Optimal Capacity Decision-Making of Omnichannel Catering Merchants Considering the Service Environment Based on Queuing Theory

Abstract

Many catering merchants are currently adopting omnichannel sales methods. At a restaurant, customers can place their orders, and while waiting for the food to be completed, they can enjoy the restaurant’s service environment. Online customers, by placing orders on various apps, can skip the queue, increasing their utility by reducing the opportunity cost, but they cannot enjoy the restaurant’s service environment. For merchants, it is important to adequately address how to choose the optimal capacity to provide services to customers in different channels and obtain optimal profits based on the service environment. In this paper, we establish a stylized theoretical model to study the impact of the service environment on customers’ shopping behavior online and offline, and we study how merchants can formulate the optimal capacity. Our main results are as follows: First, improving the service environment was conducive to reducing the merchants’ safety capacity and could help them obtain higher profits. In short-term omnichannel operations, merchants obtained higher profits than in traditional offline single-channel operations. Improving the service environment increased profits, but the profit difference between short-term omnichannel and offline single-channel operations gradually decreased. Finally, in long-term operations, the process of online customers transferring to offline channels based on a high-quality service environment might be detrimental to merchants’ profits.

1. Introduction

1.1. Backgrounds and Motivation

The development of Internet technologies and e-commerce has promoted the development of online channels for many businesses. Online channels not only bring more customers to firms, but also bring convenience to customers. In the traditional restaurant environment, customers enter a restaurant and place orders with the help of front-end servers. Then, the orders are sent to the kitchen and the cooks prepare the food, while the customers wait in the restaurant for the food to be finished. The quality of the environment in the restaurant and the ability of the service staff greatly affect the perceived utility for customers. As studied by Yuan et al. [1], a high-quality service environment can provide customers with a better experience, which reduces their delay sensitivity and makes them more willing to wait in the queue. However, a poor waiting environment will make customers leave or reduce their desire to shop. This type of service environment is composed of two categories. The first is the physical environment, such as the decor of the restaurant, the atmosphere, etc. The world’s top restaurants, such as Sublimotion in Spain, provide an unparalleled dining environment. The second category is social; that is, the attitude and ability of the service staff are the factors [2,3,4]. The most notable example is the Haidilao hot pot restaurants, which are famous for the unique and high-quality service their staff provides. In actual operation, a high-quality physical environment and the ability and attitude of the service staff are indispensable, and customers will have a comprehensive perception of these while waiting and dining. Therefore, in this paper, we did not distinguish environments with different attributes, but instead unified them into the service environment.

Today’s catering industry is focused on developing online channels, especially with the development of platforms such as Meituan, Ele.me, and Uber Eats, which have brought omnichannel catering to a new level. Unlike traditional offline customers, online customers can order food on an app, then wait for a delivery or go to the store or restaurant to pick up the order (buy online and pick up in store, BOPS). Taking McDonald’s as an example, customers can choose to dine in or order online. The customers can then choose delivery service or go to the store to pick up the order themselves. As a result, online customers can skip the queue in McDonald’s at the ordering stage and can wait for the food to be delivered to their home or office, which reduces the opportunity cost for merchants in terms of customer waiting costs [5]. One of the main disadvantages of online ordering is that customers are not able to enjoy the high-quality service environment in restaurants. This shortcoming has different impacts on customers’ online shopping behavior. In the short term, online customers may not understand the environment of the restaurant; for example, they cannot know the specific service environment of a newly-opened restaurant. In the long term, various social platforms and word-of-mouth provide opportunities for online customers to learn about restaurants. Online customers can learn about the service environment from apps such as Twitter, WeChat, and Instagram, and gradually change their consumption tendency, becoming offline customers. Therefore, merchants using omnichannel catering need to fully consider the impact of the service environment on customer behavior online and offline in different time spans and manage their capacity accordingly.

1.2. Research Questions and Main Findings

The main problems that were addressed in this paper are as follows: (1) How does the service environment affect the capacity formulation of merchants when using omnichannel catering? (2) Can the service environment foster higher customer demand and bring profits to merchants using the omnichannel approach? (3) What are the differences in merchants’ capacity decisions and profits based on the service environment during operations over different time spans?

To address the above questions, we developed a stylized queue model in which a restaurant or caterer serves sensitive customers who are waiting. The restaurant’s service system is modeled as a two-stage tandem queuing network. In this system, customers can place orders through the front-end staff or through apps, and then wait while their orders are delivered to the kitchen and their food is prepared. While waiting, offline customers can enjoy the restaurant’s high-quality service environment and experience additional satisfaction. Therefore, the restaurant needs to choose the optimal capacity level at both the ordering and production stages based on the impact of the service environment in order to maximize its profit margin. In this study, we first examined the impact of the service environment on online and offline customers’ shopping behaviors in an omnichannel mode. We then analyzed the impact on the short- and long-term operations. We compared the capacity of restaurants or caterers using the omnichannel approach with those using the traditional offline single-channel approach. We studied the ways that omnichannel merchants should make optimal capacity decisions based on the service environment in different periods to obtain profits.

This paper has several interesting findings: (1) The service environment can effectively reduce the safety capacity, thereby controlling a merchant’s investment in safety capacity. (2) With an improved service environment, the profit difference between the short-term omnichannel approach and the traditional offline single-channel approach gradually decreases. (3) A better service environment is more conducive to reduced safety capacity for omnichannel merchants in long-term operation, but the resulting transfer of online customers to offline channels is detrimental to the merchants’ profit.

1.3. Contributions and Novelties

This paper makes several novel contributions. First, unlike other research, we have extended the impact of the service environment to omnichannel marketing. Based on the service environment, the operation of catering merchants using the omnichannel approach has been analyzed, and the omnichannel theory is enriched. Second, the impact of the service environment on the capacity management of merchants adopting different channels was studied, which has been a less-considered issue in the existing literature. Third, we compared the differences in the decision-making and profits of catering merchants based on the service environment in different channels and operating cycles, which is beneficial information that enables merchants to make optimal decisions to obtain profits in different situations. All of our conclusions are drawn from a theoretical analysis and rigorous proof (see Appendix A). The results are meaningful.

The follow-up research in this paper is organized as follows: In Section 2, we review the literature on omnichannel catering, the service environment, and the capacity management of queues. In Section 3, we construct a stylized base model to study the impact of the service environment on the merchant’s capacity management in the traditional catering industry. In Section 4, we study the impact of the service environment provided by omnichannel catering merchants on the online and offline shopping behavior of customers in the short term, and we analyze the optimal capacity formulation at different stages. In Section 5, we extend the omnichannel approach from short-term to long-term operations, and analyze online customers’ shopping behaviors to capture their characteristics during long-term operations. In Section 6, we perform a numerical analysis through theoretical models of offline single-channel, short-term omnichannel, and long-term omnichannel approaches. In Section 7, we present our conclusions. A flow diagram of the work is shown in Figure 1.

2. Literature Review

The study in this paper is related to omnichannel catering, the service environment, and capacity management of queuing. In omnichannel catering, Namkung and Jang [6] believed that food quality and appearance are the main factors affecting customer satisfaction. In addition, Ha and Jang [2] and Kisang et al. [3] believed that the service ability and attitude of servers also greatly affected customer satisfaction and loyalty. In the process of ordering and being served food, customers pay special attention to the attitudes of service staff. Bowden-Everson et al. [7] used structural equation modeling to study the relevant factors affecting customer loyalty in the catering industry and found that the experience customers have with the service process greatly affect their loyalty. The study of Yuan et al. [1] also observed that this kind of service experience comes not only from the service staff, but also from the environment. For this purpose, we collectively refer to this as the service environment.

