Cloud-Kitchens: Value Creation Through Co-Location

Abstract

“Cloud Kitchens” are delivery-only facilities that house multiple restaurants. Food-delivery platforms operate such kitchens to exploit two advantages: (a) Location advantage, arising due to a cloud-kitchen’s central location—this enables lower delivery times to customers. (b) Consolidation advantage, which accrues when multiple restaurants choose to co-locate at the cloud kitchen—this enables the platform to use a common pool of delivery drivers, thereby reducing costs. However, a cloud-kitchen’s eventual impacts on both the restaurants and the platform are intricately connected through their respective decisions—namely, the restaurants’ location decisions and the platform’s delivery capacity and delivery time. We examine conditions under which a cloud kitchen simultaneously benefits the primary stakeholders: delivery platform, restaurants, and customers. Our game-theoretic analysis considers two restaurants and a delivery platform. The restaurants simultaneously decide whether to stay at their initial (extreme) locations or relocate to a centrally located cloud kitchen. The platform decides the driver headcount and the delivery times for customers. In line with industry trends, we show that as population density increases beyond a threshold, the restaurants co-locating at the cloud kitchen is first a Pareto-dominant equilibrium and then the unique equilibrium. The platform and customers also prefer this equilibrium, leading to a win-win-win for the stakeholders. A cloud-kitchen’s benefit to the platform further increases as the drivers’ operational environment becomes more constrained, i.e. drivers’ carry-limit and speed decrease, and driver cost increases.

1 Motivation

“We are constantly working on innovations [cloud kitchens] that help merchants [restaurants] find new, meaningful ways to reach customers and run their businesses more efficiently.”—Fuad Hannon, Head of New Business Verticals at DoorDash [delivery platform], on the benefits of a cloud kitchen for restaurants and customers. (PR Newswire, 2019) “We were impressed by the overall partnership and scale DoorDash could reach with this concept, and we found the notion of a delivery-only kitchen in Redwood City very appealing as it helps us test out demand in new markets, reaching new customers and areas quickly.”—Min Park, CFO of Rooster & Rice [restaurant], on the benefits of a cloud kitchen for DoorDash [delivery platform], restaurants, and customers. (PR Newswire, 2019)

The restaurant industry is a vital part of economies worldwide—for instance, in the U.S., it contributes 4% to the annual GDP and employs 10% of the workforce (Lew, 2020). One of the main challenges that this industry faces is low profit margins—industry experts estimate that an average restaurant’s profit margin typically ranges from 3% to 5%, with successful restaurants earning a profit margin of about 10% (Stabiner, 2016). Over the last decade, restaurants have explored food delivery as an alternate sales channel to cater to a broader base of consumers, increase volumes, and thereby profitability. However, independent restaurants and small/mid-sized chains are too small to own and operate a delivery fleet. The business opportunity to provide delivery logistics to such restaurants has led to the emergence of smartphone-based online food-delivery platforms (or, in short, delivery platforms) such as Uber Eats, Deliveroo, Swiggy, Grubhub, Ele.me, Meituan, and DoorDash. These delivery platforms have grown rapidly and now command a large share of the food-delivery market; for instance, in the U.S., the combined market share of the top three delivery platforms exceeds 80% (Yeo, 2021; Edison Trends, 2020). This growth has also led to a change in consumer behavior, with an increasing number of customers getting accustomed to the idea of getting their food delivered through such platforms. The coronavirus disease 2019 (COVID-19) pandemic further highlighted the importance of delivery platforms for customers, who were often left with only the food-delivery option due to lock-downs or restaurant shutdowns.

The partnership between restaurants and delivery platforms has certainly benefited customers, who now have more options to choose from at their fingertips and the convenience of food delivered to their homes. For restaurants and delivery platforms, however, it is a different story. Although delivery platforms have been successful in bringing more customers and revenue to restaurants, this additional revenue comes at a steep price for restaurants. In particular, delivery platforms charge a high service fee for managing delivery logistics; this fee usually ranges from 15% to 30% of the order dollar value and puts additional pressure on restaurants’ low profitability (Forman, 2021). Given their high service fees, it would seem that delivery platforms themselves are profitable; however, recent financial suggest otherwise. For instance, in 2020, Deliveroo disclosed a loss of 309 million USD and Grubhub’s losses were 155.9 million USD (Lunden, 2021). These losses were attributed to the high costs of maintaining delivery logistics and customer acquisitions. Indeed, the low profitability of delivery platforms has been an enduring challenge due to two reasons. First, these platforms are unable to increase the current high service fees that they charge restaurants since restaurants already suffer from low-profit margins. Second, delivery platforms are unable to pass on the costs to customers in the form of an increase in delivery fees as this leads to a decrease in demand. Consequently, it is imperative for platforms to explore other drivers of profitability or radically change the current business model to sustain their business.

As a recent innovation in the restaurant industry, delivery platforms have started building “cloud kitchens” to increase their profitability. ¹ Cloud kitchens typically house multiple restaurants and offer only delivery services (i.e., no dine-in facilities) (Ye et al., 2020). For delivery platforms, the cloud-kitchen business model differs from the traditional business model in the following aspect. Traditionally, delivery platforms have built and managed online portals for customers to place orders and provide the delivery logistics to fulfill these orders, with restaurants themselves owning or renting their facilities. With the cloud-kitchen, in addition to providing the online portal and the delivery logistics, the delivery platform rents the space for the facility and leases it to restaurants. ² Cloud kitchens are often situated in “central” locations that make it convenient for the delivery platform to access customers and facilitate faster deliveries (Alexander-Erber, 2020; Nicholas Upton, 2020; Escher, 2019; Bromwich, 2019). Since customers’ propensity to order from a platform depends on delivery time, faster deliveries from a centrally located cloud kitchen can increase customer demand (Bradshaw, 2019). However, this increase is contingent on whether or not one or more restaurants choose to relocate at the cloud kitchen, which in turn depends on the rental costs at the cloud kitchen as well as the platform’s ability to optimize its delivery operations—namely, delivery capacity and delivery time, both of which are costly. Thus, the cloud-kitchen’s eventual impacts on both the restaurants and the platform are intricately connected through their respective decisions—namely, the location decisions of the restaurants and the platform’s chosen delivery capacity and delivery time. The goal of our analysis in this paper is to identify the conditions under which cloud kitchens offer a win-win-win scenario for the platform, the restaurants, and the customers.

1.1 Overview of Model and Results

We study a game-theoretic model consisting of two restaurants and a delivery platform operating in a linear city. ³ The restaurants are situated at the two ends of the city, and customers place orders through the delivery platform’s online portal. At the city center, the delivery platform operates a cloud kitchen, which the restaurants may rent for a fixed rental rate; the fixed rental rate at the two ends of the city is normalized to zero. The higher rental rate at the cloud kitchen reflects the reality that rents are higher at locations that are closer to customers or in areas with higher population density. We model this fixed rental rate as an increasing function of the population density. The platform charges the restaurants a proportion of their revenue for delivery services and employs drivers with a “carry limit,” which is the maximum number of orders a driver can carry simultaneously.

The model has two stages. In the first stage, the restaurants decide whether to remain at their current extreme locations or relocate to the cloud kitchen. The restaurants face the following trade-off: moving to the cloud kitchen entails higher rent, but it provides a central location that may allow lower delivery-time guarantees, leading to increased customer demand. The restaurants’ location decisions result in four scenarios—a “co-located” scenario in which the restaurants are co-located at the cloud-kitchen and three “dedicated” scenarios in which the restaurants choose separate locations. The platform observes the first-stage location decisions and makes the following decisions in the second-stage. For every point in the city, the platform determines the delivery-time guarantee—this guarantee is a function of the distance between the point and the restaurant. In addition, under a dedicated scenario, the platform decides the driver headcount assigned to each restaurant. Under the co-located scenario, the platform uses the same driver pool to serve both restaurants and decides the driver headcount in this pool. The platform faces the following trade-off: increasing driver head-count allows for lower delivery-time guarantees, but it also increases the cost incurred by the platform.

For each scenario, the delivery platform’s problem is to determine the driver headcount and the delivery-time guarantees, with the objective of maximizing its profit. However, this optimization problem is challenging due to two factors. First, the delivery-time guarantee is a function of the distance between the customer and the restaurant. Second, the relationships between the platform’s decision variables are nonlinear. For example, the lower bound on the delivery-time guarantee is nonlinear in the driver headcount. To tackle this problem, we use two transformations to obtain a relaxation of the original problem. Although the relaxed problem is a non-convex, non-linear optimization problem, its decision variables are real numbers. When the city’s population density is sufficiently high, the optimal solution of the relaxed problem exhibits certain structural properties, which we establish and exploit to solve the delivery platform’s optimization problem in closed form. This closed-form solution allows us to (a) derive the equilibria of the location game of the restaurants, and (b) investigate the impact of the driver’s carry-limit, speed, and cost, on the delivery platform’s profit rates under these equilibria.

Our analysis shows that a high population density combined with the advantages of the cloud kitchen make it an attractive business model for all the three primary stakeholders, i.e. the restaurants, the delivery platform, and the customers. Specifically, we show that as the population density increases beyond a certain threshold, the restaurants co-locating at the cloud kitchen is first a Pareto-dominant Nash equilibrium and then the unique Nash equilibrium. This “cloud-kitchen equilibrium” is driven by two advantages of the cloud-kitchen. First, due to its central location, the cloud kitchen provides better access to customers, which leads to increased demand. We refer to this advantage as the cloud-kitchen’s location advantage. Second, the cloud kitchen allows the delivery platform to consolidate its delivery capacity and thereby reduce cost. Consequently, the delivery platform is able to pass on the benefit of this cost reduction to the restaurants and the customers by decreasing the delivery-time guarantees. We refer to this advantage as the cloud-kitchen’s consolidation advantage; the benefit from this advantage accrues only when multiple restaurants co-locate at the cloud kitchen. Also, under the cloud-kitchen equilibrium, we show that (a) the delivery platform’s and the restaurants’ profit rates are greater than those under the status-quo equilibrium where the two restaurants stay at their current locations, and (b) the delivery-time guarantees are lower than those under the status-quo equilibrium. Consequently, cloud kitchens can indeed create a win-win-win scenario for the three stakeholders in dense cities.

