Optimal Hub Selection for Rapid Medical Deliveries using

Unmanned Aerial Vehicles with Consideration of Flight Dynamics

Jose Escribano Maciasa,*, Panagiotis Angeloudisa, Washington Ochienga

a Centre for Transport Studies, Department of Civil and Environmental Engineering, Imperial College London, SW7 2BU,

UK

* Corresponding author.Email address: jose.escribano-macias11@imperail.ac.uk (J. Escribano)

ARTICLE INFO ABSTRACT

Keywords:

UAV

Selection-routing problem

Trajectory optimisation

Humanitarian relief distribution

Heuristic

Unmanned Aerial Vehicles are being increasingly considered for deployment in emergencies to enhance humanitarian response operations. Beyond regulation, vehicle range and integration with the underlying humanitarian supply chains inhibit their deployment in practice. To address these technical issues, we present a novel bi-level approach consisting of a trajectory optimisation algorithm, and hub selection-routing algorithm. A new trajectory optimisation model is presented that considers UAV battery consumption at each stage of flight. Such trajectories are provided as inputs to a bi-level hub selection-routing optimisation model that seeks to determine the optimal location of supporting infrastructure in the distribution supply chain.

1. Introduction

Humanitarian logistics operations aim to rapidly distribute relief resources and save lives in immediate peril.

With the increasing frequency of natural hazards expected due to climate change, the total expenditure on

humanitarian operations will increase (UNISDR, 2008), which for 2016 amounted to US$5,367 million for the

World Food Programme (WFP) alone (WFP 2017a). Given this level of expenditure, it is vital that respondents

have access to efficient systems that minimise risk and cost. A critical part of such systems is transportation

infrastructure. Due to deficiencies of conventional transport infrastructure in post-disaster scenarios, new

transportation modes are required to access affected regions. To this end, the use of Unmanned Aerial Vehicles

(UAVs) is increasingly being considered within the humanitarian community for logistics purposes among

others (Soesilo et al. 2016).

Commercially available UAVs are small to medium sized aircraft that can be either remotely controlled

or operate autonomously without human intervention. Recent advances in manufacturing processes,

communications, control and battery technology have facilitated applications in freight transport (Keane & Carr

2013; Heutger et al. 2014; Griffin 2016). Amazon was among the first major companies in the logistics sector

to announce plans for large-scale civil deployment of UAVs (CBS News 2013). Since then, many commercial

initiatives, particularly in the humanitarian and emergency response sectors, have been announced seeking to

deploy UAVs for remote sensing, mapping and last-mile cargo distribution (Soesilo & Sandvik 2016).

A list of high-profile UAV initiatives is provided in Table 1. A common classification is based on

vehicle body structure, with multi-rotors, fixed-wing and hybrids being the three main classes. Multi-rotor

vehicles behave like helicopters with multiple propellers and are capable of Vertical Take-Off and Landing

(VTOL). Fixed-wing UAVs resemble conventional aircraft and fly by persistently exerting forward thrust.

Finally, hybrids combine the features above, with higher control authority than fixed-wing UAVs and greater

range and payload capacity than multi-rotor vehicles, features that render them particularly suitable for logistics

applications. This is also reflected in Table 1, which indicates that most civilian UAV-logistics initiatives use

hybrid platforms.

Table 1 List of existing UAV-based delivery projects.

Organisation Technology Status Country Project type Reference

Amazon Hybrid Planned US & UK Commercial (Griffin 2016)

Zipline Fixed-wing Ongoing Rwanda Commercial (Simmons 2016)

RedLine Fixed-wing Planned Rwanda Humanitarian (Afrotech EPFL 2016)

Google Hybrid Planned US & UK Commercial (BBC News 2015)

DHL Hybrid Ongoing Germany Commercial (DHL 2015)

Zookal Multi-rotors Planned Australia Commercial (Welch 2015)

MSF Multi-rotors Ongoing PNG Humanitarian (Meier & Soesilo 2014)

UNICEF Multi-rotors Trial completed Malawi Humanitarian (UNICEF 2016)

Wings For Aid Fixed-wing Planned Belgium Humanitarian (Schretlen 2015)

WeRobotics Fixed-wing Trial completed Peru Humanitarian (Meier & Bergelund 2017)

Partial or complete damage to transport infrastructure as a result of natural disasters or armed conflict is

a common cause of community isolation and an impediment to relief efforts (Meier et al. 2016). Furthermore,

it exposes field workers and volunteers to significant safety risks. Autonomous airborne delivery systems have

the potential to address these issues with minimal infrastructure (limited to recharging and loading/unloading

stations) and less capital investment requirements compared to conventional transport methods. Despite the

presence of several commercial initiatives and pilot studies (Table 1), there is a lack of research on the dynamics

of large-scale UAV-deployment operations.

Haidari et al. (2016) evaluate the cost-effectiveness of UAV logistics for vaccine deliveries which, given

the value/weight ratio, perishability and temperature requirements represent an ideal initial use case for UAVs.

Factors that may affect UAV performance and operating costs such as weather, malfunctions, and maintenance

are not considered. Nedjati et al. (2015) studied the deployment of UAV for relief distribution ignoring the

variations in endurance due to payload, while D’Andrea (2014) developed a coarse method for the estimation

of costs to operate UAVs for delivery. Finally, Dorling et al. (2017) focused on urban good distribution using

multi-rotors considering battery and payload relationships. However, they studied the single depot routing

problem, which limits coverage and opportunities for multi-depot optimisation.

Compared to the broader spectrum of research in freight transportation, UAV operations are unique due

to the lightweight nature of the vehicles compared to the intended payloads, and the highly restrictive range

limitations, necessitating a bespoke family of freight distribution and supply chain design models. This paper

seeks to address this gap by developing a UAV distribution network design framework that determines optimal

UAV trajectories for relief distribution, supported by optimally determined operational hubs. In structuring this

framework, we identified two operational components driving the decision considered in this paper:

UAV operations component, focusing on the selection of aircraft trajectories, considering aerodynamic

effects, varying payloads, energy requirements and terrain topography. Travel times and energy

requirements are used to calculate variable transportation costs at the humanitarian mission design stage.

Humanitarian operations component, focusing on the design of the relief cargo supply chain and the

selection of optimal warehouse depots. The location of depots and structure of deliveries is determined with

the objective of minimising supply times and the overall duration of the humanitarian mission.

The relationship between the two components and the resulting models developed in this paper are captured in

Fig. 1.

A trajectory optimisation model is used to represent the decisions involved in the UAV Operations

component. It estimates UAV endurance for a range of payloads, determines flight paths that are clear from any

obstructions in the terrain and considers the variable energy demands associated with changes in flight height.

Humanitarian operations are modelled as a hub selection and vehicle routing problem. Optimal delivery

arrangements consider shipment quantities, the spatial arrangement of hubs, and the associated UAV itineraries.

The resulting decision framework can be used to quantify the performance of UAVs as an alternative means of

humanitarian cargo distribution in terms of operational costs and the volumes of material shipped over a fixed

mission time.

Humanitarian Operations

Relief Distribution Requirements

Aircraft Dynamics Modelling

Energy Use EstimationDemand Estimation

Debris collection, evacuation, etc.

Hub Location

Vehicle Routing

Location Routing Problem

UAV Operations

Battery Discharge Estimation

Aircraft trajectories

Damage & Population

Trajectory Optimisation

Optimised UAV-based humanitarian operations

Geographical and weather

data

Fig. 1. Framework flowchart indicating the dependency of two distinct systems in this paper

In their simplest form, humanitarian response operations use a fixed set of warehouses, also acting as

vehicle depots. Models developed for such operations commonly use a Vehicle Routing Problem (VRP)

formulation and derivative forms that can accommodate vehicle capacity constraints, time windows, budget

limits, network accessibility and fleet size limitations.

Barbarosoğlu et al. (2002) developed a two-step model to optimise the use of helicopters in disaster

logistics, minimising travelled distance. The main limitation of their approach is its lack of scalability, being

able to solve very small-scale operations (6 nodes). De Angelis et al. (2007) addressed a similar problem

presenting a multi-depot multi-vehicle routing and scheduling model for food provision operations by WFP in

Angola. The model is solved as a conventional VRP, achieving exact solutions in 2 hours. Özdamar (2011)

increases scalability by developing a static integer network flow problem to coordinate helicopter operations

for aid distribution and evacuation assistance using a bi-level heuristic. Coelho et al. (2017) formulate and solve

the Green Vehicle Routing Problem (G-VRP), that considers a fleet of UAVs with limited range that requires

refuelling or recharging in smart cities. A limitation of this study is that vehicle trajectories are disregarded, a

significant simplification given the several obstacles present in urban environments. Additionally, the

optimisation of potential recharging points or warehouse location is not considered.

Closer to the problem addressed in this paper, Nedjati et al. (2015) formulated an integer problem to

deploy medium-sized UAVs in Tehran, Iran. The problem is solved using an exact method for different mission

horizons and demand requirements but does not account for aerodynamic effects on UAVs during flight and the

relationship between energy consumption and payload carried. Other relevant studies have solved the VRP for

UAV systems without accounting for endurance as a limiting factor (Weinstein & Schumacher 2007; Ho et al.

2008; Chen & Cruz 2003). Furini et al. (2016) developed two Mixed Integer Linear Problems (MILP)

formulations for a variation of the VRP, the time-dependent Travelling Salesman Problem (TSP) in controlled

airspace. Conflict zones are used to enforce minimum separation rules, with individual UAVs allowed to hold

their position to avoid conflicts. Murray & Chu (2015) formulate and solve the Flying Sidekick Travelling

Salesman Problem (FSTSP), in which a UAV supports a conventional ground vehicle that acts as a mobile depot

for UAV deliveries to customers along its route. The UAV is assumed to have a fixed range regardless of its

payload, and replenishes its energy stocks aboard the vehicle.

The Location Routing Problem (LRP) is a relaxed version of the VRP that in addition to optimal

shipments also determines the location of depots used, therefore integrating aspects of the Hub Location

Problem (HLP). Several solution techniques have been developed for the LRP, including exact algorithms

(Laporte et al. 1986; Perl & Daskin 1985; Albareda-Sambola et al. 2005) and heuristics for larger problem

instances over 30 nodes and multiple time windows (Yi & Özdamar 2007; Wasner & Zäpfel 2004; Sarıçiçek &

Akkus 2013).

Yi & Özdamar (2007) solve the capacitated LRP for evacuation and field support in disaster response

using a two-stage modelling approach that consists of an integer dynamic flow problem and a detailed routing

method. To improve the performance, the model considers aggregate commodity flows instead of distinct

vehicle movements similar to Özdamar (2011). Sarıçiçek & Akkuş (2014) study the use of UAVs to monitor

national borders in Turkey. A two-stage algorithm is used, consisting of a p-hub median problem to determine

hub locations, and a routing model to determine UAV itineraries. An exact solution method is presented, which

however does not consider terrain obstructions and variable aircraft altitudes. Dorling et al. (2017) develop a

mixed-integer programming problem to solve the single-depot single-visit vehicle routing problem using UAVs.

Assuming the use of multi-rotors, Dorling et al. (2017) investigate the impact of energy consumption in relation

to the payload, the varying budget and network size.

Solution techniques for trajectory optimisation problems can be categorised based on their use of either

direct or indirect method. The former does not depend upon full analytic formulations and operates by

transforming the model into a nonlinear program (NLP) through discretisation (Betts 1998). The indirect method

uses calculus of variations and provides higher accuracy. However, it results in a small region of convergence

requiring accurate initial solutions (not always available) and the derivation of the full analytic expression.

Despite the availability of software capable of derivation, the direct method is more commonly used due

to its higher applicability (Betts 1998). It operates by transforming the trajectory optimisation method to an NLP

through discretisation, which is then solved based on an initial solution. Geiger (2009) used NLP and neural

networks to define routes that maximise surveillance time. Direct collocation is used to solve the states and

controls for every node. A similar methodology is used to analyse and avoid conflicts, that may occur within

the European Free Route Airspace environment (Raghunathan et al. 2004; Ruiz & Soler 2015; Arrieta-Camacho

et al. 2007). Further aspects of trajectory design are explored by Chakrabarty & Langelaan (2010), who created

a methodology that found the optimal energy efficient paths assuming known wind field data. A technique

known as regenerative soaring is used, which consists of converting wind’s kinetic energy to electrical energy.

