courier dispatch in on-demand delivery
TRANSCRIPT
![Page 1: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/1.jpg)
Courier Dispatch in On-Demand Delivery
Mingliu ChenThe Department of Industrial Engineering and Operations Research, Columbia University, [email protected]
Ming HuRotman School of Management, University of Toronto, [email protected]
We study a courier dispatching problem in an on-demand delivery system where customers are sensitive
to delay. Specifically, we evaluate the effect of temporal pooling by comparing systems using the dedicated
strategy, where only one order is delivered per trip, vs. the pooling strategy, where a batch of consecutive
orders is delivered on each trip. We capture the courier delivery system’s spatial dimension by assuming
that following a Poisson process, demand arises at a uniformly generated point within a service region, as
a generalization of the circular city model. With the same objective of revenue maximization, we find that
the dispatching strategy depends critically on customers’ patience level, the size of the service region, and
whether the firm can endogenize the demand. We obtain concise but informative results when there is a
single courier and customers’ underlying arrival rate is large enough, meaning a crowded market such as rush
hour delivery. In particular, when the firm has a growth target and needs to achieve an exogenously given
demand rate, using the pooling strategy is optimal if the service area is large enough to fully exploit the
pooling efficiency. Otherwise, using the dedicated strategy is optimal. In contrast, if the firm can endogenize
the demand rate by varying the delivery price, using the dedicated strategy is optimal for a large service
area, and vice versa. The reason is that it is optimal for the firm to sustain a relatively low demand rate by
charging a high price for a large service radius: within this large area, the pooling strategy would lead to a
long wait time because the multiple orders required for pooling accumulate slowly. Moreover, under market
penetration with exogenous demand, customers’ patience level has no impact on the dispatch strategy, but
when the demand rate can be endogenized, the dedicated strategy is preferable if customers are impatient,
and vice versa. Furthermore, we extend our model to account for social welfare maximization, a hybrid
delivery policy, a general arrival rate that does not have to be large, a non-uniform distribution of orders in the
service region, and multiple couriers. We also conduct numerical analysis and simulations to complement our
main results and find that most observations in our base model still hold in these extensions and numerical
studies.
1. Introduction
On-demand delivery of food and groceries has gained traction nowadays. Given the prevalence of
smart devices and the existence of a flexible labor force of independent contractors, many food and
grocery stores have started on-demand delivery for relatively small orders. For example, Starbucks
plans to expand their coffee delivery services across the United States and has already established
delivery services in China in 30 cities and more than 2000 stores (Jargon 2018). Unlike traditional
package delivery services, coffee delivery involves spontaneous orders for small quantities. Typically,
customers who order consumables like coffee do not order in advance and expect the coffee to still
be hot on arrival. A customer may choose not to order if the expected delivery time is too long.
1
![Page 2: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/2.jpg)
2 Chen, Hu: Courier Dispatch in On-Demand Delivery
In the domain of hyper-fast (or so-called instant) delivery, delivery companies offer a wait time
expectation coupled with a price tag, e.g., 10-minute grocery delivery for $2 by Gorillas and 30-
minute grocery and food delivery for $1.95 by Gopuff with additional markups on product prices.
To meet such a promise of rapid delivery, companies such as Gorillas and Gopuff employ and
staff couriers who are dedicated to multi-hourly shifts, fulfilling orders from “dark” warehouses or
micro-fulfillment centers. The Covid-19 pandemic has solidified this trend. Many more vendors are
hiring dedicated couriers to deliver their goods. According to Rana and Haddon (2021b), about a
half of 150 registered restaurant on Spread, a start-up delivery platform hires dedicated drivers to
their own deliveries. Furthermore, in order to compete with large platforms, they set much lower
delivery prices to carve out a market share. This trend is particularly significant in pizza delivery.
Melton (2021) reports that Domino’s has established market penetration by using dedicated drivers
and offering cheaper than market price pies. During the pandemic, Domino’s market share has
increased by 31%.
Since on-demand deliveries are by nature sensitive to delay, many delivery systems dispatch a
courier whenever an order arrives. Thus, the couriers can serve only one order per trip, in the hope
of reducing delivery time for each customer. The empirical analysis of Mao et al. (2019) shows
that delivery delay significantly reduces future orders. However, there are still many occasions
on which a firm can utilize batch delivery if multiple orders are placed around the same time in
the same area. That is, a courier may deliver multiple orders per trip; we refer to this as the
temporal pooling strategy. In this strategy, a courier is not necessarily dispatched as soon as an
order arrives; orders are allowed to accumulate over time and then a batch of sequential orders
is delivered in one trip. We show that this strategy achieves delivery efficiency in the form of
a shorter expected travel distance per order and lower variability in traveling distance per trip.
However, while this pooling strategy benefits the supply side, it undoubtedly affects customers’
service experiences on the demand side, which may deter customers from using the service or may
require the firm to compensate customers monetarily for the longer wait and hence reduce the
strategy’s attractiveness. Therefore, each delivery strategy has its own advantages: the dedicated
delivery system may mean a shorter wait for each customer, while batch delivery appears more
efficient from the firm’s perspective.
The on-demand courier dispatch problem differs from traditional delivery problems (such as the
celebrated Traveling Salesman Problem) where there are many stops per trip. Orders containing
on-demand supplies (such as coffee, food, and medicine) typically have short delivery windows.
According to Rana and Kang (2021), food delivery platforms such as DoorDash and Uber are
researching on bundling orders together but unlike traditional delivery services, they also plan to
deliver all orders in an hour. Thus, on-demand delivery services cannot deliver with large batch
![Page 3: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/3.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 3
sizes consistently. In particular, according to an internal study conducted by one of the largest
delivery platforms in China, for food delivery their couriers carry less than two orders on average
per trip, even during peak lunch and dinner hours (see Figure 1).
(a) (b)
Figure 1 The distribution of orders per courier during peak (a) lunch and (b) dinner hours
Another important factor in the operations of delivery systems is whether the actual demand
is exogenous or can be endogenized through pricing. On one hand, an emerging delivery platform
needs to maintain growth and carve out its market share by sustaining a certain demand rate,
also known as market penetration (Rana and Haddon 2021a).1 Studies on market penetration can
be traced back to Buzzell et al. (1975), followed by empirical evidence (see, e.g., Szymanski et al.
1993), stating there is a positive correlation between the market share and (long-term) profitability.
Thus, for a vendor in its early stage of operations under market penetration, the demand can be
exogenously determined to achieve certain market share. On the other hand, a delivery platform,
who has already established a stable market base, can endogenize the demand by varying prices or
fees to further optimize its revenue.
In this paper, we take the perspective of a vendor providing delivery service and attempt to
address the following research questions: when is temporal pooling beneficial and when should
a courier be dedicated to one order per trip? More specifically, we consider scenarios where the
delivery system with dedicated couriers has exogenous and endogenous demand, respectively, and
identify the key factors affecting its operating strategy. In either scenario, we use the vendor’s
revenue as the performance measure. For simplicity, we refer to the delivery strategy with temporal
pooling as the batch or pooling strategy and the one serving a single order per trip as the dedicated
strategy. In the exogenous demand case, depending on the expected wait time associated with each
strategy, the vendor sets the price to achieve the targeted demand rate. In the endogenous demand
case, the vendor has the full freedom in varying the price to moderate the demand rate.
We build a stylized model capturing the spatial aspect of delivery systems under different dis-
patch strategies. Following a Poisson process, demand arises at a uniformly distributed point in a
1 See also https://gadallon.substack.com/p/premature-scaling-will-gorillas-go.
![Page 4: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/4.jpg)
4 Chen, Hu: Courier Dispatch in On-Demand Delivery
Exogenousdemand
Endogenousdemand
Small area Dedicated BatchLarge area Batch Dedicated
Patient customers — BatchImpatient customers — Dedicated
Table 1 Optimal delivery strategy according to nature of demand
service region. By using a disk-shaped service area and recognizing the similarities between delivery
and queueing systems, we obtain concise but informative analytical results. We find that whether
the demand is endogenized critically affects the vendor’s optimal dispatch strategy. In our base
model, we assume there is a single courier for dispatch (which is relaxed in an extension). We first
analyze a large market where customers’ potential arrival rate is large (which is relaxed in another
extension). We show that in such a crowded market, if the demand rate is exogenously given as
under market penetration, there is a threshold size for the service area below which it is optimal to
use the dedicated strategy and above which it is optimal to use the pooling strategy. We find that
whichever strategy that produces a shorter expected wait time for the exogenously given fraction
of customers is optimal for the vendor. Thus, customers’ patience level does not directly impact
the decision on the delivery strategy because it does not affect the length of wait time itself.
Results are very different if the firm can endogenize the demand rate as under revenue max-
imization. With endogenized demand, there is a threshold size for the service area below which
it is optimal for the firm to deliver in batches and above which it is optimal to adopt dedicated
delivery. This result is in stark contrast to the one for exogenous demand, and runs counter to
popular belief that serving in batches leads to higher delivery efficiency in a large service area than
dedicated delivery (which is likely gained under the assumption that the demand rate is exoge-
nously given). The intuition of our finding is that, in a relatively large service area, both strategies
involve substantial travel distances, leading to long wait times. By maintaining a high demand
rate, the firm needs to sacrifice a lot of profit margin to ensure customers joining the service. As a
result, the firm favors a relatively small endogenized demand rate for both strategies. The pooling
strategy loses its efficiency edge in this case since it takes a long time to accumulate multiple orders
with a low demand rate. The dedicated strategy is more efficient since its optimal demand rate is
lower than the one under the pooling strategy. Furthermore, we also find that there is a threshold
on customers’ patience level below which the pooling strategy is optimal and above which the
dedicated strategy is optimal. We summarize these results in Table 1.
We then examine a variety of extensions of the base model. First, we consider social welfare
maximization as the vendor’s objective. We find that all the insights in our base model carry over.
![Page 5: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/5.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 5
Second, based on the dedicated and batch strategies in the base model, we consider a contingent
hybrid policy. To be more specific, the courier uses batch strategy as long as there are more than
one outstanding orders and use dedicated strategy otherwise. We show that this hybrid policy
dominates the dedicated strategy. However, the trade-off between the dedicated and batch strategies
still persists between the contingent and batch strategies.
Third, we relax the large-market assumption by studying general customer arrival rates. We find
that all results for an exogenous demand rate still hold. When the firm can endogenize the demand,
there are still thresholds on the size of the service area and customers’ patience level above which
it is optimal to use the dedicated strategy.
Fourth, we relax the batch-of-two assumption in our base model. In our numerical calculations,
we find that our main insights in the base model still hold. However, with a larger batch size, there
are other issues that need to be addressed, which we leave for future research.
Fifth, we consider an extension in which the demand is not uniformly dispersed inside the service
area. Specifically, we consider a circular city structure in which all demands are distributed only on
the edge of the disk. Not only do we confirm that all results in our base model still hold, but also we
compare the thresholds to those in the base model. We find that the thresholds for both the service
radius and customers’ patience level, above which it is optimal to use the dedicated strategy and
vice versa, are lower in this setting, compared with those in the base model. Under this setting,
the pooling strategy is at a disadvantage since the courier has to travel a longer distance to the
edge of the service region before exploiting the efficiency created by pooling.
Finally, we generalize the base model to allow for multiple couriers. Again, we obtain analytical
results and numerical observations consistent with the base model of a single courier.
2. Literature Review
Two of the papers most closely related to ours are Cao et al. (2020a) and Yildiz and Savelsbergh
(2019). Cao et al. (2020a) study the optimal deployment strategy for vendors with high mobility,
often referred to as the stall economy. Although their main focus is on using the combination
of the analytical model and machine learning algorithms to explain the scalability of the stall
economy, the authors also empirically evaluate the benefit of demand pooling. They divide the
service area into several subregions and consider demand pooling that serves orders arriving within
the same time window in the same subregion together, before moving to the next subregion. In
their empirical study, they find that such demand pooling is more beneficial when customers are
patient, which is consistent with our analytical results under the endogenous demand rate. Yildiz
and Savelsbergh (2019) also consider a circular delivery area similar to that in our model, where
a single restaurant located at the center of the disk serves the entire disk-shaped area. They only
![Page 6: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/6.jpg)
6 Chen, Hu: Courier Dispatch in On-Demand Delivery
consider the dedicated strategy. Their focus is on the optimal service radius and compensation for
crowdsourced couriers, whereas ours is on evaluating the benefit of temporal demand pooling.
Our paper belongs to the stream of research on spatial queueing models. This literature typically
considers a logistical setting in which vehicles are modeled as servers and their traveling time to
serve customers equals the service time. Berman et al. (1985) and Berman et al. (1987) focus on
finding one or multiple service hubs in a network to minimize the expected response time to random
demand. They model the service system using queueing models incorporating the spatial features
of the network. Bertsimas and Van Ryzin (1990, 1992) consider stochastic and dynamic routing of
vehicles to serve service requests that are randomly generated over a service region. The authors
evaluate the performances of various policies and identify optimal and near-optimal policies under
light and heavy traffic. Recently, spatial queueing models are also utilized in smart city design
(see, e.g., He et al. 2017, Mak 2020). In contrast to these papers, we focus on the comparison of
the dedicated and the pooling strategy and also incorporate the pricing decision to examine the
interactions between the demand side’s pricing decision and the supply side’s dispatch decision.
Our paper is also related to the recent research that uses queueing models to study the on-demand
economy. Taylor (2018) and Bai et al. (2019) treat freelancers in the on-demand economy as servers
in queueing models. They approximate the customers’ wait time with the help of M/M/k queues.
Frazelle et al. (2020) examine different contracts between a delivery platform and a single restaurant
and compare their performance to that in a centralized setting where the restaurant controls prices.
Chen et al. (2020a) study a similar research question by examining a setting with two streams of
customers, tech-savvy and traditional. Both papers model the food-serving restaurant as a stylized
M/M/1 queue. Cui et al. (2019, 2020) model line-sitting and queue-scalping, respectively, based on
M/M/1 queues. They treat line-sitting and queue-scalping as innovative service models, as opposed
to traditional First-Come-First-Serve, and compare their performances in equilibria.