For the delivery platform, a natural metric that represents the relative attractiveness of the cloud-kitchen business model is the ratio of its profit rate under the cloud-kitchen equilibrium to that under the status-quo equilibrium. We analyze the impact of drivers’ carry-limit, speed, and cost, on this ratio. The relative attractiveness of the cloud-kitchen increases when (a) driver carry-limit is below a threshold and decreases, (b) driver speed decreases, and (c) driver cost increases. Succinctly put, as the operational environment for drivers’ carry-limit, speed, and cost, becomes more constrained, the cloud-kitchen becomes more attractive for the delivery platform. This increased attractiveness can be attributed either to the cloud-kitchen’s location advantage or to its consolidation advantage. As driver carry-limit decreases, the location advantage increases but the consolidation advantage decreases; the former effect is stronger, leading to the cloud kitchen becoming more attractive for the delivery platform. As driver speed decreases, both the location and the consolidation advantages increase, resulting in an improved attractiveness for the cloud kitchen. As driver cost increases, the cloud-kitchen’s location advantage increases while the consolidation advantage remains unaffected, again leading to the cloud kitchen becoming more attractive.

Our main findings are in line with the current trends in the restaurant industry. The Euromonitor group predicts that the worldwide cloud-kitchen market will grow to 1 trillion USD by 2030 (Guszkowski, 2020). The growth in the number of cloud kitchens has also been steady over the past years, with currently about 7500 cloud kitchens in China, more than 3500 in India, and around 1500 in the United States (Lock, 2021). Further, the largest delivery platforms such as Grubhub and DoorDash in the United States, Ele.me and Meituan in China, JustEat and Deliveroo in Europe, and Swiggy and Zomato in India, have been expanding their cloud-kitchen footprint across large cities.

The remainder of the paper is organized as follows. The next section reviews the related literature. Section 3 defines our setting and model. Sections 4 and 5 characterize the conditions under which restaurants co-locating at the cloud kitchen is an equilibrium. In Section 6, we investigate the impact of drivers’ carry-limit, speed, and cost on the relative attractiveness of the cloud-kitchen equilibrium for the delivery platform. Finally, Section 7 concludes the paper.

2 Related Literature

The phenomenon of the co-location of firms, formally known as “clusters” in the economics and management literature, is common in the manufacturing and retail industries (see, e.g., Rosenthal and Strange, 2020; Kukalis, 2010; Klepper, 2007; Bozarth et al., 2007). One can observe such clusters, for instance, in the form of industry blocks and supplier parks in the manufacturing industry and in the form of shopping centers in the retail industry. A common theme across the literature that studies this phenomenon is that clusters benefit the co-locating firms as well as other stakeholders in the industry (Habermann et al., 2015; Konishi, 2005; Rosenthal et al., 2004; Duranton and Puga, 2004). For instance, when suppliers co-locate at a supplier park, the synergy and cooperation increase product quality and the pace of innovation (Alcácer and Delgado, 2016; Alcácer and Chung, 2007). Further, a supplier park close to a manufacturer’s production facility allows the manufacturer to focus on its core competencies and build supplier capabilities. In the retail industry, clustering is primarily attributed to consumers’ valuation or price uncertainty, which is realized once the consumer visits the retailer. This uncertainty increases consumers’ transportation costs as they may need to visit multiple retailers before purchasing a product. The co-location of retailers benefits both consumers and retailers since consumers may now visit a single location, decreasing their transportation cost and increasing the probability of sale (Zhao et al., 2019; Takahashi, 2013; Konishi, 2005).

We contribute to this literature in the following aspects. We show that, akin to clusters in manufacturing and retail, a cloud kitchen housing multiple restaurants benefits all the three primary stakeholders, namely, the restaurants, the delivery platform and the customers. Specifically, the co-location of restaurants at a cloud kitchen increases the profits of both the restaurants and the delivery platform while the customers benefit due to faster deliveries. An important aspect in which cloud kitchens differ from the clustering phenomenon in manufacturing and retail is the presence of the delivery platform and its multi-sided network. Further, unlike a manufacturer who may provide direct incentives to suppliers, such as long-term contracts, for co-locating at a supplier park, the delivery platform does not provide any direct incentives to restaurants for co-locating at a cloud kitchen. Rather, as in our model, a restaurant incurs a higher rental rate at a cloud kitchen compared to that at a non-central location. The co-location of restaurants at a cloud kitchen also differs from a retail cluster in that consumers who order from a cloud kitchen have access to detailed reviews of the restaurants and prices of the menu items on the platform and, therefore, do not have much valuation or price uncertainty.

Our paper also contributes to the growing literature on on-demand platforms, including ride-hailing services (Uber, Lyft), grocery delivery services (Instacart), food delivery services (DoorDash, Uber Eats), and handyman services (Task Rabbit). This stream of literature identifies operational mechanisms and factors that improve co-ordination between supply and demand and, hence, have a positive impact on the platform’s efficiency and the welfare of stakeholders associated with the platform. Examples include platform pricing (see, e.g., Gangwar and Bhargava, 2022; Lin et al., 2020; Guda and Subramanian, 2019; Bai et al., 2019) and capacity management (Chakravarty, 2021; Gurvich et al., 2019), labor welfare (Benjaafar et al., 2022) and incentives to drivers by the platform (Bai et al., 2019; Kabra et al., 2016), reputation mechanisms (Kokkodis and Ipeirotis, 2016) and online dispute resolution (Burtch et al., 2021), geographic proximity of buyers and sellers (Cullen and Farronato, 2021), platform’s facilitation of the exploration of new products (Hagiu and Wright, 2020), and partner search on matching platforms (Kanoria and Saban, 2021). In a similar spirit, a delivery platform’s endeavor to build a centrally located cloud kitchen allows for capacity consolidation, which improves the profits of both the platform and the restaurants and, at the same time, facilitates faster deliveries to customers.

An important feature of an on-demand platform is its multi-sided network. A promising area of research is to understand how the payoffs and incentives of certain stakeholders in such a network affect other stakeholders; see Benjaafar and Hu (2020). A recent stream of literature focuses on the restaurant industry and addresses this gap, examining aspects such as the performance of contracts between a delivery platform and a restaurant (Chen et al., 2022; Feldman et al., 2022), the impact of delivery platforms on the composition of customers for a restaurant (Chen et al., 2022), and the impact of delivery platforms on the ability of quick service restaurants to accurately forecast demand (Karamshetty et al., 2020). Since these studies examine settings in which the delivery platform does not operate a cloud kitchen, their analytical models and assumptions are not pertinent to our setting. Feldman et al. (2022) argue that a no-contract relationship can outperform a one-way sharing contract and Chen et al. (2022) show that a delivery platform can change the composition of dine-in and delivery customers at a restaurant. However, a no-contract relationship is not feasible in the cloud kitchen setting since the delivery platform provides the kitchen space for a restaurant to prepare its menu items. Further, since a cloud kitchen only serves delivery customers, the issue of change in the composition of a restaurant’s customers does not arise in our setting. Finally, since cloud kitchens are similar to local micro-fulfillment centers and urban consolidation centers in the e-commerce industry, our paper is also naturally connected to the notion of a Smart City Operations; see, for example, Deng et al. (2021) and Hasija et al. (2020).

3 Model

Consider a linear city modeled by the unit interval $[0,1]$ . There are two restaurants, $R_A$ and $R_B$ . The current location of $R_A$ is $0$ and that of $R_B$ is $1$ . A delivery platform provides food-delivery logistics to these restaurants. The delivery platform also owns a cloud kitchen at the city center, i.e. at location $1/2$ . Typically, a cloud kitchen consists of restaurants that specialize in different types of cuisines (to avoid competition and serve a wider base of consumers). Accordingly, it is useful to view restaurants $R_A$ and $R_B$ as specializing in different types of cuisines—each restaurant serves customers who prefer its cuisine over the other. Thus, the restaurants do not compete for customers. We also analyze a setting in which the restaurants compete with each other and show that our main insights remain qualitatively the same. We briefly discuss this model with competition in Remark 2 below; the analysis of the model is in Appendix B.

Customers place orders for their preferred restaurant through the delivery platform; thus, all orders are delivery orders and customers do not travel to the restaurants. ⁴ The restaurants simultaneously decide whether to stay at their current locations or move to the cloud kitchen ⁵ , i.e. restaurant $R_A$ chooses between locations $0$ and $1/2$ and $R_B$ chooses between $1$ and $1/2$ . Each restaurant incurs a fixed rental cost per unit time (fixed rental rate) if it moves to the cloud kitchen. We normalize the fixed rental rate at the restaurants’ current locations to zero. ⁶ The primary goal of our analysis is to understand how the population density of the city and the two advantages of the cloud kitchen—namely, better access to food delivery for customers (location advantage) and cost reduction due to capacity consolidation for the delivery platform (consolidation advantage)—affect restaurants’ location decisions. To isolate these effects and for clarity of exposition, it is ideal to consider a setting in which the two restaurants are symmetric with respect to the maximum demand rate and the revenue per order. Accordingly, we consider the following setting. The maximum-possible arrival rate of orders (maximum order rate) for each restaurant is the same and is denoted by $\lambda$ . It is useful to view $\lambda$ as an indicator of the population density in the city—a higher population density implies a higher value of $\lambda$ . The revenue per order for each restaurant is denoted by $p$ and each restaurant pays a $\beta$ proportion of the revenue per order to the delivery platform and retains the remaining $(1-\beta )$ proportion. We assume that $p$ is exogenously determined since, in practice, delivery platforms typically serve small and mid-sized restaurants with limited pricing power. Further, we restrict attention to the percentage-revenue-sharing contract, which is common in the restaurant industry as it aligns the incentives of the platform and the restaurants, and is easy to implement. We express the fixed rental rate at the cloud kitchen as a continuous function of $\lambda$ ; specifically, $F(\lambda ):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ .

Since each restaurant has two location choices, we have the following four scenarios: $\mathcal{S}\in \{{\mathcal{S}}_1:=(0,1),{\mathcal{S}}_2:=(0,1/2),{\mathcal{S}}_3:=(1/2,1),{\mathcal{S}}_4:=(1/2,1/2)\}$ . We will refer to ${\mathcal{S}}_4$ as the co-located scenario and each of the other scenarios as a dedicated scenario, a label that will soon become meaningful. For the dedicated scenarios, i.e. for $\mathcal{S}\in \{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3\}$ , the delivery platform makes the following decisions for restaurant $R_j$ , $j\in \{A,B\}$ : the driver headcount (i.e., the number of drivers) assigned to restaurant $R_j$ and the delivery-time guarantee function, i.e. the delivery-time guarantees for points in the city from $R_j$ . We denote the driver headcount as $N_j(\mathcal{S})$ . The delivery-time guarantee for customers located at a distance $u$ from $R_j$ is denoted as ${\tau }_j(u;\mathcal{S})$ . In the co-located Scenario ${\mathcal{S}}_4$ , a common pool of drivers serves both restaurants—the delivery platform decides the driver headcount, denoted as $N_{AB}({\mathcal{S}}_4)$ , in this pool. Further, in ${\mathcal{S}}_4$ , the delivery-time guarantee (which is identical from both restaurants) for customers at a distance $u$ from $1/2$ is denoted as ${\tau }_{AB}(u;{\mathcal{S}}_4)$ .

Figure 1.

The locations of the restaurants in Scenario ${\mathcal{S}}_1$ .

Customers are time-sensitive and are willing to wait for a maximum of one unit of time. We assume that a restaurant’s demand rate (number of orders per unit time) from a point in the city decreases with an increase in its delivery-time guarantee. Specifically, given a delivery-time guarantee of $\tau (u;\mathcal{S})$ , the demand rate for a restaurant from customers located at a distance $u$ is

$$\lambda (1-\tau (u;\mathcal{S}){)}^{+}.$$

(1)

Remark 1

Notice that the dependence of the demand rate on price is not explicit in (1). Since the revenue per order $p$ for both the restaurants is identical and fixed throughout our analysis, $\lambda$ can be considered as the demand rate corresponding to  $p$ .

We now turn our attention to the cost of drivers that the platform employs to make the deliveries. The delivery platform incurs a cost of $c$ per unit time for each driver. We assume that $c$ is exogenously specified, since food-delivery platforms typically have limited influence on driver wages due to the increasingly competitive labor market and the low profit margins of small and mid-sized restaurants. Further, each driver operates at a speed of $\omega >1$ . This ensures that, if the delivery platform hires a sufficiently high number of drivers, then it can serve every point in the city from a restaurant within $1$ unit of time, thus ensuring a positive demand rate for each restaurant from any point in the city. Each driver also has a carry-limit of $m$ orders that the driver can carry simultaneously. We provide more specifics on the delivery platform’s decisions regarding drivers when we analyze the scenarios in detail.

We use the elements above to build a two-stage game-theoretic model in which the objective of each party (namely, the restaurants and the delivery platform) is to maximize its individual profit. In the first stage, $R_A$ and $R_B$ anticipate the delivery-platform’s decisions and simultaneously choose whether to stay at their current locations or relocate to the cloud kitchen at the center. In the second stage, the delivery platform observes the restaurants’ location decisions and decides the driver headcount and the delivery-time guarantees for the orders that arise from that market. Since, in practice, cloud kitchens typically operate in densely populated cities ⁷ , we are primarily interested in the outcome under sufficiently high values of the population density $\lambda$ . Specifically, we consider $\lambda \ge {\lambda }_0$ , where ${\lambda }_0$ is as specified in Definition 1. We also assume that the driver carry-limit $m>\frac{2c}{\beta p\omega }$ : this inequality ensures that a driver’s carry-limit is sufficiently high so that the platform generates a profit from a driver operating at her full capacity.

Remark 2

(Competing Restaurants). In the model we discussed in this section, the demand rate for a particular restaurant in a given scenario depends (in addition to price) on the delivery-time guarantees from that restaurant alone. In Appendix B, we consider a model with competition in which a restaurant’s demand rate from an interval served by both the restaurants depends on the delivery-time guarantees to this interval from both the restaurants. Further, in the model with competition, the delivery platform also decides the proportion of the city to serve from a restaurant. In addition, for analytical tractability, the competitive model assumes that the delivery-time guarantee from a restaurant is the same for any point in the city that is served from that restaurant, and the drivers are un-capacitated, i.e. the carry-limit is infinite.

To determine the outcome of the simultaneous location decisions of the restaurants, we need the profit rates for both the restaurants in each scenario $\mathcal{S}\in \{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\}$ . Since the profit rates of the restaurants depend on the delivery platform’s decisions, we first formulate the delivery platform’s profit rate in each of the four scenarios.

Figure 2.

The locations of the restaurants in Scenario ${\mathcal{S}}_2$ .

3.1 The Profit Rates of the Delivery Platform and the Restaurants

We consider each of the four scenarios.

Scenario ${\mathcal{S}}_1=(0,1)$ : In this scenario, $R_A$ is located at $0$ and $R_B$ at $1$  (see Figure 1). We first explain the delivery platform’s decisions with respect to $R_A$ . The delivery platform decides the driver headcount $N_A\ge 0$ and provides a delivery-time guarantee ${\tau }_A(u)\ge 0$ to customers located at a distance $u$ from $R_A$ . The delivery platform’s decisions satisfy the following inequalities:

$$\begin{array}{rl}{\tau }_A(u)& \ge \frac{u}{\omega }+\frac{2}{\omega N_A},\text{ for }u\in [0,1],\text{ and }\end{array}$$

(2)

$$\begin{array}{rl}& \lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du\le \frac{m\omega N_A}{2}.\end{array}$$

(3)

Here, we briefly discuss the inequalities in (2) and (3); see Appendix C for a detailed explanation of each of these inequalities. The right-hand side of (2) represents the minimum feasible delivery-time guarantee that the delivery platform can provide from $R_A$ to customers located at a distance of $u$ from this restaurant. This guarantee can be derived as follows. The length of each two-way trip for a driver from $R_A$ to the farthest point in the city (location $1$ ) is $2$ . Hence, a driver takes $\frac{2}{\omega }$ units of time to complete this two-way trip, and a new delivery trip starts from $R_A$ every $\frac{2}{\omega N_A}$ units of time. Further, it takes an additional $\frac{u}{\omega }$ units of time for a driver to reach customers located at a distance $u$ from $R_A$ . The left-hand side of (3) represents the demand rate for restaurant $R_A$ from the entire city. Since each driver makes $\frac{\omega }{2}$ number of trips per unit of time and each driver has a carry-limit of $m$ , the right-hand side in (3) represents the delivery platform’s total driver capacity per unit time for restaurant $R_A$ . The delivery platform’s decisions pertaining to $R_B$ are $N_B$ and ${\tau }_B(u)$ . The relationships between $N_B$ and ${\tau }_B(u)$ can be derived by replacing the subscript $A$ in (2) and (3) with $B$ .

We can now compute the delivery platform’s profit rate and the profit rates of the restaurants in this scenario. Let us denote the vector of the delivery platform’s decisions as $\theta :=(N_A,{\tau }_A(u),N_B,{\tau }_B(u))$ .

The delivery platform’s profit rate $\mathrm{\Pi }(\theta ;{\mathcal{S}}_1)$ is

$$\begin{array}{rl}\mathrm{\Pi }(\theta ;{\mathcal{S}}_1)& =\beta p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du-c{N}_A+\\ & \beta p\lambda {\int }_0^1(1-{\tau }_B(u){)}^{+}du-c{N}_B.\end{array}$$

The delivery platform’s profit rate $\mathrm{\Pi }(\theta ;{\mathcal{S}}_1)$ is the sum of the delivery platform’s profit rates from the two restaurants. Further, the delivery platform’s profit rate from a particular restaurant is the difference between the revenue rate that the delivery platform earns from this restaurant and the cost per unit time of the drivers that the delivery platform employs to fulfill orders for this restaurant. Given $\theta$ , the profit rates for $R_A$ and $R_B$ are, respectively,

$$(1-\beta )p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du\text{ and }(1-\beta )p\lambda {\int }_0^1(1-{\tau }_B(u){)}^{+}du.$$

We note that, in this scenario, the restaurants $R_A$ and $R_B$ are located at $0$ and $1$ , respectively. Since both these locations have zero rental costs, the restaurants’ profit rates are the same as their respective revenue rates.

Scenarios ${\mathcal{S}}_2=(0,1/2)$ and ${\mathcal{S}}_3=(1/2,1)$ : In Scenario ${\mathcal{S}}_2$ , $R_A$ is located at $0$ and $R_B$ at $1/2$  (see Figure 2). Since $R_A$ ’s location is the same as that in Scenario ${\mathcal{S}}_1$ , the delivery platform’s decisions pertaining to $R_A$ and its profit rate from $R_A$ , remain the same as in Scenario ${\mathcal{S}}_1$ . Accordingly, we focus on $R_B$ . For $R_B$ , the delivery platform provides a delivery-time guarantee of ${\tau }_B(u)$ to customers located at a distance of $u$ from $R_B$ , and assigns $\frac{N_B}{2}$ drivers to each side. The following relationships hold between $N_B$ and ${\tau }_B(u)$ in this scenario:

$$\begin{array}{rl}& {\tau }_B(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_B},\text{ for }u\in [0,\frac{1}{2}],\text{ and}\\ & 2\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_B(u){)}^{+}du\le m\omega N_B.\end{array}$$

Since $R_B$ is located at $1/2$ , each two-way trip for a driver from $R_B$ to locations $0$ or $1$ is of length $1$ . Hence, a driver takes $\frac{1}{\omega }$ units of time to complete a two-way trip from $R_B$ , and a new delivery trip starts from $R_B$ every $\frac{2}{\omega N_B}$ units of time. Further, it takes an additional $\frac{u}{\omega }$ units of time for a driver to reach customers located at a distance $u$ from $R_B$ . Therefore, the minimum feasible delivery-time guarantee for customers located at a distance of $u$ from $R_B$ is $\frac{u}{\omega }+\frac{2}{\omega N_B}$ . Further, since each driver makes $\omega$ number of trips per unit of time and driver’s carry-limit is $m$ , the total driver capacity for $R_B$ in this scenario is $m\omega N_B$ . The delivery platform’s profit rate $\mathrm{\Pi }(\theta ;{\mathcal{S}}_2)$ , where  $\theta :=(N_A,{\tau }_A(u),N_B,{\tau }_B(u))$ , is

$$\begin{array}{rl}\mathrm{\Pi }(\theta ;{\mathcal{S}}_2)& =\beta p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du-c{N}_A+\\ & 2\beta p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_B(u){)}^{+}du-c{N}_B.\end{array}$$

In this scenario, $R_A$ is located at $0$ and incurs a zero rental cost while $R_B$ is located at $1/2$ and incurs a rental rate of $F(\lambda )$ . Thus, the profit rates for $R_A$ and $R_B$ are, respectively,

$$\begin{array}{rl}& (1-\beta )p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du\text{ and }\\ & 2(1-\beta )p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_B(u){)}^{+}du-F(\lambda ).\end{array}$$

This completes our formulation for Scenario ${\mathcal{S}}_2$ .