Hosseini & Mesbahi (2013) solve the optimal path planning and power allocation problems for a solar-powered

UAV. Finally, linearised path planners have been developed by Forsmo (2012) as a Mixed Linear Problems

(MLP). However, the linearisation of the problem results in significant loss in accuracy and lack of scalability.

The presence of critical time constraints is common in most optimisation models employing the VRP and

LRP models discussed thus far. The presence of an additional time-dimension makes these models challenging

to solve, with the algorithms developed usually being applicable to only small instances. Bi-level models are

commonly used to improve solution times, often at the expense of optimality levels. In the case of UAV-specific

studies, all the routing algorithms employ predetermined aircraft ranges that do not account for the complex

energy consumption patterns observed in real life, which are affected by payloads, aerodynamics and flight

stages. A list of the papers and features reviewed is provided in

. Table 2 Classification of reviewed literature needs more detail

Decision-making

Focus Author Features Solution Method

(Chen & Cruz 2003) Genetic Algorithm(Weinstein & Schumacher 2007) Exact(Ho et al. 2008) Genetic Algorithm(Nedjati et al. 2015)

Scheduling; Capacity

Exact

(Özdamar 2011)Capacity; Refuelling; Split Delivery

Bi-stage exactOperational Routing

(De Angelis et al. 2007)Scheduling; Capacity; Time Windows

Exact

(Chakrabarty & Langelaan 2010) Wind; Energy Estimation A* Algorithm

(Raghunathan et al. 2004; Arrieta-Camacho et al. 2007; Pannequin et al. 2007)

Avoidance Direct Collocation

(Geiger 2009) Wind Direct Collocation

(Hosseini & Mesbahi 2013) Battery Modelling Direct Collocation

Tactical Trajectory

(Forsmo 2012) Avoidance; Waypoints

Mixed Linear Problem

(Barbarosoğlu et al. 2002) Bi-stage hierarchical

(Gulczynski et al. 2011)

Capacity; Split Delivery HeuristicRouting

(Dorling et al. 2017) Energy; Simulated Annealing

(Yi & Özdamar 2007) Pick-up & Delivery Bi-stage exactSimulated Annealing

Location-Routing (Wu et al. 2002; Sarıçiçek &

Akkus 2013) AllocationBi-stage exact

(Laporte et al. 1986; Albareda-Sambola et al. 2005) Capacity Exact

Strategic Location-Routing

(Wasner & Zäpfel 2004) Pick-up and Delivery Iterative Heuristic

The remainder of this paper describes the two models that form our proposed trajectory-location

optimisation framework for humanitarian relief distribution. The trajectory optimisation formulation is

described in Section 2, followed by the hub location-routing problem in Section 3. A numerical case study of

the 1999 Chi-Chi earthquake in Taiwan is presented in Section 4 with outputs compared to the results of the

study by Sheu (2010) on optimal ground-based response, used as a benchmark case.

2. Trajectory optimisation

Given the overwhelming adoption of hybrid UAVs for logistics applications (Table 1), we opted to derive

the trajectory optimisation model presented in this section with this class of aircraft, considering both stages of

flight (VTOL and cruise).

2.1. Modelling Demand

As the model specifically targets hybrid UAVs, take-off and landing operations are assumed to be carried

out vertically using four rotors. For the purposes of this study, we assume that the aircraft is battery powered,

with the variable weight of the vehicle, therefore, being only dependent on the payload. Nevertheless, we

intended this formulation to be as generic as possible, making it easy to incorporate further operational aspects

in the future (such as liquid fuels, aerodynamic and kinematic constraints).

Given the low aircraft velocities during VTOL, we consider the effects of air resistance to be insignificant

during this stage and will not be accounted for in this study. Following from Stoney (1993), we can assume the

desired acceleration during VTOL is equal to (where is the vehicle mass and the gravity coefficient), 1.2𝑚𝑔 𝑚 𝑔

with deceleration fixed at . We use a predetermined take-off altitude of 50 metres for each journey, 0.8𝑚𝑔

providing sufficient clearance for most obstacles and ensuring compliance with the majority of national drone

regulations on maximum altitudes (100-150 meters) for low weight UAVs (Stöcker et al. 2017). Due to the long

flight distances considered in this study, the impact of these assumptions on the total duration (and total energy

consumption) are negligible compared to the total flight time.

A range of efficiency considerations applies to trajectory planning. The propeller efficiency changes

continuously during the flight, as it is dependent upon aircraft velocity, as well as the diameter, design and

revolutions of the propellers (Brandt & Selig 2011). Furthermore, the efficiency of the motor ranges between

0% and 80% according to torque, rotation speed (RPM) and current (Gottlieb 1997). Additionally, battery

performance varies with cell chemistry, temperature, discharge rates and cycle life among others (Linden 1984).

We regard the definition of accurate mathematical models for the aircraft propeller, engine and battery to lie

beyond the scope of this study, and instead adopt an aggregate efficiency coefficient , with an assumed constant 𝜇

value of 0.5. The latter provides a worst case scenario for energy expenditure by a drone delivery aircraft

(D’Andrea 2014). We also consider a parasite drag coefficient and the drag polar of value 0.015 and 0.03 𝐶𝐷0 𝐾

respectively provided by Hall (2016). Both are determined by the shape of the vehicle and require the derivation

from experimental data as numerical evaluation for both is challenging (Ostler et al. 2009).

Finally, we assume that UAVs are allowed to fly over urban environments and Beyond Visual Line of

Sight (BVOS). At the time of writing, EASA proposes that UAVs of the size described in this paper cannot

operate BVOS in civilian settings, but we can reasonably expect a relaxation of this rule for emergency response

(EASA 2015). Furthermore, there is a record of such application in Rwanda, where Zipline was permitted to

provide UAV-based cargo deliveries (Simmons 2016).

2.2. Model Formulation

A point-mass and force balance model (Fig. 2) was used to derive energy consumption relationships and

the trajectory optimisation model. The following notation is used in the remainder of this chapter:

Indices: Sets:

𝑡 mission time step 𝒯 mission time horizon

𝑣 vehicle (UAVs) 𝒱 vehicle (UAV) fleet

State Variables: Control Variables:

𝑈 airspeed [𝑚𝑒𝑡𝑟𝑒𝑠/𝑠𝑒𝑐𝑜𝑛𝑑] 𝑇 thrust force [𝑁𝑒𝑤𝑡𝑜𝑛]

𝑆𝑂𝐶 state of charge [𝐴𝑚𝑝𝑠 ‒ 𝐻𝑜𝑢𝑟] Ψ heading angle [ ]𝑟𝑎𝑑𝑖𝑎𝑛𝑠

𝑥,𝑦,𝑧 Cartesian coordinates [𝑘𝑚] 𝛾 flight path angle [𝑟𝑎𝑑𝑠]𝐶𝐿 lift coefficient [ ‒ ]

𝐶𝐷 drag coefficient [ ‒ ]

Force Parameters: Battery Parameters:

𝐷 drag force [𝑁] 𝑃 power [𝑊𝑎𝑡𝑡𝑠]

𝐿 lift force [𝑁] 𝐼 current [𝐴𝑚𝑝𝑠]

𝑊 weight [𝑁] 𝑉 voltage [𝑉𝑜𝑙𝑡𝑠]

𝑚 mass [𝑘𝑖𝑙𝑜𝑔𝑟𝑎𝑚𝑠] 𝐵 battery capacity [𝐴ℎ]

𝑔 acceleration due to gravity [𝑚/𝑠2] Δ𝑡 time step [𝑠]

𝑎 vehicle acceleration [𝑚/𝑠2] 𝜇 propeller efficiency [ ‒ ]

𝐸𝑐 total battery capacity [ ‒ ]

Aerodynamic Parameters

𝐾 drag polar [ ‒ ]

𝜌 air density [𝑘𝑔/𝑚3]

𝑆 wing area [𝑚2]

𝑞 dynamic pressure: 12𝜌𝑈2 [𝑘𝑔/𝑚𝑠]

𝐶𝐷0 parasite drag coefficient [ ‒ ]

Fig. 2. Point mass model during cruise flight and vertical take-off and landing.

The following relationship holds true in a state of vertical equilibrium:

𝑇 = 𝑚(𝑎 ‒ 𝑔) (1)where denotes the thrust force, represents the mass of the vehicle, is the vehicle’s acceleration rate, and 𝑇 𝑚 𝑎

defines the acceleration due to gravity. The propulsive power required to generate a given thrust is 𝑔 𝑃 𝑇

determined from the following:

𝑃 = 𝑈𝑇/𝜇(2)

where determines the instantaneous speed of the vehicle, and is the efficiency parameter. The electric current 𝑈 𝜇

required for is determined by:𝑃

𝐼 = 𝑃/𝑉 (3)

where represents the battery voltage. The above relationships can be used to determine power consumption 𝑉

rates at any point in the flight. Over any discrete timestep , the resulting change in the state of charge ( ) Δ𝑡 𝑆𝑂𝐶

of the vehicle is:

Δ𝑆𝑂𝐶 =‒ 𝐼Δ𝑡/𝐵 (4)

where is the capacity of the battery1. 𝐵

The trajectory optimisation model used to determine cruising activities is defined as an Optimal Control

Problem (OCP), in which a set of state and control variables alter the system state. The following system of

equations describes aircraft movements in a Cartesian coordinate system (Fig. 4):

𝑑𝑥/𝑑𝑡 = 𝑈cos (Ψ)cos (𝛾) (5.1)

1 Battery capacity, , is expressed in , so a battery of discharging at a current of would take an hour to 𝐵 𝐴ℎ 12𝐴ℎ 12𝐴discharge fully.

𝑑𝑦/𝑑𝑡 = 𝑈sin (Ψ)cos (𝛾) (5.2)𝑑𝑧/𝑑𝑡 = 𝑈sin (γ) (5.3)

Since we aim to minimise battery consumption, we assume that any changes to the heading angle are Ψ

negligible. The resulting formulation (in two dimensions) based on the point-mass model in Fig. 4 is as follows:

Objective:𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒 1 ‒ 𝑆𝑂𝐶𝑣(𝑡𝑓) ∀ 𝑣 ∈ 𝒱 (6)

Subject to: 𝑑𝑠𝑣/𝑑𝑡 = 𝑈𝑣(𝑡)cos (𝛾𝑣(𝑡)) ∀ 𝑣 ∈ 𝒱 (6.1)

𝑑𝑧𝑣/𝑑𝑡 = 𝑈𝑣sin (𝛾𝑣(𝑡)) ∀ 𝑣 ∈ 𝒱 (6.2) 𝑑𝑈𝑣/𝑑𝑡 = 𝑇𝑣(𝑡)/𝑚𝑣 ‒ 𝐷𝑣(𝑡)/𝑚𝑣 + 𝐿𝑣(𝑡)sin (𝛾𝑣(𝑡))/𝑚𝑣 ∀ 𝑣 ∈ 𝒱 (6.3)

𝐷𝑣(𝑡) = 𝑞𝑆𝐶𝐷𝑣(𝑡) ∀ 𝑣 ∈ 𝒱 (6.4)

𝐶𝐷𝑣(𝑡) = 𝐶𝐷0 + 𝐾𝐶 2

𝐿𝑣(𝑡) ∀ 𝑣 ∈ 𝒱 (6.5)

𝐶𝐿𝑣(𝑡) =

𝐿𝑣(𝑡)

𝑞𝑆 ∀ 𝑣 ∈ 𝒱 (6.6)

𝐿𝑣(𝑡) = 𝑚𝑔/cos (𝛾𝑣(𝑡)) ∀ 𝑣 ∈ 𝒱 (6.7) 𝑑𝑆𝑂𝐶𝑣/𝑑𝑡 =‒ 𝑇𝑣(𝑡)𝑈𝑣(𝑡)/𝜇𝐵𝑉 ∀ 𝑣 ∈ 𝒱 (6.8)

|𝛾𝑣| ≤𝜋2, 0 ≤ 𝑇𝑣 ≤ 𝑇𝑚𝑎𝑥, 𝑈𝑣 = 𝑈𝑐𝑟, 𝑧𝑣 ≥ 𝑓(𝑥,𝑦) ∀ 𝑣 ∈ 𝒱 (6.9)

