Pricing for Multi-modal Pickup and Delivery Problems with Heterogeneous Users
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Pricing for Multi-modal Pickup and Delivery Problems with
Heterogeneous Users
Mark Beliaev, Negar Mehr, Ramtin Pedarsani
Abstract— In this paper we study the pickup and delivery
problem with multiple transportation modalities, and address
the challenge of efficiently allocating transportation resources
while price matching users with their desired delivery modes.
Precisely, we consider that orders are demanded by a hetero-
arXiv:2303.10253v1 [eess.SY] 17 Mar 2023
geneous population of users with varying trade-offs between
price and latency. To capture how prices affect the behavior of
heterogeneous selfish users choosing between multiple delivery
modes, we construct a congestion game taking place over a star
network with independent sub-networks composed of parallel
links connecting users with their preferred delivery method.
Using the unique geometry of this network we prove that
one can define prices explicitly to induce any desired network
flow, i.e, given a desired allocation strategy we have a closed-
form solution for the delivery prices. In connection with prior
works that consider non-atomic congestion games, our result
shows that one can simplify the Linear Program formulations
used to solve for edge prices by first finding the path prices
combinatorially. We conclude by performing a case study on a
meal delivery problem with multiple courier modalities using
data from real world instances.
I. I NTRODUCTION
Fig. 1. We represent the pickup and delivery problem as a congestion game
As the world continues to integrate with digital technology, played over a star network. Each sub-network is an independent source–sink
we become more reliant on e-commerce services such as pair denoted by i ∈ I, which can be viewed as a population of users at
some location demanding a particular order at a certain rate. Each source–
food delivery and ride-hailing. The global food delivery sink pair is connected by a set of parallel edges j ∈ J , which can be
market has seen exponential growth, with the most mature viewed as the set of delivery modes the users choose from. Note that we
markets becoming four to seven times larger from 2018 are not concerned with how the couriers are routed to the pickup or delivery
location, and instead focus on how we allocate the different delivery modes
to 2021 [1]. In 2022, Uber reported a 19% year-over-year for each order. Specifically, our goal is to induce an optimal allocation
increase in online bookings, marking a daily average of 23 of transportation modalities by appropriately setting prices for each order-
million trips on their platform [2]. Despite this growth, many modality pair.
pickup and delivery services operate under low profit margins
due to high driver wages [3]. To fulfill market demands and
mitigate these costs, recent efforts have been made to intro- This problem is analogous to a congestion game taking place
duce autonomous transportation methods for food delivery over a star network, as depicted in Figure 1, where the
and ride hailing, such as drones or air taxis [4] and robot source–sink pairs represent independent sub-networks com-
couriers [5] or taxis [6]. In light of these developments, we posed of parallel links connecting users with their preferred
address the challenge of efficiently allocating transportation delivery method. This unique network structure enables us
resources between customers while price matching them with to show that we can explicitly define prices to induce any
their desired delivery modes. desired network flow, i.e, given a desired allocation strategy
This paper examines the pickup and delivery problem with we have a closed-form solution for the delivery prices.
multiple transportation modalities, and demonstrates how one The main contributions of this work are:
can achieve a desired allocation strategy for a set of orders by • We construct a congestion game that captures how
appropriately setting prices for each modality. Specifically, prices affect the behaviour of heterogeneous selfish
we consider orders demanded by a heterogeneous population users choosing between multiple delivery modes.
of users with varying trade-offs between price and latency. • Building on results from prior works, we prove that in
M. Beliaev is with Graduate School of Electrical & Computer Engineer- this setting the set of prices can be explicitly defined
ing, University of California Santa Barbara, Santa Barbara, CA, USA. for a desired network flow.
N. Z. Mehr is with the Faculty of Aerospace Engineering, University of • We demonstrate our model with a case study on a meal
Illinois at Urbana-Champaign, Champaign, IL, USA.
R. Pedarsani is with the Faculty of Electrical & Computer Engineering, delivery problem with multiple courier modalities, using
University of California Santa Barbara, Santa Barbara, CA, USA. real world instances provided by Grubhub [7].Related Work. The application of emerging transportation subsequent Section II, we formally introduce the problem
modalities such as unmanned aerial vehicles or drones has setting and show how it is analogous to a congestion game.
drawn a lot of attention. Many works looks at how drones can Following this, in Section III we describe our main theoret-
be utilized in logistic operations such as delivery systems [8], ical result in the general framework of the aforementioned
urban air taxi [9], on-demand meal delivery [10], as well congestion game. We go on to apply these results in Sec-
as many other applications [4]. Other works look at safety tion IV, modeling the meal delivery problem with multiple
verification for dynamical systems utilizing drones to account courier modalities by using the public Grubhub dataset [7].
for factors such as collision avoidance [11] and schedule Lastly, we conclude our work in Section V, listing potential
feasibility [12]. The pickup and delivery vehicle routing avenues for improvement and further research.
problem with drones has also been considered by some,
II. P ROBLEM F ORMULATION
where mixed integer linear programming models are used to
find routing solutions for optimizing various objectives [13], We model our pickup and delivery problem using a static
[14]. Unlike these works, our research lies in the broader system, where during a given time interval1 , there is a
field of congestion games, specifically building on previous set of orders I demanded by a population of users. We
works that consider pricing in non-atomic congestion games. consider that each order originates from a unique neighbor-
Congestion games aim to allocate traffic over transporta- hood composed of a heterogeneous population represented
tion networks represented by graphs, where each road cor- by the interval [0, 1], where each point a ∈ [0, 1] is a
responds to an edge with a latency function representing the non-cooperative and infinitesimal unit referred to as a user.
