Analysis of Bullwhip Effect and Inventory Cost in an Omnichannel Supply Chain

Abstract

This paper explores the optimization of the bullwhip effect (BWE) and inventory costs considering price information symmetry in an omnichannel environment, offering novel insights into managing supply chain dynamics. We examine the pick-up lead time in the “buy online and pick up in store” (BOPS) channel as a critical operational factor, analyzing how the interaction with the ordering lead time affects omnichannel supply chain performance. The research highlights the impacts of the BOPS strategy on demand and inventory information, developing a comparative examination of the BWE and inventory expenses within various supply chain contexts. We discover that the interplay between ordering lead time and pick-up lead time significantly affects both inventory costs and the BWE of omnichannel retailers, with these impacts presenting an inverse relationship. While numerous studies have validated that product returns can restrain the information distortion in supply chains, our findings reveal that this relationship holds true in omnichannel retail only within specific supply chain contexts. This comprehensive approach offers valuable insights for omnichannel supply chain managers seeking to optimize the BOPS strategy and improve overall operational efficiency.

1. Introduction

In an era marked by the convergence of digital and physical commerce, omnichannel enterprises have emerged as pioneers in integrating diverse retail channels to cater to the evolving demands of consumers. These enterprises strive to provide a seamless and comprehensive experience that encompasses shopping, entertainment, and social interaction across various platforms, including brick-and-mortar stores, online marketplaces, and digital media [1]. A significant portion of the retail industry has embraced omnichannel strategies to enhance their service offerings and remain competitive [2]. Prominent examples include Walmart, Uniqlo, and Amazon, which have successfully implemented the “buy online and pick up in store” (BOPS) service. The rise of omnichannel retailing has had a profound impact on traditional retail methods and the associated supply chains. This new retail paradigm has been propelled by several key factors, including the increasing ease of accessing information, the growing social interaction needs of shoppers, and the rising demand for personalized experiences [3]. Omnichannel retail offers a more convenient and personalized shopping experience by leveraging data analytics, customer insights, and advanced technologies. In contrast, single-channel retail, whether online or offline, can no longer fulfill the multifaceted demands of modern consumers who seek a seamless transition between different shopping environments. The complexity and variability of supply chain systems have increased significantly with the advent of omnichannel retailing. This new business model requires a high degree of integration and coordination across various channels. The challenges posed by this complexity are not only substantial but also present opportunities for innovation and optimization in supply chain management [4].

The adoption of various omnichannel retail strategies by retailers can have a significant impact on the bullwhip effect (BWE) and inventory costs [2]. In omnichannel supply chains, there is no price information asymmetry between online and offline retail channels [5]. When price information is symmetric, price-conscious consumers tend to refer to and compare prices across different sales channels, which in turn influences their purchasing decisions [6]. This cross-channel price comparison can lead to more intense price fluctuations, thereby exacerbating the variability of demand information in omnichannel retail markets. As fluctuating order information is transmitted upstream in the supply chain by retailers, it inevitably causes misdirection and distortion, thereby intensifying the BWE in omnichannel supply chains [7].

Currently, the BOPS strategy is among the most extensively implemented omnichannel retail strategies [8]. This strategy allows consumers to purchase goods online and subsequently opt to collect their items from physical stores. However, returns in the BOPS strategy, which occur after customer experience, have multifaceted impacts on operations management. These impacts include increased costs, fluctuations in inventory levels, and increased complexity in inventory management [9,10]. Additionally, the pick-up lead time in the BOPS channel introduces uncertainty into the supply chain. This is primarily manifested in increased difficulty in demand forecasting, greater complexity in inventory management, higher logistics and operational costs, and potential negative impacts on customer experience [11]. Given these challenges, retailers need to implement effective strategies to optimize omnichannel management and mitigate the influence of uncertainty on the supply chain [12]. Considering the price information symmetry in omnichannel retail, this study investigates the BWE and inventory costs in omnichannel supply chains under different scenarios. Specifically, this study examines the impact of the return rate in the omnichannel retail experience on ordering decisions and inventory levels. It also analyzes the formation mechanisms and mitigating measures of the BWE in omnichannel supply chains. The focus of the research is to offer new insights and theoretical support for the coordination of the BWE in omnichannel supply chains. The exploration of the BWE and inventory costs under the omnichannel retail model provides valuable support for managers’ ordering decisions and cost optimization across different supply chain scenarios.

The research endeavors to explore the following three questions:

(1)

How to optimize the BWE and inventory costs for omnichannel retailers across various supply chain contexts?

(2)

What are the formation mechanisms and mitigation measures of the BWE in omnichannel supply chains, considering their differences in the characteristics?

(3)

How do the pick-up lead time and return rate in the BOPS channel influence the BWE and inventory costs in omnichannel supply chains?

The contributions of this paper are manifested in three primary areas. First, this study originally explores the optimization of the BWE and inventory costs in omnichannel supply chains under the condition of price information symmetry. The research provides novel insights into managing supply chain dynamics in an omnichannel environment. Second, this paper examines the pick-up lead time in the BOPS channel as a critical operational determinant. It explores how the interaction between the pick-up lead time and the ordering lead time influences the overall efficiency of the omnichannel supply chain. This analysis offers a deeper understanding of the operational challenges and opportunities associated with BOPS strategies. Third, this study analyzes the effect of the return rate in omnichannel retail on inventory and demand information. It further conducts a comparative analysis of the BWE and inventory costs across various supply chain contexts. This comprehensive approach provides valuable insights for supply chain managers seeking to restrain the negative impacts of returns and optimize supply chain performance.

The rest of this paper is organized as follows. Section 2 offers a thorough examination of the pertinent literature. Section 3 develops the omnichannel retail market demand model under the BOPS strategy. Section 4 examines the ordering strategies for the omnichannel retailers. Section 5 quantifies the BWE and expected costs under the omnichannel strategy and conducts a comparative analysis across different scenarios. Finally, Section 6 encapsulates the primary insights derived from this study.

2. Literature Review

The BWE has been broadly acknowledged among researchers as a significant element contributing to reduced operational efficiency in supply chains [13,14,15,16]. As market demand information becomes distorted while being transmitted upstream through the supply chain, it leads to excessive inventory levels, increased inventory costs, inaccurate production planning, and substantial economic losses [17,18,19]. This research focuses on optimizing the BWE and inventory costs in omnichannel supply chains across various scenarios. The theoretical contributions of this study to the existing literature are primarily reflected in two key areas: (1) the bullwhip effect and (2) omnichannel retailing.

2.1. Bullwhip Effect

The bullwhip effect, a crucial metric to evaluate supply chain efficiency, is characterized by the mutation and magnification of market information as it propagates upstream through the various nodes of the supply chain [20,21,22]. Specifically, fluctuations in downstream market demand are magnified in upstream ordering information, resulting in significant corporate profit losses, excessive inventory levels, inaccurate production plans, and disordered capacity planning [23]. Over the past two decades, the BWE has attracted substantial attention from both academic researchers and industry practitioners. Consequently, an increasing number of scholars have delved into the BWE by examining more complex supply chain environments [24,25,26].