With the development of Internet technologies and various food delivery platforms, omnichannel catering has developed and received widespread attention [8]. In online channels, customer reviews and online advertisements can often affect the behavior of other customers and profits of merchants [9]. Simonson and Rosen’s research [10] reported that 30% of American consumers first start online shopping by browsing product information and reviews on Amazon. Some scholars have also paid attention to the delivery process when using online channels. Yan et al. [11] constructed a food delivery network (foodnet) through spatial crowdsourcing (SC), where customers are extremely sensitive to the delivery time; hence, fast delivery is important. The profit that can be obtained by using different delivery methods was also the focus of study [12]. Pei et al. [13] used a BP neural network model to evaluate customer experiences with online channels in the catering industry, providing a new method for measuring the degree of customer experience. Zhao and Yang [12] conducted a related study on food delivery services in the catering industry using an O2O business model. Their results showed that food delivery platforms can achieve higher total profits by adopting self-delivery. In contrast to this line of research, we focused on analyzing the impact of the service environment on the optimal capacity of omnichannel catering merchants at different stages using a theoretical model. This makes the specific capacity formulation more realistic, thereby helping catering merchants develop their businesses better in the omnichannel mode.

The research on the service environment originated from Kotler’s study [14], in which he defined the physical environment that can enhance the customer’s purchase intention as “atmospherics”, which is an important tool in marketing. Bitner [15] and Siew et al. [16] analyzed the impact of the physical attributes of changes in the service environment on customers’ use of the system and willingness to pay. Arnould et al. [4] added social environment elements to the definition of service environment derived from the previous studies. Ha and Jang [2] and Kisang et al. [3] introduced social environment factors such as the attitude and ability of servers in the catering industry to this theory, which provided a basis for the catering industry to improve service quality in order to attract customers. Dong and Siu [17] used a service scenario scale to measure the intensity of two types of service environments, substantive and communicative, and to measure the service environment’s intensity and effect. Yuan et al. [1] studied the impact of the service environment on customer waiting sensitivity based on the queuing model; they also analyzed the service providers’ optimal decision-making processes and profits based on the service environment under a monopoly or duopoly.

It can be seen from the above review that most recent studies have focused on measuring the service environment and its impact on the customer waiting process, while we have paid more attention to the role of the service environment in the omnichannel context. Therefore, the contribution of this paper to the literature is mainly an extension of the impact of the service environment on omnichannel marketing. At the same time, we have analyzed the relationship between online and offline customers’ shopping behaviors and the service environment in the catering industry, especially with regard to the proportion of online and offline customers in long-term operations. This further expands the scenarios for application in the service environment.

One of the core issues in this paper is the merchant’s formulation of capacity, and many studies have been published works regarding this problem. Mendelson [18] and Chen and Frank [19] studied the impact of wait-sensitive customers on the capacity of queuing systems. Another aspect to note is that capacity management is related to the labor force. As a result, scholars have also conducted studies on labor management in the on-demand economy [20,21]. Bassamboo et al. [22] studied a staffing problem in which the average arrival time to jobs was random and found that simple capacity specifications derived from a relevant newsvendor problem were very accurate. Barrera and Garcia [23] studied the problem of limited capacity allocation, focusing on supplier capacity allocation in congested service systems such as medical care and catering. Gao and Su [24] analyzed the formulation of omnichannel catering capacity with the use of online ordering and offline self-ordering technologies and developed a stylized theoretical model to study the impact of these technologies on the customer demand, employment level, and merchants’ profits. Because changes in customer demand have a certain seasonality [25], the methods of formulating reasonable business capacity to hedge such changes is a concern. Ju et al. [26] took high-demand products during the COVID-19 pandemic as the research object and analyzed government subsidies under the impact of production capacity. Unlike these types of studies, we chose to contribute to the literature by studying the impact of the service environment in brick-and-mortar stores on the optimal capacity decision of merchants using the omnichannel approach. Because the service environment affects the shopping behavior of customers, and because shopping behavior affects the capacity management of merchants, knowing how to manage the capacity according to the service environment is an urgent problem for merchants. Our research is innovative precisely because it covers the impact of the service environment on the management of capacity, which is of great significance for the development of related theories.

3. Base Model

In this section, we first build the base capacity model for the catering industry using the traditional offline single-channel system. Assuming that the catering merchant only provides offline food service for customers, the service system is a two-stage tandem queueing network. In stage one, customers can place orders through front-end servers, and the orders are sent to the kitchen for food preparation in stage two. The catering merchant can obtain net revenue $r$ from each customer, and they provide services for customers at the rates of ${\mu }_1$ and ${\mu }_2$ as a representation of the service and production capacity at stage one and two, respectively. We assume that the cost the merchant pays to maintain capacity ${\mu }_i$ at stage $i=1, 2$ is $c_i{\mu }_i$, where the unit capacity cost $c_i>0$ [24]. In order to ensure that it is profitable for the merchant to serve each customer, we assume that $r-c_1-c_2>0$. In addition, traditional offline customers who dine at the restaurant can experience its environment while waiting. Here, it is assumed that the quality of the service environment as perceived by the customer is $\gamma$, and the customer’s basic expectation of the service environment is ${\gamma }_0$. When $\gamma >{\gamma }_0$, customers receive positive utility from the high-quality service environment; otherwise, they receive negative utility.

We can set the scale of customers in the entire market as 1, i.e., normalized processing, in which the proportion of offline customers is $\theta$ and the proportion of online customers is $1-\theta$. (Although there are many customers who prefer online shopping in the current market, there are still some merchants who choose not to sell online, which makes them lose the online customers. Thus, we did not study the shopping behavior of online customers and decision-making in the traditional offline single-channel mode for the time being). Each customer can be regarded as infinitesimal relative to the overall market, meaning a single customer has no influence on the market. Customers must wait at two stages before receiving their food: $w_i\left({\mu }_i,\mathsf{\Lambda }\right)$ is the waiting time at stage $i=1,2$ given the capacity ${\mu }_i$ and total demand $\mathsf{\Lambda }$. Each customer’s shopping rate is assumed to linearly decrease with increased waiting time, but to linearly increase while waiting if the service environment is improved. Each customer’s shopping rate $\lambda$ in the offline single-channel can be expressed as follows:

$$\lambda ={\left[\alpha -\beta \left(w_1\left({\mu }_1,\Lambda \right)+w_2\left({\mu }_2,\Lambda \right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,\Lambda \right)+w_2\left({\mu }_2,\Lambda \right)\right)\right]}^{+}.$$

(1)

where coefficients $\alpha \mathrm{and}\beta (>0)$ represent the base shopping rate and waiting sensitivity, respectively, and $\Delta \gamma =\gamma -{\gamma }_0$ is the service environment effect, that is, the utility obtained by the customer’s perception of the service environment per unit time. Kotler [14] referred to the physical environment that can improve customer utility and shopping intention as “atmospherics”, while Yuan et al. [1] noted that the service environment can effectively reduce the waiting cost of customers in the waiting process, thereby improving their utility. Essentially, the service environment effect refers to the change in customer utility brought about by the service environment; a high-quality service environment will bring an increase in utility for customers, while a poor-quality service environment will bring a decrease in utility. At the same time, this paper expresses the service environment effect in a linear way. In the traditional offline single-channel mode, only offline customers make purchases, so the total demand $\Lambda =\lambda \theta$. In the M/M/1 queue, the waiting time of customers can be denoted as $w_1\left({\mu }_1,\lambda \theta \right)=1/\left({\mu }_1-\lambda \theta \right)$ and $w_2\left({\mu }_2,\lambda \theta \right)=1/\left({\mu }_2-\lambda \theta \right)$. According to the shopping rate function, the merchant chooses the capacity at stage $i=1,2$ to maximize profit, i.e.,

$$\begin{gathered} \underset{\lambda <{\mu }_1,\lambda <{\mu }_2}{\max }\left\{r\theta \lambda -c_1{\mu }_1-c_2{\mu }_2\right\} \\ s.t. \lambda ={\left[\alpha -\beta \left(w_1\left({\mu }_1,\lambda \theta \right)+w_2\left({\mu }_2,\lambda \theta \right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,\lambda \theta \right)+w_2\left({\mu }_2,\lambda \theta \right)\right)\right]}^{+} \end{gathered}$$

(2)

Proposition 1. 

