Strategic Inventory Management with Private Brands: Navigating the Challenges of Supply Uncertainty

Abstract

In the context of globalized and complex supply chains, supply uncertainty occurs frequently. To reduce dependence on suppliers, retailers often consider holding strategic inventory and introducing private brands. To explore the relationship between private brands and strategic inventory strategies, and to determine the optimal strategic decisions, this paper constructs a two-stage supply chain model. Using game theory methods, we calculate the equilibrium outcomes of the supply chain under two scenarios: one with only national brands and the other with the introduction of private brands. The main findings are as follows. First, we identify the optimal decisions for both suppliers and retailers in each scenario. The influencing factors include perceived quality, inventory costs, and supply stability. Second, we find that there are constraints for retailers to activate strategic inventory, but these constraints are less restrictive when private brands are introduced. Finally, introducing private brands benefits retailers in implementing strategic inventory, although the extent of this impact depends on the conditions under which the strategic stockpile is implemented. These findings fill the gap in the existing literature on the impact of private brand introductions on strategic inventory under supply uncertainty and highlight valuable implications for business decision-makers.

1. Introduction

In today’s deeply integrated global economy, the compounded impacts of geopolitical conflicts, trade barrier restructuring, frequent extreme weather events, and disruptive technological changes have significantly amplified the vulnerabilities of supply chains [1,2]. From the manufacturing shutdowns triggered by semiconductor shortages to global inflation caused by the energy crisis, supply uncertainty has become a constant risk in business operations [3,4]. This is particularly true for the retail industry, which often faces the dilemma of being “out of stock” due to supply disruptions, as it heavily relies on external suppliers for national brand products (labeled NBs) [5,6]. For example, during the pandemic, many supermarket shelves were left empty, with only a limited supply of local products available.

In the face of supply uncertainty, traditional response strategies, such as excessive stockpiling of strategic inventory, may partially alleviate the risk of supply disruptions [5,6]. Researches show that retailers can mitigate the risk of supply instability and enhance supply chain resilience through strategic inventory reserves [7]. Furthermore, strategic inventory can also improve the retailer’s bargaining power with suppliers and reduce dependence on them [8]. However, in the real business environment, adopting strategic inventory comes with high storage costs and capital occupation pressures, posing challenges for retailers in making informed decisions.

In addition to strategic inventory, another option for retailers to enhance their bargaining power with suppliers is the introduction of private brand products (labeled PBs) [9,10]. PBs, also known as private labels or store brands, are products that carry a brand name chosen by the retailer and are completely owned, controlled, and marketed by the retailer. In practice, PBs are typically segmented by product category, for instance, grocery, non-food, and apparel. Research shows that, compared to NBs, PBs offer advantages such as higher cost-effectiveness, flexible pricing, and a reliable supply [11,12]. They can help retailers increase profit margins, boost customer loyalty, and reduce dependency on suppliers [13]. In recent years, PBs have developed rapidly worldwide. According to PLMA’s report, in the U.S., PBs have achieved all-time record highs in both unit and dollar shares in 2023. For example, PBs’ dollar sales increased by 4.7% to reach approximately USD 236.3 billion, with a significant presence across nearly all food and non-food categories. Moreover, the report highlights that U.S. consumers—driven by high market awareness—exhibit a strong acceptance of PBs (https://www.plma.com/about_industry/research_reports_publications/consumer-research/plmas-2024-private-label-report accessed on 20th December 2024). Products like Walmart’s “Great Value”, Tesco’s “Tesco Value”, and Marks & Spencer’s “M&S” have become core competencies in attracting consumers. However, some scholars have found that the introduction of PBs may lead to a deterioration in the retailer–supplier relationship, which, in turn, affects retailers’ profits [14]. Although existing studies have extensively explored strategic inventory and PBs, most focus independently on the impact of each strategy on retailers’ profitability. There is a theoretical gap regarding the interaction between PBs and strategic inventory which limits the ability to fully guide retailers who must simultaneously make decisions about both strategies in practice. The coexistence of strategic inventories and PBs is common in practice, as exemplified by Wumart. This leading Chinese supermarket chain maintains a large amount of strategic inventory without any supply chain disruption, while also operating PBs such as “Liangshiji”. When the COVID-19 pandemic broke out, Wumart ensured a stable supply of goods. Therefore, studying the impact of PBs on retailers’ strategic inventory under conditions of supply instability holds significant practical importance. This paper aims to address the gaps in existing theories by investigating the following questions:

• Under what conditions will a retailer stock strategic inventory?
• How does the introduction of PBs impact the retailer’s strategic inventory decisions?
• What impact does the introduction of PBs have on the various entities in the supply chain under conditions of supply uncertainty?

To address these questions, we construct a game model with supply uncertainty. By calculating the equilibrium decisions of the supplier and retailer under the scenarios of having only NBs and introducing PBs, and then comparing and analyzing the results numerically, we found the following. First, holding strategic inventory is not always beneficial for retailers; the conditions for activating strategic inventory depend on inventory costs, supplier wholesale prices, and supply stability. Second, the introduction of PBs allows retailers to make flexible strategic choices. When strategic inventory cannot be implemented, PBs can help retailers secure profits and maintain supply chain operations. Finally, retailers’ decisions should be scientifically formulated based on the perceived quality of PBs, inventory costs, and supply stability.

The main contribution of this paper is to fill the theoretical gap in research on the impact of introducing PBs on strategic inventory in the context of supply uncertainty. It also reveals how PBs influence strategic inventory decisions. The main conclusions provide valuable managerial insights and offer guidance for real-world retail businesses with regard to making informed decisions.

