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Greedy Algorithms for Scheduling Package Delivery with Multiple Drones

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Unmanned Aerial Vehicles (or drones) can be used for a myriad of civil applications, such as search and rescue, precision agriculture, or last-mile package delivery. Interestingly, the cooperation between drones and ground vehicles (trucks) can even enhance the quality of service. In this paper, we investigate the symbiosis among a truck and multiple drones in a last-mile package delivery scenario, introducing the Multiple Drone-Delivery Scheduling Problem (MDSP). From the main depot, a truck takes care of transporting a team of drones that will be used to deliver packages to customers. Each delivery is associated with a drone's energy cost, a reward that characterizes the priority of the delivery, and a time interval representing the launch and rendezvous times from and to the truck. The objective of MDSP is to find an optimal scheduling for the drones that maximizes the overall reward subject to the drone's battery capacity while ensuring that the same drone performs deliveries whose time intervals do not intersect. After showing that MDSP is an NP-hard problem, we devise an optimal Integer Linear Programming (ILP) formulation for it. Consequently, we design a heuristic algorithm for the single drone case and two more heuristic algorithms for the multiple drone case. Finally, we thoroughly compare the performance of our presented algorithms on different synthetic datasets.
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Greedy Algorithms for Scheduling Package Delivery with
Multiple Drones
Francesco Betti Sorbelli
University of Perugia,
Italy
francesco.bettisorbelli@unipg.it
Federico Corò
Missouri University of Science and
Technology, United States
federico.coro@mst.edu
Sajal K. Das
Missouri University of Science and
Technology, United States
sdas@mst.edu
Lorenzo Palazzetti
University of Florence,
Italy
lorenzo.palazzetti@uni.it
Cristina M. Pinotti
University of Perugia,
Italy
cristina.pinotti@unipg.it
ABSTRACT
Unmanned Aerial Vehicles (or drones) can be used for a myriad
of civil applications, such as search and rescue, precision agricul-
ture, or last-mile package delivery. Interestingly, the cooperation
between drones and ground vehicles (trucks) can even enhance
the quality of service. In this paper, we investigate the symbiosis
among a truck and multiple drones in a last-mile package deliv-
ery scenario, introducing the Multiple Drone-Delivery Scheduling
Problem (MDSP). From the main depot, a truck takes care of trans-
porting a team of drones that will be used to deliver packages to
customers. Each delivery is associated with a drone’s energy cost,
a reward that characterizes the priority of the delivery, and a time
interval representing the launch and rendezvous times from and
to the truck. The objective of MDSP is to nd an optimal sched-
uling for the drones that maximizes the overall reward subject to
the drone’s battery capacity while ensuring that the same drone
performs deliveries whose time intervals do not intersect. After
showing that MDSP is an NP-hard problem, we devise an optimal
Integer Linear Programming (ILP) formulation for it. Consequently,
we design a heuristic algorithm for the single drone case and two
more heuristic algorithms for the multiple drone case. Finally, we
thoroughly compare the performance of our presented algorithms
on dierent synthetic datasets.
CCS CONCEPTS
Mathematics of computing
Combinatorial optimization;
Computing methodologies Optimization algorithms
;
The-
ory of computation Scheduling algorithms.
This work was partially supported by NSF grants CNS-1818942, OAC-1725755, OAC-
2104078, and SCC-1952045; and also partially supported by "HALY-ID" project funded
by the European Union’s Horizon 2020 under grant agreement ICT-AGRI-FOOD no.
862665, no. 862671, and from MIPAAF .
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fee. Request permissions from permissions@acm.org.
ICDCN 2022, January 4–7, 2022, New Delhi, India
©2022 Association for Computing Machinery.
ACM ISBN xxxx. . . $15.00
https://doi.org/xxxx
KEYWORDS
Drone; truck; last-mile delivery system
ACM Reference Format:
Francesco Betti Sorbelli, Federico Corò, Sajal K. Das, Lorenzo Palazzetti,
and Cristina M. Pinotti. 2022. Greedy Algorithms for Scheduling Package
Delivery with Multiple Drones. In Proceedings of 23st International Confer-
ence on Distributed Computing and Networking (ICDCN 2022). ACM, New
York, NY, USA, 9 pages. https://doi.org/xxxx
1 INTRODUCTION
Since when the Unmanned Aerial Vehicles (drones) have been
developed, both the research and industrial communities have
started to exploit this new technology in a plethora of applica-
tions. In particular, drones can be eciently and eectively used
for search and rescue operations over disastrous areas hit by an
earthquake [
5
,
27
], for observing and monitoring crops in precision
agriculture contexts [
14
,
21
], for localizing and monitoring missing
people [
3
,
16
], or for shipping parcels to customers to accelerate
the delivery process in a package delivery context [
2
,
15
,
25
,
26
,
36
].
Interestingly, the collaboration between drones and trucks, rely-
ing on their respective capabilities and mobility, can signicantly
improve the quality of service, especially in the last-mile delivery
scenario [10, 13].
In this work, we investigate the cooperation between a truck
and one or multiple drones for last-mile package delivery. With the
help of drones, delivery companies can perform more deliveries and
hence extend their revenue and business [
24
]. In fact, drones can
deliver small packages quickly as they can easily traverse dicult
terrain [
1
], often using shorter routes not otherwise possible for
trucks. In this scenario, the drones y (starting from the truck) to
the assigned locations and deliver the packages, then leave and
meet again with the truck to perform new deliveries. However, sig-
nicant challenges arise due to such constraints as the drones’ high
energy consumption and their current inability to simultaneously
serve multiple customers at a time [
11
,
18
]. The premise under con-
sideration is that a delivery company has to make several deliveries
to the customers in a city by relying on a truck carrying a eet of
drones with the same capabilities. In this paper, we study this prob-
lem by assuming that the main depot knows the locations of the
customers to visit and the consequent roads to travel for the truck.
Therefore, before leaving the depot, the delivery company plans the
ICDCN 2022, January 4–7, 2022, New Delhi, India Francesco Bei Sorbelli, Federico Corò, Sajal K. Das, Lorenzo Palazzei, and Cristina M. Pinoi
drones’ ying sub-routes to accelerate the deliveries, taking into
account not only the energy used by the drones but also the revenue
generated. Since drones have limited battery capacity, a delivery has
a cost in terms of the energy required, and this limits the number
of deliveries that a drone can perform. Furthermore, there can be
conicts among deliveries if they cannot be accomplished by the
same drone, i.e., if the corresponding sub-routes, and hence their
time intervals, intersect each other. In addition, each delivery has an
associated reward that characterizes its importance (e.g., a higher
reward means higher priority). Hence, given the truck’s route in
the city, the goal is to plan a scheduling for the drones such that the
total reward is maximized subject to the constraints of limited drone’s
battery capacity and the conicts among deliveries.
