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Service and capacity planning in crowd-sourced delivery
Transportation Research Part C 100 (2019) 177–199
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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Service and capacity planning in crowd-sourced delivery Baris Yildiza, , Martin Savelsberghb ⁎
a b
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Department of Industrial Engineering, Koc University, Istanbul, Turkey H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, USA
ABSTRACT
The success of on-demand service platforms, e.g., Uber and Lyft to obtain a ride and Grubhub and Eat24 to get a meal, which rely on crowd-sourced transportation capacity, has radically changed the view on the potential and benefits of crowd-sourced transportation and delivery. Many retail stores, for example, are examining the pros and cons of introducing crowd-sourced delivery in their omni-channel strategies. However, few models exist to support the analysis of service area, service quality, and delivery capacity planning, and their interaction, in such environments. We introduce a model that seeks to do exactly that and can answer many fundamental questions arising in these settings. Using on-demand meal delivery platforms as an example, we investigate, among others, the relation between service area and profit and delivery offer acceptance probability and profit, and the benefits of integrating delivery service of multiple restaurants, and generate many valuable insights.
1. Introduction Enabled by the recent advances in communication and GPS technologies, on-demand service platforms are reshaping our way of life, witnessed by the fact that Uber (www.uber.com) and Lyft (www.lyft.com) have become household names. On-demand service platforms connect waiting-time sensitive customers with independent service providers (Bai et al., 2016). We focus on on-demand meal delivery platforms, which target a massive market. As of 2015, about $210 billion worth of food is ordered for delivery or takeout on an annual basis solely in the US (Morgan Stanley Research, 2016), whereas the two leading on-demand meal delivery platforms, Grubhub (www.Grubhub.com) and Eat24 (www.eat24.com), generated a combined $2.6 billion in food sales capturing only a tiny fraction of the total market (Bakker, 2016). However, succeeding and thriving in this market requires looking beyond traditional last mile delivery practices. Meal delivery is arguably the ultimate challenge in last mile logistics: a typical order is expected to be delivered within an hour (much less if possible), and within minutes of the food becoming ready. Restaurant-operated delivery services can no longer meet the high diner expectations of fast service, low cost, and near-continuous availability. Ondemand meal delivery platforms, benefiting from the delivery density that comes from delivering orders from multiple restaurants and relying on crowd-sourced delivery, are meeting diner expectations – most of the time. In a market with high order arrival rate fluctuations throughout the day (soaring during the meal times and dropping to very low levels in between) the advantage of crowd-sourced delivery, i.e., contracting individuals to deliver meals, is obvious (Reuters, 2016). However, even though such couriers (the term we will use throughout the paper to refer to individuals contracted to deliver meals) may provide the desired flexibility in delivery capacity, they also introduce many challenges (Allon et al., 2012; Farber, 2015; Chen and Sheldon, 2016; Gurvich et al., 2016; Taylor, 2018; Cachon et al., 2017). Ensuring that a sufficient number of couriers is available to provide the service quality that is critical to sustain an online meal delivery service platform is difficult, in part because couriers do not necessarily accept all delivery tasks offered to them (Yildiz and Savelsbergh, 2018; Sampaio et al., 2019). Novel approaches for service coverage and delivery capacity planning are needed to exploit the benefits and to limit the downsides of crowd-sourced delivery for successful business implementations. However, mostly due to the novelty of the practice, there is little research on crowd-
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Corresponding author. E-mail address: [email protected] (B. Yildiz).
https://doi.org/10.1016/j.trc.2019.01.021 Received 16 August 2018; Received in revised form 21 January 2019; Accepted 21 January 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.
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sourced meal delivery operations providing insights that can help practitioners (more) effectively manage demand (delivery orders) and supply (delivery capacity) so as to achieve their strategic goals related to profitability, market share, and customer satisfaction. Our study is meant to be a first step in that direction. One key characteristic of the crowd-sourced meal delivery environment is the self-scheduling delivery capacity, i.e., couriers independently choosing the time and duration of their service. Consistent with the existing literature and empirical findings, we model this feature as a market with a large pool of couriers making decisions based on a rational expectation about their earnings (Snir and Hitt, 2003; Moreno and Terwiesch, 2014; Farber, 2015; Chen and Sheldon, 2016; Gurvich et al., 2016; Bai et al., 2016; Banerjee et al., 2015; Bimpikis et al., 2016; Taylor, 2018; Cachon et al., 2017; Kabra et al., 2017). In such a market, an online meal delivery platform has to carefully coordinate capacity and demand in order to maximize its profit while ensuring a target service quality. An online meal delivery platform has two ways to influence diner demand. First, by adjusting the price they charge restaurants/ diners for deliveries. Second, by adjusting the service coverage area associated with a restaurant. In the literature, only the former has been investigated. However, in practice, many on-demand service platforms avoid real time pricing, in part, because of diner resistance to the practice (MacMillan, 2015; Taylor, 2018), and consider dynamic service coverage and demand management strategies (directing diner demands to restaurants that are easier to serve) to achieve high diner satisfaction levels which is crucial for a sustainable growth in the market. Therefore, we focus on service coverage planning as the main leverage to manipulate demand (even though our model can be used to analyze dynamic pricing). An online meal delivery platform has two ways to manage delivery capacity. First, by adjusting courier compensation, which indirectly controls the crowd-sourced delivery capacity. Second, by employing salaried drivers, they directly control the companyprovided delivery capacity. Here, we use “salaried drivers” to indicate any form of compensation that gives the company more direct control over its delivery capacity; it also covers minimum-pay guarantees in return for a courier’s commitment to make deliveries in a specified block of time (as is done by some online platforms). Using compensation to manipulate capacity has been investigated in the expanding on-demand service platforms literature (Gurvich et al., 2016; Bai et al., 2016; Bimpikis et al., 2016; Taylor, 2018; Cachon et al., 2017), but, to the best of our knowledge, we are the first to investigate the use of hybrid delivery capacity, composed of crowdsourced and company-provided capacity, to ensure service quality. Self-scheduling is not the only challenge associated with crowd-sourced delivery capacity. The fact that couriers do not have to accept delivery tasks offered to them also complicates capacity planning and ensuring service quality. For various reasons, e.g., expectations about the likelihood of future delivery task offers or expectations about the tips associated with future delivery task offers, couriers may decide to reject a delivery task offer. A distinctive feature of our study is that we investigate the impact of this “self-selecting” aspect of crowd-sourced delivery capacity as well as the potential of company-provided delivery capacity to mitigate any negative effect on service quality. In this paper, we develop a theoretical foundation for conducting analyses and deriving results that shed light on some fundamental questions related to supply and demand management in crowd-sourced meal delivery. In particular, we present a stylized model to investigate and answer, among others, the following questions:
• Which restaurants should be included in the network? • How much should restaurants be charged for the deliveries? • Which diners should be served? • How much should couriers be paid for the deliveries? • Should salaried drivers be used alongside couriers? We start by investigating a setting with a single restaurant serving diners with a delivery location inside a circle with the restaurant at its center. This setting captures (most of) the critical aspects of the on-demand meal delivery environment while being simple enough to obtain analytic results, and simplifies the exposition. Most of the analytical results for the “single restaurant” setting readily extend to the multiple restaurant setting (and more general diner location distributions) assuming that the average time it takes a courier to execute a delivery task can be estimated with a reasonable level of accuracy. We focus on analyzing and optimizing profit as a function of service, i.e., the size of the area served and the time it takes to serve diners in that area, and capacity, i.e., the number of couriers delivering to diners (which depends, among others, on the compensation offered to couriers). The contributions of this study can be summarized as follows.
• We introduce service and capacity planning problems in the context of crowd-sourced meal delivery and present a theoretical framework to analyze these problems. • We explicitly consider the autonomous decision making of courier and demonstrate the benefits of supplementing crowd-sourced delivery capacity with company-provided delivery capacity. • We derive results concerning the optimal service coverage area, courier compensation, and delivery capacity composition that maximizes the profit of an online meal delivery platform given a target service quality level. A few insights resulting from our analyses are that: – The service area that maximizes profit depends on the revenue per delivery, the average time of a delivery, the service quality targets, and courier earning expectations; – The service area has a significant impact on profit; – A large diner base close to the restaurants not only increases profits, but also enables serving a larger area; 178
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– Company provided delivery capacity not only increases service reliability, but can also increase profits; and – Often, a small sacrifice in profit, can result in a substantial improvement in service quality. The remainder of the paper is organized as follows. In Section 2, we briefly discuss related literature. In Section 3, we introduce our modeling framework and present our analyses. We conclude, in Section 4, with a discussion of practical implications of our findings. For ease of exposition, all mathematical proofs are provided in the appendix. 2. Literature review Crowd-sourced delivery falls into the broad class of innovative business practices called crowd-logistics (Sampaio et al., 2019). The more general term crowd-sourcing was first used by Howe (2006) to define the business practices that occur in the intersection of outsourcing and shared economy (Zervas et al., 2014; Jiang and Tian, 2016). In a recent study, Rai et al. (2017) provides a review of these disruptive business applications, which have started to attract significant research interest. Crowd-sourced delivery, or alternatively crowd-shipping, constitutes a major class of crowd-logistics. Relying on the public (rather than employees or third-party providers) to deliver goods or packages distinguishes crowd-shipping from delivery practices relying new technologies for deliveries, e.g., delivery drones or bots (Hong et al., 2018; Perboli and Rosano, 2019). Within crowd-sourced delivery two distinct types of services can be distinguished: door-to-door and store-to-door (Sampaio et al., 2019). The former is a P2P model in a two-sided market (Economides and Katsamakas, 2006), in which the crowd (providers) pickup and deliver cargo for and between individuals (customers). Hitch (acquired by Lyft) and Roadie are examples of platform providers facilitating door-to-door delivery services. In these platforms, customers post their requests for deliveries and providers declare their journeys. A matching problem is solved to bring together customers and providers (Allon et al., 2012; Moreno and Terwiesch, 2014; Arnosti et al., 2014; Hu and Zhou, 2015). Store-to-door crowd-delivery, on the other hand, is considered as B2C practice (of which crowd-sourced meal delivery is an example). There are different store-to-door crowd-delivery practices, such as using in-store customers to deliver on-line orders in their neighborhood. Walmart has considered such a crowd-shipping practice to improve its online retail operations (Barr and Wohl, 2013). A different business model is employed by the food and grocery delivery networks that serve a three-sided-market where a retailer/restaurant and couriers (two complementary providers) jointly offer a service to customers. In this model, online platforms function as a store front where customers can select the retailer/restaurant from which they want to purchase (Sampaio et al., 2019). Meal delivery network providers, such as Grubhub, Eat24, UberEats and Foodora all fall into this group. Much of the thinking and discussion about crowd-shipping has developed from the practitioner’s side and the literature that analyzes crowd-shipping is still rather limited (Rai et al., 2017; Carbone et al., 2015). In a recent study, Paloheimo et al. (2016) present a case study in Finland to investigate the possible benefits of crowd-shipping practices in library deliveries. The authors argue that crowd-shipping has the potential to reduce carbon emissions, to lower operational costs, and to improve social cohesion in the society. Introducing a new variant of the Vehicle Routing Problem, Archetti et al. (2016) investigate the Vehicle Routing Problem with Occasional Drivers, which is motivated by the crowd-shipping concept of using in-store customers to deliver on-line orders. The authors emphasize the complexity of deciding a compensation scheme that ensures sufficient delivery capacity to achieve a certain service quality. A more in-depth and detailed analysis (in a dynamic context) of this crowd-shipping concept can be found in Dayarian and Savelsbergh (2017). Introducing a different perspective, Kafle et al. (2017) suggest a two-echelon distribution system in which distribution in the second echelon is performed by the crowd. Relay (or exchange) points connect the first and the second echelons. The use of exchange points in city logistics is also promoted by the Physical Internet initiative (Montreuil, 2011). In a recent study, Chen et al. (2017) introduce the Multi-Driver Multi-Parcel Matching Problem (MDMPMP), in which parcels can be transported by a single or by multiple drivers and at each relay point a matching problem is solved to take advantage of transfer opportunities to improve system performance. In the context of public transport, integration of ride-sourcing (i.e., on-demand ride-sharing) and scheduled transport services to expand coverage and improve efficiency is also attracting attention, see for example Stiglic et al. (2018) and Yan et al. (2018). Reyes et al. (2018) is the first, to the best of knowledge, to study the application of crowd-shipping in meal delivery operations. Introducing the Meal Delivery and Routing Problem (MDRP), the focus of the study is the development of an online dispatching algorithm that makes the best use of the available crowd-delivery capacity to improve a variety of system performance measures. Yildiz and Savelsbergh (2018) propose an exact solution algorithm to solve the static version of the MDRP in order to be able to assess the quality of online dispatching algorithms and to extract valuable insights from the characteristics of optimal solutions. In none of the aforementioned studies an analytical framework has been put forward to study the service coverage, courier compensation, and delivery capacity planning problems that arise in crowd-delivery applications, which is the focus of our research. Our research also relates to the burgeoning on-demand service platforms literature, which mostly assumes a two-sided market structure and seeks to determine optimal service price and provider (courier) compensation. Recent studies Banerjee et al. (2015), Gurvich et al. (2016), Bai et al. (2016), Gurvich et al. (2016), Cachon et al. (2017), Taylor (2018) focus on providers’ (couriers’) participation decisions in an uncertain demand environment. Banerjee et al. (2015), Bai et al. (2016) and Cachon et al. (2017) compare static versus system-state-dependent (dynamic) pricing policies. Cachon et al. (2017) suggest that “surge pricing” can be beneficial for all the stakeholders in a self-scheduling capacity environment, and Bai et al. (2016) show that platforms can increase their profits with dynamic pricing. In contrast, Banerjee et al. (2015) argues that under a broad range of conditions the platform cannot improve throughput, revenue, or welfare by employing a dynamic pricing policy and the main strength of dynamic pricing is its robustness. Taylor (2018) studies static pricing in settings where customer resistance or other strategic reasons do not allow 179
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dynamic pricing. Proposing an augmented newsvendor formula that captures the tradeoffs for the platform and the providers, Gurvich et al. (2016) find that for a fixed provider pool, provider autonomy reduces the number of working providers and increases the optimal price. The latter result does not agree with the findings of Taylor (2018) which shows that demand is sensitive to waiting times, where as the model of Gurvich et al. (2016) (and also the model of Cachon et al. (2017)) does not relate the demand directly with the service quality. In this study, we consider exogenous prices (even though our model can be used to analyze optimal dynamic pricing to improve profitability) to focus on the service coverage and delivery capacity composition decisions (which have not been studied in-depth before). Similar to our study, Banerjee et al. (2015), Bai et al. (2016), and Taylor (2018) consider time-sensitive customers. In these papers, customer demand, therefore, depends on the waiting time. To obtain insights from a rich queuing model, Banerjee et al. (2015) and Bai et al. (2016) employ approximations and large system asymptotics, where as Taylor (2018) presents a stylized model which allows for exact analysis. Using a different approach, we do not employ utility functions to represent the time sensitivity of the customers, but use service quality targets set by the meal delivery platform to capture customers’ time sensitivity. 3. Model In this section, we present the basic setting we consider as well as the basic model we use for analyzing service coverage and delivery capacity design. There is a single restaurant with a circular service area with radius r. Orders arrive in the service area according to a random process with rate (r ) . We assume Euclidean distances and a fixed speed v for all movements. The delivery-time of an order is defined to be the time of the out-and-back trip from the restaurant to the diner location (i.e., we assume instantaneous delivery at the diner location). Deliveries are performed by couriers, i.e., individuals that sign into the meal delivery network to receive delivery offers. Couriers can sign in and out whenever they want and they are allowed to reject delivery offers. For each delivery, the on-demand meal delivery platform earns a revenue of c dollars (the amount charged to the restaurant). Couriers are paid a flat fee of p dollars for each delivery they perform. Couriers are assumed to be at the restaurant when they sign in and to return to the restaurant after they complete a delivery. Couriers arrive at the restaurant according to a random process. The arrivals include couriers signing in as well as couriers returning from a delivery. We consider a time interval T during which the parameters of the system, e.g., order arrival and courier arrival rates, are assumed to be stable. We denote the mean delivery-time during this interval by 1/ µ (r ) and measure the service quality by the mean ready-to-pickup (RtP) time during this interval, where RtP is the difference between the order pickup time and the order ready time. We will take the order arrival time to be the order ready time, whereas in practice, the order ready time has to account for meal preparation time. RtP time is closely related to click-to-door (CtD) time, i.e., the difference between order delivery time and order placement time, but RtP time is a more relevant measure for capacity planning purposes, as it excludes meal preparation time and the travel time from the restaurant to the delivery location, which are not affected by the available delivery capacity. Thus, an online meal delivery platform sets a RtP time target and plans its delivery capacity so as to meet this target in order to assure diner satisfaction. We note that Reyes et al. (2018) and Yildiz and Savelsbergh (2018) found that RtP times are small compared to CtD times, on the average RtP time is only 5% of CtD. In Fig. 1, we present a timeline with pertinent events (diner related as well as courier related) and important concepts, e.g., CtD and RtP. We define a willingness-to-wait function (p) , which we assume to be strictly increasing in p, to capture the average time a courier stays in the system without receiving a delivery offer when the payment per delivery is p. That is, on the average, a courier signs out, when he or she is not offered a delivery job for (p) 1/ µ (r ) time units. The findings of Chen and Sheldon (2016), who analyzed 25 million Uber trips, suggest that this is a reasonable model. Note that the willingness-to-wait function can be viewed as (a proxy for) the courier’s utility function, as it reflects the courier’s actions based on his earning rate expectations. Next, we use the equilibrium behavior of the system to derive the number of couriers in the system. Because for the remainder of this section we assume the radius r is given, we will use, for ease of presentation, and µ instead of (r ) and µ (r ) . At the equilibrium, for a per-delivery payment p, the number of couriers in the system, m (p) , stabilizes at (p) . The reasoning is as follows. When the courier earning rate rises above p = p / (p) , more couriers will stay in the system and, as a result, the number of deliveries per courier per time unit starts to decrease. This, in turn, will drive the courier earning rate back down to p . Similarly, if the courier earning rate falls below p , more couriers will leave the system and, as a result, the number of deliveries per courier per time unit starts to increase. This, in turn, will drive the courier earning rate back up to p . Thus, we assume that the courier earning rate remains in a band around p . This agrees with the findings of Hall and Krueger (2016), who present a comprehensive analysis of the labor market for Ubers driver-partners, based on both survey and administrative data.
