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How to incorporate unmanned aerial vehicles to improve delivery of door-to-door air cargo?

Valentine GOLFIER

Master Thesis

Civil Engineering / Traffic Engineering

February 2018

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Acknowledgements I would like to express my gratitude to my supervisors from UC Berkeley and the ETH Zürich, Prof.

Mark Hansen and Prof. Monica Menendez for their useful comments, remarks and engagement

through the learning process of this Master Project. Furthermore, I would like to thank Mogeng Yin,

Frank Ketcham and my husband Thomas Vetterli for their constant accessibility, kindness and

encouragements. I had a lot of pleasure to carry out this project.

Thank you,

Valentine

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Master Thesis

How to incorporate unmanned aerial vehicles to improve delivery of door-to-door air cargo?

Valentine GOLFIER

IVT ETH Zurich

Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland

Phone: +41 (0) 78 695 19 37

Fax: +1 (510) 495 79 11

E-Mail: [email protected]

February 2018

Abstract

Flight delay in the airline industry is a big challenge in the United States. Whereas it is a well-known issue for the transportation of passengers, it also represents an important headache for air cargo carriers. Indeed, flight delay leads to a time and cost pressure for the last-mile ground delivery system in logistics operations. Besides the delay costs due to late delivery and lateness, the shortened available delivery time window generates additional operational costs. This study aims at assessing the potential of the use of commercial UAVs in last-mile delivery systems to decrease the specific problem of additional costs caused by cargo flight delay. Although there is still a lot of unknown variables concerning the scalability and feasibility of the system, its numerous advantages and its quick development present a promising future. Two different truck-drone combination concepts are modeled and analyzed: the first one with the drones launched from the depot and the other with the drones launched from the trucks. For both concepts, a grid search approach for the vehicle routing model with stochastic time windows (VRMSTW) was used to model and estimate both operational and delay costs to cargo operators under different distributions of flight delay. For each flight delay scenario, it is captured how cargo operators should adapt their operational strategy to minimize the total costs. When an optimal strategy is chosen with the second concept, the total delivery costs per day are drastically reduced, from up to 45%. In contrast, the insertion of drones with the second concept does not bring any added value due to the limited range of drones of 10 miles. For the first

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concept to become significantly efficient, the technical performance of drones must be improved to a minimum range of 10 miles.

Keywords

Cargo flight delay ; Vehicle routing problem with stochastic time window; Grid-Search method; Unmanned Aerial Vehicles; Delivery costs

Preferred citation style

Golfier, V. (2018) How to incorporate unmanned aerial vhicles to improve delivery of door-to-door air cargo?,/Master Thesis, IVT, ETH Zurich, Zurich.

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Table of contents

1 Unmanned Aerial Vehicles: a solution to decrease the costs of ground delivery systems? .. 10

2 Literature review ................................................................................................................ 12

2.1 The current state of commercial drones ....................................................................... 12

2.2 Relationship between cargo flight delay and deliveries delay ...................................... 15

3 The vehicle routing problem with a combination of trucks and drones................................ 22

3.1 A comparison of three delivery-system concepts ......................................................... 22

3.2 General assumptions ................................................................................................... 23

3.3 Methodology: the grid search method ......................................................................... 25

3.4 Model 1: Trucks and drones launched from the depot ................................................. 27

3.5 Model 2: Trucks and drones launched from the trucks ................................................. 33

4 Results ............................................................................................................................... 40

4.1 Input parameters ......................................................................................................... 40

4.2 Numerical results ........................................................................................................ 44

5 Sensitivity Analysis ........................................................................................................... 48

5.1 Number of parcels carried simultaneously ................................................................... 48

5.2 Range of the drone ...................................................................................................... 49

6 Conclusion ......................................................................................................................... 52

7 Reference list ..................................................................................................................... 55

8 Appendix ........................................................................................................................... 57

8.1 Code - Python ............................................................................................................. 57