The analytical methods for examining the BWE primarily encompass statistical methods, system simulation methods, and empirical and case study approaches [27,28]. Wang and Disney’s research has identified five key elements in the statistical modeling of the BWE: the demand dynamics, estimating techniques, inventory policies, lead times, and information sharing [27]. First, the demand process is a fundamental driver of retailers’ ordering decisions and the BWE in supply chains. The selection of an appropriate demand function depends on factors such as the supply chain intricacy, the characteristics of the market, and the aims of the research. Commonly used demand models in BWE studies include the Auto-Regressive Moving Average (ARMA) model, price-sensitive model, independent and identically normally distributed (Normal i.i.d.) model, and hybrid model. Among these, the AR(1) model, a particular variant of the ARMA model, is the most widely used demand function by researchers [22]. However, in recent years, a significant upward trend has been observed among scholars to analyze the fluctuations in supply chain ordering information using price-sensitive demand models [29]. Second, the three predominant estimating techniques employed in BWE modeling are exponential smoothing (ES), minimum mean squared error (MMSE), and moving average (MA) [6]. Comparative analyses have been conducted to assess the effect of these estimating techniques on the BWE. Findings indicate that the BWE exhibits distinct characteristics under different forecasting methods [21]. The MMSE technique offers the smallest forecasting error and highest accuracy. However, its complex model construction increases the complexity of the BWE expression, making it challenging to analyze the properties of the BWE and inventory cost. As a result, it is optimal for relatively simple supply chain scenarios [24]. In contrast, MA is widely used in complex supply chains due to its simplicity and ease of implementation. ES is more commonly applied in corporate decision-making analyses. Overall, MMSE is preferred when minimizing prediction error is the primary objective, while MA and ES are favored for forecasting lead-time demand in highly complex supply chain settings. Third, commonly employed inventory policies in BWE research encompass the (s, Q) policy, (s, S) policy, and order-up-to policy. The order-up-to policy is a frequently adopted and well-established approach in supply chain analysis. This policy has consistently demonstrated its efficacy as a locally optimal solution for inventory control, which minimizes total discounted holding and shortage costs [30]. Fourth, lead times are a primary factor contributing to the variation in demand information within the supply chain. Lee et al. demonstrated through statistical methods that lead times significantly amplify the BWE [23,31]. This finding has been corroborated by other researchers who have examined the BWE in complex supply chain environments [29]. Finally, the efficacy of information sharing has been widely substantiated in reducing the BWE and expected costs [32,33]. Empirical studies and mathematical analyses have shown that information sharing among different organizational entities in the supply chain is a powerful means of enhancing operational efficiency [34,35,36]. Incentivizing information sharing can mitigate market information distortion within the supply chain system.

The recent literature highlights that the BWE has been extensively examined in complex supply chain contexts, with researchers focusing on a variety of critical issues. These include the application of artificial intelligence [37], internet of things technologies [38], information sharing [39], advancements in demand forecasting techniques [13], product portfolio adjustments [15], improvements in inventory strategies [16], recycling and remanufacturing [31], and the role of servitized manufacturers [40]. Despite these diverse explorations, most of the current research on the BWE has primarily centered on single-channel retail and the associated supply chain analysis [41]. Consequently, there are few studies regarding the BWE within omnichannel supply chain contexts. The paper addresses this focus by considering the impact of price information symmetry from different retail channels on ordering decisions and inventory levels in omnichannel supply chains.

This research assumes normal i.i.d. demand and price processes in the omnichannel market and employs the MMSE method for lead-time demand forecasting. The assumptions are justified in the analysis of the BWE in omnichannel supply chains [42,43,44,45,46], as illustrated in

Table 1. Classification of forecasting techniques.

Forecasting Technique	Model Formulation	Advantages of the Technique
MMSE	${\widehat{d}}_{t+i}=E(d_{t+i}\left|d_{t-1},\right.d_{t-2}\dots )$	Lowest prediction error and highest accuracy [18,21,23,24,27,29,30,34,35]
MA	${\widehat{d}}_{t+i}={\displaystyle {\sum }_{j=1}^K{d}_{t-j}/\psi }$ *	Simple calculation and clear trends [20,22,27,32]
ES	${\widehat{d}}_{t+i}=\overline{\omega }{d}_{t-1}+(1-\overline{\omega }){\widehat{d}}_{t-1}$ *	Recent data focus and quick adaptation [14,26,27,28]

* $\overline{\omega }$ and $\psi$ denote the smoothing coefficient and the historical observation periods, respectively.

and

Table 2. Demand and price models in bullwhip effect research.

Studies on Bullwhip Effect	Demand Model	Price Model	Supply Chain Contexts
Tai et al. [43,45]	Price-sensitive AR(1) demand	Normal i.i.d	Simplified supply chain with a single retail channel
Ma et al. [42]	Price-sensitive demand	VAR(1)	Complex supply chain with multiple retail channels
Giri et al. [44]	Price-sensitive demand	ARMA(1,1)	Complex supply chain with a single retail channel
Gao et al. [24]	Normal i.i.d	\	Complex supply chain with a single retail channel
Ponte et al. [35], Papanagnou [38]	Normal i.i.d	\	Complex supply chain with a single retail channel
Wang et al. [46]	Price-sensitive demand	AR(1)	Complex supply chain with multiple retail channels
This study	Price-sensitive demand	Normal i.i.d	Simplified supply chain with omnichannel

. Therefore, this study aims to offer novel perspectives and theoretical support for the coordination of the BWE in omnichannel supply chains.

2.2. Omnichannel Retailing

Omnichannel retailing signifies that interactions between retailers and customers are no longer confined to a single channel [47]. Retailers leverage a diverse array of interactive channels, including physical stores, online shops, mobile terminals, and social media, to engage with consumers. This model has emerged as the dominant trend in the retail industry, fully integrating the strengths of various retail channels [48]. Under the omnichannel marketing model, retailers are required to integrate the resources of physical stores, virtual shops, and multiple media channels. This integration ensures that consumers have access to comprehensive product information and an enhanced shopping experience [49].

Currently, omnichannel retail strategies mainly include Buy Online and Pick Up in Store (BOPS), Reserve Online and Pick Up and Pay in Store (ROPS), Showrooming, Buy Offline and Ship from Online, Buy Online and Return in Store, and Buy Online and Ship from Store [50]. The omnichannel retail models have attracted considerable interest from both the academic and business communities [51]. Scholars have conducted extensive research on various aspects of omnichannel retailing, including retailing, pricing, returns, warehousing, and route design [52]. These studies explored how retailers can enhance the effectiveness of omnichannel supply chains with different retail strategies.