travel time experienced by users on that edge [15]. In these We sort these users by money sensitivity, viewing αi :
settings, one aims to find the optimal network flow that mini- [0, 1] → (0, ∞) as an unbounded, non-decreasing function
mizes a social cost, such as the aggregate latency experienced representing the trade-off between price and time for users in
by all users. However, if we assume that users are self– the population corresponding to order i. Thus, when placing
interested and choose their routes selfishly by minimizing an order i ∈ I, each user chooses one of the J delivery
their individual latency, the resulting flow follows a network modes j ∈ J : {1, . . . , J} based on delivery time `i,j , dollar
equilibrium [16], [17]. One area of research is focused on price τi,j , and their money/time valuation αi (a). Finally, we
categorizing the trade-off in social cost between the optimal assume users placing order i ∈ I have inelastic demands,
network flow and the equilibrium network flow [18], [19], i.e., they will not switch their demand to a different order
[20]. Many works specifically look at how tolling can be used and will always choose one of the J delivery modes. Our
to price network edges such that the equilibrium network goal is then to find the set of delivery prices which would
flow corresponds to the desired optimal flow [21], [22], induce some desired allocation of users between the delivery
[23]. In our work, we make the distinction that users are modes j ∈ J for each order i ∈ I.
heterogeneous in their trade-off between price and time. This problem is analogous to a congestion game played
While in the homogeneous case it has been long known over a star network as portrayed in Figure 1, where each
that marginal cost pricing can guarantee that the equilibrium order i ∈ I corresponds to a source–sink pair connected by
flow equals the optimal flow [24], this strategy does not a set of parallel edges J representing the different delivery
hold for heterogeneous populations. More recent research has options. Each source–sink pair i ∈ I has an associated
demonstrated that for directed graphs with one source–sink demand of traffic flow at the sink which represents the
pair, optimal tolls exist and can be found by solving a poly- population of users a ∈ [0, 1] requesting deliveries. Although
nomial size set of linear inequalities, given that the number we model this flow demand with the unit interval to simplify
of users in the heterogeneous population is finite [25]. In this notation, we can allow for an arbitrary demand ri at each
seminal paper, it was assumed that the model was nonatomic, source–sink pair i. The edge corresponding to modality
meaning that each user corresponded to an infinitesimal unit j ∈ J for source–sink pair i ∈ I has a congestion dependent
of flow, and inelastic, meaning that the demand could not latency `i,j , which represents the time needed to complete
change as a function of the road parameters. Following the order, and a price issued to control congestion τi,j , which
this work, others have improved the result by considering represents the dollar price payed by the user. Note that when
multicommodity networks [26], [27], allowing user demand we drop index j from the notation of terms like `i,j by
to be elastic [28], and addressing the atomic setting [29]. In writing `i , we refer to the set of latencies {`i,j }j∈J over
this paper, we keep the assumption of a nonatomic model all the edges J for a given source–sink pair i.
with inelastic demands, but consider a graph structure which With this approach, we can view network flow as an
is unique to the pickup and delivery problem considered. By allocation of users over the delivery modes. To represent
exploiting this graph structure, we can define prices explicitly such allocation strategies, we define 0 ≤ xi,j ≤ 1 as the
to induce any desired network flow without limiting it to flow of users on edge jP∈ J corresponding to source–
an optimal flow. Whereas prior works directly use Linear sink pair i ∈ I, where j∈J xi,j = 1 must be satisfied.
Program (LP) formulations to find edge prices in general More precisely, for each source–sink pair i we view this
directed graphs, our main result implies that one can first find 1 Without loss of generality, we can define the time interval during which
path prices combinatorially to simplify the LP formulation. the orders I are demanded as one hour, using the same unit of time for all
The rest of the paper is organized as follows. In the variables and constants throughout our formulation.flow as a Lebesgue-measurable function xi : [0, 1] → J if for all source–sink pairs i ∈ I, the corresponding flow
which corresponds to a flow over the edges {xi,j }j∈J . We xi : [0, 1] → J is an equilibrium flow satisfying Eq. (1).
use notations x = {xi,j }i∈I,j∈J and τ = {τi,j }i∈I,j∈J
As we show in the subsequent section, any network flow
to denote the entire set of edge flows and edge prices
x is a stable allocation strategy for some set of prices τ .
respectively. As we will later show in Section IV when
performing our case study, we can use x as a decision III. M AIN R ESULTS
variable to find an optimal allocation strategy for a given Before stating our main result, we need to elaborate
objective, and explicitly define prices τ that induce this on one more property of equilibrium flows that applies to
desired strategy. For now, we continue to detail how latency individual source–sink pairs. Intuitively, we expect Nash
and user equilibrium are considered in our framework. flows to exhibit a structure where users a ∈ [0, 1] close to 0,
Congestion. We first describe the congestion element of who value time more than money, will choose an option with
our framework, namely the latency function defined for small latency but large price. Similarly, users further away
each edge. Specifically, we assume that each edge j ∈ J from 0 will choose an option with a relatively larger latency
corresponding to source–sink pair i ∈ I has a nonnegative but a smaller price. Finally, users close to 1 will choose an
and continuous latency `i,j as a function of the entire option with very large latency in order to pay a very small
network flow x. Each latency function `i,j describes the time price. We encapsulate this notion below.
