploring a two-phase newsvendor model with a retailer selling to strategic customers, our work determines the downside-risk-averse retailer’s equilibrium ordering level and selling price. On this basis, effects of the downside risk aversion and the strategic behavior on the retailer’s optimum decisions and profit are analyzed. We find that the reverse effect of the strategic behavior can be mitigated by the retailer’s downside risk constraint. We also extend the model to a decentralized supply chain case. It is found that a low (high) downside risk aversion would mean that the supply chain profit in the decentralized case can (cannot) dominate the centralized under some (any) wholesale price contracts when customers are strategic. In addition, for different risk aversions, we also construct contracts to optimize the supply chain profit. Our results will provide reference evidence of making operational management decisions for the downside-risk-averse retailer in the case of strategic customers.

1. Introduction

To attract more customers, retailers usually create markdown sales [1]. Moreover, many e-commerce platforms (e.g., Amazon, Tmall and JD) provide customers with easier and faster access to the promotion information on markdown. Therefore, a growing number of customers wait for price markdowns [2]. This leads to an obvious decrease in sales quantity after the price-off promotion. To alleviate this situation, the retailer has to make more generous discounts in the regular periods. This generates a vicious cycle. It implies that the strategic customer behavior (abbr. SCB) can obviously affect the retailer’s profit [3]. As a result, the SCB needs to be taken into consideration in the process of the retailer’s operation management.

In addition to the SCB, the retailer’s downside risk (abbr. DR) constraint is also an important factor influencing the retailer’s profit. In fact, many retailers are DR-averse [4,5]. That is, when making retailing decisions, the retailer wants the maximal expected profit, as well as hoping that the DR (i.e., the likelihood that the actual profit does not exceed the given profit target) is not too high. In fact, to reduce the supply chain (abbr. SC) risk, the manager usually sets profit targets for the staff in charge of making ordering and pricing decisions to assess their performance [5]. It leads many staff to make their inventory decisions by “working backward” from the targets [6]. This is because they are well-aware of the fact that ordering too little could lead to no chance of meeting the target, and ordering too much could lead to too much backlogged inventory. Therefore, the profit target constraint is a critical factor impacting the retailer’s decisions. That is, the DR aversion needs to be considered when making retail decisions.

From the discussion so far, both the SCB and the DR-averse retailer exist in practice. Intuitively, each is closely related to a retailer’s decisions in the process of pricing and stock management. The retailer’s manager would certainly be interested in the related pricing and inventory management problem when the SCB and the DR-averse retailer present simultaneously. However, among the optimal decision problems related to the DR-averse retailer (e.g., Gan et al. [4]; Deng and Zheng [5]), studies on the effects of the SCB are extremely limited. To fill this gap, we focus on the DR-averse retailer’s integrated stock and pricing problem with the strategic customers. The aim of this paper is to test how the retailer’s DR aversion and the SCB impact the decisions and profits of the retailer and the SC profit using a single case study. Thus, we concentrate on the following two questions.

(i)

For the centralized SC, what is the DR-averse retailer’s equilibrium price and stock policy when the SCB is considered? How do the DR aversion and the SCB influence the retailer’s policy and profit?

(ii)

For the decentralized SC, how does the DR aversion affect the SC profit when the SCB is considered? How does the DR aversion affect the design of the contract which maximizes the SC profit?

To address these research questions, we develop the DR-averse retailer’s two-phase newsvendor model with the SCB (see Porteus [7] for the definition of newsvendor model). The retailer wants the expected profit optimization, where the probability cannot be too high that the actual profit does not exceed the predetermined target level. Customers are allowed to either purchase in phase 1 with the regular price or purchase in phase 2 at the salvage price. They maximize their utilities by deciding when to buy. We assume that the retailer’s and customers’ beliefs concur with the actual outcomes. We present the rational expectation equilibrium (see Muth [8] for the definition of this equilibrium), and derive the retailer’s equilibrium ordering level and selling price. Based on this, effects of the DR aversion and the SCB on the optimum policy and profit are analyzed using the convex analysis and the monotonicity analysis. In addition, we also extend the setting of the model to a decentralized SC. Under this setting, for different DR aversions, we compare the centralized and decentralized SC profits and construct contracts maximizing the SC profit.

We now present our contributions below by comparing our work with Gan et al. [4] and Su and Zhang [9], considering that their studies are most related to our work.

• About the model, we build the integrated stock and pricing model with the DR aversion and the strategic customers. However, ref. [4] only considers the DR-averse retailer but ignores the price decision and the SCB, and ref. [9] only focuses on the strategic behavior but ignores the DR aversion.
• About effects of the DR constraint, we argue that the DR constraint could boost the retailer’s profit if DR aversion is neither too low nor too high. This does not hold in the model of Gan et al. [4] since they ignore the SCB.
• Regarding the centralized and decentralized SC cases, we compare the profits of two cases. We determine that a low (high) DR aversion can mean that the SC profit in the decentralized case can (cannot) dominate the centralized under some (any) WPC when customers are strategic. This is not considered by Gan et al. [4], though they also consider the DR-averse retailer.
• The effects of DR aversion on the design of the SC contracts are not considered by Su and Zhang [9]. We find that for a low DR aversion, there exists a WPC which ensures the decentralized SC profit maximization. However, for a high DR aversion, any WPC cannot maximize the SC profit. Fortunately, we can construct a modified WPC (i.e., a hybrid version of a WPC plus a return contract) to ensure the SC profit maximization. In addition, we can also construct a modified WPC under which each supply chain member’s profit is higher than under the WPC. Though Gan et al. [4] also design a modified WPC to maximize the SC profit, it can only be applied when the retailing price is exogenous and customers are not strategic.