We make the following contributions in this paper.
We propose an optimization problem, called Multiple Drone-
Delivery Scheduling Problem (MDSP), and prove its
NP
-
hardness, devising also the optimal Integer Linear Program-
ming (ILP) formulation which is only suitable for small-sized
instances.
We design a heuristic algorithm for the single drone case,
and two heuristic algorithms for the multiple drones case,
which can be run for larger instances.
We extensively compare the performance of our algorithms
on synthetic datasets.
The remainder of this paper is structured as follows. Section 2
surveys the related works. Section 3 introduces the MDSP show-
ing its
NP
-hardness, and proposes the optimal ILP formulation.
Sections 4 devises algorithms for MDSP for the single and multi-
ple drones cases. Section 5 evaluates the algorithms on synthetic
datasets. Finally, Section 6 oers conclusions and future research
directions.
2 RELATED WORK
This section reviews the literature related to the problem of deliv-
ering packages with the help of trucks and drones. We categorize
the existing works based on whether the deliveries are done via
both trucks and drones, or only by drones in which case the truck
is only a carrier.
2.1 Delivery by Both Trucks and Drones
The truck-drone cooperation in a last-mile delivery scenario has
been investigated for the rst time when the ying sidekicks trav-
eling salesman problem (FSTSP) has been introduced in [
22
]. The
FSTSP is a particular case of the original TSP, where drones take o
from the truck, y to deliver packages to customers, and then go
back to the truck in another place. Both the vehicles can perform de-
liveries, but each has to stop waiting for the other at the rendezvous
location. The authors proposed an optimal mixed-integer linear
programming formulation (MILP) and two heuristic solutions for
solving the FSTSP. A similar approach is also studied by the same
authors in [23] for multiple drones.
A greedy heuristic algorithm for the truck-and-drone delivery
system is proposed in [
8
]. The heuristic starts with a solution for
the truck only, which comprises nodes (i.e., locations) provided by a
TSP instance. Then it greedily builds short sub-paths for the drones
by excluding some nodes from the truck’s path in order to reduce
the overall tour’s length in terms of time. Importantly, such a tour
must serve all the customers’ locations. Finally, the truck’s route
preserves the order of the nodes from the initial solution, and the
drone ies between the neighboring nodes.
In [
9
], the authors considered a scenario with multiple recharge-
able drones. Drones can only carry one package at a time and have
to return to the truck’s roof to charge their battery after each deliv-
ery. It is assumed that the speed is the same for both drones and
trucks, but their mobility is dierent – the drones move according
to the Euclidean metric, while the trucks follow the Manhattan
metric. The authors presented a heuristic to solve the problem of
nding a good schedule for all drones and trucks that minimizes
the average delivery time of the packages.
In the delivery system in [
29
], the truck has a potentially large
capacity for carrying packages, but it travels at slow speeds in
urban areas due to road intersections or trac jam. On the other
hand, drones are faster and not restricted to street networks, but
their range and carrying capacity are limited. So, to minimize the
total tour duration to serve all customers the authors proposed an
optimal MILP formulation based on timely synchronizing the truck
and drone ows. They also introduced a dynamic programming
recursion based on an exact branch-and-price approach capable of
optimally solving small instances.
The delivery with a truck and drones relying on a rechargeable
station for the drones is presented in [
17
]. The station can furnish
a large number of drones, and it is located near customer areas and
away from the distribution center. The authors showed that this
problem can be divided into TSP and parallel identical machine
scheduling (PMS) problems. Through this approach, they success-
fully reduce the complexity of the problem and obtain an exact
solution. A hybrid approach is presented in [
35
], where the authors
propose to simultaneously employ trucks, truck-carried drones,
and independent drones to construct a more ecient truck-drone
parcel delivery system. A novel routing and scheduling algorithm
are proposed to solve the hybrid parcel delivery problem.
2.2 Delivery by Drones Only
A similar work to ours is proposed in [
4
], where the authors consider
a predened route for the truck, and the problem is to optimize
the planning of the drone’s ights to/from the truck while serving
the customers. Unlike us, however, their goal is to determine the
launch and meeting points between drones and trucks. On the
other hand, in our work these points are provided in input, and
our goal is to determine a subset of deliveries to maximize the
revenue. Furthermore, these points are calculated in such a way
that the intervals generated do not intersect with each other, thus
making all deliveries conict-free. Importantly, in [
4
] the authors
assume that drones do not have a battery constraint. While their
goal is to reduce total makespan (i.e., the time dierence between
the start and end of a delivery sequence), our goal is to maximize
the revenue/reward for deliveries.
The authors in [
28
] studied the cooperation between trucks and
drones from a dierent point of view. In this work, the truck is
only concerned with transporting the drones in order to improve
the delivery system and make it more ecient. The objective is to
determine a valid combination of the drone’s characteristics (i.e.,
Multiple Drone-Delivery Scheduling Problem ICDCN 2022, January 4–7, 2022, New Delhi, India
speed and range) to synchronously let them cooperate with the
truck avoiding useless stops at locations. Only an optimal MILP
formulation is proposed for solving this problem.
In the work in [
20
], the truck-drone tandem is proposed for
cooperatively performing all the required deliveries. Once any de-
livery has been performed, a drone can immediately recharge its
battery. This means that all deliveries are feasible and, accordingly,
no scheduling is required.
In [
7
] a clustering is proposed to nd truck’s intermediate lo-
cations in which it can stop and send drones to deliver packages
to near locations. Such clustering aims at minimizing the overall
makespan.
3 PROBLEM FORMULATION
In this section, we rst introduce the system and delivery models,
and then dene the Multiple Drone-Delivery Scheduling Problem
(MDSP). We also show that MDSP is NP-hard.