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List of tables Table 1: Amazon-type drone performance (Pogue, 2016) ............................................................ 15 Table 2: Input parameters ............................................................................................................ 41 Table 3: Optimal time window Tp and (associated total costs) under different available time window distributions (own results) ........................................................................................................... 45

List of figures Figure 1: Airspace design for small drone operations (Leswing, 2015) ........................................ 13 Figure 2: Drone prototypes a) Amazon Prime Air (Stern, 2013) b) DHL (Mendoza, 2014) c) UPS (Macdonald, 2017) ...................................................................................................................... 13 Figure 3: Available time window to deliver with and without flight delay (own representation) .. 16 Figure 4: Illustration of the trade-off between operational costs and delay costs based on the distribution of the available time window Tav (adapted from Yin, Hansen &Shen (2016) ) .......... 19 Figure 5: Predicted percentage of package delay vs. flight delay (Yin, Liu, & Hansen, 2014) ...... 20 Figure 6: Drone delivery concepts (own representation) .............................................................. 23 Figure 7: Geometry of the delivery area and its service zones (adapted from (Yin, Hansen, & Shen, 2016)) ......................................................................................................................................... 24 Figure 8: Grid-search method (own representation) ..................................................................... 27 Figure 9: The three possible states of a ring (Yin, Hansen, & Shen, 2016) .................................. 31 Figure 10: Illustration of one truck-drone loop in the delivery region (own representation) ......... 34 Figure 11: Flight delay distribution at destination airports (FFA ASPM) ..................................... 43 Figure 12: Triangular distribution of available time windows (Yin, Hansen, & Shen, 2016) ........ 43 Figure 13: Total cost vs planned time windows under different available time window distributions (own representation) ................................................................................................................... 44 Figure 14: Optimal solutions of programing and simulation models (own results) ....................... 45 Figure 15: Proportion of costs associated to trucks and drones (own results) ............................... 46 Figure 16: Proportion of operational and delay costs (own representation) .................................. 47 Figure 17: Sensibility analysis for the number of parcels carried simultaneously by a drone within one loop (own results) ................................................................................................................. 49 Figure 18: Sensibility analysis for the range of the drones (own results) ...................................... 50 Figure 19: Optimal solutions for the toal costs under different technical performances of drones (own results) ............................................................................................................................... 51

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List of abbreviations FAA: Federal Aviation Administration

FAA ASPM: Federal Aviation Administration, Aviation Systems Performance Metrics

UAR: Unmanned aerial rotorcraft

UAS: Unmanned Aerial Systems

UAV: Unmanned Aerial Vehicles

VRPSTW: Vehicle routing (problem) model with Stochastic Time-Windows

VRP: Vehicle Routing Problem

VTOL: Vertical Take-Off and Landing

List of notations T time window [h]

Tav available time window [h]

R radius or the delivery region [mi]

∂(r) customer density at radius r, simplified as ∂ under uniform distribution [customer/mi2]

Vtr cruise speed of the trucks [mph]

Vdr cruise speed of the drones [mph]

ttr service time per customer for the trucks [h]

tdr service time per customer for the drones [h]

Str,i total service time in the ith ring for each truck driver [h/veh]

Sdr,i total service time in the ith ring for each drone [h/veh]

ri outer radius of the ith ring [mi]

ri-1 inner radius of the ith ring [mi]

Li length (radial distance) of the ith ring (and thus of all its zones) [mi]

ix

Ni number of vehicles needed in the ith ring [veh]

𝐷"# traverse (along the ring) distance travelled by one vehicle with a zone in the ith ring [mi]

𝐷"$ line-haul (along the radius) distance travelled by one vehicle with a zone in the ith ring [mi]

𝐷"$%&'$ local (within the zone) distance travelled by one vehicle with a zone in the ith ring [mi]

𝐷"#%#'$ distance travelled by one vehicle with a zone in the ith ring [mi]