The remaining content of this research is organized as below. Section 2 is the literature review. Section 3 includes two subsections. Section 3.1 is the model formulation and analysis for the centralized SC. Section 3.2 is the related analysis for the decentralized SC. Finally, Section 4 presents the research conclusions. Proofs are in the Appendix A.

2. Literature Review

Our study is associated with the literature on SCB and the related research is rich. Under the setting of the deterministic customer demand, Lin et al. [2] develop a manufacturer–retailer SC problem related to strategic customers, and argue that the SCB can benefit the retailer and the whole SC. Chen et al. [10] explore the retailer’s pricing model with the SCB and the reference price effect. It is verified that these two factors can weaken the double marginalization. In the setting of multiple periods, Farshbaf-Geranmayeh and Zaccour [11] study a pricing problem and find the fixed pricing policy is better than the markdown policy. These three papers only focus the deterministic customer demand and ignore influences of demand randomness on the SC members’ decisions. Lobel et al. [12] consider the strategic customers and a firm which releases new products over time. They find that it can profit the firm to commit the information of the new product in advance. For the stochastic demand case, Hu et al. [13] analyze the seller’s multiple-period stock and markdown model with the markdown price policy. They find that the optimal markdown decision relies on the amount of the remaining inventory in the preceding period. Gershkov et al. [14] study a retailer’s dynamic pricing model, where customers can choose when to buy and the retailer can predict customer demand. They find some dynamic pricing strategies could be applied to boost the retailer’s profit. Dong and Wu [15] develop a two-period dynamic pricing problem assuming that the customer demand must be satisfied. They develop some conditions to ensure that the dynamic pricing policy is optimal when customers are strategic. Under the two-period price and stock model, Zhao et al. [16] analyze the impacts of reference price effect and the quick response policy. They argue that the quick response policy can help the seller reduce the initial stock as well as improve the profit when the customer is strategic. Wu et al. [17] compare three pricing policies including the fixed price policy, high–low policy and strategic high pricing policy. In addition, they present the conditions to ensure that each policy is optimal when the customers are strategic. All the models mentioned above assume that the seller’s decisions cannot affect the probability that the customer demand is met in the markdown selling period. In the process of the actual operation, this probability, however, can be influenced by the retailer’s inventory decision. For example, if the inventory is insufficient, customers buying late may not obtain the products; hence, their expected surpluses may decrease.

Different from the abovementioned studies on strategic customers, Su and Zhang [9] consider the case that the probability that customer demand is satisfied can be influenced by the retailer’s inventory decision. They establish a seller’s two-period newsvendor model, and determine the equilibrium pricing and inventory policy. They argue that the SCB can negatively affect the seller’s profit. Additionally, this effect could be alleviated by both price commitment and quantity commitment. On the basis of the model of Su and Zhang [9], several papers also explore some policies to decrease the negative influence of the SCB. For example, capacity rationing and the customer reimbursement contract have been verified to be effective methods to alleviate the negative effect by Liu and Van Ryzin [18] and Du et al. [19], respectively. In addition, Song et al. [20] and Xu and Duan [21] prove the quick replenishment policy can also decrease the reverse impact of the SCB. Note that the above studies associated with strategic customers have ignored the case where the retailer is DR-averse. In contrast to these studies, we consider the DR-averse retailer and find that the reverse influence of the SCB can be mitigated by the retailer’s DR constraint.

We now review the literature stream on the DR-averse retailer’s decision model. The retailer’s risk aversion has been characterized by different methods, such as the concave utility function [22], piecewise linear utility function [23,24], mean–variance objective [25,26], conditional value at risk [27,28], and risk measure [5,29,30]. We characterize the retailer’s risk aversion with the DR measure, i.e., the DR-averse retailer is considered. Specifically, the retailer’s decisions must ensure that the probability with the actual profit being less than some threshold is not too high. There are some studies that consider the DR-averse retailer’s stock decision. For example, Gan et al. [29] studied the SC coordination with the DR-averse retailer and constructed the contract related to a pareto-optimal solution. Gan et al. [4] consider a DR-averse retailer’s inventory decision problem based on a newsvendor model. They find that the DR constraint can hurt the retailer. Deng and Zheng [5] develop a DR-averse seller’s newsvendor problem in which both the demand and the supply are random. They find that the DR-averse seller cannot order too much or too little. All the above studies on the DR-averse retailer ignore the influence of the SCB on the retailer’s policy.