Fig. 1. Timeline of events. 180
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Table 1 Notation used throughout the paper. r:
The radius of the service area The order arrival rate in a service area of radius r The mean delivery-time for orders in a service area of radius r The payment to a courier for a delivered order The average time a courier stays in the system without receiving a delivery offer when the payment per delivery is p The number of couriers in the system when the payment per delivery is p The minimum number of couriers needed in the system The diner density at distance x 0 from the restaurant Service-quality-aspiration parameter The revenue received for a delivered order The average vehicle speed The courier waiting-rate
(r ) : µ (r ) : p: (p) : m ( p) : m: g (x ) : : c: v: b:
Since the courier earning rate stabilizes at p , each courier, on average, makes Tp / p deliveries during T. Consequently, we have
m (p ) =
T Tp p
=
p = p
(p). (1)
Equality (1) tells us that the number of couriers in the system is a function of the order arrival rate and the per-delivery payment p. Interestingly, the number of couriers in the system only indirectly depends on the delivery-time. The average delivery-time 1/ µ has to be less than (p) , otherwise no courier would remain in the system, but given that, a courier stays in the system as long as he does not have to wait too much between consecutive deliveries. On the other hand, the delivery-time directly affects the service quality. With the same number of couriers in the system, higher delivery-times will result in worse service quality. Note that for a given number of couriers and a given delivery-time distribution, the service quality, i.e., the mean RtP, is fixed. We next investigate how the mean RtP is affected by the number of couriers in the system. First, note that if the system parameters were deterministic, the number of couriers needed, m , can be determined by finding the number of couriers that equalizes the total delivery capacity, m T , with the total delivery demand, T / µ , which gives m = /µ . However, the stochasticity in the system implies that some “excess capacity” is required to ensure a certain service quality. Therefore, we define a service-quality-aspiration parameter > 1, and use the following relation in or models:
m =
µ
.
(2)
Note that higher values of will provide more reliable and faster service, possibly at the expense of higher operational costs. In our analyses, we investigate the impacts of the service-quality-aspiration parameter value on the mean RtP and the operational costs. For convenience, we summarize the notation used throughout the paper in Table 1. Note that we assume that diner demand is exogenous and does not depend on the service quality, i.e., the mean ready-to-pickup time. A mean ready-to-pickup time of a few minutes is critical to the success of online meal delivery platforms, and, thus, constitutes only a very small fraction of the click-to-door time. As a consequence, assuming that diner demand is not impacted by small variations in ready-to-pickup time is reasonable. Observe that when the order arrival rate, the number of couriers in the system, and the delivery-time distribution are given, the mean RtP value can determined by viewing the system as a multi server queue with general arrival and service time distributions (G/ G/ m) . Although most performance metrics for this queuing system are not known, we can obtain close approximations of the waiting time, which corresponds to the mean RtP value, using simulation or approximation approaches (see, for example, Whitt, 1993; Kimura, 1994). In our models, we do not consider order bundling, i.e., assigning more than one order to a courier. Bundling of orders, i.e., consolidation, may lead to higher utilization of the delivery capacity. However, the high service expectations of diners, and, as a result, the high self-imposed RtP targets, severely limit the opportunity for order bundling. As a consequence, in practice, only a small fraction of orders are bundled by on-demand meal delivery platforms. (Note too that order bundling leads to a loss of freshness for at least one of the orders). Therefore, the insights obtained from our analysis, even though we do not consider order bundling, are still meaningful in practice. 3.1. Analyses Using the framework introduced above, we now derive analytical results that can help answer several relevant questions about supply and demand management in crowd-sourced meal delivery operations. One such question is “What is the coverage (defined by the radius of the service area) and the courier compensation that maximizes the profit for a meal delivery network while also achieving its service target?”. Assuming that the order arrival rate is a function of the service area radius r, we will analyze the relationships between coverage, profit, and service quality. We will first assume that couriers always accept the delivery offers they receive and the service platform does not supplement the delivery capacity provided by couriers with company-employed drivers. Then, subsequently, we will analyze the effects of couriers rejecting some of the delivery orders they receive and supplementing 181
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crowd-sourced delivery capacity with company-employed delivery capacity. Let c be the revenue obtained (the amount charged to the restaurant) by the on-demand meal delivery platform for each delivery. We will refer to the parameter c as the unit-revenue. Then, we can define the (rate of) profit as = (c p) (r ) and consider the following optimization problem to find the optimal coverage for a given service-quality-aspiration level :
P : max c
p
r >0
(r )
(3)
(r ) . µ (r )
s.t. (r ) (p) =
(4)
The objective is to maximize the profit for the meal delivery platform and the constraint enforces that a sufficient number of couriers is active in the system. Next, we show that formulation P can be transformed into an unconstrained optimization problem that can easily be solved by d calculating the derivative dr (r ) and setting it equal to zero. Assuming that (r ) > 0 for all r > 0 , we can rewrite (4) as (p) = / µ (r ) . Let R be the distance to the furthest possible diner location (or some (physical) limit on the radius of the service area). Then replacing p with 1 ( / µ (r )) , we obtain the unconstrained optimization problem 1
P : max c
(r ).
µ (r )
r (0, R]
(5)
Given (5), we can analyze the optimal radius of the service area by considering specific forms of the functions , µ , and . 0 units of distance away from the Let g : + +, g (0) = 0 be a Lebesgue integrable function that gives the diner density at x d d restaurant. We assume g (x ) > 0 , for all x > 0 . For the notational convenience, we define g (x ) = dx G (x ) and G (x ) = dx G¯ (x ) . Let > 0 be the order arrival rate, i.e., the average frequency with which orders are placed, which implies that the order arrival rate for a r radius r is given by (r ) = 0 g (x ) dx . For presentational convenience, we assume a linear willingness-to-wait function, (p) = p/ b , where we will refer to b > 0 as the courier waiting-rate. The following proposition shows that for a large class of problems there is a closed form expression for the optimal radius. Proposition 1. Given unit-revenue c, speed v, courier waiting-rate b, and service-quality-aspiration , the optimal radius r of the service area is cv /2b . Interestingly, Proposition 1 shows that the optimal radius of the service area does not depend on the diner density function g and order frequency . Furthermore, the optimal radius r is linearly increasing in the unit-revenue c and the speed v, and linearly decreasing in the courier waiting-rate b and service-quality-aspiration . Note that solving the equation (r ) (p) =
( ), for p, (r ) µ (r )
gives the optimal courier compensation p that maximizes the profit. Although the optimal radius does not depend on the diner density function g, it is not hard to see that profit and courier compensation are affected by the diner density distribution in the service area. Therefore, we continue our analysis with the following class of diner density functions
g (x ) =
(k + 1) x k Rk + 1
0,
, if x
R (6)
o.w.,
with k > 1 and R > 0 . If diners are uniformly distributed in the service area, we set k = 1. For any k < 1, the diner density decreases with the distance from the restaurant, with k = 0 corresponding to a linear and k < 0 corresponding to a super-linear decrease in the density. For the diner density functions of type (6), we can find the optimal service area as defined in Proposition 1 and obtain the meal delivery platform’s profit as follows:
(r ) =
c (k + 2)
(c
( )
k+1 cv , 2 bR 2 bR (k + 1) v (k + 2)
if r
),
R
if r = R.
(7)
Inspecting (7), we see that the profit (r ) is increasing in unit-revenue c and vehicle speed v, and decreasing in courier waitingrate b and service-quality-aspiration , as expected. The profit, however, differs significantly depending on whether it is optimal to offer service to all diners (r = R ) or not (r < R ). In the former situation, the unit-revenue c has the largest impact on profit; for all values of k, the profit increases super-linear in c. In that situation, it is also clear that the parameter k, i.e., the diner distribution, is a dominant factor in determining the profitability of providing delivery service. On the other hand, we see that when it is optimal to serve all diners, the impact of the diner distribution is less pronounced. Even though our focus is not on pricing decisions, i.e., we assume that other factors dominate the determination of the per-delivery charge c, we make a few relevant observations. First, we note that (7) enables a straightforward way to calculate the optimal per-delivery charge even when the order frequency is sensitive to this value (i.e., when depends on c). Replacing with (c ), one can conduct a bisection search to find the value c that maximizes (7). Second, we observe that when order frequency is insensitive to the per-delivery charge (e.g., for an increase/decrease in the perdelivery charge, the change order frequency is less than or equal to ), (7) suggests that it is optimal to set c = 2R b/ v , i.e., to service 182
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Fig. 2. Impact of service area on profit.
the entire region. Throughout this section, we illustrate our analyses with numerical examples. In all these numerical examples, we use the diner density distributions (6) with k ( 1, 3], furthest diner location R = 5000 meters, order frequency = 2 per minute, waiting-rate b = $0. 33 per minute (which implies a $20 per hour earning rate expectation for the couriers), and service-quality-aspiration = 1.075 (which, based on preliminary experiments, provides RtP times of less than 5 min). We start by illustrating the impact of service area on the delivery platform’s profit. In Fig. 2, assuming speed v = 100 m/min, we show the change in the ratio (r )/ (r ) for k {0, 1, 2} and c {8, 16, 24} . The graphs in Fig. 2 clearly show that the choice of service area has a dramatic impact on profit. Interestingly, profit is less impacted by service area as unit-revenue increase and diner locations tend to be closer to the restaurant (k decreases). Next, in Figs. 3a and b, we illustrate the change in profit for c {8, 16, 24} and v {100, 200, 400} , respectively. Since our analytical results shows that an increase in speed has the same affect as an decrease in b, we do not provide a separate graph to illustrate the effects of changing b. Both figures illustrate how strongly the profitability of a delivery service depends on the location of the restaurant relative to its diner base. Confirming the interpretation of (7), Fig. 3 shows that a restaurant’s relative location to its diner base might dominate other factors (e.g., unit-revenue, vehicle speed, and courier waiting-rate) in the case that it is optimal to serve only part of the service area. A closer look at Fig. 3a also shows that until the unit-revenue c gets large enough to ensure that serving all diners is optimal, an increase in unit-revenue c has a bigger impact for smaller values of k. This suggests that when the diner base is dispersed and farther away from the restaurant, the unit-revenue c plays a crucial role in determining the profitability of providing delivery service. Examining Fig. 3b, we see that when it is not optimal to serve the entire region, the impact of higher vehicle speeds on profit is less when diners tend to be either close to the restaurant or far away from the restaurant. When delivery distances are short (k is small) the vehicle speed does not make much of difference, because it will mostly affect a courier’s idle time between orders. When delivery distances are long (k is large), its negative effect on profit dominates any positive effect of higher
Fig. 3. Profit as a function of unit-revenue and vehicle speed. 183
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Fig. 4. Trade-off between profit and service quality.
vehicle speeds. Service quality is another critical aspect of a meal delivery operation. Recall that service quality is captured by RtP, or waiting time, which we approximate using the formula proposed by Gans et al. (2003) (see Appendix B for details). Reviewing the trade-off curves between profit and service quality shown in Fig. 4, where we plot the profit (left vertical axis) and mean RtP values (right [0.01, 1.2] and diner density distribution parameter k {0, 1, 2} and unit-revenue vertical axis) for service-quality-aspiration c {8, 16, 24} , provides valuable insights. Before interpreting the trade-off curves, we note that when the number of couriers required to deliver meals in a service area with (r ) (optimal) radius r is fractional, i.e., µ (r ) , we take the ceiling to calculate RtP times. This causes the discontinuities in the RtP graphs. The effects of rounding are more pronounced when the mean RtP is high, i.e., when there is congestion at the restaurant, because in such situations having an additional courier has a larger impact. The trade-off curves in Fig. 4 show that high service quality (i.e., low RtP values) can be achieved by modest sacrifices in profit. For example, when unit revenue, c, is $8 and the diner density distribution parameter, k, is 0, then choosing = 1.075 instead of = 1.010 , reduced RtP by more than 50% while the reduction in the profit is less than 8%. A similar trend is observable for all unitrevenue values and diner distributions. However, Fig. 4 also shows that improving RtP beyond a certain level becomes costly, i.e., reduces profit significantly. Therefore, meal delivery platforms should carefully balance profit and service quality (customer satisfaction). Moreover, we observe that as diner locations tend to be further away from the restaurant (i.e., parameter k increases), it becomes more costly to improve service quality. This observation shows, again, that the location of the restaurant (relative to the diner locations) plays a critical role in profitability and service quality. It is also informative to examine the courier compensation (per delivery) associated with an optimal service area. By Proposition 1, we can derive the courier compensation for diner density distribution functions of the form (6) as follows:
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Fig. 5. Courier compensation as a function of unit-revenue and vehicle speed.
p (r ) =
c (k + 1) , (k + 2)
if r < R
2 bR (k + 1) , (k + 2) v
if r = R
(8)
We see that the courier compensation linearly increases with unit-revenue c as long as the radius of the optimal service area is less than R, but after that, the unit-revenue no longer affects courier compensation. Furthermore, we see that the courier compensation does not depend on ordering frequency , i.e., the per-delivery payment to couriers does not increase or decrease as diner demand changes. Observe that when unit-revenue c increases, the radius r of the optimal service area increases, and, thus, so does the mean delivery time, which results in an increase in number of couriers, each receiving a higher per-delivery payment. Note that as long as the radius of the optimal service area is less than R, a constant fraction k + 1 of the unit-revenue is paid to the couriers, and this k+2 fraction only depends on the diner density distribution. As diners are located closer to the restaurant, the fraction of unit-revenue paid to the couriers reduces. When it is optimal to serve all diners (i.e., r = R ), courier compensation is independent of the unit-revenue c and determined by the size of the service area R, the vehicle speed v, the courier waiting-rate b, and the service-quality-aspiration . In Fig. 5, we present a numeric example to illustrate the effect on courier compensation when the unit-revenue c and the vehicle speed v change. Fig. 5 shows that courier compensation differs substantially depending the on the optimal service area (the blue line, in both graphs, corresponds to a situation in which it is optimal to serve the entire area, and the orange and green lines correspond to situations in which it is optimal to serve a smaller area (r < R )). We see that when the setting becomes more favorable, e.g., higher unit-revenue, higher vehicle speed, and higher diner density near the restaurant, it is primarily the meal delivery platform that benefits, as the fraction of unit-revenue going to the couriers decreases. 185
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3.1.1. Per-mile compensation So far, we have assumed a per-delivery compensation for couriers. Next, we extend the analysis to include a per-mile compensation in addition to the per-delivery compensation. Specifically, we assume that a courier is paid > 0 per time unit spent delivering an order (i.e., driving out and back from the restaurant to the delivery location). After updating the profit function (3) to include the delivery-time compensation, i.e.,
= c
p
µ (r )
(r ),
(9)
we can derive the following result. Proposition 2. Given unit-revenue c, speed v, courier waiting-rate b, delivery-time compensation , and service-quality-aspiration , the optimal radius r of the service area is cv /2( b + ) . Using Proposition 2, we obtain the maximum profit
(r ) =
c (k + 2)
(
k+1 cv , 2R ( b + )
)
if r < R
c
2 R ( b + )(k + 1) , v (k + 2)
if r = R,
(r ) and the associated per-delivery compensation p (r ) :
(10)
and
p (r ) =
c (k + 1) , (k + 2)
if r < R
2R ( b + )(k + 1) , v (k + 2)
if r = R.