Through comparative analyses of various omnichannel retail strategies, several scholars have identified the benefits and limitations of the BOPS strategy. Gallino and Moreno [53] employed an empirical approach to examine the effects of BOPS on demand levels across different channels for dual-channel retailers. They found that while BOPS reduced online sales volume, it increased offline sales by attracting more consumers to physical stores and driving the sales of related products. Their research indicated that BOPS could enhance retailers’ total revenue by leveraging the strengths of both offline and online channels. However, other studies have highlighted the potential inefficiencies associated with BOPS. Kong et al. [54] developed a theoretical framework to assess the influence of BOPS adoption on omnichannel vendors, comparing different pricing tactics to determine their effectiveness. Their findings suggested that BOPS is not universally beneficial. The effectiveness depends on factors such as unit operating costs within the BOPS channel, customer hassle costs, and cross-selling profits. Zhang et al. [55] analyzed the influence of customer returns and order cancellations during offline experiences under the BOPS strategy on inventory decisions and product pricing. They concluded that high operational costs in the offline pick-up process could negatively affect retailers’ sales interests. Furthermore, scholars have compared the impacts of BOPS and ROPS strategies on omnichannel merchants’ profits and supply chain efficiency. Jin et al. [2] analyzed how BOPS affects the service area size in the offline channel and found that the optimal service area depends on the proportion of unit product stockholding expense to customer arrival frequency. They also compared the optimal profits of omnichannel retailers under BOPS and ROPS strategies, highlighting significant differences in supply chain efficiency. Additionally, researchers have explored the effect of different pricing strategies on supply chain efficiency. Gao and Su [50] assumed that omnichannel retailers adopt a uniform pricing strategy and that consumer return behavior is limited to the online retail channel. They compared various information strategies and analyzed their impact on retailers’ inventory levels and profits. The findings indicated that different pricing strategies, such as uniform pricing and differentiated pricing, will result in substantial differences in supply chain efficiency.

Price symmetry and uniform pricing are prevalent assumptions in omnichannel studies [56]. Price symmetry posits that identical products are priced uniformly across various sales channels, including physical stores, online platforms, and mobile applications. This assumption streamlines the analysis of omnichannel supply chains by mitigating the complexity associated with channel-specific pricing strategies. In the contemporary retail environment, where the widespread use of mobile Internet enables consumers to easily compare prices across channels, uniform pricing can mitigate perceptions of inequity and price discrimination [9]. Despite the growing prevalence of dynamic pricing, many retailers strive to maintain consistent pricing across channels, ensuring a coherent and predictable pricing strategy [57]. Overall, the assumption of uniform pricing across all channels is widely adopted by researchers and is a common practice in omnichannel retailing [12].

Table 3. Assumption of price information symmetry in omnichannel research.

Studies on Omnichannel	Price Information Symmetry	Pricing Strategy in Omnichannel
Jin et al. [2], Silbermayr and Waitz [3], Goedhart et al. [9,10], Gao et al. [12], Gao and Su [50], Cocco and Demoulin [56]	Symmetry	Uniform pricing
Basak et al. [5], Kim et al. [8]	Symmetry	Differentiated pricing
Zhao et al. [7]	Asymmetry	Differentiated pricing
This study	Symmetry	Uniform pricing

categorizes the price information symmetry and diverse pricing strategies employed in omnichannel studies.

Building on the widely adopted BOPS omnichannel retail strategy, this paper examines the influence on the BWE and inventory cost. We explore the coordination measures of the BWE in omnichannel supply chains across various supply chain scenarios, thereby offering valuable references for decision optimization for omnichannel managers.

3. Demand Model

The omnichannel supply chain system is composed of a manufacturer, an omnichannel retailer, and consumers. The interactions and dissemination dynamics of information and logistics flows among the entities within the omnichannel supply chain network are illustrated in Figure 1.

Omnichannel retailers employing the BOPS strategy leverage digital and intelligent technologies to provide a richer and more convenient customer experience. The customer experience is one of the key features distinguishing omnichannel retailing from single-channel online retailing. However, the offline experience in the BOPS channel also presents several challenges, including the inevitable occurrence of returns and the significant pick-up lead time. When retailers adopt the BOPS strategy, a portion of consumers may cancel their orders due to dissatisfaction with the products after experiencing them offline. Consequently, part of the reserved demand reverts to the retailer’s inventory. The total demand observed by omnichannel retailers under the BOPS strategy can be expressed as follows: $D_t^{\prime }=d_t^1+d_t^2+d_t^3$. Herein, $d_t^1$ represents the demand in the online retail channel, $d_t^2$ denotes the demand in the BOPS retail channel, and $d_t^3$ signifies the demand in the offline retail channel. While the actual total demand of the omnichannel retailer is $D_t=d_t^1+d_t^2+d_t^3-\theta d_{t-l-1}^2$, where $\theta$ represents the return rate of the BOPS channel and $l$ denotes the average pick-up lead time in the BOPS channel. In an omnichannel model, the demand for each channel is delineated as follows: $d_t^i={\alpha }_{i}a-b_i{p}_t+{\epsilon }_t^i,i=1,2,3$. Also, $a>0$ stands for the market demand scale, and $0\le {\alpha }_i\le 1$ indicates the market share of each retail channel, with ${\displaystyle \sum {\alpha }_i}=1$. $b_i>0$ denotes the price sensitivity coefficient, and $b_i{p}_t$ represents the portion of demand in each channel influenced by price, where demand is negatively correlated with price. Set $\pi =b_1+b_2+b_3$. ${\epsilon }_t^i$ is the identically and independently normally distributed demand disturbance, characterized by a mean of 0 and a standard deviation of ${\sigma }_i$, i.e., ${\epsilon }_t~N\left(0,{\sigma }_i^2\right)$. ${\epsilon }_t^i$ bears no correlation to the market price $p_t$, and the covariance between the market price and the demand disturbance is $Cov\left(p_t,{\epsilon }_{t^{\prime }}^i\right)=0,\left(\forall t,t^{\prime }\right)$.

In an omnichannel market environment, a retailer sells a single product in a perfectly competitive market, where the market price results from the interplay of market supply and demand. This study posits that due to symmetric price information across channels, a synchronized pricing strategy is adopted, expressed as $p_t=\mu +{\eta }_t$, where $\mu$ represents the constant term in the price process, and ${\eta }_t$ denotes the price disturbance term, which follows an independently and identically normal distribution characterized by a mean of zero and a variance of ${\xi }^2$. The relationship between the market price and price disturbance in terms of covariance is given by $Cov\left(p_t,{\eta }_{t^{\prime }}\right)=0,\left(t<t^{\prime }\right)$. If the price process follows a normal distribution and the demand is a price-sensitive model with a normal disturbance term, then the demand process will also be normally distributed. Therefore, the price-sensitive demand in this study can be described as independent and identically normally distributed (Normal i.i.d) with a mean of ${\alpha }_{i}a-b_i\mu$ and a variance of $b_i^2{\xi }^2+{\sigma }_i^2$, i.e., ${\epsilon }_t~N\left({\alpha }_{i}a-b_i\mu ,b_i^2{\xi }^2+{\sigma }_i^2\right)$. The normal i.i.d demand model has been extensively utilized in studies on the BWE [24].