In summary, existing studies on the SCB primarily focus on the risk-neutral retailer’s stock and pricing model. Moreover, most of existing studies on the DR-averse retailer ignore the SCB. Therefore, effects of the SCB and the DR aversion have not been considered simultaneously in the SC papers. In this study, a DR-averse retailer’s two-phase selling model with SCB is developed. We assume that the retailer’s profit is optimized under the DR constraint that the chance of the actual profit not being above a given target level is no greater than a given threshold [4]. Customers are strategic and need to maximize their utilities by deciding when to buy [9].

Table 1. Summary of existing studies on the SCB.

Au (Year)	Demand Randomness	Price and Stock Decisions	Do the Retailer’s Decisions Affect the Chance that Demand Is Met?	Is the Retailer DR-Averse?	How to Alleviate Negative Effects of Strategic Customers
Lin et al. (2018)	×	Price and stock	×	×	―
Chen et al. (2020)	×	Price	×	×	―
Lobel etal. (2016)	×	Price	×	×	―
Farshbaf-Geranmayeh and Zaccour (2021)	×	Pricing and advertising	×	×	―
Hu et al. (2016)	√	stock and markdown	×	×	―
Dong and Wu (2019)	√	Price and stock	×	×	―
Zhao et al. (2022)	√	Price and stock	×	×	―
Wu et al. (2022)	√	Price and stock	×	×	―
Su and Zhang (2008)	√	Price and stock	√	×	Price commitment, quantity commitment
Liu and van Ryzin (2008)	×	Stock	√	×	Capacity rationing
Du et al. (2015)	√	Price and stock	√	×	Customer reimbursement contract
Song et al. (2021)	√	Price and stock	√	×	Partial-coverage strategy
Xu and Duan (2020)	√	Price and stock	√	×	Quick replenishment policy
This paper	√	Price and stock	√	√	DR constraint

presents relations between the existing literature on the SCB and this study.

3. Model and Analysis

This section discusses the effects of the SCB and the DR aversion in the setting of a supplier–retailer–customers SC using a single case study method. Section 3.1 assumes that the supplier and the retailer want the whole SC profit maximization; hence, it is reduced to the retailer–customer SC, i.e., the centralized case. Section 3.2 assumes that supply chain members are independent and want their own profit maximization; hence, it could be deemed as the decentralized case [9].

3.1. The Centralized SC

In this subsection, we first develop a DR-averse retailer’s two-phase selling model with SCB. Next, by solving the model, we determine the retailer’s stock and price policy. Finally, we explore influences of the DR aversion when the SCB is considered, and also analyze impacts of the SCB when the DR-averse retailer is considered.

3.1.1. Model Setup

We explore a DR-averse retailer’s two-phase newsvendor model. The retailer wants the maximal total expected profit, in which the probability, which is not above a predetermined level $\gamma \in (0,1)$, of the actual profit is not above a predetermined target $\alpha$. Customer demand $\epsilon$ is nonnegative and random, and its probability density and distribution functions are $\varphi$ and $\mathsf{\Phi }$, respectively. Similar to Su and Zhang [9], the failure rate function $\varphi (x)/\overline{\mathsf{\Phi }}(x)$ increases. Customers, where they are homogenous in product valuation, maximize their utilities by deciding when to buy.

In phase 1, the retailer first privately generates an expectation $\widehat{r}$ on customers’ reservation price $r$ (i.e., a customer will not buy in phase 1 if the selling price exceeds $r$). Based on the belief $\widehat{r}$, the retail price $p$ and order quantity $y$ are chosen to optimize the two-phase expected profit. Additionally, $y$ products are ordered with a unit ordering cost $c$. Assume that the price $p$ is announced but the stock $y$ is not 9. Thus, a customer generates belief $\widehat{\theta }$ on the chance of getting the product in phase 2. Additionally, based on their belief $\widehat{\theta }$, they generate their reservation price $r$. Next, customer demand is realized. In phase 2, all the remaining stock is dealt with at a salvage price $s$.

We now present the customers’ decision problem. Based on the observed product valuation $v$ and product price $p$, a customer forms the belief $\widehat{\theta }$. Then, the related product utilities of phase 1 and 2 are $v-p$ and $(v-s)\widehat{\theta }$, respectively. Note that if $p\le v-(v-s)\widehat{\theta }$, then the customer will purchase in phase 1. Thus, $r=v-(v-s)\widehat{\theta }$.

Given the belief $\widehat{r}$ over the reservation price $r$, the retailer’s problem is

$$\begin{array}{l}\widehat{u}=\underset{y\ge 0,p\le \widehat{r}}{\max }[(p-s)E(y\wedge \epsilon )-(c-s)y]\\ s.t. \mathrm{Pr}\{(p-s)(y\wedge \epsilon )-(c-s)y\le \alpha \}\le \gamma .\end{array}$$

(1)

In this problem, $x\wedge y=\min \{x,y\}$. Let $x\vee y=\max \{x,y\}$, which will be used later.

To simplify the constraint condition of problem (1), we present Lemma 1 in the following.

Lemma 1. 

If$y\le y^0=\alpha /(p-c)$, then the DR$\mathrm{Pr}\{(p-s)(y\wedge \epsilon )-(c-s)y\le \alpha \}=1$; if$y>y^0$, then the constraint in problem (1) equals

$$\mathsf{\Phi }\left(\frac{\alpha +(c-s)y}{p-s}\right)\le \gamma$$

(2)

With the condition of equivalence in Lemma 1, we could obtain the DR-averse retailer’s optimum inventory and pricing decision in problem (1) for the given belief $\widehat{r}$ over the reservation price.