To solve Formula (2) when the service environment level$\gamma$is not so high, i.e.,$\gamma <\beta +{\gamma }_0$, there is a unique optimal solution that enables the merchant to obtain a positive optimal profit:

$${\mu }_1^b=\theta {\lambda }^b+\sqrt{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_1},$$

(3)

$${\mu }_2^b=\theta {\lambda }^b+\sqrt{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_2},$$

(4)

where

$${\lambda }^b=\alpha -\left(\beta +{\gamma }_0-\gamma \right)\sqrt{\frac{c_1}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}-\left(\beta +{\gamma }_0-\gamma \right)\sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}>0.$$

(5)

The above results provide a theoretical basis for catering merchants to choose the optimal capacity when considering the service environment at the ordering and food preparation stages. The expression of the optimal capacity follows the rule of thumb for capacity planning [22]; that is, the optimal capacity should include the base capacity $\theta {\lambda }^b$ to meet the base demand and the safety capacity ${\mu }_{safe,i}^b$ to hedge against changes in demand; that is, ${\mu }_{safe,i}^b=\sqrt{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_i}$ at stage $i=1,2$. More attention is paid to changes in the safety capacity as this involves the ability of merchants to hedge against changes in the market demand. The following lemmas are proposed.

Lemma 1.

Sensitivity analysis of the safety capacity ${\mu }_{safe,i}^b$yields$\frac{\partial {\mu }_{safe,i}^b}{\partial c_i}<0$,$\frac{\partial {\mu }_{safe,i}^b}{\partial \beta }>0$,$\frac{\partial {\mu }_{safe,i}^b}{\partial \gamma }<0$,$\frac{\partial {\mu }_{safe,i}^b}{\partial {\gamma }_0}>0$.

From Lemma 1, we know that a lower unit cost $c_i$ can enable the merchant to choose a higher safety capacity. When customers are more concerned about the waiting time, i.e., when $\beta$ increases, the merchant should then improve the corresponding safety capacity to hedge against changes in customer demand. In addition, if the environment of the restaurant is of high quality, i.e., $\gamma$ is high, then the merchant does not need to set too high of a safety capacity, which can save more cost. However, if customers care more about the quality of the environment, there will be a higher ${\gamma }_0$, so it is necessary to set a higher safety capacity to ensure a shorter waiting time, especially in the case of ${\gamma }_0>\gamma$.

Lemma 2.

The relationship between the customer’s waiting time and the merchant’s capacity at stage $i=1,2$can be expressed as$w_1^b=\frac{1}{{\mu }_{safe,1}^b}=\sqrt{\frac{c_1}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}$,$w_2^b=\frac{1}{{\mu }_{safe,2}^b}=\sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}$.

That is, the waiting time of customers at different stages is actually directly affected by the safety capacity. When the merchant chooses a higher safety capacity to hedge against changes in market demand, customers will experience less waiting time. This directly inverse relationship between waiting time and safety capacity provides a new explanation for the role of the safety capacity in queuing systems. That is, the safety capacity not only acts to hedge against changes in market demand, but also directly affects the waiting time of customers in the system. The service environment will therefore indirectly affect the customer’s wait time by directly affecting the safety capacity. Compared with the base capacity ${\lambda }^b$ which is used to meet the base demand, ${\lambda }^b$ does not affect the customer’s wait time. Therefore, throughout this paper, we will focus on the analysis of the safety capacity.

According to Proposition 1, the optimal profit of the merchant in this case can be expressed as follows:

$${\pi }^b=\alpha \theta \left(r-c_1-c_2\right)-2\sqrt{c_1\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}-2\sqrt{c_2\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}.$$

(6)

The term $\sqrt{c_i\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)},i=1,2$ in the above formula is the cost $c_i{\mu }_i^{safe}$ required to maintain the corresponding safety capacity at each stage. When customers pay more attention to the waiting time, i.e., when $\beta$ increases, the merchant will improve the safety capacity to quickly respond to customer demand, which will increase the merchant’s cost at each stage and lower the profit. When the actual service environment level $\gamma$ gradually rises above the expected ${\gamma }_0$, that is, when the service environment effect $\Delta \gamma$ gradually increases, the merchant’s profit can subsequently be increased. However, when $\gamma$ gradually falls below the expected ${\gamma }_0$, i.e., when $\Delta \gamma$ gradually decreases, the merchant’s profit will gradually decrease. Therefore, when merchants provide a higher quality service environment relative to customers’ expectations, the merchants can obtain higher profit.

When the actual service environment quality is much higher than the customer’s expectation, i.e., when $\gamma \ge \beta +{\gamma }_0$, although there is no optimal mathematical solution, it still has a certain practical significance. That is, when the service environment quality is much higher than customers’ expectations and waiting sensitivity, customers will receive higher utility from the increased waiting time, which enables merchants to minimize capacity to reduce costs. In this case, obtaining food may not be the customers’ main purpose, but rather to experience a high-quality casual service environment. This situation mostly occurs in restaurants such as Sublimotion and other top restaurants that provide an unparalleled dining environment. As these luxury restaurants do not offer omnichannel services and are not universal, they were not the focus of this paper. Thus, the assumption that $\gamma <\beta +{\gamma }_0$ was held throughout the study.

4. Decision-Making in the Short-Term Omnichannel Mode

4.1. Decision-Making

In this section, we consider decision-making for the omnichannel merchant. In contrast to traditional catering services, the development of online platforms and Internet technology and the expansion of online channels has become an important driving force in the development of food service firms. For example, KFC, McDonald’s, and other corporations have opened online channels, allowing customers to skip the ordering queue and not waste time during the ordering process in the restaurant. Especially during the COVID-19 outbreak, contactless delivery methods provided a boost for the expansion of online channels, making more customers willing to join. Omnichannel catering merchants operate online and offline channels at the same time, expanding their markets and providing better services to more customers. In the omnichannel mode, online customers (compared with offline customers) can skip the offline ordering queue and reduce their waiting time at the ordering stage; because they are waiting for food at home or in the office, they can engage in other work to reduce the opportunity cost. Offline customers care more about enjoying the restaurant’s high-quality environment and good service.

At the same time, merchants must pay attention to differences and mutual conversions between online and offline customers in short-term and long-term operations. In this section we prioritize the decision-making processes for omnichannel merchants in the short term. In this case, online customers do not perceive the quality of the restaurant’s environment, so their shopping rate and proportion are not affected by the service environment. In long-term operations, online customers can learn about the specific service environment of restaurants through various social platforms (TikTok, WeChat, Twitter, and Instagram) and advertising, which may change their behavior and scale of their shopping. Decisions about long-term operations are discussed in the next section.