3.1 System Model
Let
𝐴
be the 2-D delivery area representing our last-mile delivery
(application) scenario. Let
𝜓𝐴
be the depot from where the
deliveries start. The position of the depot is located at
(𝑥𝜓, 𝑦𝜓)
,
assumed to be at the origin of the Cartesian coordinate system
in
(
0
,
0
)
. Such a delivery area also comprises roads and customer
positions. At the depot, a truck is in charge of transporting a eet of
𝑚
drones
𝑑1, . . . , 𝑑𝑚
with the same capabilities used for deliveries in
𝐴
. The truck does not perform deliveries; it only carries the drones
within 𝐴.
Let
𝜌𝑗
be a straight road delimited by two endpoints
𝜆𝑗
and
𝜆𝑗+1
, where
𝑗=
0
, . . . , 𝑟
and
{𝜆1, . . . , 𝜆𝑟} ⊂ 𝐴
. The truck leaves
the depot
𝜓
visiting the rst endpoint
𝜆1
along the straight road
segment
𝜌0=𝜓𝜆1
, then the next endpoint
𝜆2
along the segment
𝜌1=𝜆1𝜆2
, and so on, up to the last endpoint
𝜆𝑟
, and eventually
going back to the depot
𝜓
. These segments make a closed path (cycle)
𝐶
formed by the sequence of endpoints
𝜆0, 𝜆1, . . . , 𝜆𝑟, 𝜆𝑟+1
such that
𝜆0=𝜆𝑟+1=𝜓
. Clearly, the
𝑟+
1contiguous roads
𝜌0, . . . , 𝜌𝑟
dene
𝐶.
Let
𝐷={𝛿1, . . . , 𝛿𝑛} 𝐴
be the set of
𝑛
distinct points or
locations to be served (i.e., the customers) by the drones. Each cus-
tomer’s (delivery) point
𝛿𝑖
has a pair of coordinates
(𝑥𝛿𝑖, 𝑦𝛿𝑖) ∈ 𝐴
,
for
𝑖=
1
, . . . , 𝑛
. Figure 1 illustrates the delivery area
𝐴
with roads
and customers.
3.2 Delivery Model
In this paper we assume that drones can deliver a single package
at a time due to stringent payload constraints. A drone’s delivery
is performed by planning a sub-ight that passes through three
points. Specically, the drones take o (with packages) from the
truck which continues to drive in the city, delivers packages to the
customers which are waiting for them, and return to the rendezvous
locations with the truck again. For each customer’s location
𝛿𝑖
, let
𝛿𝐿
𝑖
and
𝛿𝑅
𝑖
be respectively the launch point and rendezvous point of
the drone. Note that
𝛿𝐿
𝑖
and
𝛿𝑅
𝑖
can lie on dierent roads. In general,
𝛿𝐿
𝑖
lies on road
𝜌𝑗
while
𝛿𝑅
𝑖
lies on road
𝜌𝑧
, where 0
𝑗𝑧𝑟
(see Figure 1). For instance, the truck in Figure 2 travels from
𝜆𝑗
to
truck
depot
drone
drone
Figure 1: An example delivery area 𝐴with
6
roads 𝜌𝑗and
8
customers 𝛿𝑖to serve. The depot 𝜓is at (
0
,
0
); The truck’s
path is the solid line, while the drones’ paths are the dashed
lines.
λj
λj+1
λz
λz+1
ρjρz
δi
δL
iδR
i
Figure 2: Two non-adjacent roads with a delivery to do: the
launch and rendezvous points are highlighted.
𝛿𝐿
𝑖
carrying the drone, which in turn takes o at
𝛿𝐿
𝑖
ying towards
𝛿𝑖
delivering the package, and nally continues ying towards
𝛿𝑅
𝑖
.
In the meanwhile, the truck continues its ground route reaching
other endpoints. When both the truck and the drone arrive at
𝛿𝑅
𝑖
,
the truck gathers again the drone and they continue to travel up to
point
𝜆𝑧+1
. Note that, in order to save time, the truck do not travel
back and forth for picking up the drones.
Let
𝑤𝑖
0be the energy cost (or weight) for a drone in terms
of energy spent to perform a single delivery
𝛿𝑖
ying from/to the
truck. Let
𝐵
0be the drone’s energy budget in terms of battery
capacity that limits the number of possible deliveries that it can
do. Note that if the drone’s current residual energy is not enough
for additional ights, it is not possible to perform further deliveries
unless a new battery is swapped. Nevertheless, we assume that
drones have to rely only on the initial single battery charge.
Let
𝑝𝑖
0be the reward for executing a delivery
𝛿𝑖
. The meaning
of the reward characterizes premium users having higher priority
than regular users. For instance, delivery companies oer dierent
subscriptions according to the following rule: the more you pay, the
faster you receive your parcels. In our context, the higher the priority
of delivery
𝛿𝑖
, the larger is the reward value
𝑝𝑖
. Hence, in order to
satisfy the premium users, the scheduling for the drones needs to
prioritize the deliveries, guaranteeing rst the ones belonging to
ICDCN 2022, January 4–7, 2022, New Delhi, India Francesco Bei Sorbelli, Federico Corò, Sajal K. Das, Lorenzo Palazzei, and Cristina M. Pinoi
the premium users (because they have more reward) and second
the ones from regular users.
Let
𝑡
0denote the time. Let
𝑡0=
0be the initial time when the
truck leaves the depot at location
𝜆0=𝜓
, for performing deliveries
within the area
𝐴
. Similarly, let
𝑡𝑟+1
be the nal time of the delivery
application at
𝜆𝑟+1=𝜓
. It is important to observe that the truck
travels its route along a predened sequence of endpoints to be
visited exactly once, in a specic direction. Hence, if the truck plans
to travel through the endpoints
𝜆𝑖
and
𝜆𝑗
, with
𝑖<𝑗
, then
𝑡𝑖<𝑡𝑗
,
with 0
𝑖<𝑗𝑟+
1. For any delivery
𝛿𝑖
, let
𝑡𝐿
𝑖
be the launch
time for the drone from point
𝛿𝐿
𝑖
, and let
𝑡𝑅
𝑖
be the rendezvous time
at point
𝛿𝑅
𝑖
, where
𝑡0𝑡𝐿
𝑖<𝑡𝑅
𝑖𝑡𝑟+1
. Finally, let
𝜏𝑖=𝑡𝑅
𝑖𝑡𝐿
𝑖
be the span time of delivery in
𝛿𝑖
. Consequently, for a given
𝛿𝑖
, let
𝐼𝑖=[𝑡𝐿
𝑖, 𝑡𝑅
𝑖]
dene the drone’s delivery interval time that determines
its ying time-window for delivery.