Proposition 1. 

For the given belief$\widehat{r}$over the reservation price, if$\gamma >\mathsf{\Phi }(\alpha /(\widehat{r}-c))$, then the DR-averse retailer’s pricing level$p$and inventory level$y$satisfy that$p=\widehat{r}$and

$$y={\mathsf{\Phi }}^{-1}\left(1-\frac{c-s}{\widehat{r}-s}\right)\wedge \frac{(\widehat{r}-s){\mathsf{\Phi }}^{-1}(\gamma )-\alpha }{c-s}$$

3.1.2. Rational Expectation Equilibrium (RE)

We first consider RE equilibrium and derive the equilibrium solution. Next, we develop the property of the optimum pricing and stock policy. Finally, we explore the impacts of the SCB and the DR aversion.

Definition 1. 

The RE equilibrium$(p,y,r,\widehat{r},\widehat{\theta })$meets that

(i)

$r=v-(v-s)\widehat{\theta }$;

(ii)

$p=\widehat{r}$;

(iii)

$y={\mathsf{\Phi }}^{-1}\left(1-\frac{c - s}{\widehat{r} - s}\right)\wedge \frac{(\widehat{r} - s){\mathsf{\Phi }}^{-1}(\gamma ) - \alpha }{c - s}$;

(iv)

$\widehat{\theta }=\mathsf{\Phi }(y)$;

(v)

$\widehat{r}=r$.

The first three conditions are the retailer’s and customers’ rational decisions as regards maximizing their utilities for given expectations $\widehat{r}$ and $\widehat{\theta }$. They can be obtained through the discussion in Section 3.1.1. Conditions (iv) and (v) show that expectations are in accordance with the actual situations. In the following, we verify that condition (iv) holds. Condition (iv) shows that the expectation $\widehat{\theta }$ is in accordance with the actual likelihood that a customer can obtain the product in phase 2. In fact, in equilibrium at $p=r$, all consumers would purchase in phase 1. It implies that if a customer wants to purchase the product in phase 2, then the likelihood that they could obtain the product is equal to there being remaining products in phase 2. It further equals that the ordering quantity $y$ exceeds the actual customer demand $\epsilon$, i.e., $\mathrm{Pr}\{\epsilon <y\}=\mathsf{\Phi }(y)$. Therefore, condition (iv) holds.

To simplify the subsequent analysis on the equilibrium $(p,y,r,\widehat{r},\widehat{\theta })$ in Definition 1, we develop results in Lemma 2 as below. Specifically, Lemma 2 analyzes the property of the minimum of two decreasing functions.

Lemma 2. 

If functions$f_1(x)$and$f_2(x)$are strictly decreasing, and$f_1(x_1)=f_2(x_2)=M$, then$f_1(x)\wedge f_2(x)>M$if$x<x_1\wedge x_2$, and$f_1(x)\wedge f_2(x)=M$if$x=x_1\wedge x_2$.

Based on the five conditions of RE equilibrium $(p,y,r,\widehat{r},\widehat{\theta })$ and the above Lemma, we characterize the DR-averse retailer’s inventory and pricing policy in the RE equilibrium. For simplicity, denote parameters $y^{bind}\in \left\{y:\mathsf{\Phi }\left(\frac{\alpha + (c - s)y}{(v - s)\overline{\mathsf{\Phi }}(y)}\right)=\gamma \right\}$ and ${\gamma }^{sa}=\mathsf{\Phi }\left(\frac{\alpha + (c - s){\mathsf{\Phi }}^{-1}\left(1 - \sqrt{(c - s)/(v - s)}\right)}{\sqrt{(v - s)(c - s)}}\right)$.

Proposition 2. 

If the RE equilibrium solution$(y^{sa},p^{sa})$exists, then it is unique. Moreover, the DR-averse retailer’s ordering quantity

$$y^{sa}=y^{bind}\wedge {\mathsf{\Phi }}^{-1}\left(1-\sqrt{(c-s)/(v-s)}\right)=\{\begin{array}{ll}{\mathsf{\Phi }}^{-1}\left(1-\sqrt{(c-s)/(v-s)}\right),\hfill & \mathrm{if} {\gamma }^{sa}<\gamma <1,\hfill \\ y^{bind},\hfill & \mathrm{if} 0<\gamma \le {\gamma }^{sa},\hfill \end{array}$$

(3)

the optimal price$p^{sa}=(v-s)\overline{\mathsf{\Phi }}(y^{sa})+s$, and the expected profit

$$u^{sa}=(v-s)\overline{\mathsf{\Phi }}(y^{sa})E(y^{sa}\wedge \epsilon )-(c-s)y^{sa}$$

(4)

In addition, the condition ensuring the existence of the equilibrium solution is$y^{sa}-\alpha /(p^{sa}-c)>0$.