The following boundary conditions are assumed:

Initial: , , , , 𝑥𝑖(𝑡0) = 𝑥0 𝑦𝑖(𝑡0) = 𝑦0𝑖𝑧(𝑡0) = 𝑧0𝑖

𝑆𝑂𝐶𝑖(𝑡0) = 1 𝛾𝑖(𝑡0) = 𝛾0𝑖(7.1)

Final: , , , , 𝑥𝑖(𝑡𝑓) = 𝑥𝑓𝑖𝑦(𝑡𝑓) = 𝑦𝑓𝑖

𝑧𝑖(𝑡𝑓) = 𝑧𝑓𝑖𝑆𝑂𝐶𝑖(𝑡𝑓) = 𝑓𝑟𝑒𝑒 𝛾𝑖(𝑡0) = 𝑓𝑟𝑒𝑒 (7.2)

The objective of the resulting model is to maximise the state of charge by the end of the flight, with

values of 1 and 0 representing a fully charged and a depleted battery, respectively. Equations (6.1)-(6.2) 𝑆𝑂𝐶

represent the movement of the aircraft in Cartesian coordinates, governed by the flight path angle. Equation 𝛾

(6.3) is derived from the balance of horizontal forces acting upon the aircraft:𝑚𝑎 = 𝑇 ‒ 𝐷 + 𝐿sin (𝛾) = 𝑇 ‒ 𝑞𝑆𝐶𝐷 + 𝑚𝑔tan (𝛾) (8)

Constraint (6.5) describes the behaviour of the coefficient of drag , which is dependent on the drag 𝐶𝐷

polar , parasite drag coefficient and lift coefficient . is proportional to the UAV mass as indicated 𝐾 𝐶𝐷0 𝐶𝐿 𝐶𝐿 𝑚

by equation (6.6) and (6.7), where the vertical component of the lift force can be assumed to equate to the 𝐿

weight force . Finally, constraint (6.8) is a simplified version of the battery consumption derivation described 𝑊

through equations (2)- (4). The flight path angle is limited to 90 degrees. We assume that at any time the thrust 𝛾

is positive and lower than the maximum level of thrust provided by the motors. 𝑇

We assume that a speed target is predetermined and fixed throughout the cruise stage of the flight. 𝑈𝑐𝑟

The elevation has a lower-bound constraint derived by the function , which models ground elevation of 𝑧 𝑓(𝑥,𝑦)

the terrain. We considered true airspeed (actual movement of the aircraft relative to the air mass) to be equal to

indicated airspeed (measured speed of the vehicle). The impact of this assumption is minimal, as instrumental

errors are negligible during clean flights, the vehicle travels at low speeds (below sonic speed), and air density

variations with sea level can be ignored (Gracey 1980). Additional boundary conditions enforce the values of

heading and flight path angles for the initial and final points of the trajectory. UAVs are assumed to loiter Ψ 𝛾

and decelerate once they reach their destinations.

The finite difference method (Grossmann & Roos 2007) is used to discretise the problem into discrete

elements, therefore transforming the OCP into the following nonlinear optimisation problem:

Objective:𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒 1 ‒ 𝑆𝑂𝐶𝑣,𝑡𝑓 ∀ 𝑣 ∈ 𝒱 (9)

Subject to: 𝑠𝑣,𝑡 + 1 = 𝑠𝑣,𝑡 + Δ𝑡 𝑈𝑣,𝑡cos (𝛾𝑣,𝑡) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.1) 𝑧𝑣,𝑡 + 1 = 𝑧𝑣,𝑡 + Δ𝑡 𝑈𝑣.𝑡sin (𝛾𝑣,𝑡) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.2)

0 = 𝑇𝑣,𝑡 ‒ 𝐷𝑣,𝑡 + 𝐿𝑣,𝑡sin (𝛾𝑣,𝑡) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.3) 𝐷𝑣,𝑡 = 𝑞𝑆𝐶𝐷𝑣,𝑡 ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.4)

𝐶𝐷𝑣,𝑡= 𝐶𝐷0 + 𝐾𝐶 2

𝐿𝑣,𝑡∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.5)

𝐶𝐿𝑣,𝑡= 𝐿𝑣,𝑡/𝑞𝑆 ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.6)

𝐿𝑣,𝑡 = 𝑚𝑔/cos (𝛾𝑣.𝑡) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.7) SOC𝑣,𝑡 + 1 = SOC𝑣,𝑡 ‒ Δ𝑡 𝑇𝑖,𝑡𝑈𝑖,𝑡/𝜇𝐵𝑉 ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.8)

|𝛾𝑣,𝑡| ≤𝜋2, 0 ≤ 𝑇𝑣,𝑡 ≤ 𝑇𝑚𝑎𝑥, 𝑈𝑣,𝑡 = 𝑈𝑐𝑟, 𝑧𝑣,𝑡 ≥ 𝑓(𝑥,𝑦) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (9.9)

Constraints (9.1)-(9.9) follow from (6.1)-(6.9), with the size number of discrete steps in being a 𝑡 𝒯

compromise between computational time and results accuracy. The formulation presented in Equation (9)-(9.9)

was implemented using the pyomo.dae modelling framework and solved using the IPOPT engine (Hart et al.

2011; Hart et al. 2012). An illustrative example of a trajectory is presented in Fig. 3.

-100040

0

40

Z (m

)

Y (km)

1000

X (km)

20 20

2000

0 0 0 500 1000 1500 2000Time (seconds)

400

500

600

700

800

900

1000

Altit

ude

(met

res)

Fig. 3 Example trajectory of 10 elements between the Kinyi and Renai regions of the Nanto County in Taiwan.

Table 3Flight stages during trajectory optimisation.

Flight Stage Controls State Variables Thrust Use Condition

Take-Off 𝑇 𝑈, 𝜇, 𝑔, 𝑉, 𝐵, 𝑚 𝑇 = 𝑚(𝑎 ‒ 𝑔) 𝑎 > 0

Climb 𝑇, 𝛾 𝑠, 𝑧, 𝐶𝐷0, 𝑈, 𝜇, 𝑔, 𝑉, 𝐵, 𝑞, 𝑆, 𝑚 𝑇 = 𝐷 + 𝑚𝑈 ‒ 𝐿sin (𝛾) 𝛾 > 0

Cruise 𝑇, 𝛾 𝑠, 𝑧, 𝐶𝐷0, 𝑈, 𝜇, 𝑔, 𝑉, 𝐵, 𝑞, 𝑆, 𝑚 𝑇 = 𝐷 + 𝑚𝑈 ‒ 𝐿sin (𝛾) 𝛾 = 0

Descend 𝑇, 𝛾 𝑠, 𝑧, 𝐶𝐷0, 𝑈, 𝜇, 𝑔, 𝑉, 𝐵, 𝑞, 𝑆, 𝑚 𝑇 = 𝐷 + 𝑚𝑈 ‒ 𝐿sin (𝛾) 𝛾 < 0

Landing 𝑇 𝑈, 𝜇, 𝑔, 𝑉, 𝐵,𝑚 𝑇 = 𝑚(𝑎 ‒ 𝑔) 𝑎 < 0

Fig. 4. The coordinate system of the trajectory optimisation problem.

The nonlinear trajectory optimization presented in this section is used to design UAV routes accounting

for geographical boundaries and payload variation. Flight ranges and times will be provided as inputs to the

next stage of the model, focusing on the determination of optimal warehouse locations and UAV itineraries.

3. Depot location and routing for UAV operations

In addition to determining the exact routes that UAVs will follow, the models presented in the previous

section can be used to calculate costs and flight times for a range of payloads and vehicle specifications. These

are key inputs to the models considered in this section, where we seek to determine the optimal selection of

vehicle depots and calculate deployment plans for relief cargo distribution. These decisions are modelled as a

hub selection-routing problem, which combines aspects of the discrete un-capacitated hub location problem and

the time-dependent Green VRP with multiple hubs and split deliveries.

Relief operations commonly rely on standardised supply kits that are developed for specific

combinations of incident type (earthquake, flooding, displacement), demography (diet, religion) and climate.

This practice simplifies the logistics of disaster response, as only a single commodity needs to be stocked,

transported and distributed (WFP 2017b). As such, it is a sensible assumption to consider a single unitised

commodity in the models used in this study.

We assume that the energy source aboard UAVs is replaceable, with fully charged batteries always

available in depots. Battery replacement times are considered to be negligible compared to overall mission

durations.

3.1. Model formulation

In this section, we introduce a formulation that seeks to minimise the number of hubs and required mission time

for the distribution of a predefined amount of relief cargo. Given the multi-objective nature of this problem, we

express a combined objective function which considers the costs of the resulting configurations to be negligible

given the small payload size and battery weight. The following notation is used in the remainder of this chapter:

Indices: Sets:

𝑖,𝑗 nodes in the network 𝒩 nodes in the network

𝑡 mission time steps ℋ hubs in the network

𝑣 vehicles (UAVs) 𝒯 mission time horizon

𝒱 vehicle set

Parameters:

𝐹 fixed setup costs of a hub [£]

𝑃𝑒𝑛 penalty for not delivering a unit of demand [£]

𝑄 vehicle payload capacity [𝑘𝑔]

𝐶𝑖,𝑗 time cost between nodes and 𝑖 𝑗 [£]

𝐴𝑖 aid demand at node , a negative demand indicates supply 𝑖 [𝑘𝑔]

𝐸𝑖,𝑗 percentage battery consumption for travel between nodes to 𝑖 𝑗 [ ‒ ]

Decision variables:𝑟𝑖,𝑗,𝑣,𝑡 Boolean variable determining if vehicle travels from to at time 𝑣 𝑖 𝑗 𝑡

𝑐𝑖,𝑗,𝑣,𝑡 amount of cargo carried from to by vehicle over time step 𝑖 𝑗 𝑣 𝑡

ℎ𝑖 Boolean variable determining whether node becomes a hub𝑖

𝑢𝑖 unfulfilled demand to node 𝑖

𝑠𝑖,𝑝𝑖 slack variable for unfulfilled demand to node 𝑖

𝑒𝑣,𝑡 current percentage battery charge

Objective:

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒 𝑍 = ∑𝑖 𝜖 𝒩

ℎ𝑖𝐹 + 𝑀𝑎𝑥(∑𝑡 ∈ 𝒯

∑𝑖 ∈ 𝒩

∑𝑗 ∈ 𝒩

𝑟𝑖,𝑗,𝑣,𝑡 𝐶𝑖,𝑗) + ∑𝑖 ∈ 𝒩

𝑢𝑖𝑃𝑒𝑛(10)

Subject to:

∑𝑗 ∈ 𝒩

𝑟𝑖,𝑗,𝑣,𝑡 ‒ ∑𝑗 ∈ 𝒩

𝑟𝑗,𝑖,𝑣,(𝑡 + Δ𝑡𝑖,𝑗) = 0 ∀ 𝑖 ∈ 𝒩, ∀ 𝑣 ∈ 𝒱 ∀ 𝑡 ∈ 𝒯 (10.1)

∑𝑡 ∈ 𝒯

∑𝑣 ∈ 𝒱

∑𝑗 ∈ 𝒩

𝑐𝑖,𝑗,𝑣,𝑡 + ∑𝑡 ∈ 𝒯

∑𝑣 ∈ 𝒱

∑𝑗 ∈ 𝒩

𝑐𝑗,𝑖,𝑣,𝑡 + (𝑠𝑖 ‒ 𝑝𝑖) = 𝐴𝑖 ∀ 𝑖 ∈ 𝒩 (10.2)𝑠𝑖 ‒ 𝑝𝑖 = 𝑢𝑖 ∀ 𝑖 ∈ 𝒩 (10.3)‒ |𝐴𝑖| ≤ 𝑝𝑖 ≤ 0 ∀ 𝑖 ∈ 𝒩 (10.4)

0 ≤ 𝑠𝑖 ≤ |𝐴𝑖| ∀ 𝑖 ∈ 𝒩 (10.5)𝑟𝑖,𝑗,𝑣,𝑡𝑄 ≥ 𝑐𝑖,𝑗,𝑣,𝑡

∀ 𝑖 ∈ 𝒩, ∀ 𝑗 ∈ 𝒩∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (10.6)