it takes for an order i delivered by modality j to arrive at
the customer’s location from the moment it was placed. We Definition 3. For a given source–sink pair i ∈ I, a flow xi
note that in order to claim our main theoretical result, we do at Nash equilibrium is canonical if:
not need any further restrictions on the latency functions `i,j . • For any edge j ∈ J , the users assigned to j form a
We leave further discussion regarding latency to Section IV, possibly empty or degenerate subinterval of [0, 1].
where we provide a case study for the meal delivery problem • If a1 < a2 , then `i,xi (a1 ) (x) ≤ `i,xi (a2 ) (x).
and model latency using concepts from queuing theory. Until • If a1 < a2 , then τi,xi (a1 ) ≥ τi,xi (a2 ) .
then, we stick with the aforementioned assumptions and In other words, a canonical Nash flow xi splits [0, 1] into
simply use notation `i,j (x) when defining edge latency. at most J potentially degenerate sub intervals, inducing an
User Equilibrium. We are now ready to discuss how users ordering over the edges to which xi assigns users that is
choose between the different delivery modes. When con- nondecreasing in latency and nonincreasing in prices. Using
fronted with a set of prices τi and latencies `i for the varying results from prior work which proposed this definition [25],
edge options j ∈ J , user a ∈ [0, 1] will choose the shortest we can state the following existence property.
edge relative to lengths `i,j (x) + αi (a)τi,j . Essentially, every
source–sink pair i ∈ I corresponds to its own nonatomic Proposition 2. For a given source–sink pair i ∈ I, every
game in which users a ∈ [0, 1] choose between the j ∈ J instance (αi , `i , τi ) admits a canonical Nash flow.
pure strategies available. The noncooperative behaviour of With these properties, we can say that for a given source–
users results in a Nash equilibrium, which is a stable point sink pair i ∈ I and instance (αi , `i , τi ), there exists a
where no user has an incentive to unilaterally alter their canonical Nash flow x̃i : [0, 1] → J . This canonical Nash
chosen strategy. Specifically, we let pai,j (x, τi ) = `i,j (x) + flow represents the flow {x̃i,j }j∈J , where users in interval
αi (a)τi,j represent the evaluation user a ∈ [0, 1] assigns to [aj−1 , aj ] ∈ [0, 1] are routed on edge j for some correspond-
edge j for source–sink pair i. ing set a0 ≤ a1 ≤ . . . ≤ aJ , with a0 = 0 and aJ = 1. In the
Definition 1. For a given source–sink pair i ∈ I, we call the pickup and delivery setting, we can assume that the delivery
flow xi : [0, 1] → J an equilibrium or Nash flow for instance provider already has a set of flows {xi,j }j∈J representing
(αi , `i , τi ) if for any user a ∈ [0, 1] and edge j ∈ J : the desired allocation strategy for order i, and wants to find a
corresponding set of prices {τi,j }j∈J such that the induced
pai,xi (a) (x, τi ) ≤ pai,j (x, τi ). (1) equilibrium flow {x̃i,j }j∈J is equal to the desired flow.
The existence of such Nash flows is a well known and a Building on top of the aforementioned results, we find a
general result [30]. closed-form solution to this problem.
Proposition 1. For a given source–sink pair i ∈ I, any Theorem 1. For a given source–sink pair i ∈ I, any desired
instance (αi , `i , τi ) admits a Nash flow xi : [0, 1] → J flow {xi,j }j∈J is an equilibrium flow for instance (αi , `i , τi ),
satisfying Eq. (1). where the set J : {1, . . . , J} orders the edges by non-
decreasing latency, αi : [0, 1] → (0, ∞) is a non-decreasing
Note that the above results not make any claims about the distribution function, `i is the set of corresponding edge
existence of a network flow x for which all source sink pairs latencies, and τi is the set of prices defined by:
exhibit Nash equilibria. We use the term stable allocation J−1
strategy to encompass this notion, formally defining it below.
X `i,k+1 − `i,k
τi,j = τi,J + , (2)
αi (ak )
Definition 2. For a given star network defined by the source– k=j
sink pairs i ∈ I and edges j ∈ J , we call the network flow for all j ∈ J , where τi,J is any predefined price for the
x : {xi }i∈I a stable allocation strategy for instance (α, `, τ ) cheapest option.Proof. The proof strategy is as follows: using a subset of Essentially, the above Eq. (4) splits the delivery time `i,j
the inequalities defined for Nash equilibrium in Eq. (1), we into three components: service time si,j , travel time ti,j , and
first show that for some desired Nash flow {xi,j }j∈J there pickup time ui,j for modality j of order i.
is only one set of valid prices τi that satisfies this subset We view the service time si,j as a constant representing
of inequalities. We complete the proof by showing that the the time spent at the pickup and drop-off locations when
corresponding set of prices τi does indeed satisfy all of the completing order i using delivery mode j. Some examples
inequalities defined in Eq. (1). We provide a full proof of of this include parking for vehicle couriers, landing for aerial
this result in Appendix I. couriers, loading, and unloading. Similarly, we define the
travel time ti,j as the time it takes to physically travel
It follows directly that given any network flow x : {xi }i∈I between pickup ri and drop-off di locations using delivery
representing a desired allocation strategy over all orders, one mode j. The travel time ti,j between locations can be pre-
can independently set prices τ : {τi }i∈I for each source–sink computed separately for each modality j and order i using
pair to make x a stable allocation strategy. some known functions. Lastly, we view the pick-up time ui,j
as the time it takes for a courier of delivery mode j to arrive
Corollary 1. For a given star network defined by source–sink
at pick-up location ri . Unlike the other two components, the
pairs i ∈ I and edges j ∈ J , any network flow x : {xi }i∈I
time required for pickup ui,j should depend on our decision
is a stable allocation strategy for instance (α, `, τ ) when the
variable x by varying based on the availability, as well as the
set of prices τ is defined according to Eq. (2).
expected travel time between the pickup location and nearest
We highlight an interesting point about our main result. available courier.