Similar to our paper, a stream of literature in operations management also uses couriers’ travel
distances to quantify the cost of delivery. These papers typically deal with a large number of orders
per delivery trip and resort to asymptotic analysis of variants of the Traveling Salesman Problem
(TSP) to quantify the expected travel distance (see, e.g., Cachon 2014, Carlsson and Song 2017,
Qi et al. 2018, Cao et al. 2020b). In contrast, we assume that a courier delivers no more than
two orders per trip, supported by empirical evidence (see Figure 1). Furthermore, with a spatial
queueing formulation, our analysis is anchored by not only the expected travel distance, but also
the variability in traveling during delivery trips. More recently, He et al. (2020) also recognize that
using TSP may not accurately depict the trip length in food delivery as couriers and the platform
may not share the same information. For example, the couriers may have additional information
![Page 7: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/7.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 7
on the road condition, driving pattern, etc., which are ignored in the TSP formulation. Thus, they
propose prediction models on travel time using machine learning models.
Many papers also discuss the impact of dispatch policies on operational efficiency. Klapp et al.
(2018a,b) consider the dynamic dispatch wave problem. In their setting, dispatch decisions are
made at pre-determined times of a day, and the decision maker decides on which orders to be
delivered in each wave. The major trade-off in whether to deliver an order or not is between
reducing the number of outstanding orders so they can be delivered by the end of the day versus
waiting for nearby orders to show up so the delivery efficiency can be improved. Voccia et al. (2019)
also consider a multi-vehicle dynamic pickup and delivery problem with same-day delivery as the
time constraint. Other papers such as Azi et al. (2012) and Ulmer et al. (2018) also study the
optimal order assignment along with the optimal timing for vehicle departure in a single-depot
setup. Unlike these papers, we consider the pricing decision besides the dispatching policies.
Finally, our spatial modeling approach relates to Hotelling’s circular city model in economics.
The original model has suppliers and consumers evenly dispersed on a circle, and consumers have
preferences over suppliers based on their relative locations. We extend the original circular city
model (see, e.g., Salop 1979) to have the supplier sitting at the center of the circle and customers
located inside the circle, forming a disk-shaped service area. In one of our extensions, we also
investigate the extreme case where customers only reside on the edge of the disk. Some recent papers
in operations management also use spatial models based on a circular city. Chen et al. (2020b)
consider a matching problem in ride-sharing where drivers and riders depart from the center of a
circle going to different locations on the edge of the circle. They use the circular angle induced by
the circular city to characterize the mismatch between drivers and riders. Feng et al. (2020) also
use a circular city to study ride-hailing where drivers travel clockwise or counterclockwise with
a constant speed, picking up riders on the circle. Unlike our spatial model, none of these papers
consider areas inside the circle as part of the service region.
3. Model
Consider a vendor who has a facility located at the center of a disk-shaped region with radius r > 0
and hires a single courier serving customers in the area. We relax the single-courier assumption
and consider multiple couriers in Section 6.6. The structure of our service area is a generalization
of the “Circular City Model” (see, e.g., Salop 1979), in the sense that customers also occupy areas
inside the disk, whereas the original model only considers the edge of the disk. The centrally
located facility can be a store, urban warehouse, restaurant or ghost kitchen. We assume the arrival
process of customers is Poisson with rate Λr2, which scales with the area πr2 of the service region.2
2 We can also assume that the arrival rate scales with the circumference of the circle, which is linear in r. That is,the arrival rate is Λr. Our results still hold.
![Page 8: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/8.jpg)
8 Chen, Hu: Courier Dispatch in On-Demand Delivery
Upon arrival, each customer’s location is independent and uniformly distributed on the disk. Each
customer is also subjected to a wait cost with rate c per unit of time. Furthermore, we assume that
each customer has a valuation v for the delivery service, which follows a general distribution with
the cumulative density function (c.d.f.) F . Without loss of generality, we normalize the support
of F to [0,1]. The vendor can decide the charge for each delivery service at price p. We assume
that customers are sensitive towards the “virtual” in-line delay (see, e.g., Liu and Huang 2021),
which represents the time between an order is placed and the delivery courier is en-route. That is,
a customer would be satisfied once a courier is on the way to make a delivery of her order. Thus,
given a wait time w (from the order time to the start of making the dedicated delivery, even in
a batch delivery), a customer’s utility from using the delivery service is simply v − p− cw. Note
that the above utility expression assumes that the wait cost is linear in time. This is indeed a
simplification of the reality but also a reasonable one. In practice, most instant delivery services
offer a “soft promise” in wait time (e.g., 10 minutes for Gorillas, 30 minutes for Gopuff, and 1 hour
for Instacart). This implies that the wait cost by customers tends to be an increasing function of
wait time. A linear approximation of this function may be more appropriate than a step function
in which there is no cost if the wait time is below a cutoff and a constant one otherwise. Indeed,
if a delivery service like Instacart can consistently deliver in 30 minutes, the company would alter
its sales pitch by emphasizing 30 minutes as the expected wait time rather than 1-hour delivery.
Before moving on, we want to provide an alternative interpretation of our model, which accounts
for location-based valuation and pricing. We can consider each customer has valuation V = v+ cd
for the delivery service, where v is still the valuation of the service, and d is the expected travel
time required for a courier to reach the focal customer. Thus, our setup can be interpreted as that
besides the base valuation of the service, customers are willing to pay for a higher delivery price if
customers are sensitive to the wait time until they receive the order, and their location is further
away from the expected location of a courier when she embarks on the dedicated delivery trip.
The vendor can decide the charge for each delivery service at price P = p+ cd, so that the price
is also location dependent. Given a wait time w (from the order time to the start of making the
dedicated delivery), a customer’s utility from using the delivery service is V −P − cw= v−p− cw.
Therefore, the effects of the location-dependent valuation and price cancel out in our model, which
means that customers’ ordering decision is independent of their locations on the disk. Readers can
make either interpretation of the model based on the applications.
Obviously, only customers with nonnegative utilities will use the delivery service from the vendor.
Customers with negative utilities may choose to pick up the orders by themselves, or not order at
all. Denote by λ ∈ [0,Λr2] the effective demand rate of the delivery service. Since each customer’s
![Page 9: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/9.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 9
valuation v follows a distribution with the c.d.f. F , the demand rate λ satisfies λ/Λr2 = 1−F (p+
cw), which implies that for all λ∈ (0,Λr2], we have
p= F−1
(1− λ
Λr2
)− cw, (1)
where function F−1 is the inverse function of the c.d.f. F . Thus, there is a one-to-one mapping
between price p and positive demand rate λ. Note that if the demand rate λ is exogenous, then
it is possible to have p < 0, as the vendor needs to subsidize customers for the service, which may
happen when the vendor wants to grow a market. This would not happen when the demand rate
is endogenized. Using the expression in (1) for positive demand rate, the vendor’s revenue function
can be written as
V (λ,w) = λp= λ
(F−1
(1− λ
Λr2
)− cw
), λ > 0. (2)
The vendor makes operational decisions based on the revenue it generates according to (2).
We emphasize that wait time for each customer, w, in a steady state also depends on the effective
demand rate λ. In later sections, when comparing the vendor’s revenue functions under different
delivery modes, we replace w by the expected wait time for each customer, which is a function of
the demand rate λ. The underlying assumption is that customers anticipate a wait time and use
it to anticipate and decide on whether to adopt the service. In equilibrium, their anticipated wait
time is consistent with their actual experiences over repeated interactions.
We consider and compare two delivery strategies, the dedicated strategy and the pooling strategy.
On one hand, with the dedicated strategy, the courier serves orders one by one in the First-Come-
First-Serve fashion (referred to as “dedicated delivery”). On the other hand, with the pooling
strategy, the courier is not en-route for delivery until exactly two3 orders are accumulated, which
can be interpreted as serving orders in batches of two (referred to as “batch delivery”). Figure
2 illustrates the differences between the two strategies, using the example of restaurant delivery.
When serving dedicated delivery, a courier leaves the restaurant immediately when an order arrives.
After delivering the food, the courier comes back to the restaurant to pick up or to wait for the
next order. When serving batch, the courier does not leave the restaurant until two orders have
arrived. Then, the courier delivers both orders in a single delivery trip before coming back to the
restaurant for the next batch. We do not specify the fulfillment sequence within a batch, as long as
the resulting order is purely random; for example, the sequence can follow the time order of arrivals
or a spatial order, such as always traveling clockwise. The fulfillment sequence within a batch does
not affect the total travel distance of a courier, but may affect the wait time of a specific order.
![Page 10: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/10.jpg)
10 Chen, Hu: Courier Dispatch in On-Demand Delivery
Figure 2 Serving dedicated v.s. serving batch
If the resulting fulfillment order is purely random, customers would still have the same expected
wait time over repeated interactions with the system.
We recognize the similarity between our delivery system and a single-server queue where a courier
acts as the server and customers’ orders queue up. Since potential customer arrivals follow a Poisson
process and a fraction of the customers choose the delivery service based on the expected wait,
the arrival process of orders is also Poisson with the rate equal to the effective demand rate λ. As
for the service process, we assume that the courier picks up the delivery goods at the centrally
located facility instantly and spends no time at each customer’s location. Thus, the service time
only consists of the courier’s traveling time between the facility and customers’ location(s). We
define a delivery trip as the process starting when the courier picks up the delivery goods at the
facility and ending when he returns. We utilize the results in the queueing literature to derive the
expected wait time of customers under each delivery strategy in the next two subsections.
3.1. Dedicated Delivery
Suppose the courier uses the dedicated delivery strategy to serve customer orders, i.e., dedicated
delivery. As mentioned, orders arrive following a Poisson process with rate λ in equilibrium. The
service time is the time that the courier spends in delivering each order. When dedicated delivery is
adopted, each delivery trip is simply the round trip between the facility and a random customer’s
location. By assuming a constant travel speed and normalizing it to 1, the service time equals the
travel distance per delivery trip.
Denote by a random variable XD the shortest Euclidean distance of a delivery trip when serving
orders under dedicated delivery. That is, XD is two times the distance between the center of the
3 According to an internal study by one of the largest delivery platforms in China, their couriers carry less than twoorders per trip on average, see Figure 1.
![Page 11: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/11.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 11
disk with radius r and a uniformly distributed point on the disk. According to the “Disk Point
Picking” literature (see, e.g., Solomon 1978), we have
E[XD] =1
2πr2
∫ r2
0
∫ 2π
0
2√xdθdx=
4
3r, and E[X2
D] =1
2πr2
∫ r2
0
∫ 2π
0
4xdθdx= 2r2. (3)
Note that the first moment of random variable XD represents the expected distance of the delivery
trip, which is also the expected service time under our normalization of the travel speed. Then we
can treat this delivery system as an M/G/1 queue with the service rate and load factor equal to
µD =1
E[XD]=
3
4r, and ρD =
λ
µD=
4
3λr, (4)
respectively.
We define the expected wait time by WD when serving orders under dedicated delivery as a
function of demand rate, service rate, and the coefficient of variation of the arrival and service
processes. That is, we have
WD(λ,µ,C) :=λ
µ(µ−λ)
C
2, ∀λ,C ≥ 0, µ > 0, (5)
where the term represents in-line delay of an M/G/1 queue (see, e.g., Gross et al. 2008, p. 222).
Note that the summation of coefficients of variation of our M/G/1 queue’s arrival and service
processes is
CD = 1 +E[X2
D]− (E[XD])2
(E[XD])2=
9
8. (6)
Thus, according to (5), WD(λ,µD,CD) represents the expected wait time for each customer when
the courier serves orders following dedicated delivery. Therefore, we can rewrite the revenue function
in (2) as
VD(λ,WD(λ,µD,CD)) = λ
[F−1
(1− λ
Λr2
)− cWD(λ,µD,CD)
], (7)
representing the revenue rate of the delivery service when the vendor adopts dedicated delivery.
3.2. Batch or Pooling Strategy
Instead of serving orders with dedicated delivery, the courier can also deliver orders using batch
delivery. In this paper, we assume that each batch consists of two orders and that inside each batch,
orders are delivered following a predetermined rule. That is, the courier does not leave the facility
until two orders have arrived. Thus, when comparing our delivery system to a queueing system, we
consider orders entering the queue in pairs of two. That is, an arriving order does not technically
enter the queue if all outstanding orders in the system are already in pairs of two. Instead, it waits
and enters the queue together with the next order that arrives. Therefore, when the demand rate
![Page 12: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/12.jpg)
12 Chen, Hu: Courier Dispatch in On-Demand Delivery
is λ, we can effectively treat the inter-arrival time as being Erlang distributed with order 2 and
having a mean of 2/λ (with the arrival rate being λ/2).
Next, we analyze the service process of the delivery system using batch. A delivery trip needs to
include three parts: travel between the facility and the first order’s location, between the first and
second orders’ locations, and finally, back to the facility from the second order’s location. Denote
by random variable XB the shortest distance a courier needs to travel per trip. According to the
“Disk Line Picking” literature (see, e.g., Solomon 1978), we have4
E[XB] =1
πr4
∫ r2
0
∫ r2
0
∫ π
0
(√x+ y− 2
√xy cos(θ) +
√x+√y
)dθdxdy=
(128
45π+
4
3
)r,
E[X2B
]=
1
πr4
∫ r2
0
∫ r2
0
∫ π
0
(√x+ y− 2
√xy cos(θ) +
√x+√y
)2
dθdxdy≈ 5.428r2. (8)
Since the travel speed is normalized to 1, the travel distance in each delivery trip is the service
time for the courier. Using the first moment of XB, we can derive the service rate and load factor
of this service queue as
µB =1
E[XB]=
45π
4r(32 + 15π), and ρB =
λ
2µB=
2λr(32 + 15π)
45π, (9)
respectively. With both arrival and service processes characterized, we recognize that our batch
service can be analyzed through an E2/G/1 queue.
Since the inter-arrival time follows an Erlang-2 distribution, combining the first and second
moments of XB, we have the summation of the coefficients of variation for arrival and service
processes as
CB =1
2+
E[X2B]− (E[XB])2
(E[XB])2≈ 0.583. (10)
Unfortunately, we do not have a closed-form expression for the expected in-line delay of the E2/G/1
queues. Seeking analytical results, we use Kingman’s formula (see, e.g., Gross et al. 2008, p. 344)
to approximate the in-line delay of this E2/G/1 queue as a G/G/1 queue. That is, we have
Wq ≈1
2µB
ρB1− ρB
CB =CB2
λ
µB(2µB −λ), (11)
where CB is defined in (10). The Kingman’s formula we adopt serves as an upper bound (see,
e.g., Kingman 1962) on the in-line delay and is asymptotically exact in the heavy traffic regime.
All of our results in favor of the pooling strategy can be refined to be analytically exact, as we
use the upper bound of the in-line delay under batch delivery in comparison with the dedicated
4 Note that the only approximation in equation (8) is on the coefficient in the second moment, which is computedaccurately using numerical integration.