The rest of the paper is organized as follows. Section 2 presents a comprehensive literature review. Section 3 describes the construction problem, develops the model framework, and outlines the basic assumptions. Section 4 computes the equilibrium solutions of the model for different scenarios. Section 5 provides a comparative model analysis and numerical simulations. Section 6 discusses how our results answer the research questions. Finally, Section 7 concludes the paper with managerial implications and future perspectives.

2. Literature Review

This paper investigates the impact of PBs on retailers’ strategic inventories in the face of supply uncertainty. Two main types of literature are closely related to this paper: strategic inventory in supply chains and PB introduction.

2.1. Strategic Inventory

Holding strategic inventory in supply chains is a business strategy where companies acquire goods over time and retain inventory to mitigate supply chain disruptions and production fluctuations [15]. Anand et al. found that retailers holding inventory can lower the average wholesale price, alleviating the double marginalization effect [16]. Arya et al. investigated the impact of manufacturers’ rebate contracts on strategic inventory and found that these contracts suppress retailers’ strategic inventory behavior but bring more profits to supply chain members [17]. Arya et al. explored the impact of strategic inventory on a supply chain consisting of a single supplier and a retailer with multiple divisions and elaborated on enterprise inventory management in centralized versus decentralized decision making [18]. Roy et al. investigated the impact of strategic inventory on supply chain members under the condition of unobservability of retailer inventory levels, and the results indicate that retailers may voluntarily disclose their inventory-level information [19]. Li et al. examined strategic inventory decisions in competitive supply chains and found that intensified competition may induce retailers to order more inventory [20]. Guan et al. found that manufacturers’ channel encroachment suppresses retailers’ strategic inventory behavior [21]. Martin et al. examined the impact of retailers using strategic inventory when product quality declines [22]. Graves et al. discussed the role of safety stock as a crucial component of supply chain resilience strategies. They highlighted that increasing safety stock levels for critical items can help buffer against demand uncertainty and supply disruptions, especially in the wake of major supply chain disturbances such as those observed during the COVID-19 pandemic. Their findings support the notion that holding additional inventory can serve as a resilience mechanism [23].

In recent years, scholars have examined various supply chain structures related to the retail industry in the context of strategic inventory issues. Saha et al. examined the impact of strategic inventory [24]. The results show that all supply chain members can achieve higher profits if the holding cost is within a certain range, allowing the retailer to maintain strategic inventory, and, while cooperation between two manufacturers can lead to better outcomes without SI, this is not always the case when the retailer holds strategic inventory. Dong et al. employed a two-period dynamic model to explore the impact of manufacturers’ strategic inventories on supply chain decisions and profits. They found that the manufacturer may hold a positive inventory level at equilibrium, which influences the retailer to carry more strategic inventory at a higher wholesale price in the first period. While the manufacturer’s strategic inventory always hurts the retailer’s profit, it may enhance channel profits, consumer surplus, and social welfare [25]. Yang et al. constructed a two-period dynamic model to explore the relationship between supplier encroachment and the retail platform’s strategic inventory withholding behavior. The results show that the retail platform’s strategic inventory decisions depend on the holding cost without encroachment and are moderated by the commission rate when the holding cost is intermediate with encroachment [26]. Most of the above studies are based on situations where the supply is stable and strategic inventories are constructed mainly for bargaining purposes with suppliers. This paper introduces the strategic inventory problem into an environment of unreliable supplies and considers the impact of PB strategies in a fashion which is more in line with real-world scenarios.

2.2. PB Introduction

Supply chain management, with the introduction of PBs, is another important area of research. Earlier scholars mainly focused on the impact aspects of introducing PBs. For example, Mills found that the introduction of PBs by retailers can not only achieve higher profits, but also mitigate the double marginal effect in the supply chain [27]. Chintagunta et al. found that the introduction of PBs has a significant impact on the supply chain pricing and profitability levels by considering price elasticity factors through an empirical study of oatmeal category products from Dominicks Finer Foods in the U.S. [28]. Mandhachitara et al.’s research shows that, in developed countries such as the U.S., PBs are more widely accepted, owing to the higher level of market awareness among consumers in these areas [29]. Wu et al. analyzed 364 articles covering 43 countries; notably, 82 articles involved U.S. data—the highest among the regions—while Spain accounted for 54 articles, making it the second largest. Moreover, cross-country studies on PBs predominantly used data from the U.S. and Europe [30]. Ru et al. found that the introduction of PBs can produce a win–win situation in a retailer-dominated situation [31]. In recent years, academics have been focusing on the impact of product quality on PB introduction. Choi et al. considered the impact of PB introduction on the supply chain under different quality positioning and showed that retailers can benefit manufacturers when they introduce high-quality PBs [32]. Hara et al. studied the impact of PB introduction when a retailer cooperates with an NB supplier and suggested that the introduction of high-end PBs can be a win–win situation for retailers [33]. Li et al. studied how retailers should make PB quality decisions [34].

Meanwhile, with the transformation of retail business models, scholars have paid attention to channel changes. Li et al. considered the model of introducing PBs under different sales models under the platform model [35]. Xu et al. investigated the logistics and distribution of fresh PBs under the platform model and found that the introduction of fresh PBs is conducive to promoting the acceptance of platform logistics services by merchants [11]. Li et al. explored the situation where manufacturers create new products in response to the invasion of platform PBs and found that the introduction of new low-priced products by manufacturers is conducive to market competition [14]. Huang et al. analyzed the optimal pricing strategy under the introduction of PBs based on the return perspective [13]. Shen et al., based on the perspective of different agency contracts, explored the relationship between the perceived value of PBs and the introduction strategy [36]. However, most of the above literature has not yet considered the category problem of PBs, nor has it focused on the product-ordering problem under the retailer–supplier competition and gaming which cannot provide comprehensive support for the enterprise’s realistic decision making. Balasubramanian et al. employed a two-period game-theoretic framework to investigate the impact of PB competition on a retailer’s strategic inventory decisions. The analysis revealed that it is never optimal for the retailer to hold PBs as strategic inventory, and, while PB competition can sometimes worsen the retailer’s situation, low holding costs can make strategic inventory and PB competition complementary and beneficial to the retailer [37].