4. Ordering Decision

4.1. Inventory Strategy and Forecasting Technique

Within the ordering lead time $L$, the omnichannel retailer observes the total demand $D_{t-1}$ at the end of period $t-1$ and calculates the order-up-to inventory level $y_t$ of period $t$. Concurrently, at the beginning of period $t$, the retailer submits an order $q_t$ to the upstream manufacturer and is scheduled to obtain the goods from the manufacturer at period $t+L$. To forecast the order-up-to inventory level, the retailer employs demand forecasting techniques. In this section, an order-up-to inventory strategy coupled with an MMSE estimating technique is adopted.

The order-up-to strategy is among the most extensively utilized approaches in supply chain management. Prior studies have established that this strategy is locally optimal, as it can effectively minimize the overall discounted shortage and holding costs [6,27]. Given its proven efficacy, the omnichannel retailer in this study employs the optimal order-up-to strategy. The ordering quantity can be formulated as follows:

$$q_t=y_t-\left(y_{t-1}-D_{t-1}\right)$$

(1)

where the order-up-to level $y_t$ is composed of two components: the anticipated inventory to satisfy the forecasted lead-time demand, along with a safety inventory level to offset unforeseen fluctuations in demand. Thus, the omnichannel retailer’s order-up-to level for each period can be given by:

$$y_t={\widehat{D}}_t^L+z{\widehat{\sigma }}_t^L$$

(2)

Herein, ${\widehat{D}}_t^L$ denotes the forecasted average lead-time demand. $z$ is a fixed value determined to achieve the targeted service level, namely the safety factor and ${\widehat{\sigma }}_t^L$ represents the demand estimating error during $L$ periods with ${\widehat{\sigma }}_t^L=\sqrt{Var\left(D_t^L-{\widehat{D}}_t^L\right)}$.

Substituting Equation (2) into Equation (1), the order quantity $q_t$ can be expressed in the following form:

$$q_t={\widehat{D}}_t^L-{\widehat{D}}_{t-1}^L+d_{t-1}+z\left({\widehat{\sigma }}_t^L-{\widehat{\sigma }}_{t-1}^L\right)$$

(3)

4.2. Ordering Decision of the Omnichannel Retailer

The omnichannel retailer employs the MMSE forecasting technique to determine the lead-time demand. The demand estimate under the MMSE method is the conditional expectation based on historical demand information. If ${\widehat{D}}_{t+i}$ denotes the omnichannel retailer’s demand estimate for period $t+i$ made at period $t-1$, the expected demand ${\widehat{D}}_{t+i}$ under the MMSE approach will be ${\widehat{D}}_{t+i}=E\left(D_{t+i}\left|D_{t-1}\right.\right)$. Also, the price forecast under the MMSE technique is ${\widehat{p}}_{t+i}=E\left(p_{t+i}\left|p_{t-1}\right.\right)$.

When the ordering lead time exceeds the pick-up lead time under the BOPS strategy, the lead-time demand in the omnichannel retailer’s ordering decision encompasses only the unknown information requiring prediction for the retailer. In contrast, with a shorter ordering lead time relative to the pick-up lead time, the lead-time demand incorporates partial known demand and price information ${\epsilon }_{t+i-l-1}$ and $p_{t+i-l-1}$, which are available to the omnichannel retailer. Thus, the dynamic interplay between the two lead times can result in completely distinct decisions regarding order placement for the omnichannel retailer and significantly influence supply chain efficiency, encompassing the BWE and expected costs. When $i>l$, ${\epsilon }_{t+i-l-1}$ and $p_{t+i-l-1}$ represent the unknown information requiring prediction. Whereas when $i\le l$, ${\epsilon }_{t+i-l-1}$ and $p_{t+i-l-1}$ are the known information for the retailer.

Therefore, the forecasted price and the predicted demand disturbance for period $t+i-l-1$ are as follows:

$${\widehat{p}}_{t+i-l-1}=E\left(p_{t+i-l-1}\left|p_{t-1}\right.\right)=\left\{\begin{array}{ll}\mu ,& i>l\\ p_{t+i-l-1},& i\le l\end{array}\right.$$

(4)

$${\widehat{\epsilon }}_{t+i-l-1}=E\left({\epsilon }_{t+i-l-1}\left|{\epsilon }_{t-1}\right.\right)=\left\{\begin{array}{ll}0,& i>l\\ {\epsilon }_{t+i-l-1},& i\le l\end{array}\right.$$

(5)

When a synchronized pricing strategy is employed, the total demand for the omnichannel retailer at period $t+i$ under the BOPS strategy is given by:

$$D_{t+i}=\left(1-\theta {\alpha }_2\right)a-\left(b_1+b_2+b_3\right)p_{t+i}+\theta b_2{p}_{t+i-l-1}+{\epsilon }_{t+i}^1+{\epsilon }_{t+i}^2+{\epsilon }_{t+i}^3-\theta {\epsilon }_{t+i-l-1}^2$$

(6)

Consequently, the forecasted average lead-time demand during periods $\left[t,t+L\right)$ for the omnichannel retailer is given by:

$$\begin{array}{c}{\widehat{D}}_t^L={\displaystyle \sum _{i=0}^{L-1}{\widehat{D}}_{t+i}}\hfill \\ =\left\{\begin{array}{ll}L\left(1-\theta {\alpha }_2\right)a-L\left(b_1+b_2+b_3\right)\mu +\left(L-l-1\right)\theta b_2\mu +\theta b_2{\displaystyle \sum _{i=0}^l{p}_{t+i-l-1}}-\theta {\displaystyle \sum _{i=0}^l{\epsilon }_{t+i-l-1}^2},& L>l\\ L\left(1-\theta {\alpha }_2\right)a-L\left(b_1+b_2+b_3\right)\mu +\theta b_2{\displaystyle \sum _{i=0}^{L-1}{p}_{t+i-l-1}}-\theta {\displaystyle \sum _{i=0}^{L-1}{\epsilon }_{t+i-l-1}^2},& L\le l\end{array}\right.\end{array}$$

(7)

Lemma 1. 

When the omnichannel retailer employs the BOPS strategy, the variance of the forecast error in the omnichannel supply chain is a fixed constant and remains unchanged over time:

When $L>l$,

$$\begin{array}{l}{\left({\widehat{\sigma }}_t^L\right)}^2=L{\sigma }_1^2+L{\sigma }_3^2+\left(L-l-1\right){\left(1-\theta \right)}^2{\sigma }_2^2+\left(l+1\right){\sigma }_2^2\\ +\left(L-l-1\right){\left(\pi -\theta b_2\right)}^2{\xi }^2+\left(l+1\right){\pi }^2{\xi }^2\end{array}$$

(8)

When $L\le l$,

$${\left({\widehat{\sigma }}_t^L\right)}^2=L{\sigma }_1^2+L{\sigma }_3^2+L{\sigma }_2^2+L{\pi }^2{\xi }^2$$

(9)

Proof. 