𝜃" half of the nominated circular angle in the ith ring served by one driver [rad]

𝜔" half of the nominated circular distance in the ith ring served by one driver [mi]

Ai nominated rectangular area in the ith ring served by one driver [mi2]

K total number of rings [-]

Ntr total number of trucks needed [veh]

Ndr total number of drones needed [veh]

Dtr total distance travelled by trucks [mi]

Ddr total distance travelled by drones [mi]

Wmax maximum payload that a drone can carry at a time [lbs]

𝑤 average weight of a parcel [lbs]

Dmax range of the drone, back and forth [mi]

𝐶,,#/ cost per truck (vehicle-based cost) [$/veh]

𝐶,,0/ cost per drone (vehicle-based cost) [$/veh]

𝐶1,#/ cost per truck per mile (mileage cost) [$/veh/mi]

𝐶1,0/ cost per drone per mile (mileage cost) [$/veh/mi]

𝐶2,#/ cost per missed delivery [$/delivery]

𝐶$,#/ cost per hour of lateness [$/h]

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1 Unmanned Aerial Vehicles: a solution to decrease the costs of ground delivery systems?

Flight delay in the airline industry is a big challenge in the United States. Whereas it is a well-

known issue for the transportation of passengers, it also represents an important headache for

air cargo carriers. Indeed, flight delay leads to a time and cost pressure for the last-mile ground

delivery system in logistics operations. Besides the delay costs due to late delivery and lateness,

the shortened available delivery time window generates additional operational costs.

A study from Yin & al. (2016) proposed a continuous approximation of the vehicle routing

model with stochastic time windows (VRMSTW) to estimate both operational and delay costs

to cargo operators under different distributions of flight delay. For each flight delay scenario,

they captured how cargo operators should adapt their operational strategy to minimize the total

costs.

The improvement of truck fleets management and the optimization of ground delivery systems

in general is not a new concern in the transportation field. Facing the burden of commercial

traffic in cities, some innovative solutions are studied. One trend that has been emerging for a

few years is the use of unmanned aerial vehicles (UAVs) for delivery operations. While

Amazon completed its first drone delivery to a customer in Britain in November 2016 (Halzack,

2016), FedEx, UPS, Google and other competitors are actively researching to develop their own

prototype of flying delivery systems. Although there is still a lot of unknown variables

concerning the scalability and feasibility of the system, its numerous advantages and its quick

development present a promising future.

This study aims at assessing the potential of the use of commercial UAVs in last-mile delivery

systems to decrease the specific problem of additional costs caused by cargo flight delay. Their

great speed and their independence from the ground traffic could offset the time pressure due

to flight delay and thus avoid the additional operational and delay costs. Section 2 reviews the

current legal and technical state of drones as well as the previous research on establishing the

link between cargo flight delay and additional operation and delay costs of ground delivery

systems. The main results of the vehicle routing problem with stochastic time windows

(VRPSTW) heuristic model developed by Yin & al. (2016) are summed-up. Section 3 extends

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this model with the insertion of drones in the vehicle fleet. Two different truck-drone

combination concepts are analyzed: one with the drones launched from the depot and the other

with the drones launched from the trucks. For each on them, the model is formulated and the

solution method is presented. The numerical results obtained by the application of the models

are provided in Section 4 and section 5 summarizes the main points of the study.

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2 Literature review

2.1 The current state of commercial drones

Facing the burden of commercial traffic in cities, a new trend has been emerging for a

few years in the transportation field: the use of unmanned aerial vehicles (UAV) for delivery

operations. While Amazon completed its first drone delivery to a customer in Britain in

November 2016 (Halzack, 2016), FedEx, UPS, Google and other competitors are actively

researching to develop their own prototype of flying delivers. Although there is still a lot of

unknown variables concerning the scalability and feasibility of the system, its numerous

advantages and its quick development present a promising future.