(11)
Again, we see that the optimal service area (radius) does not depend on the diner density distribution function g and the order frequency . In fact, the optimal radius, maximum profit, and courier compensation implied by Proposition 2 are identical to those implied by Proposition 1, when we replace b by ( b + ) . This points out an interesting insight about the courier compensation structure in the presence of a per-mile component. Note that the courier wait-rate b is the unit time cost of enticing a courier and it is paid regardless of the courier’s activity level (i.e., the time the courier spends on making deliveries) and is inflated using to create excess capacity to handle stochasticity. On the other hand, the per-mile compensation is directly related with the activity level of the courier, since it is not paid when a courier is idle. As such, in the summation ( b + ) , the first term represents a fixed cost to attract a courier, whereas the second term accounts for variable costs. To further examine the effect of a per-mile payment on profit and courier compensation, we conduct numerical experiments with diner density functions (6). In Fig. 6, we illustrate how profit changes for different per-mile payments. We compare the profit when there is no per-mile payment (i.e., = 0 ) and the profit with a 10 and 20 cents payment for each minute of driving (i.e., = 0.10 and = 0.20 ). As expected, we see that the profit reduces for higher per-mile pay rates (i.e., higher values of ). Furthermore, we see that the impact is more pronounced, when diner locations tend to be further away from the restaurant. Thus, the reduction in profit as a result of per-mile payments is largest for settings in which it is optimal to serve the entire region. For low unit-revenue values, the effect on profit of offering a per-mile payment or not offering a per-mile payment is relatively small. This is due to the fact that when unitrevenue is low, the optimal service area becomes smaller and, as a consequence, the per-mile payments to couriers become a smaller portion of courier compensation. Presenting the information is a slightly different form, in Fig. 7 we show the fraction of unit-revenue paid to the courier, where we note that, on average, the courier compensation is p = p (r ) + / µ (r ) per delivery. Interestingly, we see that when it is optimal not
Fig. 6. Effect of per-mile pay on profit. 186
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Fig. 7. The fraction of unit-revenue paid to the courier for different per-mile pay rates.
to serve the entire region (i.e., r < R ) the fraction of the unit-revenue paid to a courier is the same regardless of the per-mile payment (r < R for unit-revenue c $16). In that case, the courier compensation is c (k + 1)/ k + 2 , i.e., a fraction k + 1/ k + 2 of unit-revenue (the orange line in Fig. 7). This shows that when courier compensation includes a per-mile component, the meal delivery platform reduces the service area (to maximize its profit), but the fraction of the unit-revenue paid to the courier is the same as in the case when there is no per-mile component (i.e., (8)). 3.1.2. Delivery offer rejections So far, we have assumed that a courier always accepts a delivery offer. However, in practice, this is not the case. One possible strategy to ensure service reliability is to complement the delivery capacity provided by couriers with delivery capacity provided by company-employed drivers. These drivers can be used when couriers reject delivery offers. In this subsection, we investigate the implications of delivery offer rejections on the optimal service area as well as delivery capacity planning in settings with both (selfemployed) couriers and (company-employed) drivers. Let p¯ be the unit-time driver compensation. We assume that drivers are more expensive than couriers, but accept all delivery offers. Let m ¯ be the number of drivers in the system during the time interval T. Before presenting our analysis, we observe that from a cost minimization perspective, it is always sub-optimal to offer a delivery that is acceptable to a courier to a driver. The reasoning is as follows. Since the number of couriers that stay in the system depends on the order arrival rate, a delivery that is acceptable to a courier should not be offered to a driver, since doing so effectively decreases the order arrival rate for couriers and would therefore result in a decrease in the number of couriers, say a decrease of e > 0 couriers. This implies that to achieve the target service quality, e more drivers need to be employed. Since drivers are more expensive than couriers, this shows that deliveries that are acceptable to couriers should not be offered to drivers; drivers should only be offered those deliveries that are rejected by couriers. Consequently, the order arrival process can be divided into two distinct arrival processes, each served separately by couriers and by drivers. Before proceeding with our analyses, we need to clarify the operational setting we have in mind and what we imply when we say a delivery offer is rejected by a courier. When an order arrives, it is offered to a courier. If the courier rejects, the order can, of course, be offered to another courier, and so on, before offering/assigning it to a driver. A possible decision rule is to offer an order n times to a courier before offering/assigning it to a driver. In this case, the order acceptance probability is the probability that at least one of the n couriers accepts the delivery offer. Note that in practice, the response to a delivery offer is not instantaneous (in fact, it may take a non-negligible amount of time). As a result, to ensure the target service quality, large values of n may be unacceptable. Moreover, it is likely that there is high correlation between courier order acceptance/rejection decisions, which also suggests that smaller values of n are more appropriate. Regardless of these considerations, and to simplify the exposition, we will assume n = 1. We start by investigating the case in which all couriers accept a delivery offer with probability > 0 . If one considers the various characteristics of an order, e.g., distance from current location to restaurant, distance from restaurant to delivery location, delivery location, time of day, and expected tip amount, and assumes a heterogeneous utility function for its evaluation, the net effect is an offer rejection probability as considered here. We modify formulation P, i.e., (3) and (4), to incorporate the probability of delivery offer rejection as follows:
P ( ): max c (r )
p (r )
r , p, m ¯ 0
s.t.
(r ) (p) =
(1 m ¯ = ¯
pm ¯¯
(12)
(r ) µ (r )
(13)
) (r ) µ (r )
(14)
The objective is to maximize profit, i.e., the revenue minus payments for deliveries to couriers and payments for deliveries to drivers. As before, the per-delivery payment p for couriers has to be chosen such that the number of couriers is sufficient to achieve 187
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the target service quality, which is ensured by Constraint (13). Constraint (14) ensures that a sufficient number of drivers is available to perform deliveries of orders rejected by couriers. We can replace p with 1 ( b / µ (r )) to obtain the unconstrained optimization problem: 1
P ( ): max c (r ) r [0, R]
b µ (r )
p¯ ¯
(r )
(1
) (r ) µ (r )
(15)
Proposition 3. Given unit-revenue c, speed v, courier waiting-rate b, courier acceptance-probability , and service-quality-aspiration , the optimal radius r of the service area is
cv 2( b + ¯p¯ (1
))
.
(16)
Proposition 3 reveals that the effect of the possibility of delivery offer rejection is a modification of the term b; it is replaced with ) . This makes sense, as we can consider b as the unit-time cost of enticing a courier and this cost the weighted sum b + ¯p¯ (1 ) of the orders. Clearly, as the acceptance-probability decreases and the unit-time cost of cannot be applied to a fraction (1 drivers p¯ increases, the radius of the optimal service area decreases. Using the diner density functions (6), we can derive the associated profit and courier compensation as follows
(r ) =
c (k + 2)
c
(
cv 2R ( b + ¯p¯ (1
))
)
k+1
2 R (k + 1) v ( b + ¯p¯ (1 ))(k + 2)
, if r < R if r = R,
(17)
and
p (r ) =
c b (k + 1) , ( b + ¯p¯ (1 ))(k + 2)
if r < R
2R b (k + 1) v (k + 2)
if r = R.
(18)
To better assess the effect acceptance-probabilities, we analyze the results of a numerical experiment conducted with diner density functions (6). In Fig. 8, we illustrate how the profit changes for different courier offer acceptance-probabilities. We compare the profit when couriers always accept delivery offers ( = 1) to lower acceptance-probabilities = 0.9 and 0.8. In these experiments, we use = ¯ = 1.075 and assume that the unit-time cost of employing a driver is twice as expensive as courier; specifically, we use b = 1/3 and p¯ = 2/3. Fig. 8 shows that the negative impact of delivery offer rejections is more pronounced when the unit-revenue is higher. The reason is that a higher unit-revenue increases the radius of the optimal service area, which, in turn, implies that the number of “expensive” drivers needed increases and the profit decreases. We also investigate the benefits of employing drivers to provide service reliability. Fig. 9 shows the profit for three settings: 100% delivery offer acceptance, 90%, 80%, and 70% delivery offer acceptance without drivers to perform the rejected deliveries, and 90%, 80%, and 70% delivery offer acceptance with drivers to perform the rejected deliveries. For simplicity, there is no penalty for not serving an order in the settings in which couriers can reject delivery offers and there are no drivers. Fig. 9 reveals an interesting insight about the use of company-employed drivers. For all settings, we see that if diners are located relatively close to the restaurant, using drivers to recover a portion of the lost revenue due to delivery offer rejections is worthwhile and increases profit. However, if diner locations tend to be further away from the restaurant using drivers may actually reduce profit
Fig. 8. Impact of delivery off acceptance-probability on profit. 188
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Fig. 9. Benefit of employing drivers to perform deliveries of orders rejected by couriers.
(this, of course, ignores any long-term effects of not serving placed orders). Not surprisingly, the benefit of employing drivers gets larger when the courier acceptance-probability is lower and when the unit-revenue increases. Note that in this case, the ratio between ) due to the fact that regardless of the service area, couriers are the number of couriers and drivers is fixed and equal to / ¯ (1 expected, on average, to serve percent of the orders while drivers need to serve the remaining 1 percent of the orders and the mean delivery-times are the same for the orders served by couriers and drivers. As we show next, when the courier order acceptance probabilities have more complex forms, calculating the ratio between the number of couriers and drivers becomes more involved. For obvious reasons, the delivery distance is likely to be a significant factor in a courier’s accept-reject decision, and deserves special attention. Therefore, we extend our analysis by considering the case where the delivery offer acceptance-probability for couriers is a function of the distance from the restaurant to the order drop-off location. More formally, we assume that the probability of accepting a delivery offer for a drop-off location that is x 0 units away from the restaurant is given as (x ) [0, 1], where is a nondecreasing function of the distance x. Note that in this case the expected delivery-time for the orders that are accepted by the couriers can be different (possibly smaller) than the expected delivery time of orders accepted by drivers. r We define the arrival rate of orders that are to be delivered by couriers as 1 (r ) = G1 (r ) , where G1 (r ) = 0 (x ) g (x ) dx . Similarly, we define the arrival rate of orders to be delivered by drivers as delivery-time for couriers and drivers can be calculated as µ1 = the profit maximization problem as
P ( ): max c (r ) r , p, m ¯ 0
s.t. 1 (r ) (p) =
p 1 (r )
2 (r ) = G1 (r )
G2 (r ) , where G2 (r ) =
r 2xg (x ) (x ) dx 0 v
pm ¯¯
and µ 2 =
G 2 (r )
r 2xg (x )(1 0 v
r
0 )(x )
(1
dx
(x )) g (x ) dx . The average
, respectively. We can rewrite
(19)
1 (r ) µ1 (r )
(20) 189
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¯ = ¯ m
2 (r ) . µ 2 (r )
(21)
As before, we restate this optimization problem as an unconstrained one, i.e., 1
P ( ): max c (r )
µ1 (r )
r 0
1 (r )
p¯
¯ 2 (r ) , µ 2 (r )
(22)
and use it to derive the following proposition. Proposition 4. Given unit-revenue c, speed v, courier waiting-rate b, courier acceptance-probability function , and service-quality-aspiration [0, R]): , the radius r of the service area that maximizes the profit is the solution to the following equation (for r
r ( b (r ) + ¯p¯ (1
(r ))) =
cv 2
(23)
One interesting acceptance-probability function to examine is the one in which there is some threshold radius r up to which a courier always accepts a delivery offer, and after which a courier always rejects a delivery offer (which implies that the delivery will be performed by a driver), i.e.,
1, if x 0, o.w.
(x ) =
r,
(24)
In this setting, the radius of the service area maximizing profit is
r¯ =
min r ,
cv 2 b
cv ¯¯ 2Bp
, if
cv ¯¯ , 2Bp
r,
o.w.
(25)
and the number of drivers required is
m ¯ =
0, ¯
if r¯ r¯ r
2ax g (x ) dx , v
r,
o.w.
(26)
Note that this case can also represent the situation in which the couriers are homogeneous in their valuations of an offer (they either all consider an offer worthwhile to accept or not). Furthermore, the above result can be easily extended situations in which certain (types of orders) are unacceptable to (most of the) couriers and have to be performed drivers. To further investigate the impact of distance-dependent delivery offer acceptance by couriers, we conduct numerical experiments using the diner density functions (6) and offer acceptance-probability functions: t
(x ) = 1
x , t R
0.
(27)
Note that as parameters t and grow, the delivery offer rejection probability diminishes. In Fig. 10, we compare the profit for the situations in which couriers always accept delivery offers and in which couriers reject delivery offers depending on their distance from the restaurant according to the above probability function with = 5 and t {0.5, 1.0, 2.0} . Again, Fig. 10 shows that the effect
Fig. 10. Profit with distance-dependent delivery offer rejection probabilities. 190
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Fig. 11. Benefit of employing drivers to perform deliveries of orders rejected by couriers (distance-dependent orffer acceptance-probabilities).
of offer rejections on profit is becomes more pronounced when the unit-revenue (and hence the service area) increases. Nor surprisingly, as order drop-off locations tend to be further away from the restaurant, distance-dependent delivery offer rejection has a higher impact. We also see that the profit is sensitive to the parameter t in the acceptance-probability function and that profit drops significantly for t < 1. Next, we repeat the analysis where investigate the impact of employing drivers to provide service reliability on profit for three settings: 100% delivery offer acceptance, delivery offer acceptance without drivers to perform the rejected deliveries, and delivery offer acceptance according with drivers to perform the rejected deliveries, but now with distance-dependent order acceptanceprobabilities for couriers. (Again, there is no penalty for not serving an order in the settings in which couriers can reject delivery offers and there are no drivers.) The results, shown in Fig. 11, demonstrate, as before, that the difference between the three settings becomes less pronounced as the unit-revenue, average distance from restaurant to drop-off locations, and offer acceptance-probability decreases (t parameter increases). Interestingly, the benefit of employing drivers to perform deliveries of orders rejected by couriers is less when the rejection probability depends on the distance of the drop-off location from the restaurant. The reason is that few short-distance delivery offers are rejected, and we saw earlier that those are exactly the ones for which it is beneficial to have drivers as a backup option. Using Proposition 4 to find the radius r of the optimal service area and considering the acceptance-probability functions (27), we next examine the composition of the delivery capacity (i.e., couriers versus drivers). For different diner distributions, Fig. 12 shows
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Fig. 12. Percentage of couriers in the total delivery capacity.
the percentage of couriers for a given acceptance-probability function (i.e., for a given value of t). In each graph, the courier percentages are shown for unit-revenue values c {8, 16, 24} . We see that the courier percentage drops when diner locations get further from the restaurant and when courier order rejection probabilities increase (small t values). Furthermore, we see that when unitrevenue increases and, as a consequence, the optimal service area gets larger and more drivers are needed, the percentage of couriers drops. Finally, the graphs demonstrate that when acceptance-probability is high (large t values) and the overwhelming majority of the delivery capacity consists of couriers, the delivery capacity composition becomes less sensitive to changes in the acceptance probability. 3.2. Multiple service areas and multiple restaurants Our analyses up to now have assumed a single service area for a single restaurant. To enhance the practical value of our analyses and insights, in this section, we consider two extensions: multiple service areas and multiple restaurants. First, we consider a single restaurant with the option to deliver meals to a finite set of service areas, N. If it is decided to deliver meals in service area i N , then the order arrival rate is ai , and the average delivery-time is 1/ µi . Next, we define a binary decision variable x i , i N to indicate whether to deliver meals in service area i ( x i = 1) or not ( x i = 0 ). Thus, vector x B N , whose ith entry corresponds to the value of binary variable x i , characterizes a possible service coverage decision. For a given service coverage decision x B N , the resulting order arrival rate is denoted by (x) and the resulting mean delivery-time is denoted by 1/ µ (x) . We define the following integer programming model to find optimal service coverage:
IP: max c
p
x BN
s.t.