In omnichannel operation, there are two types of customers in the market who have a similar perception of the service environment: (1) online customers, who place orders through apps and wait for the food at home or in the office, thus unable to perceive the restaurant’s service environment; and (2) offline customers, who place orders and wait for their food in a restaurant, and thus can perceive the service environment. As in the previous section, the proportion of offline customers is represented by $\theta$ and the proportion of online customers is $1-\theta$. The proportion of customers in different channels remains fixed in the short term model.

Our model can reflect the characteristics of online and offline customers in different channels. First, after online customers order food on apps, they can wait for the food at home or in the office and perform other tasks while they wait. According to a study by Kostecki [5], the waiting cost is made up of opportunity cost and anxiety cost; anxiety cost is described as the physical or psychological discomfort and impatience experienced while waiting. Because online customers can still perform other tasks, they only bear the anxiety cost. Assuming that the proportion of anxiety cost to the overall waiting cost is $\delta$, and $\delta \in \left(0,1\right]$, then the waiting sensitivity of online customers is $\delta \beta \le \beta$. This decrease in waiting sensitivity can be called the convenience effect. Online customers only need to wait for the food to be completed, so their shopping rate can be expressed as follows:

$${\lambda }_{\mathrm{o}}={\left[\alpha -\delta \beta w_2\left({\mu }_2,\Lambda \right)\right]}^{+}$$

(7)

In short-term operations, offline customers still need to place orders at stage one and wait for the food at stage two. Therefore, for offline customers, the total waiting time is still the sum of the waiting time of both stages. However, at stage one, only offline customers need to wait; the demand satisfied at stage one is represented by $\theta {\lambda }_s$, where ${\lambda }_s$ represents the offline customers’ shopping rate. Thus, we can obtain the waiting time of offline customers at stage one using $w_1\left({\mu }_1,{\Lambda }_1\right)=1/\left({\mu }_1-\theta {\lambda }_s\right)$. At stage two, production in the kitchen needs account for the needs of both online and offline customers, which means that the total demand can be expressed as ${\Lambda }_2=\theta {\lambda }_s+\left(1-\theta \right){\lambda }_o$ and the waiting time can be expressed as $w_2\left({\mu }_2,{\Lambda }_2\right)=1/\left({\mu }_2-\left(\theta {\lambda }_s+\left(1-\theta \right){\lambda }_o\right)\right)$. Finally, the offline customers’ shopping rate can be expressed as follows:

$${\lambda }_s={\left[\alpha -\beta \left(w_1\left({\mu }_1,{\mathsf{\Lambda }}_1^{OS}\right)+w_2\left({\mu }_2,{\mathsf{\Lambda }}_2^{OS}\right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,{\mathsf{\Lambda }}_1^{OS}\right)+w_2\left({\mu }_2,{\mathsf{\Lambda }}_2^{OS}\right)\right)\right]}^{+},$$

(8)

where ${\mathsf{\Lambda }}_1^{OS}=\theta {\lambda }_s$ and ${\mathsf{\Lambda }}_2^{OS}=\theta {\lambda }_s+\left(1-\theta \right){\lambda }_o$. Merchants choose the corresponding optimal capacity ${\mu }_1$ and ${\mu }_2$ to maximize profits, i.e.,

$$\begin{gathered} \underset{\theta {\lambda }_s<{\mu }_1,\theta {\lambda }_s+\left(1-\theta \right){\lambda }_{\mathrm{o}}<{\mu }_2}{\max }\left\{r\left(\theta {\lambda }_s+\left(1-\theta \right){\lambda }_{\mathrm{o}}\right)-c_1{\mu }_1-c_2{\mu }_2\right\} \\ s.t. {\lambda }_{\mathrm{o}}={\left[\alpha -\delta \beta w_2\left({\mu }_2,\left(\theta {\lambda }_s+\left(1-\theta \right){\lambda }_o\right)\right)\right]}^{+}, \\ {\lambda }_s={\left[\alpha -\beta \left(w_1\left({\mu }_1,{\mathsf{\Lambda }}_1^{OS}\right)+w_2\left({\mu }_2,{\mathsf{\Lambda }}_2^{OS}\right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,{\mathsf{\Lambda }}_1^{OS}\right)+w_2\left({\mu }_2,{\mathsf{\Lambda }}_2^{OS}\right)\right)\right]}^{+}, \end{gathered}$$

(9)

where ${\mathsf{\Lambda }}_1^{OS}=\theta {\lambda }_s$ and ${\mathsf{\Lambda }}_2^{OS}=\theta {\lambda }_s+\left(1-\theta \right){\lambda }_o$. From the solution of the above model, the following propositions can be made.

Proposition 2.

Under the assumption of $\gamma <\beta +{\gamma }_0$, there is a unique optimal solution to optimize the merchant’s profit:

$${\mu }_1^{OS}=\theta {\lambda }_s+\sqrt{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_1},$$

(10)

$${\mu }_2^{OS}=\theta {\lambda }_s+\left(1-\theta \right){\lambda }_{\mathrm{o}}+\sqrt{\left(\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)\right)/c_2},$$

(11)

where${\lambda }_s^{OS}=\alpha -\sqrt{\frac{c_1\left(\beta +{\gamma }_0-\gamma \right)}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}-\left(\beta +{\gamma }_0-\gamma \right)\sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)}}>0$and${\lambda }_o^{OS}=\alpha -\delta \beta \sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)}}>0.$

The safety capacity at stage $i=1,2$ is ${\mu }_{safe,1}^{OS}=\sqrt{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_1}$ and ${\mu }_{safe,2}^{OS}=\sqrt{\left(\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)\right)/c_2}$, respectively. (Note: Superscript $OS$ indicates that the omnichannel operation is in the short term.)

Lemma 3. 

A sensitivity analysis of safety capacity${\mu }_{safe,1}^{OS}$

and ${\mu }_{safe,2}^{OS}$yields (i)$\frac{\partial {\mu }_{safe,1}^{OS}}{\partial c_i}<0$,$\frac{\partial {\mu }_{safe,1}^{OS}}{\partial \beta }>0$,$\frac{\partial {\mu }_{safe,1}^{OS}}{\partial \gamma }<0$,$\frac{\partial {\mu }_{safe,1}^{OS}}{\partial {\gamma }_0}>0$. (ii)$\frac{\partial {\mu }_{safe,2}^{OS}}{\partial c_i}<0$,$\frac{\partial {\mu }_{safe,2}^{OS}}{\partial \beta }>0$,$\frac{\partial {\mu }_{safe,2}^{OS}}{\partial \gamma }<0$,$\frac{\partial {\mu }_{safe,2}^{OS}}{\partial {\gamma }_0}>0$,$\frac{\partial {\mu }_{safe,2}^{OS}}{\partial \delta }>0$.

It can be seen from Lemma 3 that whether the result is ${\mu }_{safe,1}^{OS}$ or ${\mu }_{safe,2}^{OS}$, they both change the same way in response to $c_i$, $\beta$, $\gamma ,$ and ${\gamma }_0$. Moreover, but this change is identical to Lemma 1, which suggests that the safety capacity is similar in offline single-channel and short-term omnichannel modes. The difference in the short-term omnichannel mode is that ${\mu }_{safe,2}^{OS}$ will also increase with an increased proportion of anxiety cost $\delta$. This is because once online customers are more anxious in the process of waiting, they are likely to be dissatisfied with the merchant’s service and reduce their shopping rate, which prompts the merchant to improve the safety capacity to ensure that their waiting time will not be perceived as too long.

4.2. Comparative Analysis

By comparing short-term omnichannel and offline single-channel operations, we can obtain the following theorem:

Lemma 4. 