2.1.1 Flying delivery-drones in the US: legal situation

Due to the steady increase in amateur and commercial drone sales in the past ten years,

the Federal Aviation Administration (FAA) has been driven to define regulations for the use of

UAVs in the United States.

Concerning the commercial operation of small drones (between 250mg to 25kg), the last 2016

regulations are still very restrictive, prohibiting for instance drone flights over populated areas

and requiring line-of-sight flights (Associated Press, 2016). For now, these restrictions

considerably limit the potential use of drones for package delivery in the United States, subject

on which all big American courier delivery companies are currently conducting active research.

The FAA seeks above all to guarantee public safety. To that extent, the failure, crashes, hacking,

or privacy invasion risks of UAV are still problematic. But the biggest concern of the FAA is

the cohabitation of drones with manned-aircrafts. A report of drones sightseeing from 2016

shows that despite a new registration scheme, near misses between unmanned and manned

aircrafts (mainly commercial passenger aircrafts) are rising (Harris, 2016).

Willing to find an agreement with the FAA to fly their package-delivery-drones, Amazon

proposed in 2015 an airspace division design (Figure 1), aiming at ensuring Amazon Prime Air

safety (Leswing, 2015). Their design dedicates the airspace above 500ft to manned-aircrafts

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and separates it from the drone’s airspace with a safety buffer zone between 400ft and 500ft.

Under 400ft would be the drones’ airspace, with a high-speed transit zone from 200ft to 400ft

and a take-off and landing zone under 200ft.

Figure 1: Airspace design for small drone operations (Leswing, 2015)

Even if the FAA is opened to discussions to find a working system including drones in the

airspace, Amazon and its competitors will have to provide more solid evidence in the following

years to convince the FAA of the drone’s safety before they will be allowed to perform drone-

deliveries.

2.1.2 Delivery drone capabilities

By completing its first drone delivery to a customer in Britain in November 2016,

(Halzack, 2016) performed the first concrete test and application of the research on commercial

drone-delivery. The “Prime Air” drone was also the first delivery prototype presented to the

public. DHL, UPS and other competitors also developed their own delivery-drone prototypes.

a) b) c)

Figure 2: Drone prototypes a) Amazon Prime Air (Stern, 2013) b) DHL (Mendoza, 2014) c)

UPS (Macdonald, 2017)

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The current state of the performances of the delivery-drones are summed-up in the Table 1

below. In 2016, Amazon’s vice president Paul Misener announced that the Prime Air UAV will

reach a range of 10 miles, a maximum payload of 5lbs and a speed of 40mph (Pogue, 2016).

However, the potential use of drones in delivery systems does not only depend on their technical

capabilities; a good reliability is also essential. Their high sensibility to bad weather conditions

(icy rain showers, excessive winds…) or the security risks they can face (hacking, crash, theft)

are still a serous limitation for a potential commercial use in the United Stated.

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PERFORMANCE ADVANTAGES LIMITATIONS

Technical • Speed • Unmanned = reduced

labor cost

• Limited payload

(5lbs = 2.3kg)

• Small battery autonomy

(20 min)

• Limited range

(~7 mi=~10km)

Infrastructure • Independent from ground traffic

• Delivery in areas with limited transportation infrastructure

• Landing infrastructure requirements (2m2 area)

Environment • Green (battery) • Noise • Extremely weather

dependent

Security • Unmanned • Hacking, theft risk • Crash risk • Plausible invasion of

privacy

Table 1: Amazon-type drone performance (Pogue, 2016)

2.2 Relationship between cargo flight delay and deliveries delay

Yin & al. (2016) proposed both a heuristic approach and a continuous approximation of

the vehicle routing model with stochastic time windows (VRMSTW) to estimate operational

and delay costs to cargo operators under different distributions of flight delay. For each flight

delay distribution scenario, they captured how cargo operators should adapt their operational

strategy to minimize the total costs. This section explains the main concept behind their model.

