Designing a transportation network for
a UAV delivery service
Author:
Amirali Ghahari
aghahari@uark.edu
(479)-595-6928
University of Arkansas
Master of Science in Industrial Engineering
Currently is Ph.D. Candidate in Industrial Engineering Department
ABSTRACT
Recently, the use of Unmanned Ariel Vehicle (UAV) for delivery services has become a topic of interest
and research for large commercial service providers such as Google and Amazon. The delivery speed of
UAVs provides such companies with significant advantage in their market. Although utilizing the UAVs in
product delivery has received tremendous excitement, there are several issues that need to be resolved
prior to real world implementation. The first issue with using UAVs as a transportation mode is their
limited flying range. Small UAVs are not able to fly long distances due to their limited battery life. Flight
range can be improved by using battery swapping stations in the planning horizon. UAVs can change
their depleted batteries at these stations and continue their flights. In our research, we attempt to
develop a model to construct a network of those stations in order to enable the UAVs to fly long
distances and make deliveries to different demand points.
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INTRODUCTION
Recently, the application of Unmanned Aerial Vehicles (UAVs) for delivery has become a topic of
increasing interest. Two well-known companies, Google and Amazon, are exploring this type of delivery
mechanism as part of their newest technology innovation strategy (Google X (2) Amazon Prime Air (1)).
Other applications of UAVs include disaster relief, border security, and uses in agriculture. In the
undeveloped countries with road deficiencies or for developed countries with congestion problems,
UAVs are a qualified mechanism of transport. When a disaster occurs, the roads might become
unpassable and the UAVs can become an appropriate way to deliver medicine and first aid packages.
Small UAVs can fly at low altitudes and can easily avoid obstacles at low altitudes. However,
they are limited in the size of the payloads they can carry and they can only fly for short distances due to
their battery capacity. For example, Micro UAV’s can fly for only one hour and carrying a maximum load
of 5 kilo grams (3). Long operations in terms of distance or time will require a system of UAVs and
automated service stations to swap batteries making the UAVs capable of traveling longer distances.
These kinds of stations currently exist and they operate automatically without the need for any labor (9).
In order to use the UAVs as delivery equipment, the locations of battery swap stations in the
supply chain network should be determined. This problem could be categorized as a facility location
problem in a supply chain system while the objective is minimizing the total cost of supporting the
mission of deliveries. Once the battery swap stations are located in the intended region in a cost
effective manner, we have a network of stations that enable the UAVs to fly longer distances and deliver
the payloads efficiently and effectively.
The next section briefly reviews the related literature. Section 3 discusses the mathematical
optimization model that is developed for this problem. The preliminary result that is collected from
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solving the constructed model is shown in section 4. In the last section a summary of the problem
results along with a conclusion is provided.
LITERATURE REVIEW
Several research studies have begun to explore the design of UAV systems for a variety of
situations.. Godzdanker et al. in (4) propose to solve a p-median problem to locate a fixed number of
stations. Godzdanker et al. in (5) study the station location problem while considering the UAV flight
path. Suzuki et al. (9) develop a Petri net to find the required number of UAVs and service stations to
maintain a desired number of UAVs in flight. Kim et al. (7) develop a mixed integer linear program (MILP)
model to handle the problem of scheduling a system of UAVs with multiple shared bases in disparate
geographic locations. Like Russel and Lamont (8), the authors solve the problem by using a genetic
algorithm. Kim and Morrison (6) develop a mixed integer linear program to describe the joint design and
scheduling of the UAV problem. The authors assumed returning flights, deterministic demand, multiple
tasks conducted by an UAV. Recently published research (10), considers refueling stations for electric
vehicles (EV). In this study, EVs act as delivery vehicles and should meet the demand points that have
deterministic demands
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MODEL FORMULATION
Problem Description:
In this paper, we study a delivery system in which a UAV acts as the only form of delivery
vehicle. In this problem, multiple depots and stochastic demand points are considered. Each
UAV can fly a limited range since it has a finite battery capacity. The battery swap stations can
be used to improve the flying range of the UAVs. The battery swap stations carry limited
number of batteries and they get replenished with new batteries at pre-determined time
points. There is fixed cost per battery swap associated with utilizing the battery swap stations.