See Appendix A. □

The difference in the lead-time demand forecasts of the omnichannel retailer for period $t$ and period $t-1$ is as follows:

$${\widehat{D}}_t^L-{\widehat{D}}_{t-1}^L=\left\{\begin{array}{l}\theta b_2\left(p_{t-1}-p_{t-l-2}\right)-\theta \left({\epsilon }_{t-1}^2-{\epsilon }_{t-l-2}^2\right), L>l\\ \theta b_2\left(p_{t+L-l-2}-p_{t-l-2}\right)-\theta \left({\epsilon }_{t+L-l-2}^2-{\epsilon }_{t-l-2}^2\right), L\le l\end{array}\right.$$

(10)

From Lemma 1, we have ${\widehat{\sigma }}_t^L={\widehat{\sigma }}_{t^{\prime }}^L,\left(\forall t,t^{\prime }\right)$. Substituting ${\widehat{\sigma }}_t^L-{\widehat{\sigma }}_{t-1}^L=0$ into Equation (3), the order quantity of the omnichannel retailer under the BOPS strategy is reformulated as follows:

$$q_t=\left\{\begin{array}{ll}\left(1-\theta {\alpha }_2\right)a-\left(b_1+\left(1-\theta \right)b_2+b_3\right)p_{t-1}+{\epsilon }_{t-1}^1+\left(1-\theta \right){\epsilon }_{t-1}^2+{\epsilon }_{t-1}^3,& L>l\\ \left(1-\theta {\alpha }_2\right)a+\theta b_2{p}_{t+L-l-2}-\left(b_1+b_2+b_3\right)p_{t-1}+{\epsilon }_{t-1}^1+{\epsilon }_{t-1}^2+{\epsilon }_{t-1}^3-\theta {\epsilon }_{t+L-l-2}^2,& L\le l\end{array}\right.$$

(11)

5. Bullwhip Effect and Expected Cost

5.1. Bullwhip Effect of the Omnichannel Retailer

The BWE is measured by the proportion of the variance in the retailer’s orders to the variance in market demand. A ratio greater than one indicates the presence of the BWE in supply chains, which typically leads to potential costs such as excessive inventory buildup, profit loss, and disruption in capacity planning. Therefore, the variance of the order quantity is frequently employed as a metric for assessing the BWE and information variability in supply chains.

Theorem 1. 

When an omnichannel retailer employs the BOPS strategy, the expressions for the order variance ${\sigma }_q^2=Var\left(q_t\right)$ in two supply chain scenarios are as follows:

When $L>l$,

$$Var\left(q_t\right)={\left(b_1+\left(1-\theta \right)b_2+b_3\right)}^2{\xi }^2+{\sigma }_1^2+{\left(1-\theta \right)}^2{\sigma }_2^2+{\sigma }_3^2$$

(12)

When $L\le l$,

$$Var\left(q_t\right)={\left(b_1+b_2+b_3\right)}^2{\xi }^2+{\theta }^2{b}_2^2{\xi }^2+{\sigma }_1^2+\left(1+{\theta }^2\right){\sigma }_2^2+{\sigma }_3^2$$

(13)

Proof. 

See Appendix B. □

5.2. Inventory Cost of the Omnichannel Retailer

When the omnichannel retailer employs the BOPS strategy, the expected inventory cost $C_t$ of the retailer can be expressed in the following formula:

$$\begin{array}{cc}{C}_t& =E_{{\epsilon }_t^i,{\eta }_t}\left[P{\displaystyle {\int }_{y_t}^{\infty }\left(D_t^L-y_t\right)}d{\overline{F}}_t\left(D_t^L\right)+H{\displaystyle {\int }_{-\infty }^{y_t}\left(y_t-D_t^L\right)}d{\overline{F}}_t\left(D_t^L\right)\right]\hfill \\ & ={\widehat{\sigma }}_t^L\left[\left(H+P\right)L\left(z\right)+Hz\right]\hfill \end{array}$$

(14)

where ${\overline{F}}_t\left(D_t^L\right)$ denotes the cumulative distribution function of the lead-time demand. y_t represents the optimal order-up-to level for omnichannel retailer and is expressed as $y_t={\widehat{D}}_t^L+z{\widehat{\sigma }}_t^L$. $H$ and $P$ denote the unit holding cost and unit penalty cost of the retailer, respectively. $L\left(x\right)$ is the standard normal distribution function, where $L\left(x\right)={\displaystyle {\int }_x^{\infty }\left(\mathrm{y}-x\right)}d\Phi \left(y\right)$ decreases with $x$ and satisfies the property that $H\left(z+L\left(z\right)\right)+PL\left(z\right)\le H\left(x+L\left(x\right)\right)+PL\left(x\right),\forall x\ge z$. ${\widehat{\sigma }}_t^L$ is the forecast error of the lead-time demand, which is detailed by Lemma 1.

From Equation (14), it can be observed that under the BOPS strategy, the expected inventory cost of the omnichannel retailer is associated with the subsequent parameters: the ordering lead time $L$, the price sensitivity coefficient $b_i$, the pick-up lead time $l$, the return rate $\theta$, the variances of the market demand and price disturbance terms.

5.3. Comparative Analysis

In this section, we analyze the impacts of key parameters on the omnichannel supply chain and examine the differential effects of these parameters on the BWE and inventory costs.

To derive Proposition 1, we conduct an analysis of the relationship between price sensitivity and the BWE in omnichannel supply chains under two distinct scenarios: when the ordering lead time is greater than the pick-up lead time and when it is less than the pick-up lead time.

Proposition 1. 

The influence of the price sensitivity coefficient $b_i$ on $BWE$ in an omnichannel supply chain under the BOPS strategy is characterized as follows:

(a) 

When $L>l$, in the interval $\left(0,+\infty \right)$, ${\displaystyle \frac{\mathsf{\partial }BWE}{\mathsf{\partial }{b}_i}}\ge 0$. Therefore, $BWE$ increases with the price sensitivity coefficient $b_i$.

(b) 

When $L\le l$, in the interval $\left(0,+\infty \right)$, ${\displaystyle \frac{\mathsf{\partial }BWE}{\mathsf{\partial }{b}_i}}\ge 0$. Therefore, $BWE$ increases with the price sensitivity coefficient $b_i$ .

Proof. 

See Appendix C. □

Proposition 1 indicates that regardless of the interplay between the ordering lead time and the pick-up lead time, the BWE in the omnichannel supply chain consistently increases with the price sensitivity coefficient across different retail channels. Products with higher demand elasticity tend to cause more pronounced demand information fluctuations in the omnichannel supply chain.

Figure 2 shows how different levels of price sensitivity coefficients affect the BWE in omnichannel supply chains. The x-axis represents the price sensitivity coefficient in different retail channels, while the y-axis shows the corresponding BWE. The red dashed line in the figure illustrates the BWE when the ordering lead time is greater than the pick-up lead time. Conversely, the green line depicts the BWE when the ordering lead time is less than the pick-up lead time. Figure 2 corroborates Proposition 1, showing that under different supply chain scenarios, the price sensitivity of each channel for the omnichannel retailer is positively correlated with the BWE. This is because a higher price sensitivity coefficient means that consumers react more strongly to price changes, leading to greater fluctuations in demand. Retail channels with higher price sensitivity coefficients, such as the online and BOPS channels, contribute more significantly to the BWE, as consumers in these channels are more likely to adjust their purchasing behavior through price comparison.