![Page 13: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/13.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 13
strategy. All of our results still hold for a numerical verification in which the expected in-line delay
is computed from a simulated system of the E2/G/1 queue. In Online Appendix D, we provide
simulation results on the accuracy of all the approximations in this paper. In summary, all the
closed-form approximations considered in this paper are fairly accurate.
Note that the batch delivery has a shorter in-line delay compared to a hypothetical M/G/1
dedicated delivery system where the arrival rate is λ/2. The reason is that the batch system has a
lower coefficient of variation, i.e., CB ≤CD, which means there is less variability in both the arrival
and service processes of the batch system. More specifically, the variability in the arrival process
is reduced from 1 in the dedicated system to 1/2 in the batch system due to temporal pooling of
orders, and the variability in the service process is reduced from 1/8 in the dedicated system to
about 0.083 in the batch system due to spatial pooling of two delivery trips into one.
Recall that when using an E2/G/1 queue to analyze our batch system, a single order does not
enter the queue until a second order arrives. In other words, the in-line delay does not include the
time to form a batch of two orders, which is on average 1/λ. We assume that the customer does
not know the exact state of the system, as is the case in practice. That is, she has no information
on her position in the queue. Thus, from a customer’s perspective, her expected wait time consists
of three parts: the expected wait time for a second order to arrive if her order does not enter the
queue immediately, the average in-line delay once her batch enters the queue, and if she is the
second in her batch to be served, the time it takes to serve the first. Define the expected wait time
WB as a function of the demand rate, service rate, and the coefficient of variation. That is, we have
WB(λ,µ,C) :=1
2λ+
λ
µ(2µ−λ)
C
2+
1
2
E[XD]
2=
1
2λ+
λ
µ(2µ−λ)
C
2+r
3, ∀λ,C ≥ 0, µ > 0, (12)
where the components correspond to the three parts in the customer’ s expected wait time, respec-
tively. In particular, the last term E[XD]/4, represents the expected extra delay if the courier serves
her order in second. So in a half of the time, she needs to wait the courier delivers the other order
first (taking E[XD]/2 time in expectation) before en-route with her order. Thus, WB(λ,µB,CB) rep-
resents a customer’s expected wait time when the courier is serving batch. Note that WB(λ,µB,CB)
approaches infinity as λ goes to 0. The reason is that the courier never leaves the facility with a
single order, so a customer may need to wait for a long time when a second order takes some time
to arrive. Thus, the revenue function in (2) becomes
VB(λ,WB(λ,µB,CB)) = λ
[F−1
(1− λ
Λr2
)− cWB(λ,µB,CB)
], λ∈ (0,2µB) . (13)
It is worth pointing out that limλ→0
VB(λ,WB(λ,µB,CB)) = − c2< 0 as the expected wait time
WB(λ,µB,CB) approaches infinity when λ approaches 0. Thus, in batch serving, if the vendor needs
![Page 14: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/14.jpg)
14 Chen, Hu: Courier Dispatch in On-Demand Delivery
to maintain a low demand rate close to 0, the vendor has a negative revenue rate. In other words,
maintaining a low demand rate in batch serving is unprofitable for the vendor, because it requires
a significant subsidy to customers. However, we only use this limit case to provide intuitions on a
disadvantage of batch serving, since to gain profitability, the vendor can simply serve dedicated,
which generates nonnegative revenue when the demand is very low.
4. Exogenous Demand Rate
In this section, we evaluate the performance of adopting the dedicated delivery and pooling (batch)
strategies when the demand is exogenous. Throughout the base model, we use the vendor’s revenue
as the performance measure. That is, although the demand rate is exogenous, the vendor can still
make the decision on which delivery mode to operate, coupled with the corresponding price to
achieve the targeted demand rate, in order to attain a higher revenue. This is the case when the
firm has an exogenously given demand segment to cover, due to the needs of growing or penetrating
a market or other goals that are not directly related to revenue creation from delivery services,
e.g., the need of matching the delivery capacity with the kitchen capacity. We observe immediately
that serving batch can sustain a higher demand rate than serving dedicated delivery, since when
comparing the load factors in (4) and (9), we have ρB < ρD, if λ > 0 is fixed. Furthermore, since
both ρD and ρB are linearly increasing in r, we also observe that serving batch allows the delivery
service to handle a larger service region than serving dedicated delivery.
When comparing the revenue functions in (7) and (13), if the demand rate λ is exogenous,
the delivery strategy that has the shorter expected wait time leads to higher revenue. That is,
the operating strategy with exogenous demand is efficiency driven. Therefore, in the following
two propositions, we compare the revenues generated via the two delivery strategies and their
corresponding expected wait times.
Proposition 1. Suppose the demand rate is exogenously given. There exists a threshold on the
demand rate below which serving dedicated leads to a shorter expected wait time and thus higher
revenue, and above which serving batch leads to a shorter expected wait time and thus higher
revenue.
Proposition 1 states that when the exogenous demand rate is low, operating dedicated delivery
is better than batch. The intuition is that when the demand rate is low, it takes a very long time
to accumulate two orders so that the courier can make a batch delivery trip. Figure 3(a) provides
a visual representation of the wait times. As an extreme case, when the demand rate goes to zero,
the expected wait time for each customer will approach infinity under batch. However, adopting
dedicated delivery leads to a much shorter expected wait time.
![Page 15: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/15.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 15
As the average time to accumulate two orders drastically decreases when the demand rate
increases, the overall expected wait time under batch decreases as well. When the demand rate
becomes very high, the in-line delay of customers dominates the average wait time for a pair of two
orders to accumulate. Thus, the expected wait time increases with a sufficiently high demand rate.
As mentioned, serving batch can handle a higher demand rate than serving dedicated because the
average travel distance associated with delivering an order is shorter. In Figure 3(a), we observe
that the expected wait time under dedicated delivery approaches infinity faster when λ becomes
sufficiently large than that under batch delivery does.
Not only is there a threshold on the demand rate that changes the vendor’s delivery strategy,
the next proposition states that there is such a threshold on the size of the service region as well.
Proposition 2. Suppose the demand rate is exogenously given. There exists a threshold on
the service radius, below which serving dedicated leads to a shorter expected wait time and thus
higher revenue, and above which serving batch leads to a shorter expected wait time and thus higher
revenue.
Proposition 2 states that operating dedicated delivery is better if the service radius is small and
serving batch is better when the service radius is large. This result appears to be intuitive as one
may think that when the service radius is large, serving batch can reduce the total travel distance
of the courier. However, the first moments of the lengths of delivery trips under both dedicated
delivery and batch scale with r with other parameters fixed in (3) and (8), respectively. Thus, one
can verify that for any service radius, compared with dedicated delivery, serving batch leads to
a longer average total travel distance but a shorter distance per order, i.e., E[XB]/2 ≤ E[XD] ≤
E[XB]. The main reason behind Proposition 2 is that when the service radius is small, the time to
accumulate two orders when serving batch is much longer than the actual travel time. On the other
hand, if the service radius is large, the travel time becomes longer than the time to accumulate
two orders, which is independent of the service radius when the demand rate is exogenous. Thus,
serving batch is more beneficial when the service radius is large. Figure 3(b) provides a visual
illustration of the expected wait time of a customer when the courier serves dedicated delivery and
batch, respectively.
Corollary 1. Suppose the demand rate is exogenously given.
(i) There exist thresholds in demand rate and service radius (which are the same as those in
Propositions 1 and 2, respectively) such that below which, the price is higher when using
dedicated delivery and above which, serving batch leads to a higher price.
![Page 16: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/16.jpg)
16 Chen, Hu: Courier Dispatch in On-Demand Delivery
(ii) There exist thresholds in demand rate and service radius (which are the same as those in
Propositions 1 and 2, respectively) such that below which, the expected wait time per order
is shorter when using dedicated delivery and above which, serving batch leads to a shorter
expected wait time per order.
Corollary 1 extends the results in Propositions 1 and 2 to price and delivery efficiency. Corollary
1 is straightforward since when the demand rate is exogenous, the price is non-increasing with
respect to the wait time. Furthermore, as we use the expected wait time per order as the measure
of delivery efficiency, serving batch is more efficient when either the demand rate is high enough
or the service radius is large enough. Otherwise, dedicated delivery is more efficient as it bypasses
the order accumulation time.
(a) (b)
Figure 3 Expected wait time when serving dedicated or batch. (a) r= 1, (b) λ= 1
As mentioned, the case with an exogenous demand rate can describe the market penetration
stage experienced by many start-up companies or applications in public or other business settings
with rigid demand requirements. For example, consider a newly formed ghost kitchen in a mega
city, which hires a given number of kitchen staff (so the maximum kitchen throughput is given)
at the operational level, or tries to carve our a targeted market share in the local takeaway food
market at the tactic level. Thus the kitchen needs to maintain a targeted demand rate through
methods such as offering delivery promotions, which greatly limits its pricing decision. If the
service area is fixed, using dedicated delivery outperforms serving batch, if and only if the targeted
demand rate is relatively low. Serving batch is only beneficial if a relatively high demand rate
needs to be maintained, so temporal pooling can add efficiency en route without losing too much
time accumulating orders. Furthermore, using dedicated delivery leads to a shorter expected wait
time for customers and higher revenue if the service area is relatively small. However, with a
predetermined larger service area, it is better to serve batch taking advantage of the efficiency en
route.
![Page 17: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/17.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 17
We conclude this section by pointing out that if the demand rate is exogenously determined,
only the effective demand rate λ and the service radius r impact the vendor’s delivery decision,
since we only need to compare the expected wait times for customers under the two strategies.
That is, the underlying arrival rate of customers Λ, wait cost parameter c, and the distribution
function F of customer valuations do not affect the delivery strategy once the targeted demand
rate is determined. In the next section, we compare and contrast the results of this section to the
case where the demand rate λ can be optimized.
5. Endogenous Demand Rate
The previous section covers the scenario with an exogenous demand rate that needs to be sustained.
In this section, the vendor aims at maximizing its revenue with an endogenized demand rate. That
is, there is no exogenous constraint on the demand rate and the vendor maximizes its revenue by
designing the optimal demand rate. Therefore, unlike Section 4 where the vendor can only choose
which delivery mode to operate in with a given demand rate, in this section, the vendor also chooses
the optimal demand rate in each mode which can be achieved via the freedom in varying the price.
Seeking for tractable analytical results, we first take advantage of a crowded market setting
where the underlying arrival rate of customers is high enough. Suppose the arrival rate scales with a
density factor n∈N. As n increases, the arrival rate nΛ increases as well, meaning that the market
gets more and more crowded. Thus, with customer valuations drawn from the c.d.f. F (with its
support normalized to [0,1]), the revenue function in (2) can be modified to
Vn(λ,w) = λ
(F−1
(1− λ
nΛr2
)− cw
), λ≥ 0. (14)
As in this section the vendor maximizes the revenue rate by choosing the demand rate, λ= 0 will
not be the optimal choice.
Define function
V∞(λ,w) := limn→∞
Vn(λ,w) = λ (1− cw) , λ≥ 0, (15)
where the equality follows from the fact that the upper bound on customer valuations has been
normalized to 1. The expression in (15) represents the limiting revenue when the density factor
n goes to infinity. According to (15), when the underlying arrival of customers goes to infinity,
the vendor only serves those who have a valuation almost equal to 1, the upper bound. Thus,
at the limit, the vendor’s revenue is independent of customer valuation distribution. In general,
under a given delivery strategy, when the targeted demand rate λ increases, two terms in (14)
change: (i) the base price F−1
(1− λ
nΛr2
)needs to be adjusted downwards to incentivize more
adoption, and (ii) the expected wait time w increases as a result of a higher joining rate and thus
![Page 18: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/18.jpg)
18 Chen, Hu: Courier Dispatch in On-Demand Delivery
more discount cw needs to be paid to compensate customers for the longer wait. The crowded
market assumption assumes away the first effect which is verified by Lemma 1. We will relax this
assumption in Subsection 6.3.
First, we present a lemma on utilizing the expression in (15), which greatly simplifies our analysis
for a crowded market.
Lemma 1. Consider n∈N and a c.d.f. F such that F−1 is Lipschitz continuous. There is
limn→∞
maxλ∈[0,µD)
Vn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)
V∞(λ,WD(λ,µD,CD)), (16)
and
limn→∞
maxλ∈[0,2µB)
Vn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)
V∞(λ,WB(λ,µB,CB)). (17)
Lemma 1 implies that we can simply optimize the demand rates for serving dedicated and batch
using the limiting revenue function in (15) when n approaches infinity. Therefore, the vendor’s
demand-rate decision is independent of the customer valuation distribution. Since function V∞
has a much more concise expression than the non-limiting revenue function, it is much easier
to be analyzed and used for comparing optimal solutions under different delivery strategies. In
particular, the next two propositions summarize the results for a crowded market when the vendor
can endogenize the demand rate.
Proposition 3. Assume a large market and suppose the demand rate can be endogenized.
(i) There exists a threshold c∞ on customers’ wait cost parameter c, below which serving batch
leads to higher revenue and above which serving dedicated leads to higher revenue.
(ii) As c crosses the threshold c∞ such that the optimal strategy switches from serving batch
to serving dedicated delivery, the optimal demand rate has a discontinuous drop, i.e.,
limc→c∞− λ∗(c)> limc→c∞+ λ∗(c), where λ∗(c) is the optimal demand rate as a function of the
wait cost coefficient c, and the corresponding optimal price has a discontinuous surge.
Proposition 3(i) states that if the vendor can optimize the revenue rate by endogenizing the
demand rate, serving dedicated is better if customers are impatient (i.e., c is sufficiently high).
With patient customers, it is optimal to serve batch (i.e., c is sufficiently low). This is in contrast
to the result in Section 4: when the demand rate is fixed, the wait cost parameter c has no impact
on the vendor’s delivery decision, since it does not affect the expected wait time. Proposition 3(ii)
says that there is a sudden drop in the optimal demand rate and a surge in the optimal price
when the cost of waiting crosses the threshold such that the optimal delivery strategy changes
from batch to dedicated. When customers are impatient, it is better for the vendor to have a less
![Page 19: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/19.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 19
crowded system with a relatively low demand rate, which gives an edge to dedicated fulfillment.
If customers are patient, it is better to sustain a higher demand rate while implementing batch
strategy, which shortens the time needed to accumulate two orders and hence the overall expected
wait time. This intuition is consistent with Proposition 1 that a low (resp., high) demand rate
favors dedicated (resp., batch).
Proposition 4. Assume a large market and suppose the demand rate can be endogenized.