Each demand point has a probability distribution for its demand which determines the number
of units demanded at that location. While the demand points might need multiple units of a
product, each UAV could carry exactly one unit of product. This means that satisfying a demand
point requires multiple visits of UAVs. As discussed earlier there are multiple depots that could
supply the products therefore the UAVs go to the closest depot after satisfying a demand point.
The demand should be satisfied by the delivery due date determined by the demand point
otherwise a lost sale will be incurred to the supply system. The problem attempts to minimize
the cost of constructing the battery swap stations considering the lost sales. There are different
types of stations depending on the capacity of holding batteries while the fixed construction
and operating costs increase as the capacity increases.
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Notation:
Below is the list of parameters and variables that are utilized in the mathematical model:
Sets:
𝑆: Set of all suppliers
𝐷: set of all Demand points
𝐵: set of all candidate locations for battery swap stations
A: set of all arcs between locations
𝑊: set of all scenarios
𝑇: set of all time slots
𝐾: set of all types of battery swap stations
Parameters:
𝑑𝑖𝜔 ∶ Amount of demands of demand point 𝑖 in scenario 𝜔
𝑏𝑘 : The number of batteries available at battery swaps stations type 𝑘
𝑐𝑙 ∶ Penalty cost for each unsatisfied demand
𝑐𝑠𝑘 : Cost of constructing a battery swap station type 𝑘.
𝑐𝑜𝑘 : Operational cost of a battery swap station in one period type 𝑘.
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Variables:
𝜔,𝑡
𝑋𝑖,𝑗
: Number of full travels between location 𝑖 and location 𝑗 in scenario 𝜔 at period 𝑡.
𝜔,𝑡
𝑌𝑖,𝑗
∶ Number of empty travels between location 𝑖 and location 𝑗 in scenario 𝜔 at period 𝑡.
𝑉𝑖𝜔,𝑡 ∶ Number of satisfied demands/supplies for location 𝑖 in scenario 𝜔 in period 𝑡.
𝐿𝜔,𝑡
: Number of lost sale for demand point 𝑖 in scenario 𝜔 in period 𝑡.
𝑖
𝑍𝑖𝑘
∶ Binary variable that is equal to 1 if batteries swap station 𝑖 construced
𝑂𝑖𝑘,𝑡 ∶ Binary variable that is equal to 1 if the batteries swap station type 𝑘 in period 𝑡 is
functional
Mathematical Model:
The following optimization model aims at minimizing the total cost of constructing the battery
swap stations considering the expected lost sale and the supply chain system requirement. One
of the critical requirements of this system is the flying limitation of each UAV. In order to
incorporate this limitation in our mathematical model, a network 𝐺 = (𝑉, 𝐸) is considered
where 𝑉 = 𝐵 ∪ 𝐷 ∪ 𝑆, union of all depots, demand points and candidate locations for the
battery swap stations, and 𝐸 is the set of all possible arcs between the nodes that exist in 𝑉.
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𝑀𝑖𝑧 𝑍 = ∑ 𝑐𝑠𝑘 ∗ 𝑍𝑖𝑘 +
∑
𝑖∈𝑏,𝑘∈𝑘
𝑖∈𝑑,𝑘∈ 𝑤,𝑡∈ 𝑡
𝑐𝑙 ∗ 𝐿𝑘,𝑡
𝑖 +
∑
𝑐𝑜𝑘 ∗ 𝑂𝑖𝑘,𝑡
(1)
𝑖∈𝐵,𝑘∈𝐾,𝑡∈𝑇
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑇𝑜:
∑
𝜔,𝑡
𝑋𝑖,𝑗
𝑗∈𝐹𝑆(𝑖)
∑
− ∑
𝜔,𝑡
𝑋𝑗,𝑖
𝑉𝑖𝜔,𝑡
= {−𝑉 𝜔,𝑡
𝑖
𝑗∈𝑅𝑆(𝑖)
𝜔,𝑡
𝑌𝑖,𝑗
𝑗∈𝐹𝑆(𝑖)
− ∑
0
𝑌𝑗,𝑖𝜔,𝑡
𝑗∈𝑅𝑆(𝑖)
𝑉𝑖𝜔,𝑡
{−𝑉 𝜔,𝑡
𝑖
0
∑ 𝑉𝑖𝜔,𝑡 = ∑ 𝑉𝑖𝜔,𝑡
𝑖∈ 𝐷
,∀𝑖 ∈ 𝑆, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