In short-term omnichannel operations, the following observations hold: (i) At stage one, ${\mu }_{safe,1}^{OS}={\mu }_{safe,1}^b$; at stage two, ${\mu }_{safe,2}^{OS}>{\mu }_{safe,2}^b$; and if the merchant provides a better service environment, i.e., if $$increases, then ${\mu }_{safe,2}^{OS}-{\mu }_{safe,2}^b$ also increases. (ii) $w_1^{OS}=1/{\mu }_{safe,1}^{OS}$ and $w_2^{OS}=1/{\mu }_{safe,2}^{OS}$. Thus, $w_1^{OS}=w_1^b$ at stage one, and $w_2^{OS}<w_2^b$ at stage two, and $w_2^b-w_2^{OS}$ increases as $\gamma$increases.

According to Lemma 4 (i), we can know that regardless of the service environment, the optimal safety capacity of the merchant in the short-term omnichannel mode at stage one will be the same as that of the traditional offline single-channel mode. Therefore, from Lemma 4 (ii), it can also be known that no matter what mode the offline customer is in, the waiting time at stage one will be fixed. This is mainly because at stage one, the main service object of the front-end staff is the offline customer base, and in short-term operations, the introduction of online customers will not change the proportion of offline customers.

At stage two, the entry of online customers increases the production burden of the kitchen, so ${\mu }_{safe,2}^{OS}$ in the omnichannel mode is greater than ${\mu }_{safe,2}^b$ to better hedge the changing demands of online and offline customers. Once the quality of the service environment is improved, offline customers can enjoy a better service environment, and the safety capacity level chosen by the omnichannel merchant will be significantly higher than in offline single-channel mode, i.e., $\frac{\partial \left({\mu }_{safe,2}^{OS}-{\mu }_{safe,2}^b\right)}{\partial \gamma }>0$. This indicates that although the increase in $\gamma$ can promote a simultaneous decrease in ${\mu }_{safe,2}^{OS}$ and ${\mu }_{safe,2}^b$, the decrease in ${\mu }_{safe,2}^{OS}$ in the omnichannel mode is relatively smaller. Unlike offline single-channel merchants, who only need to consider offline customers, omnichannel merchants must consider the demands of both online and offline customers at the food production stage. As a result, although improving the service environment can prompt merchants to reduce the safety capacity, omnichannel merchants must also consider the demands of online customers, meaning that they cannot reduce the safety capacity as quickly as in the offline single-channel mode. This also leads to the result of the customer waiting time $w_2^{OS}$ being less in the omnichannel model, compared with the offline single-channel mode. As observed with merchants that operate both offline and online channels (e.g., KFC and McDonald’s), once online customers join, the model inevitably increases the production capacity burden on the kitchen. However, if there is a high-quality service environment, such as one that provides rest areas, entertainment areas, etc. [1], it will effectively reduce the waiting cost of customers, and thus they will be more willing to wait. The kitchen cannot increase production capacity excessively. Compared with the offline single-channel mode, the waiting cost of online customers in the omnichannel mode will not be reduced due to the service environment; thus, omnichannel merchants cannot quickly reduce the safety production capacity.

According to Proposition 2, the merchant’s profit in short-term omnichannel operations is expressed by:

$$\begin{gathered} {\pi }^{OS}=\alpha \theta \left(r-c_1-c_2\right)+\alpha \left(1-\theta \right)\left(r-c_2\right)-2\sqrt{c_1\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)} \\ -2\sqrt{c_2\left(\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)\right)}. \end{gathered}$$

(12)

Lemma 5.

(i) Compared with the traditional offline single-channel mode, the offline customer shopping rate and total demand also increase in the omnichannel mode, i.e.,${\lambda }_s^{OS}>{\lambda }^b$ and ${\Lambda }^{OS}>{\Lambda }^b$. (ii) With an improved service environment level$\gamma$, the merchant’s profit in the short-term omnichannel mode will increase accordingly, i.e.,$\frac{\partial {\pi }^{OS}}{\partial \gamma }>0$. (iii) In comparison to the traditional offline single-channel mode,${\pi }^{OS}>{\pi }^b$in the short-term omnichannel mode. The value of${\pi }^{OS}-{\pi }^b$will decrease with the gradual improvement of the service environment, i.e.,$\frac{\partial \left({\pi }^{OS}-{\pi }^b\right)}{\partial \gamma }=\frac{\partial {\pi }^{OS}}{\partial \gamma }-\frac{\partial {\pi }^b}{\partial \gamma }<0$.

Combining the analysis of Lemma 4 and 5, we can know that in the short-term omnichannel mode, the merchant’s safety capacity at stage one is the same as that of the offline single-channel mode. At stage two, the merchant chooses a higher level of safety capacity to meet the demand of offline and online customers, which will also ensures a shorter waiting time for customers in the short-term omnichannel mode at stage two, i.e., $w_2^{OS}=\sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\delta \beta \left(r-c_2\right)}}<w_2^b=\sqrt{\frac{c_2}{\theta \left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)}}$. Customers in the queueing system are wait-sensitive, thus the shorter waiting time can improve the offline customer shopping rate at stage two. In addition, merchants in the short-term omnichannel mode will also experience demand from online customers, meaning that the total demand will be greatly increased compared with the offline single-channel mode. This also reveals that in the short term, an omnichannel mode can increase the customer shopping rate. According to Lemma 5 (iii), this increase in demand can bring a higher profit to the omnichannel merchant. We also know that the continuous improvement of service environment level $\gamma$ increases both ${\pi }^{OS}$ and ${\pi }^b$, but the increase of ${\pi }^b$ will be greater than that of ${\pi }^{OS}$. Therefore, the merchant’s profit in the offline single-channel mode will approach that of the short-term omnichannel mode. This is also true in reality; restaurants that pay special attention to the service environment can often obtain high profits from the offline channel. For example, at Sublimotion in Spain, the high-quality service environment attracts more customers and brings more profit to the restaurant. Therefore, even if it has opened an online channel, the offline channel remains its main profit channel.

5. Decision-Making in the Long-Term Omnichannel Mode

5.1. Decision-Making

In this section, we analyze the capacity decisions of omnichannel merchants in the long term. Compared with the short term, online customers learn relevant information about the service environment of the restaurants from social platforms such as WeChat, Twitter, and Instagram in long-term operations. Some online customers will gradually be attracted to restaurants with a high-quality service environment and become offline customers. In particular, customers will engage in rapid information exchange through various social media channels, such as through social networking sites, instant messaging tools, and virtual communities [27]. This method of social exchange can help customers better obtain information about the service environment of restaurants and helps to increase visibility for merchants. Therefore, with the gradual strengthening of the interaction of information, customers may convert from online to offline channels and vice versa.