Let
𝐼={𝐼1, . . . , 𝐼𝑛}
be the interval set, where 1
𝑖𝑛
, associated
with the deliveries within
𝐴
. Two intervals
𝐼𝑖
and
𝐼𝑗
are said to be
compatible if their intersection is empty, i.e.,
𝐼𝑖𝐼𝑗=
, for
𝑖𝑗
;
otherwise, the two intervals are in conict. In other words,
𝐼𝑖
and
𝐼𝑗
are compatible if
𝑡𝑅
𝑖<𝑡𝐿
𝑗
or
𝑡𝑅
𝑗<𝑡𝐿
𝑖
. A subset
𝑆𝐼
is said to be
compatible if
𝐼𝑖𝐼𝑗=
for any pair
𝐼𝑖, 𝐼 𝑗𝑆
. This means that a
drone can perform any subset of such deliveries as long as it has
enough battery. Precisely, a given compatible
𝑆𝐼
is feasible if the
energy cost
C(𝑆)=Í𝐼𝑖𝑆𝑤𝑖𝐵
(the energy budget). The reward
of a feasible set 𝑆is P(𝑆)=Í𝐼𝑖𝑆𝑝𝑖.
Recall that
𝑚
is the number of drones. Two feasible subsets
𝑆𝑝, 𝑆𝑞𝐼
can be assigned to two drones
𝑑𝑝
and
𝑑𝑞
, with 1
𝑝
𝑞𝑚
, if
𝑆𝑝𝑆𝑞=
. Assuming that
𝑆={𝑆1, . . . , 𝑆𝑚}
consists
of
𝑚
feasible sets assigned to the drones
{𝑑1, . . . , 𝑑𝑚}
, the overall
reward
P(𝑆)
is dened as the sum of the reward of each drone, i.e.,
P(𝑆)=Í𝑚
𝑗=1Í𝐼𝑖𝑆𝑗𝑝𝑖.
3.3 Problem Denition
Let us now formally dene the delivery scheduling problem.
Problem 1 (
Multiple Drone-Delivery Scheduling Problem
(MDSP))
.Let
𝛿1, . . . , 𝛿𝑛
be the set of
𝑛
deliveries,
𝑚
the number of
drones, and
𝐵
the drone’s battery budget. The objective of MDSP is
to nd a family
𝑆={𝑆
1, . . . , 𝑆
𝑚} ⊆ 𝐼
of
𝑚
feasible subsets with
𝑆
𝑝𝑆
𝑞=
, for 1
𝑝𝑞𝑚
, such that the overall reward
P(𝑆)
is maximized. Specically,
𝑆=arg max
𝑆={𝑆1,...,𝑆𝑚} ⊆𝐼
P(𝑆)
such that C (𝑆𝑖) ≤ 𝐵𝑖=1, . . . , 𝑚
In the following we show the
NP
-hardness of MDSP, even for
the single drone case.
An alternative way to dene the set of deliveries and constraints
is to observe that the set of intervals
𝐼
can be described as an interval
graph, which is a special case of chordal graphs. In MDSP, the set of
deliveries
𝐷
determines the set of intervals
𝐼
that can be pairwise
compatible or in conict with respect to their starting and ending
times. Given an instance of MDSP, we can visualize the intervals
and their compatibility along a temporal line. Figure 3 illustrates
the intervals corresponding to Figure 1, in which
𝐼2
is in conict
with both 𝐼1and 𝐼3, whereas 𝐼1and 𝐼3are compatible.
t0
I1
I2
I3
I6
I4I5
I7
I8
Figure 3: Delivery intervals corresponding to Fig. 1.
A simple way to gure out if two intervals are in conict or
not is by considering the interval graph representation. Formally,
in the undirected interval graph,
𝐺=(𝑉 , 𝐸)
, the vertex-set
𝑉=𝐼
,
where each vertex uniquely corresponds to an interval
𝐼𝑖𝐼
, for
𝑖=
1
, . . . , 𝑛
; and an edge
(𝐼𝑖, 𝐼 𝑗) ∈ 𝐸
indicates that the deliveries
𝛿𝑖
and
𝛿𝑗
are not compatible, implying
𝐼𝑖𝐼𝑗
. For example, in Figure 3,
𝐼1
and
𝐼3
can be executed without conicts, i.e.,
(𝐼1, 𝐼3)𝐸
; while
𝐼2
is not compatible with both
𝐼1
and
𝐼3
, implying
(𝐼2, 𝐼1),(𝐼2, 𝐼3) 𝐸
.
Note that vertex 𝐼6has four conicts because its degree is 4.
I1I4I5I8
I2I3I6I7
Figure 4: Interval graph 𝐺for the example in Fig. 1.
In the following we demonstrate that MDSP is an
NP
-hard prob-
lem by showing that the classic 0–1 Knapsack Problem (KP) is a
special case of MDSP when intervals are conict-free.
Theorem 1. MDSP is NP-hard.
Proof (Sketch).
Our approach is by reduction from KP, which
is known to be NP-hard [19] and dened as follows.
Given a set
𝑋={
1
, . . . , 𝑛 }
of
𝑛
items, each associated with a cost
𝑤𝑖
and reward
𝑝𝑖
, and a knapsack of capacity
𝑐
, the KP is to nd
a subset of
𝑋
that maximizes the sum of the rewards, satisfying
the capacity constraint. Given an instance of KP, we translate it as
an instance of MDSP as follows: we rst set the energy budget of
MDSP equal to the capacity constraint of the knapsack, i.e.,
𝐵=𝑐
.
Then we create a set of deliveries
𝐷
starting from the set of items
𝑋
as follows: for each item
𝑥𝑖𝑋
, create a delivery
𝛿𝑖𝐷
with
equal cost and reward. Hence, we can observe that the deliveries are
pairwise compatible in the corresponding MDSP instance. Thus, a
solution for KP is a solution for MDSP and vice versa. This reduction
takes polynomial time. Hence, proven.