(x) (p)
(x)
(28)
(x) . µ (x)
(29)
As before, the objective is to maximize profit, and Constraint (29) ensures that sufficient delivery capacity is available. Replacing (x) with i N ai xi and the mean delivery-time 1/ µ (x) with i N ai xi / µi / i N ai xi , we can rewrite IP as:
max
c
x BN
p ai x i
(30)
i N
s.t.
(p) i N
µi
ai x i
0
(31)
= { i N : (p ) / µi 0} be the set of service areas for which the mean delivery-time multiplied by is less than or equal to Let / µi ) a i . the courier’s willingness-to-wait time, and let N = N N+ and = i N + ( (p) It is clear that meals should be delivered to service areas i N+, because (c p) ai 0, i N , and setting x i = 1 increases the value of the left hand side of Constraint (31). Thus, IP can be transformed into the following knapsack problem: N+
KN : max
x BN
s.t. i N
c
p ai x i
(32)
i N
µi
(p ) a i x i
.
(33)
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Fig. 13. Service area selection examples.
Let x¯ be the optimal solution to KN, then we have that x , with
xi =
{
1if i N+ x¯i o.w.,
(34)
is an optimal solution to IP. Note that this assumes no special form for the courier’s willingness-to-wait function , and so this result can trivially be extended to situations in which each service area i N has its own unit-revenue ci and per-delivery payment pi . We observe that the optimal service coverage expands as the per-delivery payment p increases (assuming is a nondecreasing function of p) and shrinks as the service-quality-aspiration and the mean delivery-times increase. Note too that this result reinforces the earlier observation regarding the benefit of having a large diner base close to restaurant. The diner base close to the restaurant is captured in , and a large value of allows expanding coverage to accommodate diners in service areas further away from the restaurant. Consider, for example, the two settings illustrated in Fig. 13a and b. The diamond shape in the figures indicates the location of the restaurant and the circles represent the locations of the four candidate service areas
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Fig. 14. Profit increase as a function of mean delivery-time reduction.
(the same in the two settings). The delivery-times (in minutes) 1/ µi , i {1, 2, 3, 4} and order arrival rates i, i {1, 2, 3, 4} are shown below each service area; the size of the circles are proportional to the order arrival rates. We assume c = $12, p = $10, (p) = 22 minutes, and = 1.1. The first setting represents a situation in which there is a large diner base close to the restaurant, and the second setting illustrates a situation where this is not the case. (Note that in the second setting, the order arrival rate in service area 1 is decreased from 4 to 1 and the order arrival rate in service area 2 is increased from 1 to 4.) In the first setting, it is optimal to deliver meals in service areas 1, 2, and 4, whereas in the second setting, it is optimal to deliver meals only to service areas 1 and 2. This example clearly shows that having a large diner base close to restaurant enables a meal delivery platform to expand service to other areas. It is also interesting to observe in the first setting that even though service area 3 is closer to the restaurant, compared to the service area 4, it is optimal to deliver meals in service area 4 rather than service area 3 because it has a larger order arrival rate. In some practical settings, where home deliveries from a single store or single distribution center are considered, our analyses involving a single restaurant may assist in assessing the advantages and disadvantage of crowd-sourced delivery. On the other hand, in many environments, e.g., our motivating on-demand meal delivery platform, there are multiple restaurants serving diners. Fortunately, our analyses do not depend strongly on the assumption that there is a single restaurant. To extend our results to a multirestaurant setting, all that is required is to redefine r , (r ) , and µ (r ) as follows:
• r: the maximum direct driving time for orders that are accepted by the online platform (from the restaurant to the delivery location); • (r ): the arrival rate of orders that are accepted by the online platform; and • µ (r ) the mean delivery-time of orders that are accepted by the online platform. Note that in a multi-restaurant setting, the delivery time not only depends on the restaurant where the order needs to be picked up and the location where the meal needs to be dropped off, but also on the (quality of the) routing decisions. In the single-restaurant setting, a courier always returns to the same restaurant for his next order pickup. In a multi-restaurant setting, a courier may go to a different restaurant for his next order pickup. Clearly, dispatching decisions, i.e., deciding where to send couriers after they complete a delivery, impacts the mean delivery-time. (More complex settings can be envisioned, e.g., in which a courier picks up meals from different restaurants before delivering these meals to diner locations, but in practice it is most common to have couriers serve single orders.) The mean delivery-time depends on the environment (the number and location of restaurants, the order arrival patterns, the dispatching technology, the street network, the traffic pattern, etc.), but as long as a reasonably accurate estimate of mean deliverytime is available, our analyses for the single-restaurant setting can be used to answer various strategic and operational questions about a multi-restaurant environment. On-demand meal delivery platforms may, for example, consider investing resources to improve their dispatching technology in order to reduce and mean delivery-time. Or, they may consider employing demand management strategies encouraging diners to place orders at certain restaurants to reduce mean delivery-time. In the remainder of this section, we analyze the effects of reductions in mean delivery-time considering both continuous and discrete diner density distributions (i.e., single delivery area and multiple delivery areas). First, we consider the case with multiple restaurants serving a single delivery area. Fig. 14 shows the relation between profit and mean delivery-time, where we have assumed that couriers receive a per-mile payment = 0.1 in addition to a per-deliver
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Fig. 15. The increase in profit as a function of the reduction in mean delivery-time.
compensation and unit-revenue c = $8 (the results are similar for other per-mile payments and unit-revenue values). Note that a reduction in mean delivery-time is akin to an increase in vehicle speed, and, therefore, we model an percent reduction in the mean delivery-time as an percent increase in vehicle speed. We see that modest reductions in mean delivery-time can produce significant gains in profit, especially when the diner locations are further from the restaurant, e.g., a 5% reduction in mean delivery-time results in an increase in profit of more than 20% (diner density function with k = 3). Next, we consider the case with multiple restaurants serving multiple delivery areas. Fig. 15 shows the relation between profit and mean delivery-time. In the considered setting, there are 20 candidate service areas with different order arrival rates and mean delivery-times. The order arrival rates are drawn from a uniform random distribution between 0 and 1 orders per minute and the mean delivery-times is drawn from a uniform distribution between 15 and 25 min (details can be found in Appendix C). We assume = 1.1, (p) = 20 , and c > p, which means that profit is maximized by serving as many diners as possible without violating the target service quality. Fig. 15 shows that as the mean delivery-times decrease, meals are delivered to more service areas and, hence, the profit increases (significantly and rapidly) until meals are delivered to all service areas (which occurs when mean delivery-time is reduced by 12%). We observe that there are points where a small reduction in mean delivery-times results in a big jump in profit. This happens when a small reduction in mean delivery-times means that it has become feasible to add another service area or that it has become feasible to replace a small service area with a larger one. Both examples indicate the benefits of reducing the mean delivery-time, for example by more effective dispatching technology or demand management. 4. Discussion We presented a framework to study service coverage and capacity planning problems in the context of crowd-sourced mealdelivery. This setting represents a three-sided market in which providers with complementary capabilities (restaurants and couriers) offer a service to customers (diners) that interact with the providers through an online platform. Although meal delivery is a natural fit for this three-sided market framework, there are many other settings in which providers with complementary capabilities offer (or can offer) services to customers. The same business model can be employed to take advantage of opportunities offered by the shared economy and advancing communication technologies. As such, we believe that the results of our study can be applicable to a much wider set of businesses than on-demand meal delivery platforms. On-demand meal delivery platforms highlight the novel challenges associated with managing crowd-sourced delivery capacity which self-schedules and self-selects the delivery orders to perform. We propose a modeling approach that, for the first time, captures (i) service coverage decisions, (ii) self-selecting behavior of couriers, and (iii) hybrid delivery capacity (composed of crowd-sourced couriers and company-employed drivers). Deciding the service coverage and service quality is critical to the success and profitability of an on-demand meal delivery platform. There are many situations in which the results and insights generated in this research may guide and support decisions. A
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prime example is entering a new market, i.e., providing on-demand meal delivery services in a new city. Decisions have to be made regarding the restaurants that will be part of the network, regarding the areas served by these restaurants, regarding the courier compensation scheme, to name just a few. But our results and insights may also guide and support decisions in an established market, for example in helping the sales team identify restaurants to target for inclusion in the network, or in helping the operations team to adjust the service area of restaurants in the network that are less profitable, or in helping the operations team assess the potential benefits of incentive schemes to reduce delivery offer rejection rates of couriers. The results and insights generated in this research may also indirectly guide and support decisions. A possible demand management scheme that might be considered is to dynamically adjust the radius of the service area of a restaurant to respond to changes in order arrival rates or delivery capacity (planned or unplanned). A better understanding of how the size of a service area relates to profit and service quality, and how the distribution of diners in the service area affects these relations, may help design and implement an effective version of such a demand management scheme. However, our results and insights extend beyond on-demand meal delivery platforms. Planning crowd-sourced delivery capacity is fundamentally different from the traditional delivery capacity planning. In the latter, the focus is on determining an optimal fleet size and mix and maximizing the utilization of the vehicles in the fleet. In the former, the focus is on ensuring a sufficient number of couriers in the system at any time and ensuring a high courier order acceptance probability. Our research provides one possible framework and some initial insights into effective planning of crowd-sourced delivery capacity and related issues concerning compensation of crowd-sourced delivery capacity and ensuring service quality when relying on crowd-sourced delivery capacity. Our analyses have provided insights into the interaction between courier satisfaction (i.e., couriers’ willingness-to-wait), courier compliance (i.e., couriers’ offer acceptance-probability) and profit. Our analyses show that couriers’ earning expectations play a critical role in the optimal service area and profit. Of course, couriers make their decisions based on their perceptions about the current and future system state, which are likely inaccurate. The meal-delivery platform provider, on the other hand, knows the current system state and employs (or can employ) technology to predict the future system state, and it is therefore an interesting question whether sharing some or all of this information with the couriers can be beneficial. Our analyses also show that the self-scheduling and self-selecting behavior of couriers has a significant impact on optimal service area and profit. Our numerical experiments suggest that hybrid delivery capacity (i.e., couriers and drivers) can both improve reliability as well as profit. However, the use of our model is not limited to investigating delivery capacity composition questions. It can be also used to evaluate and compare various strategies to improve couriers compliance. For example, offering extra compensation for orders that are more likely to be rejected by couriers. Our models may help to determine how large the extra compensation should be to achieve certain goals. Acknowledgment This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant No. 2219. Appendix A. Mathematical proofs A.1. Proof of Proposition 1
µ
1
Note that with the given customer density function the expected delivery-time for a given service radius r can be calculated r 2x 2G¯ (r ) 1 2r 1 = G (r ) 0 v g (x ) dx = v . Using (p) = b p , profit function (5) can be rewritten as follows. G (r ) v
(r ) = c
b
(r ) = c G (r ) Then setting
d dr
2G¯ (r ) G (r ) v
2r v
G (r )
(35)
2 b G¯ (r )
2 b rG (r )
(36)
v
(r ) = 0 we get:
d (r ) = c g (r ) dr
2ab rg (r ) + 2 b G (r ) v
2 b G (r )
=0
(37)
cv 2b
as desired. Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (37). Hence, the optimal solution which yields r = is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.2. Proof of Proposition 2 Let g , G, G ,
(r ) = c
and µ be defined as in Section 3.1. Since we assume (p) = b p , substituting p = 1
2r b v
2 bG¯ (r ) G (r ) v
2r v
2G¯ (r ) G (r ) v
G (r ),
1
( ) in (9) gives: µ (r )
(38)
which can be simplified as: 196
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2 b rG (r )
(r ) = c G (r ) Then setting
d dr
bG¯ (r )
2
2
2 G¯ (r )
rG (r )
v
v
.
(39)
(r ) = 0 we get:
2
d (r ) = c g (r ) dr
brg (r ) + 2
bG (r ) v
2
bG (r )
2
rg (r ) + 2
G (r ) v
2
G (r )
=0
(40)
cv . 2( b + )
which yields r = Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (40). Hence, the optimal solution is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.3. Proof of Proposition 3 Let g , G, G , and µ be defined as in Sectionsec:basic. Taking the derivative of objective function 1 (p) = b p we get: d dr
(r ) = cg (r ) r 2xg (x )
g (r ) 0 v G (r )
b
g (r )
b G (r ) dx
2 ¯pr ¯ (1 v
r 2xg (x ) dx 0 v G (r ) 2
(r ) with respect to r and using
2rg (r ) vG (r )
+
) g (r )
(41)
That simplifies to:
d (r ) = dr
g (r )( 2b r + cv + 2 2 p2 r (
1))
v
.
(42)
Then setting (42) equal to zero, we get the desired result:
r =
cv 2( b + ¯p¯ (1
(43)
))
Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (43). Hence, the optimal solution is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.4. Proof of Proposition 4 Taking the derivative of objective function
d (r ) = dr Since we have
g (r )(cv + 2 ¯pr ¯ ( (r ) v
1)
1
(r ) with respect to r and using (p) = b p we get:
2 br (r ))
(44)
> 0 and g (r ) > 0 for all r > 0 this equation simplifies to
r ( b (r ) + ¯p¯ (1
(r ))) =
cv 2
(45)
as desired. Appendix B. RtP approximation Let EW (M /M / m) be the expected waiting times of customers in a (M /M / m) queue, where m is the number of servers (drivers) in the system. The mean RtP times can be modeled as the mean waiting times in a M/G/m queue with poisson arrivals and general service time distributions. From the queuing theory we have: 1
EW M / M /m =
() m ! mµ (1
m
m 1
µ
mµ
)
2
j=0
() µ
j!
j
+
()
m
µ
m! 1
mµ
(46)
We consider the following approximation of EW (M /G / m) proposed by Gans et al. (2003).
EW M / G/ m = where,
C2
(1 + C 2) EW (M / M /m ) 2
(47)
is the coefficient of variation of the service time distribution.
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Appendix C. Problem data for multiple service regions See Table 2. Table 2 Base case service times and order rates. Region ID (i ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Mean service time (1/µi )
Order rate ( i)
18 23 19 20 18 23 20 22 16 20 17 25 16 21 25 19 25 24 21 23
1.15057 1.53001 0.13265 0.06455 0.71748 1.39642 1.30933 0.12539 0.12718 1.53665 1.32674 0.04354 1.55416 1.61243 0.17639 1.13934 1.11323 0.38830 0.47310 1.53714
References Allon, G., Bassamboo, A., Çil, E.B., 2012. Large-scale service marketplaces: the role of the moderating firm. Manage. Sci. 58 (10), 1854–1872. Archetti, C., Savelsbergh, M., Speranza, M.G., 2016. The vehicle routing problem with occasional drivers. Eur. J. Oper. Res. 254 (2), 472–480. Arnosti, N., Johari, R., Kanoria, Y., 2014. Managing congestion in decentralized matching markets. In: Proceedings of the Fifteenth ACM Conference on Economics and Computation. ACM, pp. 451. Bai, J., So, K., Tang, C., Chen, X., Wang, H., 2016. Coordinating Supply and Demand on an On-demand Service Platform: Price, Wage, and Payout Ratio [PhD thesis]. working paper, 2016. Bakker, E., 2016. The on-demand meal delivery report: Sizing the market, outlining the business models, and determining the future market leaders, (accessed: 2017-11-02). Banerjee, S., Johari, R., Riquelme, C., 2015. Pricing in ride-sharing platforms: a queueing-theoretic approach. In: Proceedings of the Sixteenth ACM Conference on Economics and Computation. ACM, pp. 639. Barr, A., Wohl, J., 2013. Exclusive: Walmart may get customers to deliver packages to online buyers, Forbes. (accessed: 201711-02). Bimpikis, K., Candogan, O., Daniela, S., 2016. Spatial pricing in ride-sharing networks, Working Paper. Cachon, G.P., Daniels, K.M., Lobel, R., 2017. The role of surge pricing on a service platform with self-scheduling capacity. Manuf. Service Oper. Manage. Carbone, V., Rouquet, A., Roussat, C., 2015. Carried away by the crowd: what types of logistics characterise collaborative consumption. In: 1st International Workshop on Sharing Economy, pp. 1–21. Chen, M.K., Sheldon, M., 2016. Dynamic pricing in a labor market: surge pricing and flexible work on the uber platform, In Working Paper. Chen, W., Mes, M., Schutten, M., 2017. Multi-hop driver-parcel matching problem with time windows. Flexible Services Manuf. J. 1–37. Dayarian, I., Savelsbergh, M., 2017. Crowd-based City Logistics, Optimization Online. Economides, N., Katsamakas, E., 2006. Two-sided competition of proprietary vs. open source technology platforms and the implications for the software industry. Manage. Sci. 52 (7), 1057–1071. Farber, H.S., 2015. Why you cant find a taxi in the rain and other labor supply lessons from cab drivers. Q. J. Econ. 130 (4), 1975–2026. Gans, N., Koole, G., Mandelbaum, A., 2003. Telephone call centers: tutorial, review, and research prospects. Manuf. Service Oper. Manage. 5 (2), 79–141. Gurvich, I., Lariviere, M., Moreno, A., 2016. Operations in the On-demand Economy: Staffing Services with Self-scheduling Capacity. Northwestern University, Evanston, IL.. Hall, J.V., Krueger, A.B., 2016. An analysis of the labor market for ubers driver-partners in the united states, Working Paper 22 843, National Bureau of Economic Research, November 2016. Hong, I., Kuby, M., Murray, A.T., 2018. A range-restricted recharging station coverage model for drone delivery service planning. Transport. Res. Part C: Emerg. Technol. 90, 198–212. Howe, J., 2006. The rise of crowdsourcing. Wired Mag. 14 (6), 1–4. Hu, M., Zhou, Y., 2015. Dynamic matching in a two-sided market, Available at SSRN. Jiang, B., Tian, L., 2016. Collaborative consumption: Strategic and economic implications of product sharing. Manage. Sci. Kabra, A., Elena, B., Karan, G., 2017. The efficacy of incentives in scaling marketplaces, Technical report, Working paper. Kafle, N., Zou, B., Lin, J., 2017. Design and modeling of a crowdsource-enabled system for urban parcel relay and delivery. Transport. Res. Part B: Methodol. 99, 62–82. Kimura, T., 1994. Approximations for multi-server queues: system interpolations. Queueing Syst. 17 (3), 347–382. MacMillan, D., 2015. The $50 billion question: can uber deliver? Wall Street J. (June 16), A1–A12. Montreuil, B., 2011. Toward a physical internet: meeting the global logistics sustainability grand challenge. Log. Res. 3 (2–3), 71–87. Moreno, A., Terwiesch, C., 2014. Doing business with strangers: reputation in online service marketplaces. Inf. Syst. Res. 25 (4), 865–886. Morgan Stanley Research, 2016. Food delivery: What if all food could be delivered as easily as pizza?.