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2.2.1 Distribution costs

The Figure 3 below represents the chain of events of the last-mile ground delivery system

operated by cargo companies such as FedEx and UPS and the impact of flight delay on the

logistic chain and its operational costs. After being freighted by plane to the regional airport,

the goods are transported by truck to the different distribution centers to be sorted out and

finally delivered by truck to the end customer.

As most service providers, cargo companies seek constantly to improve the quality of

their service while keeping some cost-effective logistics operations. For last-mile delivery

operations, the two challenges are operational and delay costs. In case of flight delay, all

segments of the distribution chain will get affected, experiencing some delay as well. The

assumption is made here that the flight delay is transferred exactly to other segments. Any

Figure 3: Available time window to deliver with and without flight delay (own

representation)

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additional gain or loss of time due to packages sorting or traffic is neglected. The last segment

of the distribution chain (from the distribution center to the end-customer) experiences some

time pressure as the initial time-window available Tav,1 to deliver all parcels on time is reduced

to Tav,2 (Figure 3). If the companies keep the same size of the truck fleet, some packages will

not be delivered on time. This will induce some delay costs as customers can ask for a full

refund for any late delivery and will get disappointed by the service. In a flight delay situation,

cargo companies have one option to limit their delay costs: increasing the size of their truck

fleet. They will be able to deliver the parcels faster and thus on time. However, this method

induces more trucks and drivers and thus more operational costs. Hence, to minimize the

distribution costs, operators must find a trade-off between operational and delay costs:

Operational costs:

• Veh-based operational costs [$/truck]

• Mile-based operational costs [$/mile]

Delay costs:

• Late delivery costs [$/late delivery]

• Lateness costs [$/hour of lateness]

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2.2.2 Stochastic flight delay

Flight delay is a stochastic phenomenon which leads to a stochastic arrival of trucks to

the local distribution center and thus to a stochastic available time window to deliver the

packages. As described above, the operators can adjust their strategy (size of the fleet) to cope

with the time-pressure induced by flight delay. However, a delivery strategy is a medium- or

even long-term strategy which cannot be changed daily. This means that the operators must

find an optimal strategy which must be robust enough to absorb a certain amount of delay but

which also allows a certain proportion of missed deliveries and its corresponding costs.

The approach used by Yin & al. (2016) to solve this problem is to optimize the delivery strategy

(i.e. the size of the fleet, i.e. the time-window Tp needed to deliver all packages from the time

the trucks leave the distribution center) based on the distribution of the available time windows

Tav, in order to minimize the distribution costs. The Figure 4 illustrates the link between the

costs and the distribution of available time windows. If the operator chooses a very small and

conservative time-window Tp (yellow line on the left), it means that the strategy aims for a large

fleet of trucks that can deliver all packages in a very small amount of time and so absorb even

the biggest delays. On the one hand, the delay costs will be minimal but on the other hand, due

to the numerous trucks and drivers, the operational costs will rise considerably. Moreover, the

fleet will be over-sized for most of the Tav (i.e. flight delay) scenarios. On the contrary, if the

operator chooses a very large time window Tp (yellow line on the right), it means that the

strategy aims for a very small fleet of trucks with smaller operational costs but that needs a lot

of time to deliver all parcels, leading to major delay costs. Indeed, there is a high probability

that the needed time-window T is larger than the available one Tav, causing numerous missed

deliveries. The optimal time window (i.e. size of the fleet) drawn as a red line in Figure 4.a.

will be located somewhere in the middle of those two extremes.

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Figure 4: Illustration of the trade-off between operational costs and delay costs based on the

distribution of the available time window Tav (adapted from Yin, Hansen &Shen (2016) )

2.2.3 The impact of cargo flight delay on deliveries delay in figures

It is known that cargo flight delay leads to delayed deliveries. But which proportion of

deliveries miss their due-time per day? How much of this delay is due to cargo flight delay?

Which costs are induced by these additional delayed packages?