3.4 The Optimal Algorithm
For optimally solving MDSP in the general case with
𝑚
drones,
we can use an ILP formulation. We enumerate the deliveries as
N={
1
, . . . , 𝑛 }
, and drones as
M={
1
, . . . , 𝑚}
. Let
𝑥𝑖 𝑗 ∈ {
0
,
1
}
be
a decision variable that is 1if the delivery
𝑗∈ N
is accomplished
by the drone
𝑖∈ M
; otherwise it is 0. Hence, the ILP formulation is
formed by an objective function and a few constraints as follows:
Multiple Drone-Delivery Scheduling Problem ICDCN 2022, January 4–7, 2022, New Delhi, India
max
𝑚
Õ
𝑖=1
𝑛
Õ
𝑗=1
𝑝𝑗𝑥𝑖 𝑗 (1)
subject to:
𝑛
Õ
𝑗=1
𝑤𝑗𝑥𝑖 𝑗 𝐵, 𝑖∈ M (2)
𝑚
Õ
𝑖=1
𝑥𝑖 𝑗 1,𝑗∈ N (3)
𝑥𝑖 𝑗 +𝑥𝑖𝑘 1,𝑖∈ M;𝑗, 𝑘 ∈ N𝑠.𝑡 . 𝐼𝑗𝐼𝑘(4)
𝑥𝑖 𝑗 ∈ {0,1},𝑖∈ M,𝑗∈ N (5)
The objective function in Expression
(1)
maximizes the overall
reward using a eet of
𝑚
drones. Constraint
(2)
states that each
drone has an energy budget
𝐵
in terms of battery; Constraint
(3)
enforces each delivery to be performed by no more than one drone;
and Constraint
(4)
ensures that deliveries can be performed by the
same drone only if they are not in conict.
Since the above ILP formulation, for solving MDSP, is only suit-
able for small instances in input, in the following we propose three
time-ecient heuristic algorithms suitable for larger instances in
input, in scenarios involving a single or multiple drones. From now
on, we will interchangeably use the terms ‘delivery’ and ‘interval’.
4 PROPOSED ALGORITHMS
This section proposes a heuristic algorithm for MDSP with a single
drone, namely, Max Ratio Single drone (Mr-S), and two heuristic
algorithms for MDSP with multiple drones, namely, Max Cliqe
Multiple drones (Mc-M) and Max Ratio Multiple drones (Mr-
M).
4.1 The Mr-S Algorithm
The Mr-S is a heuristic to solve MDSP with a single drone, requiring
O(𝑛log 𝑛)
time and
O(𝑛)
space. The pseudo-code of Mr-S is given
in Algorithm 1.
Algorithm 1: The Mr-S Algorithm
1sort(𝐼)s.t. 𝑝1
𝑤1. . . 𝑝𝑛
𝑤𝑛
2𝑆𝑂𝐿 ← ∅
3for 𝑖1, . . . ,𝑛 do
4if is-augmentable(𝐼𝑖, 𝑆 𝑂𝐿)then
5𝑆𝑂𝐿 𝑆𝑂𝐿 𝐼𝑖
6return 𝑆𝑂𝐿
In Mr-S we rst sort the intervals in non-increasing order by
the ratio
𝑝𝑖
𝑤𝑖
of the reward to the energy cost (Line 1, Algorithm 1).
Then, we add to the current solution
𝑆𝑂𝐿
the best possible interval
𝐼𝑖
(i.e., with the largest ratio) that does not create conicts with
the already chosen intervals and also evaluating if the residual
energy budget is sucient. This is done via the
is-augmentable
procedure (Line 4).
Note that the greedy technique in Mr-S that solves MDSP can
eciently compute approximated solutions of the KP (i.e., instances
of MDSP without interval conicts) by exploiting the submodularity
property. Recall that a set function
F
: 2
𝑋R
is said to be
submodular if given a nite set
𝑋={𝑥1, . . . , 𝑥𝑛}
, for any
𝑆𝑇𝑋
and
𝑥𝑋\𝑇
it holds
F (𝑆𝑥) F (𝑆) F (𝑇𝑥) − F (𝑇)
.
Unfortunately, our objective function
P(𝑆)
is not submodular due
to the presence of intervals that overlap/intersect each other. If
P(𝑆)
had been submodular, it would have been possible to obtain a
solution that guarantees at least 1
𝑒1
0
.
63 of the optimum [
33
]
taking overall
O(𝑛log 𝑛)
time. For MDSP, due to the presence of
conicts among intervals, the submodularity property does not hold,
and hence the approximation ratio of an algorithm that exploits the
strategy in Mr-S is unbounded. We can explain this issue by taking
into account the example in Figure 5, showing that the solution can
be arbitrarily bad with respect to the optimum. Indeed, if we use Mr-
Sto solve MDSP, in presence of two intervals
𝐼1
and
𝐼2
in conict,
where the span of
𝐼1
is very large, and assuming reward
𝑝2=Δ>
1
and 0
<𝜖<
1,Mr-S would return
𝐼1
(because
1
1𝜖>Δ
Δ+𝜖
) gaining
a reward of only 1. Therefore, the ratio of the reward obtained by
Mr-S to the reward obtained by the optimum is
1
Δ
. Such a ratio can
be arbitrarily small, and hence unbounded. This example can easily
be extended to
𝑛
intervals such that
𝐼2, . . . , 𝐼𝑛
are compatible with
each other, and 𝐼1is in conict with the other 𝑛1intervals.
The Mr-S algorithm requires
O(𝑛log 𝑛)
time due to the time
required for sorting, and O (𝑛)space.
4.2 The Mc-M Algorithm
The heuristic Mc-M solves MDSP with multiple drones in
O(𝑚(𝑛log 𝑛+
(𝑛))
time and
O(𝑛)
space, where
(𝑛)
is the time required by a
subroutine used in it. The pseudo-code of Mc-M is given in Algo-
rithm 2.
Algorithm 2: The Mc-M Algorithm
1ˆ
𝐼𝐼, 𝑚𝑚 , 𝑆𝑂𝐿 ← ∅
2while ˆ
𝐼do
3{𝑆1, . . . ,𝑆𝜔} ← create-subsets(ˆ
𝐼)
4sort(𝑆𝑖)s.t. P ( 𝑆𝑖) P ( 𝑆𝑖+1)
5𝑆𝑂𝐿 𝑆𝑂𝐿 ∪ {𝑆1, . . . , 𝑆min{𝜔,𝑚}}
6ˆ
𝐼ˆ
𝐼\Ð𝑆𝑖}
7𝑚𝑚min{𝜔 ,𝑚}
8return 𝑆𝑂𝐿
In Mc-M we sequentially perform a partitioning, depending on
the size of the maximum clique
𝜔
, of the current graph induced
by the residual intervals. Then, we assign a drone for each sub-
partition. This is done via the
create-subsets
procedure (Line 3,
Algorithm 2). Finally, we nd a global solution comprising the
previously computed solutions.