We assume that the conversion rate in the long term for customers in both the online and offline channels is $\eta$ and $\eta \in \left[-1,1\right]$, where $\eta >0$ means that the online customers transfer to offline channels, and the number of transfers is $\left(1-\theta \right)\eta$. When $\eta <0$, it indicates the proportion of online customers who transferred from offline channels, and the number of online customers added to the online customer base is $\left(1-\theta \right)\eta$. Therefore, in the long-term equilibrium, the total number of online customers is $\left(1-\theta \right)\left(1-\eta \right)$, and the total number of offline customers is $\theta +\left(1-\theta \right)\eta$; the conversion rate $\eta$ may be affected by factors such as price, delivery capacity, geographic location, etc. Certain studies [28,29] have stated that the offline experience is used as a main marketing method to hedge against the impact of online channels. The offline experience enhances the attractiveness of physical stores, which makes online customers shift to the offline channel. A high-quality service environment also affects the offline experience of customers by attracting more customers to enter the offline channel. Therefore, there must be $\frac{\partial \eta }{\partial \gamma }>0$; that is, a better service environment will prompt online customers to transfer to offline channels. According to this observation, we can obtain the shopping rate of customers in different channels as follows:

The shopping rate of online customers is expressed by:

$${\lambda }_{\mathrm{o}}={\left[\alpha -\delta \beta w_2\left({\mu }_2,{\Lambda }^{O\mathrm{L}}\right)\right]}^{+}$$

(13)

and the shopping rate of offline customers is expressed by:

$${\lambda }_s={\left[\alpha -\beta \left(w_1\left({\mu }_1,{\Lambda }_1^{OL}\right)+w_2\left({\mu }_2,{\Lambda }_2^{OL}\right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,{\Lambda }_1^{OL}\right)+w_2\left({\mu }_2,{\Lambda }_2^{OL}\right)\right)\right]}^{+},$$

(14)

where ${\Lambda }_1^{OL}=\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s$, ${\Lambda }_2^{OL}=\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s+\left(1-\theta \right)\left(1-\eta \right){\lambda }_{\mathrm{o}}$.

Therefore, in the long term, the omnichannel merchant should choose the optimal capacity at each stage to maximize profit, i.e.,

$$\begin{gathered} \underset{\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s<{\mu }_1,{\Lambda }^{O\mathrm{L}}<{\mu }_2}{\max }\left\{r\left(\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s+\left(1-\theta \right)\left(1-\eta \right){\lambda }_{\mathrm{o}}\right)-c_1{\mu }_1-c_2{\mu }_2\right\} \\ s.t. {\lambda }_{\mathrm{o}}={\left[\alpha -\delta \beta w_2\left({\mu }_2,{\Lambda }^{O\mathrm{L}}\right)\right]}^{+}, \end{gathered}$$

(15)

$${\lambda }_s={\left[\alpha -\beta \left(w_1\left({\mu }_1,{\Lambda }_1^{OL}\right)+w_2\left({\mu }_2,{\Lambda }_2^{OL}\right)\right)+\left(\gamma -{\gamma }_0\right)\left(w_1\left({\mu }_1,{\Lambda }_1^{OL}\right)+w_2\left({\mu }_2,{\Lambda }_2^{OL}\right)\right)\right]}^{+}.$$

Proposition 3.

Under the assumption$\gamma <\beta +{\gamma }_0$, the above capacity decision model has a unique optimal solution, as follows:

$${\mu }_1^{OL}=\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s+\sqrt{\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_1},$$

(16)

$$\begin{gathered} {\mu }_2^{OL}=\left(\theta +\left(1-\theta \right)\eta \right){\lambda }_s+\left(1-\theta \right)\left(1-\eta \right){\lambda }_{\mathrm{o}} \\ +\sqrt{\left(\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\left(1-\eta \right)\delta \beta \left(r-c_2\right)\right)/c_2}, \end{gathered}$$

(17)

where${\lambda }_s^{OL}=\alpha -\sqrt{\frac{c_1\left(\beta +{\gamma }_0-\gamma \right)}{\left(\theta +\left(1-\theta \right)\eta \right)\left(r-c_1-c_2\right)}}-\sqrt{\frac{c_2{\left(\beta +{\gamma }_0-\gamma \right)}^2}{\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\left(1-\eta \right)\delta \beta \left(r-c_2\right)}}>0$and${\lambda }_o^{OL}=\alpha -\delta \beta \sqrt{\frac{c_2}{\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\left(1-\eta \right)\delta \beta \left(r-c_2\right)}}>0.$

According to Proposition 3, we know that in long-term operations, the safety capacity at each stage is ${\mu }_{safe,1}^{OL}=\sqrt{\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)/c_1}$, ${\mu }_{safe,2}^{OL}=\sqrt{\left(\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\left(1-\eta \right)\delta \beta \left(r-c_2\right)\right)/c_2}$. (Note: Superscript $OL$ indicates that the omnichannel operation is long-term.)

Sensitivity analysis of safety capacity ${\mu }_{safe,1}^{OL}$ and ${\mu }_{safe,2}^{OL}$ gives us the following lemma.

Lemma 6. 

(i)$\frac{\partial {\mu }_{safe,1}^{OL}}{\partial c_i}<0$, $\frac{\partial {\mu }_{safe,1}^{OL}}{\partial \beta }>0$,$\frac{\partial {\mu }_{safe,1}^{OL}}{\partial \gamma }<0$,$\frac{\partial {\mu }_{safe,1}^{OL}}{\partial {\gamma }_0}>0$, and$\frac{\partial {\mu }_{safe,1}^{OL}}{\partial \eta }>0$. (ii)$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial c_i}<0$,$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial \beta }>0$,$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial \gamma }<0$,$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial {\gamma }_0}>0$, and$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial \delta }>0$. When$\gamma \le \overline{\gamma }$, we have$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial \eta }>0$; and when$\gamma >\overline{\gamma }$, we have$\frac{\partial {\mu }_{safe,2}^{OL}}{\partial \eta }<0$, where$\overline{\gamma }=\beta +{\gamma }_0-\frac{\delta \beta \left(r-c_2\right)}{r-c_1-c_2}$.

In long-term omnichannel operations, whether it is ${\mu }_{safe,1}^{OL}$ or ${\mu }_{safe,2}^{OL}$, the changes of both with $c_i$, $\beta$, $\gamma$, and ${\gamma }_0$ are the same, and are similar to Lemma 1 and 3. As the conversion rate $\eta$ increases, the safety capacity at the ordering stage inevitably increases, i.e., $\frac{\partial {\mu }_{safe,1}^{OL}}{\partial \eta }>0$. As more customers flock to offline channels, it inevitably increases crowding during the ordering stage. Therefore, merchants must subsequently improve the safety capacity ${\mu }_{safe,1}^{OL}$ to hedge against changes in the number of offline channel customers. However, for ${\mu }_{safe,2}^{OL}$, the increase in conversion rate $\eta$ does not necessarily lead to an improvement in ${\mu }_{safe,2}^{OL}$; this change relationship will be affected by the service environment level. When the service environment level is low, i.e., $\gamma \le \overline{\gamma }$, the increased conversion rate will bring more offline customers. The offline customers cannot receive more utility from the service environment, which requires merchants to provide a higher safety capacity to reduce the waiting cost. Once the service environment level is high, more offline customers will be likely to consider the value of the service environment, meaning that merchants should also reduce the safety capacity so that customers can stay for longer periods of time. This facilitates a greater perceived utility from the service environment for customers. In the real world, many top restaurants with high-quality service environments tend to produce at a slower rate, which is also a contrast to fast-food restaurants.

5.2. Comparative Analysis

Compared with the safety capacity in short-term omnichannel operations, there are the following theorems:

Lemma 7. 

(i) In long-term omnichannel operations, when $\eta >0$, we have ${\mu }_{safe,1}^{OL}>{\mu }_{safe,1}^{OS}$, and when$\eta \le 0$, we have${\mu }_{safe,1}^{OL}\le {\mu }_{safe,1}^{OS}$. (ii) At stage two,$\overline{\gamma }=\beta +{\gamma }_0-\frac{\delta \beta \left(r-c_2\right)}{r-c_1-c_2}$. When$\gamma \le \overline{\gamma }$, we have${\mu }_{safe,2}^{OL}\ge {\mu }_{safe,2}^{OS}$; when$\gamma >\overline{\gamma }$, we have${\mu }_{safe,2}^{OL}<{\mu }_{safe,2}^{OS}$.