, ∀𝑖 ∈ 𝐷, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇 (2)
, ∀𝑖 ∈ 𝐵, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
, ∀𝑖 ∈ 𝑆, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
, ∀𝑖 ∈ 𝐷, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇 (3)
, ∀𝑖 ∈ 𝐵, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
(4)
, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
(5)
, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
(6)
𝑖∈𝑆
𝜔,𝑡
≥ 𝑑𝑖𝜔
∑ 𝐿𝜔,𝑡
𝑖 + 𝑉𝑖
𝑡∈𝑇
∑ 𝑉𝑖𝜔,𝑡 ≤ 𝑑𝑖𝜔
𝑡∈𝑇
𝜔,𝑡
𝜔,𝑡
+ 𝑌𝑖,𝑗
≤ ∑ 𝑏𝑘 ∗ 𝑂𝑖𝑘,𝑡
∑ 𝑋𝑖,𝑗
𝑗∈𝐹𝑆(𝑖)
∑ 𝑍𝑖𝑘 ≤ 1
, ∀𝑖 ∈ 𝐵 , ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇 (7)
𝑘∈𝐾
, ∀𝑖 ∈ 𝐵
(8)
, ∀𝑖 ∈ 𝐵
, ∀𝑘 ∈ 𝐾
(9)
𝑘∈𝐾
∑ 𝑂𝑖𝑘,𝑡 ≤ |𝑇| ∗ 𝑍𝑖𝑘
𝑡∈ 𝑇
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𝜔,𝑡
𝜔,𝑡
𝑋𝑖,𝑗
, 𝑌𝑖,𝑗
𝐼𝑛𝑡𝑒𝑔𝑒𝑟
𝑂𝑖𝑘,𝑡 ∈ {0,1}
𝑍𝑖𝑘 ∈ {0,1}
𝑉𝑖𝜔,𝑡
𝐼𝑛𝑡𝑒𝑔𝑒𝑟
,∀𝑖 ∈ 𝑆 ∪ 𝐷 ∪ 𝐵, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
, ∀𝑖 ∈ 𝐵, ∀𝑘 ∈ 𝐾, ∀𝑡 ∈ 𝑇
(10)
, ∀𝑖 ∈ 𝐵, ∀𝑘 ∈ 𝐾
,∀𝑖 ∈ 𝑆 ∪ 𝐷 ∪ 𝐵, ∀𝜔 ∈ 𝑊, ∀𝑡 ∈ 𝑇
Equation (1) is the objective function that consists of three parts, the first part is the total cost
of positioning and constructing the stations, the second part is the penalty function for any lost
sales, and the third part is the operational cost for the logistic network. Constraint set (2) and
(3) are flow balance constraints for loaded and empty flights respectively. Constraint set (4) is
the feasibility condition which must be held in order to have a feasible solution. Constraint set
(5) and (6) are upper and lower bounds for the satisfied demand and lost sale. Constraint set (7)
is the upper bound of number of batteries in each period based on the type of the stations.
Constraint set (8) assures that in each location only one type of station is established.
Constraint set (9) is the upper bound for the maximum number of operational periods. Finally,
constraint set (10) is the non-negativity constraint.
COMPUTATIONAL RESULTS
Problem instance
In this section, a problem instance is created to examine the effectiveness of the constructed
model for solving small size problems. In this instance, we used the locations of two large drug store
companies, CVS and Walgreens, around the Chicago metropolitan area. One Hundred and twenty
demand points (60 demand points for each drug store company) are identified and used for this
problem instance. Two supplier depots are randomly located around the Chicago area in order to supply
the demand points. 90 different locations are considered as candidates for battery swap stations which
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forces the optimization model to select the solution set among the candidate locations. The candidate
locations are randomly placed on a geographical grid with a distance interval of 3 miles. The flying range
of UAVs is considered to be 4 miles. We considered a penalty cost of $220 for each lost sale occurrence.
As mentioned before, battery swap stations could have different capacity sizes. In this problem instance,
5 different station sizes are considered as the possible alternatives. Lastly, each battery swap station is
assumed to be replenished 6 times during the planning horizon.
In the following table, the parameters associated with each type of stations are
presented.