Yin & al. (2014) analyzed a set of proprietary delivery records from a freight auditing company

as well as flight data obtained from a FAA ASPM (Federal Aviation Administration, Aviation

Systems Performance Metrics) database. They focused on the “due to next day” packages which

are the most vulnerable to disruption caused by flight delay. The figures presented below

describe the “Priority service” delivering packages by 10:30 am.

They showed that most packages are due by 10:30 am (85%) while 6% are due by 12:00am and

8% by 4:40pm. Among the earliest packages, about 14% of them are delayed and 23% of this

delay is due to air cargo flight delay. In the major markets, these numbers can be even higher:

flight delay accounts for 22% to 38% of package delay.

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Figure 5: Predicted percentage of package delay vs. flight delay (Yin, Liu, & Hansen, 2014)

2.2.4 Motivation for this study

The consequences of cargo flight delays on the last-mile delivery system are now clearly

identified and quantified. The next step is to find some solutions to counter these effects. As

most service providers, cargo companies constantly seek to improve the quality of their service

while keeping some cost-effective logistics operations. Concerning the last-mile delivery, a lot

of solutions to gain time are currently analyzed, from autonomous ground vehicles to droids or

unmanned aerial vehicles. This last solution is the focus of this study. The goal is to assess is a

truck-drone delivery system could be more robust to flight delays than the current traditional

truck delivery system. Could the distribution costs be decreased? Based on the vehicle routing

problem with stochastic time windows (VRPSTW) developed by Yin & al (2016), two models

will be formulated to determine the best strategy for cargo companies to apply to decrease their

total costs under different distribution of flight delay.

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3 The vehicle routing problem with a combination of trucks and drones

This section presents the two vehicle routing problem with a truck-drone combination

models developed in the study. After setting the general assumptions and the methodology used,

the formulation of the models and their solution are described.

3.1 A comparison of three delivery-system concepts

Three concepts for package-delivery were considered in this work, the traditional one with

only trucks and two with a combination of drones and trucks. For each scenario, the optimal

strategy (i.e. size and zones assignment of the fleet) to minimize the distribution costs was

determined and compared one with another. In the cases where technical performances of the

drone represented a limitation for the improvement of the system with trucks only, some

minimal required technical performances were defined.

The first scenario, called Concept 0 (Figure 6a), corresponds to the traditional delivery system

with trucks serving a defined delivery area around a distribution center. The routing model with

stochastic time windows developed by Yin & al. (2016) for this scenario will serve as the basis

for the development of the two alternative models (Concepts 1 & 2) and for the comparison of

their performance.

The first alternative (Concept 1, Figure 6b) relates to the project currently being developed by

Amazon. The solution consists of single small drones delivering one (or several) parcels at a

time from the depot to the client (Pogue, 2016). The drones do not completely replace trucks

but are assigned a part of the deliveries. Which portion of these deliveries is the optimal one

will be defined with the model.

It will be shown in section 4 that the limited range (i.e. the battery autonomy) of small delivery-

drones is a big constraint for their use as delivery vehicles. The second alternative (Concept 2, Figure 6c) seeks to overcome this limitation by using trucks as launching platforms. The truck

leaves the depot with the drone “parked” on its roof. Once it has reached the assigned delivery

zone, the drone takes off from the roof to deliver its pre-loaded parcels, while the truck keeps

completing its own deliveries in the zone. By flying back and forth between the truck and the

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customers, the drone can be reloaded with new parcels and have its battery changed by the

driver of the truck, till all deliveries of the zone are completed. UPS realized its first residential

drone delivery with this concept in February 2017 (Kastrenakes, 2017).

a) b)

c)

Figure 6: Drone delivery concepts (own representation)

a) Concept 0: Trucks only b) Concept 1: Trucks and drones launched from the depot

c) Concept 2: Trucks and drones launched from the trucks

3.2 General assumptions

The assumptions used in the models are based on previous research. First, the design of the

delivery region into zones assigned to each vehicle follows the research of Langevin & Soumis

(1989). Furthermore, the models 1 and 2 incorporating drones in the delivery process are an

adaptation of the vehicle routing model (model 0) with stochastic time-windows (VRPSTW)

developed by Yin & al. (2016). Whereas some new assumptions were made for the introduction

of drones in the concepts 1 and 2, the basic assumptions are similar to the ones made by Yin &

al. (2016) for model 0.