Unlike Proposition 1 and 2, the safety capacity at stage one will be affected by the conversion rate $\eta$. As online customers convert to offline customers, i.e., when $\eta >0$, merchants should choose a higher level ${\mu }_{safe,1}^{OL}$ to meet the customers’ needs. If the conversion rate $\eta <0$, offline customers will enter the online channel; thus, merchants should choose a lower level of safety capacity to serve fewer offline customers and reduce the corresponding costs.

Offline customers place orders in the restaurant and wait for the food to be prepared, and while waiting, they can enjoy the service environment. In contrast, online customers do not experience the service environment, but can enjoy the convenience of the process and reduce the opportunity cost by waiting for food at home or in the office. In Section 4, we referred to this as the convenience effect. Because the convenience effect mainly reduces the opportunity cost of the online customer waiting, it can be measured by $1-\delta$. Therefore, $\overline{\gamma }=\beta +{\gamma }_0-\frac{\delta \beta \left(r-c_2\right)}{r-c_1-c_2}$ in Lemma 7 (ii) can be regarded as the equilibrium between the environmental effect perceived by the offline customers and the convenience effect perceived by the online customers. With an improved convenience effect, i.e., with a decreased $\delta$, $\overline{\gamma }$ should subsequently increase to achieve equilibrium.

When the service environment quality is lower than the above equilibrium, that is, when $\gamma \le \overline{\gamma }$, the service environment effect will be lower than the convenience effect. Therefore, merchants should choose a higher safety capacity at stage two in long-term operations compared with short-term operations. Once $\gamma >\overline{\gamma }$, that is, when the service environment level is relatively high, merchants should choose a lower safety capacity for the long term. That is, in long-term operations, a higher service environment level can more effectively reduce the safety capacity. The conversion rate $\eta$ between customers interestingly does not affect the equilibrium $\overline{\gamma }$. This indicates that regardless of how the conversion rate changes, omnichannel merchants can more effectively reduce the cost of maintaining safety capacity in the case of $\gamma >\overline{\gamma }$ within the context of long-term operations.

Because $w_1^{OL}=1/{\mu }_{safe,1}^{OL}$, $w_2^{OL}=1/{\mu }_{safe,2}^{OL}$. Thus, the following theorem about waiting time must be examined:

Lemma 8. 

(i) In long-term omnichannel operations, when $\eta >0$, we have $w_1^{OL}<w_1^{OS}$; and when$\eta \le 0$, we have$w_1^{OL}\ge w_1^{OS}$. (ii) At stage two,$\overline{\gamma }=\beta +{\gamma }_0-\frac{\delta \beta \left(r-c_2\right)}{r-c_1-c_2}$. When$\gamma \le \overline{\gamma }$, we have$w_2^{OL}\le w_2^{OS}$; and when$\gamma >\overline{\gamma }$, we have$w_2^{OL}>w_2^{OS}$.

In long-term operations, if the conversion rate $\eta$ is greater than zero, this indicates that more customers will enter the offline channel. Therefore, the waiting time at stage one will be shorter than in short-term operations. Once customers are more inclined to choose online channels, then $\eta \le 0$. At this time, merchants should choose a lower safety capacity at stage one, causing longer wait times for offline customers. Therefore, in the current omnichannel context, many restaurants reduce the service staff at the front desk to save the cost input at stage one, which is also confirmed in the study of Gao and Su [24].

At stage two, because there is an equilibrium $\overline{\gamma }$, when the service environment level is lower than this equilibrium, customers will experience a shorter wait in long-term operations. Conversely, if the service environment quality is higher, customers will have a longer wait and can better perceive it.

Lemma 9. 

There is an equilibrium of $\overline{\gamma }=\beta +{\gamma }_0-\frac{\delta \beta \left(r-c_2\right)}{r-c_1-c_2}$ between the service environment effect and convenience effect, where there must be the following: (i) For the shopping rate of offline customers, when $\eta >0$and$\gamma <\overline{\gamma }$, we have${\lambda }_s^{OL}>{\lambda }_s^{OS}$; when$\eta <0$and$\gamma >\overline{\gamma }$, we have${\lambda }_s^{OL}<{\lambda }_s^{OS}$; and when$\eta =0$and$\gamma =\overline{\gamma }$, we have${\lambda }_s^{OL}={\lambda }_s^{OS}$. (ii) For the shopping rate of online customers, when$\gamma <\overline{\gamma }$, we have${\lambda }_o^{OL}>{\lambda }_o^{OS}$; when$\gamma >\overline{\gamma }$, we have${\lambda }_o^{OL}<{\lambda }_o^{OS}$; and when$\gamma =\overline{\gamma }$, we have${\lambda }_o^{OL}={\lambda }_o^{OS}$.

According to Lemma 9 (i), the relative shopping rate of customers in offline channels in long-term and short-term operations is affected by the $\eta$ and $\overline{\gamma }$. If online customers shift to offline channels and the service environment quality is lower than equilibrium, the offline customer shopping rate will be higher in long-term operations, compared with short-term operations. Conversely, the offline customers’ shopping rate in long-term operations will be lower than in short-term operations. According to Lemma 9 (ii), only when the service environment level is higher than equilibrium $\overline{\gamma }$ will the shopping rate of online customers be higher in the long-term operations over short-term operations. Otherwise, the shopping rate will be lower in the long term. This is especially interesting, as when the service environment effect is not so high, i.e., when $\gamma <\overline{\gamma }$, the online customer shopping rate will increase in the long term instead.

For the merchant, acquiring profit is the purpose of operation, which means that according to Proposition 3, the omnichannel merchant’s profit in long-term operations can be obtained as follows:

$$\begin{gathered} {\pi }^{OL}=\alpha \left(\theta +\left(1-\theta \right)\eta \right)\left(r-c_1-c_2\right)-2\sqrt{c_1\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)} \\ +\alpha \left(1-\theta \right)\left(1-\eta \right)\left(r-c_2\right)-2\sqrt{c_2\left(\left(\theta +\left(1-\theta \right)\eta \right)\left(\beta +{\gamma }_0-\gamma \right)\left(r-c_1-c_2\right)+\left(1-\theta \right)\left(1-\eta \right)\delta \beta \left(r-c_2\right)\right)}. \end{gathered}$$

(18)

Lemma 10. 

The omnichannel catering merchant’s profit in long-term operations decreases with an increased conversion rate $\eta$, i.e.,$\frac{\partial {\pi }^{OL}}{\partial \eta }<0$.

$r-c_2$ and $r-c_1-c_2$ represent the unit net revenue that the merchant obtains from online and offline customers, respectively, which indicates that serving more online customers will enable the merchant to obtain more unit net revenue by reducing the cost at stage one. Therefore, according to Lemma 10 and $\frac{\partial \eta }{\partial \gamma }>0$, in the long term, if the merchant provides a better service environment and more customers move to the offline channel, the profit will be lower than it would be in the short term with the same service environment quality. Improving the service environment is conducive to lowering the safety capacity to reduce the related cost. Therefore, the merchant needs to reduce the rate of converting online customers to the offline channel while improving the service environment, which requires providing additional utility for the customers in other ways, such as greater benefits, better packaging, etc. In particular, artfully designed take-out packaging is adopted by many businesses as a way to attract online customers (Din Tai Fung, Haidilao, etc.), and adding messages of blessings to the packages can often evoke good feelings in customers. In the long term, merchants should still build up the service environment in consideration of the word-of-mouth effect and spreading awareness. In this paper, we only compare the profit of long-term and short-term operations based on the customer shift generated by the service environment. The word-of-mouth effect will be studied in future works.