Figure 3.

The locations of the restaurants in Scenario ${\mathcal{S}}_3$ .

Figure 4.

The locations of the restaurants in Scenario ${\mathcal{S}}_4$ .

Scenario ${\mathcal{S}}_3$ is symmetric to ${\mathcal{S}}_2$  (see Figure 3). Hence, we directly state all expressions of interest in Scenario ${\mathcal{S}}_3$ .

The delivery platform’s profit rate is

$$\begin{array}{rl}\mathrm{\Pi }(\theta ;{\mathcal{S}}_3)& =2\beta p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_A(u){)}^{+}du-c{N}_A+\\ & \beta p\lambda {\int }_0^1(1-{\tau }_B(u){)}^{+}du-c{N}_B.\end{array}$$

The profit rates for $R_A$ and $R_B$ are, respectively,

$$\begin{array}{rl}& 2(1-\beta )p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_A(u){)}^{+}du-F(\lambda )\text{ and }\\ & (1-\beta )p\lambda {\int }_0^1(1-{\tau }_B(u){)}^{+}du.\end{array}$$

Scenario ${\mathcal{S}}_4=(1/2,1/2)$ : In this scenario, both $R_A$ and $R_B$ locate at $1/2$  (see Figure 4). Thus, the delivery platform uses a common pool of drivers to serve orders from both restaurants. Note that each order is composed of items entirely from one restaurant. The remainder of the arguments mirror those applied to $R_B$ in ${\mathcal{S}}_2$ , since it was located at $1/2$ in that scenario. We denote the driver headcount as $N_{AB}$ and the delivery-time guarantee to customers located at a distance $u$ from $1/2$ as ${\tau }_{AB}(u)$ . The delivery platform assigns $\frac{N_{AB}}{2}$ drivers to each side. Arguing as before, we have the following relationships between $N_{AB}$ and ${\tau }_{AB}(u)$ :

$$\begin{array}{rl}& {\tau }_{AB}(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_{AB}},\text{ for }u\in [0,\frac{1}{2}],\text{ and }\\ & 4\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_{AB}(u){)}^{+}du\le m\omega N_{AB}.\end{array}$$

We denote the delivery platform’s decision in this scenario as $\theta :=(N_{AB},{\tau }_{AB}(u))$ . The delivery platform’s profit rate $\mathrm{\Pi }(\theta ;{\mathcal{S}}_4)$ is

$$\mathrm{\Pi }(\theta ;{\mathcal{S}}_4)=4\beta p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_{AB}(u){)}^{+}du-c{N}_{AB}.$$

The profit rate for either restaurant is

$$2(1-\beta )p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_{AB}(u){)}^{+}du-F(\lambda ).$$

We now proceed to consider the outcome of the simultaneous location decisions of the restaurants.

3.2 The Location Game of the Restaurants

Restaurants $R_A$ and $R_B$ simultaneously choose their respective locations to maximize their individual profit rate. Since the profit rates of the restaurants depend on the delivery platform’s decision $\theta$ , let ${\pi }_A(\theta ;\mathcal{S})$ and ${\pi }_B(\theta ;\mathcal{S})$ denote the profit rates of restaurants $R_A$ and $R_B$ in Scenario $\mathcal{S}$ , respectively. Then, the simultaneous location decisions of the restaurants can be represented as a static game; see Table 1.

The rows and columns of Table 1 represent, respectively, the location choices of restaurants $R_A$ and $R_B$ . The first and the second terms in each cell of the table represent, respectively, the profit rates for $R_A$ and $R_B$ . To determine these rates, however, we first need to obtain the optimal decisions of the delivery platform in each scenario. We next solve for these decisions.

Table 1.

The Simultaneous Location Game of Restaurants $R_A$ and $R_B$ .

	$R_B$
$R_A$	1	1/2
$0$	${\pi }_A(\theta ;{\mathcal{S}}_1)$	${\pi }_A(\theta ;{\mathcal{S}}_2)$
	${\pi }_B(\theta ;{\mathcal{S}}_1)$	${\pi }_B(\theta ;{\mathcal{S}}_2)$
$1/2$	${\pi }_A(\theta ;{\mathcal{S}}_3)$	${\pi }_A(\theta ;{\mathcal{S}}_4)$
	${\pi }_B(\theta ;{\mathcal{S}}_3)$	${\pi }_B(\theta ;{\mathcal{S}}_4)$

4 Optimal Decisions of the Delivery Platform

In this section, we derive the delivery platform’s optimal decision vector, denoted as ${\theta }^{*}(\lambda ;\mathcal{S})$ , in Scenario $\mathcal{S}\in \{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\}$ . We denote the delivery platform’s optimal profit rate in Scenario $\mathcal{S}$ as ${\mathrm{\Pi }}^{*}(\lambda ;\mathcal{S}):=\mathrm{\Pi }({\theta }^{*}(\lambda ;\mathcal{S}),\mathcal{S})$ . Given the delivery platform’s optimal decision vector in Scenario $\mathcal{S}$ , we also compute the profit rates of restaurants $R_A$ and $R_B$ , denoted by ${\pi }_A(\lambda ;\mathcal{S}):={\pi }_A({\theta }^{*}(\lambda ;\mathcal{S}),\mathcal{S})$ and ${\pi }_B(\lambda ;\mathcal{S}):={\pi }_B({\theta }^{*}(\lambda ;\mathcal{S}),\mathcal{S})$ , respectively.

4.1 Scenario ${\mathcal{S}}_1$

Recall that, in this scenario, the delivery platform’s decisions with respect to $R_A$ and $R_B$ are decoupled. Moreover, the symmetry in the location choices of $R_A$ and $R_B$ implies that the expressions we derive for the various quantities pertaining to $R_A$ apply to $R_B$ simply by replacing the subscript $A$ with $B$ . Accordingly, we focus on $R_A$ and optimize the delivery platform’s decisions, namely, the driver headcount $N_A$ and the delivery-time guarantee function ${\tau }_A(\cdot )$ . Let ${\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)$ denote the delivery platform’s optimal profit from $R_A$ in Scenario ${\mathcal{S}}_1$ . The delivery platform solves:

$$\begin{array}{rl}{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)& :=\underset{N_A,{\tau }_A(\cdot )}{max}\beta p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du-c{N}_A\end{array}$$

(P_A1)

$$\begin{array}{rl}\text{s.t.}& N_A\ge 0,\\ & {\tau }_A(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_A},\text{ for }u\in [0,1],\text{ and }\end{array}$$

(4)

$$\begin{array}{rl}& \lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du\le \frac{m\omega N_A}{2},\end{array}$$

(5)

where the second constraint ensures that, for a given $u$ , the delivery-time guarantee is at least the minimum feasible delivery-time guarantee that can be provided from restaurant $R_A$ to customers located at a distance $u$ . The third constraint ensures that $R_A$ ’s demand rate is at most the total driver capacity per unit time for $R_A$ . Note that $\frac{m\omega N_A}{2}$ is also the maximum possible demand rate that can be fulfilled using $N_A$ drivers operating at a speed of $\omega$ and with a carry-limit of $m$ .

The delivery platform’s trade-offs with respect to the driver headcount $N_A$ and the delivery-time guarantee ${\tau }_A(u)$ are as follows. As $N_A$ decreases, the cost incurred by the delivery-platform decreases. However, decreasing the driver headcount increases the time between availability of drivers, which in turn, increases the minimum feasible delivery-time guarantee that the delivery platform can provide to customers. Additionally, decreasing the driver headcount also decreases the total driver capacity per unit time. We show in our analysis that at least one of the constraints (4) or (5) is tight at the optimum of ( P A1 ). Thus, as the driver headcount decreases, the delivery platform’s revenue rate decreases. As ${\tau }_A(u)$ decreases, the delivery platform’s revenue rate increases. However, to decrease ${\tau }_A(u)$ , the delivery platform needs to increase the driver headcount, which increases cost.

Our analysis of ( P A1 ) consists of four steps. First, we define the following relaxation of ( P A1 ); we refer to this relaxed version as ( P A2 ).

$$\begin{array}{rl}{\mathrm{\Pi }}_{P_{A2}}^{*}(\lambda )& :=\underset{N_A,{\tau }_A(\cdot )}{max}\beta p\lambda {\int }_0^1(1-{\tau }_A(u){)}^{+}du-c{N}_A\\ \text{s.t.}& N_A\ge 0,\\ & {\int }_0^1(1-{\tau }_A(u){)}^{+}du\le {\int }_0^1{(1-\frac{u}{\omega }-\frac{2}{\omega N_A})}^{+}du,\\ \text{and}& {\int }_0^1(1-{\tau }_A(u){)}^{+}du\le \frac{m\omega N_A}{2\lambda }.\end{array}$$

(P_A2)

Second, we let $z_A={\int }_0^1(1-{\tau }_A(u){)}^{+}du$ . We can then express ( P A2 ) as the following optimization problem over non-negative real numbers $N_A$ and $z_A$ ; we refer to this problem as ( P A3 ).

$$\begin{array}{rl}{\mathrm{\Pi }}_{P_{A3}}^{*}(\lambda )& :=\underset{N_A,z_A}{max}\beta p\lambda z_A-c{N}_A\\ \text{s.t.}& N_A\ge 0,\\ & z_A\le {\int }_0^1{(1-\frac{u}{\omega }-\frac{2}{\omega N_A})}^{+}du,\text{ and}\\ & z_A\le \frac{m\omega N_A}{2\lambda }.\end{array}$$

(P_A3)

Third, we show in Lemma 1 that the optimal solution of ( P A3 ), denoted as $\{{\hat{N}}_A,{\hat{z}}_A\}$ , exhibits two structural properties that allow us to solve ( P A3 ) in closed-form. (The proofs of all the technical results have been placed in Appendix A.) Lemma 1

The unique optimal solution $\{{\hat{N}}_A,{\hat{z}}_A\}$ to ( P A3 ) has the following properties:

1. (1)

$\mathrm{\exists }{\lambda }_{A0}$ such that $\mathrm{\forall }\lambda \ge {\lambda }_{A0}$ , we have ${\hat{N}}_A\ge \frac{2}{\omega -1}$ .