(i) There exists a threshold r∞ on the service radius r, below which serving batch leads to higher
revenue and above which serving dedicated leads to higher revenue.
(ii) As r crosses the threshold r∞ such that the optimal strategy switches from serving batch
to dedicated, the optimal demand rate has a discontinuous drop, i.e., limr→r∞− λ(r) >
limr→r∞+ λ(r), where λ(r) is the optimal demand rate as a function of the service radius r,
and the corresponding optimal price has a discontinuous surge.
Proposition 4 states that when the market is crowded, the vendor should serve dedicated if the
service radius r is large enough. Instead, serving batch is optimal if the service radius is sufficiently
small. This result contrasts with Proposition 2, in which the demand rate is exogenous. With a
large service radius, the courier’s travel time is long under either dedicated or batch, which leads
to a relatively long expected wait time for customers. Thus, it is better for the vendor to sustain
a relatively small demand rate, otherwise the compensation for the long wait would be significant.
Again, there is a sudden drop in the optimal demand rate and a surge in the optimal price when
the service radius crosses the threshold such that the optimal delivery strategy changes from batch
to dedicated. Recall that serving batch is less profitable than serving dedicated when the demand
rate is low, since serving batch has a much longer expected wait time. That is, an order may have
to wait for a long time for another order to arrive and form a batch before it is en route for delivery.
When the service radius is small, it is beneficial to operate under a relatively high demand rate, as
the average travel distance is shorter under either delivery strategy than it is with a large service
radius. As mentioned, serving batch is more profitable for a relatively high demand rate.
Next, we discuss the practical implications of our results by discussing a few examples. During
rush hour for a delivery system, the vendor may have far more potential customers than it has the
capacity to serve. Customers ordering a cup of coffee may be impatient because hot coffee will be
cold if not delivered in time, whereas a grocery vendor or restaurant that only serves cold dishes
like sushi may have more patient customers. Thus, as implied by Proposition 3, even though the
two businesses have the same service area, the coffee shop may prefer the dedicated strategy, and
the grocery vendor or sushi restaurant, the pooling strategy. As an implication of Proposition 4,
even if their customers have the same patience level, a restaurant serving only a 10-block radius in
![Page 20: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/20.jpg)
20 Chen, Hu: Courier Dispatch in On-Demand Delivery
Midtown Manhattan may prefer the batch strategy but a restaurant with similar characteristics
delivering throughout Midtown Manhattan may want to use the dedicated strategy, since the
latter has a much bigger service area. This implication may seem counterintuitive at first glance,
as a larger service area may require more emphasis on delivery efficiency that may be achieved
by the pooling strategy (as conveyed in Proposition 2). The key to understanding this seemingly
counterintuitive insight is that for a large service area, the dedicated strategy is coupled with a
high delivery price, while the pooling strategy needs to keep the delivery price relatively low to
compensate customers for the wait. With the profit margin being taken into account as the demand
rate is endogenized, the dedicated strategy becomes optimal for a large service area.
We conclude this section by summarizing the results and contrasting them with those when the
demand rate is exogenously given. First, we observe that with an endogenous demand rate, it is
optimal to serve dedicated if the service area is large. This result directly contrasts with the one
for an exogenous demand rate, where it is optimal to serve batch for a large service area. Second,
customers’ patience level, which has no impact if the demand rate is exogenous, greatly affects the
vendor’s delivery strategy for the endogenized demand rate. With the demand rate endogenously
determined, if customers are patient, the vendor should serve batch. However, if customers are
impatient, serving dedicated generates higher revenue. Finally, for a crowded market, we are able
to identify the optimal delivery strategy analytically for the entire spectrum of customers’ patience
level and the service area’s size, respectively.
6. Extensions
In this section, we consider three extensions of our base model. We investigate each one and examine
the robustness of our results and intuitions obtained from Sections 4 and 5.
6.1. Social Welfare
Another objective of interest is the social welfare generated by the delivery system. We define the
social welfare generated per order as the summation of the vendor’s revenue and the customer’s
profit, i.e., v − cw, where w is the expected wait time, in view of that the price is an internal
transfer between the vendor and a customer. Thus, the social welfare generated per order is
SW (λ,w) = Λr2P(v≥ p+ cw)E[v− cw |v≥ p+ cw] = Λr2
∫ 1
F−1(1− λΛr2
)(v− cw)dF (v). (18)
The next proposition characterizes the impacts on the social welfare when the vendor focuses on
market penetration or maximizing revenue, respectively.
Proposition 5. (i) Suppose the demand rate is exogenous, there exist thresholds on the
demand rate and service radius, below which serving dedicated leads to higher social welfare
and above which serving batch leads to higher social welfare.
![Page 21: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/21.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 21
(ii) Suppose the demand rate is endogenous and the market is crowded. There exist thresholds
on the service radius and customers’ patience level, below which serving batch leads to higher
social welfare and above which serving dedicated leads to higher social welfare.
Essentially, we recover the results in Sections 4 and 5 in Proposition 5. Thus, even when the
performance measure changes from the vendor’s revenue to social welfare, our major insights in
the previous sections still hold. When the demand rate is exogenous, the key factor in operations is
the delivery efficiency. On the other hand, when the demand can be endogenized, the vendor needs
to consider the optimal demand rate to sustain, which has a tremendous impact on the system
efficiency.
6.2. Contingent Policy
Another natural extension to our base model is to consider a contingent policy alternating between
serving dedicated and batch depends on the size of the queue. Suppose the courier serves the orders
in batch if and only if there are more than one outstanding orders in the queue and serves dedicated
otherwise (i.e., when there is a single unfilled order). At the first glance, it seems this contingent
policy takes advantage of both delivery methods considered in this paper. In the next proposition,
we show its relationship with dedicated and batch delivery.
Proposition 6. For any demand rate λ > 0, the contingent policy leads to a shorter expected
wait time for customers, compared to that of dedicated delivery. But there still exists a trade-off
between this contingent policy and batch delivery.
Proposition 6 states that in terms of the expected wait time, the contingent policy always
dominates dedicated delivery. Thus, we can conclude that the contingent policy indeed outperforms
dedicated delivery. However, the major trade-off between dedicated and batch delivery still persists
between this contingent policy and batch delivery. As batch serving always waits to accumulate
two orders before dispatch, it can take advantage of a large demand rate setting where the expected
wait time to accumulate another order is shorter than a delivery trip with a single order. On the
other hand, the contingent policy is better suited when the demand rate is relatively low, providing
the flexibility to avoid long wait time for order accumulation.
To better analyze the performance of the contingent policy considered here or any other state-
dependent delivery policy, we believe a dynamic program model is needed and this is beyond the
scope of this paper. We hope our discussion on the contingent policy can stimulate future research
in this direction.
![Page 22: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/22.jpg)
22 Chen, Hu: Courier Dispatch in On-Demand Delivery
6.3. General Arrival Rate
First, we relax the large-market assumption in Section 5. In this subsection, we consider the general
arrival rate of customers. We investigate whether observations such as Propositions 3 and 4 still
hold without the arrival rate being at the limit. To keep our results concise and informative, we also
assume that customer valuations are uniformly distributed on [0,1]. That is, F (v) = v for v ∈ [0,1]
and F (v) = 0 otherwise. Note that our result does not anchor on the uniform distribution assump-
tion. Statements in this subsection can be generalized to more general valuation distributions as
well. We leave the detailed discussion to Online Appendix C.2.
When the courier serves dedicated, the revenue maximization problem for the vendor is
maxλ∈[0,µD)
VD(λ,WD(λ,µD,CD)), (19)
where the constraint on the demand rate λ reflects the load factor ρD < 1 so that the system is
stable. Similarly, when the courier serves batch, the maximization problem is
maxλ∈[0,2µB)
VB(λ,WB(λ,µB,CB)), (20)
where functions VD and VB are defined in (7) and (13), respectively; the constraint on λ reflects
ρB < 1. Note that we do not include constraint λ≤Λr2 in either (19) or (20). The reason is that for
any demand rate that is greater than Λr2 (which is still mathematically possible), the corresponding
revenue function has a negative value, so it cannot be optimal. The next two propositions summarize
the results when the vendor optimizes its revenue according to (19) and (20).
Proposition 7. Fix r,Λ> 0. Consider F (v) = v for v ∈ [0,1] and F (v) = 0 otherwise. With the
demand rate endogenized, there exists a threshold cen on the customers’ wait cost parameter c, such
that for all c≥ cen, it is optimal to serve dedicated.
Proposition 7 complements the results in Proposition 3 while assuming that each customer’s
valuation follows an independent standard uniform distribution. Even with general arrival rates
of customers, it is still optimal to serve dedicated when customers are impatient (i.e., c is large
enough). Unfortunately, it is challenging to demonstrate analytically that it is optimal with general
arrival rates to serve batch when customers are very patient, unlike the case in the limiting regime.
With general arrival rates, both the distribution of customers’ valuations and the expected wait
time affect the overall revenue as mentioned in Section 5. The distribution of valuations determines
the optimal base price, which, unlike the crowded market, is not independent of the demand
anymore. Furthermore, finite arrival rates may prevent the delivery system from achieving the
optimal demand rate when customers are patient. This hurts serving batch specifically since the
pooling strategy shines under a high demand rate and its efficiency may not be fully exploited
![Page 23: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/23.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 23
in this case. Moreover, the price compensation would have to be significant in order to sustain a
large demand rate with finite arrivals. However, we can still numerically verify that there exists a
threshold on wait cost parameter c below which it is optimal to serve batch. Figure 4(a) provides
such a visual illustration: the optimal revenue functions of serving dedicated and batch only cross
once.
Proposition 8. Fix c,Λ> 0 and constant L such thatΛ
c3>L (with the exact expression of con-
stant L provided in the online appendix). Consider F (v) = v for v ∈ [0,1] and F (v) = 0 otherwise.
With the demand rate endogenized, there exists a threshold ren on the service radius r, such that
for all r≥ ren, it is optimal to serve dedicated.
Proposition 8 extends the result in Proposition 4 when each customer’s valuation follows an
independent standard uniform distribution. We show that with general customer arrival rates, it
is still optimal to serve dedicated when the service radius is large enough. We only require an
extra minor condition that either the arrival rate of customers is high enough or their wait cost
parameter is low enough. Similar to Proposition 7, it is very difficult to establish optimal conditions
for serving batch. In fact, in our numerical experiments, we find counterexamples where it may
not be optimal to serve batch when the radius is small. Instead, as in the counterexample shown
in Figure 4(b), it is only optimal to serve batch when the service radius is medium. For sufficiently
small or large service radii, it is always better to serve dedicated. As mentioned, serving batch
has the edge over dedicated when the demand rate is relatively high. When the service radius
is sufficiently small, it is beneficial to sustain a high demand rate for both dedicated and batch.
However, due to the finite arrival rate of customers, the demand rate cannot reach the magnitude
at which serving batch outperforms serving dedicated; otherwise, the price discount to sustain a
high demand rate for batch delivery would be too great. This also explains why we only observe a
single threshold on the service radius in Proposition 4 in the large market limiting regime.
(a) (b)
Figure 4 Optimal revenue under dedicated and batch delivery. (a) r= 1, Λ = 25, (b) c= 0.03, Λ = 25
![Page 24: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/24.jpg)
24 Chen, Hu: Courier Dispatch in On-Demand Delivery
6.4. Batch Size Greater Than Two
In our base model, we consider batches with size of two, in view of applications in food delivery,
to better illustrate the main trade-offs in our delivery policies. Here we extend the model to a
batch size greater than two and conduct numerical studies. Namely, we consider each batch has
size of three or more. In Figure 5, we provide the empirical cumulative distribution functions on
the courier’s travel time (service time) per order when using batch with different sizes. As we can
see, increasing the batch size can reduce the chance of inducing long service times. However, this
margin of improvement gets smaller as the batch size increases. In addition, smaller batch sizes
have higher chances to induce very short service time. In addition to these observations, we choose
not to consider batch sizes greater than three for the following two reasons. First, for any batch
greater than two, we need to consider proper routing policies in delivery, which is not the focus of
our paper. When the batch size equals to three, in the following, we consider the courier delivers
orders with a purely random ordering. But it will be observed that the gap between a random
fulfillment policy versus delivery policy based on the shortest path gets larger when the batch
size increases. Second, as the batch size increases, it may be in the best interest of the vendor to
consider contingent policies as in Section 6.2, which we leave as a future research direction as they
should be analyzed using non-stationary models.
Figure 5 Empirical cumulative distribution functions with various batch sizes. r= 1
Now, we analyze this batch system using an Erlang-3 arrival process. That is, an arriving order
does not enter the dispatch queue until a batch of three is formed. Then, batches have the arrival
rate of λ/3, with the inter-batch time following the Erlang-3 distribution. Following similar deriva-
tion to (8), we have
E[X3B] =
∫x,y,z∈[0,r2], θ,φ∈[0,π]
(√x+ y− 2
√xy cos(θ) +
√y+ z− 2
√yz cos(φ)
+√x+√z
)dUx dUy dUz dUθ dUφ,
![Page 25: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/25.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 25
E[X2
3B
]=
∫x,y,z∈[0,r2], θ,φ∈[0,π]
(√x+ y− 2
√xy cos(θ) +
√y+ z− 2
√yz cos(φ)
+√x+√z
)2
dUx dUy dUz dUθ dUφ, (21)
where Ux, Uy, Uz are independent uniform distributions on interval [0, r2], and Uθ, Uφ are indepen-
dent uniform distributions on interval [0, π]. Using the above first and second moments, we can get
the service rate and load factor as
µ3B =1
E[X3B]and ρ3B =
λ
3µ3B
. (22)
Furthermore, the coefficient of variation is
C3B =1
3+
E [X23B]− (E [X3B]
2)
(E [X3B])2. (23)
As a result, the expected wait time can be calculated using
W3B(λ,µ,C) =1
λ+
λ
µ(3µ−λ)
C
2+
1
3
(4
3+
128
45π
)r, (24)
where the first term is the average wait time an order has to wait to form a batch (an order needs
to wait for 0, 1, or 2 more orders with equal probability to form a batch), the second term is the
in-line delay, and the last term is the extra delay if other order(s) in the batch needs to be delivered
firstly. Then, W3B(λ,µ3B,C3B) is the expected wait time.
(a) (b)
Figure 6 Revenue functions when serving dedicated versus batch of size 3 under a large market.
(a) c= 0.1, (b) r= 1
In our numerical calculations, we always observe that there are single thresholds in the service
radius r and wait cost c, respectively, such that below which, the vendor should serve with batches,
and above which, dedicated delivery generates more revenue. This is consistent with our findings
in the case with a batch size of two. Thus, our main insights are not limited by the simplification
of considering batches with the size of two.