𝑒𝑣,𝑡 + 1 = (1 ‒ ∑𝑖 ∈ 𝒩

∑𝑗 ∈ ℋ

𝑟𝑖𝑗𝑣𝑡)(𝑒𝑣,𝑡 + ∑𝑖 ∈ 𝒩

∑𝑗 ∈ 𝒩

𝐸𝑖𝑗𝑡𝑟𝑖𝑗𝑣𝑡) ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (10.7)

1 ≥ 𝑒𝑣,𝑡 + ∑𝑖 ∈ 𝒩

∑𝑗 ∈ 𝒩

𝐸𝑖𝑗𝑡𝑟𝑖𝑗𝑣𝑡 ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (10.8)0 ≤ 𝑒𝑣,𝑡 ≤ 1 ∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (10.9)∑

𝑖 ∈ 𝒩ℎ𝑖 ≥ 1 (10.10)

𝑟𝑖,𝑗,𝑣,𝑡 ∈ [0,1], 𝑐𝑖,𝑗,𝑣,𝑡 ∈ [0,1,…] ∀ 𝑖 ∈ 𝒩, ∀ 𝑗 ∈ 𝒩∀ 𝑣 ∈ 𝒱, ∀ 𝑡 ∈ 𝒯 (10.11)

ℎ𝑖 ∈ [0,1], 𝑢𝑖 ∈ [0,1,…, 𝐴𝑖], 𝑠𝑖 ∈ [0,1,…], 𝑝𝑖 ∈ [0, ‒ 1,…] ∀ 𝑖 ∈ 𝒩 (10.12)

The model introduces a numerical penalty associated with the loss of a life that could be attributed 𝑃𝑒𝑛

to a late or missed delivery of planned relief cargo units. In this study, we consider to act as the Big M 𝑃𝑒𝑛

method, such that the model prioritises mission completion (Griva et al. 2009). Constraint (10.1) ensures that

the number of vehicles entering and exiting a node is equal, while (10.2) enforces a mass balance of goods at

each node. Two slack variables are introduced to linearise the behaviour of absolute values as shown in equation

(10.3). Constraints (10.4) and (10.5) ensure that unsatisfied demand is bound by original demand. Constraint

(10.6) limits cargo flows to assigned vehicle capacities, while (10.7)-(10.8) ensures that the battery capacity is

not violated between every hub visit. Batteries can only be replaced at a hub – the time needed to do so is

considered to be negligible. Equation (10.9) limits the total battery expenditure to a maximum capacity 𝐸𝑐.

Constraint (10.10) ensures that at least one hub is selected. Constraints (10.11-10.12) limit the decision variables

, and slack variables , to integer values, while and are limited to Boolean values.𝑐𝑖,𝑗,𝑣,𝑡 𝑢𝑖 𝑠𝑖 𝑡𝑖 𝑟𝑖,𝑗,𝑣,𝑡 ℎ𝑖

Table 4Model Runtime in sample problems. Those with no result were run for 24 hours without providing a result.

Instance Demand (kg) Payload (kg) Mission time (mins) Cargo delivered (kg) Runtime (s)

50 300 0.25

100 600 0.432

150 876 3.46

50 450 0.5

100 879 0.89

4 nodes 1000

3

126 1000 1.2

2 50, 100, 150 No result N/A7 nodes 800

3 50, 100, 150 No result N/A

50 500 0.842

100, 150 No result N/A10 nodes 1450

3 50, 100, 150 No result N/A

The main limitation of the formulation above is the large number of decision variables that need to be

considered – this results in exceptionally long computational time requirements for any instance of practical

size. The model was solved using CPLEX for small instances, with runtimes found to depend upon time-step

resolution and network size. To overcome this problem, we present in the following section a multi-stage

approximate heuristic.

3.2. Hub flow sub-problem

The first component of our approximate heuristic is based on a static, integer-based hub-flow assignment model,

which is a relaxation on the VRP-specific aspects of the original problem. A formulation yielding optimal

vehicle and cargo flows is provided below:

Objective:

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒 𝑍 = ∑𝑖 ∈ 𝒩

ℎ𝑖𝐹 + ∑𝑖 ∈ 𝒩

∑𝑗 ∈ 𝒩

𝑟𝑖,𝑗 𝐶𝑖,𝑗 + ∑𝑖 ∈ 𝒩

𝑢𝑖 𝑃𝑒𝑛 (11)

Subject to:

∑𝑗 ∈ 𝒩

𝑟𝑖,𝑗 ‒ ∑𝑗 ∈ 𝒩

𝑟𝑗,𝑖 = 0 ∀ 𝑖 ∈ 𝒩 (11.1)

∑𝑗 ∈ 𝒩

𝑐𝑖,𝑗 + ∑𝑗 ∈ 𝒩

𝑐𝑗,𝑖 + 𝑢𝑖 = 𝐴𝑖∀ 𝑖 ∈ 𝒩 (11.2)

𝑠𝑖 ‒ 𝑝𝑖 = 𝑢𝑖 ∀ 𝑖 ∈ 𝒩 (11.3)‒ |𝐴𝑖| ≤ 𝑝𝑖 ≤ 0 ∀ 𝑖 ∈ 𝒩 (11.4)

0 ≤ 𝑠𝑖 ≤ |𝐴𝑖| ∀ 𝑖 ∈ 𝒩 (11.5)𝑟𝑖,𝑗𝑄 ≥ 𝑐𝑖,𝑗 ∀ 𝑖 ∈ 𝒩, ∀ 𝑗 ∈ 𝒩 (11.6)∑

𝑖 ∈ 𝒩ℎ𝑖 ≥ 1 (11.7)

𝑟𝑖,𝑗 ∈ [0,1,…], 𝑐𝑖,𝑗 ∈ [0,1,…] ∀ 𝑖 ∈ 𝒩, ∀ 𝑗 ∈ 𝒩 (11.8) ℎ𝑖 ∈ [0,1], 𝑢𝑖 ∈ [0,1,…], 𝑠𝑖 ∈ [0,1,…], 𝑝𝑖 ∈ [0, ‒ 1,…] ∀ 𝑖 ∈ 𝒩 (11.9)

The main difference with the formulation in Section 3.1 is the removal of the time and vehicle dimensions.

The objective function (11) seeks to minimise the total travel time of all deployed UAVs, the fixed costs of hubs

and the penalty related to loss of life – as before, the various objectives are collectively accommodated by

considering costs. Many of the constraints from the earlier formulation are carried over: (11.1) ensures vehicle

balance is satisfied, (11.2) considers the balance of cargo, (11.3-11.5) limit unmet demand, (11.6) ensures that

the payload capacity is satisfied, and (11.7) enforces the creation of a hub. The constraint relating to the battery

capacity of the UAVs have been removed from this formulation, as it is enforced at the second stage of the

heuristic. The aggregate flows represent a lower-bound solution to the original problem and can act as proxies

for the number of vehicles and volume of cargo transported along every link.

3.3. Route assignment algorithm

The approximate algorithm presented in this section builds upon earlier work from Özdamar (2011) and Dorling

et al. (2017), and can be used to transform downscaled aggregate flows into distinct UAV itineraries. The

algorithm can be further decomposed into four distinct decision stages, namely route continuity, cargo

assignment, energy monitoring, and vehicle assignment.

Route creation and continuity: This stage generates vehicle tours that begin and end at a hub. Starting with the

total number of vehicles to operate from each hub, the algorithm iterates through outgoing material flows

(obtained from the hub-flow stage) in hubs, and creates tours by selecting the next destination randomly. This

process is continued until a node is encountered with no onwards flows – in this case, the vehicle is randomly

directed to either an unvisited node or a hub that terminates the tour. The latter does not need to be the original

hub were the tour began. An implementation of this process using pseudocode is provided below:

1. Algorithm RouteContinuity has2. Input: vehicle flows 𝑥𝑖𝑗3. Output: routes 𝑁𝑟4.5. for each in hub set do𝑖 𝐻6. 𝑁𝑟 = ∑

𝑗𝑥𝑖,𝑗7. for each in do𝑟𝑣 𝑁𝑟8. add random initial flow to 𝑟9. randomly𝑟𝑣(1) ←𝑥 ∗

𝑖,𝑗

10. block selected 𝑥 ∗𝑖,𝑗

11. if exists and if 𝑥𝑗,𝑘 𝑘 ≠ ℎ𝑢𝑏

12. randomly𝑟𝑣(2) ←𝑥 ∗𝑗,𝑘

13. block 𝑥 ∗𝑗,𝑘

14. else15. randomly𝑟𝑣(2) ←𝑥 ∗

𝑗,𝑘16. break17. while 𝑘 ≠ ℎ𝑢𝑏18. if node is connected to hub𝑙19. 𝑟𝑣(𝑒𝑛𝑑) ←𝑥 ∗

𝑘,𝑙20. else21. randomly𝑟𝑣(𝑒𝑛𝑑) ←𝑥 ∗

𝑘,𝑙22. break23. return 𝑁𝑟

Pseudocode 1. Route Continuity.

Cargo assignment: In this stage, cargo flows are again assigned to vehicle tours with considerations of indicative

flow optimality, vehicle capacity, and total unmet demand of visited nodes. Once all routes have been assigned

with cargo, the algorithm checks whether any material flow remains unassigned and calculates whether all

destination points have their total demand for relief aid satisfied. If this is not the case, the algorithm activates

a reassignment process that identifies positive or negative deviations from set demand targets and adjusts the

flows that were assigned to each vehicle. The pseudocode for both operations is provided below:

1. Algorithm CargoAssignment has2. Input: routes , flows , demand , payload capacity 𝑁𝑟 𝑐𝑖,𝑗 𝐴𝑖 𝑄3. Output: route assignments 𝑁𝑟𝑐4.5. for each in 𝑟𝑣 𝑁𝑟6. for each node in 𝑛 𝑟𝑣7. cargo assigned to route ,𝑟𝑐(𝑛)←min (𝑐𝑖,𝑗 𝑄, 𝐴𝑖)8. 𝑁𝑟𝑐←𝑟𝑐9. for each node in 𝑖 𝐴𝑖10. if 𝐴𝑖 ≠ Sum(𝑟𝑐(𝑟𝑣(𝑛) = 𝑖))11. 𝑁𝑟𝑐 = ReassignCargo(𝑁𝑟𝑐, 𝑐𝑖,𝑗, 𝐴𝑖, 𝑄)12. return 𝑁𝑟𝑐

Pseudocode 2. Cargo Assignment Pseudocode.

1. Algorithm ReassignCargo has2. Input: routes , flows , demand , payload capacity 𝑁𝑟 𝑐𝑖,𝑗 𝐴𝑖 𝑄3. Output: route assignments 𝑁𝑟𝑐4. for each node in 𝑖 𝐴𝑖5. while 𝐴𝑖 > Sum(𝑟𝑐(𝑟𝑣(𝑛) = 𝑖))6. Find(𝑟𝑐(𝑟𝑣(𝑛) = 𝑖) < 𝑄7. 𝑟𝑐(𝑟𝑣(𝑛) = 𝑖) += 18. while 𝐴𝑖 < Sum(𝑟𝑐(𝑟𝑣(𝑛) = 𝑖))9. Find(𝑟𝑐(𝑟𝑣(𝑛) = 𝑖) > 010. 𝑟𝑐(𝑟𝑣(𝑛) = 𝑖) ‒= 111. return 𝑁𝑟𝑐

Pseudocode 3. Cargo Reassignment Pseudocode.

Energy monitoring: Once route continuity and cargo assignments have been addressed, we proceed to an energy

monitoring stage that ensures that battery capacity is respected by all tours. We calculate battery consumption

at each stop along a tour. Payload variations are also considered, with consumption for an average trip expected

to reduce as cargo is being discharged. If energy consumption in a given tour exceeds the allowance afforded

by battery capacity, we inject a visit to a hub where the vehicle’s battery is swapped. This visit is placed ahead

of the tour leg that violates capacity, and proceeds as follows: in the first instance, we opt for the hub closest to

the last feasible node. If this hub is too far to ensure compliance and the battery would have been depleted before

the vehicle arrives for replenishment, we revert to the original route and inject the stop at an earlier step. This

process is repeated at later points in the tour where the battery capacity is expected to be depleted again.