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3.2.1 Geometry of the delivery area

The delivery area is modelled as a circle centered around a sorting center and partitioned into

zones assigned to each vehicle (Figure 7a).

• The region of radius 𝑅 contains 𝑖 rings, which are divided into zones approximated as

triangles for the inner ring and as rectangles for the other rings (Figure 7b).

• Each zone, as the one shaded on the Figure 7a, has a length of 𝐿" and a width of 2𝑤" to

make one round trip during the delivery tour. As demonstrated by Newell & al. (Newell

Gordon, 1986a), 𝑤" = 8 9:;(/)

with 𝛿(𝑟) = 𝛿

• For a matter of simplification, the customer density 𝛿 (number of customer per unit area) among the delivery area is assumed under uniform distribution meaning that 𝛿 is a constant.

• The zones of a same ring are assigned to only one type of vehicles, either trucks or

drones.

Figure 7: Geometry of the delivery area and its service zones (adapted from (Yin, Hansen, &

Shen, 2016))

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3.2.2 Delivery network

• Whereas the trucks serve the area using a radial ring road network, the drones make diagonal movements between each customer location as they fly. Their service routes are shown in the Figure 7c.

• Each zone is served by one vehicle, drone or truck, and each vehicle is assigned to a unique zone. Thus, each vehicle makes one and only round trip from the depot.

3.2.3 Delivery timing

• To simplify the analysis, a one-to-one correspondence between the flight delay and the time window change for the last-mile delivery segment from the depot to the end-customer is assumed. We do not consider any other delay that can occur during the parcel-handling or sorting, or due to traffic congestion if the delivery time was shifted in the morning peak-hour because of the delay.

• The time window Tp is defined as the time between the departure of the vehicle from the depot and the time it completes its last delivery in the assigned zone. It is the time needed by the truck to deliver all customers of its assigned zone.

• The available time-window Tav is the time between the departure of the vehicle from the depot and the delivery time asked by the customer. It is the time available for the truck to deliver all customers of its assigned zone on time.

• If the vehicle is to complete all its deliveries on time (i.e. late deliveries are unacceptable), the time window must be equal or smaller than the available time window Tp ≤ Tav.

3.3 Methodology: the grid search method

The three models were formulated based on the grid-search method and then coded in

Python. With defined input parameters, they give the best strategy in term of fleet size to

minimize total costs under different distributions of available time windows.

A vehicle routing problem (VRP) model for the concept 0 was established by Yin & al. (2016).

They used two different approaches for the optimization problem: a grid-search method and a

nonlinear mixed integer programming (MIP) formulation. Although the MIP formulation is

more exact (and more complex) than the grid-search method, they demonstrated that the results

26

given by the two approaches were very similar. Therefore, the simpler grid-search method was

chosen to develop the new VRP models for concepts 1 and 2.

The grid-search method (Figure 8) developed by Yin & al. (2016) consists of first, resolving

the deterministic VRP for different planned time-windows Tp. The “planned time window”

refers to the aimed time window within which the operator wants to be able to complete all

deliveries. For each planned time widow Tp, the configuration of the service area (e.g. the radial

distance Li of the rings and the size of each delivery zone) which minimizes the total number

of vehicles (e.g. the total travelled distance) is calculated. Then, based on this geometry, the

associated number of delayed packages p and the total lateness d are calculated under different

distributions of the available time window Tav (e.g. actual available time window, depending

on the amount of flight delay). Thus, the total daily costs (operational and delay costs) for each

planned time window Tp can be determined. The optimal planned time window Tp* will be the

one which minimizes the total delivery costs.