To account for the availability of couriers, we use the con-
Remark 1. Although Equation 2 defines prices for parallel
cept of server utilization from queuing theory. Specifically,
edges, this is equivalent to finding prices for paths in
we use the M/M/c queue as an approximate model for
more general graphs composed of one source–sink pair.
the availability of couriers since we can obtain closed form
While prior works derive Linear Program (LP) formulations
formulas for the average order arrival and order completion
to directly find edge prices which induce the equilibrium
rates. For a given modality j, we set c to the total number of
flow, Theorem 1 implies that one can first find path prices
couriers NP j , approximate the rate at which users are placing
combinatorially to simplify the LP formulation. Since users
orders as i∈I xi,j , and define the rate at which an order is
choosing between delivery modes can be represented by
completed by these types of couriers as µj . Note that we can
parallel edges, we forego defining paths in our formulation
define the order completion rate µj as a constant provided
to be concise.
by historical data, or estimate it using the parameters of
our problem instance as we will later show. Drawing these
IV. C ASE S TUDY: M EAL D ELIVERY P ROBLEM
analogies allows us to define the utilization ρj of our queuing
To show the usability of our model, we apply our theoret- system for couriers of modality j as:
ical framework to the meal delivery problem with multiple P
xi,j
courier types. Our goal is to find the optimal allocation ρj = i∈I . (5)
Nj µj
strategy with respect to some objective, where we will
use Theorem 1 to set the prices which induce this desired In our regime of interest, the rate of order arrivals is
strategy. Our objective will be to find the optimal values of magnitudes larger than the rate of order completions, and
x which minimize the expected latency over all orders: hence the number of available couriers c needs to be large.
Using the M/M/c latency function, one can easily show
1 XX that in this regime of interest the time spent waiting for
L(x) = `i,j (x)xi,j . (3)
|I| an available server is negligible unless we are close to the
i∈I j∈J
capacity limit [31]. For example, given a system with c = 50
Before we set up and solve this optimization problem, we servers and a demand of 100 requests per hour, when the
first specify how delivery time is measured, and how cost is server utilization is high at ρ = 0.99, the average time spent
accounted for. in the system is 84 minutes, with 55 minutes in the queue.
Once we lower the utilization to ρ = 0.9, the average time
A. Formulation spent in the system is 29 minutes, with only 2 minutes spent
Latency Model. We begin by characterizing each order i ∈ I in the queue. This means that the expected waiting time for
by a 2–tuple hri , di i, consisting of a pick-up and drop-off a courier to be available is relatively small compared to the
location respectively. We would like our system to model delivery time, given that the utilization parameter ρj is below
the time it takes for an order to arrive at the customer’s a reasonable threshold. Thus to make sure that customers are
location from the moment it was placed. We refer to this as not experiencing long wait times for couriers to respond, we
the delivery time `i,j for order i ∈ I and modality j ∈ J , can upper-bound the utilization parameter ρj for all courier
computing it as: types, and ignore the affect of availability.
To model the time a courier must spend traveling to the
`i,j (x) = si,j + ti,j + ui,j (x). (4) pick-up location ri , we take a probabilistic approach bycalculating the expected travel time of the nearest available we bound our decision variable between the domain of [0, 1]
courier. Specifically, we assume that for modality j, some in Eq. (12) so that there are no negative values in the solution.
portion βi,j ∈ (0, 1] of available couriers are distributed The optimization problem defined above is non-linear
around the pick-up location ri such that their travel times and non-convex, and we use a public implementation of
are uniform in [0, kj ]. Note that we can choose kj as some the interior-point filter line-search algorithm [32] to solve
constant unit of time from which βi,j is estimated based on it. As aforementioned, many choices can be made for the
the pick-up location and delivery mode. Since we know that formulation of the latency functions `i,j , cost constraint
the expected number of available couriers will be (1−ρj )Nj , C, and the optimization objective L. To efficiently use the
we can define the pick-up time as the expected travel time interior point method, it is desired for the objective function
of the nearest courier: and constraints to be twice differentiable so that the Hessian
kj can be defined. We note that apart from this consideration,
ui,j (x) = , (6) we also kept the prices τ independent of the objective
1 + βi,j Nj (1 − ρj )
function because of the permutations required to compute
where we used the fact that the expected minimum value of them. Though neither of these restrictions are necessary, they
1
n independent uniform random variables in [0, 1] is n+1 . allow us to efficiently implement the optimization problem
Cost Model. Before setting up our optimization problem, and demonstrate our main result.
we need to model the cost of operating this delivery system. We can now discuss how we setup our case study, in which
We define the dollar cost of completing order i ∈ I by a we model a meal delivery problem with three transportation
courier of modality j as the delivery cost ci,j . This way, we modalities: cars, drones, and robots. To define the problem
can define the total cost of running our delivery system given parameters for our optimization formulation, we used real
the allocation strategy x: world instances from Grubhub [7], which list information
J
XX about the orders placed and car couriers available throughout
C(x) = ci,j xi,j , (7) a given time interval. Although there is no consideration of
i∈I j=1 other modalities, we use the provided information as a basis
and define our remaining parameters to be consistent. We
where C(x) is units of dollars per hour because xi,j is a rate
list how these parameters are defined below, and provide our
of orders per hour. Since we expect the delivery cost ci,j to
full implementation in our code available online [33].