Summarizing the above literature, it can be observed that the introduction of PBs has a significant impact on the supply chain operation. However, most of the existing studies have not yet considered supply chain disruption scenarios or the structural changes in product ordering introduced by strategic inventory, a fact which limits their applicability for real-world decision making by enterprises. This paper examines how strategic inventory changes when retailers introduce PBs in the context of supply uncertainty, providing robust theoretical support for enterprises with regard to making supply chain decisions within a complex and dynamic market environment.

3. Problem Description and Model Setup

Here, we consider a two-stage supply chain model consisting of a supplier and a retailer. The supplier sells NBs to the retailer at wholesale price, and the retailer subsequently sells it to consumers at a retail price (referred to as Scenario N). Under the Stackelberg game, the supplier acts as the leader, and the retailer acts as the follower. Both participants are risk-neutral and aim to maximize profits in their decision making, an approach which is consistent with [38,39,40].

We first outline the timeline of the two-stage supply chain. In the first stage, the supply is stable. The supplier determines the wholesale price of NBs, and the retailer purchases a quantity of $K(K>0)$ units as strategic inventory, with a holding cost of $h$ per unit. The holding cost refers to the storage and disposal costs incurred for the strategic inventory and is consistent with [1,21,24]. In the second stage, the retailer orders a quantity of $K$ for selling, but unpredictable supply disruptions may occur, such as production capacity interruptions caused by diseases, natural disasters, geopolitical issues, or the supplier prioritizing other channels. Supply uncertainty can be represented in the model as a package of multiple scenarios, including stochastic production, all-or-nothing supply, and stochastic capacity [40]. In this paper, we assume that supply uncertainty is consistent with an all-or-nothing supply model [41,42,43]. If a disruption occurs, the retailer will not receive any products. For example, during peak sales seasons, NB suppliers may prioritize their own online channels, leading to supply disruptions for retailers. Consistent with the literature [43], we assume that the probability of a normal supply in the second stage is $\lambda (0<\lambda <1)$, which serves as an indicator of the supply stability. Thus, the probability of a supply disruption is $1-\lambda$.

In this paper, we investigate the impact of PBs on the supply chain (referred to as Scenario P). In our study, we focus specifically on the PB category that comprises products which are easy to store and not prone to spoilage. This segment is particularly relevant for our analysis of strategic inventory decisions under supply uncertainty, as these products allow for more predictable inventory management. The retailer can introduce PBs and we assume that the supply of PBs is always stable due to the significant control from the retailer. We assume that the market consists of one unit of consumers. Consumers exhibit heterogeneous quality preferences, which influence their willingness to pay $V$. Consistent with the assumption made by Ru et al. [31], Guo et al. [44], Ru et al. [45], Li et al. [10], and Alan et al. [46], we assume that consumer willingness to pay follows a uniform distribution over the interval [0, 1]. We set the perceived quality of the NBs to 1 and denote the perceived quality of the PBs as $\mu (1/2<\mu <1)$. This is because PBs are still in the early stages of market presence compared to NBs and lag behind in terms of technology, production experience, and quality control. This hypothesis is very common in PB-related research, such as [47,48,49].

The sequence of the game in two scenarios is shown in Figure 1. First, the supplier decides the wholesale price of NBs and the retailer decides the strategic inventory level. Second, the retailer decides the order quantity of NBs, and the order quantity of PBs if PBs are introduced. Finally, the consumer chooses to buy the product based on the actual supply of the product and products will be sold.

When there are only NBs on the market, for a given price, the consumer’s utility after purchasing NBs is $u_1^N=V-p_1^N$. When both NBs and PBs exist on the market, the utility of consumers purchasing NBs is $u_1^P=V-p_1^P$, while the utility of consumers purchasing PBs is $u_2^P=\mu V-p_2$. Thus, the inverse demand function of two products under different scenarios and cases can be summarized as follows:

$$p_1^{ij}=1-(q_1^{ij}+K^i)-\mu q_2^{ij}$$

(1)

$$p_2^j=\mu (1-(q_1^{Pj}+K^P)-q_2^j)$$

(2)

where $i=N,P$ denotes scenario N and scenario P, $k=D,N$ denotes the case of supply disruption and not, and $q_2^{Nj}=0$ indicates no introduction of PBs. Similar inverse demand functions for PBs and NBs have been adopted in the literature, such as [32,44,50]. All the notations are summarized in

Table 1. Notations and definitions.

Notations	Definition
Indices
$t$	Subscript, index of product, $t=1$ for NBs and $t=2$ for PBs
$i$	Superscript, index of scenario, $i=\{N,P\}$
$j$	Superscript, index of case, $j=\{D,N\}$, e.g., $D$ for disruption
Parameters
$h$	The unit holding cost for strategic inventory
$\lambda$	Supply stability: the probability of supply stabilization
$\mu$	The perceived quality of PBs
Decision variables
$q_t^{ij}$	The order quantity for product $t$ in scenario $i$ with case $j$
$w^i$	The wholesale price for NBs in scenario $i$
$K^i$	The strategic inventory in scenario $i$
Dependent variables
$p_t^{ij}$	The retail price for product $t$ in scenario $i$ with case $j$
$E^i$	The consumer surplus in scenario $i$
${\pi }_M^i$	The supplier’s profit in scenario $i$
${\pi }_R^i$	The retailer’s profit in scenario $i$

. Without loss of generality, we assume that the marginal cost of production is equal to zero for all products, an approach which is the same as that adopted in other studies [11,14,51].