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Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Service and capacity planning in crowd-sourced delivery Baris Yildiza, , Martin Savelsberghb ⁎
a b
T
Department of Industrial Engineering, Koc University, Istanbul, Turkey H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, USA
ABSTRACT
The success of on-demand service platforms, e.g., Uber and Lyft to obtain a ride and Grubhub and Eat24 to get a meal, which rely on crowd-sourced transportation capacity, has radically changed the view on the potential and benefits of crowd-sourced transportation and delivery. Many retail stores, for example, are examining the pros and cons of introducing crowd-sourced delivery in their omni-channel strategies. However, few models exist to support the analysis of service area, service quality, and delivery capacity planning, and their interaction, in such environments. We introduce a model that seeks to do exactly that and can answer many fundamental questions arising in these settings. Using on-demand meal delivery platforms as an example, we investigate, among others, the relation between service area and profit and delivery offer acceptance probability and profit, and the benefits of integrating delivery service of multiple restaurants, and generate many valuable insights.
1. Introduction Enabled by the recent advances in communication and GPS technologies, on-demand service platforms are reshaping our way of life, witnessed by the fact that Uber (www.uber.com) and Lyft (www.lyft.com) have become household names. On-demand service platforms connect waiting-time sensitive customers with independent service providers (Bai et al., 2016). We focus on on-demand meal delivery platforms, which target a massive market. As of 2015, about $210 billion worth of food is ordered for delivery or takeout on an annual basis solely in the US (Morgan Stanley Research, 2016), whereas the two leading on-demand meal delivery platforms, Grubhub (www.Grubhub.com) and Eat24 (www.eat24.com), generated a combined $2.6 billion in food sales capturing only a tiny fraction of the total market (Bakker, 2016). However, succeeding and thriving in this market requires looking beyond traditional last mile delivery practices. Meal delivery is arguably the ultimate challenge in last mile logistics: a typical order is expected to be delivered within an hour (much less if possible), and within minutes of the food becoming ready. Restaurant-operated delivery services can no longer meet the high diner expectations of fast service, low cost, and near-continuous availability. Ondemand meal delivery platforms, benefiting from the delivery density that comes from delivering orders from multiple restaurants and relying on crowd-sourced delivery, are meeting diner expectations – most of the time. In a market with high order arrival rate fluctuations throughout the day (soaring during the meal times and dropping to very low levels in between) the advantage of crowd-sourced delivery, i.e., contracting individuals to deliver meals, is obvious (Reuters, 2016). However, even though such couriers (the term we will use throughout the paper to refer to individuals contracted to deliver meals) may provide the desired flexibility in delivery capacity, they also introduce many challenges (Allon et al., 2012; Farber, 2015; Chen and Sheldon, 2016; Gurvich et al., 2016; Taylor, 2018; Cachon et al., 2017). Ensuring that a sufficient number of couriers is available to provide the service quality that is critical to sustain an online meal delivery service platform is difficult, in part because couriers do not necessarily accept all delivery tasks offered to them (Yildiz and Savelsbergh, 2018; Sampaio et al., 2019). Novel approaches for service coverage and delivery capacity planning are needed to exploit the benefits and to limit the downsides of crowd-sourced delivery for successful business implementations. However, mostly due to the novelty of the practice, there is little research on crowd-
⁎
Corresponding author. E-mail address: [email protected] (B. Yildiz).
https://doi.org/10.1016/j.trc.2019.01.021 Received 16 August 2018; Received in revised form 21 January 2019; Accepted 21 January 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.
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sourced meal delivery operations providing insights that can help practitioners (more) effectively manage demand (delivery orders) and supply (delivery capacity) so as to achieve their strategic goals related to profitability, market share, and customer satisfaction. Our study is meant to be a first step in that direction. One key characteristic of the crowd-sourced meal delivery environment is the self-scheduling delivery capacity, i.e., couriers independently choosing the time and duration of their service. Consistent with the existing literature and empirical findings, we model this feature as a market with a large pool of couriers making decisions based on a rational expectation about their earnings (Snir and Hitt, 2003; Moreno and Terwiesch, 2014; Farber, 2015; Chen and Sheldon, 2016; Gurvich et al., 2016; Bai et al., 2016; Banerjee et al., 2015; Bimpikis et al., 2016; Taylor, 2018; Cachon et al., 2017; Kabra et al., 2017). In such a market, an online meal delivery platform has to carefully coordinate capacity and demand in order to maximize its profit while ensuring a target service quality. An online meal delivery platform has two ways to influence diner demand. First, by adjusting the price they charge restaurants/ diners for deliveries. Second, by adjusting the service coverage area associated with a restaurant. In the literature, only the former has been investigated. However, in practice, many on-demand service platforms avoid real time pricing, in part, because of diner resistance to the practice (MacMillan, 2015; Taylor, 2018), and consider dynamic service coverage and demand management strategies (directing diner demands to restaurants that are easier to serve) to achieve high diner satisfaction levels which is crucial for a sustainable growth in the market. Therefore, we focus on service coverage planning as the main leverage to manipulate demand (even though our model can be used to analyze dynamic pricing). An online meal delivery platform has two ways to manage delivery capacity. First, by adjusting courier compensation, which indirectly controls the crowd-sourced delivery capacity. Second, by employing salaried drivers, they directly control the companyprovided delivery capacity. Here, we use “salaried drivers” to indicate any form of compensation that gives the company more direct control over its delivery capacity; it also covers minimum-pay guarantees in return for a courier’s commitment to make deliveries in a specified block of time (as is done by some online platforms). Using compensation to manipulate capacity has been investigated in the expanding on-demand service platforms literature (Gurvich et al., 2016; Bai et al., 2016; Bimpikis et al., 2016; Taylor, 2018; Cachon et al., 2017), but, to the best of our knowledge, we are the first to investigate the use of hybrid delivery capacity, composed of crowdsourced and company-provided capacity, to ensure service quality. Self-scheduling is not the only challenge associated with crowd-sourced delivery capacity. The fact that couriers do not have to accept delivery tasks offered to them also complicates capacity planning and ensuring service quality. For various reasons, e.g., expectations about the likelihood of future delivery task offers or expectations about the tips associated with future delivery task offers, couriers may decide to reject a delivery task offer. A distinctive feature of our study is that we investigate the impact of this “self-selecting” aspect of crowd-sourced delivery capacity as well as the potential of company-provided delivery capacity to mitigate any negative effect on service quality. In this paper, we develop a theoretical foundation for conducting analyses and deriving results that shed light on some fundamental questions related to supply and demand management in crowd-sourced meal delivery. In particular, we present a stylized model to investigate and answer, among others, the following questions:
• Which restaurants should be included in the network? • How much should restaurants be charged for the deliveries? • Which diners should be served? • How much should couriers be paid for the deliveries? • Should salaried drivers be used alongside couriers? We start by investigating a setting with a single restaurant serving diners with a delivery location inside a circle with the restaurant at its center. This setting captures (most of) the critical aspects of the on-demand meal delivery environment while being simple enough to obtain analytic results, and simplifies the exposition. Most of the analytical results for the “single restaurant” setting readily extend to the multiple restaurant setting (and more general diner location distributions) assuming that the average time it takes a courier to execute a delivery task can be estimated with a reasonable level of accuracy. We focus on analyzing and optimizing profit as a function of service, i.e., the size of the area served and the time it takes to serve diners in that area, and capacity, i.e., the number of couriers delivering to diners (which depends, among others, on the compensation offered to couriers). The contributions of this study can be summarized as follows.
• We introduce service and capacity planning problems in the context of crowd-sourced meal delivery and present a theoretical framework to analyze these problems. • We explicitly consider the autonomous decision making of courier and demonstrate the benefits of supplementing crowd-sourced delivery capacity with company-provided delivery capacity. • We derive results concerning the optimal service coverage area, courier compensation, and delivery capacity composition that maximizes the profit of an online meal delivery platform given a target service quality level. A few insights resulting from our analyses are that: – The service area that maximizes profit depends on the revenue per delivery, the average time of a delivery, the service quality targets, and courier earning expectations; – The service area has a significant impact on profit; – A large diner base close to the restaurants not only increases profits, but also enables serving a larger area; 178
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– Company provided delivery capacity not only increases service reliability, but can also increase profits; and – Often, a small sacrifice in profit, can result in a substantial improvement in service quality. The remainder of the paper is organized as follows. In Section 2, we briefly discuss related literature. In Section 3, we introduce our modeling framework and present our analyses. We conclude, in Section 4, with a discussion of practical implications of our findings. For ease of exposition, all mathematical proofs are provided in the appendix. 2. Literature review Crowd-sourced delivery falls into the broad class of innovative business practices called crowd-logistics (Sampaio et al., 2019). The more general term crowd-sourcing was first used by Howe (2006) to define the business practices that occur in the intersection of outsourcing and shared economy (Zervas et al., 2014; Jiang and Tian, 2016). In a recent study, Rai et al. (2017) provides a review of these disruptive business applications, which have started to attract significant research interest. Crowd-sourced delivery, or alternatively crowd-shipping, constitutes a major class of crowd-logistics. Relying on the public (rather than employees or third-party providers) to deliver goods or packages distinguishes crowd-shipping from delivery practices relying new technologies for deliveries, e.g., delivery drones or bots (Hong et al., 2018; Perboli and Rosano, 2019). Within crowd-sourced delivery two distinct types of services can be distinguished: door-to-door and store-to-door (Sampaio et al., 2019). The former is a P2P model in a two-sided market (Economides and Katsamakas, 2006), in which the crowd (providers) pickup and deliver cargo for and between individuals (customers). Hitch (acquired by Lyft) and Roadie are examples of platform providers facilitating door-to-door delivery services. In these platforms, customers post their requests for deliveries and providers declare their journeys. A matching problem is solved to bring together customers and providers (Allon et al., 2012; Moreno and Terwiesch, 2014; Arnosti et al., 2014; Hu and Zhou, 2015). Store-to-door crowd-delivery, on the other hand, is considered as B2C practice (of which crowd-sourced meal delivery is an example). There are different store-to-door crowd-delivery practices, such as using in-store customers to deliver on-line orders in their neighborhood. Walmart has considered such a crowd-shipping practice to improve its online retail operations (Barr and Wohl, 2013). A different business model is employed by the food and grocery delivery networks that serve a three-sided-market where a retailer/restaurant and couriers (two complementary providers) jointly offer a service to customers. In this model, online platforms function as a store front where customers can select the retailer/restaurant from which they want to purchase (Sampaio et al., 2019). Meal delivery network providers, such as Grubhub, Eat24, UberEats and Foodora all fall into this group. Much of the thinking and discussion about crowd-shipping has developed from the practitioner’s side and the literature that analyzes crowd-shipping is still rather limited (Rai et al., 2017; Carbone et al., 2015). In a recent study, Paloheimo et al. (2016) present a case study in Finland to investigate the possible benefits of crowd-shipping practices in library deliveries. The authors argue that crowd-shipping has the potential to reduce carbon emissions, to lower operational costs, and to improve social cohesion in the society. Introducing a new variant of the Vehicle Routing Problem, Archetti et al. (2016) investigate the Vehicle Routing Problem with Occasional Drivers, which is motivated by the crowd-shipping concept of using in-store customers to deliver on-line orders. The authors emphasize the complexity of deciding a compensation scheme that ensures sufficient delivery capacity to achieve a certain service quality. A more in-depth and detailed analysis (in a dynamic context) of this crowd-shipping concept can be found in Dayarian and Savelsbergh (2017). Introducing a different perspective, Kafle et al. (2017) suggest a two-echelon distribution system in which distribution in the second echelon is performed by the crowd. Relay (or exchange) points connect the first and the second echelons. The use of exchange points in city logistics is also promoted by the Physical Internet initiative (Montreuil, 2011). In a recent study, Chen et al. (2017) introduce the Multi-Driver Multi-Parcel Matching Problem (MDMPMP), in which parcels can be transported by a single or by multiple drivers and at each relay point a matching problem is solved to take advantage of transfer opportunities to improve system performance. In the context of public transport, integration of ride-sourcing (i.e., on-demand ride-sharing) and scheduled transport services to expand coverage and improve efficiency is also attracting attention, see for example Stiglic et al. (2018) and Yan et al. (2018). Reyes et al. (2018) is the first, to the best of knowledge, to study the application of crowd-shipping in meal delivery operations. Introducing the Meal Delivery and Routing Problem (MDRP), the focus of the study is the development of an online dispatching algorithm that makes the best use of the available crowd-delivery capacity to improve a variety of system performance measures. Yildiz and Savelsbergh (2018) propose an exact solution algorithm to solve the static version of the MDRP in order to be able to assess the quality of online dispatching algorithms and to extract valuable insights from the characteristics of optimal solutions. In none of the aforementioned studies an analytical framework has been put forward to study the service coverage, courier compensation, and delivery capacity planning problems that arise in crowd-delivery applications, which is the focus of our research. Our research also relates to the burgeoning on-demand service platforms literature, which mostly assumes a two-sided market structure and seeks to determine optimal service price and provider (courier) compensation. Recent studies Banerjee et al. (2015), Gurvich et al. (2016), Bai et al. (2016), Gurvich et al. (2016), Cachon et al. (2017), Taylor (2018) focus on providers’ (couriers’) participation decisions in an uncertain demand environment. Banerjee et al. (2015), Bai et al. (2016) and Cachon et al. (2017) compare static versus system-state-dependent (dynamic) pricing policies. Cachon et al. (2017) suggest that “surge pricing” can be beneficial for all the stakeholders in a self-scheduling capacity environment, and Bai et al. (2016) show that platforms can increase their profits with dynamic pricing. In contrast, Banerjee et al. (2015) argues that under a broad range of conditions the platform cannot improve throughput, revenue, or welfare by employing a dynamic pricing policy and the main strength of dynamic pricing is its robustness. Taylor (2018) studies static pricing in settings where customer resistance or other strategic reasons do not allow 179
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dynamic pricing. Proposing an augmented newsvendor formula that captures the tradeoffs for the platform and the providers, Gurvich et al. (2016) find that for a fixed provider pool, provider autonomy reduces the number of working providers and increases the optimal price. The latter result does not agree with the findings of Taylor (2018) which shows that demand is sensitive to waiting times, where as the model of Gurvich et al. (2016) (and also the model of Cachon et al. (2017)) does not relate the demand directly with the service quality. In this study, we consider exogenous prices (even though our model can be used to analyze optimal dynamic pricing to improve profitability) to focus on the service coverage and delivery capacity composition decisions (which have not been studied in-depth before). Similar to our study, Banerjee et al. (2015), Bai et al. (2016), and Taylor (2018) consider time-sensitive customers. In these papers, customer demand, therefore, depends on the waiting time. To obtain insights from a rich queuing model, Banerjee et al. (2015) and Bai et al. (2016) employ approximations and large system asymptotics, where as Taylor (2018) presents a stylized model which allows for exact analysis. Using a different approach, we do not employ utility functions to represent the time sensitivity of the customers, but use service quality targets set by the meal delivery platform to capture customers’ time sensitivity. 3. Model In this section, we present the basic setting we consider as well as the basic model we use for analyzing service coverage and delivery capacity design. There is a single restaurant with a circular service area with radius r. Orders arrive in the service area according to a random process with rate (r ) . We assume Euclidean distances and a fixed speed v for all movements. The delivery-time of an order is defined to be the time of the out-and-back trip from the restaurant to the diner location (i.e., we assume instantaneous delivery at the diner location). Deliveries are performed by couriers, i.e., individuals that sign into the meal delivery network to receive delivery offers. Couriers can sign in and out whenever they want and they are allowed to reject delivery offers. For each delivery, the on-demand meal delivery platform earns a revenue of c dollars (the amount charged to the restaurant). Couriers are paid a flat fee of p dollars for each delivery they perform. Couriers are assumed to be at the restaurant when they sign in and to return to the restaurant after they complete a delivery. Couriers arrive at the restaurant according to a random process. The arrivals include couriers signing in as well as couriers returning from a delivery. We consider a time interval T during which the parameters of the system, e.g., order arrival and courier arrival rates, are assumed to be stable. We denote the mean delivery-time during this interval by 1/ µ (r ) and measure the service quality by the mean ready-to-pickup (RtP) time during this interval, where RtP is the difference between the order pickup time and the order ready time. We will take the order arrival time to be the order ready time, whereas in practice, the order ready time has to account for meal preparation time. RtP time is closely related to click-to-door (CtD) time, i.e., the difference between order delivery time and order placement time, but RtP time is a more relevant measure for capacity planning purposes, as it excludes meal preparation time and the travel time from the restaurant to the delivery location, which are not affected by the available delivery capacity. Thus, an online meal delivery platform sets a RtP time target and plans its delivery capacity so as to meet this target in order to assure diner satisfaction. We note that Reyes et al. (2018) and Yildiz and Savelsbergh (2018) found that RtP times are small compared to CtD times, on the average RtP time is only 5% of CtD. In Fig. 1, we present a timeline with pertinent events (diner related as well as courier related) and important concepts, e.g., CtD and RtP. We define a willingness-to-wait function (p) , which we assume to be strictly increasing in p, to capture the average time a courier stays in the system without receiving a delivery offer when the payment per delivery is p. That is, on the average, a courier signs out, when he or she is not offered a delivery job for (p) 1/ µ (r ) time units. The findings of Chen and Sheldon (2016), who analyzed 25 million Uber trips, suggest that this is a reasonable model. Note that the willingness-to-wait function can be viewed as (a proxy for) the courier’s utility function, as it reflects the courier’s actions based on his earning rate expectations. Next, we use the equilibrium behavior of the system to derive the number of couriers in the system. Because for the remainder of this section we assume the radius r is given, we will use, for ease of presentation, and µ instead of (r ) and µ (r ) . At the equilibrium, for a per-delivery payment p, the number of couriers in the system, m (p) , stabilizes at (p) . The reasoning is as follows. When the courier earning rate rises above p = p / (p) , more couriers will stay in the system and, as a result, the number of deliveries per courier per time unit starts to decrease. This, in turn, will drive the courier earning rate back down to p . Similarly, if the courier earning rate falls below p , more couriers will leave the system and, as a result, the number of deliveries per courier per time unit starts to increase. This, in turn, will drive the courier earning rate back up to p . Thus, we assume that the courier earning rate remains in a band around p . This agrees with the findings of Hall and Krueger (2016), who present a comprehensive analysis of the labor market for Ubers driver-partners, based on both survey and administrative data.