depend on the distance traveled by courier j to complete
For service time si,j , we directly used the given pickup
order i, modeling ci,j as a constant is a practical choice.
and dropoff times for car couriers, and scaled them by 0.2
Alternatively, one can define a cost model using wages for
for drones and robots. Similarly for travel time ti,j , we used
different courier types, making ci,j dependent on the delivery
the real distances between restaurants and order locations,
time and hence a function of the allocation strategy. We leave
converting them to time by using constant speeds for all
this extension for future work, as the defined model is still
modalities. For cars we set the speed to 19.2 km/h according
useful for many applications.
to the dataset, and scaled the speed of drones and robots to
Optimization Problem. We are now ready to set up the
be 38.4 km/h and 5.76 km/h respectively. For pickup time
overall optimization problem.
ui,j , we computed all the parameters required in Eq. (6).
1 XX The number of couriers N was directly chosen for each
min L(x) = `i,j (x)xi,j (8)
x |I| instance so that the problem was feasible under the utilization
i∈I j∈J
XX capacity of ρ̄ = 0.9. We then generated the locations of
subject to C(x) = τi,j xi,j , (9) all three courier modalities, and computed the portion of
i∈I j∈J
available couriers βi,j that were at most k = 10 minutes
ρj (x) ≤ ρ̄ ∀j ∈ J , (10) away from the restaurant corresponding to order i. For car
X
xi,j = 1, ∀i ∈ I, (11) couriers, we directly sampled from the provided locations,
j∈J while for drone couriers, we sampled uniformly from a
and 0 ≤ xi,j ≤ 1, ∀i ∈ I, j ∈ J . (12) grid spanning the restaurant locations. To capture robot
couriers delivering from restaurants closer to downtown, we
For this case study, we want to find the allocation strategy sampled their locations uniformly from a grid centered in
x which minimizes expected delivery time L, as shown in the middle of the restaurant locations, with length and width
Eq. (8). To make this problem more practically-interesting, equal to their coordinate’s respective standard deviations.
we constrain the operational cost in Eq. (9) to equal the total Using these parameters, we estimated the mean rate µj of
compensation received from all deliveries. Note that because order completions as the inverse of expected delivery time
Theorem (1) allows us to arbitrarily set price for the cheapest Ei [`i,j ]−1 for each modality j, assuming load was equally
delivery option, this is equivalent to finding the minimum distributed across the orders. We use the same cost per order
price τ̄ that satisfies the constraint. We also constrain courier ci,j of $10 for car deliveries, and set it to $5 for drone and
utilization in Eq. (10) by choosing an appropriate upper robot deliveries. For user trade-off between price and time
bound ρ̄ for all delivery modes j ∈ J . We use the constraint α(a), we use a linear function with the lowest evaluation
in Eq. (11) to satisfy demands for each order i ∈ I. Finally, α(1) set to 10 dollars per hour, and the highest α(0) setCars Drones Robots Total making the total operational cost smaller, and (2) the faster
Orders (%) 100 0 0 100 speed of drones allows us to charge users who favor shorter
Cost ($ per hour) 2121.00 0 0 2121.00 delivery times more than users who favor cheaper delivery
Latency `j (min) 21.50 0 0 21.50
Price τj ($) 10.00 0 0 10.00 prices. We show the result corresponding to the most efficient
Distance (km) 2.36 0 0 2.36 allocation strategy in Table II, where we can see that the
TABLE I
delivery time has decreased, and note that the minimum order
R ESULTS FOR A MEAL DELIVERY SYSTEM WITH 100 CAR COURIERS .
price of $5.42 is almost half of the previous $10. The total
M INIMUM ORDER PRICE IS $10.00.
operational cost of 1593.85 dollars per hour is lower than
before, while the average price for drone deliveries at $11.29
is high compared to the other delivery modes.
Cars Drones Robots Total In the final case we replace a larger portion of car couriers
with drones, and consider that there are 20 car, 20 drone,
Orders (%) 50 24 26 100
Cost ($ per hour) 1061.90 255.85 276.10 1593.85 and 35 robot couriers available. We again show the results
Latency `j (min) 21.33 5.66 27.18 19.12 corresponding to the most efficient allocation in Table III,
Price τj ($) 6.64 11.29 5.63 7.52 where we note that the minimum delivery price has dropped
Distance (km) 2.45 2.53 2.03 2.36
further to $3.95. As expected, the total operational cost and
TABLE II average delivery time have also decreased. We can also tell
R ESULTS FOR A MEAL DELIVERY SYSTEM WITH 50 CAR , 10 DRONE , more clearly from the distances reported that drones are used
AND 35 ROBOT COURIERS AVAILABLE . M INIMUM ORDER PRICE IS $5.42 to complete orders for customers furthest away from their
chosen restaurant, while robots are used for customers closest
to their chosen restaurant. This is due to the travel speeds
Cars Drones Robots Total of the different transportation modalities, as drones can
Orders (%) 20 57 23 100 travel efficiently between distant destinations, while robots
Cost ($ per hour) 431.90 599.45 244.95 1276.30
Latency `j (min) 15.55 6.29 15.19 10.23
are restricted to operate in a smaller range as they have a
Price τj ($) 5.34 6.97 4.29 6.02 pedestrian pace. Lastly, we point out that the delivery price
Distance (km) 2.15 2.96 1.08 2.36 for drones no longer needs to be as high compared to other
TABLE III modes. Since drones can now support more orders overall,
R ESULTS FOR A MEAL DELIVERY SYSTEM WITH 20 CAR , 20 DRONE , AND the premium charged for faster delivery can be lowered.