From Proposition 2, when the equilibrium solution exists, if the predetermined bound $\gamma$ of the DR is large, the optimum inventory $q^{sa}={\mathsf{\Phi }}^{-1}\left(1-\sqrt{(c-s)/(v-s)}\right)$. This is also the risk-neutral retailer’s inventory decision via Proposition 1 in Su and Zhang [9]. If the predetermined upper bound $\gamma$ of the DR is small, then the retailer’s inventory $q^{sa}=q^{bind}$. At this point, the retailer’s DR is equivalent to the given threshold $\gamma$. Note that results in Proposition 2 are obtained from the RE equilibrium. Thus, results in Proposition 2 and the proceeding results hold only under the premise that the retailer’s and customers’ expectations accord with the actual outcomes. Otherwise, these results would not be applied.

According to Proposition 2, if the random demand $\epsilon$ follows the uniform distribution, i.e., $\epsilon \sim U[0,1]$, then a more simplified and direct characterization of the equilibrium solution $(y^{sa},p^{sa})$ could be derived as below. The proofs of Corollary 1 and Corollary 2 are omitted since they can be immediately obtained from Proposition 2.

Corollary 1. 

Assume that the random demand$\epsilon \sim U[0,1]$. If the DR-averse retailer’s equilibrium price and order quantity exist, then they satisfy

$$y^{sa}=\left(1-\sqrt{\frac{c-s}{v-s}}\right)\wedge \frac{(v-s)\gamma -\alpha }{(v-s)\gamma +(c-s)}=\{\begin{array}{ll}1-\sqrt{(c-s)/(v-s)},\hfill & \mathrm{if} {\gamma }^{sa}<\gamma <1,\hfill \\ \frac{(v - s)\gamma - \alpha }{(v - s)\gamma + (c - s)},\hfill & \mathrm{if} 0<\gamma \le {\gamma }^{sa},\hfill \end{array}$$

$$p^{sa}=s+\{\begin{array}{ll}\sqrt{(v-s)(c-s)},\hfill & \mathrm{if} {\gamma }^{sa}<\gamma <1,\hfill \\ \frac{(v - s)(c - s + \alpha )}{(v - s)\gamma + c},\hfill & \mathrm{if} 0<\gamma \le {\gamma }^{sa},\hfill \end{array}$$

where ${\gamma }_{sa}=\frac{\alpha + c - s}{\sqrt{(v - s)(c - s)}}-\frac{c - s}{v - s}$.

Both Proposition 2 and Corollary 1 present the DR-averse retailer’s equilibrium price and stock policy when the SCB is considered. That is, they answer the first half of the first research question presented in the Introduction.

Using Proposition 2, we can also obtain the impacts of the DR aversion and the cost parameters on the DR-averse retailer’s decisions and profit as below.

Corollary 2. 

For the DR-averse retailer faced with SCB, the following results hold.

(a) 

The optimal inventory level $y^{sa}$decreases with the target profit $\alpha$and increases with the probability $\gamma$.

(b) 

The optimal pricing level$p^{sa}=(v-s)(1-y^{sa})+s$is increasing in$\alpha$and decreasing in $\gamma$.

(c) 

The optimal profit $u^{sa}$first decreases and then increases in $\alpha$, and is first increasing and then decreasing in $\gamma$when $0<\gamma \le {\gamma }^{sa}$;$u^{sa}$does not change with $\alpha$and$\gamma$when ${\gamma }^{sa}<\gamma <1$.

(d) 

The optimal inventory level$y^{sa}$increases with $v$and decreases with $c$; the pricing level $p^{sa}$increases with $c$.

According to Gan et al. (2005), when the parameter $\alpha$ increases or the parameter $\gamma$ decreases, the retailer’s DR aversion increases. Thus, Corollary 2(a) and Corollary 2(b) state that if DR aversion is high, the ordering quantity should be lowered and the retail price should be raised. According to Corollary 2(c), for a low $\gamma$ (i.e., a high DR aversion), the retailer’s profit first decreases and then increases with the DR aversion. For a high $\gamma$ (i.e., a low DR aversion), the retailer’s profit is invariant. In addition, using Corollary 2(d), when a customer’s valuation is higher and the unit ordering cost is lower, increasing the ordering quantity will benefit the retailer. Corollary 2 show how the DR aversion and the SCB influence the retailer’s policy. They answer the second half of the first research question presented in the Introduction.

After analyzing the effects of risk aversion, we focus on the effects of the SCB. To carry this out, we introduce three benchmarks: the case with myopic customers and a DR neutral retailer, the case with strategic customers and a DR neutral retailer, and the case with myopic customers and a DR-averse retailer. For ease of description, we represent the three cases with subscripts $T$, $S$, $A$, respectively (See

Table 2. Notations.

Subscript	Description
$T$	The case with myopic customers and a RN retailer
$S$	The case with strategic customers and a RN retailer
$A$	The case with myopic customers and a DRA retailer
$sa$	The case with strategic customers and a DRA retailer

for notations). The retailer’s decisions under the four cases in Table 2 can be obtained according to the studies of Su and Zhang [9], Gan et al. [4], and Proposition 1 of this paper, and are presented in

Table 3. The retailer’s optimum decisions in four cases when the random demand $\epsilon \sim U[0,1]$.