4. Model Solution and Equilibrium Analysis

The retailer can choose whether or not to introduce PBs, and, based on this, there are two scenarios. In this section, we explore the equilibrium results under different scenarios. All proofs are presented in Appendix A.

4.1. Only NBs (Scenario N)

When no PBs exist, in case N, the retailer’s sales volume is $K^N+q_1^{NN}$; in case D, the retailer’s sales volume is $K^N$. Thus, the retailer’s and supplier’s profit function are:

$$\begin{array}{cc}\hfill \max {\pi }_R^N(q_1^{NN},K^N)& =\lambda {\pi }_R^{NN}+(1-\lambda ){\pi }_R^{ND}\hfill \\ & =\lambda (p_1^{NN}(q_1^{NN}+K^N)-w^N{q}_1^{NN})+(1-\lambda )p_1^{ND}{K}^N-h{K}^N\hfill \end{array}$$

(3)

$$\max {\pi }_M^N(w^N)=\lambda q_1^{NN}{w}^N+w^N{K}^N$$

(4)

Using backward recursion, the optimal solution for the retailer under unconstrained conditions is obtained as follows: $(q_1^{NN}(w^N),K^N(w^N))=\left({\displaystyle \frac{h}{2-2\lambda }},{\displaystyle \frac{h+\lambda -\lambda w^N+w^N-1}{2(\lambda -1)}}\right)$. The constraint for the strategic inventory is $K\ge 0$, and, if, $K=0$ the retailer will not maintain strategic inventory. Therefore, the sign of $h+\lambda -\lambda w^N+w^N-1$ affects the implementation of the strategic inventory strategy, which, in turn, influences the structure of the supply chain. By discussing the range of values for $w^N$, we can have Lemma 1 as follows.

Lemma 1.

When PBs are not introduced, for a given $w^N$, the optimal solutions for the retailer are:

$$(q_1^{NN}(w^N),K^N(w^N))=\left\{\begin{array}{ll}\left({\displaystyle \frac{h}{2-2\lambda }},{\displaystyle \frac{h+\lambda -\lambda w^N+w^N-1}{2(\lambda -1)}}\right)& if{w}^N<{\displaystyle \frac{h+\lambda -1}{\lambda -1}}\\ \left({\displaystyle \frac{1-w^N}{2}},0\right)& if{w}^N\ge {\displaystyle \frac{h+\lambda -1}{\lambda -1}}\end{array}\right.$$

(5)

Lemma 1 indicates that, when only NBs are available, the retailer’s optimal order quantity is solely dependent on inventory costs and supply stability. Furthermore, if the wholesale price set by the supplier is too high, the retailer will not maintain strategic inventory. Substituting Lemma 1 and Equation (5) into Equation (4) yields equilibrium results.

Lemma 2.

When PBs are not introduced, there exists a $h^N$ such that the equilibrium prices, quantities, and profits are given in

Table 2. Equilibrium results under scenario N.

Items	Equilibrium Results
	$h<h^N$	$h\ge h^N$
$w^N$	${\displaystyle \frac{1-h}{2}}$	${\displaystyle \frac{1}{2}}$
$q_1^N$	${\displaystyle \frac{h}{2-2\lambda }}$	${\displaystyle \frac{1}{4}}$
$K^N$	${\displaystyle \frac{h\lambda +h+\lambda -1}{4(\lambda -1)}}$	0
$p_1^{NN}$	${\displaystyle \frac{3-h}{4}}$	${\displaystyle \frac{3}{4}}$
$p_1^{ND}$	${\displaystyle \frac{(h-3)\lambda +h+3}{4-4\lambda }}$	0
$E^N$	${\displaystyle \frac{h+3}{4}}$	${\displaystyle \frac{4-3\lambda }{4}}$
${\pi }_M^N$	${\displaystyle \frac{1}{8}}{(h-1)}^2$	${\displaystyle \frac{\lambda }{8}}$
${\pi }_R^N$	${\displaystyle \frac{-h(3h\lambda +h+2\lambda -2)+\lambda -1}{16(\lambda -1)}}$	${\displaystyle \frac{\lambda }{16}}$

, where $h^N=1-\sqrt{\lambda }$.

Lemma 2 demonstrates that the retailer’s strategic inventory strategy is closely related to inventory costs. Only when inventory costs are sufficiently low will the retailer hold strategic inventory. Furthermore, by maintaining strategic inventory, the retailer weakens the supplier’s monopoly position, allowing the retailer to flexibly adjust the ordering strategy during the selling period based on the supplier’s reliability.

Proposition 1.

When PBs are not introduced, the values of the decision variables, consumer surplus, and profits vary with $h$ as follows:

• When $h<h^N$, $\partial w^N/\partial h<0$, $\partial q_1^N/\partial h>0$, $\partial K^N/\partial h<0$, $\partial E^N/\partial h>0$, and $\partial {\pi }_M^N/\partial h>0$
• When $h^{N1}<h<h^N$, $\partial {\pi }_R^N/\partial h>0$; when $h^{N1}>h$, $\partial {\pi }_R^N/\partial h<0$, where $h^{N1}={\displaystyle \frac{1-\lambda }{3\lambda +1}}$ decrease with $\lambda$.