35 ROBOT COURIERS AVAILABLE . M INIMUM DELIVERY PRICE IS $3.95 Overall, our case study shows that by setting prices
according to users’ trade-offs between money and time, one
can implement a desired allocation strategy over multiple
delivery modalities while improving their profit margins.
to 100 dollars per hour. We go on to discuss the results of
our case study for an instance with 505 unique orders, each V. C ONCLUSION
demanded with a rate of 0.42 deliveries per hour. We model the pickup and delivery problem with multiple
transportation modalities as a congestion game played over
B. Results a star network, and show that we can explicitly define prices
We first consider the case when there are only 100 car to induce any desired network flow. With this framework, we
couriers available, with no other transportation modality. We construct a case study of the meal delivery problem and use
show the result in Table I, listing the portion of orders real historical data to define our parameters. We show that
delivered as a percentage, the the operational cost in dollars by utilizing autonomous transportation methods which are
per hour, the latency or delivery time `j in minutes, the more efficient, one can set prices according to users’ trade-
delivery price in dollars, and the distance between customer offs between money and time to induce a desired allocation
and restaurant in kilometers. We note that the statistics strategy while improving their profit margins. We go over
corresponding to drones and robots are set to 0 as they are not some of the implications of our work, pointing out limitations
applicable in this case. Since car couriers have an operational and directions for improvement.
cost of $10 per order, we need an average delivery price We first note that in the setting of non-atomic congestion
of $10 to satisfy it. Note that although in this setting the games taking place on graphs composed of one source–
minimum delivery price can be set arbitrarily for all orders sink pair, prior works have asked if a feasible solution
since users have no choice to make, when we introduce other can be found to compute optimal prices for edges com-
delivery modalities this is no longer the case as the prices binatorially, without relying on LP formulations [25]. Our
must follow Eq. (2) to satisfy Nash conditions. main theoretical result states that in these settings, one can
In the next case, we consider that there are 50 car, 10 define optimal prices for paths combinatorially, implying that
drone, and 35 robot couriers available. Other than improving the LP formulation used to find prices for edges can be
the average delivery time, we expect the inclusion of drones simplified. This points to the possibility that other network
and robots to lower the minimum delivery price required due structures inherit properties which allow one to find prices
to two factors: (1) drones and robots are cheaper to operate efficiently, and we leave this direction for future works.Further, we point out that our case study is only one [10] Y. Liu, “An optimization-driven dynamic vehicle routing algorithm
example where such a model is useful. Due to the general for on-demand meal delivery using drones,” Computers & Operations
Research, vol. 111, pp. 1–20, 2019. [Online]. Available: https:
construction of the congestion game defined, our analysis is //www.sciencedirect.com/science/article/pii/S0305054819301431
practical for any application that utilizes a platform to price [11] C. Llanes, M. Abate, and S. Coogan, “Safety from fast, in-the-
match customers with different transportation methods. Some loop reachability with application to uavs,” in 2022 ACM/IEEE 13th
International Conference on Cyber-Physical Systems (ICCPS), 2022,
examples include ride-hailing, airport taxis which provide pp. 127–136.
transportation via UAVs, and delivery for services other than [12] Q. Wei, G. Nilsson, and S. Coogan, “Scheduling of urban air mobility
food. Since our formulation poses little restriction on the services with limited landing capacity and uncertain travel times,” in
2021 American Control Conference (ACC), 2021, pp. 1681–1686.
latency function defined, one can construct a model that is
[13] J. B. Gacal, M. Q. Urera, and D. E. Cruz, “Flying sidekick travel-
suitable for the desired application. ing salesman problem with pick-up and delivery and drone energy
Of course our framework gives no guarantees on finding optimization,” in 2020 IEEE International Conference on Industrial
the optimal allocation strategy, and instead provides a method Engineering and Engineering Management (IEEM), 2020, pp. 1167–
1171.
by which prices can be set to induce a desired strategy. One [14] Y. Lu, C. Yang, and J. Yang, “A multi-objective humanitarian pickup
important restriction made in our formulation is the omission and delivery vehicle routing problem with drones,” Annals of Opera-
of prices from the main objective function. Due to the or- tions Research, pp. 1–63, 2022.
[15] S. C. Dafermos and F. T. Sparrow, “The traffic assignment problem
dering permutations required to solve for the delivery prices, for a general network,” Journal of Research of the National Bureau
one can not directly compute the derivatives of the objective of Standards B, vol. 73, no. 2, pp. 91–118, 1969.
and constraints if they depended on these delivery prices. [16] J. G. Wardrop, “Some theoretical aspects of road traffic research,”
Proceedings of the Institution of Civil Engineers, vol. 1, no. 3,
In this case, one can compute the derivative information pp. 325–362, 1952. [Online]. Available: https://doi.org/10.1680/ipeds.
empirically, derive heuristics which treat the permutations 1952.11259
directly, or utilize other optimization techniques. [17] Y. Sheffi, Urban Transportation Networks: Equilibrium Analysis With
Mathematical Programming Methods. Prentice Hall, 01 1985.
Lastly, we want to comment on the ethical implications
[18] T. Roughgarden, Selfish Routing and the Price of Anarchy. MIT press
of our case study. On the positive side, our results show Cambridge, 05 2005.
that by utilizing more autonomous transportation methods [19] D. A. Lazar, S. Coogan, and R. Pedarsani, “Routing for traffic
one can improve profit margins. However, this is because networks with mixed autonomy,” IEEE Transactions on Automatic
Control, vol. 66, no. 6, pp. 2664–2676, 2021.
our model considers car couriers operated by humans as less [20] ——, “Capacity modeling and routing for traffic networks with mixed
cost efficient. While one may have financial incentives to autonomy,” in 2017 IEEE 56th Annual Conference on Decision and
substitute part of their current workforce with autonomous Control (CDC), 2017, pp. 5678–5683.