By examining the relationship between the return rate and the BWE in omnichannel supply chains under two distinct scenarios, Proposition 2 is derived.

Proposition 2. 

The influence of the return rate $\theta$ on $BWE$ in an omnichannel supply chain under the BOPS strategy has the following property:

(a) 

When $L>l$, in the interval $\left(0,1\right)$, ${\displaystyle \frac{\mathsf{\partial }BWE}{\mathsf{\partial }\theta }}\le 0$. Therefore, $BWE$ decreases with the return rate $\theta$.

(b) 

When $L\le l$, in the interval $\left(0,1\right)$, ${\displaystyle \frac{\mathsf{\partial }BWE}{\mathsf{\partial }\theta }}\ge 0$. Therefore, $BWE$ increases with the return rate $\theta$.

Proof. 

See Appendix D. □

Proposition 2 indicates that with a shorter ordering lead time relative to the pick-up lead time, the BWE in the omnichannel supply chain increases with the return rate. Products with higher return rates tend to cause more pronounced BWE, thereby reducing supply chain efficiency. Conversely, when the ordering lead time exceeds the pick-up lead time, the BWE in an omnichannel supply chain diminishes with the return rate.

Figure 3 illustrates how varying return rates impact the BWE in omnichannel supply chains. The x-axis denotes the return rate in the BOPS channel, and the y-axis represents the corresponding BWE. Figure 3 validates Proposition 2, revealing that different supply chain scenarios lead to different impacts of the return rate on the BWE in the omnichannel supply chain. Notably, when the ordering lead time is relatively longer, the return behavior of customers after experiencing the product in the BOPS channel can actually suppress the BWE. Numerous studies have demonstrated the mitigating effects of product returns on the BWE in supply chains [24,30]. However, the results of this study show that product returns can mitigate the BWE of the omnichannel retailer only when the ordering lead time exceeds the pick-up time delay. In this scenario, the returned goods are retained in the retailer’s inventory and can meet a portion of the market demand, thereby partially offsetting the fluctuations in current market demand and reducing information distortion. Another possible reason is that when the pick-up time delay is incorporated into the ordering cycle, the omnichannel retailer can promptly coordinate demand forecasts in the future, thereby improving the precision of order placement decisions. However, when the pick-up time delay exceeds one ordering cycle, product returns will magnify the BWE for the retailer. Accordingly, the returned goods are retained in the retailer’s stock and will satisfy a portion of future market demand rather than current demand and thus cannot effectively balance the current demand information variation. Product returns cannot fully offset the current demand fluctuations and therefore can actually amplify the BWE under specific supply chain scenarios. Moreover, the interplay between lead times substantially influences supply chain efficiency, including the BWE and expected costs. Overall, when the ordering lead time exceeds the pick-up lead time, allowing consumer returns can actually suppress information distortion and the BWE for the retailer.

By comparing the magnitude of the BWE in omnichannel supply chains across two distinct scenarios, Proposition 3 is established.

Proposition 3. 

Under the BOPS strategy, $BWE$ in the omnichannel supply chain when $L>l$ is less than that when $L\le l$.

Proposition 3 highlights that the interplay between the ordering lead time and the pick-up lead time in omnichannel supply chains significantly impacts the BWE experienced by the retailer. Specifically, if the ordering lead time exceeds the pick-up lead time, demand information fluctuations in the omnichannel supply chain are more moderate, leading to higher supply chain efficiency. Conversely, with a shorter ordering lead time relative to the pick-up lead time, the omnichannel retailer faces a more pronounced BWE. Therefore, omnichannel supply chain managers should strive to control and reduce the pick-up lead time for customer experiences as much as possible.

Figure 4 demonstrates the impact of the relationship between pick-up lead time and ordering lead time on the BWE in omnichannel supply chains. The x-axis represents the price sensitivity coefficient, while the y-axis shows the corresponding BWE. The red dashed line in the figure indicates the BWE when the ordering lead time exceeds the pick-up lead time, whereas the green line represents the BWE when the ordering lead time is shorter than the pick-up lead time. Figure 4 supports Proposition 3, illustrating that the BWE in supply chains is lower when the ordering lead time exceeds the pick-up lead time. This is because a longer ordering lead time provides the retailer with additional time to adjust orders, thereby reducing demand information fluctuations. Additionally, the retailer can more accurately forecast demand, minimizing inventory overstock and stockout occurrences. In contrast, with a shorter ordering lead time relative to the pick-up lead time, although the retailer can respond more quickly to demand changes, this also increases demand information volatility. As a result, inventory management becomes more complex, leading to higher inventory costs and stockout risks.

By investigating the differential impact of the return rate on inventory costs in omnichannel supply chains across two distinct scenarios, Proposition 4 is formulated.

Proposition 4. 

The influence of the return rate $\theta$ on inventory cost $C_t$ in an omnichannel supply chain under the BOPS strategy can be described as follows:

(a) 

When $L>l$, in the interval $\left(0,1\right)$, ${\displaystyle \frac{\mathsf{\partial }{C}_t}{\mathsf{\partial }\theta }}\le 0$. Therefore, the inventory cost $C_t$ diminishes with the return rate $\theta$.

(b) 

When $L\le l$, in the interval $\left(0,1\right)$, the inventory cost $C_t$ is not correlated with the return rate $\theta$.

Proof. 

See Appendix D. □

Proposition 4 reveals that when the ordering lead time exceeds the pick-up lead time, inventory cost in an omnichannel supply chain decreases as the return rate increases, with products having lower return rates incurring higher inventory costs. Conversely, with a shorter ordering lead time relative to the pick-up lead time, there is no significant correlation between the return rate and the inventory cost.

Figure 5 illustrates the impact of the return rate in the BOPS channel on the inventory costs of an omnichannel retailer when the ordering lead time exceeds the pick-up lead time. The x-axis represents the return rate in the BOPS channel, while the y-axis shows the corresponding inventory costs. The blue dashed line in the figure indicates the inventory costs associated with a longer ordering lead time, whereas the orange dashed line represents the inventory costs with a shorter ordering lead time. Figure 5 corroborates Proposition 4, showing that if the ordering lead time exceeds the pick-up lead time, the return rate will be negatively correlated with expected cost in the omnichannel supply chain. As products with higher return rates exhibit better liquidity within the supply chain, returned items reduce the actual inventory needed for sales and enable the retailer to lower inventory levels and costs. In contrast, products with lower return rates require higher inventory levels to cope with demand fluctuations, thereby incurring higher inventory costs.

By comparing the impact of the pick-up lead time on inventory costs under two distinct omnichannel supply chain scenarios, we formulate Proposition 5.

Proposition 5. 