I2
I1
Figure 5: Instance where the submodularity property does
not hold for Mr-S. Here, we have 𝑝1=
1
, 𝑤1=
1
𝜖for 𝐼1, and
𝑝2=Δ, 𝑤2=Δ+𝜖for 𝐼2.
ICDCN 2022, January 4–7, 2022, New Delhi, India Francesco Bei Sorbelli, Federico Corò, Sajal K. Das, Lorenzo Palazzei, and Cristina M. Pinoi
The set
ˆ
𝐼
and the number
𝑚
of available drones are rst ini-
tialized (Line 1). Next, until
ˆ
𝐼
is not empty (Line 2), an optimal
partitioning invoking the
create-subsets
procedure is performed
to generate
𝑆1, . . . , 𝑆𝜔
(Line 3). Since
ˆ
𝐼
is an interval graph and
therefore a chordal graph, the size of the maximum clique
𝜔
can
be computed in polynomial time [
12
]. With this knowledge, we
divide the set of intervals into several conict-free subsets based
on the number
𝜔
. In fact, on a clique of size
𝑘
all the vertices are in
conict, and hence
𝑘
drones are needed for conict-free deliveries.
We remark that nding the size of the maximum clique
𝜔
in general
graphs is
NP
-hard. Nevertheless, if the graph
𝐺
is specically an
interval graph, 𝜔can be quickly computed in polynomial time.
The most important part of Mc-M resides in the procedure
create-subsets
(Line 3). To eciently implement
create-subsets
,
we initially sort the deliveries (and hence the time intervals) by
the launch and rendezvous times in non-decreasing order, into two
dierent ordered sets. Numbers assigned to drones can be labeled
as 1
, . . . , 𝜔
. Then, we consider a pointer
𝑖
at the beginning of each
ordered set. We also create a min-heap data structure initialized
with only one element, which is the rst number. If the element
indexed by the pointer is a launch time, we extract the minimum
number from the heap and associate such value with interval
𝑖
and then move the pointer one position, otherwise, if the element
indexed by the pointer is a rendezvous time, we insert the number
of the interval
𝑖
in the heap and move the pointer one position.
After the extraction, if the heap is empty, we then insert a new
number (i.e., the next available integer not used until now). Having
said this, the performance of
create-subsets
can further be im-
proved by taking into account the eventual residual energy budget.
Since the returned solution belongs only to one subset, say
𝑆𝑖𝐶𝑖
,
the sum of energy costs in
𝑆𝑖
may be less than
𝐵
. This means that
the solution can be augmented by including intervals from other
subsets. This approach can be applied selectively by ensuring that
the interval to be added to the current solution does not create
any conict. A possible strategy is to greedily pick the compatible
interval having the maximum ratio between the reward and cost.
Once the partitioning is completed, for each subset
𝐶𝑖
, nd a
subset
𝑆𝑖𝐶𝑖
with maximum reward such that
C(𝑆𝑖) ≤ 𝐵
. Note
that, subsets
𝑆𝑖
for each
𝑖
are feasible solutions. It is important to
observe that nding an optimal set
𝑆𝑖
is equivalent to solving a
KP with budget
𝐵
on the elements
𝐶𝑖
, which is an
NP
-hard prob-
lem. However, in our implementation, we take into account only
approximated solutions of KP in polynomial time exploiting the
submodularity property.
Then, the
{𝑆1, . . . , 𝑆𝜔}
are sorted in non-decreasing order by the
total reward (Line 4). Now, depending on the number of current
available groups
𝜔
and drones
𝑚
, we assign the best
min{𝜔, 𝑚 }
groups to drones (Line 5). After that, we update the current solution
𝑆𝑂𝐿
and the number of available drones (Lines 6–7). Finally, the
solution is returned (Line 8).
4.2.1 Random Interval Graphs. In this section, we provide a few
interesting considerations about the value of
𝜔
in random graphs.
Since the delivery may occur anywhere in the delivery area
𝐴
, we
may consider random interval graphs associated with the delivery
intervals
𝐼
, and study the expected value of
𝜔
. If the interval graph is
dense, i.e., the number of edges
|𝐸|=𝑂(𝑛2)
, where
𝑛
is the number
of vertices, then
𝜔=𝑂(𝑛)
[
31
]. In the general case of random
graphs in which the intervals and their associated lengths have
equal probability and they are all independent from each other,
𝜔=𝑛
2+𝑜(𝑛)
[
30
]. Other estimations of
𝜔
can also be determined
with respect to the length of the intervals. For example, if these
interval lengths are uniformly distributed in
[
0
, 𝑟 ]
, there exist some
results. In particular, if
𝑟=𝑓(
1
/𝑛)
, then
𝜔=(
1
±𝜖)log 𝑛
log log 𝑛
, and if
𝑟=𝑓(log 𝑛
𝑛), then 𝜔=𝑂(log 𝑛)[31].
The Mc-M algorithm takes
O(𝑚(𝑛log 𝑛+(𝑛)))
time for invok-
ing 𝑚times the create-subsets algorithm, and O (𝑛)space.
4.3 The Mr-M Algorithm
The Mr-M is a heuristic algorithm to solve MDSP with multi-
ple drones, and requires
O(𝑚(𝑛log 𝑛))
time and
O(𝑛)
space. The
pseudo-code of Mr-M is given in Algorithm 3.
Algorithm 3: The Mr-M Algorithm
1ˆ
𝐼𝐼
2for 𝑖1, . . . ,𝑚 do
3𝑆𝑖Mr-S(ˆ
𝐼)
4ˆ
𝐼ˆ
𝐼\𝑆𝑖
5return 𝑆𝑂𝐿 ← {𝑆1, . . . , 𝑆𝑚}
In Mr-M we sequentially perform Mr-S (for each drone) on the
current set of intervals not assigned to drones yet. After that, we
return a global solution which is the union of all the determined
solutions in the previous steps.