Proposition 1 indicates that, under moderate inventory costs, as inventory costs increase, the supplier will lower the wholesale price. This is because the supplier is concerned that higher inventory costs will lead the retailer to reduce order quantities, thereby resulting in lower profits. As a result, the retailer will increase the order quantity during the selling period but will reduce the strategic inventory reserve. This situation, ultimately, leads to an increase in the supplier’s profit and consumer surplus. Interestingly, we found that, when inventory costs are within a certain range, the retailer’s profit increases as inventory costs rise. However, when inventory costs fall below a critical threshold, the retailer’s profit decreases as inventory costs increase. This critical value is negatively correlated with supply stability.

4.2. Both NBs and PBs (Scenario P)

When both PBs and NBs are present on the market, in case N, the retailer’s sales volume for NBs is $K^P+q_1^{PN}$, while the volume for PBs is $q_2^N$; in case D, the retailer’s sales volume for NBs is $K^P$ and the volume for PBs is $q_2^N$. In scenario P, the retailer’s and supplier’s profit functions are:

$$\begin{array}{cc}\hfill \max {\pi }_R^P(q_1^{PN},q_2^P,K^P)& =\lambda (p_1^{PN}(q_1^{PN}+K^N)-w^P{q}_1^{PN})+(1-\lambda )p_1^{PD}{K}^P-(w^P+h)K^P\hfill \\ & +\lambda p_2^{PN}{q}_2^P+(1-\lambda )p_2^{PD}{q}_2^P\hfill \end{array}$$

(6)

$$\max {\pi }_M^P(w^P)=\lambda q_1^{PN}{w}^P+w^P{K}^P$$

(7)

Using backward recursion, the optimal solution for the retailer under unconstrained conditions is obtained as follows:

$$(q_1^{PN}(w^P),q_2^P(w^P),K^P(w^p))=\left({\displaystyle \frac{h}{2-2\lambda }},{\displaystyle \frac{h+w^P}{2-2\mu }},{\displaystyle \frac{h\lambda \mu -h+\lambda \mu -\lambda -\mu +\lambda w^P-w^P+1}{2(\lambda -1)(\mu -1)}}\right)$$

Similar to Lemma 1, by discussing the range of values for $w^P$, we can have Lemma 3.

Lemma 3.

When PBs are introduced, for a given $w^P$, the optimal solutions for the retailer are:

$$q_1^{PN}(w^P)=\left\{\begin{array}{ll}{\displaystyle \frac{h}{2-2\lambda }}& if{w}^P<{\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\\ {\displaystyle \frac{\mu +w^P-1}{2\lambda \mu -2}}& if{w}^P\ge {\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\end{array}\right.$$

(8)

$$q_2^P(w^P)=\left\{\begin{array}{ll}{\displaystyle \frac{h+w^P}{2-2\mu }}& if{w}^P<{\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\\ {\displaystyle \frac{1+\left(-1+w^P\right)\lambda }{2-2\lambda \mu }}& if{w}^P\ge {\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\end{array}\right.$$

(9)

$$K^P(w^P)=\left\{\begin{array}{ll}{\displaystyle \frac{h\lambda \mu -h+\lambda \mu -\lambda -\mu +\lambda w^P-w^P+1}{2(\lambda -1)(\mu -1)}}& if{w}^P<{\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\\ 0& if{w}^P\ge {\displaystyle \frac{-\lambda h\mu +h-\lambda \mu +\lambda +\mu -1}{\lambda -1}}\end{array}\right.$$

(10)

Lemma 3 indicates that, even when the retailer introduces PBs, if the supplier’s wholesale price is too high, similar to Lemma 1, the retailer will also not maintain strategic inventory. Furthermore, regardless of whether the retailer initially holds strategic inventory, as the supplier’s wholesale price increases, the retailer will produce more PBs. By substituting Lemma 3 and Equations (8)–(10) into Equation (7), we can have the equilibrium results under scenario P as follows.

Lemma 4.

When PBs are introduced, there exists a $h^P$ such that the equilibrium prices, quantities, and profits are given in

Table 3. Equilibrium results under scenario P.

Items	Equilibrium Results
	$h<h^P$	$h\ge h^P$
$w^P$	${\displaystyle \frac{1}{2}}\left(1-h-\mu \right)$	${\displaystyle \frac{1-\mu }{2}}$
$q_1^P$	${\displaystyle \frac{h}{2-2\lambda }}$	${\displaystyle \frac{1-\mu }{4-4\lambda \mu }}$
$q_2^P$	${\displaystyle \frac{h-\mu +1}{4-4\mu }}$	${\displaystyle \frac{2-\lambda -\lambda \mu }{4-4\lambda \mu }}$
$K^N$	${\displaystyle \frac{2h\lambda \mu -h\lambda -h+\lambda \mu -\lambda -\mu +1}{4(\lambda -1)(\mu -1)}}$	0
$p_1^{NN}$	${\displaystyle \frac{1}{4}}\left(3-h-\mu \right)$	${\displaystyle \frac{3-\mu }{4}}$
$p_1^{ND}$	${\displaystyle \frac{1}{2}}\left(1-h\right)\mu$	0
$p_2^{PN}$	${\displaystyle \frac{h(-\lambda )-h-\lambda \mu +3\lambda +\mu -3}{4(\lambda -1)}}$	${\displaystyle \frac{\mu \left(1+\lambda +\mu -3\lambda \mu \right)}{4-4\lambda \mu }}$
$p_2^{PD}$	${\displaystyle \frac{(h-1)\lambda \mu +\mu }{2-2\lambda }}$	${\displaystyle \frac{-4+2\mu +3\lambda \mu -\lambda {\mu }^2}{4\left(-1+\lambda \mu \right)}}$
$E^P$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$
${\pi }_M^P$	${\displaystyle \frac{{(h+\mu -1)}^2}{8(1-\mu )}}$	${\displaystyle \frac{\lambda {\left(-1+\mu \right)}^2}{8-8\lambda \mu }}$
${\pi }_R^P$	${\displaystyle \frac{A}{16(\lambda -1)(\mu -1)}}$	${\displaystyle \frac{\lambda +4\mu -2\lambda \mu -3\lambda {\mu }^2}{16-16\lambda \mu }}$

$A=1-2h+h^2-\lambda +2h\lambda +3h^2\lambda +2\mu +2h\mu -2\lambda \mu -2h\lambda \mu -4h^2\lambda \mu -3{\mu }^2+3\lambda {\mu }^2$.