Fig. 1. Timeline of events. 180
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Table 1 Notation used throughout the paper. r:
The radius of the service area The order arrival rate in a service area of radius r The mean delivery-time for orders in a service area of radius r The payment to a courier for a delivered order The average time a courier stays in the system without receiving a delivery offer when the payment per delivery is p The number of couriers in the system when the payment per delivery is p The minimum number of couriers needed in the system The diner density at distance x 0 from the restaurant Service-quality-aspiration parameter The revenue received for a delivered order The average vehicle speed The courier waiting-rate
(r ) : µ (r ) : p: (p) : m ( p) : m: g (x ) : : c: v: b:
Since the courier earning rate stabilizes at p , each courier, on average, makes Tp / p deliveries during T. Consequently, we have
m (p ) =
T Tp p
=
p = p
(p). (1)
Equality (1) tells us that the number of couriers in the system is a function of the order arrival rate and the per-delivery payment p. Interestingly, the number of couriers in the system only indirectly depends on the delivery-time. The average delivery-time 1/ µ has to be less than (p) , otherwise no courier would remain in the system, but given that, a courier stays in the system as long as he does not have to wait too much between consecutive deliveries. On the other hand, the delivery-time directly affects the service quality. With the same number of couriers in the system, higher delivery-times will result in worse service quality. Note that for a given number of couriers and a given delivery-time distribution, the service quality, i.e., the mean RtP, is fixed. We next investigate how the mean RtP is affected by the number of couriers in the system. First, note that if the system parameters were deterministic, the number of couriers needed, m , can be determined by finding the number of couriers that equalizes the total delivery capacity, m T , with the total delivery demand, T / µ , which gives m = /µ . However, the stochasticity in the system implies that some “excess capacity” is required to ensure a certain service quality. Therefore, we define a service-quality-aspiration parameter > 1, and use the following relation in or models:
m =
µ
.
(2)
Note that higher values of will provide more reliable and faster service, possibly at the expense of higher operational costs. In our analyses, we investigate the impacts of the service-quality-aspiration parameter value on the mean RtP and the operational costs. For convenience, we summarize the notation used throughout the paper in Table 1. Note that we assume that diner demand is exogenous and does not depend on the service quality, i.e., the mean ready-to-pickup time. A mean ready-to-pickup time of a few minutes is critical to the success of online meal delivery platforms, and, thus, constitutes only a very small fraction of the click-to-door time. As a consequence, assuming that diner demand is not impacted by small variations in ready-to-pickup time is reasonable. Observe that when the order arrival rate, the number of couriers in the system, and the delivery-time distribution are given, the mean RtP value can determined by viewing the system as a multi server queue with general arrival and service time distributions (G/ G/ m) . Although most performance metrics for this queuing system are not known, we can obtain close approximations of the waiting time, which corresponds to the mean RtP value, using simulation or approximation approaches (see, for example, Whitt, 1993; Kimura, 1994). In our models, we do not consider order bundling, i.e., assigning more than one order to a courier. Bundling of orders, i.e., consolidation, may lead to higher utilization of the delivery capacity. However, the high service expectations of diners, and, as a result, the high self-imposed RtP targets, severely limit the opportunity for order bundling. As a consequence, in practice, only a small fraction of orders are bundled by on-demand meal delivery platforms. (Note too that order bundling leads to a loss of freshness for at least one of the orders). Therefore, the insights obtained from our analysis, even though we do not consider order bundling, are still meaningful in practice. 3.1. Analyses Using the framework introduced above, we now derive analytical results that can help answer several relevant questions about supply and demand management in crowd-sourced meal delivery operations. One such question is “What is the coverage (defined by the radius of the service area) and the courier compensation that maximizes the profit for a meal delivery network while also achieving its service target?”. Assuming that the order arrival rate is a function of the service area radius r, we will analyze the relationships between coverage, profit, and service quality. We will first assume that couriers always accept the delivery offers they receive and the service platform does not supplement the delivery capacity provided by couriers with company-employed drivers. Then, subsequently, we will analyze the effects of couriers rejecting some of the delivery orders they receive and supplementing 181
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crowd-sourced delivery capacity with company-employed delivery capacity. Let c be the revenue obtained (the amount charged to the restaurant) by the on-demand meal delivery platform for each delivery. We will refer to the parameter c as the unit-revenue. Then, we can define the (rate of) profit as = (c p) (r ) and consider the following optimization problem to find the optimal coverage for a given service-quality-aspiration level :
P : max c
p
r >0
(r )
(3)
(r ) . µ (r )
s.t. (r ) (p) =
(4)
The objective is to maximize the profit for the meal delivery platform and the constraint enforces that a sufficient number of couriers is active in the system. Next, we show that formulation P can be transformed into an unconstrained optimization problem that can easily be solved by d calculating the derivative dr (r ) and setting it equal to zero. Assuming that (r ) > 0 for all r > 0 , we can rewrite (4) as (p) = / µ (r ) . Let R be the distance to the furthest possible diner location (or some (physical) limit on the radius of the service area). Then replacing p with 1 ( / µ (r )) , we obtain the unconstrained optimization problem 1
P : max c
(r ).
µ (r )
r (0, R]
(5)
Given (5), we can analyze the optimal radius of the service area by considering specific forms of the functions , µ , and . 0 units of distance away from the Let g : + +, g (0) = 0 be a Lebesgue integrable function that gives the diner density at x d d restaurant. We assume g (x ) > 0 , for all x > 0 . For the notational convenience, we define g (x ) = dx G (x ) and G (x ) = dx G¯ (x ) . Let > 0 be the order arrival rate, i.e., the average frequency with which orders are placed, which implies that the order arrival rate for a r radius r is given by (r ) = 0 g (x ) dx . For presentational convenience, we assume a linear willingness-to-wait function, (p) = p/ b , where we will refer to b > 0 as the courier waiting-rate. The following proposition shows that for a large class of problems there is a closed form expression for the optimal radius. Proposition 1. Given unit-revenue c, speed v, courier waiting-rate b, and service-quality-aspiration , the optimal radius r of the service area is cv /2b . Interestingly, Proposition 1 shows that the optimal radius of the service area does not depend on the diner density function g and order frequency . Furthermore, the optimal radius r is linearly increasing in the unit-revenue c and the speed v, and linearly decreasing in the courier waiting-rate b and service-quality-aspiration . Note that solving the equation (r ) (p) =
( ), for p, (r ) µ (r )
gives the optimal courier compensation p that maximizes the profit. Although the optimal radius does not depend on the diner density function g, it is not hard to see that profit and courier compensation are affected by the diner density distribution in the service area. Therefore, we continue our analysis with the following class of diner density functions
g (x ) =
(k + 1) x k Rk + 1
0,
, if x
R (6)
o.w.,
with k > 1 and R > 0 . If diners are uniformly distributed in the service area, we set k = 1. For any k < 1, the diner density decreases with the distance from the restaurant, with k = 0 corresponding to a linear and k < 0 corresponding to a super-linear decrease in the density. For the diner density functions of type (6), we can find the optimal service area as defined in Proposition 1 and obtain the meal delivery platform’s profit as follows:
(r ) =
c (k + 2)
(c
( )
k+1 cv , 2 bR 2 bR (k + 1) v (k + 2)
if r
),
R
if r = R.
(7)
Inspecting (7), we see that the profit (r ) is increasing in unit-revenue c and vehicle speed v, and decreasing in courier waitingrate b and service-quality-aspiration , as expected. The profit, however, differs significantly depending on whether it is optimal to offer service to all diners (r = R ) or not (r < R ). In the former situation, the unit-revenue c has the largest impact on profit; for all values of k, the profit increases super-linear in c. In that situation, it is also clear that the parameter k, i.e., the diner distribution, is a dominant factor in determining the profitability of providing delivery service. On the other hand, we see that when it is optimal to serve all diners, the impact of the diner distribution is less pronounced. Even though our focus is not on pricing decisions, i.e., we assume that other factors dominate the determination of the per-delivery charge c, we make a few relevant observations. First, we note that (7) enables a straightforward way to calculate the optimal per-delivery charge even when the order frequency is sensitive to this value (i.e., when depends on c). Replacing with (c ), one can conduct a bisection search to find the value c that maximizes (7). Second, we observe that when order frequency is insensitive to the per-delivery charge (e.g., for an increase/decrease in the perdelivery charge, the change order frequency is less than or equal to ), (7) suggests that it is optimal to set c = 2R b/ v , i.e., to service 182
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Fig. 2. Impact of service area on profit.
the entire region. Throughout this section, we illustrate our analyses with numerical examples. In all these numerical examples, we use the diner density distributions (6) with k ( 1, 3], furthest diner location R = 5000 meters, order frequency = 2 per minute, waiting-rate b = $0. 33 per minute (which implies a $20 per hour earning rate expectation for the couriers), and service-quality-aspiration = 1.075 (which, based on preliminary experiments, provides RtP times of less than 5 min). We start by illustrating the impact of service area on the delivery platform’s profit. In Fig. 2, assuming speed v = 100 m/min, we show the change in the ratio (r )/ (r ) for k {0, 1, 2} and c {8, 16, 24} . The graphs in Fig. 2 clearly show that the choice of service area has a dramatic impact on profit. Interestingly, profit is less impacted by service area as unit-revenue increase and diner locations tend to be closer to the restaurant (k decreases). Next, in Figs. 3a and b, we illustrate the change in profit for c {8, 16, 24} and v {100, 200, 400} , respectively. Since our analytical results shows that an increase in speed has the same affect as an decrease in b, we do not provide a separate graph to illustrate the effects of changing b. Both figures illustrate how strongly the profitability of a delivery service depends on the location of the restaurant relative to its diner base. Confirming the interpretation of (7), Fig. 3 shows that a restaurant’s relative location to its diner base might dominate other factors (e.g., unit-revenue, vehicle speed, and courier waiting-rate) in the case that it is optimal to serve only part of the service area. A closer look at Fig. 3a also shows that until the unit-revenue c gets large enough to ensure that serving all diners is optimal, an increase in unit-revenue c has a bigger impact for smaller values of k. This suggests that when the diner base is dispersed and farther away from the restaurant, the unit-revenue c plays a crucial role in determining the profitability of providing delivery service. Examining Fig. 3b, we see that when it is not optimal to serve the entire region, the impact of higher vehicle speeds on profit is less when diners tend to be either close to the restaurant or far away from the restaurant. When delivery distances are short (k is small) the vehicle speed does not make much of difference, because it will mostly affect a courier’s idle time between orders. When delivery distances are long (k is large), its negative effect on profit dominates any positive effect of higher
Fig. 3. Profit as a function of unit-revenue and vehicle speed. 183
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Fig. 4. Trade-off between profit and service quality.
vehicle speeds. Service quality is another critical aspect of a meal delivery operation. Recall that service quality is captured by RtP, or waiting time, which we approximate using the formula proposed by Gans et al. (2003) (see Appendix B for details). Reviewing the trade-off curves between profit and service quality shown in Fig. 4, where we plot the profit (left vertical axis) and mean RtP values (right [0.01, 1.2] and diner density distribution parameter k {0, 1, 2} and unit-revenue vertical axis) for service-quality-aspiration c {8, 16, 24} , provides valuable insights. Before interpreting the trade-off curves, we note that when the number of couriers required to deliver meals in a service area with (r ) (optimal) radius r is fractional, i.e., µ (r ) , we take the ceiling to calculate RtP times. This causes the discontinuities in the RtP graphs. The effects of rounding are more pronounced when the mean RtP is high, i.e., when there is congestion at the restaurant, because in such situations having an additional courier has a larger impact. The trade-off curves in Fig. 4 show that high service quality (i.e., low RtP values) can be achieved by modest sacrifices in profit. For example, when unit revenue, c, is $8 and the diner density distribution parameter, k, is 0, then choosing = 1.075 instead of = 1.010 , reduced RtP by more than 50% while the reduction in the profit is less than 8%. A similar trend is observable for all unitrevenue values and diner distributions. However, Fig. 4 also shows that improving RtP beyond a certain level becomes costly, i.e., reduces profit significantly. Therefore, meal delivery platforms should carefully balance profit and service quality (customer satisfaction). Moreover, we observe that as diner locations tend to be further away from the restaurant (i.e., parameter k increases), it becomes more costly to improve service quality. This observation shows, again, that the location of the restaurant (relative to the diner locations) plays a critical role in profitability and service quality. It is also informative to examine the courier compensation (per delivery) associated with an optimal service area. By Proposition 1, we can derive the courier compensation for diner density distribution functions of the form (6) as follows:
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Fig. 5. Courier compensation as a function of unit-revenue and vehicle speed.