(2)

$\mathrm{\exists }{\lambda }_{A1}$ such that $\mathrm{\forall }\lambda \ge max\{{\lambda }_{A0},{\lambda }_{A1}\}$ , we have

$${\hat{z}}_A={\int }_0^1{(1-\frac{u}{\omega }-\frac{2}{\omega {\hat{N}}_A})}^{+}du=\frac{m\omega {\hat{N}}_A}{2\lambda },$$

(6)

where ${\hat{N}}_A$ solves

$${\int }_0^1{(1-\frac{u}{\omega }-\frac{2}{\omega N_A})}^{+}du=\frac{m\omega N_A}{2\lambda }.$$

The first result in Lemma 1 implies that, $\mathrm{\forall }\lambda \ge {\lambda }_{A0}$ and $u\in [0,1]$ , the integrand on the right-hand side of the first equality in (6) is positive. Then, using the second result in Lemma 1, we derive the optimal driver headcount $N_A^{*}$ and the delivery-time guarantee function ${\tau }_A^{*}(\cdot )$ that solve ( P A1 ). We formally state the result below. Lemma 2

In Scenario ${\mathcal{S}}_1$ , $\mathrm{\forall }\lambda \ge max\{{\lambda }_{A0},{\lambda }_{A1}\}$ , the delivery platform’s optimal decisions corresponding to restaurant $R_A$ are:

$$\begin{array}{rl}& N_A^{*}={\hat{N}}_A=\lambda \left(\frac{(2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}}{2m{\omega }^2}\right)\\ & \text{and }{\tau }_A^{*}(u)=\frac{u}{\omega }+\frac{2}{\omega N_A^{*}}.\end{array}$$

The results in Lemmas 1 and 2 show that, for sufficiently large $\lambda$ , the optimal driver headcount is such that the optimal delivery-time guarantee for customers located at a distance $u$ from $R_A$ is equal to the minimum feasible delivery-time guarantee that can be provided to these customers, given that driver headcount. Additionally, the optimal driver headcount and the optimal delivery-time guarantees are such that the corresponding demand rate for $R_A$ is equal to its total driver capacity per unit time. In other words, at optimality, the delivery platform offers the best feasible delivery-time guarantee and uses all available driver capacity. The symmetry in the location choices of restaurants $R_A$ and $R_B$ implies that the delivery platform’s optimal decisions for $R_A$ in Lemma 2 are also applicable for $R_B$ . The delivery platform’s optimal profit rate is ${\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_1)={\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)+{\mathrm{\Pi }}^{*}(\lambda ;B,{\mathcal{S}}_1)$ . The following result states the platform’s profit rate and the profit rates for $R_A$ and $R_B$ in Scenario ${\mathcal{S}}_1$ . Recall that $m>\frac{2c}{\beta p\omega }$ .

Theorem 1

For Scenario ${\mathcal{S}}_1$ , we have:

1. (1)

The optimal profit rate for the delivery platform is

$$\begin{array}{rl}{\mathrm{\Pi }}^{*}(\lambda ;S_1)& =\lambda (\frac{\beta pm\omega }{2}-c)\times \\ & \left(\frac{(2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}}{m{\omega }^2}\right).\end{array}$$

(2)

The profit rates for the restaurants $R_A$ and $R_B$ are

$$\begin{array}{rl}{\pi }_A(\lambda ;{\mathcal{S}}_1)& ={\pi }_B(\lambda ;{\mathcal{S}}_1)=\frac{(1-\beta )p\lambda }{4\omega }\times \\ & ((2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}).\end{array}$$

4.2 Scenarios ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$

From our definition of Scenario ${\mathcal{S}}_2$ in Section 3, it is evident that the results for restaurant $R_A$ from Scenario ${\mathcal{S}}_1$ directly carry over to this scenario. We therefore focus on restaurant $R_B$ and solve for the delivery platform’s optimal decisions. Recall that, in this scenario, $R_B$ is located at $1/2$ and incurs a rental cost of $F(\lambda )$ . The delivery platform uses $\frac{N_B}{2}$ drivers on each side of $R_B$ and provides a delivery-time guarantee of ${\tau }_B(u)$ to customers located at a distance of $u$ from $R_B$ . Let ${\mathrm{\Pi }}^{*}(\lambda ;B,{\mathcal{S}}_2)$ denote the delivery platform’s optimal profit rate. The delivery platform solves:

$$\begin{array}{rl}{\mathrm{\Pi }}^{*}(\lambda ;B,{\mathcal{S}}_2)& :=\underset{N_B,{\tau }_B(\cdot )}{max}2\beta p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_B(u){)}^{+}du-c{N}_B\\ \text{s.t.}& N_B\ge 0,\\ & {\tau }_B(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_B},\text{ for }u\in [0,\frac{1}{2}],\text{ and }\\ & 2\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_B(u){)}^{+}du\le m\omega N_B.\end{array}$$

(P_B1)

Our analysis of ( P B1 ) is analogous to that of ( P A1 ). Hence, we directly state the optimal solution to ( P B1 ), denoted as $N_B^{*}$ and ${\tau }_B^{*}(\cdot )$ , and provide the details in the appendix.

Lemma 3

In Scenario ${\mathcal{S}}_2$ , $\mathrm{\exists }{\lambda }_{B0},{\lambda }_{B1}$ such that $\mathrm{\forall }\lambda \ge max\{{\lambda }_{B0},{\lambda }_{B1}\}$ , the delivery platform’s optimal decisions corresponding to restaurant $R_B$ are:

$$\begin{array}{rl}& N_B^{*}=\lambda \left(\frac{(4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{128m{\omega }^2}{\lambda }}}{8m{\omega }^2}\right)\\ & \text{and }{\tau }_B^{*}(u)=\frac{u}{\omega }+\frac{2}{\omega N_B^{*}}.\end{array}$$

Using the results in Lemmas 2 and 3, Theorem 2 formally states the delivery platform’s optimal profit rate and the profit rates of the two restaurants in Scenario ${\mathcal{S}}_2$ .

Theorem 2

For Scenario ${\mathcal{S}}_2$ , we have:

1. (1)

The optimal profit rate for the delivery platform is

$$\begin{array}{rl}& {\mathrm{\Pi }}^{*}(\lambda ,{\mathcal{S}}_2)=\lambda (\frac{\beta pm\omega }{2}-c)\times \\ & \left(\frac{(2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}}{2m{\omega }^2}\right)+\\ & \lambda (\beta pm\omega -c)\left(\frac{(4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{128m{\omega }^2}{\lambda }}}{8m{\omega }^2}\right).\end{array}$$

(2)

The profit rates for the restaurants $R_A$ and $R_B$ are

$$\begin{array}{rl}& {\pi }_A(\lambda ;{\mathcal{S}}_2)=\frac{(1-\beta )p\lambda }{4\omega }\times \\ & ((2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}),\\ & {\pi }_B(\lambda ;{\mathcal{S}}_2)=\frac{(1-\beta )p\lambda }{8\omega }\times \\ & ((4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{128m{\omega }^2}{\lambda }})-F(\lambda ).\end{array}$$

Recall that Scenario ${\mathcal{S}}_3$ is identical to Scenario ${\mathcal{S}}_2$ with the roles of $R_A$ and $R_B$ reversed. Thus, all the results for Scenario ${\mathcal{S}}_2$ hold for Scenario ${\mathcal{S}}_3$ with the subscripts $A$ and $B$ swapped.

4.3 Scenario ${\mathcal{S}}_4$

Restaurants $R_A$ and $R_B$ are co-located at $1/2$ in this scenario. The delivery platform allocates $\frac{N_{AB}}{2}$ drivers on each side of $1/2$ and provides a delivery-time guarantee of ${\tau }_{AB}(\cdot )$ (for both the restaurants) on each of these sides. Let ${\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_4)$ denote the delivery platform’s optimal profit rate. Then, the delivery platform’s optimization problem is:

$$\begin{array}{rl}{\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_4)& :=\underset{N_{AB},{\tau }_{AB}(\cdot )}{max}4\beta p\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_{AB}(u){)}^{+}du-c{N}_{AB}\\ \text{s.t.}& N_{AB}\ge 0,\\ & {\tau }_{AB}(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_{AB}},\text{ for }u\in [0,\frac{1}{2}],\text{ and}\\ & 4\lambda {\int }_0^{\frac{1}{2}}(1-{\tau }_{AB}(u){)}^{+}du\le m\omega N_{AB}.\end{array}$$

(P_AB1)

We state the delivery platform’s optimal decisions $N_{AB}^{*}$ and ${\tau }_{AB}^{*}(\cdot )$ in the following lemma.

Lemma 4

In Scenario ${\mathcal{S}}_4$ , $\mathrm{\exists }{\lambda }_{AB0},{\lambda }_{AB1}$ such that $\mathrm{\forall }\lambda \ge max\{{\lambda }_{AB0},{\lambda }_{AB1}\}$ , the delivery platform’s optimal decisions are:

$$\begin{array}{rl}& N_{AB}^{*}=\lambda \left(\frac{(4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{64m{\omega }^2}{\lambda }}}{4m{\omega }^2}\right)\\ & \text{and }{\tau }_{AB}^{*}(u)=\frac{u}{\omega }+\frac{2}{\omega N_{AB}^{*}}.\end{array}$$

The following result states the delivery platform’s optimal profit rate and the restaurants’ profit rates in this scenario.