, where $h^P=1-\mu -\sqrt{{\displaystyle \frac{-\lambda +3\lambda \mu -3\lambda {\mu }^2+\lambda {\mu }^3}{-1+\lambda \mu }}}$.

Lemma 4 indicates that, after the retailer introduces PBs, the strategic inventory strategy is still dependent on inventory costs. Only when inventory costs are sufficiently low (under $h^P$) will the retailer maintain strategic inventory. Notably, after the introduction of PBs, consumer surplus tends to stabilize, suggesting that the presence of PBs significantly enhances consumer choice and provides a stable purchasing environment for consumers.

Proposition 2.

When PBs are introduced, the values of the decision variables, consumer surplus, and profits vary with $h$ as follows:

• When $h<h^P$, $\partial w^P/\partial h<0$, $\partial q_1^P/\partial h>0$, $\partial q_2^P/\partial h>0$, $\partial K^P/\partial h<0$, and $\partial {\pi }_M^N/\partial h>0$.
• There exist a ${\lambda }^P$ and $h^{P1}$, when $\mu =3/4$ and $h>h^{P1}$; $\mu <3/4$, $\lambda <{\lambda }^P$, and $h<h^{P1}$ or $\lambda >{\lambda }^P$ and $h>h^{P1}$; $\mu <3/4$ and $h>h^{P1}$ or $\lambda >{\lambda }^P$ and $h>h^{P1}$, where $\partial {\pi }_R^P/\partial h>0$. Otherwise, $\partial {\pi }_R^P/\partial h<0$, where ${\lambda }^P={\displaystyle \frac{1}{-3+4\mu }}$ and $h^{N1}={\displaystyle \frac{-1+\lambda +\mu -\lambda \mu }{-1-3\lambda +4\lambda \mu }}$.

As indicated by Proposition 2, similar to Proposition 1, under moderate inventory costs, as inventory costs increase, the supplier will lower the wholesale price. The retailer will then increase the order quantities for both products and reduce the strategic inventory reserve. Additionally, in the case of introducing PBs, the strategic inventory strategy cannot guarantee an increase in the retailer’s profit. There exists a critical inventory cost threshold, which is related to the perceived quality of PBs and supply stability, which influences the variation in the retailer’s profit.

Proposition 3.

When PBs are introduced and the strategic inventory strategy is implemented, the decision variables and profits vary with the perceived quality of PBs, as follows: $\partial w^P/\partial \mu <0$, $\partial q_1^P/\partial \mu >0$, $\partial q_2^P/\partial \mu >0$, $\partial K^P/\partial \mu <0$, $\partial {\pi }_M^P/\partial \mu <0$, $\partial {\pi }_R^P/\partial \mu >0$.

Proposition 3 indicates that the perceived quality of products plays a crucial role in shaping a supply chain. As the consumer-perceived quality of PBs improves, the competitive pressure on NBs intensifies, leading to a decline in the wholesale price of NBs as the supplier attempts to maintain its market position. Simultaneously, both the retailer’s PBs and the NBs experience higher order quantities, reflecting a shift in consumer demand toward PB products. From a profitability perspective, the supplier’s profit declines due to price reductions and intensified competition, while the retailer benefits from increased sales and improved profit margins as the PBs become more competitive. Additionally, the retailer reduces its strategic inventory holdings of NBs as the reliance on a single NB supplier diminishes. Overall, an increase in PB perceived quality enhances the retailer’s bargaining power, reduces dependence on the NB supplier, and reshapes inventory and pricing strategies, ultimately influencing supply chain equilibrium.

5. Comparison Analysis and Numerical Simulation

In this section, we discuss the comparative equilibrium results across different scenarios, exploring the impact of strategic inventory and the introduction of PBs on the supply chain. We use $△$ as the difference value between equilibrium results across different scenarios. For example, the optimal profit difference between scenario N and scenario P for the retailer is given by $△{\pi }_R={\pi }_R^N-{\pi }_R^P$; if $△{\pi }_R<0$, it means that the introduction of PBs is better for the retailer’s profit. We also provide numerical simulations to support these findings.

5.1. The Impacts on Strategic Inventories

According to Lemma 2 and Lemma 4, regardless of whether the retailer introduces PBs, there exists a critical inventory cost threshold under both scenarios that causes the retailer not to maintain a strategic inventory. By comparing the critical inventory cost thresholds under these two scenarios, we can determine the extent of the limitations on the activation of the strategic inventory.

Proposition 4.

When the retailer introduces PBs, it becomes easier for the retailer to implement a strategic inventory strategy, as $h^P<h^N$. Furthermore, the optimal level of strategic inventory reserve is lower.

Proposition 4 indicates that the introduction of PBs enables the retailer to flexibly respond to supply chain disruptions. When inventory costs are high, the retailer can choose to increase the quantity of PBs to replace the strategic inventory, thereby maintaining a stable profit level despite the unstable supply. When inventory costs are low, the retailer can reserve strategic inventory and supplement it with the production of some PBs to maximize profit. Additionally, the introduction of PBs can reduce the retailer’s strategic inventory purchase volume, thereby alleviating the pressure on the retailer’s capital accumulation.