[21] S. C. Dafermos, “Toll patterns for multiclass-user transportation
machines, other decisions can be made that improve wages networks,” Transportation Science, vol. 7, no. 3, pp. 211–223, 1973.
and work conditions for employees. Such a discussion is [Online]. Available: http://www.jstor.org/stable/25767702
beyond the scope of our work, and is a topic that should be [22] R. Cole, Y. Dodis, and T. Roughgarden, “How much can taxes
carefully addressed by policy makers before corporations are help selfish routing?” in Proceedings of the 4th ACM Conference
on Electronic Commerce, ser. EC ’03. New York, NY, USA:
allowed to make decision that greedily improve their profits. Association for Computing Machinery, 2003, p. 98–107. [Online].
Available: https://doi.org/10.1145/779928.779941
R EFERENCES [23] P. N. Brown and J. R. Marden, “The robustness of marginal-cost taxes
in affine congestion games,” IEEE Transactions on Automatic Control,
[1] K. Ahuja, V. Chandra, V. Lord, and C. Peens, “Ordering in: The rapid vol. 62, no. 8, pp. 3999–4004, 2017.
evolution of food delivery,” McKinsey & Company, vol. 22, 2021.
[24] M. Beckmann, C. Mcguire, and C. Winsten, “Studies in the economics
[2] “Uber announces results for fourth quarter and
of transportation, yale university press,” New Haven, Connecticut,
full year 2022,” Feb 2023. [Online]. Avail-
USA, 1956.
able: https://www.businesswire.com/news/home/20230208005139/en/
Uber-Announces-Results-for-Fourth-Quarter-and-Full-Year-2022 [25] R. Cole, Y. Dodis, and T. Roughgarden, “Pricing network edges
[3] A. Shetty, J. Qin, K. Poolla, and P. Varaiya, “The value of pooling for heterogeneous selfish users,” in Proceedings of the Thirty-Fifth
in last-mile delivery,” in 2022 IEEE 61st Conference on Decision and Annual ACM Symposium on Theory of Computing, ser. STOC ’03.
Control (CDC), 2022, pp. 531–538. New York, NY, USA: Association for Computing Machinery, 2003, p.
[4] M. Moshref-Javadi and M. Winkenbach, “Applications and research 521–530. [Online]. Available: https://doi.org/10.1145/780542.780618
avenues for drone-based models in logistics: A classification and [26] G. Karakostas and S. G. Kolliopoulos, “Edge pricing of multicom-
review,” Expert Systems with Applications, vol. 177, p. 114854, 2021. modity networks for heterogeneous selfish users,” 45th Annual IEEE
[Online]. Available: https://www.sciencedirect.com/science/article/pii/ Symposium on Foundations of Computer Science, pp. 268–276, 2004.
S0957417421002955 [27] L. Fleischer, K. Jain, and M. Mahdian, “Tolls for heterogeneous selfish
[5] “Starship food delivery app,” Feb 2023. [Online]. Available: users in multicommodity networks and generalized congestion games,”
https://www.starship.xyz/starship-food-delivery-app/ in 45th Annual IEEE Symposium on Foundations of Computer Science,
[6] K. Heineke, R. Heuss, K. Philipp, A. Kelkar, and M. Kellner, “The 2004, pp. 277–285.
road to affordable autonomous mobility,” McKinsey & Company, 2022. [28] G. Karakostas and S. G. Kolliopoulos, “Edge pricing of multicom-
[7] D. Reyes, A. Erera, M. Savelsbergh, S. Sahasrabudhe, and R. O’Neil, modity networks for selfish users with elastic demands,” Algorithmica,
“The meal delivery routing problem,” Optimization Online, vol. 6571, vol. 53, pp. 225–249, 2006.
2018. [29] D. Fotakis, G. Karakostas, and S. G. Kolliopoulos, “On the existence
[8] M. Beliaev, N. Mehr, and R. Pedarsani, “Congestion-aware bi-modal of optimal taxes for network congestion games with heterogeneous
delivery systems utilizing drones,” in 2022 European Control Confer- users,” in Algorithmic Game Theory, 2010.
ence, ECC 2022. Institute of Electrical and Electronics Engineers [30] D. Schmeidler, “Equilibrium points of nonatomic games,” Université
Inc., 2022, pp. 1944–1951. catholique de Louvain, Center for Operations Research and
[9] H. Ale-Ahmad and H. S. Mahmassani, “Factors affecting demand Econometrics (CORE), LIDAM Reprints CORE 146, 1973. [Online].