Theorem 3

For Scenario ${\mathcal{S}}_4$ , we have:

1. (1)

The optimal profit rate for the delivery platform is

$$\begin{array}{rl}& {\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_4)=\lambda (\beta pm\omega -c)\times \\ & \left(\frac{(4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{64m{\omega }^2}{\lambda }}}{4m{\omega }^2}\right).\end{array}$$

(2)

The profit rate for the restaurants $R_A$ and $R_B$ are

$$\begin{array}{rl}& {\pi }_A(\lambda ;{\mathcal{S}}_4)={\pi }_B(\lambda ;{\mathcal{S}}_4)=\frac{(1-\beta )p\lambda }{8\omega }\times \\ & ((4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{64m{\omega }^2}{\lambda }})-F(\lambda ).\end{array}$$

Definition 1 (Threshold ${\lambda }_0$ )

Recall from Section 3 that we are interested in the outcome of the two-stage game under sufficiently high values of the population density $\lambda$ . In Section 5, we analyze the restaurants’ location game for $\lambda \ge {\lambda }_0$ . Using Lemmas 2–4, we define

$${\lambda }_0:=max\{{\lambda }_{A0},{\lambda }_{A1},{\lambda }_{B0},{\lambda }_{B1}\}.$$

5 Analysis of the Location Game of the Restaurants

Given the delivery platform’s optimal decisions in each scenario, restaurants $R_A$ and $R_B$ simultaneously decide whether to stay at their current locations or relocate to the cloud kitchen at the center. Next, we analyze the static game between $R_A$ and $R_B$ and characterize the equilibria of this game.

From Theorems 1–3, we have

$$\begin{array}{rl}& {\pi }_A(\lambda ;{\mathcal{S}}_1)={\pi }_A(\lambda ;{\mathcal{S}}_2)={\pi }_B(\lambda ;{\mathcal{S}}_1)={\pi }_B(\lambda ;{\mathcal{S}}_3)=\\ & \frac{(1-\beta )p\lambda }{4\omega }((2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{16m{\omega }^2}{\lambda }}),\\ & {\pi }_A(\lambda ;{\mathcal{S}}_3)={\pi }_B(\lambda ;{\mathcal{S}}_2)=\\ & \frac{(1-\beta )p\lambda }{8\omega }((4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{128m{\omega }^2}{\lambda }})-F(\lambda ),\\ & {\pi }_A(\lambda ;{\mathcal{S}}_4)={\pi }_B(\lambda ;{\mathcal{S}}_4)=\\ & \frac{(1-\beta )p\lambda }{8\omega }((4\omega -1)+\sqrt{(4\omega -1{)}^2-\frac{64m{\omega }^2}{\lambda }})-F(\lambda ).\end{array}$$

Since ${\pi }_A(\lambda ;{\mathcal{S}}_4)>{\pi }_A(\lambda ;{\mathcal{S}}_3)$ and ${\pi }_B(\lambda ;{\mathcal{S}}_4)>{\pi }_B(\lambda ;{\mathcal{S}}_2)$ , scenarios ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ cannot define Nash equilibria. Thus, if it is optimal for one restaurant to locate at the cloud kitchen, then it is also optimal for the other restaurant to locate there. Thus, the possible Nash equilibria are:

1. Status-Quo Nash Equilibrium: The restaurants choose to stay at their current locations.

2. Cloud-Kitchen Nash Equilibrium: The restaurants choose to co-locate at the cloud kitchen.

The following lemma provides necessary and sufficient conditions for these equilibria. Recall that the parameter $\lambda$ can be viewed as an indicator of the population density of a city.

Lemma 5

The characterizing conditions for the status-quo and cloud-kitchen equilibria are:

1. (1)

The status-quo equilibrium exists if and only if $\lambda$ satisfies:

$$\begin{array}{rl}\frac{2F(\lambda )}{(1-\beta )p\lambda }\ge & \frac{1}{4\omega }+\sqrt{{(1-\frac{1}{4\omega })}^2-\frac{8m}{\lambda }}-\\ & \sqrt{{(1-\frac{1}{2\omega })}^2-\frac{4m}{\lambda }}.\end{array}$$

(2)

The cloud-kitchen equilibrium exists if and only if $\lambda$ satisfies:

$$\begin{array}{rl}\frac{2F(\lambda )}{(1-\beta )p\lambda }\le & \frac{1}{4\omega }+\sqrt{{(1-\frac{1}{4\omega })}^2-\frac{4m}{\lambda }}-\\ & \sqrt{{(1-\frac{1}{2\omega })}^2-\frac{4m}{\lambda }}.\end{array}$$

To examine the dependence of these equilibria on the parameter $\lambda$ , we identify a practically relevant functional form of the rental cost $F(\lambda )$ based on the Urban Economics literature. This literature studies factors that determine city structures, urban land use, wages in urban labor markets and agglomeration economies; see, e.g. Duranton and Puga (2015) for a review. This literature also provides both the theoretical foundation and the empirical evidence for mono-centric and poly-centric cities—in particular, this understanding facilitates the calculation of the rent gradient (change in rent with respect to the distance from a central business district), the density elasticity of rent (percentage change in rent with respect to percentage change in density), and the density elasticity of housing prices (Duranton and Puga, 2020; Drennan and Kelly, 2010). Relevant to our purpose, this literature finds that commercial rent is concave in the population density of a city and also identifies a common form: $F(\lambda )$ is proportional to ${\lambda }^{\alpha }$ , where $\alpha <1$ represents the density elasticity of commercial rent. Further, several studies empirically estimate $\alpha$ : Rosenthal and Strange (2020) use datasets of commercial leases in 89 urban U.S. areas during 2019 and 2020 to show that $\alpha$ ranges from 0.0458 to 0.1338; Liu et al. (2018) estimate $\alpha$ in the urban U.S. to be about 0.1068 using datasets for 18 metropolitan cities; Koster et al. (2014) estimate $\alpha$ in the Netherlands to be around 0.082. Using $F(\lambda )=k\cdot {\lambda }^{\alpha }$ , where $\alpha <1$ and $k$ is a constant, the ratio $\frac{F(\lambda )}{\lambda }$ is a decreasing function of $\lambda$ and $\underset{\lambda \to \mathrm{\infty }}{lim}\frac{F(\lambda )}{\lambda }=0$ . Using these properties, the result below characterizes the Nash equilibria of the restaurants’ location game; see Figure 5 for a pictorial representation of this result. Theorem 4

Let ${\lambda }_0$ be as specified in Definition 1 and assume that $\lambda \ge {\lambda }_0$ . Then, there exist thresholds ${\lambda }_1$ and ${\lambda }_2$ with ${\lambda }_1<{\lambda }_2$ such that the following statements hold for the location game of the restaurants:

1. (1)

If ${\lambda }_0\le {\lambda }_2$ , then for all $\lambda \in [max\{{\lambda }_0,{\lambda }_1\},{\lambda }_2]$ , both the status-quo equilibrium and the cloud-kitchen equilibrium exist. The cloud-kitchen equilibrium is the Pareto-dominant Nash equilibrium, i.e. the profit rates for both the restaurants under the cloud-kitchen equilibrium are greater than those under the status-quo equilibrium. For all $\lambda >{\lambda }_2$ , only the cloud-kitchen equilibrium exists.

(2)

Otherwise (i.e., if ${\lambda }_0>{\lambda }_2$ ), only the cloud-kitchen equilibrium exists.

Figure 5.

Pictorial Representation of Theorem 4.

Remark 3

In the model with competition that we analyze in Appendix B, we establish Theorem 6 which parallels Theorem 4 above and conveys the same message: as the population density increases beyond a threshold, restaurants co-locating at the cloud kitchen is first a Pareto-dominant equilibrium and then the unique equilibrium.

Theorem 4 establishes that the profitability of both the restaurants is higher under the cloud-kitchen equilibrium. A natural question arises: Are the other two primary stakeholders, namely the delivery platform and the customers, also better off under this equilibrium? The following result answers this question in the affirmative. Proposition 1

When both the status-quo and the cloud-kitchen equilibria exist, the following statements hold.

1. (1)

The delivery platform’s profit rate under the cloud-kitchen equilibrium is greater than that under the status-quo equilibrium.

(2)

The delivery-time guarantees for customers under the cloud-kitchen equilibrium are lower than those under the status-quo equilibrium.

Together, Theorem 4 and Proposition 1 imply that cloud kitchens represent a win-win-win for the three primary stakeholders, namely, the delivery platform, the restaurants, and the customers. We make two observations.

• The maximum order rate $\lambda$ , which is representative of the population density of the city, plays a central role in driving the Nash equilibrium of the restaurants’ location game. Specifically, as $\lambda$ increases, either the cloud-kitchen equilibrium dominates the status-quo equilibrium when both equilibria exist or only the cloud-kitchen equilibrium exists. This result is in line with current industry trends: Delivery platforms have predominantly opened cloud kitchens in cities or areas with high population density. ⁸ Further, over the past year, the number of online orders has increased due to COVID-19 pandemic, thus increasing the maximum order rate and, in turn accelerating the growth of cloud kitchens.

• The cloud-kitchen equilibrium is driven by two advantages. The first advantage is due to a cloud-kitchen’s “central location”, which allows for better access to customers. Here, better access refers to the ability of the delivery platform to provide lower delivery-time guarantees to customers—this increases profitability for both the delivery platform and the restaurants. We refer to this advantage as the cloud kitchen’s location advantage. The second advantage accrues from the ability of the delivery platform to consolidate its driver capacity when the restaurants co-locate at the cloud kitchen. This consolidation increases the delivery platform’s profitability as it can use the same set of drivers to deliver orders for multiple restaurants. We refer to this advantage of the cloud kitchen as its consolidation advantage. An important observation from our analysis is that the delivery platform and the restaurants can both exploit the location and consolidation advantages if and only if the restaurants choose to co-locate at the center. These two advantages drive the delivery platform’s decisions and also the restaurants’ location choices in such a way that both the delivery platform and the restaurants are better off when the restaurants co-locate at the cloud kitchen.

The observation that co-location of restaurants at the cloud kitchen benefits all stakeholders is similar in spirit to the benefit offered by clustering in other industries such as manufacturing and retail; see, e.g. Bozarth et al. (2007). To facilitate a better understanding of our analysis, we now present two numerical examples.

5.1 Numerical Examples

The first example illustrates the increase in the profits of the delivery platform and the restaurants as the maximum demand rate increases. The second example demonstrates the validity of the result in Theorem 4 for the “triangular” demand function, where the maximum demand rate at the cloud-kitchen is greater than that at the extreme locations.