The influence of the pick-up lead time $l$ on inventory cost $C_t$ in an omnichannel supply chain under the BOPS strategy is characterized as follows:

(a) 

When $L>l$, in the interval $\left(0,+\infty \right)$, ${\displaystyle \frac{\mathsf{\partial }{C}_t}{\mathsf{\partial }l}}\ge 0$. Therefore, the inventory cost $C_t$ is positively correlated with the pick-up lead time $l$.

(b) 

When $L\le l$, in the interval $\left(0,+\infty \right)$, the inventory cost $C_t$ is not correlated with the pick-up lead time $l$.

Proof. 

See Appendix E. □

Proposition 5 suggests that reducing the pick-up lead time in the BOPS channel can significantly mitigate inventory costs within the omnichannel supply chain. By optimizing the pick-up lead time, omnichannel retailers can enhance overall supply chain efficiency.

Figure 6 shows the impact of the pick-up lead time in the BOPS channel on the inventory costs of an omnichannel retailer when the ordering lead time exceeds the pick-up lead time. The x-axis represents the pick-up lead time in the BOPS channel, while the y-axis shows the corresponding inventory costs. The assertion in Proposition 5 is supported by Figure 6, which illustrates a positive correlation between the pick-up lead time in the BOPS channel and inventory costs in the omnichannel supply chain. Specifically, a shorter pick-up lead time enables consumers to retrieve their orders more promptly, thereby decreasing the duration that products remain in inventory and subsequently reducing inventory costs. Additionally, a reduced pick-up lead time accelerates inventory turnover rates, allowing retailers to manage inventory resources more efficiently. This increased turnover not only diminishes inventory costs but also improves the efficiency of capital utilization.

Through a comparative analysis of inventory costs within two distinct omnichannel supply chain contexts, we derive Proposition 6.

Proposition 6. 

Under the BOPS strategy, inventory cost $C_t$ in an omnichannel supply chain when $L>l$ is higher than that when $L\le l$.

Proposition 6 reveals that the interplay between ordering lead time and pick-up lead time in omnichannel supply chains significantly impacts retailers’ inventory costs, which is diametrically opposed to the characteristics of the BWE. When the ordering lead time surpasses the pick-up lead time, inventory costs escalate, diminishing supply chain efficiency. Whereas, with a shorter ordering lead time, inventory costs for omnichannel retailers are mitigated. Given this, omnichannel supply chain managers must strategically prioritize their focus areas, directing efforts toward reducing the ordering lead time if the emphasis is on optimizing inventory costs or controlling the pick-up lead time if the priority is on coordinating and mitigating the BWE.

Figure 7 describes the impact of the relationship between pick-up lead time and ordering lead time on the inventory costs of an omnichannel retailer. The x-axis represents the return rate in the BOPS channel, while the y-axis shows the corresponding inventory costs. The red dashed line in the figure indicates the inventory costs when the ordering lead time is greater than the pick-up lead time, and the green line represents the inventory costs when the ordering lead time is less than the pick-up lead time. Figure 7 substantiates Proposition 6, illustrating that a longer ordering lead time relative to the pick-up lead time results in higher inventory costs. As retailers are compelled to maintain elevated inventory levels to buffer against demand variability, extended ordering lead times hamper their capacity to quickly respond to changes in market demand, thus resulting in excess inventory, increased capital tie-up, and storage expenses.

By characterizing the impact of the price sensitivity coefficient on inventory costs in omnichannel supply chains under two scenarios, Proposition 7 is established.

Proposition 7. 

The influence of the price sensitivity coefficient $b_i$ on inventory cost $C_t$ in an omnichannel supply chain under the BOPS strategy can be described as follows:

(a) 

When $L>l$, in the interval $\left(0,+\infty \right)$, ${\displaystyle \frac{\mathsf{\partial }{C}_t}{\mathsf{\partial }{b}_i}}\ge 0$. Therefore, expected cost $C_t$ increases with price sensitivity coefficient $b_i$.

(b) 

When $L\le l$, in the interval $\left(0,+\infty \right)$, ${\displaystyle \frac{\mathsf{\partial }{C}_t}{\mathsf{\partial }{b}_i}}\ge 0$. Therefore, expected cost $C_t$ increases with price sensitivity coefficient $b_i$.

Proof. 

See Appendix C. □

Proposition 7 reveals that the inventory costs in omnichannel supply chains consistently rise with the price sensitivity coefficient, irrespective of the interplay between the ordering lead time and the pick-up lead time. Products with higher demand elasticity typically incur greater inventory costs within the omnichannel supply chain. This correlation stems from the fact that a higher price sensitivity coefficient implies that consumers are more responsive to price changes, resulting in more pronounced demand fluctuations. To manage this volatility, retailers must maintain higher inventory levels, which in turn raises inventory costs. Additionally, a higher price sensitivity coefficient complicates demand forecasting, compelling retailers to hold more inventory to mitigate demand uncertainty and further escalating inventory costs.

Through an analysis of the impact characteristics of ordering lead time on inventory costs within two distinct omnichannel supply chain scenarios, Proposition 8 is established.

Proposition 8. 

The impact of the ordering lead time L on the expected inventory cost in an omnichannel supply chain fulfills the subsequent characteristic:

In the interval $\left(0,\infty \right)$, $\mathsf{\partial }{C}_t/\mathsf{\partial }L>0$. Therefore, the expected cost has a positive relationship with the lead time L in an omnichannel supply chain.

Proof. 

See Appendix E. □

Proposition 8 suggests that reducing the ordering lead time can significantly mitigate inventory costs in the omnichannel supply chain. Shortening lead times is a proven strategy for lowering inventory costs within supply chains. However, lead times are often dictated by the production capacity and scheduling of upstream manufacturers. Consequently, retailers can take an active role by engaging in information sharing with upstream partners to negotiate shorter lead times or secure economic benefits, such as price discounts. These measures can enhance omnichannel supply chain efficiency and lead to mutually beneficial outcomes.

6. Conclusions and Limitations

6.1. Conclusions

This study develops an omnichannel supply chain network comprising one manufacturer and one omnichannel retailer. It is posited that the retailer adopts an order-up-to strategy and utilizes the MMSE estimating method to determine ordering decisions. This study derives expressions for the BWE and expected costs for the omnichannel retailer under symmetric price information. Additionally, it examines the influence of the return rate and pick-up lead time under the BOPS strategy on retailer’s ordering decisions and inventory levels. By contrasting the distinct characteristics of the BWE and expected costs across various scenarios, this study explores the formation mechanism and coordination measures of the BWE in omnichannel supply chains. The findings of this study offer crucial support for omnichannel supply chain managers to control costs and mitigate information fluctuations under different supply chain scenarios.