According to Younis et al.’s empirical research data on 800 questionnaires, the average perceived quality of PBs is roughly 0.8 [52]. Based on that, we assume the perceived quality of PBs to be 0.8 and observe the change in strategic inventory under different supply stability scenarios. Figure 2 reveals two key insights. First, when supply stability is held constant, the introduction of PBs significantly reduces the level of strategic inventory required. Notably, the critical threshold for activating strategic inventory is lower with PBs in place. This indicates that retailers can adopt a more flexible approach to inventory management.

Second, the figure shows that the level of strategic inventory is sensitive to the stability of the supply. As the supply becomes more stable, retailers tend to reduce their strategic inventory holdings. In contrast, under conditions of poor supply stability, the demand for strategic inventory increases sharply, reflecting the need to buffer against higher risks of supply disruption.

Overall, these results suggest that incorporating PBs not only lowers the barrier for activating strategic inventory, but also enables retailers to better adjust their inventory levels in response to varying supply conditions. This more flexible strategic approach could help retailers manage costs more effectively while maintaining service levels in uncertain environments.

Furthermore, Proposition 4 also indicates that, under the same $h$, a retailer operating without PBs would need to hold a larger buffer, thereby incurring higher financing costs. For smaller retailers, who typically face financial constraints such as limited cash flow, we suggest several practical financial strategies to implement strategic inventory more effectively. First, warehouse receipt financing, which uses strategic inventory as collateral, can help secure short-term financing. Second, factoring allows retailers to convert receivables into cash, thereby easing cash flow constraints. Third, negotiating extended payment periods with the suppliers, known as supplier credit terms, can provide additional working capital relief.

5.2. The Impacts on Profit

5.2.1. Supplier’s Profit

Based on Lemma 2, Lemma 4, and Proposition 4, the value of $△{\pi }_M$ depends on the different intervals of $h$, resulting in three distinct forms:

$$△{\pi }_M=\left\{\begin{array}{ll}{\displaystyle \frac{\mu \left(h^2+\mu -1\right)}{8(\mu -1)}}& if0<h<h^P\\ {\displaystyle \frac{1}{8}}{(h-1)}^2+{\displaystyle \frac{\lambda {(\mu -1)}^2}{8\lambda \mu -8}}& if{h}^P\le h<h^N\\ {\displaystyle \frac{\lambda \mu (\lambda +\mu -2)}{8\lambda \mu -8}}& if{h}^N\le h\end{array}\right.$$

(11)

By comparing the sign of $△{\pi }_M$, we can assess the differences in the supplier’s optimal profit across various scenarios.

Proposition 5.

Regardless of the value of $h$, the introduction of PBs will always lead to a decrease in the supplier’s profit, i.e., $△{\pi }_M>0$.

According to Proposition 5 as the retailer introduces PBs, the retailer’s dependence on the supplier decreases, leading to a reduction in the supplier’s profit. We consider the supply chain parameters with four cases as $\mu \in \{0.55,0.8\}$, $\lambda \in \{0.1,0.8\}$ to show the changes in the profit difference for the supplier under different inventory costs and supply stabilities (Figure 3).

As shown in Figure 3, when PBs are not introduced, the supplier’s profit is higher. Moreover, as inventory costs decrease, the profit loss caused by the introduction of PBs becomes more significant for the supplier. The higher the perceived quality of the PBs, the greater the profit loss for the supplier. On the other hand, Proposition 5 also indicates that, when the retailer introduces PBs, the supplier may lower the stability of the supply in order to protect their profit, in turn affecting the retailer’s strategic choices.

5.2.2. Retailer’s Profit

Based on Lemma 2, Lemma 4, and Proposition 4, the value of $△{\pi }_R$ depends on the different intervals of $h$, resulting in three distinct forms:

$$△{\pi }_R=\left\{\begin{array}{ll}{\displaystyle \frac{\mu \left(h^2-3\mu +3\right)}{16(\mu -1)}}& if0<h<h^P\\ {\displaystyle \frac{B}{16\left(-1+\lambda \right)\left(-1+\lambda \mu \right)}}& if{h}^P\le h<h^N\\ {\displaystyle \frac{\left(4+\lambda \left(-2+\lambda -3\mu \right)\right)\mu }{16\left(-1+\lambda \mu \right)}}& if{h}^N\le h\end{array}\right.$$

(12)

where $B=-2h\left(-1+\lambda \right)\left(-1+\lambda \mu \right)-h^2\left(1+3\lambda \right)\left(-1+\lambda \mu \right)-\left(-1+\lambda \right)\left(1-4\mu +\lambda \left(-1+\mu +3{\mu }^2\right)\right)$. By comparing the sign of $△{\pi }_R$, we can assess the differences in the supplier’s optimal profit across various scenarios.

Proposition 6.

Regardless of the value of $h$, the introduction of PBs will always lead to an increase in the retailer’s profit i.e., $△{\pi }_R<0$.

According to Proposition 6, the introduction of PBs can guarantee that the retailer’s profit will always increase. When PBs are introduced, the retailer can build a more resilient supply chain by setting a combination of strategic inventories, order quantities for both NBs and PBs, and by adjusting prices based on supply stability conditions. However, the improvement in profit is not fixed, and the difference is based on the specific inventory costs, perceived quality of PBs, and supply stability. We use numerical simulations to explain this phenomenon; the parameters are the same as those used in Proposition 5. Then, Figure 4 illustrates the interaction between exogenous variables and retailer’s profit change.