We consider the following values of the parameters. The length of the city is $25$ miles and the maximum demand rate is $\lambda =50$ orders per hour per mile. The revenue per order is $p=40$ USD and the revenue-share percentage (that the delivery platform charges from the restaurants) is $\beta =25\mathrm{\%}$ . The cost rate of drivers is $c=8$ USD per hour, the driver carry-limit is $m=2$ orders, and drivers travel at a speed of $\omega =1.2$ city lengths per hour; since the length of the city is 25 miles, $\omega =1.2$ corresponds to a speed of 30 miles per hour. Recall from our discussion of the Urban Economics literature (Section 5) that a common functional form for the fixed rental rate is $F(\lambda )=K\cdot {\lambda }^{\alpha }$ with $\alpha <1$ . For the purpose of our examples, we let $K=50$ and $\alpha =0.5$ .

Example 1

(Impact of Increase in the Maximum Demand Rate ( $\lambda$ ) on the Profits of the Platform and the Restaurants). Using the parameter values above, along with Definition 1, Lemma 5, and Theorem 4, it is straightforward to compute the three thresholds used in Theorem 4, namely ${\lambda }_0,{\lambda }_1$ , and ${\lambda }_2$ . We have ${\lambda }_0=34$ orders per hour, ${\lambda }_1=51.4$ orders per hour, and ${\lambda }_2=80.4$ orders per hour. Since the maximum demand rate of the city is $\lambda =50$ orders per hour, Theorem 4 implies that the status-quo equilibrium exists and the cloud-kitchen equilibrium does not exist.

We now consider a $10\mathrm{\%}$ increase in $\lambda$ , i.e. $\lambda =55$ orders per hour. For this value of $\lambda$ , both the status-quo and the cloud-kitchen equilibria exist. If restaurants chose the status-quo equilibrium, i.e. they continue to stay at their current extreme locations, then this $10\mathrm{\%}$ increase in $\lambda$ results in an $11\mathrm{\%}$ increase in the profits of both restaurants and the platform. If, however, the restaurants choose to relocate to the cloud-kitchen, the profits of both the restaurants increase by $13\mathrm{\%}$ and the platform’s profit increases by $224\mathrm{\%}$ . Thus, the cloud kitchen equilibrium is Pareto dominant and preferred by both the restaurants and the platform.

Example 2

(Triangular Demand Rate Function). Recall that, in our base model, the demand rate is uniform and equal to $\lambda$ throughout the city. We now consider a setting in which the demand rate is a “triangular” function, with its peak at the cloud-kitchen. Specifically, for a constant $\eta \ge 1$ , the demand rate at a point located at a distance of $u$ from $0$ is

$$\overline{\lambda }(u)=\{\begin{array}{ll}(2(\eta -1)u+1)\lambda ,& u\le \frac{1}{2}\\ (2(1-\eta )u+2\eta -1)\lambda ,& u>\frac{1}{2}.\end{array}$$

Note that for $\eta =1$ , this demand rate reduces to the uniform rate in the base model.

The delivery platform’s optimization problem pertaining to Restaurant $R_A$ in Scenario ${\mathcal{S}}_1$ is as follows:

$$\begin{array}{rl}{\overline{\mathrm{\Pi }}}^{*}(\lambda ,\eta ;A,{\mathcal{S}}_1)& :=\underset{N_A,{\tau }_A(\cdot )}{max}\beta p{\int }_0^1\overline{\lambda }(u)(1-{\tau }_A(u){)}^{+}du-c{N}_A\\ \text{s.t.}& N_A\ge 0,\\ & {\tau }_A(u)\ge \frac{u}{\omega }+\frac{2}{\omega N_A},\text{ for }u\in [0,1],\text{ and }\\ & {\int }_0^1\overline{\lambda }(u)(1-{\tau }_A(u){)}^{+}du\le \frac{m\omega N_A}{2}.\end{array}$$

Using arguments analogous to those used to establish Lemmas 1 and 2, it is straightforward to verify that for sufficiently large $\lambda$ , the platform’s optimal decisions for $R_A$ in ${\mathcal{S}}_1$ are

$$\begin{array}{rl}& {\overline{N}}_A^{*}=\frac{(\eta +1)\lambda }{2}\left(\frac{(2\omega -1)+\sqrt{(2\omega -1{)}^2-\frac{32m{\omega }^2}{(\eta +1)\lambda }}}{2m{\omega }^2}\right)\\ & \text{and }{\overline{\tau }}_A^{*}(u)=\frac{u}{\omega }+\frac{2}{\omega {\overline{N}}_A^{*}}.\end{array}$$

As expected, for $\eta =1$ , the expressions above coincide with those in Lemma 2. For brevity, we do not provide the solutions to the platform’s optimization problems in the other three scenarios. Instead, for this setting, we verify the result in Theorem 4 numerically. As in Example 1, we consider the following values of the parameters: $\lambda =50$ , $p=40$ , $\beta =25\mathrm{\%}$ , $c=8$ , $m=2$ , $\omega =1.2$ , $K=50$ , and $\alpha =0.5$ . Further, we set $\eta =1.2$ . Given these parameter values, we have that ${\lambda }_0=34$ , ${\lambda }_1=44$ , and ${\lambda }_2=72$ ; both equilibria exist for $\lambda \in [44,72]$ and only the cloud-kitchen equilibrium exists for $\lambda \ge {\lambda }_2=72$ . Recall that, in our base model, we have that ${\lambda }_1=51.4$ and ${\lambda }_2=80.4$ ; see Example 1. Intuitively, the higher demand rate at the cloud-kitchen (relative to the extreme locations) makes it more attractive for the restaurants. Therefore, under this setting, the values of ${\lambda }_1$ and ${\lambda }_2$ in Theorem 4 are lower than those in our base model.

The extent of the relative benefit offered by the cloud-kitchen equilibrium depends on several factors, including the carry-limit, speed, and cost of drivers. In the next section, we offer insights on the behavior of this relative benefit.

6 Insights on the Relative Attractiveness of Cloud-Kitchens

Here, we investigate the impact of carry-limit, speed, and cost of drivers on the delivery platform’s profit rates under the cloud-kitchen and status-quo equilibria. We focus on the more interesting case when both these equilibria exist and consider the ratio of the delivery platform’s profit rate under the cloud-kitchen equilibrium to that under the status-quo equilibrium, i.e. the ratio

$$\frac{{\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_4)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)+{\mathrm{\Pi }}^{*}(\lambda ;B,{\mathcal{S}}_1)}.$$

(7)

This ratio can naturally be interpreted as the relative attractiveness of the cloud-kitchen equilibrium for the delivery platform. Thus, investigating the impact of the above driver-related parameters on this ratio allows us to identify conditions under which the attractiveness of the cloud-kitchen business model increases for the delivery platform. The following result summarizes the key insights from this investigation. Proposition 2

The relative attractiveness of a cloud kitchen for the delivery platform increases when (a) driver carry-limit is below a threshold and decreases, (b) driver speed decreases, and (c) driver cost increases.

To aid our discussion of the results in Proposition 2, we express the ratio in (7) as

$$\begin{array}{rl}\frac{{\mathrm{\Pi }}^{*}(\lambda ;{\mathcal{S}}_4)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)+{\mathrm{\Pi }}^{*}(\lambda ;B,{\mathcal{S}}_1)}=& \underset{\text{Location Advantage}}{\underbrace{\left(\frac{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)}\right)}}\times \\ & \underset{\text{Consolidation Advantage}}{\underbrace{\left(\frac{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_4)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)}\right)}}.\end{array}$$

(8)

Recall that $R_A$ is located at $0$ in Scenario ${\mathcal{S}}_1$ and at $1/2$ in Scenarios ${\mathcal{S}}_3$ and ${\mathcal{S}}_4$ . Since ${\mathcal{S}}_1$ and ${\mathcal{S}}_3$ are dedicated scenarios, the difference between the delivery platform’s profit rates from restaurant $R_A$ in ${\mathcal{S}}_3$ and ${\mathcal{S}}_1$ can be attributed to change in $R_A$ ’s location. Since location $1/2$ provides better access to customers, the ratio ${\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)/{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)$ represents the location advantage. This advantage can be precisely expressed as follows:

$$\frac{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_1)}=\underset{\begin{array}{c}\text{Ratio of profit rate per driver}\\ \text{in }{\mathcal{S}}_3\text{ to that in }{\mathcal{S}}_1\end{array}}{\underbrace{\left(\frac{\beta pm\omega -c}{\frac{\beta pm\omega }{2}-c}\right)}}\times \underset{\begin{array}{c}\text{Ratio of driver headcount}\\ \text{in }{\mathcal{S}}_3\text{ to that in }{\mathcal{S}}_1\end{array}}{\underbrace{\left(\frac{N_A^{*}({\mathcal{S}}_3)}{N_A^{*}({\mathcal{S}}_1)}\right)}}.$$

(9)

We now turn to the consolidation advantage. Since ${\mathcal{S}}_4$ is the consolidated scenario, the difference between the delivery platform’s profit rates from restaurant $R_A$ in Scenarios ${\mathcal{S}}_4$ and ${\mathcal{S}}_3$ can be attributed to consolidation of driver capacity. Hence, the ratio ${\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_4)/{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)$ can be viewed as the cloud-kitchen’s consolidation advantage, which can be expressed as:

$$\frac{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_4)}{{\mathrm{\Pi }}^{*}(\lambda ;A,{\mathcal{S}}_3)}=\underset{\begin{array}{c}\text{Ratio of driver headcounts in }{\mathcal{S}}_4\text{ and }{\mathcal{S}}_3\end{array}}{\underbrace{\frac{1}{2}\left(\frac{N_{AB}^{*}({\mathcal{S}}_4)}{N_A^{*}({\mathcal{S}}_3)}\right)}}.$$

(10)

With these observations and expressions, we now discuss the impact of carry-limit, speed, and cost of drivers on a cloud-kitchen’s location and consolidation advantages.

• Impact of the driver carry-limit $\mathit{m}$ : As the driver carry-limit $m$ decreases, both terms $(\beta pm\omega -c)$ and $(\frac{\beta pm\omega }{2}-c)$  decrease. However, the ratio of these two terms increases; intuitively, each driver becomes more profitable in ${\mathcal{S}}_3$ relative to that in ${\mathcal{S}}_1$ . Further, as $m$ decreases, the optimal driver headcounts $N_A^{*}({\mathcal{S}}_3)$ and $N_A^{*}({\mathcal{S}}_1)$ both increase, and so does the ratio $N_A^{*}({\mathcal{S}}_3)/N_A^{*}({\mathcal{S}}_1)$ . Thus, from (9), the location advantage increases as $m$ decreases. However, as the driver carry-limit $m$ decreases, the ratio $N_{AB}^{*}({\mathcal{S}}_4)/N_A^{*}({\mathcal{S}}_3)$ decreases, and thus from (10), the consolidation advantage decreases. We show in Proposition 2 that, when the driver-carry limit $m$ is below a threshold, the increase in the location advantage in (9) dominates the decrease in the consolidation advantage in (10). Thus, from (8), the cloud-kitchen’s relative attractiveness for the delivery platform increases when the driver-carry limit $m$ is below a threshold and decreases.