![Page 26: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/26.jpg)
26 Chen, Hu: Courier Dispatch in On-Demand Delivery
Consistent with Section 6.3, for general arrival rates, when the radius is large enough, the vendor
should use dedicated delivery. We can again find numerical examples such that there are two
thresholds on the service radius such that only between these thresholds, serving batch outperforms
dedicated delivery in terms of revenue maximization. However, we can also find extreme parameters
such that batch delivery is completely dominated by dedicated delivery for all service radii. We
believe that the possible inferior performance of batch delivery with a size greater than two is
contributed to the random routing policy and lack of contingent policies as mentioned earlier,
which are beyond the scope of this paper and left for future research.
6.5. Circular Service Area
So far we have compared serving dedicated and batch on a disk-shaped service area where customers
are uniformly located inside the disk. In this subsection, we consider a service area that only
constitutes the edge of the disk, i.e., the circumference of the circle. That is, we still have the facility
located at the center of the disk but orders are only coming from locations that are uniformly
distributed on the edge of the disk with radius r. This type of city structure has been examined
by many researchers before, most notably by Salop (1979). The so-called “circular city” model
has a lot of practical implications because many major cities have this kind of circular or ring
structure (e.g., Beijing and Moscow). These cities have massive business areas in the inner rings,
with residential areas surrounding the city center in an outer ring. The circular city model also
captures scenarios where the storage warehouse is in a relatively remote area and couriers have
to travel long distances in each direction to reach the nearest residential area. Furthermore, it
also serves as an extreme case where customers’ locations are not uniformly distributed inside the
service area.
As we do in the previous sections, we propose an appropriate queueing system and analyze the
delivery strategies for it. When serving dedicated, the service time for each order is deterministic
since the time travel from the center to any point on the edge of the circle is fixed. Thus, this
delivery system can be treated as an M/D/1 system under dedicated delivery. We still denote the
effective demand rate for this system by λ, under the arrival rate Λr2. Then this M/D/1 queue has
the service rate and load factor as
µD,C =1
2r, and ρD,C =
λ
µD,C= 2λr, (25)
respectively, where the subscript represents dedicated in a circular service area. The wait time for
a customer in this system is simply characterized by function WD(λ,µD,C ,CD,C) in (5) with
CD,C = 1, (26)
![Page 27: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/27.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 27
since the arrival process is Poisson and the service process is deterministic. Furthermore, the revenue
function can be written as
VD(λ,WD(λ,µD,C ,CD,C)) = λ
[F−1
(1− λ
Λr2
)− cWD(λ,µD,C ,CD,C)
].
Next, we consider serving batch in a circular service area with radius r. Again, the arrival process
has the inter-arrival time following an Erlang-2 distribution. The service time needs to include
three parts: first, the travel time from the center of the circle to a random point on its edge; second,
the travel time between two uniformly distributed points on the edge of the circle; and finally
the travel time from the edge of the circle back to the center. Denote by random variable Y the
distance a courier needs to travel per trip. Then we have random variable Y following a uniform
distribution on [2r,2r+πr], with
E[Y ] = 2r+πr
2, and σ2
Y =(πr)2
12.
Thus, this queueing system has the service rate and load factor as
µB,C =1
E[Y ]=
1
2r+ πr2
, and ρB,C =λ
2µB,C=λr(4 +π)
4, (27)
respectively, where the subscript represents batch in the circular service area.
Next, again, we use Kingman’s formula to approximate the average wait time for each customer.
The average wait time for each order follows from WB(λ,µB,C ,CB,C), with
CB,C =1
2+
σ2Y
(E[Y ])2=
1
2+
π2
3(4 +π)2. (28)
Furthermore, the revenue function in (13) incorporates the adjusted wait time as
VB(λ,WB(λ,µB,C ,CB,C)) = λ
[F−1
(1− λ
Λr2
)− cWB(λ,µB,C ,CB,C)
], λ∈ (0,2µB,C) .
We find that all the major results in Sections 4 and 5 still hold even if we change the service
area from a disk to a circle. We relegate the formal statements and detailed derivations to Online
Appendix C.3 to avoid repetition.
Due to the change in the city’s geometry, there are no orders coming from areas inside the disk.
Thus, the courier always needs to travel a fair distance before reaching the delivery area (the outer
ring). As a result, the thresholds for switching delivery strategy also change, though the threshold
structure remains. The next proposition provides the relationship between thresholds in a circular
city and those in the base model.
Proposition 9. Assume a crowded market and suppose that the demand rate can be endoge-
nized.
![Page 28: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/28.jpg)
28 Chen, Hu: Courier Dispatch in On-Demand Delivery
(i) In a circular service area, there exist thresholds c∞ and r∞ such that the vendor should serve
batch if the wait cost and radius parameters c and r fall below the thresholds, respectively.
Otherwise, the vendor should serve dedicated.
(ii) We have
c∞ > c∞, and r∞ > r∞, (29)
where c∞ and r∞ are thresholds in Propositions 3 and 4, respectively, for the counterpart of
the base model serving the entire disk.
Proposition 9 shows that both thresholds on the wait cost coefficient and service radius are
lower if orders only come from the edge of the disk. The reason is that the courier needs to travel
to the edge of the disk before benefiting from the pooling effect of batch delivery. Thus, serving
dedicated has more advantage in this setting. This implies that dedicated delivery will more likely
be beneficial when the orders tend to be distributed on the outskirts of a service region than when
they have a more uniform distribution inside the region.
6.6. Multiple Couriers
So far we have focused on cases with a single courier. Suppose the vendor hires k couriers to serve
the disk-shaped delivering area at the same time. We reassess the performance of dedicated vs.
batch delivery.
First, note that having k couriers does not change the service process for each courier individually.
Thus, when serving dedicated, the arrival process is still Poisson and the service rate remains µD
for each courier. However, the load factor is different since we have k couriers instead of one. That
is, we have
ρD,k =λ
kµD=
4λr
3k,
where the subscript of the load factor represents serving dedicated with k couriers. Thus, this
service system can be analyzed through an M/G/k queue. In order to obtain a tractable expected
wait time, we utilize two approximations together. Recall that the summation of coefficients of
variation of the arrival and service processes is CD = 9/8. We approximate the in-line delay of an
M/G/k queue as
Wq {M/G/k} ≈ CD2Wq {M/M/k} ≈ CD
2
ρ
√2(k+1)
D,k
λ(1− ρD,k), (30)
where we first use an M/M/k queue with the same input to approximate the in-line delay of the
M/G/k counterpart (see, e.g., Gross et al. 2008, p. 345), and then use a well-studied approximation
![Page 29: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/29.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 29
for the M/M/k queue itself (see, e.g., Sakasegawa 1977). This approximation is consistent with
recent studies on on-demand economy, see, e.g., Bai et al. (2019), Taylor (2018), and Benjaafar
et al. (2020). As the result of such an approximation, the expected wait time for each customer is
WF,k(λ)≈ CD2
ρ
√2(k+1)
D,k
λ(1− ρD,k)=
CD2(kµD−λ)
(λ
kµD
)√2(k+1)−1
. (31)
After these setups, the revenue function is simply V (λ,WF,k(λ)).
Next, we consider serving batch. Similar to serving dedicated, each courier’s service rate µB =45π
4r(32 + 15π)remains the same but the load factor needs to take k couriers into consideration.
That is, we have
ρB,k =λ
2kµB=
2λr(32 + 15π)
45kπ.
Using the same approximation method as in (30), which can be applied to G/G/k systems as well,
the expected wait time for each customer is
WB,k(λ)≈ 1
2λ+
CB2(2kµB −λ)
(λ
2kµB
)√2(k+1)−1
+r
3. (32)
It is worth pointing out that under this approximation scheme, the queueing systems reduce to
those in Section 3 where k= 1, both for dedicated and for batch.
With multiple couriers, we can still show analytically that there is a threshold on the exogenous
demand rate below which serving dedicated generates higher revenue and above which serving
batch is more profitable. This result is consistent with Proposition 1. If the vendor can endogenize
the demand, there is still a threshold on customers’ wait cost parameter above which it is optimal to
serve dedicated, consistent with Proposition 7. We leave the formal statements of these analytical
results in Online Appendix C.4. Other results in Sections 4 and 5, such as Propositions 2 and 8, are
very difficult to prove analytically with multiple couriers. However, we still observe these results
in our numerical experiments.
Figure 7 provides two examples. Figure 7(a) shows the relationship between the expected wait
time when serving dedicated and that when serving batch. Similar to Proposition 2(i) and Figure
3(a), there appears to be a threshold on the service radius r below which serving dedicated leads
to a shorter wait time than serving batch when the demand rate λ is fixed (and vice versa).
Furthermore, Figure 7(b) gives an example of the revenue function with respect to the service
radius. As we can see, serving dedicated still outperforms batch when the service radius is large
enough, just as in Proposition 8 in Section 5.
![Page 30: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/30.jpg)
30 Chen, Hu: Courier Dispatch in On-Demand Delivery
(a) (b)
Figure 7 Expected wait time and revenue functions with multiple couriers.
(a) k= 5, and λ= 0.5, (b) k= 3, c= 0.1, and Λ = 10
7. Conclusion
In this paper, we compare and contrast the fundamentals of using dedicated vs. pooling delivery
strategy. We model the two strategies as queueing systems serving dedicated and batch, respec-
tively. In addition, we incorporate a spatial feature in these systems using a generalized circular
city model. This spatial feature makes our service system relevant to the daily operations of the
on-demand delivery industry. We highlight the scenarios in which dedicated or pooling delivery
strategy is optimal, and our results remain robust in a variety of extensions.
Our research contributes to the literature of innovative operations and smart cities. One of the
major managerial insights is that, contrary to the common belief, temporal pooling, such as serving
batch, may not always increase delivery efficiency in a large service area and lead to higher revenue
for the vendor. When the vendor can endogenize the demand, pooling should only be used when
the vendor can profitably sustain a relatively large demand rate. With impatient customers or a
large service area, it is better for the vendor to use the dedicated strategy, but charge a relatively
high delivery price. This paper also contributes to the spatial queueing literature by providing an
analytically tractable framework using a generalized circular city model, which is relevant to many
practical applications. In particular, our model accurately depicts delivery systems with a small
number of orders per trip.
This paper can shed light on operational strategies for on-demand delivery services in the emerg-
ing markets such as food or grocery delivery where dedicated couriers or robots/drones are deployed
to make deliveries. Although our focus is mainly on investigating the benefit of temporal and spa-
tial pooling in delivery, there are many other interesting research questions in the delivery business.
Our modeling framework can potentially serve as a building block for future research in areas such
as, but not limited to, contracting and compensation for couriers and incentive management with
freelancers, e.g., by endogenizing the number of dedicated couriers or freelancers in a shift through
a wage decision or a payout contract, which is currently missing in the model.
![Page 31: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/31.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 31
Our work is not without limitations. First, given a general arrival rate, for some parameters, e.g.,
when the wait cost parameter or the service radius has a sufficiently low value, we cannot obtain
an unambiguous preference for the dedicated or pooling delivery strategy. For these parameters
and a general valuation distribution, one needs to resort to a numerical comparison. Second, the
empirical demand distribution is most likely not a uniform distribution. A data-driven approach
needs to be adopted to prescribe the best strategy for a specific practical setting. Third, we assume
that the couriers are employees of the vendor and their delivery speed is independent of their
workload. As mentioned, the compensation and behavioral issues for couriers may also need to be
examined. Finally, we assume the firm commits to either the dedicated or pooling strategy, as the
resulting pricing and response time could be easily conveyed to consumers. In practice, the firm
can further improve its performance by making optimal contingent decisions about dispatching
and routing depending on the realized locations of outstanding orders, which are outside the scope
of our stylized model.
References
Azi, N., Gendreau, M., and Potvin, J.-Y. (2012). A dynamic vehicle routing problem with multiple delivery
routes. Ann. Oper. Res., 199(1):103–112.
Bai, J., So, K., Tang, C., Chen, X., and Wang, H. (2019). Coordinating supply and demand on an on-demand
service platform with impatient customers. Manufacturing Service Oper. Management, 21(3):556–570.
Benjaafar, S., Ding, J.-Y., Kong, G., and Taylor, T. (2020). Labor welfare in on-demand service platforms.
Manufacturing Service Oper. Management. Forthcoming.
Berman, O., Larson, R., and Chiu, S. (1985). Optimal server location on network operating as an M/G/l
queue. Oper. Res., 33:746–771.
Berman, O., Larson, R., and Parkan, C. (1987). The stochastic queue p-median problem. Transportation
Sci., 21(3):207–216.
Bertsimas, D. and Van Ryzin, G. (1990). A stochastic and dynamic vehicle routing problem in the euclidean
plane. Oper. Res., 39(4):601–615.
Bertsimas, D. and Van Ryzin, G. (1992). Stochastic and dynamic vehicle routing in the euclidean plane with
multiple capacitated vehicles. Oper. Res., 41(1):60–76.
Buzzell, R., Gale, B., and Sultan, R. (1975). Market share—a key to profitability. Harvard Business Review,
pages 97–106.
Cachon, G. (2014). Retail store density and the cost of greenhouse gas emissions. Management Sci.,
60(8):1907–1925.
![Page 32: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/32.jpg)
32 Chen, Hu: Courier Dispatch in On-Demand Delivery
Cao, J., Ma, C., and Qi, W. (2020a). Stall economy: The value of mobility and precision deployment of retail
on wheels. Working paper.
Cao, J., Olvera-Cravioto, M., and Shen, Z.-J. (2020b). Last-mile shared delivery: A discrete sequential
packing approach. Math. Oper. Res. Forthcoming.
Carlsson, J. and Song, S. (2017). Coordinated logistics with a truck and a drone. Management Sci.,
64(9):4052–4069.
Chen, M., Hu, M., and Wang, J. (2020a). Food delivery service and restaurant: Friend or foe? Management
Sci. Forthcoming.
Chen, M., Sun, P., and Wan, Z. (2020b). Matching supply and demand with mismatch-sensitive players.
Working paper.
Cui, S., Wang, Z., and Yang, L. (2019). The economics of line-sitting. Management Sci., 66(1):227–242.
Cui, S., Wang, Z., and Yang, L. (2020). A model of queue-scalping. Management Sci. Forthcoming.
Feng, G., Kong, G., and Wang, Z. (2020). We are on the way: Analysis of on-demand ride-hailing. Manu-
facturing Service Oper. Management. Forthcoming.