p (r ) =
c (k + 1) , (k + 2)
if r < R
2 bR (k + 1) , (k + 2) v
if r = R
(8)
We see that the courier compensation linearly increases with unit-revenue c as long as the radius of the optimal service area is less than R, but after that, the unit-revenue no longer affects courier compensation. Furthermore, we see that the courier compensation does not depend on ordering frequency , i.e., the per-delivery payment to couriers does not increase or decrease as diner demand changes. Observe that when unit-revenue c increases, the radius r of the optimal service area increases, and, thus, so does the mean delivery time, which results in an increase in number of couriers, each receiving a higher per-delivery payment. Note that as long as the radius of the optimal service area is less than R, a constant fraction k + 1 of the unit-revenue is paid to the couriers, and this k+2 fraction only depends on the diner density distribution. As diners are located closer to the restaurant, the fraction of unit-revenue paid to the couriers reduces. When it is optimal to serve all diners (i.e., r = R ), courier compensation is independent of the unit-revenue c and determined by the size of the service area R, the vehicle speed v, the courier waiting-rate b, and the service-quality-aspiration . In Fig. 5, we present a numeric example to illustrate the effect on courier compensation when the unit-revenue c and the vehicle speed v change. Fig. 5 shows that courier compensation differs substantially depending the on the optimal service area (the blue line, in both graphs, corresponds to a situation in which it is optimal to serve the entire area, and the orange and green lines correspond to situations in which it is optimal to serve a smaller area (r < R )). We see that when the setting becomes more favorable, e.g., higher unit-revenue, higher vehicle speed, and higher diner density near the restaurant, it is primarily the meal delivery platform that benefits, as the fraction of unit-revenue going to the couriers decreases. 185
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3.1.1. Per-mile compensation So far, we have assumed a per-delivery compensation for couriers. Next, we extend the analysis to include a per-mile compensation in addition to the per-delivery compensation. Specifically, we assume that a courier is paid > 0 per time unit spent delivering an order (i.e., driving out and back from the restaurant to the delivery location). After updating the profit function (3) to include the delivery-time compensation, i.e.,
= c
p
µ (r )
(r ),
(9)
we can derive the following result. Proposition 2. Given unit-revenue c, speed v, courier waiting-rate b, delivery-time compensation , and service-quality-aspiration , the optimal radius r of the service area is cv /2( b + ) . Using Proposition 2, we obtain the maximum profit
(r ) =
c (k + 2)
(
k+1 cv , 2R ( b + )
)
if r < R
c
2 R ( b + )(k + 1) , v (k + 2)
if r = R,
(r ) and the associated per-delivery compensation p (r ) :
(10)
and
p (r ) =
c (k + 1) , (k + 2)
if r < R
2R ( b + )(k + 1) , v (k + 2)
if r = R.
(11)
Again, we see that the optimal service area (radius) does not depend on the diner density distribution function g and the order frequency . In fact, the optimal radius, maximum profit, and courier compensation implied by Proposition 2 are identical to those implied by Proposition 1, when we replace b by ( b + ) . This points out an interesting insight about the courier compensation structure in the presence of a per-mile component. Note that the courier wait-rate b is the unit time cost of enticing a courier and it is paid regardless of the courier’s activity level (i.e., the time the courier spends on making deliveries) and is inflated using to create excess capacity to handle stochasticity. On the other hand, the per-mile compensation is directly related with the activity level of the courier, since it is not paid when a courier is idle. As such, in the summation ( b + ) , the first term represents a fixed cost to attract a courier, whereas the second term accounts for variable costs. To further examine the effect of a per-mile payment on profit and courier compensation, we conduct numerical experiments with diner density functions (6). In Fig. 6, we illustrate how profit changes for different per-mile payments. We compare the profit when there is no per-mile payment (i.e., = 0 ) and the profit with a 10 and 20 cents payment for each minute of driving (i.e., = 0.10 and = 0.20 ). As expected, we see that the profit reduces for higher per-mile pay rates (i.e., higher values of ). Furthermore, we see that the impact is more pronounced, when diner locations tend to be further away from the restaurant. Thus, the reduction in profit as a result of per-mile payments is largest for settings in which it is optimal to serve the entire region. For low unit-revenue values, the effect on profit of offering a per-mile payment or not offering a per-mile payment is relatively small. This is due to the fact that when unitrevenue is low, the optimal service area becomes smaller and, as a consequence, the per-mile payments to couriers become a smaller portion of courier compensation. Presenting the information is a slightly different form, in Fig. 7 we show the fraction of unit-revenue paid to the courier, where we note that, on average, the courier compensation is p = p (r ) + / µ (r ) per delivery. Interestingly, we see that when it is optimal not
Fig. 6. Effect of per-mile pay on profit. 186
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Fig. 7. The fraction of unit-revenue paid to the courier for different per-mile pay rates.
to serve the entire region (i.e., r < R ) the fraction of the unit-revenue paid to a courier is the same regardless of the per-mile payment (r < R for unit-revenue c $16). In that case, the courier compensation is c (k + 1)/ k + 2 , i.e., a fraction k + 1/ k + 2 of unit-revenue (the orange line in Fig. 7). This shows that when courier compensation includes a per-mile component, the meal delivery platform reduces the service area (to maximize its profit), but the fraction of the unit-revenue paid to the courier is the same as in the case when there is no per-mile component (i.e., (8)). 3.1.2. Delivery offer rejections So far, we have assumed that a courier always accepts a delivery offer. However, in practice, this is not the case. One possible strategy to ensure service reliability is to complement the delivery capacity provided by couriers with delivery capacity provided by company-employed drivers. These drivers can be used when couriers reject delivery offers. In this subsection, we investigate the implications of delivery offer rejections on the optimal service area as well as delivery capacity planning in settings with both (selfemployed) couriers and (company-employed) drivers. Let p¯ be the unit-time driver compensation. We assume that drivers are more expensive than couriers, but accept all delivery offers. Let m ¯ be the number of drivers in the system during the time interval T. Before presenting our analysis, we observe that from a cost minimization perspective, it is always sub-optimal to offer a delivery that is acceptable to a courier to a driver. The reasoning is as follows. Since the number of couriers that stay in the system depends on the order arrival rate, a delivery that is acceptable to a courier should not be offered to a driver, since doing so effectively decreases the order arrival rate for couriers and would therefore result in a decrease in the number of couriers, say a decrease of e > 0 couriers. This implies that to achieve the target service quality, e more drivers need to be employed. Since drivers are more expensive than couriers, this shows that deliveries that are acceptable to couriers should not be offered to drivers; drivers should only be offered those deliveries that are rejected by couriers. Consequently, the order arrival process can be divided into two distinct arrival processes, each served separately by couriers and by drivers. Before proceeding with our analyses, we need to clarify the operational setting we have in mind and what we imply when we say a delivery offer is rejected by a courier. When an order arrives, it is offered to a courier. If the courier rejects, the order can, of course, be offered to another courier, and so on, before offering/assigning it to a driver. A possible decision rule is to offer an order n times to a courier before offering/assigning it to a driver. In this case, the order acceptance probability is the probability that at least one of the n couriers accepts the delivery offer. Note that in practice, the response to a delivery offer is not instantaneous (in fact, it may take a non-negligible amount of time). As a result, to ensure the target service quality, large values of n may be unacceptable. Moreover, it is likely that there is high correlation between courier order acceptance/rejection decisions, which also suggests that smaller values of n are more appropriate. Regardless of these considerations, and to simplify the exposition, we will assume n = 1. We start by investigating the case in which all couriers accept a delivery offer with probability > 0 . If one considers the various characteristics of an order, e.g., distance from current location to restaurant, distance from restaurant to delivery location, delivery location, time of day, and expected tip amount, and assumes a heterogeneous utility function for its evaluation, the net effect is an offer rejection probability as considered here. We modify formulation P, i.e., (3) and (4), to incorporate the probability of delivery offer rejection as follows:
P ( ): max c (r )
p (r )
r , p, m ¯ 0
s.t.
(r ) (p) =
(1 m ¯ = ¯
pm ¯¯
(12)
(r ) µ (r )
(13)
) (r ) µ (r )
(14)
The objective is to maximize profit, i.e., the revenue minus payments for deliveries to couriers and payments for deliveries to drivers. As before, the per-delivery payment p for couriers has to be chosen such that the number of couriers is sufficient to achieve 187
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the target service quality, which is ensured by Constraint (13). Constraint (14) ensures that a sufficient number of drivers is available to perform deliveries of orders rejected by couriers. We can replace p with 1 ( b / µ (r )) to obtain the unconstrained optimization problem: 1
P ( ): max c (r ) r [0, R]
b µ (r )
p¯ ¯
(r )
(1
) (r ) µ (r )
(15)
Proposition 3. Given unit-revenue c, speed v, courier waiting-rate b, courier acceptance-probability , and service-quality-aspiration , the optimal radius r of the service area is
cv 2( b + ¯p¯ (1
))
.
(16)
Proposition 3 reveals that the effect of the possibility of delivery offer rejection is a modification of the term b; it is replaced with ) . This makes sense, as we can consider b as the unit-time cost of enticing a courier and this cost the weighted sum b + ¯p¯ (1 ) of the orders. Clearly, as the acceptance-probability decreases and the unit-time cost of cannot be applied to a fraction (1 drivers p¯ increases, the radius of the optimal service area decreases. Using the diner density functions (6), we can derive the associated profit and courier compensation as follows
(r ) =
c (k + 2)
c
(
cv 2R ( b + ¯p¯ (1
))
)
k+1
2 R (k + 1) v ( b + ¯p¯ (1 ))(k + 2)
, if r < R if r = R,
(17)
and
p (r ) =
c b (k + 1) , ( b + ¯p¯ (1 ))(k + 2)
if r < R
2R b (k + 1) v (k + 2)
if r = R.
(18)
To better assess the effect acceptance-probabilities, we analyze the results of a numerical experiment conducted with diner density functions (6). In Fig. 8, we illustrate how the profit changes for different courier offer acceptance-probabilities. We compare the profit when couriers always accept delivery offers ( = 1) to lower acceptance-probabilities = 0.9 and 0.8. In these experiments, we use = ¯ = 1.075 and assume that the unit-time cost of employing a driver is twice as expensive as courier; specifically, we use b = 1/3 and p¯ = 2/3. Fig. 8 shows that the negative impact of delivery offer rejections is more pronounced when the unit-revenue is higher. The reason is that a higher unit-revenue increases the radius of the optimal service area, which, in turn, implies that the number of “expensive” drivers needed increases and the profit decreases. We also investigate the benefits of employing drivers to provide service reliability. Fig. 9 shows the profit for three settings: 100% delivery offer acceptance, 90%, 80%, and 70% delivery offer acceptance without drivers to perform the rejected deliveries, and 90%, 80%, and 70% delivery offer acceptance with drivers to perform the rejected deliveries. For simplicity, there is no penalty for not serving an order in the settings in which couriers can reject delivery offers and there are no drivers. Fig. 9 reveals an interesting insight about the use of company-employed drivers. For all settings, we see that if diners are located relatively close to the restaurant, using drivers to recover a portion of the lost revenue due to delivery offer rejections is worthwhile and increases profit. However, if diner locations tend to be further away from the restaurant using drivers may actually reduce profit
Fig. 8. Impact of delivery off acceptance-probability on profit. 188
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Fig. 9. Benefit of employing drivers to perform deliveries of orders rejected by couriers.
(this, of course, ignores any long-term effects of not serving placed orders). Not surprisingly, the benefit of employing drivers gets larger when the courier acceptance-probability is lower and when the unit-revenue increases. Note that in this case, the ratio between ) due to the fact that regardless of the service area, couriers are the number of couriers and drivers is fixed and equal to / ¯ (1 expected, on average, to serve percent of the orders while drivers need to serve the remaining 1 percent of the orders and the mean delivery-times are the same for the orders served by couriers and drivers. As we show next, when the courier order acceptance probabilities have more complex forms, calculating the ratio between the number of couriers and drivers becomes more involved. For obvious reasons, the delivery distance is likely to be a significant factor in a courier’s accept-reject decision, and deserves special attention. Therefore, we extend our analysis by considering the case where the delivery offer acceptance-probability for couriers is a function of the distance from the restaurant to the order drop-off location. More formally, we assume that the probability of accepting a delivery offer for a drop-off location that is x 0 units away from the restaurant is given as (x ) [0, 1], where is a nondecreasing function of the distance x. Note that in this case the expected delivery-time for the orders that are accepted by the couriers can be different (possibly smaller) than the expected delivery time of orders accepted by drivers. r We define the arrival rate of orders that are to be delivered by couriers as 1 (r ) = G1 (r ) , where G1 (r ) = 0 (x ) g (x ) dx . Similarly, we define the arrival rate of orders to be delivered by drivers as delivery-time for couriers and drivers can be calculated as µ1 = the profit maximization problem as
P ( ): max c (r ) r , p, m ¯ 0
s.t. 1 (r ) (p) =
p 1 (r )
2 (r ) = G1 (r )
G2 (r ) , where G2 (r ) =
r 2xg (x ) (x ) dx 0 v
pm ¯¯
and µ 2 =
G 2 (r )
r 2xg (x )(1 0 v
r
0 )(x )
(1
dx
(x )) g (x ) dx . The average
, respectively. We can rewrite
(19)
1 (r ) µ1 (r )
(20) 189
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¯ = ¯ m
2 (r ) . µ 2 (r )
(21)
As before, we restate this optimization problem as an unconstrained one, i.e., 1
P ( ): max c (r )
µ1 (r )
r 0
1 (r )
p¯
¯ 2 (r ) , µ 2 (r )
(22)
and use it to derive the following proposition. Proposition 4. Given unit-revenue c, speed v, courier waiting-rate b, courier acceptance-probability function , and service-quality-aspiration [0, R]): , the radius r of the service area that maximizes the profit is the solution to the following equation (for r
r ( b (r ) + ¯p¯ (1
(r ))) =
cv 2
(23)
One interesting acceptance-probability function to examine is the one in which there is some threshold radius r up to which a courier always accepts a delivery offer, and after which a courier always rejects a delivery offer (which implies that the delivery will be performed by a driver), i.e.,
1, if x 0, o.w.
(x ) =
r,
(24)
In this setting, the radius of the service area maximizing profit is
r¯ =
min r ,
cv 2 b
cv ¯¯ 2Bp
, if
cv ¯¯ , 2Bp
r,
o.w.
(25)
and the number of drivers required is
m ¯ =
0, ¯
if r¯ r¯ r
2ax g (x ) dx , v
r,
o.w.
(26)
Note that this case can also represent the situation in which the couriers are homogeneous in their valuations of an offer (they either all consider an offer worthwhile to accept or not). Furthermore, the above result can be easily extended situations in which certain (types of orders) are unacceptable to (most of the) couriers and have to be performed drivers. To further investigate the impact of distance-dependent delivery offer acceptance by couriers, we conduct numerical experiments using the diner density functions (6) and offer acceptance-probability functions: t
(x ) = 1
x , t R
0.