• Impact of the driver speed $\mathit{\omega }$ : As the driver speed $\omega$ decreases, both terms $(\beta pm\omega -c)$ and $(\frac{\beta pm\omega }{2}-c)$ decrease, but their ratio increases. Further, the ratio $N_A^{*}({\mathcal{S}}_3)/N_A^{*}({\mathcal{S}}_1)$ increases as $\omega$ decreases, and, hence, from (9), the location advantage increases. In addition, decreasing $\omega$ also increases the ratio $N_{AB}^{*}({\mathcal{S}}_4)/(2N_A^{*}({\mathcal{S}}_3))$ , which is the consolidation advantage. Consequently, from (8), a cloud kitchen becomes relatively more attractive for the delivery platform as the driver speed $\omega$ decreases.

• Impact of the driver cost $\mathit{c}$ : Recall from our analysis in Section 4 that, for a sufficiently high population density (which exceeds a threshold that depends on the driver cost $c$ ), the optimal driver headcount itself is independent of driver cost. Therefore, the ratio $N_A^{*}({\mathcal{S}}_3)/N_A^{*}({\mathcal{S}}_1)$ in (9) and the ratio $N_{AB}^{*}({\mathcal{S}}_4)/(2N_A^{*}({\mathcal{S}}_3))$ in (10) are independent of driver cost. Consequently, from (8), (9), and (10), it follows that the behavior of the relative attractiveness of a cloud kitchen is determined by the behavior of the ratio $(\beta pm\omega -c)/(\frac{\beta pm\omega }{2}-c)$ . Since this ratio increases as $c$ increases, a cloud-kitchen’s relative attractiveness for the delivery platform also increases as the driver cost increases.

The analysis above shows the impact of driver-related parameters on the relative benefit from the location and consolidation advantages of the cloud-kitchen. Roy et al. (2022) also highlight the importance of a restaurant’s location decision and discuss the factors that affect this decision. We now present a numerical example that showcases an interesting lever through which the delivery platform can attract restaurants to the cloud kitchen. Example 3

(Revenue-Share Percentage: A Lever for Delivery Platforms). The platform can exploit the revenue-share percentage as an important lever to increase its profits. To illustrate, we consider the impact of reducing the revenue-share percentage from $\beta =25\mathrm{\%}$ to $\beta =20\mathrm{\%}$ on the profits of the platform and the restaurants. Recall that, with $\beta =25\mathrm{\%}$ , we have ${\lambda }_1=51.4$ orders per hour. Since ${\lambda }_1=51.4$ is greater than $\lambda =50$ , the restaurants choose to stay at their current extreme locations. With $\beta$ reduced to $20\mathrm{\%}$ from $25\mathrm{\%}$ , the value of ${\lambda }_1$ decreases to $42.1$ , thus making the cloud-kitchen equilibrium the preferred choice of the restaurants. However, note that by reducing $\beta$ , the delivery platform’s per-order revenue also decreases. Thus, by reducing $\beta$ , while the delivery platform can incentivize the restaurants to relocate to the cloud kitchen and thereby benefit from an increase in the sales volume, this benefit may be offset by a reduced per-order revenue. When $\beta$ decreases from $25\mathrm{\%}$ to $20\mathrm{\%}$ , it is straightforward to verify that, while the delivery platform’s revenue per order decreases by $20\mathrm{\%}$ , its total sales rate increases by $46.3\mathrm{\%}$ and its delivery cost decreases by $26.8\mathrm{\%}$ . Thus, using Theorems 1–3, the delivery platform’s profit increases by $105\mathrm{\%}$ . Further, the profit of each restaurant also increases by $9\mathrm{\%}$ .

The key takeaway from the example above is as follows. In practice, given that restaurants already operate at low-profit margins, delivery platforms are often unable to increase their revenue-share percentages. This example illustrates that the cloud-kitchen business model can allow a delivery platform to, in fact, decrease its revenue-share percentage and improve its own profit as well as the profits of the restaurants.

Remark 4

(Incorporating Dine-In Customers). In our base model, all orders at the two restaurants are delivery orders. Consider a setting where each restaurant also serves dine-in customers. Suppose that each restaurant has the option of moving its online delivery business to the cloud-kitchen at the city center (to serve delivery customers) and continue to operate its dine-in facility at its current location. For this setting, suppose that a restaurant currently located at an extreme location pays a rental rate of $F_1(\lambda )$ to serve both dine-in and delivery customers from this location. If this restaurant chooses to relocate its delivery business to the cloud kitchen, then it incurs a rental rate of $F_1(\lambda )+F(\lambda )$ , where $F(\lambda )$ is the rental rate at the cloud kitchen. Thus, the increase in the restaurant’s rental rate by relocating its delivery business to the cloud kitchen is $(F_1(\lambda )+F(\lambda ))-F_1(\lambda )=F(\lambda )$ , which is the same as in our base model. Consequently, our analysis and results for the base model remain applicable for this setting as well.

Remark 5

(Incorporating Delivery Fee). In practice, delivery platforms typically charge a delivery fee in addition to the price of the order. We can incorporate this in our model as follows. Suppose that the delivery fee is ${\hat{p}}_f$ , the order price is $\hat{p}$ , and the customer is charged $\hat{p}+{\hat{p}}_f$ . The platform retains the delivery fee ${\hat{p}}_f$ and a $\hat{\beta }$ proportion of the order price $\hat{p}$ , while the restaurant keeps the remaining $1-\hat{\beta }$ proportion. To incorporate this structure, we define $p:=\hat{p}+{\hat{p}}_f$ and $\beta$ such that $(1-\beta )p=(1-\hat{\beta })(p-{\hat{p}}_f)$ or, equivalently, $\beta =\hat{\beta }+\frac{{\hat{p}}_f}{p}(1-\hat{\beta })$ in our model.

Remark 6

(Surge Pricing). In practice, delivery platforms typically employ surge pricing to better match supply and demand. Specifically, surge pricing (a) helps drivers earn more during high-demand periods, thereby increasing the supply of drivers and (b) reduces the demand for orders (less people want to order food at a higher price). At an intuitive level, the impact of the introduction of surge pricing by the delivery platform is roughly similar to the impact of increasing the parameter $p$ (revenue per order) in our model. The impact of increasing $p$ on the platform’s and the restaurants’ decisions is quite complex. As $p$ increases, the maximum demand rate $\lambda$ decreases—this decrease in $\lambda$ in turn leads to a decrease in the number of drivers hired and thus the number of orders fulfilled by the platform. However, since $p$ is higher, the profit from each order is also higher. Thus, the eventual impact on the platform’s profits and the restaurants’ location decisions depends on which of these effects (increase in the profit from each order and decrease in the number of orders fulfilled by the platform) is stronger.

7 Concluding Remarks

Over the last decade, the convenience of food delivery combined with increasing traffic congestion in high-density cities has resulted in delivery orders replacing not only dine-in and drive-through orders at restaurants but also home-cooked meals (Cookpad, 2018). There are two important environmental implications of this trend, as we now discuss.

First, since cloud kitchens lead to an increase in the number of delivery orders in a city, an increase in the number of cloud kitchens can have a positive impact on the environment via a reduction in carbon emissions. Cloud kitchens are similar to local micro-fulfillment centers, in the sense that both lead to a decrease in number of last-mile delivery trips and promote the usage of electric vehicles (Specter, 2020; Accenture, 2021). Local micro-fulfillment centers, a recent innovation in the e-commerce industry, allow the placement of inventory closer to customers. This leads to an increase in the success rate of last-mile deliveries and, hence, a reduction in the number of last-mile delivery trips. Further, since these centers are located closer to customers, electric vehicles can be used to make deliveries. A recent report (Accenture, 2021) finds that local micro-fulfillment centers have the potential to lower last-mile emissions between 17% and 26% by 2025 in cities such as London, Sydney, and Chicago. Similar to local micro-fulfillment centers, cloud kitchens can lead to a decrease in the number of delivery trips. Typically, there are limited opportunities for a delivery platform to consolidate orders from multiple restaurants as they are not necessarily co-located. A cloud kitchen, however, houses multiple restaurants and, hence, allows for multiple orders to be delivered in a single delivery trip. Further, like local micro-fulfillment centers, cloud kitchens are located closer to customers and well-suited to adopt electric vehicles for delivery (Descant, 2020). Given these benefits of cloud kitchens, another potential research direction is to understand how cloud kitchens can impact last-mile delivery emissions in dense cities.

Second, an increase in the number of delivery orders may lead to a decrease in food wastage. Belavina (2021) shows that increasing the grocery-store density in a city (up to a certain threshold) improves consumers’ access to groceries, and can lead to a decrease in food wastage. In a similar vein, cloud kitchens provide consumers with better access to food delivery since they allow for faster deliveries in high-density areas. Thus, an increase in the density of cloud kitchens can result in improved food delivery and thereby a higher replacement of home-cooked meals, leading to less stocking by consumers at home and, ultimately, a reduction in food wastage. However, such a change in consumer behavior also has important health implications which can be incorporated in a model that analyzes the impact of cloud kitchen on food wastage. Further, multiple restaurants co-locating at a cloud kitchen can also lead to a reduction in food wastage at the restaurant level, since restaurants can consolidate their supply of fresh groceries required for preparing menu items. Given our main result, namely that co-locating at a cloud kitchen is either the Pareto dominant Nash equilibrium or the unique Nash equilibrium as the population density of a city increases beyond a threshold, understanding the impact of an increase in the density of cloud kitchens on food wastage is an interesting and important research direction.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478231224950 - Supplemental material for Cloud-Kitchens: Value Creation Through Co-Location

Supplemental material, sj-pdf-1-pao-10.1177_10591478231224950 for Cloud-Kitchens: Value Creation Through Co-Location by Arun Rout, Milind Dawande and Ganesh Janakiraman in Production and Operations Management