Figure 4 shows that, when inventory costs are too high and the retailer is unable to adopt a strategic inventory strategy, the introduction of PBs significantly increases the retailer’s profit level. Furthermore, under the same conditions, the higher the perceived quality of PBs, the greater the increase in the retailer’s profit. Under the condition of introducing PBs of the same quality, if inventory costs are high and supply stability is poor, the profit increase after introducing PBs will be greater.

6. Discussions

In this section, we discuss how our research results answer the questions raised in the previous section.

In addressing the conditions under which a retailer opts to hold a strategic inventory, our analysis indicates that such a decision is economically justified only when the cost structure—specifically, the wholesale price and inventory holding expenses—remains sufficiently low. Our model demonstrates that only under these favorable cost conditions does the benefit of buffering against supply uncertainty outweighs the costs, enabling the retailer to optimize its inventory levels. This optimality is derived by balancing the marginal cost of additional inventory against the anticipated gains from mitigating potential supply disruptions.

The introduction of PBs significantly reshapes the retailer’s inventory strategy. Our findings reveal that the presence of PBs effectively lowers the threshold required to initiate strategic inventory practices. In other words, PBs provide an alternative supply source that enhances the retailer’s flexibility and responsiveness. This additional option not only alleviates reliance on external suppliers, but also enables retailers to maintain a more adaptable and proactive inventory policy, especially in environments characterized by supply uncertainty.

Under conditions of supply uncertainty, the incorporation of PBs exerts a multifaceted influence across the supply chain. Our simulations suggest that while the introduction of PBs can lead to competitive adjustments—such as a reduction in wholesale prices and altered order quantities—the overall effect is a stabilization of supply chain dynamics. Specifically, PBs help maintain a more consistent consumer surplus and support a resilient supply chain structure, even if this comes at the cost of reduced supplier margins. These dynamics highlight the role of the complex interplay among market perceptions, cost structures, and supply stability in shaping the strategic decisions of all entities involved.

7. Conclusions and Future Research

In the current development of the retail industry, strategic inventory and PB introduction are two widely adopted strategies. While previous studies have examined the impact of each strategy on profitability independently, limited attention has been paid to the interaction among these strategies, especially under supply uncertainty [1,37]. However, with the advancement of technology and changes to the retail environment, supply chain structures have evolved significantly, leading to more frequent supply disruptions. This paper addresses this gap by proposing a novel retailer strategic inventory model that integrates PB introduction in an unstable supply context. By examining the strategic inventory and profits of various supply chain stakeholders under different scenarios, we draw the following main conclusions.

Firstly, regardless of whether PBs are introduced, the retailer will only maintain the strategic inventory if the wholesale price and inventory costs are sufficiently low. The introduction of PBs can lower the critical threshold for activating strategic inventory, allowing the retailer to make more flexible strategic decisions. We also identify the optimal strategic inventory levels under different scenarios. Secondly, the introduction of PBs lowers the barriers to implementing strategic inventory strategies, thereby indirectly enhancing supply chain resilience. In scenarios where PBs are introduced, consumer surplus remains stable, though lower than when only NBs are present. The supply level across the entire supply chain remains stable. Even when strategic inventory costs are too high to implement, PBs can still support the retailer. Finally, the impact of PBs on the profits of retailers and suppliers is multifaceted, depending on factors such as the perceived quality of PBs, inventory costs, and supply stability. Some previous pieces of literature suggest that the introduction of PBs benefits suppliers as well. However, this dynamic changes when supply instability and strategic inventory are taken into account.

This study makes several key contributions to the literature. First, it is the first paper to develop a theoretical model that comprehensively considers supply uncertainty, PB introduction, and strategic inventory decisions. Second, through comparative model analysis, we identify the boundary conditions under which a retailer engages in strategic inventory and examine the changes brought by the introduction of PBs under different scenarios. Finally, our findings yield several important managerial insights that can serve as valuable references for business decision-makers.

For retail enterprises, both strategic inventory and PBs are valuable strategic options. These strategies can help retailers strengthen their bargaining power with suppliers. It is important to note that the implementation of these strategies is not static; they need to be applied flexibly, considering factors such as the perceived quality of PBs, inventory costs, and supply stability. For instance, in highly competitive environments, retailers can adopt a more aggressive PB strategy. By leveraging the cost advantages associated with PBs, retailers can secure lower wholesale prices and invest in quality improvements to build consumer trust. This, in turn, permits them to lower their strategic inventory levels without compromising service levels, thus achieving a more efficient and responsive supply chain. Conversely, in markets with lower competitive pressures or more stable supplier relationships, a more cautious approach may be warranted. In these settings, retailers might opt to gradually introduce PBs, maintaining a relatively higher level of strategic inventory as a safeguard against supply disruptions.

This study employs a two-stage Stackelberg game model to analyze strategic inventory decisions under supply uncertainty. While this approach offers valuable theoretical insights, it also involves several simplifying assumptions. For instance, our model assumes a binary supply scenario (either full supply or complete disruption) and a monopoly-like supplier–retailer relationship. In reality, supply disruptions can be partial, such as delayed shipments or quality issues, and supply chains often involve multiple competing suppliers and retailers [40]. For future research, we suggest extending our framework by incorporating more advanced modeling techniques, such as dynamic programming or stochastic optimization, which can better account for partial supply disruptions and logistical challenges like warehouse limitations and transportation delays. We also aim to broaden our sensitivity analysis by testing various market conditions (e.g., changes in consumer demand elasticity and supplier pricing) and by integrating empirical data with our theoretical model. These enhancements will help bridge the gap between theoretical assumptions and real-world complexities, ultimately leading to more tailored and actionable managerial recommendations.