1. Algorithm EnergyMonitoring has2. Input: routes , battery consumption 𝑁𝑟 𝑏𝑖,𝑗3. Output: feasible routes 𝑁𝑏4.5. for each in 𝑟𝑣 𝑁𝑟6. 𝑟𝑏𝑣

←𝑟𝑣7. 𝑟𝑏𝑐

←𝑟𝑐8. for each in 𝑛 𝑟𝑏𝑣

9. = 𝑒 𝑒 + 𝑏𝑖,𝑗(𝑐𝑖,𝑗)10. while > 𝑒 𝑅11. hub 𝑟𝑏𝑣

(𝑛 ‒ 1)← ℎ12. 𝑟𝑏𝑐

(𝑛 ‒ 1)← 𝑟𝑐(𝑛 ‒ 2)1. if > 𝑒 𝑅13. 𝑟𝑏𝑣

,𝑟𝑏𝑐←DeleteRouteStop(𝑟𝑣(𝑛 ‒ 1), 𝑟𝑐(𝑛 ‒ 1))

14. 𝑛 = 𝑛 ‒ 115. 𝑁𝑟←𝑟𝑏𝑐

, 𝑟𝑏𝑣

16. return 𝑁𝑏

Pseudocode 4. Energy Monitoring Pseudocode.

Vehicle assignment: This stage evaluates the time required to complete each route, which is then assigned to a

vehicle that minimises the overall mission time. UAVs are initially allocated to hubs in proportion to the total

cargo delivered from each hub. Trip assignments are determined with the objective of minimising relocation

time, vis-à-vis the time for a UAV to travel between two hubs to pick up the cargo. This process is repeated for

every UAV until the total mission time equals the expected mission time. Should any trips remain, they are

assigned individually to minimise the total mission time.

1. Algorithm VehicleAssignment has2. Input: feasible routes , time matrix , drones 𝑁𝑏 𝑡𝑖,𝑗 𝐷3. Output: route for each UAV set 𝑁𝑑4.5. for each in 𝑟 𝑁𝑏6. route time 𝑡𝑟 = 07. for each in 𝑛 𝑟8. 𝑡𝑟 = 𝑡𝑟 + 𝑡𝑛,𝑛 + 19. drone route time 𝑡𝑑 = Zeros(𝐷)10. for each in 𝑟 𝑁𝑏11. for each drone in 𝑑 𝐷12. 𝑡𝑚𝑖𝑛 = min (𝑡𝑑)13. if 𝑡𝑑 = 𝑡𝑚𝑖𝑛14. 𝑟𝑑← 𝑟15. 𝑡𝑑 = 𝑡𝑑 + 𝑡𝑟16. 𝑁𝑑←Add(𝑟𝑑)17. return 𝑁𝑑

Pseudocode 5. Vehicle Assignment Algorithm Pseudocode.

3.4. Metaheuristic solution algorithm

The set of algorithms above were implemented using C#, and embedded within a custom metaheuristic solver

developed in this paper. The solver was structured as a Large Neighbourhood Search (LNS) approach, which

navigates through a constrained search space using a probabilistic search (Pisinger & Ropke 2010). Its key

strength is the ability to simultaneously explore multiple result neighbourhoods that reside in different branches

of the search space (unlike Genetic Algorithms). Furthermore, greedy algorithms (such as those presented in

section 3.3) can be considered stochastically, making it easy to avoid local maxima.

Pseudocode 6 provides the LNS developed in this paper. The search commences with a randomly

generated solution-neighbourhood , a selection stage determines which solution to manipulate in each 𝑁𝑖

iteration. A roulette method is used to ensure that all solutions have the probability to be explored, with selected

solutions manipulated using custom destroy and repair mechanisms. The exact method to be used in the latter

is determined by random selection with the probability assigned determined through calibration. The overall

workflow of the algorithm is presented in Fig. 5, with overall progress defined by the “temperature” metric,

which represents the range of the search space that can be explored from the best current solution, decreasing

with every iteration of the LNS.

Specify parameters

Trajectory optimisation Initiate heuristic

Create hub assignment

solution

Solve static integer flow

problem

Route assignment process

Solution evaluation

Is solver finished?Save final solution

No

Yes

Start

Fig. 5. Model Workflow.

1. Algorithm LNS has2. Input: temperature , absolute temperature ,𝑇 𝛼

cooling rate , neighborhood size 𝑐 𝑛3. Output: optimal solution 𝑠 ∗

4.5. neighborhood 𝑁←LNS(𝑇,𝑎,𝑐,𝑛)6. for each solution in 𝑠 𝑁7. fitness 𝑓𝑠←EvaluateSolution (𝑠)8. while 𝑇 > 𝛼9. 𝑠1←RouletteSelectionMethod (𝑁,𝑓𝑁)10. 𝑠2←Detroy (𝑠1)11. 𝑠2←Repair (𝑠2) 12. 𝑓𝑠2

←EvaluateSolution (𝑠2)

13. 𝑧←RandomNumber (0,1)

14. if or (𝑓𝑠2< 𝑓𝑠1

) (𝑧 < 𝑒

|𝑓𝑠1‒ 𝑓𝑠2

|𝑇 )

15. 𝑁𝑠1←𝑠2

16. 𝑇 = 𝑐 ⋅ 𝑇17. for each solution in 𝑠 𝑁18. integer flows 𝑠𝑥←IntegerFlowSolution (𝑠)19. routes 𝑠𝑟←RouteAssignmentAlgorithm (𝑠) 20. fitness 𝑠𝑓←EvaluateSolution (𝑠)

21. 𝑠 ∗ ←FindMinimum(𝑠𝑓)

22. return 𝑠 ∗

Pseudocode 6. Large Neighbourhood Search.

The behaviour of the LNS is controlled by two parameters. In the first instance, a “cooling rate” 𝑐

defines the rate of temperature change at each step, with values ranging between 0 and 1. The “absolute 𝑇

temperature” defines the minimum allowed temperature in the process and signals the termination of the 𝑎

search when it is reached. In every iteration, a parent solution is evaluated against a newly generated candidate 𝑠1

solution using a scalar fitness value, referred to as and respectively. If is higher, it replaces in 𝑠2 𝑓𝑠1𝑓𝑠2

𝑓𝑠2𝑠1

the neighbourhood . Otherwise, there are two possible outcomes: a) is replaced with , or b) is discarded, 𝑁 𝑠1 𝑠2 𝑠2

and a new solution is generated. Replacement with an inferior solution may appear counter-intuitive but ensures

that the algorithm does not limit itself to a local optimum. The probability of a worse solution replacing an 𝑃 𝑠2

original solution is defined as: 𝑠1

𝑃 = exp ( ‒𝛿𝐹𝑇 ) (12)

where is the difference in fitness of both solutions, and is the temperature.𝛿𝐹 𝑇

The outputs of the LNS algorithm used in this study is a set of optimal hub locations, evaluated using

combination of the objective functions from the trajectory optimisation and routing assignment models

described in Sections 3.2 and 3.3 respectively. The “Destroy” implementation used in the algorithm, described

in Pseudocode 7, eliminates hub assignments using one of three processes: random elimination (hub deletion

using random distributions), cost-based elimination (eliminates hubs that contain higher setup costs) and

utilisation-based elimination (removes hubs that deliver the least cargo). Similar aspects are also used in the

“Repair” (in Pseudocode 8) step albeit to the inverse effect: random addition, cost-based addition and utilisation-

based addition. The number of hubs added and removed at every iteration is uniformly randomised.

1. Algorithm Destroy has2. Input: solution 𝑠13. Output: new solution 𝑠24. 5. 𝑧←RandomNumber (0,1)6. if 0.4 > 𝑧7. 𝑠2←CostElimination(𝑠1)8. else if and 0.8 > 𝑧 0.4 ≤ 𝑧9. 𝑠2←UtilisationElimination(𝑠1)10. else11. 𝑠2←RandomElimination(𝑠1)12.return 𝑠2

Pseudocode 7. Destroy Algorithm.

1. Algorithm Repair has2. Input: solution 𝑠13. Output: new solution 𝑠24. 5. 𝑧←RandomNumber (0,1)6. if 0.4 > 𝑧7. 𝑠2←CostAddition(𝑠1)8. else if and 0.8 > 𝑧 0.4 ≤ 𝑧9. 𝑠2←UtilisationAddition(𝑠1)10. else11. 𝑠2←RandomAddition(𝑠1)12. return 𝑠2

Pseudocode 8. Repair Algorithm.

The problem instances considered in Section 3.1 are re-evaluated using the heuristic to determine

optimality gaps where solutions exist. The results are summarised in Table 4. Using this heuristic, all instances

become solvable in short runtime. As expected, the heuristic provides a slower response than the exact solution.

The exact solution can maximise relief distribution for small mission times, where all demand is not delivered.

Conversely, the static formulation can only optimise full missions, where all the demand is assigned. Since it is

the basis of the heuristic approach, the percentage difference between both methodologies decreases as mission

time increases.

Table 5Heuristic runtime in sample problems

Instance Demand

(kg)

Payload

(kg)

Mission time

(mins)

Cargo delivered

(kg)

Difference

(%) (-)

Runtime

(s)

50 272 9.3 2

100 568 5.3 22

150 844 3.7 2

50 404 10.2 2

100 842 4.2 2

4 nodes 1000

3

119 1000 0 2

50 50 N/A 5

100 286 N/A 52

150 546 N/A 5

50 145 N/A 5

100 536 N/A 5

7 nodes 800

3

126 800 N/A 5

50 350 30 7

100 600 N/A 72

150 749 N/A 7

50 349 N/A 7

100 747 N/A 7

10 nodes 1450

3

150 1122 N/A 7

4. Case study

The 1999 Chi-Chi earthquake in Taiwan had a magnitude of 7.3 and was one of the largest catastrophes to occur

in East Asia over the past 20 years. It caused over 2,000 deaths, 10,000 injuries, and damaged or led to the

collapse of 14,000 (Weinni 2000). The effects of the earthquake were more severe in the central county of

Nanto, which was used as a case study by Sheu (2010) of logistics of disaster response. A detailed breakdown

of the effects of the earthquake across the county’s provinces was introduced and used to develop a model for

relief demand forecasting. The model was used to create the demand dataset that has since then been used as a

benchmark case for studies on the optimisation disaster response. Given the presence of a variable terrain that

would impose long travel times on ground-based distribution, it was identified as an interesting test case for the

UAV-based developed by this paper.

For the purposes of this study, we focused on the distribution of relief resources to a frail population

composed of elders and children, that is characterised by a higher mortality rate, especially in the case of those

aged 50 and over (Liang et al. 2001). We assumed that a care package of 1 kg must be delivered to each person,

carrying essential materials such as medicines and basic sustenance.

The physical and technical specifications of the UAVs used in the case study (Table 7) were informed by

material QuestUAV Q-Pod and the X8 Cargo Drones that are publicly available online (QuestUAV 2017; UAV

Systems International 2017). Although these specific models are incapable of VTOL operation, they are similar

in size, range and payload capacity to hybrid models that are currently planned, for which the exact technical

specifications remain unavailable. A single warehouse/hub is established in each township, which is assumed

to satisfy the entirety or relief cargo demands. As such, the cost of establishing each hub was considered to be

proportional to the size of the frail population in that township.

The proposed facility selection-routing algorithm and trajectory optimisation was tested on an Intel Xeon

CPU E5-1650 processor and 32GB RAM. The trajectory model was implemented using the Pyomo and Pyomo

DAE modelling languages (Hart et al. 2011; Hart et al. 2012) was used to determine optimal trajectories for

both loaded and unloaded UAVs. All trajectories take the shortest and most direct route to their destination,

using a path profile that follows the terrain topology and overcomes geographical obstacles. The battery

consumption rate is mostly constant as the pitch is small throughout the flight. Note that the flight path angles

are very small, which yield a close to linear battery consumption rate.

An indicative trajectory for the journey between the townships of Chong Liao and Ren Ai are presented

in Fig. 7. Any increase in the payloads assigned to UAV led to a dramatic reduction in range, as indicated by

the battery consumption matrix provided in the Appendix. In this particular trajectory, the battery charge can be

reduced to 53% of its original capacity without payload, to 33% when transporting 3 kg. Further analysis of the

results, presented in the Appendix, indicates that a payload increase of 1 kg for this specific journey would

require an additional 15% energy.