Frazelle, A., Swinney, R., and Feldman, P. (2020). Can delivery platforms benefit restaurants? Working
paper.
Gross, D., Shortle, J., Thompson, J., and Harris, C. (2008). Fundamentals of Queueing Theory. John Wiley
& Sons, 4 edition.
He, L., Liu, S., and Shen, Z.-J. (2020). On-time last-mile delivery: Order assignment with travel-time
predictors. Management Sci. Forthcoming.
He, L., Mak, H., Rong, Y., and Shen, Z.-J. (2017). Service region design for urban electric vehicle sharing
systems. Manufacturing Service Oper. Management, 19(2):309–327.
Jargon, J. (2018). Starbucks to offer coffee delivery across U.S. Wall Street J. (Dec. 14).
Kingman, J. (1962). Some inequalities for the queue GI/G/l. Biometrika, pages 383–392.
Klapp, M., Erera, A., and Toriello, A. (2018a). The dynamic dispatch waves problem for same-day delivery.
Eur. J. Oper. Res., 52(2):402–415.
Klapp, M., Erera, A., and Toriello, A. (2018b). The one-dimensional dynamic dispatch waves problem.
Transportation Sci., 271(2):519–534.
Liu, Z. and Huang, H. (2021). Operating three-sided marketplace: Pricing, spatial staffing and routing in
food delivery systems. Working Paper.
Mak, H. (2020). Enabling smarter cities with operations management. Manufacturing Service Oper. Man-
agement. Forthcoming.
Mao, W., Ming, L., Rong, Y., Tang, C., and Zheng, H. (2019). Faster deliveries and smarter order assignments
for an on-demand meal delivery platform. Working paper.
![Page 33: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/33.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 33
Melton, A. (2021). The pizza business is divided on delivery. Wall Street J. (May. 30).
Qi, W., Li, L., and Shen, Z.-J. (2018). Shared mobility for last-mile delivery: Design, operational prescriptions,
and environmental impact. Manufacturing Service Oper. Management, 20(4):737–751.
Rana, P. and Haddon, H. (2021a). DoorDash and Uber Eats are hot. They’re still not making money. Wall
Street J. (May 28).
Rana, P. and Haddon, H. (2021b). Restaurants and startups try to outrun Uber Eats and DoorDash. Wall
Street J. (Feb. 21).
Rana, P. and Kang, J. (2021). For DoorDash and Uber Eats, the future is everything in about an hour.
Wall Street J. (May. 31).
Sakasegawa, H. (1977). An approximation formula Lq ≈ αρβ/(1− ρ). Annals of the Institute of Statistical
Mathematics, 29(1):67–75.
Salop, S. C. (1979). Monopolistic competition with outside goods. Bell J. Econ., 10(1):141–156.
Solomon, H. (1978). Geometric Probability. SIAM.
Szymanski, D., Bharadwaj, S., and Varadarajan, P. (1993). An analysis of the market share-profitability
relationship. J. Marketing, 57(3):1–18.
Taylor, T. (2018). On-demand service platforms. Manufacturing Service Oper. Management, 20(4):704–720.
Topkis, D. M. (1978). Minimizing a submodular function on a lattice. Oper. Res., 26(2):305–321.
Ulmer, M., Thomas, B., and Mattfeld, D. (2018). Preemptive depot returns for dynamic same-day delivery.
EURO J. Transp. Logist., 8:327–361.
Voccia, S., Campbell, A., and Thomas, B. (2019). The same-day delivery problem for online purchases.
Transportation Sci., 53(1):167–184.
Yildiz, B. and Savelsbergh, M. (2019). Service and capacity planning in crowd-sourced delivery. Transporta-
tion Res. C, 100:177–199.
![Page 34: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/34.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 1
Online Appendix to“Courier Dispatch in On-Demand Delivery”:
Supplementary Derivations and Proofs
A. Major Proofs
Proof of Propositions 1 and 2. We first show the result of Proposition 1. First, note that
we have limλ→0WB(λ,µB,CB) =∞ and limλ→µDWD(λ,µD,CD) =∞, according to (4) and (12).
Thus, we only need to show that WD(λ,µD,CD)−WB(λ,µB,CB) = 0 has a unique solution in λ to
established the desired threshold result in λ.
Consider function
f(a) :=1
λ
[a2C1
2T1(T1− a)− 1
2− a2C2
T2(T2− a)− a
3
], ∀a> 0, (A.1)
where
T1 =3
4, T2 =
45π
2(32 + 15π), C1 =
9
8, C2 = 0.583. (A.2)
Then, according to (4) and (12), we have f(λr) =WD(λ,µD,CD)−WB(λ,µB,CB). Note that f(a) =
0 is a single variable cubic equation, which can be solved using standard methods. In particular,
equation f(a) = 0 only has two positive solutions: a1 ≈ 0.5689 and a2 ≈ 1.150 (the exact symbolic
solutions with T1, T2, C1, C2 are cumbersome, thus omitted). Thus, we have 0� a1� T1 = µDr and
T1� a2, which implies that 0 =WD(λ,µD,CD)−WB(λ,µB,CB) has a unique solution λex = a1/r
on (0, µD).
We use the same technique to show Proposition 2. According to (4) and (12), with slight abuse
of notation we write WB(λ,µB,CB) and WD(λ,µD,CD) as WB(r,λ,µB,CB) and WD(r,λ,µD,CD),
respectively, to emphasize their dependence on r. have
limr→0
WB(r,λ,µB,CB) =1
2λ> 0 = lim
r→0WD(r,λ,µD,CD),
and limr→T1WD(r,λ,µD,CD) =∞. Next, since a1 is the unique solution to f(a) = 0 on a∈ (0, T1),
there is a unique rex = a1/λ such that WD(r,λ,µD,CD) = WB(r,λ,µB,CB). This completes the
proof.
Proof of Corollary 1. (i) This part follows Proposition 1 and 2 directly, where we have shown
that serving dedicated leads to shorter wait time comparing to serving batch when either the
demand is low or the radius is small, and vice versa. Thus, using the expression for the price in
(1), we reach the desired result.
![Page 35: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/35.jpg)
2 Chen, Hu: Courier Dispatch in On-Demand Delivery
(ii) We use the exactly same proof technique as in the proof of Proposition 1 and 2. We only
need to modify the definition of function f(·) is in (A.1) to
f(a) :=1
λ
[a2C1
2T1(T1− a)− 1
4− a2C2
2T2(T2− a)− a6
], ∀a> 0,
where parameters T1, T2, C1, and C2 are defined in (A.2). Then, we have f(λr) =WD(λ,µD,CD)−
WB(λ,µB,CB)/2. Again, function f(a) = 0 is a simple cubic equation, having a unique solution on
a≈ 0.5087∈ (0, T1]. We omit the details to avoid repetition.
Before proving Propositions 3 and 4, we first provide some properties of the optimal demand
rate. For notational convenience, denote
V ∞D (λ) := V∞(λ,WD(λ,µD,CD)) and V ∞B (λ) := V∞(λ,WB(λ,µB,CB)),
and the optimal solutions to the optimization problems
maxλ∈[0,µD)
V ∞D (λ), and maxλ∈[0,2µB)
V ∞B (λ), (A.3)
as λ∞F and λ∞B , respectively. Further, denote TB = 2µBr=45π
2(32 + 15π)and TD = µDr=
3
4.
Next, we present the optimal solutions and objective values to the optimization problems in
(A.3). Note that the objective functions in (A.3) are strictly concave in λ, since
d2V ∞D (λ)
dλ2=
cCDTD
r(λ− TDr
)3< 0, and
d2V ∞B (λ)
dλ2=
2cCBTB
r(λ− TBr
)3< 0,
respectively. Furthermore, by solving the first order conditions when cr < TB < 2TD, we have
dV ∞D (λ)
dλ= 1 +
crCD
(1− T2
D(TD−λr)2
)2TD
= 0, anddV ∞B (λ)
dλ= 1 +
crCB
(1− T2
B(TB−λr)2
)TB
− cr3
= 0,
respectively, which imply
λ∞F =TDr
(1− crCD√
crCD[2TD + crCD]
), λ∞B =
TBr
(1−
√3crCB√
crCB[3TB + cr(3CB −TB)]
),
(A.4)
respectively, and
V ∞D (λ∞F ) = cCD +TDr− 1
r
√crCD[2TD + crCD], (A.5)
V ∞B (λ∞B ) = c
(2CB −
1
2− 1
3TB
)+TBr− 4
r
√3crCB[3TB + cr(3B −TB)], (A.6)
where we have omitted the solutions that are outside the feasible regions.
![Page 36: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/36.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 3
Proof of Proposition 3. Fix r > 0 and define function
g(α) := r (V ∞B (λ∞B )−V ∞D (λ∞F )) = α
(2CB −
1
2− TB
3
)+TB − 4
√3αCB[3TB +α(3CB −TB)]
−[αCD +TD−
√αCD[2TD +αCD]
], (A.7)
where α := cr. By plugging in CD = 9/8, CB = 0.583, TD = 3/4, TB = 45π/(2(32+15π)) and solving
g(α) = 0, we obtain a unique solution α∗ ≈ 0.1355, which implies that function g(·) only “crosses” 0
once. Finally, one can easily verify that limα→0 g(α)> 0. Thus, we have g(α)> 0 for all α∈ (0, α∗)
and g(α)≤ 0 when α≥ α∗. Therefore, it is better for the vendor to serve dedicated if c≥ α∗/r and
to operate batch otherwise, which implies the first statements in Proposition 3.
Next, we show the second statements in Proposition 3. Recall that the threshold on the wait
cost c can be expressed as α= cr and the vendor switches from serving batch to dedicated when
α= α∗. Thus, we only need to show that λ∞B >λ∞F when cr= α= α∗, where the optimal non-zero
demands λ∞F and λ∞B are defined in (A.4).
Note that for fixed r and cr= α∗ ≈ 0.1355, we have
λ∞B −λ∞F =1
r
(TB −TD +
αCDTD√αCD[2TD +αCD]
−√
3αCB√αCBTB[3TB +α(3CB −TB)]
)> 0,
where we plug in the value of α∗, CD, CB, TD, and TB to reach the inequity. Thus, we conclude
that when switching from serving batch to dedicated, the optimal demand rate decreases. Finally,
recall the revenue function in (2) equals to demand times price, i.e., λp. When the optimal demand
switches from λB to λF < λB at cr = α∗, the corresponding optimal price surges upwards accord-
ingly. This completes the proof.
Proof of Proposition 4. This proof follows from the same steps as the proof of Proposition 3.
Fix c > 0 and define function
h(α) :=1
c(V ∞B (λ∞B )−V ∞D (λ∞F )) =
(2CB −
1
2− TB
3
)+TBα−
4√
3αCB[3TB +α(3CB −TB)]
α
−
[CD +
TDα−√αCD[2TD +αCD]
α
], (A.8)
where α= cr. By solving h(α) = 0, we obtain the same unique solution α∗ ≈ 0.1355. We omit the
details for the rest of the proof to avoid repetition.
Proof of Proposition 5. To show the first statement, we show that the social welfare function
in (18) is an non-increasing function w.r.t. w by taking the first order derivative:
∂SW (w)
∂w=−c
∫ 1
F−1(1− λΛr2
)dF (v) =−c
(1− λ
Λr2
)≤ 0.
![Page 37: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/37.jpg)
4 Chen, Hu: Courier Dispatch in On-Demand Delivery
Therefore, when fixing λ, the smaller expected wait time translates to higher social welfare. There-
fore, the first statement in Proposition 5 simply follows Propositions 1 and (2).
In order to show the second statement, we consider the large market regime and redefine the
social welfare rate as
SWn(λ,w) = Λnr2
∫ 1
F−1(1− λΛnr2
)(v− cw)dF (v). (A.9)
First, note that we have
limn→∞
SWn(λ,w) = limn→∞
Λnr2
∫ 1
F−1(1− λΛnr2
)(v− cw)dF (v) (A.10)
= λ
∫ 1
F−1(1− λΛnr2
)(v− cw)dF (v)∫ 1
F−1(1− λΛnr2
) dF (v)(A.11)
= λ limn→∞
−dF−1(1− λ
Λcr2)
dn(F−1
(1− λ
Λcr2
)− cw)f
(F−1
(1− λ
Λcr2
))−dF−1(1− λ
Λcr2)
dnf(F−1
(1− λ
Λcr2
)) (A.12)
= λ(1− cw) := SWinf(λ,w), (A.13)
where the first equality follows (1); the second equality follows L’Hopital rule and Leibniz rule.
Finally, note that we have both max{SWn(λ,WD(λ,µD,CD)),0} and
max{SWn(λ,WB(λ,µB,CB)),0} are Lipschitz continuous. To see this, take
max{SWn(λ,WD(λ,µD,CD)),0}= max
{−Λnr2
∫ 1
F−1(1− λΛnr2
)v dF (v) +λcWD(λ,µD,CD),0
},
as the example. It is easy to verify there exists a λ such that max{SWn(λ,WD(λ,µD,CD)),0}= 0
for all λ ∈ [λ, µD) since WD(λ,µD,CD) is convex, increasing and approaching infinity as λ→ µD.
Thus, we only need to focus on λ ∈ [0, λ]. Note that the derivative of −Λnr2
∫ 1
F−1(1− λΛnr2
)v dF (v)
is
Λnr2∂F−1
(1− λ
Λnr2
)∂λ
F−1
(1− λ
Λnr2
)f
(F−1
(1− λ
Λnr2
))<∞,
since F−1 (·) is a Lipschitz continuous function and thus∂F−1
(1− λ
Λnr2
)∂λ
is finite. Therefore,
the term −Λnr2
∫ 1
F−1(1− λΛnr2
)v dF (v) is also Lipschitz continuous. It is also straight forward
to verify that λcWD(λ,µD,CD) is Lipschitz continuous (see the proof of Lemma 1). Thus,
max{SWn(λ,WD(λ,µD,CD)),0} is a Lipschitz continuous function. Then one can show that
limn→0
maxλ∈[0,µD)
SWn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)
limn→0
SWn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)
λ (1−WD(λ,µD,CD))
![Page 38: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/38.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 5
limn→0
maxλ∈[0,2µB)
SWn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)
limn→0
SWn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)
λ (1−WB(λ,µB,CB)) ,
following the exactly same proof techniques in the one of Lemma 1. We omit the details to avoid
repetition.
Thus, under a crowded market, social welfare maximization over the demand rate is equivalent
to revenue maximization in Section 5, thus, producing the same results.