consolidation in urban air taxi operation,” Transportation Research Available: https://EconPapers.repec.org/RePEc:cor:louvrp:146
Record, vol. 2677, no. 1, pp. 76–92, 2023. [Online]. Available: [31] S. Kelly-Bootle and B. W. Lutek, “Chapter 5 - queueing theory,” in
https://doi.org/10.1177/03611981221098396 Probability, Statistics, and Queuing Theory with Computer Science Ap-plications, 2nd ed., ser. Computer Science and Scientific Computing,
A. O. Allen, Ed. San Diego: Academic Press, 1990, pp. 247–375.
[32] A. Wächter and L. T. Biegler, “On the implementation of an interior- `j+1 + α(a)τj+1 ≤ `j + α(a)τj ∀a ∈ [aj , aj+1 ], (17)
point filter line-search algorithm for large-scale nonlinear program-
j+1 − `j
`
ming,” Mathematical programming, vol. 106, pp. 25–57, 2006. τj − τj+1 ≥ max , (18)
[33] M. Beliaev, “Interior Point Impementation for Pickup and Delivery,” a∈[aj ,aj+1 ] α(a)
3 2023. [Online]. Available: https://github.com/mbeliaev1/mdrp
which results in the following:
A PPENDIX I `j+1 − `j
τj − τj+1 ≥ . (19)
P ROOF OF T HEOREM 1 α(aj )
Note that for sake of notation, we will drop the subscript From (16) and (19) we can see that the two inequalities
referring to orders i ∈ I, as it should be clear that the proof force the set of prices {τi,j }j∈J to follow:
applies to an individual source–sink pair. In addition, we `j+1 − `j
τj − τj+1 = ∀j ∈ {1, . . . , J − 1}, (20)
assume that indexes j ∈ J : {1, . . . , J} correspond to the α(aj )
set of edges sorted by non-decreasing latency. where if τJ is given, the rest of the prices can be found
recursively as defined in Eq (2).
τj ≥ τj+1 To complete the proof, we must show that for this set
`j ≤ `j+1 of prices τ , the desired x is indeed an equilibrium flow.
... aj−1 aj aj+1 ... Formally, x is an equilibrium flow for instance (α, `, τ ) if for
0 1
all edges j ∈ {1, . . . , J} no user a in interval a ∈ [aj−1 , aj ]
Fig. 2. A sketch depicting how a canonical Nash flow splits the population should want to switch to any other edge j 0 ∈ {1, . . . , J}:
a ∈ [a0 , aJ ] into subintervals [aj−1 , aj ) : x(a) = j, where a0 = 0,
aJ = 1, and j ∈ J .
`j + α(a)τj ≤ `j 0 + α(a)τj 0 . (21)
0
Clearly these inequalities hold when j = j , and hence we
We define two adjacent intervals that are formed by our show that they hold when j > j 0 and j < j 0 . Starting with
flow x: users a ∈ [aj−1 , aj ] on the left experience delivery the former, when j > j 0 we are considering that no user
time `j and price τj , while users a ∈ [aj , aj+1 ] on the choosing edge j will switch to any edge j 0 on the left, where
right experience delivery time `j+1 and price τj+1 . The by definition τj ≤ τj 0 and `j ≥ `j 0 . Rearranging Eq. 21, we
two intervals are portrayed in Fig. 2, where we note that have the following for all edges j > j 0 :
this definition holds for j ∈ {1, . . . , J − 1}. Using the `j + α(a)τj ≤ `j 0 + α(a)τj 0 ∀a ∈ [aj−1 , aj ],
inequalities defined in Eq. (1), we know that for x to be ` − ` 0
j j
a Nash flow for instance (α, `, τ ), no user a from the left τj 0 − τj ≥ max ,
interval a ∈ [aj−1 , aj ] should want to switch to the delivery a∈[aj−1 ,aj ] α(a)
J−1
X `k+1 − `k J−1
option corresponding to the right interval: X `k+1 − `k `j − `j 0
− ≥ ,
0
α(ak ) α(ak ) α(aj−1 )
k=j k=j
`j + α(a)τj ≤ `j+1 + α(a)τj+1 ∀a ∈ [aj−1 , aj ], (13) j−1 j−1
X `k+1 − `k X `k+1 − `k
≥ .
where we leave out denoting the flow x in latency `j (x). It α(ak ) α(aj−1 )
k=j 0 0
k=j
follows: Since α(aj−1 ) ≥ α(ak ) when j 0 ≤ k ≤ j − 1, every
summation term on the left hand side is strictly greater than
`j+1 − `j or equal to every summation term on the right hand side,
τj − τj+1 ≤ ∀a ∈ [aj−1 , aj ], (14) validating the inequalities in Eq. 21 for j > j 0 . We can do
α(a)
`
j+1 − `j
the same for j < j 0 , where now τj ≥ τj 0 and `j ≤ `j 0 :
τj − τj+1 ≤ min . (15)
a∈[aj−1 ,aj ] α(a) `j + α(a)τj ≤ `j 0 + α(a)τj 0 ∀a ∈ [aj−1 , aj ],
` 0 − `
j j
The preceding inequality can be simplified further by τj − τj 0 ≤ min ,
using the non-decreasing property of function α defining the a∈[aj−1 ,aj ] α(a)
J−1
X `k+1 − `k J−1
population’s price sensitivity: for any a1 , a2 ∈ [0, 1] such X `k+1 − `k `j − `j 0
that a1 ≤ a2 , given user a ∈ [a1 , a2 ], max α(a) = α(a2 ) and − ≤ ,
α(ak ) 0
α(ak ) α(aj )
min α(a) = α(a1 ). This comparison results in the following k=j k=j
0 0
condition which must be true for x to be a Nash flow: jX −1 −1
jX
`k+1 − `k `k+1 − `k
≤ .
`j+1 − `j α(ak ) α(aj )
τj − τj+1 ≤ . (16) k=j k=j
α(aj )
This time, since α(aj ) ≤ α(ak ) when j ≤ k ≤ j 0 − 1, every
We can repeat this process by enforcing that no user a summation term on the left hand side is strictly less than or
from the right interval a ∈ [aj , aj+1 ] should want to switch equal to every summation term on the right hand side. This
to the edge on the left: completes the proof.You can also read