Fig. 6. Nantou County map and warehouse location (indicated by a black circle).

Table 6 Nanto County post-earthquake data (Bagshaw 2012) including coordinates of each warehouse location.

Warehouse LocationNode NameLatitude Longitude

Total population

Frail population

Fatalities

1 Nanto City 23.92176 120.6808 104,777 215 1192 Tsao Tun 23.99312 120.7258 96,833 2,373 453 Ji Jii 23.82843 120.7865 12,250 8,656 1514 Pu Li 23.99367 120.9647 88,271 161 1145 Chu Shan 23.70884 120.6932 62,269 342 2066 Guo Hsin 24.01024 120.8684 24,643 567 62

Table 7 UAV parameters used

Parameter Symbol Value

Parasite Drag Coefficient [-]𝐶𝐷0 0.015

Drag Polar [-]𝐾 0.13

Payload Capacity [kg]𝑄 3

Empty Vehicular Mass [kg]𝑚 5

Wing Area [m2]𝑆 0.75

Battery Voltage [V]𝑉 11.1

Battery Capacity [Ah]𝐵 20

UAV cruise speed [m/s]𝑈𝑐𝑟𝑢𝑖𝑠𝑒 22

40-5000

0

Y (km)

20

500

10

Z (m

)

X (km)

1000

20

1500

30 40 0

UAV fligth trajectory

0 5 10 15 20 25 30time (minutes)

0.5

0.6

0.7

0.8

0.9

1

SOC

(-)

% Battery Remaining

0 5 10 15 20 25 30time (minutes)

-0.15

-0.1

-0.05

0

0.05

Path

Ang

le (r

ads)

UAV Pitch Angle

0 5 10 15 20 25 30time (minutes)

300

400

500

600

700

800

900

Z (m

)

UAV Altitude

Fig. 7. Trajectory connecting Chong Liao and Ren Ai with an empty UAV.

Using these trajectories, we applied the hub selection-routing algorithm to determine the optimal

itineraries given a location. Given an itinerary of elements, a cargo itinerary is composed of elements, 𝑛 𝑛 ‒ 1

7 Lu Gu 23.73367 120.7771 21,279 925 588 Ming Jian 23.8458 120.6764 42,754 1,126 1239 Sui Li 23.8092 120.8736 23,425 287 14

10 Yu Chi 23.87155 120.9303 17,894 327 26711 Chong Liao 23.90838 120.781 18,252 3,892 2412 Ren Ai 24.02771 121.1435 15,358 298 413 Hsin Yi 23.65008 121.0226 17,869 314 3

where each element in the cargo itinerary refers to the transported cargo between the and OD pair in the 𝑖 𝑖 ‒ 1 𝑖

destination itinerary. An indicative example is provided in Table 8.Table 8 UAV example itinerary.

UAV Type Itinerary – using node indices from Table 6

Destination Itinerary 1, 3, 11, 1, 3, 11, 1, 3, 11, 1…1

Cargo Itinerary 3, 1, 0, 3, 2, 0, 3, 1, 0…

Destination Itinerary 1, 12, 9, 12, 1, 8, 1, 8, 1…2

Cargo Itinerary 3, 1, 0, 0, 3, 0, 3, 0…

Table 9 presents the results. Nanto City was consistently selected by the models as the preferred location

of the hub. Despite Nanto’s outer position in the county, it is located at the borders of Ji Jii, Tsao Tun, Ming

Jian and Chong Liao, which are the townships of highest demand, accounting for over 80% of the total demand

within the region.

An alternative objective that could be used in the model would be one that ensures a fair and proportional

distribution of relief, such that all townships are fairly served and in accordance to demand. Such an approach

might appear incompatible with considerations of economies of scale that are used in commercial goods

distribution. However, humanitarian operations must adopt the principles of impartiality, neutrality,

independence, and humanity (Bagshaw 2012). Table 11 provides a comprehensive breakdown of distributed

resources at each individual township.

Table 9 Results with varying UAVs.

Relief Delivered in

Critical Period (kg)UAVsNumber of Hubs

(Location)

Completion Time for

persistent deliveries (days)Day 1 Day 2

35 1 (Nanto City) 3 (72 hours) 6,527 13,196

40 1 (Nanto City) 2.6 (63 hours) 7,506 15,099

45 1 (Nanto City) 2.3 (56 hours) 8,397 16,865

50 1 (Nanto City) 2.1 (50 hours) 9,377 18,701

60 1 (Nanto City) 1.7 (42 hours) 11,211 19,268

70 1 (Nanto City) 1.5 (36 hours) 12,998 19,268

80 1 (Nanto City) 1.3 (31 hours) 14,828 19,268

90 1 (Nanto City) 1.1 (28 hours) 16,700 19,268

Fig. 8. Percentage relief distributed per township with 40 drones on day 1 (left) and day 2 (right).

Table 10 Summary of earlier work.

Author Objective Methodology Indicators Scenario Results

Sheu

(2010)

Demand fore-

casting and

prioritisation of

affected groups.

Data fusion, fuzzy

clustering and

multi-criteria

group

prioritisation.

Accumulated

fatalities, relief

demand and

urgency.

Chi-Chi

earthquake -

Nanto

County.

Chong Liao, Ji

Jii and Yu Chi -

highest priority.

Sui Li, Ren Ai

and Hsin Yi -

lowest urgency.

Chang et

al. (2014)

Optimisation of

truck-based

relief

distribution.

Greedy-search

based multi-

objective genetic

algorithm.

Unsatisfied

demand, travel

time, and

transportation

cost.

Chi-Chi

earthquake -

Nanto

County

Mission

completed in 32

and 20 hours to

service 159,373

people.

Sheu

(2007)

Demand

grouping and

relief

distribution.

Fuzzy clustering

and dynamic

programming.

Time between

relief arrivals,

demand

satisfaction, and

total costs.

Chi-Chi

earthquake –

Tai Chung

County.

Total cost of

operation

US$15.8 million.

The

proposed

algorithm

Optimisation of

relief

distribution

using UAVs.

Preliminary

Trajectory

Optimisation and

Bi-level LRP.

Minimise mission

time, undelivered

demand and fixed

cost.

Chi-Chi

earthquake -

Nanto

County.

Mission

completed in 1-3

days to service

19,268 people.

We analyse our results relative to Sheu (2010), Chang et al. (2014), and Sheu (2007), a summary of each

study is provided in Table 10. Firstly, let us define the differences between our case study and Sheu (2010).

Sheu (2010) estimates fatalities over time and clusters the regions in terms of urgency for relief considering

total damages and total population. Similar to our study, the scenario considered is the Chi-Chi earthquake

response in the Nanto County, only the response time is 5 days instead of 3.

Despite the objective and methodology being different, Sheu (2010) concludes that Chong Liao, Ji Jii and

Yu Chi are of the highest priority, while Sui Li, Ren Ai, and Hsin Yi are lower. Both results mirror the scenario

considered in this study, as Chong Liao and Ji Jii contain among the highest frail populations, and Sui Li, Ren

Ai and Hsin Yi are among the smallest. The only difference resides in Yu Chi, which is of low priority in our

study due to the low percentage of frail individuals.

We compare our results with Chang et al. (2014) to assess the payload capacity limitation of UAVs with

respect to conventional transport modes. Chang et al. (2014) minimised the total time of humanitarian response

to the Chi-Chi earthquake with a Genetic Algorithm using a homogeneous fleet of 4000 kg payload trucks. They

provide a case study based on the County of Nanto, with the difference of using a total of 30 demand points

instead of 13 and 4 distribution points instead of being part of the optimisation.

According to their results, they delivered cargo for 159,373 people, each person being provided with a

care-package of 3 kg. A fleet of 97 trucks completed the mission in under 24 hours, while an unconstrained

number of trucks finished the mission in 32 hours. They propose a cost of US$ 1000 per vehicle, but research

in the market suggest that the approximate price of such units oscillates £45,000 (Trans Automobile 2017). Note

that the study does not account for possible infrastructure damage, debris, and speed limitations. Instead, a

uniform speed of 40 km/h is assumed.

In comparison with the scenario developed by Chang et al. (2014), a fleet of 90 drones provided 19,268

kg in a similar amount of time based on the framework proposed in this paper. Assuming each drone is valued

between £2,000-5,000, the resulting initial cost can vary between £180,000-450,000. A fleet of 4-10 trucks can

transport sufficient cargo for 19,716-49,290 kg in that 24-hour period. Note that the estimation provided ignores

additional labour costs from drivers and loading/unloading personnel.

Table 11 Relief distributed in kg and relative to total demand with varying UAVs.

40 UAVsDay 1 Day 2Township Cargo delivered

(kg)Proportion of total

demand (%)Cargo delivered

(kg)Proportion of total

demand (%)Nanto City - - - -

Tsao Tun 927 39 1,863 79Ji Jii 3,403 39 6,808 79

Pu Li 69 43 104 65Chu Shan 150 44 273 80Guo Hsin 150 26 414 73

Lu Gu 345 37 714 77Ming Jian 592 53 928 82

Sui Li 92 32 215 75

Finally, Sheu (2007) develops a clustering mechanism and routing approach to optimise the relief

distribution in the aftermath of the Chi-Chi earthquake using a dynamic linear programming model. Sheu (2007)

tests the model in the County of Taichung: Dongshi, Shingang and Wufeng. Instead of using a single stage

distribution as our proposed model or Chang et al. (2014), Sheu (2007) considered a bi-stage distribution: 6

Yu Chi 153 47 261 80Chong Liao 1,391 36 3,047 78

Ren Ai 114 38 226 76Hsin Yi 120 38 246 78

50 UAVsTownship Day 1 Day 1Nanto City - - - -

Tsao Tun 1,236 52 2,319 98Ji Jii 4,158 48 8,392 97

Pu Li 62 39 158 98Chu Shan 213 62 333 97Guo Hsin 282 50 549 97

Lu Gu 397 43 889 96Ming Jian 588 52 1,105 98

Sui Li 120 42 281 98Yu Chi 186 57 312 95

Chong Liao 1,850 48 3,784 97Ren Ai 133 45 280 94

Hsin Yi 152 48 299 9560 UAVs 70 UAVsTownship Day 1 Day 2

Nanto City - - - -Tsao Tun 1,485 63 1,638 69

Ji Jii 4,951 57 5,866 68Pu Li 93 58 95 59

Chu Shan 228 67 240 70Guo Hsin 279 49 387 68

Lu Gu 564 61 607 66Ming Jian 645 57 694 62

Sui Li 161 56 227 79Yu Chi 198 61 219 67

Chong Liao 2,256 58 2,602 67Ren Ai 171 57 208 70

Hsin Yi 180 57 215 6880 UAVs 90 UAVsTownship Day 1 Day 2

Nanto City - - - -Tsao Tun 1,854 78 2,103 89

Ji Jii 6,646 77 7,456 86Pu Li 129 80 128 80

Chu Shan 267 78 303 89Guo Hsin 444 78 483 85

Lu Gu 700 76 799 86Ming Jian 838 74 982 87

Sui Li 212 74 248 86Yu Chi 261 80 273 83

Chong Liao 3,031 78 3,424 88Ren Ai 219 73 250 84

Hsin Yi 227 72 251 80

supply nodes located in different counties of the country, a distribution node at each township, and a total of 24

demand points. Overall, the methodology aimed to provide 180,000 gallons of water, 25,000 meal boxes, 13,000

sleeping bags and 2,200 camps over 3 days and achieved 74% of the total. However, the time between each

arrival of relief averaged to 4.6 hours. Finally, the full 3-day response costs US$15.8 million, accounting for

the inbound and outbound cargo transportation and inventory costs. The differences between Sheu (2007) and

our proposed model is substantial as highlighted in Table 10, and comparability is limited. Such a response

would be impossible to carry out using small UAVs, which are ideal for standardised small high value

transportation and cannot accommodate the required volume for all the goods delivered in Sheu (2007).

However, as performance improves and costs reduces, the viability of using UAVs increases, and the

development of heterogeneous UAV fleets would provide payload capacity capable of high volume relief

distribution.