Case $\mathit{i}$	$\mathbf{Optimal} \mathbf{Inventory} \mathbf{Level}{\mathit{y}}^{\mathit{i}}$	$\mathbf{Optimal} \mathbf{Price} {\mathit{p}}^{\mathit{i}}$
$T$	$y^T=1-(c-s)/(v-s)$	$p^T=v$
$S$	$y^S=1-\sqrt{(c-s)/(v-s)}$	$p^S=(v-s)(1-y^S)+s$
$A$	$y^A=\left(1-\frac{c - s}{v - s}\right)\wedge \frac{(v - s)\gamma - \alpha }{c - s}$	$p^A=v$
$sa$	$y^{sa}=\left(1-\sqrt{\frac{c - s}{v - s}}\right)\wedge \frac{(v - s)\gamma - \alpha }{(v - s)\gamma + c - s}$	$p^{sa}=(v-s)(1-y^{sa})+s$

.

With the optimum decisions and expected profits presented in Table 3, we present the effect of the SCB in the following. For simplicity, we define a critical threshold point

$${\tilde{\gamma }}^{sa}=\frac{2(\alpha +c-s)}{\sqrt{4{(v-s)}^2+5(v-s)(c-s)}-\sqrt{(v-s)(c-s)}}-\frac{c-s}{v-s}$$

and denote the function

$$u(y)=(v-s)\overline{\mathsf{\Phi }}(y)E(y\wedge \epsilon )-(c-s)y$$

Proposition 3. 

Assume that the equilibrium solution$(y^{sa},p^{sa})$exists for the case$sa$and the stochastic demand$\epsilon$follows a uniform distribution, i.e.,$\epsilon \sim U[0,1]$. The following results hold.

(a)

When faced with myopic customers, the DR-averse retailer’s expected profit$u^A$is always lower than the DR neutral retailer’s profit$u^T$, i.e.,$u^A\le u^T$.

(b)

When faced with SCB, the DR-averse retailer’s profit$u^{sa}$and the DR neutral retailer’s expected profit$u^S$satisfy

$$\{\begin{array}{ll}{u}^{sa}<u^S,\hfill & \mathrm{if} \gamma \in (0,{\tilde{\gamma }}^{sa}),\hfill \\ u^{sa}>u^S,\hfill & \mathrm{if} \gamma \in ({\tilde{\gamma }}^{sa},{\gamma }^{sa}),\hfill \\ u^{sa}=u^S,\hfill & \mathrm{if} \gamma \in [{\gamma }^{sa},1)\cup \left\{{\tilde{\gamma }}^{sa}\right\}.\hfill \end{array}$$

(5)

From Proposition 3(a), when the strategic behavior is ignored, the DR-averse retailer’s expected profit is always lower than that of the DR neutral retailer, i.e., $u^A\le u^T$(See Figure 1 for illustration). This is intuitive since the DR aversion could be deemed a chance constraint hurting the retailer. In contrast, the result in Proposition 3(b) is surprising: the DR constraint could benefit the retailer faced with strategic customers, i.e., $u^{sa}>u^S$. That is, the DR-averse retailer’s expected profit strictly exceeds that of the DR neutral retailer, when customers are strategic and the DR aversion is neither too high nor too low (See Figure 1 for illustration). The reason is that the DR aversion leads to a decrease in the ordering quantity but an increase in the selling price. Moreover, the increased expected profit due to the increased price is more than the decreased expected profit due to the decreased ordering quantity. By comparing results in Propositions 3(a) and 3(b), we know that the reverse influence of the SCB can be mitigated by the retailer’s DR constraint.

Proposition 3 shows how the DR aversion influences the DR-averse retailer’s profit with the SCB, and how the SCB affects the DR-averse retailer’s profit. Thus, the second half of the first research question in the Introduction is answered.

3.2. The Decentralized Supply Chain

This subsection considers the decentralized supplier–retailer–customer SC under the WPC, where $c$ can be seen as the supplier’s purchasing cost, and $s,v,\epsilon$ have the same definitions as in Section 3.1. The sequence of events is as below. First, the WPC is negotiated first. Next, the retailer and customers make decisions. Finally, customer demand is met and surplus products are sold at price $s$. In this subsection, we aim to analyze how the DR aversion affects the decentralized SC profit when the SCB is considered. To reach it, for different DR aversions, we first compare the decentralized SC profit under the WPC with the centralized, and then design contracts to achieve the SC profit maximization. For ease of description, in all the remaining discussions, the existence of the equilibrium solution is assumed since Proposition 2 shows that we could always provide conditions to ensure the existence.

To compare the decentralized SC profit with the centralized case, we first introduce some notations and calculate each SC member’s profit in the decentralized case. The subscripts $r$,$s$, and $sc$ represent the retailer, the supplier and the SC, respectively. The superscript $wpc$ denotes the decentralized case under the WPC. Since the retailer’s unit ordering cost is $w$ under the WPC contract $w$, according to Proposition 2, the retailer’s optimum inventory in case $wpc$.