TABLE 1. Battery swap stations characteristics
Maximum # of
Type of Battery Swap
Operational Cost for each
Cost of Constructing
Batteries in the
period
Stations
Station
1
$1000
$100
5
2
$2500
$250
10
3
$3500
$350
15
4
$5000
$500
20
5
$7000
$700
30
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FIGURE 1. The solution of proposed instance.
Preliminary Results
Figure 1 illustrates the solution to the problem instance. The red dots are the demand
points, two blue dots represent the depots and green dots are selected for constructing the
battery swap stations. The demands at the all points are satisfied with the total cost of $11,200
while minimal lost sale happened (29 out of 220 total demands).
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Sensitivity analysis:
We examined the sensitivity of solutions to different parameters using the same
problem instance. The first figure shows that the minimum cost of constructing a station is
influential for the total number of stations needed to satisfy the demand points. When this cost
component increases it is more cost effective to pay the penalty cost associated with the lost
sale rather than constructing more battery swap stations.
# Station Vs Cost
25
# of Stations
20
15
10
5
0
0
200
400
600
800
1000
1200
Cost of Constructing a Station ($)
FIGURE 2. Number of total stations versus the cost of constructing each
The problem is also very sensitive to the penalty cost that is chosen for lost sale. When the
profit margin of each unit increases the optimization model would rather build more battery
swap stations in order to avoid lost sales.
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250
# of Lost sale
200
150
100
50
0
0
50
100
150
200
250
300
350
Penalty cost ($)
FIGURE 3. Amount of lost sale versus the penalty cost parameter
CONCLUSION AND FUTURE STUDY
There are several obstacles in efficiently managing the supply chains such as traffic
congestion in city areas or physical limitations when the required transportation infrastructure
does not exist. In such situations using UAVs could help to satisfy the demand effectively and
efficiently. The optimization model that is presented in this study could reduce the total cost of
using UAVs in supply chain management when it is necessary. In the future, we wish to consider
additional constraints in our problem which reflect the limitation of using UAVs in real world
situation. Such constraints could be the total number of UAVs that could be utilized or
considering the maintenance/failure requirements of battery swap stations.
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REFERENCES
[1] Amazon prime air. http://www.amazon.com/b?node=8037720011. Accessed: 2014-10-01.
[2] Inside googles secret drone-delivery program. Online. Accessed: 2014-10-01.
[3] H Bendea, Piero Boccardo, S Dequal, Fabio Giulio Tonolo, Davide Marenchino, and Marco
Piras. Low cost uav for post-disaster assessment. In Proceedings of The XXI Congress of the
International Society for Photogrammetry and Remote Sensing, Beijing (China), 3-11 July 2008,
2008.
[4] Roy Godzdanker, Matthew J Rutherford, and Kimon P Valavanis. Islands: a self- leveling
landing platform for autonomous miniature uavs. In Advanced Intelligent Mechatronics (AIM),
2011 IEEE/ASME International Conference on, pages 170-175. IEEE, 2011.
[5] Roy Godzdanker, Matthew J Rutherford, and Kimon P Valavanis. Improving endurance of
autonomous aerial vehicles through intelligent service-station placement. In Robotics and
Automation (ICRA), 2012 IEEE International Conference on, pages 3179-3184. IEEE, 2012.
[6] Jonghoe Kim and James R Morrison. On the concerted design and scheduling of multiple
resources for persistent UAV operations. Journal of Intelligent & Robotic Systems, 74(1-2):479498, 2014.
[7] Jonghoe Kim, Byung Duk Song, and James R Morrison. On the scheduling of systems of UAVs
and fuel service stations for long-term mission fulfillment. Journal of Intelligent & Robotic
Systems, 70(1-4):347-359, 2013.3
Designing a transportation network for a UAV delivery service
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[8] Matthew A Russell and Gary B Lamont. A genetic algorithm for unmanned aerial vehicle
routing. In Proceedings of the 2005 conference on Genetic and evolutionary computation, pages
1523-1530. ACM, 2005.
[9] Koji AO Suzuki, Paulo Kemper Filho, and James R Morrison. Automatic battery replacement
system for UAVs: Analysis and design. Journal of Intelligent & Robotic Systems, 65(1-4):563-586,
2012.
[10] Jun Yang and Hao Sun. Battery swap station location-routing problem with capacitated
electric vehicles. Computers & Operations Research, 55:217-232, 2015.
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