Proof of Proposition 6. We prove this result using a coupling argument. First, we recognize
that the underlying stochasticity is the Poisson arrival of orders with rate λ and their locations,
which are independent and uniformly distributed on a disk. Furthermore, in all dispatch policies,
since we normalized courier’s speed to 1, the service time is simply the travel distance.
Denote LD, and LC as the number of unfilled orders under dedicated and contingent policies,
respectively. We show that there is path-wise dominance: LD ≥ LC when the underlying random
events are coupled.
Consider an hypothetical alternative policy (short for “alternative contingent policy”), which
mimics the contingent policy and following the delivery decision for each order. To be more specific,
if two arrivals under the contingent policy are served in a batch, in the alternative policy, they
are also served in the same batch. However, unlike the original contingent policy, the courier does
not travel the shortest distance between locations when serving batch. Instead, the courier travels
to the first location but then travels back to the hub before heading to the second location. That
is, although two orders are leaving the hub together, the delivery routing is actually the same as
that of dedicated. We first argue that the alternative policy is feasible at any time t, only requiring
information up to time t of the contingent policy, and does not require clairvoyant information.
Note that the two contingent systems behave the same until the first time two orders needs to
be batched. For any two orders with location x1 and x2, the travel distance under the contingent
policy is always no greater than that of the alternative policy due to triangular inequality (i.e.
|x1|+ |x2| ≥ |x1−x2|). So whenever a batch delivery occurs, the alternative policy induces longer
service time. Thus, by an induction argument, the next delivery decision under the contingent
policy (and thus the alternative policy) is always made no later than the time either the courier
becomes idle or the arrival of the next order in the alternative policy, requiring no clairvoyant
information. Next, denote LA as the number of unfilled orders. Since the decisions of which orders
are served in batches are the same between the two systems, and the alternative policy leads to no
shorter service time, we have LA ≥LC path-wise.
Next, note that the alternative policy is identical to the dedicated policy in terms of the total
service time when coupled together, since the two policies induces the exactly same routing policy.
![Page 39: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/39.jpg)
6 Chen, Hu: Courier Dispatch in On-Demand Delivery
The only difference between the two policies is that whenever a batch decision is made in the
alternative policy, two orders leave the queue, but in the corresponding dedicated system, only one
order leave the system at a time. Therefore, we have LD ≥LA and the (strict) inequality only holds
during periods when the courier serves batch and is completing the first order in a batch of two.
Thus, we have reached LD ≥LA ≥LC . Finally, the results in expected wait time holds by Little’s
Law.
We prove the second statement by considering extreme cases. When the demand rate is close
to zero, the contingent policy leads to shorter expected wait time since the time to accumulate
orders in batch goes to infinity. On the other hand, when the demand rate is very large, which
leads to shorter order accumulation time than the travel time for an single order (1/λ� 2r), then
it is beneficial for the courier to serve batch. This completes the proof.
Before proving the rest of Propositions in Section 6, we first present a lemma on the properties
of the revenue function of serving batch, when customer valuations follow a standard uniform
distribution without the large market assumption.
Lemma 2. Consider function
g(λ, r, c,Λ) := λ
{1− λ
Λr2− c[
1
2λ+
λr2C
T (T −λr)+r
3
]}, (A.14)
with C,T > 0.
(i) Fix Λ, r > 0. Function g is submodular in (λ; c) for c > 0 and λ∈ (0, T/r).
(ii) Fix c,Λ> 0 and C < 1. Function g is submodular in (λ; r) for r >
(T 2
cCΛ
) 14
and λ∈ (0, T/r).
(iii) Fix c, r > 0. Function g is supermodular in (λ;Λ).
For the proofs of Propositions 7 and 8, denote by λ∗D and λ∗B the optimal solutions to (19) and
(20), respectively.
Proof of Proposition 7. We show that, when fixing r,Λ> 0, there exist some cen such that we
have VB(λ∗B,WB(λ∗B, µB,CB))≤ VD(λ∗D,WD(λ∗D, µD,CD)) if c≥ cen.
With slight abuse notation, denote the optimal solution to
maxλ∈[0,2µB)
λ
[1− λ
Λr2− cWB(λ,µB,CB)
], (A.15)
by λ∗B(c) for every c > 0. Also recall Proposition 1 and denote by λex the unique solution to
f(λ, r) = 0 for r > 0, which is independent of c.
Consider T =45π
2(32 + 15π)and C = CB for function g defined in (A.14). Recall CB ≈ 0.583< 1
and we recognize function g is the revenue function in (A.15) with T = TB and C =CB, so we can
![Page 40: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/40.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 7
apply the result of Lemma 2 (i) directly. Thus, we have λ∗B(c) is decreasing when c is increasing
using Topkis’s Theorem (see, e.g., Topkis 1978) since the objective function is submodular in (λ; c).
It is obvious that when c is large enough, we must have λ∗B(c) = 0. Thus, as λex > 0 is independent
w.r.t. c where λex is the threshold on which the wait times are equal under dedicated and batch in
Proposition 1, there exists some cen ∈ (0,2µB) such that for all c > cen, we have λex >λ∗B(c) since
λ∗B(c) is decreasing w.r.t. c. Therefore, when c≥ cen, we have
VB(λ∗B(c),WB(λ∗B(c), µB,CB)) ≤ λ∗B(c)
[1− λ
∗B(c)
Λr2− cWD(λ∗B(c), µD,CD)
]≤ VD(λ∗D,WD(λ∗D, µD,CD)),
where the first inequality follows from Proposition 1 since λex > λ∗B(c) and the second inequality
follows from the definition of λ∗D as the optimal solution.
Proof of Proposition 8. We show that, when considering Λ, c > 0 such thatΛ
c3> L :=
1 + 8(√
2CB + 6CB + 8√
2C3B + 8C2
B)
16CB
(2(32 + 16π)
45π
)2
≈ 13.39, there exists a threshold, ren, on the
service radius r, such that we have VB(λ∗B,WB(λ∗B, µB,CB))≤ VD(λ∗D,WD(λ∗D, µD,CD)) if r≥ ren.
Denote by λ∗B(r) the optimal solution to
maxλ∈[0,1/(2µB))
λ
[1− λ
Λr2− cWB(λ,µB,CB)
], (A.16)
when fixing r > r := 3
√5π
32 + 15π
1
(0.583Λc)14
. Note that we only need to consider the case where
λ∗B(r) solves the first order condition since any λ∗B(r) that approaches the boundaries leads
to a negative objective value, which implies VD(λ∗D,WD(λ∗D, µD,CD)) ≥ VB(λ∗B,WB(λ∗B, µB,CB))
immediately as VD(λ∗D,WD(λ∗D, µD,CD)) ≥ 0. Furthermore, note that by plugging in T =2µBr
=
45π
2(32 + 15π)and C = CB, function g in (A.14) is the revenue function of serving batch in (20).
Since we have
CB ≈ 0.583< 1, and r > r= 3
√5π
32 + 15π
1
(0.583Λc)14
=
(T 2
ΛcCB
) 14
,
we can apply Lemma 2(ii) directly and conclude that λ∗B(r) is decreasing w.r.t r when r > r.
Denote r= 45π/(c(32 + 15π)(1 + 2√
1.166)). When r > r have
λ
[1− λ
Λr2− cWB(λ,µB,CB)
]= λ
{1− λ
Λr2− c[
1
2λ+
λr2CBTB(TB −λr)
]}≤ λ
{1− c
[1
2λ+
λr2CBT (T −λr)
]}≤ λ
{1− cr1 + 2
√2CB
2T
}≤ 0,
![Page 41: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/41.jpg)
8 Chen, Hu: Courier Dispatch in On-Demand Delivery
where the second inequality follows the fact that the term1
2λ+
λr2CBT (T −λr)
is convex in λ and
attains its minimum r1 + 2
√2CB
2Twhen λ=
T (√
2CB − 1)
r(2CB − 1), and the second inequality follows r > r.
Note that there is
r=45π
c(32 + 15π)(1 + 2√
1.166)=
2T
c(1 + 2√
2CB)>
(T 2
ΛcCB
) 14
= r,
where the inequality follows fromΛ
c3>L. As λ∗B(r) is decreasing w.r.t. r, we have limr→r λ
∗B(r) = 0.
Recall that for a finite r, we have λex, the solution to WD(λ,µD,CD) =WB(λ,µB,CB), never equals
to 0. Thus, denote ` = minr∈(r,r) λex, which is strictly greater than 0. Then by definition of the
limit, there exists some ren ∈ (r, r) such that λ∗B(r)≤ `≤ λex for all r≥ ren.
Finally, consider r≥ ren. We have
VB(λ∗B,WB(λ∗B(r), µB,CB)) ≤ λ∗B(r)
[1− λ
∗B(r)
Λr− cWD(λ∗B(r), µD,CD)
]≤ VD(λ∗D,WD(λ∗D, µD,CD)),
where the first inequality follows from Proposition 1 since we have λ∗B(r)≤ λex for all r≥ ren and
the second inequality follows from that λ∗D is the optimal solution. This completes the proof.
Proof of Proposition 9. The proof of the first statement follows from the exactly same steps
as the proof of Propositions 3 and 4. We only need to substitute the values of constants by TB =
2µB,Cr=4
4 +π, TD = 2µD,Cr=
1
2, CB,C =
1
2+
π2
3(4 +π)2, and CD,C = 1. We still denote α= cr and
let α∗ be the new threshold in cr when serving a circular region. We have α∗ ≈ 0.057� 0.1809≈ α∗,
where α∗ is the optimal value of alpha when the service region is a disk. That is, when cr ≥ α∗,
serving dedicated is better than batch, and vice versa. We omit the details to avoid repetition.
B. Supplementary Proofs in Section 5
Proof of Lemma 1. We only show the argument for function Vn(λ,WD) since the same steps can
be applied to function Vn(λ,WB).
Denote function Jn := min{−Vn(λ,WD),0}. Thus, maximizing Vn(λ,WD) is equivalent to mini-
mizing Jn. Furthermore, by equation (15), denote function
J := limn→∞
Jn = min{−λ(1− cWD(λ,µD,CD)),0}.
The rest of this proof is broken into three steps:
1. We show that for each λ∈ [0, µD), there exists a sequence {λ′n} converging to λ such that
limn→∞
Jn(λ′n) = J(λ). (B.1)
![Page 42: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/42.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 9
2. We show that for every λ∈ [0, µD) and for every sequence {λ′n} converging to λ, there is
lim infn→∞
Jn(λ′n) = J(λ). (B.2)
3. Once conditions in Steps 1 and 2 are satisfied, we have that function Jn Γ−converges (see,
e.g., Dal Maso 1993) to function J , which implies that
limn→∞
minJn(λ) = min limn→∞
Jn(λ).
Thus, the desired result can be obtained.
Step 1: We show a stronger result here: for every λ ∈ [0, µD) and for every sequence {λn} con-
verging to λ, the equation in (B.1) holds. This stronger result also helps us to show the statement
in Step 2.
We begin by showing that function Jn is Lipschitz continuous for every n ∈N. First, note that
0≤ F−1
(1− λ
Λnr2
)≤ 1, since w.l.o.g., the bounded support of the valuation distribution function
F is normalized to [0,1]. From the proof of Proposition 1, we know that the wait time function
WD(λ,µD,CD) is strictly convex and increasing in λ with limλ→µDWD(λ,µD,CD) =∞. Therefore,
there exists some λ < µD, such that
F−1
(1− λ
Λnr2
)− cWD(λ,µD,CD)≤ 0, so that Jn(λ) = 0, ∀λ > λ.
Furthermore, denote KF as the Lipschitz constant for function F−1 and
KW =∂WD(λ,µD,CD)
∂λ
∣∣∣λ=λ
. (B.3)
Then we have that function WD is Lipschitz continuous when λ ∈ (0, λ) with constant KW since
function WD is strictly convex from the proof of Proposition 1. Furthermore, we have function
gn(λ) :=−λ[F−1
(1− λ
Λnr2
)− cWD(λ,µD,CD)
],
is also Lipschitz continuous with constant
Kn = µD max
{KF
Λnr2, cKW
}, (B.4)
since function gn(λ)/λ is the difference between two Lipschitz continuous functions and λ < µD.
Since there is Jn = min{0, gn}, we conclude that function Jn is Lipschitz with factor Kn.
Next, we have
|Jn(λ)−J(λ)| ≤ |gn(λ) +λ(1− cWD(λ,µD,CD))|
= λ
(1−F−1
(1− λ
Λnr2
))
![Page 43: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/43.jpg)
10 Chen, Hu: Courier Dispatch in On-Demand Delivery
< µD
(1−F−1
(1− λ
Λnr2
)), (B.5)
according to the definitions of functions Jn and J , and λ< µD.
Consider any λ∈ [0, µD) and any sequence {λ′n} converging to λ. By the definition of convergence,
consider ε > 0 and we can find N1 such that there is
|λ′n−λ|<ε
2K, ∀n≥N1, (B.6)
where K = max{Kn|n ≥ N1}. Furthermore, since F−1 is Lipschitz continuous and monotone
increasing, fixing ε > 0, we can find N2 such that
1−F−1
(1− λ
Λnr2
)≤ ε
2µD, ∀n≥N2. (B.7)
Fix n>Nε := max{N1,N2}, we have
|Jn(λn)−J(λ)| ≤ |Jn(λn)−Jn(λ)|+ |Jn(λ)−J(λ)|
≤ K|λn−λ|+µD
(1−F−1
(1− λ
Λnr2
))≤ K|λn−λ|+
ε
2
≤ ε
2+ε
2= ε,
where the first inequality follows from the triangular inequality; the second inequality follows from
(B.5) and that function Jn is Lipschitz with constant Kn ≤ K; the third inequity follows from
(B.7); and the last inequality follows from (B.6). Therefore, we conclude that (B.1) holds for every
λ∈ [0, µD) and for every sequence {λn} converging to λ.
Step 2: The stronger statement we have shown in step 1 implies the desired result in this step.
Consider any λ ∈ [0, µD) and any sequence {λn} converging to λ. Fix n >Nε = max{N1,N2} and
denote m∗ ∈ arg infm≥n Jm(xm). We have
|Jm∗(λ∗m)−J(λ)| ≤ |Jm∗(λ∗m)−Jm∗(λ)|+ |Jm∗(λ)−J(λ)|
≤ K|λ∗m−λ|+µD
(1−F−1
(1− λ
Λnr2
))≤ ε.