Initially, the set
ˆ
𝐼
that characterizes the current residual intervals
is initialized with all the deliveries
𝐼
(Line 1, Algorithm 3). After
that, we iteratively (Line 2) invoke Mr-S on
ˆ
𝐼
(Line 3) for
𝑚
times
(since the eet of drones), which contains the intervals that have
not been assigned to drones yet. At the end of the loop there are
intermediate solutions
𝑆𝑖
for drone
𝑑𝑖
. Consequently, the current set
of intervals available for the remaining drones is decreased (Line 4).
Eventually, we return the
𝑆𝑂𝐿
as a global solution which comprises
the union of the computed sub-solutions during the previous steps
(Line 5).
Table 1 compares the time and space complexities of our pro-
posed algorithms for single and multiple drones. Note that, in this
paper, we implemented Mc-M exploiting the submodularity prop-
erty, and hence its time complexity is
O(𝑚(𝑛log 𝑛))
, and hence
polynomial.
Table 1: Comparison between the algorithms.
Algorithm Time Complexity Space Complexity
Mr-S O (𝑛log 𝑛) O (𝑛)
Mc-M O (𝑚(𝑛log 𝑛+(𝑛)) ) O (𝑛)
Mr-M O (𝑚(𝑛log 𝑛)) O (𝑛)
Multiple Drone-Delivery Scheduling Problem ICDCN 2022, January 4–7, 2022, New Delhi, India
5 PERFORMANCE EVALUATION
In this section, we compare the performance, in terms of obtained
reward, of the three proposed heuristics for solving MDSP with a
single drone and multiple drones cases.
5.1 The Settings
The last-mile delivery area
𝐴
is a circle of radius 5
km
in which the
depot is located at
(
0
,
0
)
. Then, the truck’s route is randomly gen-
erated inside
𝐴
. The locations of the
𝑛={
25
,
50
,
75
,
100
}
deliveries
are uniformly generated at random in
𝐴
. The truck carries a total
of
𝑚={
1
,
3
,
5
}
drones that y at a constant speed
𝑑𝑠=
20
m/s
[
32
].
We set the drone’s battery
𝐵=
5000
kJ
[
32
]. We set the drone’s
payload to
𝑑𝑝=
5
kg
[
32
]. With respect to these parameters, the
energy cost
𝑤𝑖
for performing a delivery is computed according
to the energy model presented in [
2
,
32
], which depends on the
distance to travel, the total mass of the drone plus the payload, and
the drone’s speed. We assume for the truck a whole duration trip
of 30000
s=
8
h
20
m
, which is a reasonable amount of time for a
single working day [6].
About the rewards assigned to each delivery, we randomly gen-
erate them as an integer number between
[
1
,
100
]
according to the
Zipf distribution [
34
] varying the
𝜃
parameter in
[
0
,
0
.
4
,
0
.
8
,
1
.
0
]
.
When
𝜃=
0, the rewards are uniformly distributed in
[
1
,
100
]
; when
𝜃
0
.
8, there are a few rewards with a large probability and many
rewards occurring a few times. The energy costs and the span times
are uniformly generated according to the Uniform distribution,
assuming four span/energy congurations
Σ𝑖
(from
Σ1
with low
variability to
Σ4
with high variability). Specically, in conguration
Σ1
the maximum energy cost is 2500
kJ
and the maximum span
time is 1500
s
, in
Σ2
this pair is 5000
kJ
and 10000
s
, and so on. Note
that,
Σ1
and
Σ2
always admit feasible deliveries, while the other
two do not. Moreover,
Σ4
allows very long span times as large as
the duration of the truck’s tour. Finally, once the span length
𝜏𝑖
is
generated, the launch time
𝑡𝐿
𝑖
is uniformly generated between
𝑡0
and 𝑡𝑟+1𝜏𝑖.
Table 2 summarizes the used parameters.
Table 2: Used input parameters.
Par. Description Value Unit
𝑑𝑠drone’s speed 20 m/s
𝑑𝑝drone’s payload 5 kg
𝐵drone’s budget 5000 kJ
𝑛number of deliveries [25,50,75,100] −
𝑚number of drones [1,3,5] −
𝑡𝑟+1nal time 30000 s
𝑝max reward 100
𝑤max energy cost 2500,5000,7500,30000 kJ
𝜏max span time 1500,10000,20000,30000 s
𝜃Zipf parameter [0,0.4,0.8,1.0] −
When evaluating the performance of our algorithms presented in
Section 4, we compare Mr-S with respect to the optimum solution
in the single drone scenario, and we compare Mc-M and Mr-M with
respect to the optimum solutions in the multiple drones scenario.
Optimal solutions are obtained from the ILP formulation, which
works regardless of the number of drones.
Also, as a reference for comparison (as baseline), we propose
three greedy heuristic algorithms for single and multiple drone
scenarios. Specically, we propose the following greedy heuristics
that repeatedly select:
GeRt
Greedy Earliest Rendezvous Time: the compatible interval
with the earliest rendezvous time 𝑡𝑅
𝑖.
GSw
Greedy Smallest Weight: the compatible interval with the
smallest energy cost 𝑤𝑖.
GLp
Greedy Largest Profit: the compatible interval with the
largest reward 𝑝𝑖.
In the case of multiple drones, these greedy heuristics are repeated
on the residual subset of deliveries not yet assigned to the drones.
5.2 Experiment Results
Figure 6 illustrates the performance, in terms of collected rewards,
of our algorithms on synthetic data. Specically, in the rst row, we
present the results for MDSP with
𝑚=
1(single drone), whereas
in the other two rows we illustrate the results for MDSP with
𝑚=
3and
𝑚=
5drones, respectively. The
𝑥
-axis reports the four
dierent congurations span/energy
Σ𝑖
, while the
𝑦
-axis shows the
ratio between the reward reported by any algorithm and that by
the optimum algorithm. Clearly, the ratio is less than or equals to 1.
5.2.1 Single drone scenario. When
𝑚=
1, the best performing
algorithms are Mr-S and GLp, which both take into account the
rewards in their delivery selection rule. Instead, GSw and GeRt,
whose selection rule is unlinked to the rewards, show the worst
performance. This trend is particularly emphasized for the Zipf
parameter
𝜃=
0
.
8. In this case, there are a few large-value rewards
and many small-value rewards. Thus, GLp is facilitated in their
choice. When the rewards are uniformly distributed (
𝜃=
0), Mr-S
guarantees better performance than GLp, probably because in its
selection it considers also the cost (weight) and not only blindly
the reward. Instead, in presence of unbalanced rewards (
𝜃=
0
.