6. Numerical Analysis

In the above theoretical model analysis, we analyzed the impact of the service environment on the decision-making capacity of offline single-channel merchants, especially focusing on the role of safety capacity and profit. In this section, we explore the impact of the service environment on profit, accounting for different proportions of offline customers $\theta$ through a specific numerical analysis. Here, we assign the relevant parameters based on the literature [1] and the actual situation. Of course, for actual merchants, the simulation can be based on the above theoretical model combined with their own operational data. This also means that the theoretical model in this paper is feasible and practical. We assume that the customer’s base purchase rate is $\alpha =5$, sensitivity to waiting time is $\beta =0.5$, and basic expectation of the service environment is ${\gamma }_0=0.5$. In this system, the profit that the merchant can obtain from each customer is $r=2$, and the unit cost to maintain the capacity ${\mu }_i$ at different stages ($i=1,2$) is $c_1=0.2$ and $c_2=0.3$. The specific change of profit with the service environment level $\gamma$ is given in Figure 2.

From Figure 2, we can know that improving the service environment level will be beneficial to improving the profit for merchants in the offline single-channel mode. This also shows that in the traditional catering industry, improving the physical environment in the store and enhancing the service capability of servers can bring higher profits. The most notable example is the Haidilao hot pot restaurant chain. The dining environment and attitude of servers have attracted many customers, bringing higher profits.

It is also evident from Figure 2 that with a higher offline customer proportion $\theta$, improving the service environment can bring higher profit to offline single-channel merchants. Having more offline customers will better reflect the value generated by the service environment, as more customers will enjoy it, which in turn will bring more value.

In omnichannel operations, according to the analysis in Section 4, online customers can be regarded as only bearing the anxiety cost of waiting, so we further assume that the proportion of anxiety cost in the overall waiting cost per unit time is $\delta =0.3$. The specific change of profit with the service environment level in short-term operations is given in Figure 3.

It can also be seen from Figure 3 that as the offline customer proportion $\theta$ decreases, omnichannel merchants can obtain higher profits from the improved service environment. This is the opposite of the phenomenon shown in Figure 2, which means that in the omnichannel mode, having fewer offline customers can increase the merchant’s profit. This is because according to the net income $r-c_1-c_2$ obtained from each offline customer and net income $r-c_2$ obtained from each online customer, having more online customers can effectively reduce the merchant’s cost $c_1$ in stage one.

With a longer operation period in long-term omnichannel operations, there is a conversion rate of $\eta$ for customers in online and offline channels, and $\eta >0$ means that online customers transfer to offline channels. This conversion rate is the biggest difference between long-term and short-term omnichannel operations. In long-term omnichannel operations, we pay more attention to the impact of the conversion rate on profit. We therefore assume that the service environment level has a certain value, i.e., $\gamma =0.6$. At this time, the conversion rate $\eta$ will be affected by other exogenous factors and changes, and the specific change of profit with the conversion rate $\eta$ is given in Figure 4.

Meanwhile, it can be seen from Figure 4 that when $\gamma$ is fixed, or as online customers continue to shift to offline channels (i.e., they transfer for other reasons), profit will decrease to a certain extent. The transfer of more online customers towards the offline platform will not only cause congestion in offline channels and reduce the shopping rate of customers, but also cause merchants to invest in more capacity at the ordering stage. This will lead to increased cost at stage one, which will cause a continued decline in profit. When all online customers transfer to offline channels, i.e., when $\eta =1$, the merchant will obtain the same minimum profit.

7. Conclusions

7.1. Conclusions and Remarks

Customers in offline channels order food and wait for the food to be prepared in the restaurant, and as they wait, they can enjoy a high-quality service environment; in contrast, online customers order food through apps and wait at home or in an office, which reduces opportunity costs, as they cannot enjoy the same service environment. Due to the impact of the service environment in different channels, omnichannel catering merchants must consider the service environment at each stage when making capacity decisions. Therefore, based on the impact of the service environment on online and offline customers, this paper analyzed the optimal capacity formulation for merchants in the traditional offline single-channel, short-term omnichannel, and long-term omnichannel operations.

We first developed a decision-making model for the traditional catering merchant’s capacity management, considering the service environment. Based on this model, we obtained the optimal capacity at the ordering stage and the food production stage in the traditional offline single-channel mode. The results show that the capacity at each stage can be divided into the base capacity and safety capacity. The safety capacity plays a role in hedging the changes in demand and is a direct factor that affects customer waiting time. As the service environment improves, the safety capacity will decline and customers will wait longer.

Subsequently, we developed an optimal capacity decision model for omnichannel catering merchants and differentiated between short-term and long-term operations. In short-term operations, the omnichannel mode increased the total customer shopping rate and brought a higher profit to the merchant compared with the traditional offline single-channel mode.

In long-term operations, due to the impact of the service environment, online customers may shift to offline and vice versa. There is an equilibrium between the offline service environment effect and the online convenience effect, which is not affected by the conversion rate of online and offline customers. No matter the operation status, the service environment can effectively reduce the safety capacity, thereby controlling the merchants’ investment in capacity management. This study unexpectedly finds that with the gradual improvement of the service environment, profits in both short-term omnichannel and offline single-channel operations will increase, but the difference between the two will narrow over time. Furthermore, the online customer shift caused by the higher level of service environment will be detrimental to the acquisition of long-term profits.

7.2. Managerial Insights

In this paper, we presented the optimal capacity decision of catering merchants in different channels. However, in actual operation, management insights need to be analyzed based on each optimal decision. Therefore, the following management insights are herein provided.

Merchants should choose an omnichannel approach to attract more online customers to enter the system and increase the overall shopping rate. Moreover, omnichannel merchants should also improve the service environment, which can effectively increase profits.

In the formulation of the capacity, merchants should improve the service environment as much as possible or reduce customers’ expectations for the service environment; this allows for the setting of a lower safety capacity and reduces expenditure in this area. Improving the service environment not only involves better decor for the restaurant, but also better service attitude and ability from the service personnel. This requires catering merchants to carry out training for service personnel. In addition, merchants should also conduct sufficient market research on the proportions of online and offline customers in the market to accurately choose the capacity at each stage.

At the same time, merchants should fully grasp the information on customers transferring between online and offline channels and should reduce such transfer while improving the service environment. There are ways to ensure that customers continue to use online channels, such as providing discounts or offering artfully designed packaging for delivered products. This is beneficial to reduce the negative effects brought by the transfer of customers in a high-quality service environment.

7.3. Future Research

Future research should consider the competition among omnichannel catering merchants in the service environment, as there are often many competitors, meaning that merchants need to give their best response according to their competitors’ decisions. Meanwhile, competition in different channels is also a concern. Furthermore, we plan to use future research to consider the impact of social interactions induced by the service environment on consumption behavior in online and offline channels. Specifically, the impacts of the herd effect and word-of-mouth are of great significance for omnichannel catering merchants to choose a reasonable capacity.

Furthermore, customers within the same channel also have varying perceptions of the service environment, and the distribution of their sensitivity will satisfy a certain distribution of the customers. In a follow-up research, we will analyze the sensitivity of customers to the service environment under different distributions to more accurately improve the basis for decision-making for catering merchants. This research can effectively help omnichannel catering merchants use the service environment to attract customers, thereby bringing more profit, and is also conducive to the development of the catering industry within the omnichannel context.