(27)
Note that as parameters t and grow, the delivery offer rejection probability diminishes. In Fig. 10, we compare the profit for the situations in which couriers always accept delivery offers and in which couriers reject delivery offers depending on their distance from the restaurant according to the above probability function with = 5 and t {0.5, 1.0, 2.0} . Again, Fig. 10 shows that the effect
Fig. 10. Profit with distance-dependent delivery offer rejection probabilities. 190
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Fig. 11. Benefit of employing drivers to perform deliveries of orders rejected by couriers (distance-dependent orffer acceptance-probabilities).
of offer rejections on profit is becomes more pronounced when the unit-revenue (and hence the service area) increases. Nor surprisingly, as order drop-off locations tend to be further away from the restaurant, distance-dependent delivery offer rejection has a higher impact. We also see that the profit is sensitive to the parameter t in the acceptance-probability function and that profit drops significantly for t < 1. Next, we repeat the analysis where investigate the impact of employing drivers to provide service reliability on profit for three settings: 100% delivery offer acceptance, delivery offer acceptance without drivers to perform the rejected deliveries, and delivery offer acceptance according with drivers to perform the rejected deliveries, but now with distance-dependent order acceptanceprobabilities for couriers. (Again, there is no penalty for not serving an order in the settings in which couriers can reject delivery offers and there are no drivers.) The results, shown in Fig. 11, demonstrate, as before, that the difference between the three settings becomes less pronounced as the unit-revenue, average distance from restaurant to drop-off locations, and offer acceptance-probability decreases (t parameter increases). Interestingly, the benefit of employing drivers to perform deliveries of orders rejected by couriers is less when the rejection probability depends on the distance of the drop-off location from the restaurant. The reason is that few short-distance delivery offers are rejected, and we saw earlier that those are exactly the ones for which it is beneficial to have drivers as a backup option. Using Proposition 4 to find the radius r of the optimal service area and considering the acceptance-probability functions (27), we next examine the composition of the delivery capacity (i.e., couriers versus drivers). For different diner distributions, Fig. 12 shows
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Fig. 12. Percentage of couriers in the total delivery capacity.
the percentage of couriers for a given acceptance-probability function (i.e., for a given value of t). In each graph, the courier percentages are shown for unit-revenue values c {8, 16, 24} . We see that the courier percentage drops when diner locations get further from the restaurant and when courier order rejection probabilities increase (small t values). Furthermore, we see that when unitrevenue increases and, as a consequence, the optimal service area gets larger and more drivers are needed, the percentage of couriers drops. Finally, the graphs demonstrate that when acceptance-probability is high (large t values) and the overwhelming majority of the delivery capacity consists of couriers, the delivery capacity composition becomes less sensitive to changes in the acceptance probability. 3.2. Multiple service areas and multiple restaurants Our analyses up to now have assumed a single service area for a single restaurant. To enhance the practical value of our analyses and insights, in this section, we consider two extensions: multiple service areas and multiple restaurants. First, we consider a single restaurant with the option to deliver meals to a finite set of service areas, N. If it is decided to deliver meals in service area i N , then the order arrival rate is ai , and the average delivery-time is 1/ µi . Next, we define a binary decision variable x i , i N to indicate whether to deliver meals in service area i ( x i = 1) or not ( x i = 0 ). Thus, vector x B N , whose ith entry corresponds to the value of binary variable x i , characterizes a possible service coverage decision. For a given service coverage decision x B N , the resulting order arrival rate is denoted by (x) and the resulting mean delivery-time is denoted by 1/ µ (x) . We define the following integer programming model to find optimal service coverage:
IP: max c
p
x BN
s.t.
(x) (p)
(x)
(28)
(x) . µ (x)
(29)
As before, the objective is to maximize profit, and Constraint (29) ensures that sufficient delivery capacity is available. Replacing (x) with i N ai xi and the mean delivery-time 1/ µ (x) with i N ai xi / µi / i N ai xi , we can rewrite IP as:
max
c
x BN
p ai x i
(30)
i N
s.t.
(p) i N
µi
ai x i
0
(31)
= { i N : (p ) / µi 0} be the set of service areas for which the mean delivery-time multiplied by is less than or equal to Let / µi ) a i . the courier’s willingness-to-wait time, and let N = N N+ and = i N + ( (p) It is clear that meals should be delivered to service areas i N+, because (c p) ai 0, i N , and setting x i = 1 increases the value of the left hand side of Constraint (31). Thus, IP can be transformed into the following knapsack problem: N+
KN : max
x BN
s.t. i N
c
p ai x i
(32)
i N
µi
(p ) a i x i
.
(33)
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Fig. 13. Service area selection examples.
Let x¯ be the optimal solution to KN, then we have that x , with
xi =
{
1if i N+ x¯i o.w.,
(34)
is an optimal solution to IP. Note that this assumes no special form for the courier’s willingness-to-wait function , and so this result can trivially be extended to situations in which each service area i N has its own unit-revenue ci and per-delivery payment pi . We observe that the optimal service coverage expands as the per-delivery payment p increases (assuming is a nondecreasing function of p) and shrinks as the service-quality-aspiration and the mean delivery-times increase. Note too that this result reinforces the earlier observation regarding the benefit of having a large diner base close to restaurant. The diner base close to the restaurant is captured in , and a large value of allows expanding coverage to accommodate diners in service areas further away from the restaurant. Consider, for example, the two settings illustrated in Fig. 13a and b. The diamond shape in the figures indicates the location of the restaurant and the circles represent the locations of the four candidate service areas
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Fig. 14. Profit increase as a function of mean delivery-time reduction.
(the same in the two settings). The delivery-times (in minutes) 1/ µi , i {1, 2, 3, 4} and order arrival rates i, i {1, 2, 3, 4} are shown below each service area; the size of the circles are proportional to the order arrival rates. We assume c = $12, p = $10, (p) = 22 minutes, and = 1.1. The first setting represents a situation in which there is a large diner base close to the restaurant, and the second setting illustrates a situation where this is not the case. (Note that in the second setting, the order arrival rate in service area 1 is decreased from 4 to 1 and the order arrival rate in service area 2 is increased from 1 to 4.) In the first setting, it is optimal to deliver meals in service areas 1, 2, and 4, whereas in the second setting, it is optimal to deliver meals only to service areas 1 and 2. This example clearly shows that having a large diner base close to restaurant enables a meal delivery platform to expand service to other areas. It is also interesting to observe in the first setting that even though service area 3 is closer to the restaurant, compared to the service area 4, it is optimal to deliver meals in service area 4 rather than service area 3 because it has a larger order arrival rate. In some practical settings, where home deliveries from a single store or single distribution center are considered, our analyses involving a single restaurant may assist in assessing the advantages and disadvantage of crowd-sourced delivery. On the other hand, in many environments, e.g., our motivating on-demand meal delivery platform, there are multiple restaurants serving diners. Fortunately, our analyses do not depend strongly on the assumption that there is a single restaurant. To extend our results to a multirestaurant setting, all that is required is to redefine r , (r ) , and µ (r ) as follows:
• r: the maximum direct driving time for orders that are accepted by the online platform (from the restaurant to the delivery location); • (r ): the arrival rate of orders that are accepted by the online platform; and • µ (r ) the mean delivery-time of orders that are accepted by the online platform. Note that in a multi-restaurant setting, the delivery time not only depends on the restaurant where the order needs to be picked up and the location where the meal needs to be dropped off, but also on the (quality of the) routing decisions. In the single-restaurant setting, a courier always returns to the same restaurant for his next order pickup. In a multi-restaurant setting, a courier may go to a different restaurant for his next order pickup. Clearly, dispatching decisions, i.e., deciding where to send couriers after they complete a delivery, impacts the mean delivery-time. (More complex settings can be envisioned, e.g., in which a courier picks up meals from different restaurants before delivering these meals to diner locations, but in practice it is most common to have couriers serve single orders.) The mean delivery-time depends on the environment (the number and location of restaurants, the order arrival patterns, the dispatching technology, the street network, the traffic pattern, etc.), but as long as a reasonably accurate estimate of mean deliverytime is available, our analyses for the single-restaurant setting can be used to answer various strategic and operational questions about a multi-restaurant environment. On-demand meal delivery platforms may, for example, consider investing resources to improve their dispatching technology in order to reduce and mean delivery-time. Or, they may consider employing demand management strategies encouraging diners to place orders at certain restaurants to reduce mean delivery-time. In the remainder of this section, we analyze the effects of reductions in mean delivery-time considering both continuous and discrete diner density distributions (i.e., single delivery area and multiple delivery areas). First, we consider the case with multiple restaurants serving a single delivery area. Fig. 14 shows the relation between profit and mean delivery-time, where we have assumed that couriers receive a per-mile payment = 0.1 in addition to a per-deliver
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Fig. 15. The increase in profit as a function of the reduction in mean delivery-time.
compensation and unit-revenue c = $8 (the results are similar for other per-mile payments and unit-revenue values). Note that a reduction in mean delivery-time is akin to an increase in vehicle speed, and, therefore, we model an percent reduction in the mean delivery-time as an percent increase in vehicle speed. We see that modest reductions in mean delivery-time can produce significant gains in profit, especially when the diner locations are further from the restaurant, e.g., a 5% reduction in mean delivery-time results in an increase in profit of more than 20% (diner density function with k = 3). Next, we consider the case with multiple restaurants serving multiple delivery areas. Fig. 15 shows the relation between profit and mean delivery-time. In the considered setting, there are 20 candidate service areas with different order arrival rates and mean delivery-times. The order arrival rates are drawn from a uniform random distribution between 0 and 1 orders per minute and the mean delivery-times is drawn from a uniform distribution between 15 and 25 min (details can be found in Appendix C). We assume = 1.1, (p) = 20 , and c > p, which means that profit is maximized by serving as many diners as possible without violating the target service quality. Fig. 15 shows that as the mean delivery-times decrease, meals are delivered to more service areas and, hence, the profit increases (significantly and rapidly) until meals are delivered to all service areas (which occurs when mean delivery-time is reduced by 12%). We observe that there are points where a small reduction in mean delivery-times results in a big jump in profit. This happens when a small reduction in mean delivery-times means that it has become feasible to add another service area or that it has become feasible to replace a small service area with a larger one. Both examples indicate the benefits of reducing the mean delivery-time, for example by more effective dispatching technology or demand management. 4. Discussion We presented a framework to study service coverage and capacity planning problems in the context of crowd-sourced mealdelivery. This setting represents a three-sided market in which providers with complementary capabilities (restaurants and couriers) offer a service to customers (diners) that interact with the providers through an online platform. Although meal delivery is a natural fit for this three-sided market framework, there are many other settings in which providers with complementary capabilities offer (or can offer) services to customers. The same business model can be employed to take advantage of opportunities offered by the shared economy and advancing communication technologies. As such, we believe that the results of our study can be applicable to a much wider set of businesses than on-demand meal delivery platforms. On-demand meal delivery platforms highlight the novel challenges associated with managing crowd-sourced delivery capacity which self-schedules and self-selects the delivery orders to perform. We propose a modeling approach that, for the first time, captures (i) service coverage decisions, (ii) self-selecting behavior of couriers, and (iii) hybrid delivery capacity (composed of crowd-sourced couriers and company-employed drivers). Deciding the service coverage and service quality is critical to the success and profitability of an on-demand meal delivery platform. There are many situations in which the results and insights generated in this research may guide and support decisions. A
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prime example is entering a new market, i.e., providing on-demand meal delivery services in a new city. Decisions have to be made regarding the restaurants that will be part of the network, regarding the areas served by these restaurants, regarding the courier compensation scheme, to name just a few. But our results and insights may also guide and support decisions in an established market, for example in helping the sales team identify restaurants to target for inclusion in the network, or in helping the operations team to adjust the service area of restaurants in the network that are less profitable, or in helping the operations team assess the potential benefits of incentive schemes to reduce delivery offer rejection rates of couriers. The results and insights generated in this research may also indirectly guide and support decisions. A possible demand management scheme that might be considered is to dynamically adjust the radius of the service area of a restaurant to respond to changes in order arrival rates or delivery capacity (planned or unplanned). A better understanding of how the size of a service area relates to profit and service quality, and how the distribution of diners in the service area affects these relations, may help design and implement an effective version of such a demand management scheme. However, our results and insights extend beyond on-demand meal delivery platforms. Planning crowd-sourced delivery capacity is fundamentally different from the traditional delivery capacity planning. In the latter, the focus is on determining an optimal fleet size and mix and maximizing the utilization of the vehicles in the fleet. In the former, the focus is on ensuring a sufficient number of couriers in the system at any time and ensuring a high courier order acceptance probability. Our research provides one possible framework and some initial insights into effective planning of crowd-sourced delivery capacity and related issues concerning compensation of crowd-sourced delivery capacity and ensuring service quality when relying on crowd-sourced delivery capacity. Our analyses have provided insights into the interaction between courier satisfaction (i.e., couriers’ willingness-to-wait), courier compliance (i.e., couriers’ offer acceptance-probability) and profit. Our analyses show that couriers’ earning expectations play a critical role in the optimal service area and profit. Of course, couriers make their decisions based on their perceptions about the current and future system state, which are likely inaccurate. The meal-delivery platform provider, on the other hand, knows the current system state and employs (or can employ) technology to predict the future system state, and it is therefore an interesting question whether sharing some or all of this information with the couriers can be beneficial. Our analyses also show that the self-scheduling and self-selecting behavior of couriers has a significant impact on optimal service area and profit. Our numerical experiments suggest that hybrid delivery capacity (i.e., couriers and drivers) can both improve reliability as well as profit. However, the use of our model is not limited to investigating delivery capacity composition questions. It can be also used to evaluate and compare various strategies to improve couriers compliance. For example, offering extra compensation for orders that are more likely to be rejected by couriers. Our models may help to determine how large the extra compensation should be to achieve certain goals. Acknowledgment This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant No. 2219. Appendix A. Mathematical proofs A.1. Proof of Proposition 1
µ
1
Note that with the given customer density function the expected delivery-time for a given service radius r can be calculated r 2x 2G¯ (r ) 1 2r 1 = G (r ) 0 v g (x ) dx = v . Using (p) = b p , profit function (5) can be rewritten as follows. G (r ) v
(r ) = c
b
(r ) = c G (r ) Then setting
d dr
2G¯ (r ) G (r ) v
2r v
G (r )
(35)
2 b G¯ (r )
2 b rG (r )
(36)
v
(r ) = 0 we get:
d (r ) = c g (r ) dr
2ab rg (r ) + 2 b G (r ) v
2 b G (r )
=0
(37)
cv 2b
as desired. Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (37). Hence, the optimal solution which yields r = is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.2. Proof of Proposition 2 Let g , G, G ,
(r ) = c
and µ be defined as in Section 3.1. Since we assume (p) = b p , substituting p = 1
2r b v
2 bG¯ (r ) G (r ) v
2r v
2G¯ (r ) G (r ) v
G (r ),
1
( ) in (9) gives: µ (r )
(38)
which can be simplified as: 196
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2 b rG (r )
(r ) = c G (r ) Then setting
d dr
bG¯ (r )
2
2
2 G¯ (r )
rG (r )
v
v
.
(39)
(r ) = 0 we get:
2
d (r ) = c g (r ) dr
brg (r ) + 2
bG (r ) v
2
bG (r )
2
rg (r ) + 2
G (r ) v
2
G (r )
=0
(40)
cv . 2( b + )
which yields r = Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (40). Hence, the optimal solution is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.3. Proof of Proposition 3 Let g , G, G , and µ be defined as in Sectionsec:basic. Taking the derivative of objective function 1 (p) = b p we get: d dr
(r ) = cg (r ) r 2xg (x )
g (r ) 0 v G (r )
b
g (r )
b G (r ) dx
2 ¯pr ¯ (1 v
r 2xg (x ) dx 0 v G (r ) 2
(r ) with respect to r and using
2rg (r ) vG (r )
+
) g (r )
(41)
That simplifies to:
d (r ) = dr
g (r )( 2b r + cv + 2 2 p2 r (
1))
v
.
(42)
Then setting (42) equal to zero, we get the desired result:
r =
cv 2( b + ¯p¯ (1
(43)
))
Since we assume g (x ) > 0 , for all x > 0, r is the unique solution of (43). Hence, the optimal solution is either in the boundaries (0 or R) or r is the optimal solution. We assume the latter is the case. A.4. Proof of Proposition 4 Taking the derivative of objective function
d (r ) = dr Since we have
g (r )(cv + 2 ¯pr ¯ ( (r ) v
1)
1
(r ) with respect to r and using (p) = b p we get:
2 br (r ))
(44)
> 0 and g (r ) > 0 for all r > 0 this equation simplifies to
r ( b (r ) + ¯p¯ (1
(r ))) =
cv 2
(45)
as desired. Appendix B. RtP approximation Let EW (M /M / m) be the expected waiting times of customers in a (M /M / m) queue, where m is the number of servers (drivers) in the system. The mean RtP times can be modeled as the mean waiting times in a M/G/m queue with poisson arrivals and general service time distributions. From the queuing theory we have: 1
EW M / M /m =
() m ! mµ (1
m
m 1
µ
mµ
)
2
j=0
() µ
j!
j
+
()
m
µ
m! 1
mµ
(46)
We consider the following approximation of EW (M /G / m) proposed by Gans et al. (2003).
EW M / G/ m = where,
C2
(1 + C 2) EW (M / M /m ) 2
(47)
is the coefficient of variation of the service time distribution.
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Appendix C. Problem data for multiple service regions See Table 2. Table 2 Base case service times and order rates. Region ID (i ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Mean service time (1/µi )
Order rate ( i)
18 23 19 20 18 23 20 22 16 20 17 25 16 21 25 19 25 24 21 23
1.15057 1.53001 0.13265 0.06455 0.71748 1.39642 1.30933 0.12539 0.12718 1.53665 1.32674 0.04354 1.55416 1.61243 0.17639 1.13934 1.11323 0.38830 0.47310 1.53714
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