8),
GLp exhibits the best solution because the algorithm can select the
most rewardable intervals without any other consideration. The
performance jump of GLp is high when 𝜃=0.8.
Interestingly, Mr-S shows interesting performance with respect
to the evaluated conguration. In fact, being completely free in
the interval selection and considering both the energy and the
reward criteria, Mr-S is the best solution. This holds until the
variability is low (e.g.,
Σ1
), the length of the intervals is short, and
the greedy strategy of picking intervals with the largest reward to
cost ratio is a winning strategy. In other words, in congurations
with low variability Mr-S clearly outperforms GLp regardless of
the value of
𝜃
and the number of deliveries
𝑛
. However, with higher
variability (e.g.,
Σ4
), the intervals become larger (as well as the
number of intersections) and Mr-S could incur into particularly
bad situations like the one depicted in Figure 5. This explains the
poor performance of Mr-S when the variability increases. Observe
that
Σ1
is the conguration that most likely represents a real-world
scenario. Generally, very long intervals for drones (e.g., like those in
Σ4
) are not recommended since they create too many intersections.
Therefore, even though GLp collects more reward than Mr-S in
many congurations, Mr-S shows a very good performance in
congurations closer to the reality (
95% of the optimum solution).
ICDCN 2022, January 4–7, 2022, New Delhi, India Francesco Bei Sorbelli, Federico Corò, Sajal K. Das, Lorenzo Palazzei, and Cristina M. Pinoi
Σ1Σ2Σ3Σ4
0.2
0.4
0.6
0.8
1
ratio
m= 1, n = 50, θ = 0
Mr-S
GeRt-S
GSw-S
GLp-S
Σ1Σ2Σ3Σ4
m= 1, n = 50, θ = 0.8
Σ1Σ2Σ3Σ4
m= 1, n = 100, θ = 0
Σ1Σ2Σ3Σ4
m= 1, n = 100, θ = 0.8
Σ1Σ2Σ3Σ4
0.2
0.4
0.6
0.8
1
ratio
m= 3, n = 50, θ = 0
Mc-M
MR-M
GeRt-M
GSw-M
GLp-M
Σ1Σ2Σ3Σ4
m= 3, n = 50, θ = 0.8
Σ1Σ2Σ3Σ4
m= 3, n = 100, θ = 0
Σ1Σ2Σ3Σ4
m= 3, n = 100, θ = 0.8
Σ1Σ2Σ3Σ4
0.2
0.4
0.6
0.8
1
configuration
ratio
m= 5, n = 50, θ = 0
Mc-M
MR-M
GeRt-M
GSw-M
GLp-M
Σ1Σ2Σ3Σ4
configuration
m= 5, n = 50, θ = 0.8
Σ1Σ2Σ3Σ4
configuration
m= 5, n = 100, θ = 0
Σ1Σ2Σ3Σ4
configuration
m= 5, n = 100, θ = 0.8
Figure 6: Performance evaluation of our algorithms on a synthetic data-set. The rst row compares algorithms with a single
drone, while the other two rows compare with multiple drones. Three greedy heuristics, used as a baseline, have suxes
depending on whether they are deployed on a single drone (-S) or multiple drones (-M).
5.2.2 Multiple drones scenario. In the multiple drone scenario, i.e.,
for
𝑚={
3
,
5
}
, we observe that GeRt-M and GSw-M perform
poorly, as in the single drone scenario. Accordingly, strategies that
pick intervals (deliveries) taking into account either weight or ren-
dezvous time do not perform well at all. The Mr-M has a very
good performance collecting more than 98% in the lowest vari-
ability congurations, i.e.,
Σ1
. Dierently from the single drone
case when Mr-S was employed, Mr-M performs well even when
the variability is high. We get this behavior because most likely
intervals in conict can still be assigned to dierent drones, and the
eect of having many intersections is less important. Concerning
Mc-M, although it does not guarantee any approximation ratio, it
performs quite well, and its collected reward is always above 80%
of the optimum. It is interesting to observe that both Mr-M and
Mc-M almost always outperform GLp-M. Finally, we observe that
the performance of our proposed algorithms with
𝑚=
5drones
is slightly better than those with
𝑚=
3. This is probably because
the residual intervals can be assigned to other independent drones
without conicts.
6 CONCLUSION
This paper investigated the Multiple Drone-Delivery Scheduling
Problem (MDSP) to study the cooperation between a truck and mul-
tiple drones in a last-mile package delivery scenario. After showing
that MDSP is an
NP
-hard problem, we proposed an optimal ILP
formulation that is suitable for small instances in input. Then, for
larger instances, we provided three time-ecient heuristic algo-
rithms for the single and multiple drones. Finally, we evaluated the
performance of the proposed algorithms on synthetic datasets. As
future work, it would be worth designing other eective algorithms
as well as searching for guaranteed approximation bounds for them.
Moreover, it would be interesting to investigate multi-depot multi-
truck scenarios or to allow drones to perform multiple deliveries at
the same time or recharge their battery on charging stations. We
also plan to study a more realistic dynamic environment, dealing
with network communications between the depot and ground-air
vehicles, addressing possible delays and new/canceled deliveries.
To address these challenges, we plan to develop online strategies
to reschedule deliveries on the y.
Multiple Drone-Delivery Scheduling Problem ICDCN 2022, January 4–7, 2022, New Delhi, India
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... Unmanned aerial vehicles (e.g., drones) are being increasingly used in numerous civilian applications, such as delivery of goods [1], [2], search and rescue operations [3], [4], and precision agriculture [5]. Publications on the use of drones in agriculture started appearing around 1998 and terrifically increased in the last decade, up to more than 150 reports in 2017 [6]. ...
... In this case, O[2, 1] = max{0, 3} because with budget 2 we cannot visit any observable position in A 2 (this is the reason why we have 0) while the best solution assigning budget 2 up to the previous aisle was 3: hence the maximum is 3. For O [2,2] we have an additional comparison. In fact, we can distribute the budget as follows: 4 only for A 2 , 2 for A 1 , or 4 for A 1 . ...
... In this case, O[2, 1] = max{0, 3} because with budget 2 we cannot visit any observable position in A 2 (this is the reason why we have 0) while the best solution assigning budget 2 up to the previous aisle was 3: hence the maximum is 3. For O [2,2] we have an additional comparison. In fact, we can distribute the budget as follows: 4 only for A 2 , 2 for A 1 , or 4 for A 1 . ...
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