Step 3: Now we can conclude that function Jn Γ-Converges to function J as n approaches
infinity, since we have: 1) for each λ ∈ [0, µD), there exists a sequence {λn} converging to λ such
that equation (B.1) holds from step 1; 2) equation (B.2) holds for every λ ∈ [0, µD) and for every
sequence {λn} converging to λ from step 2. Thus, by the property of Γ-Convergence, we have
limn→∞maxλ∈[0,µD) VnD (λ) = maxλ∈[0,µD) limn→∞ V
nD (λ), and this completes the proof.
![Page 44: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/44.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 11
C. Supplementary Results and Proofs in Section 6
C.1. General Arrival Rate
Proof of Lemma 2. Noting that the function g is continuous and twice differentiable w.r.t. each
variable, we can verify the statements by taking the mixed second derivatives.
(i) Fix Λ, r > 0 and we have
∂2g(λ, r, c,Λ)
∂λ∂c=Cr
(1
T− T
(T −λr)2
)− r
3=Cλr2(λr− 2T )
T (T −λr)2− r
3< 0,
where the inequality follows from λr < T .
(ii) Fix c,Λ> 0 and there is
∂2g(λ, r, c,Λ)
∂λ∂r=
4λ
Λr3+cC
T
(1− T
2(T +λr)
(T −λr)3
)− c
3(C.1)
We verify the right-hand-side of (C.1) is decreasing in λ by taking its derivative w.r.t. λ:
4
Λr3− 2crCT (2T +λr)
(T −λr)4<
4
Λr3− 4crC
T 2< 0,
where the first inequality follows that the term2crCT (2T +λr)
(T −λr)4is increasing in λ, thus, letting
λ= 0, and the second inequality follows r >
(T 2
cCΛ
) 14
. Therefore, the expression in (C.1) reaches
its maximum −c/3 when λ= 0, suggesting submodularity.
(iii) Fix c, r > 0 and we have
∂2g(λ, r, c,Λ)
∂λ∂Λ=
2λ
Λ2r2> 0,
and this completes the proof.
C.2. Discussion of the Uniform Distribution Assumption on Customers’ Valuation
As mentioned in Section 6, the assumption that function F (v) = v for v ∈ [0,1] and F (v) = 0 is
not restrictive. According to the proof of Proposition 7 and 8, all we need is that the revenue
function when serving batch VB is submodular in (λ; c) and (λ; r), respectively. Thus, as long
as the distribution function F of customer valuations induces a inverse function F−1 leading to
submodularity, similar to Lemma 2 (i) and (ii), we can still find thresholds in wait cost c and radius
r above which, serving dedicated is optimal.
In particular, for any continuous and twice differentiable function F−1, the result in Proposition
7 still holds. To see this, note that the base price F−1
(1− λ
Λr2
)is not a function of c, so it does
not affect submodularity of the revenue function. Thus, the result in lemma 2 still holds.
![Page 45: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/45.jpg)
12 Chen, Hu: Courier Dispatch in On-Demand Delivery
Similarly, we can extend Proposition 8 under any continuous and twice differentiable function
F−1 such that
λ∂2F−1
(1− λ
Λr2
)∂λ∂r
+∂F−1
(1− λ
Λr2
)∂r
+cC
T
(2− T
2(T +λr)
(T −λr)3
)− c
3< 0,
where T = 2µB/r and C =CB.
C.3. Circular City
Proposition C.1. Consider a circular service area with radius r. When the demand rate λ> 0
is exogenous:
(i) There exists a threshold on the demand rate, below which serving dedicated leads to a shorter
wait time and thus a higher revenue, and above which serving batch is optimal.
(ii) There exists a threshold on the service radius, below which serving dedicated leads to a shorter
wait time and thus a higher revenue, and above which serving batch is optimal.
Without the large market assumption, when the demand rate λ is endogenously determined by
the vendor and customer valuations follow a standard uniform distribution:
(iii) There exists a threshold on the customers’ wait cost parameter, above which it is optimal to
serve dedicated.
(iv) There exists a threshold on the service radius, above which it is optimal to serve dedicated.
Proposition C.1 confirms that all the major results in Sections 4 and 5 still hold even if we
change the service area from a disk to a circle.
Proof of Proposition C.1. We only present a proof sketch for each statement as the details
greatly resemble the previous proofs by substituting in CD,C , and CB,C for CD, and CB:
Statements in (i) and (ii) follow from the proof of Propositions 1 and 2, with T1 = µD,Cr, T2 =
2µB,Cr, C1 = 1 and C2 =1
2+
π2
3(4 +π)2. Then we verify that the cubic equation f(a) = 0 has a
unique solution on a∈ (0, T1), which completes the proof.
Statement (iii) follows from the proof of Proposition 7, and statement (iv) follows from the proof
of Proposition 8. We omit the details.
C.4. Multiple Couriers
Proposition C.2. Consider the vendor hires k≥ 2 couriers covering the service area.
(i) When the demand rate λ> 0 is exogenous, there exists a threshold on the demand rate, below
which serving dedicated leads to a shorter wait time and thus a higher revenue, and above which
serving batch is optimal.
(ii) When the demand rate λ is endogenously determined by the vendor and customer valua-
tions follow a standard uniform distribution, there exists a threshold on the customers’ wait cost
parameter, above which it is optimal to serve dedicated.
![Page 46: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/46.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 13
As Proposition C.2 suggests, there is still a threshold on the exogenous demand rate, below
which serving dedicated leads to a smaller expected wait time and above which serving batch has
the edge when there are k couriers. Furthermore, when the wait cost parameter is relatively large,
we still have that serving dedicated dominates serving batch.
Proof of Proposition C.2. (i) For notational convenience, we denote µ1 := 2µB,k, µ2 := µF,k and
a :=√
2(k+ 1). By definition, we have µ2 >µ1. Furthermore, define function
f(λ) := WF,k−WB,k
=CD
2(kµ1−λ)
(λ
kµ1
)a−1
−
(1
2λ+
CB2(kµ2−λ)
(λ
kµ2
)a−1
+r
3
), λ∈ (0, µ1). (C.2)
Note that function f represents the difference in wait times when serving dedicated and batch.
First, we show that function f is strictly increasing w.r.t. λ by taking the first order derivative.
We have
∂f(λ)
∂λ=
1
2λ2(1 +CDh(µ1)−CBh(µ2)) ,
where
h(µ,λ) =kµ(λkµ
)a(kµ(a− 1)−λ(a− 2))
(kµ−λ)2, µ∈ (λ/k,∞). (C.3)
Note that function h is non-increasing w.r.t. µ since by taking the first order derivative, we have
∂h(µ,λ)
∂µ=
k(λkµ
)a(kµ−λ)3
[(a− 1)ak2µ2− 2(a− 2)akλµ+ (a− 1)(a− 2)λ2
]=
k(λkµ
)a(kµ−λ)3
{a(kµ−λ)[(a− 1)kµ− (a− 2)λ] +λ[akµ− (a− 2)λ]} ≥ 0,
where the inequality follows from that kµ> λ. Thus, we have
∂f(λ)
∂λ≥ 1
2λ2(1 +CDh(µ1, λ)−CBh(µ2, λ))> 0,
where the first inequality follows from that function h in non-increasing in µ together with µ2 >µ1
and the second inequality follows from CD >CB.
Next, by acknowledging function f goes to negative infinity and positive infinity when λ
approaches 0 and µ1, respectively, we can reach the first statement in Proposition C.2, since f(λ) = 0
has a unique solution.
(ii) We only need to verify that the revenue function of serving batch,
U(λ, c) := λ
{1− λ
Λr2− c
[1
2λ+
CB2(2kµ1−λ)
(λ
2kµ1
)a−1
+r
3
]}, (C.4)
![Page 47: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/47.jpg)
14 Chen, Hu: Courier Dispatch in On-Demand Delivery
is submodular in (λ, c). By taking its derivatives w.r.t. λ and c, we have
∂2U(λ, c)
∂λ∂c=−
CB
(λkµ1
)a−1
2(kµ1−λ)2(akµ− (a− 1)λ)− r
3< 0,
where the inequality follows from kµ−λ> 0.
The rest of the proof follows from the exactly same steps as in the proof of Proposition 7. Thus,
we omit the details.
D. Simulation Results
In this section, we present the simulation results on the accuracy of several approximations on
expected wait times in this paper.
First, we briefly state our simulation methods. In each sample, consider a unit disk (with radius
r= 1), and generate 1,000 arrivals uniformly distributed inside the disk using a fixed arrival rate λ.
We calculate the average wait time (sample mean) for each sample. When calculating the sample
mean, we exclude the first 200 arrivals to ensure the system has reached a steady state. In total, we
simulate 100 samples for each arrival rate and take the average of sample means as the simulated
expected wait time.
Next, we present the results by comparing the simulated expected wait times versus the approx-
imated wait times. Table D.1 demonstrates that our approximation for the expected wait time in
an E2/G/1 queue using Kingman’s formula in (12) is reasonably good. The percentage difference
is calculated as
Difference (%) :=|Simulated Time−Approx. Time|
Approx. Time.
Lastly, we choose parameters λ≤ 2µB so that the approximated system is finite and the comparisons
are meaningful.
Table D.1 Performance of approximated expected wait time with Kingman’s formula on E2/G/1 queue
λ(≤ 2µB) Simulated Time Approx. Time Difference (%)0.2 2.92 3.02 3.47%0.3 2.20 2.33 5.49%0.4 1.95 2.11 7.71%0.5 1.97 2.16 8.80%0.6 2.30 2.50 8.23%0.7 3.21 3.41 5.90%0.8 6.13 6.55 6.45%
In Table D.3, we present the expected wait time with the approximation in (31) and (32), which
are compared to their simulation counterparts, respectively. We choose courier number k= 3.
![Page 48: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/48.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 15
Table D.2 Performance of approximated expected wait time with dedicated delivery and k= 3 couriers
λ(≤ kµD) Simulated Time Approx. Time Difference (%)0.3 0.003 0.007 57.48%0.6 0.02 0.03 32.91%0.9 0.07 0.08 10.18%1.2 0.33 0.36 5.38%1.5 0.43 0.42 8.35%1.8 0.84 0.83 0.48%2.1 3.10 3.31 6.12%
Table D.3 Performance of approximated expected wait time with batch delivery and k= 3 couriers
λ(≤ 2kµB) Simulated Time Approx. Time Difference (%)0.3 2.00 2.00 0.47%0.6 1.17 1.18 1.06%0.9 0.91 0.91 0.39%1.2 0.79 0.80 0.70%1.5 0.77 0.75 1.76%1.8 0.84 0.77 9.51%2.1 1.14 0.89 27.3%
As we can see from Tables D.2 and D.3, our approximation is relatively accurate except for light
traffic dedicated systems or nearly overloaded batch systems. In the next two figures, we show that
even with approximation errors, our insights on the thresholds in wait cost c and service radius r
still hold with simulation results.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
DedicatedBatch
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
DedicatedBatch
(b)
Figure D.1 Revenue functions when serving dedicated versus batch under a large market.
(a) r= 1 k= 3, (b) c= 0.5, k= 3
E. Other Delivery Delay
In our base model, we consider that the courier does not stay for any positive amount of time at
each order location nor has any delay caused by other factors. In this section, we provide a brief
discussion on how to extend our base model to incorporate these features to the model.
![Page 49: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/49.jpg)
16 Chen, Hu: Courier Dispatch in On-Demand Delivery
Take delays occurred at the delivery locations as the example. Suppose each order incurs an
independently and identically distributed extra delay denoted as d at each order location, where
d has mean µd and variance σ2d, following a known distribution. Further assume that this extra
delay is also independent from the courier’s travel time (the time to cover the distance between
two locations). In reality, indeed there is at most a negligible correlation between the actual travel
time and the time finding a parking space at the delivery location.
Using this setup, we can rewrite the service rate and coefficient of variation as
µD =1
E[XD] +µd=
143r+µd
, CD = 1 +E[X2
D]− (E[XD])2 +σ2d
(E[XD] +µd)2= 1 +
29r2 +σ2
d(43r+µd
)2 , (E.1)
when the courier serves dedicated delivery, and
µB =1
E[XB] +µd=
14(32+15π)
45r+µd
, CB =1
2+
E[X2B]− (E[XB])2 +σ2
d
(E[XB] +µd)2=
1
2+
5.428r2−(
4(32+15π)
45r)2
+σ2d(
4(32+15π)
45r+µd
)2 ,
(E.2)
when serving batch. Then the expected wait time of the two delivery modes can be written as
WD(λ, µD, CD) and WB(λ, µB, CB), respectively.
Using this alternative setup, we can recreate the analysis in our base model. Take Section 4 with
exogenous demand for example. We can still show that there are thresholds on demand rate λ and
service radius r, respectively, such that below which serving dedicated is optimal and above which,
the courier should serve batch. We only provide a sketch of proof here, using a similar method as
the proofs for Propositions 1 and 2.
Consider function
f(λ, r) := WD(λ, µD, CD)−WB(λ, µB, CB)
=1
180πλ(−3 + 4rλ+ 3λµd)(64rλ+ 15π(−3 + 2rλ+ 3λµd))[8192r2λ2(−3 + 4rλ+ 3λµd)− 1920πrλ(−9 +λ(2r2λ− 9µd(−3 +λµd)− 12r(−2 +λµd) + 9λσ2
d))
+225π2(4(−10 + 9a)r3λ3− 54(−1 +λµd)2 + 3r2λ2(12− 9a− 28λµd + 9aλµd)
−36rλ(−3 +λ(µd +λµ2d−λσ2
d)))
]− r
3, (E.3)
where a=E[X2B]− (E[XB])2 ≈ 0.416. One can easily verify that f(λ, r) = 0 is a cubic equation with
respect to either λ or r. Furthermore, it has a unique real solution for λ or r, just as in the proof
of Proposition 1 and 2. Lastly, we only need to verify that WB(λ, µB, CB)<WD(λ, µD, CD) when
λ (or r) approaches zero, but WB(λ, µB, CB)>WD(λ, µD, CD) when λ (or r) is sufficiently large.
![Page 50: Courier Dispatch in On-Demand Delivery](https://reader034.vdocuments.site/reader034/viewer/2022051306/6279c1d7cec06875636aa4e6/html5/thumbnails/50.jpg)
Chen, Hu: Courier Dispatch in On-Demand Delivery 17
As we can see, with the addition of extra delays independent of the delivery trip travel time,
the analysis becomes a lot messier due to complicated algebra operations. Thus, we decide not to
include these features into our base model to preserve a clean yet informative analysis.
References
Dal Maso, G. (1993). An Introduction to Γ−Convergence. Birkhauser, Basel.