$y_w={\mathsf{\Phi }}^{-1}\left(1-\sqrt{\frac{w - s}{v - s}}\right)\wedge y_w^{bind}$, where $y_w^{bind}\in \left\{y:\mathsf{\Phi }\left(\frac{\alpha + (w - s)y}{(v - s)\overline{\mathsf{\Phi }}(y)}\right)=\gamma \right\}$. Additionally, the related optimal price $p_w=(v-s)\overline{\mathsf{\Phi }}(y_w)+s$. For simplicity, denote $y^{*}$ as the maximizer of function $u(y)=(v-s)\overline{\mathsf{\Phi }}(y)E(y\wedge \epsilon )-(c-s)y$, i.e., $y^{*}=\underset{y}{\arg \max }u(y)$. With the threshold $y^{*}$, we further denote $\tilde{\gamma }$ and $w^{*}$ such that

$$\tilde{\gamma }=\mathsf{\Phi }\left(\frac{\alpha +(c-s)y^{*}}{(v-s)\overline{\mathsf{\Phi }}(y^{*})}\right) \mathrm{and} w^{*}=[(v-s){\overline{\mathsf{\Phi }}}^2(y^{*})]\wedge \frac{(v-s){\mathsf{\Phi }}^{-1}(\gamma )\overline{\mathsf{\Phi }}(y^{*})-\alpha }{y^{*}}+s$$

The definition of $\tilde{\gamma }$ implies that $\tilde{\gamma }$ is the DR when the retailer’s inventory is $y^{*}$. Definition of $w^{*}$ implies that if $w^{*}\in [c,v)$, then the retailer’s optimum inventory $y_{w^{*}}=y^{*}$ under the wholesale contract $w=w^{*}$. With these preparations, we compare the decentralized SC profit and the centralized SC profit $u^{sa}$ as below.

Proposition 4. 

For the DR-averse retailer, the following two results hold.

(a)

If$\tilde{\gamma }<\gamma <1$, then the decentralized SC profit under the WPC $w\in (c,w^{*}]$exceeds the centralized case.

(b)

If$0<\gamma \le \tilde{\gamma }$, then the SC profit in the decentralized case under any WPC$w\in (c,v)$cannot exceed that in the centralized case.

Note that a larger $\gamma$ means a lower DR aversion according to Gan et al. [4]. Thus, using Proposition 4(a), if the parameter $\gamma$ is large (i.e., the DR aversion is low), then the expected profit of the decentralized SC under the WPC exceeds that of the centralized case. Otherwise, this cannot happen (See Figure 2 for illustration). In fact, for a high DR aversion, the proper increase of the wholesale price based on $c$ could lead to a decrease in the SC stock risk but an increase in the selling price. Thus, the decentralized SC profit can exceed the centralized case. When the risk aversion parameter $\gamma$ is low (i.e., the DR aversion is high), the supply quantity $y$ is too low due to the chance constraint in the centralized case. Thus, the decentralized SC profit is always below the centralized case. From the above analysis, Proposition 4 answers the first half of the second research question in the Introduction.

Following Proposition 4, we studied the contracts optimizing the SC profit by distinguishing between the case $\tilde{\gamma }<\gamma <1$ and the case $0<\gamma \le \tilde{\gamma }$.

Proposition 5. 

If the retailer’s DR aversion parameter$\tilde{\gamma }<\gamma <1$, then, under WPC contract$w\in (c,v)$, the SC in the decentralized case can achieve profit maximization with$w=w^{*}$.

Proposition 5 states that when the DR aversion is low, the SC profit maximization is achieved under the WPC contract $w=w^{*}$.

After the optimum SC contract has been constructed for the case with $\tilde{\gamma }<\gamma <1$, we now focus on the case with $0<\gamma \le \tilde{\gamma }$. When $0<\gamma \le \tilde{\gamma }$, we find that $y_c\le y^{*}$ according the proof of Proposition 4, and $y_w<y_c$ for any $w\in (c,v)$ according to the definition of $y_w$ in (15). Thus, $y_w<y^{*}$ for any $w\in (c,v)$ when $0<\gamma \le \tilde{\gamma }$, that is, the retailer’s inventory cannot be $y^{*}$ maximizing the SC profit. Hence, when $0<\gamma \le \tilde{\gamma }$, any WPC contract $w\in (c,v)$ cannot achieve the SC profit maximization. If the retailer’s ordering quantity $y$ is increased up to $y^{*}$ under the WPC contract $w$, the corresponding DR is above the predetermined upper bound $\gamma$. Thus, if we could modify the WPC contract such that the DR does not increase while the ordering quantity increases, then this contract would realize the SC profit maximization. Hence, we modify the WPC contract as a hybrid version of a WPC contract and a return contract, which is defined in the following.

Definition 2. 

The modified WPC contract$w$satisfies that

(i) 

The supplier supplies the retailer at a wholesale price w;

(ii) 

The supplier supplies at most$y^{*}$, and allows the retailer to return all the unsold products with a full refund.

Due to the full refund being allowed under the modified contract $w$ in Definition 2, when the ordering quantity $y$ increases, the sold inventory will boost the retailer’s profit but the unsold inventory will not increase the cost. Therefore, the modified WPC contract satisfies that the DR does not increase while the inventory increases.