5. Conclusion

In this paper, we address the problems of quantifying the impacts of UAVs in humanitarian logistics considering

the range variability due to payload. We propose a new facility selection-routing algorithm combined with a

trajectory optimisation method applied to UAV-based last-mile humanitarian logistics operation after the Chi-

Chi Earthquake in Taiwan 1999. A Large Neighbourhood Search algorithm envelops an MILP model that,

combined with a route assignment post-process, minimises total mission cost. Compared to earlier work by

Nedjati et al. (2015) and Murray & Chu (2015), we have considered the variability in range with respect to

payloads. This relationship is not negligible, considering range as a fixed parameter results is an unreasonable

assumption. We also consider optimal allocation of warehouse and charging points to extend the range of the

vehicles if necessary, which remains outside the scope of Dorling et al. (2017). Finally, we combine the facility-

selection algorithm with a non-linear trajectory optimisation process, which provides a first step towards the

integration of both methods.

Despite the payload and endurance limitations of UAVs, major logistical companies have announced the

development of UAV-based delivery systems. SESAR predicts a widespread adoption of UAVs in logistics

systems in the next 10 years, with operations starting by 2020 (SESAR 2016). In the humanitarian field,

organisations are increasingly depending on UAVs for a variety of tasks given their accessibility, low risk and

decreasing cost. As such, UAVs have the potential to become a key asset in the last-mile domain, especially for

low volume high-value cargo. To this effect, WeRobotics have carried field tests in Peru to determine their

suitability to delivering medical cargo (Meier et al. 2017; Meier & Bergelund 2017). Furthermore, WFP is

developing coordination mechanisms to maximise their added value and reduce safety risks emerging from their

independent use. Their integration in humanitarian logistics is imminent, particularly for last-mile operations.

The advantages of UAVs discussed above can only be realised if the barriers associated regulation and

aspects of technology are addressed. In the case of the former, efforts are currently underway by the International

Civil Aviation Organisation (ICAO), the European Aviation Safety Agency (EASA) and the Federal Aviation

Agency (FAA) to create the necessary regulation framework relating to airspace management and privacy, that

hinder widespread adoption and to integrate UAV technology in humanitarian operations (Emergency

Telecommunications Cluster 2017). The solution provided in this paper may be used to quantify the impact on

relief distribution given more lenient regulations or exemptions in the event of a natural disaster.

On the latter, current restrictions regarding payload capacities and flight endurance limit the applicability

of UAVs. Given the increasing investment on drone technology and exploration of new fuel options, we expect

improvement in the performance of UAVs.

Regarding traffic management, the trajectories developed in this paper were set at a low altitude relative

to minimise interaction with the other. However, it must be considered in further studies due to the increase in

UAV use and possible interaction with low-altitude aircraft in search and rescue duties. With the exception of

Furini et al. (2016), airspace management and collision avoidance have not been considered in any of the

reviewed literature. Such problem requires the integration of trajectory optimisation and routing algorithms,

which will be the focus of the author’s future work.

Finally, models must consider that responders adhere to the humanitarian principles of impartiality,

neutrality, independence, and humanity. Given the military origins of UAVs, a common concern in

humanitarian circles is that their use may jeopardise the perception of the organisation’s neutrality.

6. Acknowledgments

The research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC).

7. References

Afrotech EPFL, 2016. Red Line. Ecole Polytechnique Federale de Lausanne, pp.1–6. Available at: http://afrotech.epfl.ch/page-115280-en.html [Accessed July 13, 2017].

Albareda-Sambola, M., Díaz, J.A. & Fernández, E., 2005. A compact model and tight bounds for a combined location-routing problem. Computers and Operations Research, 32(3), pp.407–428.

De Angelis, V. et al., 2007. Multiperiod integrated routing and scheduling of World Food Programme cargo planes in Angola. Computers and Operations Research, 34(6 SPEC. ISS.), pp.1601–1615.

Arrieta-Camacho, J.J., Biegler, L.T. & Subramanian, D., 2007. Trajectory control of multiple aircraft: An NMPC approach. Lecture Notes in Control and Information Sciences, 358(1), pp.629–639.

Barbarosoğlu, G., Özdamar, L. & Çevik, A., 2002. An Interactive Approach for Hierarchical Analysis of Helicopter Logistics in Disaster Relief Operations An interactive approach for hierarchical analysis of helicopter logistics in disaster relief operations. European Journal of Operational Research, 140(1), pp.118–133.

BBC News, 2015. Google plans drone delivery service for 2017. BBC News - Technology, pp.1–6. Available at: http://www.bbc.co.uk/news/technology-34704868 [Accessed June 18, 2017].

Betts, J.T., 1998. Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control, and Dynamics, 21(2), pp.193–207.

Brandt, J.B. & Selig, M.S., 2011. Propeller Performance Data at Low Reynolds Numbers. 49th AIAA Aerospace Sciences Meeting, (January), pp.1–18.

CBS News, 2013. Amazon Unveils Futuristic Plan: Delivery by Drone. CBS News, pp.1–4. Available at: http://www.cbsnews.com/news/amazon-unveils-futuristic-plan-delivery-by-drone/ [Accessed July 10, 2017].

Chakrabarty, A. & Langelaan, J.W., 2010. Flight Path Planning for UAV Atmospheric Energy Harvesting Using Heuristic Search. AIAA Guidance, Navigation and Control Conference, (August), pp.1–18.

Chang, F.S. et al., 2014. Greedy-search-based multi-objective genetic algorithm for emergency logistics scheduling. Expert Systems with Applications, 41(6), pp.2947–2956.

Chen, G. & Cruz, J.B., 2003. Genetic Algorithm for Task Allocation in UAV Cooperative Control. Electrical Engineering, (August), p.13. Available at: http://www2.ece.ohio-state.edu/~cruz/Papers/C110-AIAAConf GNC 2003.pdf.

Coelho, B.N. et al., 2017. A multi-objective green UAV routing problem. Computers & Operations Research, 0, pp.1–10.

D’Andrea, R., 2014. Guest Editorial. Can Drones Deliver? IEEE Transactions on Automation Science and Engineering, 11(3), pp.647–648.

Dorling, K. et al., 2017. Vehicle Routing Problems for Drone Delivery. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(1), pp.70–85.

EASA, 2015. A proposal to create common rules for operating drones in Europe, Cologne, Germany. Available at: https://www.easa.europa.eu/system/files/dfu/205933-01-EASA_Summary of the ANPA.pdf.

Emergency Telecommunications Cluster, 2017. Humanitarian Unmanned Aerial Vehicles (UAV) Coordination Model. Emergency Telecommunications Cluster. Available at: https://www.etcluster.org/discussions/humanitarian-unmanned-aerial-vehicles-uav-coordination-model [Accessed June 1, 2017].

Forsmo, E.J., 2012. Optimal Path Planning for Unmanned Aerial Systems. Norwegian University of Science and Technology Norway.

Furini, F., Persiani, C.A. & Toth, P., 2016. The Time Dependent Traveling Salesman Planning Problem in Controlled Airspace. Transportation Research Part B: Methodological, 90, pp.38–55.

Geiger, B., 2009. Unmanned aerial vehicle trajectory planning with direct methods. Pennsylvania State University. Available at: https://etda.libraries.psu.edu/files/final_submissions/767.

Gottlieb, I., 1997. Practical Electric Motor Handbook 1st ed., Oxford, United Kingdom: Reed Educational and Professional Publishing Ltd.

Gracey, W., 1980. Measurement of Aircraft Speed and Altitude, Hampton, VA. Available at: http://www.dtic.mil/dtic/tr/fulltext/u2/a280006.pdf.

Griffin, A., 2016. Amazon Prime Air will now deliver things to people on a drone, almost straight away. Independent, pp.1–10. Available at: http://www.independent.co.uk/life-style/gadgets-and-tech/news/amazon-prime-air-drone-30-minute-deliveries-delivery-now-flying-a7474541.html [Accessed July 21, 2017].

Griva, I., Nash, S.G. & Sofer, A., 2009. Linear and Nonlinear Optimization 2nd ed., Society for Industrial and Applied Mathematics.

Grossmann, C. & Roos, H.-G., 2007. Numerical Treatment of Partial Differential Equations 3rd ed., Berlin: Springer.

Gulczynski, D., Golden, B. & Wasil, E., 2011. The multi-depot split delivery vehicle routing problem: An integer programming-based heuristic, new test problems, and computational results. Computers and Industrial Engineering, 61(3), pp.794–804.

Haidari, L.A. et al., 2016. The economic and operational value of using drones to transport vaccines. Vaccine, 34(34), pp.4062–4067.

Hall, N., 2016. The Drag Coefficient. NASA. Available at: https://www.grc.nasa.gov/www/k-12/airplane/dragco.html [Accessed May 20, 2017].

Hart, W.E. et al., 2012. Pyomo – Optimization Modeling in Python, Springer Science & Business Media.

Hart, W.E., Watson, J.P. & Woodruff, D.L., 2011. Pyomo: Modeling and solving mathematical programs in Python. Mathematical Programming Computation, 3(3), pp.219–260.

Heutger, M. et al., 2014. Unmanned Aerial Vehicle in Logistics - A DHL perspective on implication and use cases for the logistics industry, Troisdorf, Germany. Available at: http://www.dhl.com/content/dam/downloads/g0/about_us/logistics_insights/DHL_TrendReport_UAV.pdf.

Ho, W. et al., 2008. A hybrid genetic algorithm for the multi-depot vehicle routing problem. Engineering Applications of Artificial Intelligence, 21(4), pp.548–557.

Hosseini, S. & Mesbahi, M., 2013. Optimal path planning and power allocation for a long endurance solar-powered UAV. 2013 American Control Conference, pp.2588–2593.

Keane, J.F. & Carr, S.S., 2013. A Brief History of Early Unmanned Aircraft. John Hopkins APL Technical Digest, 32(3), pp.558–571. Available at: http://www.jhuapl.edu/techdigest/td/td3203/32_03-keane.pdf.

Laporte, G., Nobert, Y. & Arpin, D., 1986. An exact algorithm for solving a capacitated location-routing problem. Annals of Operations Research, 6(9), pp.293–310.

Liang, N.J. et al., 2001. Disaster epidemiology and medical response in the Chi-Chi earthquake in Taiwan. Annals of Emergency Medicine, 38(5), pp.549–555.

Linden, D., 1984. Handbook of Batteries and Fuel Cells, New York, NY, United States: McGraw-Hill Book Co.

Meier, P. et al., 2017. Cargo Drone Field Tests in the Amazon, Lima, Peru. Available at: https://werobotics.org/wp-content/uploads/2017/10/WeRobotics-Report-on-Drone-Cargo-Field-Tests-

Peru-2017.pdf?lipi=urn%3Ali%3Apage%3Ad_flagship3_pulse_read%3BEV9EAy9mThCOsgIp7NaQhA%3D%3D.

Meier, P. & Bergelund, J., 2017. Field-Testing the First Cargo Drone Deliveries in the Amazon Rainforest, Lima, Peru. Available at: http://werobotics.org/wp-content/uploads/2017/02/WeRobotics-Amazon-Rainforest-Cargo-Drones-Report.pdf.

Meier, P. & Soesilo, D., 2014. Using Drones for Medical Payload Delivery in Papua New Guinea, Geneva, Switzerland. Available at: http://drones.fsd.ch/wp-content/uploads/2016/04/Case-Study-No2-PapuaNewGuinea.pdf.

Meier, P., Soesilo, D. & Guerin, D., 2016. Cargo Drones in Humanitarian Contexts Meeting Summary, Sheffield, UK. Available at: http://drones.fsd.ch/wp-content/uploads/2016/08/CargoDrones-MeetingSummaryfinal.pdf.

Murray, C.C. & Chu, A.G., 2015. The flying sidekick traveling salesman problem: Optimization of drone-assisted parcel delivery. Transportation Research Part C: Emerging Technologies, 54, pp.86–109.

Nedjati, A., Vizvari, B. & Izbirak, G., 2015. Post-earthquake response by small UAV helicopters. Natural Hazards, 80, pp.1669–1688.

Ostler, J.N. et al., 2009. Performance Flight Testing of Small, Electric Powered Unmanned Aerial Vehicles. International Journal of Micro Air Vehicles, 1(3), pp.155–171.