(1) The interplay between ordering lead time and pick-up lead time exerts a substantial influence on both inventory costs and the BWE within an omnichannel supply chain, with these influences presenting an intriguing inverse relationship. Specifically, when the ordering lead time exceeds the pick-up lead time, inventory costs tend to rise, whereas the BWE is mitigated. In contrast, with a shorter ordering lead time relative to the pick-up lead time, inventory costs for the omnichannel retailer are diminished, but the BWE is amplified. To align with their strategic goals, supply chain managers should strategically choose between two key approaches. If the objective is to optimize inventory costs, managers should prioritize reducing the ordering lead time. By implementing advanced order management systems and automating order processing, omnichannel retailers can significantly reduce ordering lead times. A shorter ordering lead time enhances the retailer’s ability to quickly adapt to market demand fluctuations, thereby minimizing inventory overstock and associated holding costs. Conversely, if the focus is on mitigating the BWE, managers should concentrate on diminishing the pick-up lead time. Improving logistics and distribution networks can help reduce pick-up lead times. A reduced pick-up lead time enables more accurate order adjustments, which in turn diminishes demand variability and effectively reduces the BWE. Managers must carefully balance the trade-offs between inventory costs and the BWE. While reducing ordering lead times can lower inventory costs, it may also increase the BWE if not managed properly. Conversely, reducing pick-up lead times to mitigate the BWE may require additional investments in logistics and infrastructure. Therefore, a holistic approach that considers both cost and service levels is essential. By understanding the inverse relationship between ordering lead time and pick-up lead time, supply chain managers can make informed decisions that align with their strategic goals. Whether the focus is on optimizing inventory costs or mitigating the BWE, the insights provided by this research offer actionable guidance for enhancing supply chain performance in omnichannel retail environments.

(2) The return rate in the BOPS channel exerts a significant influence on the BWE and inventory costs in omnichannel supply chains, with the nature of this influence varying across different supply chain scenarios. Specifically, when the ordering lead time exceeds the pick-up lead time, both the BWE and inventory costs decrease with the return rate. While numerous studies have validated that product returns can restrain the BWE in supply chains [24,30], our findings reveal that this relationship holds true only within specific supply chain contexts. Conversely, with a shorter ordering lead time relative to the pick-up lead time, the BWE in an omnichannel model intensifies with an increase in the return rate. Products with higher return rates tend to exacerbate the BWE, as returns amplify the uncertainty of demand. The findings regarding the impact of return rates on the BWE and inventory costs in omnichannel supply chains offer valuable insights for supply chain managers and operations executives. These insights can inform strategic decisions to optimize inventory management and mitigate demand variability. Supply chain managers should tailor their strategies based on the specific context of their supply chain, particularly the relationship between ordering lead time and pick-up lead time. Managers, when faced with a longer ordering lead time, can strategically use returns to their advantage by implementing policies that encourage customers to return products if they are not satisfied. This can help reduce the need for excess inventory and mitigate the BWE. Conversely, when faced with a longer pick-up lead time, managers should focus on minimizing returns by improving product quality, providing accurate product information, and enhancing customer satisfaction. By understanding the nuanced impact of return rates on the BWE and inventory costs, supply chain managers can make informed decisions that align with their strategic goals. Whether the focus is on leveraging returns to reduce costs or minimizing returns to mitigate the BWE, the insights provided by this research offer actionable guidance for enhancing supply chain performance in omnichannel retail environments.

(3) The inventory costs and BWE in the omnichannel supply chain are consistently influenced by the price sensitivity coefficient, regardless of the interplay between the ordering lead time and the pick-up lead time. Across different retail channels, products with higher demand elasticity typically experience increased inventory costs and a more pronounced BWE. By understanding the consistent impact of the price sensitivity coefficient on inventory costs and the BWE, retailers can implement strategic measures to optimize pricing, manage inventory more effectively, and reduce demand variability. Specifically, retailers can implement several strategic measures. First, adopting optimized pricing strategies, such as dynamic pricing, can help mitigate the impact of price fluctuations on demand, thereby reducing the price sensitivity coefficient. Second, dynamic inventory management can be employed to adjust inventory levels in real-time based on the price sensitivity coefficient and demand elasticity, minimizing the risk of overstock and stockouts. Finally, demand management strategies, including promotional activities and market research, can be utilized to better manage demand variability and reduce overall uncertainty. These insights offer actionable guidance for enhancing supply chain performance in omnichannel retail environments, ultimately leading to improved efficiency and profitability.

(4) Retailers can significantly reduce expected inventory costs and enhance supply chain efficiency in the omnichannel context by shortening the pick-up lead time in the BOPS channel. Specifically, a smaller pick-up lead time helps to minimize inventory overstock, increase inventory turnover rates, and optimize inventory management, thereby improving overall supply chain efficiency. To achieve these benefits, retailers can streamline offline pick-up procedures to reduce consumer wait times and, consequently, shorten the pick-up lead time. Moreover, omnichannel retailers should also provide clear and timely information to customers regarding their order status and pick-up availability. This can reduce customer frustration and enhance overall satisfaction. By integrating these strategies, retailers can develop a more efficient pick-up process, which ultimately enhances operational efficiency and strengthens customer loyalty. In addition to optimizing pick-up lead times, managers can effectively mitigate inventory costs by strategically reducing the ordering lead time. This can be accomplished through real-time data sharing and enhanced information transparency, which help to minimize demand information variability and improve supply chain collaboration. Enhanced transparency also fosters better collaboration among supply chain partners, ensuring that all parties are aligned and can respond quickly to changes in demand. Establishing close cooperative relationships with suppliers to facilitate just-in-time replenishment is also recommended as a means of reducing the ordering lead time. These strategic initiatives collectively offer actionable guidance for retailers seeking to enhance their supply chain performance and overall operational efficiency, ultimately leading to a more agile and cost-effective omnichannel operation.

6.2. Limitation

The main limitations of this study are fivefold.

First, we focus solely on returns and cancellations in the BOPS channel, neglecting returns across other retail channels. Extending the model to include returns from all channels would require substantial modifications, but it would enable a comparative analysis of consumer return behaviors across various channels and elucidate their effects on the performance of the omnichannel supply chain. Second, we assume normal distributions for demand and price processes due to the mathematical convenience and well-understood properties, which simplify analysis. However, more complex models, such as autoregressive moving average models, may better capture real-world dynamics and should be considered in future research. Moreover, this study assumes prices are determined in a perfectly competitive market. However, real markets often exhibit imperfect competition, where firms can influence prices due to market power. This assumption may not fully reflect the strategic interactions and pricing dynamics in actual markets. Thus, integrating models of imperfect competition in future research can offer a more realistic portrayal of market dynamics. Third, we assume the use of MMSE for demand forecasting, which, despite its theoretical optimality, may not be as practical as MA or ES methods. Future research should compare various forecasting techniques to identify the most effective method for minimizing the BWE in omnichannel supply chains. Fourth, the model assumes that returns are managed efficiently without additional costs. In practice, handling returns can involve significant logistical and financial costs. Incorporating return handling costs could provide a more comprehensive understanding of the implications of returns in omnichannel retailing and represents a promising avenue for future research. Finally, this study neglects stockouts in supply chain modeling, which simplifies the analysis but may not reflect practical scenarios. Stockouts can lead to lost sales and customer dissatisfaction, impacting both revenue and customer loyalty. Examining the effects of stockouts on omnichannel supply chain performance is a worthwhile pursuit for future research.

Overall, acknowledging these limitations offers valuable direction for subsequent research endeavors.