We now show that for the case $0<\gamma \le \tilde{\gamma }$, the modified WPC contract $w$ can ensure the SC profit maximization. For preparation, denote the SC member $i$’s expected profit under the modified contract using $u_i^{mwpc}(y)$, $i=s,r$. We also denote two parameters

$$\lambda =\frac{u_s^{wpc}(y_w)}{u_r^{wpc}(y_w)} \mathrm{and} \tilde{w}=\frac{\lambda }{1+\lambda }(p^{*}-s)+\frac{1}{1+\lambda }\frac{(c-s)y^{*}}{E(y^{*}\wedge \epsilon )}+s$$

Later, we will show that under the modified WPC contract $\tilde{w}$, each SC member profit can exceed that under the WPC contract $w$.

Proposition 6. 

For the case with$0<\gamma \le \tilde{\gamma }$, under any modified WPC contract$w\in (c,v)$, if the equilibrium solution exists, then the SC profit maximization is realized. In addition, the DR-averse retailer’s ordering quantity$y=y^{*}$. Moreover, each SC member’s profit under the modified WPC$\tilde{w}$is greater than that under the WPC$w\in (c,v)$.

Proposition 6 states that the modified WPC $w\in (c,v)$ could be designed to achieve SC profit maximization. Moreover, the expected profit of each SC member under the modified WPC $\tilde{w}$ is greater than that under the WPC $w$. In addition, after the WPC contract $w$ is replaced with the modified WPC $\tilde{w}$, the ordering quantity increases from $y_w$ to $y^{*}$, while the DR does not exceed $\gamma$. This is because the modified WPC $\tilde{w}$ allows the retailer to return the unsold inventory back to the supplier with a full refund. Hence, it eliminates the retailer’s stock risk although the order increases.

In the presence of the DR aversion, Gan et al. [4] also consider a modified WPC to ensure that the SC profit is maximized. Yet, their contract is not feasible for our problem with the SCB. The modified WPC in the literature [4] allows a part of unsold products to be returned with a full refund only when the ordering quantity exceeds some predetermined level. Under their contract, they show that the retailer’s inventory decision also maximizes the SC profit, and the retailer’s DR equals $\gamma$. However, when their contract is adopted in our model and the retailer’s ordering quantity equals the quantity $y^{*}$ maximizing the SC profit, the DR exceeds the predetermined risk upper bound $\gamma$. That is, their contract cannot achieve the SC maximization for our model. In fact, when the retailer’s ordering quantity increases from $y_w$ to $y=y^{*}$, the customers’ strategic behavior leads the related selling price to decrease from $p_w$ to $p^{*}$ in our discussion. However, the price is a given external variable and is not changed with the inventory decision in the literature [4]. This naturally leads to the conclusion that the retailer’s actual profit in our problem does not exceed that in ref. [4]; hence, the DR in our model exceeds that in the literature [4]. To solve this problem, we design the modified WPC contract to allow all the unsold products to be returned, which can further lower the retailer’s stock risk. This leads the retailer’s actual profit to still be increased though the selling price decreases from $p_w$ to $p^{*}$. Hence, the related DR is lowered. Note that Propositions 5 and 6 show how to design contracts which maximize the SC profit for different DR aversion. Hence, they answer the second half of the second research problem in the Introduction.

4. Conclusions

We investigate a DR-averse retailer’s two-phase newsvendor problem with SCB. For this problem, we derive the equilibrium price and stock policy, and based on this policy, we test the effects of the SCB and the DR aversion and derive three interesting results. First, for the retailer faced with strategic customers, a higher DR aversion would lead to a low ordering quantity and a high selling price. Second, if the DR aversion is high, the retailer’s profit first decreases and then increases with the DR aversion; otherwise, the expected profit is not affected. Third, the SCB could mean that the DR constraint improves the retailer’s profit if the DR aversion is neither too low nor too high. Otherwise, the retailer’s profit is not increased.

We also extend the model to a decentralized SC. Under this case, we test the effects of DR aversion when customers are strategic, and some management implications are obtained as below. When the DR aversion is low, the decentralized SC profit could dominate that in the centralized SC. Moreover, there exists a WPC ensuring the SC profit maximization. When DR aversion is high, the decentralized SC profit under any WPC cannot exceed that of the centralized case. Moreover, any WPC cannot ensure the SC profit maximization. Fortunately, we can construct a modified WPC (i.e., a hybrid version of a WPC plus a return contract) to ensure the SC profit maximization.

Future research can extend this study to the multi-retailer case. This case is common in actual operation. Hence, it would be interesting to discuss how the number of the retailers influences the RA averse retailer’s decisions and profit with the SCB. Additionally, we also can explore whether the reverse influence of the strategic customer behavior could be alleviated when multiple DR-averse retailers are considered.

One limitation to this research is the assumption that all the customers’ valuations are the same. In fact, customer valuation varies with many factors, including age, sex and taste. However, when the heterogenous customer valuations are considered, each customer has a different reservation price. This would complicate the equilibrium solution and the subsequent discussion. Therefore, it remains an unsettled issue to analyze the newsvendor problem with the DR-averse retailer and strategic customers when the heterogenous customer valuations are considered.