Özdamar, L., 2011. Planning helicopter logistics in disaster relief. OR Spectrum, 33(3), pp.655–672.

Pannequin, J. et al., 2007. Multiple Aircraft Deconflicted Path Planning with Weather Avoidance Constraints. AIAA Guidance, Navigation and Control Conference and Exhibit, (August), pp.1–16.

Perl, J. & Daskin, M.S., 1985. A warehouse location-routing problem. Transportation Research Part B, 19(5), pp.381–396.

Pisinger, D. & Ropke, S., 2010. Large neighborhood search. In M. Gendreau & J.-Y. Potvin, eds. Handbook of Metaheuristics. Boston, MA: Springer, pp. 399–420. Available at: https://link.springer.com/chapter/10.1007/978-1-4419-1665-5_13.

QuestUAV, 2017. QuestUAV Q‐Pod Technology. , pp.1–25. Available at: https://www.geosystems.fr/images/PDF/20140613_Brochure_QPod_complete.pdf.

Raghunathan, A.U. et al., 2004. Dynamic optimization strategies for three-dimensional conflict resolution of multiple aircraft. Journal of Guidance, Control, and Dynamics, 27(4), pp.586–594.

Ruiz, S. & Soler, M., 2015. Conflict pattern analysis under the consideration of optimal trajectories in the European ATM - seminar. ATM R&D Seminar (ATM 2015), 15, pp.1–10.

Sarıçiçek, I. & Akkus, Y., 2013. Unmanned Aerial Vehicle hub-location and routing for monitoring geographic borders. Applied Mathematical Modelling, 39, pp.3939–3953.

Schretlen, J.-H., 2015. Wings for Aid: Unmanned cargo aircraft take relief supplies to disaster areas. PWC Magazine. Available at: http://www.magazine.about.pwc.nl/october2015#!/wings-for-aid-unmanned-cargo-drones [Accessed May 4, 2017].

SESAR, 2016. European Drones Outlook Study, Brussels, Belgium. Available at: http://www.sesarju.eu/sites/default/files/documents/reports/European_Drones_Outlook_Study_2016.pdf.

Sheu, J.B., 2007. An emergency logistics distribution approach for quick response to urgent relief demand in disasters. Transportation Research Part E: Logistics and Transportation Review, 43(6), pp.687–709.

Sheu, J.B., 2010. Dynamic relief-demand management for emergency logistics operations under large-scale disasters. Transportation Research Part E: Logistics and Transportation Review, 46(1), pp.1–17.

Simmons, D., 2016. Rwanda begins Zipline commercial drone deliveries. BBC News, pp.1–8. Available at:

http://www.bbc.co.uk/news/technology-37646474 [Accessed November 16, 2016].

Soesilo, D. et al., 2016. Drones in Humanitarian Action. A guide to the use of airbirne systems in humanitarian crises, Geneva, Switzerland. Available at: http://drones.fsd.ch/wp-content/uploads/2016/11/Drones-in-Humanitarian-Action.pdf.

Soesilo, D. & Sandvik, K.B., 2016. Drones in Humanitarian Action – A survey on perceptions and applications, Geneva, Switzerland. Available at: https://reliefweb.int/sites/reliefweb.int/files/resources/Drones in Humanitarian Acion Survey-Analysis-FINAL2.pdf.

Stöcker, C. et al., 2017. Review of the current state of UAV regulations. Remote Sensing, 9(5), pp.33–35.

Stoney, R.B., 1993. Design, fabrication and test of a vertical attitude takeoff and landing unmanned air vehicle. Naval Postgraduate School, Monterrey, CA, United States.

UAV Systems International, 2017. X8 Long Range Cargo Drone. Available at: http://www.uavsystemsinternational.com/product/x8-long-range-cargo-drone/ [Accessed April 4, 2017].

UNICEF, 2016. Malawi tests first unmanned aerial vehicle flights for HIV early infant diagnosis. UNICEF Press Centre. Available at: https://www.unicef.org/media/media_90462.html?p=printme [Accessed April 14, 2017].

Wasner, M. & Zäpfel, G., 2004. An integrated multi-depot hub-location vehicle routing model for network planning of parcel service. International Journal of Production Economics, 90(3), pp.403–419.

Weinni, D., 2000. Chi-Chi, Taiwan Earthquake Event Report, Menlo Park, CA. Available at: http://forms2.rms.com/rs/729-DJX-565/images/eq_chi_chi_taiwan_eq.pdf.

Weinstein, A.L. & Schumacher, C., 2007. UAV scheduling via the vehicle routing problem with time windows. In Proc. AIAA Infotech Aerospace 2007 Conference and Exhibit. Rohnert Park, CA, United States, pp. 1–14.

WFP, 2017a. Audited Annual Accounts, 2016, Rome, Italy. Available at: http://documents.wfp.org/stellent/groups/public/documents/eb/wfp291551.pdf.

WFP, 2017b. The WFP food basket. World Food Programme. Available at: https://www.wfp.org/food-assistance/kind-food-assistance/wfp-food-basket [Accessed June 14, 2017].

Wu, T.-H., Low, C. & Bai, J.-W., 2002. Heuristic solutions to multi-depot location-routing problems. Computers & Operations Research, 29(10), pp.1393–1415.

Yi, W. & Özdamar, L., 2007. A dynamic logistics coordination model for evacuation and support in disaster response activities. European Journal of Operational Research, 179(3), pp.1177–1193.

APPENDIXTable 1 Percentage battery depletion with no payload.

Nodes 1 2 3 4 5 6 7 8 9 10 11 12 131 - 0.11 0.19 0.35 0.29 0.26 0.27 0.10 0.28 0.31 0.13 0.58 0.552 0.11 - 0.23 0.29 0.39 0.17 0.36 0.21 0.30 0.29 0.13 0.50 0.583 0.18 0.23 - 0.31 0.20 0.28 0.13 0.14 0.11 0.18 0.11 0.56 0.374 0.35 0.29 0.31 - 0.51 0.14 0.42 0.40 0.27 0.17 0.25 0.23 0.465 0.28 0.39 0.19 0.50 - 0.45 0.11 0.18 0.25 0.36 0.28 0.69 0.406 0.25 0.18 0.25 0.12 0.50 - 0.39 0.32 0.28 0.20 0.17 0.33 0.517 0.27 0.35 0.13 0.41 0.11 0.38 - 0.20 0.16 0.26 0.24 0.60 0.328 0.12 0.20 0.14 0.42 0.19 0.32 0.22 - 0.24 0.31 0.15 0.61 0.499 0.28 0.31 0.11 0.28 0.25 0.27 0.16 0.24 - 0.11 0.17 0.49 0.2810 0.31 0.30 0.18 0.17 0.36 0.20 0.26 0.31 0.11 - 0.19 0.36 0.3111 0.12 0.15 0.11 0.25 0.28 0.17 0.23 0.15 0.17 0.19 - 0.49 0.4512 0.57 0.50 0.51 0.25 0.80 0.33 0.59 0.61 0.48 0.33 0.47 - 0.5213 0.55 0.59 0.42 0.47 0.41 0.58 0.32 0.49 0.28 0.31 0.45 0.52 -

Table 2 Percentage battery depletion with 1 kg payload.

Nodes 1 2 3 4 5 6 7 8 9 10 11 12 131 - 0.12 0.21 0.40 0.32 0.29 0.31 0.12 0.32 0.35 0.14 0.65 0.622 0.12 - 0.26 0.32 0.43 0.20 0.39 0.23 0.36 0.33 0.17 0.57 0.683 0.20 0.26 - 0.35 0.22 0.30 0.14 0.15 0.13 0.21 0.12 0.57 0.424 0.40 0.33 0.35 - 0.57 0.15 0.46 0.47 0.30 0.19 0.28 0.25 0.525 0.32 0.46 0.23 0.56 - 0.51 0.12 0.21 0.29 0.41 0.32 0.78 0.466 0.29 0.19 0.29 0.13 0.52 - 0.44 0.36 0.33 0.22 0.20 0.39 0.587 0.31 0.39 0.14 0.46 0.12 0.43 - 0.22 0.19 0.29 0.26 0.67 0.368 0.12 0.23 0.15 0.51 0.24 0.38 0.22 - 0.31 0.38 0.18 0.69 0.559 0.31 0.34 0.13 0.30 0.29 0.30 0.17 0.28 - 0.12 0.20 0.49 0.3510 0.35 0.33 0.21 0.19 0.40 0.22 0.29 0.35 0.13 - 0.21 0.37 0.3511 0.14 0.15 0.12 0.28 0.32 0.19 0.26 0.17 0.20 0.21 - 0.52 0.5112 0.65 0.63 0.57 0.26 0.84 0.38 0.67 0.74 0.50 0.37 0.53 - 0.5913 0.64 0.67 0.42 0.57 0.46 0.58 0.36 0.55 0.31 0.35 0.51 0.59 -

c

Table 3 Percentage battery depletion with 2 kg payload.

Nodes 1 2 3 4 5 6 7 8 9 10 11 12 131 - 0.14 0.23 0.46 0.36 0.34 0.35 0.13 0.35 0.39 0.16 0.74 0.702 0.15 - 0.30 0.37 0.50 0.23 0.45 0.27 0.42 0.43 0.25 0.65 0.743 0.23 0.29 - 0.39 0.25 0.33 0.16 0.20 0.14 0.23 0.14 0.65 0.484 0.45 0.37 0.40 - 0.64 0.15 0.54 0.51 0.34 0.21 0.32 0.28 0.605 0.37 0.48 0.25 0.64 - 0.59 0.16 0.23 0.33 0.46 0.36 0.88 0.526 0.33 0.22 0.33 0.15 0.62 - 0.50 0.41 0.34 0.25 0.22 0.44 0.657 0.35 0.46 0.16 0.53 0.15 0.53 - 0.24 0.20 0.33 0.29 0.85 0.418 0.13 0.26 0.17 0.52 0.26 0.41 0.25 - 0.31 0.41 0.20 0.79 0.649 0.36 0.39 0.14 0.34 0.33 0.34 0.20 0.31 - 0.14 0.23 0.58 0.3610 0.40 0.38 0.23 0.21 0.46 0.25 0.33 0.39 0.14 - 0.24 0.43 0.4011 0.16 0.17 0.13 0.32 0.36 0.22 0.30 0.19 0.22 0.24 - 0.60 0.5712 0.74 0.70 0.65 0.32 0.88 0.47 0.75 0.78 0.56 0.43 0.60 - 0.6713 0.70 0.74 0.47 0.59 0.52 0.65 0.45 0.64 0.36 0.40 0.60 0.67 -

Table 4 Percentage battery depletion with 3kg payload.

Nodes 1 2 3 4 5 6 7 8 9 10 11 12 131 - 0.16 0.32 0.58 0.42 0.37 0.41 0.15 0.40 0.45 0.23 0.89 0.802 0.16 - 0.33 0.43 0.55 0.25 0.51 0.37 0.45 0.43 0.30 0.74 0.843 0.26 0.33 - 0.45 0.28 0.38 0.18 0.22 0.19 0.29 0.15 0.74 0.544 0.52 0.42 0.45 - 0.74 0.19 0.81 0.58 0.39 0.24 0.38 0.32 0.675 0.41 0.61 0.28 0.73 - 0.66 0.19 0.32 0.37 0.53 0.50 1.01 0.596 0.40 0.25 0.47 0.17 0.74 - 0.56 0.49 0.39 0.29 0.25 0.49 0.767 0.41 0.52 0.21 0.62 0.15 0.57 - 0.33 0.23 0.38 0.33 0.86 0.468 0.15 0.30 0.20 0.64 0.31 0.47 0.28 - 0.35 0.45 0.22 0.89 0.729 0.44 0.45 0.16 0.47 0.37 0.39 0.22 0.37 - 0.15 0.26 0.64 0.4010 0.45 0.43 0.26 0.24 0.54 0.29 0.38 0.45 0.16 - 0.27 0.53 0.4511 0.20 0.22 0.15 0.36 0.41 0.25 0.34 0.23 0.25 0.27 - 0.75 0.6612 0.84 0.74 0.74 0.34 1 0.48 0.86 0.91 0.64 0.49 0.68 - 0.9013 0.80 0.84 0.54 0.70 0.64 0.74 0.46 0.72 0.40 0.46 0.89 0.76 -