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Figure 8: Grid-search method (own representation)

3.4 Model 1: Trucks and drones launched from the depot

The first alternative to the traditional delivery system is the combination of drones and

trucks, both departing from the distribution center. The drones do not completely replace trucks

but are assigned a part of the deliveries. Which portion of these deliveries is the optimal one

will be defined with the model.

3.4.1 Formulation

For model 1, all equations concerning the trucks come from the vehicle routing model

with stochastic time windows (VRPSTW) developed by Yin & al. (2016). The ones for drones

28

were developed for this study. They are all summarized below with the equations concerning

drones. All notations used in the formulation of models are listed in the section “List of

Notations” on the pages viii and ix.

For one vehicle serving one zone in the ith ring:

1) The local (in the service zone) travelled distance is:

𝐷#/,"$%&'$ = 𝜕𝐴"BC9+ 2𝐿" (1)

𝐷0/,"$%&'$ = 𝜕𝐴"8EBC9F:+ E G

BC;F: (2)

Where 𝐴" = 2𝜔"𝐿" is the area of a zone of the ring i. These equations are based on the

rectangular approximation of the service zone, and the fact that the traverse distance between

two uniformly distributed customers is BC9

.

2) The line-haul (from the depot to the service zone) travelled distance is:

𝐷#/,"$ = 𝐷0/,"$ = 𝑟"HG (3)

3) The total stop time needed to serve customers in the zone is:

𝑆#/," = 𝜕𝐴"𝜏#/ (4)

𝑆0/," = 𝜕𝐴"𝜏0/ (5)

The optimization problem can be formulated as:

minNO,PCQ,PCR

S [𝑋"V(𝑁#/,"∙ 𝐶,,#/ + 𝐷#/," ∙ 𝐶1,#/) + 𝑋"Y(𝑁0/," ∙ 𝐶,,0/ + 𝐷0/," ∙ 𝐶1,0/)

"]

+𝐶2 ∙ (𝑝#/ + 𝑝0/) + 𝐶$ ∙ (𝑑#/ + 𝑑0/) (6)

s. t. 𝑋"V ∙ E1C,`ab a

+ 𝐴"𝛿 ∙ 𝜏#/F ≤ 𝑇2 (7)

𝑋"Y ∙ E1C,eabea

+ 𝐴"𝛿 ∙ 𝜏0/F ≤ 𝑇2 (8)

𝑋"Y ∙ 𝐷0/,",f%gh ≤ 2 ∙ 𝐷i'j (9)

𝑋"Y ∙ 𝐴"𝛿𝑤 ≤ 𝑊i'j (10)

29

∑ (𝑋"V + 𝑋"Y) ∙ 𝐿"" = 𝑅 (11) 𝑋"mG,V ≤ maxp𝑋"V, 𝑋"Yq (12) 𝑋"mG,Y ≤ maxp𝑋"V, 𝑋"Yq (13) {𝑋"V , 𝑋"Y} ∈ {0; 1} (14)

𝐿" ≥ 0 (15)

The objective function (6) aims at minimizing the total costs. The set of independent decision

variables are Tp, Xij and Xjk where Xij and Xjk are some binary variables that define if a ring i is

served by trucks (Xij =1) or by drones (Xik =1). All the other variables can be derive from these

variables. The first term ∑ (𝑁#/,"∙ 𝐶,,#/ + 𝐷#/," ∙ 𝐶1,#/ + 𝑁0/," ∙ 𝐶,,0/ + 𝐷0/," ∙ 𝐶1,0/)"

represents the operational costs (vehicle-based costs 𝐶,,#/ and mile-based costs 𝐶1,#/)

depending on the number of trucks and drones. The second term 𝑝 ∙ p𝐶2,#/ + 𝐶2,0/q + 𝑑 ∙

p𝐶0,#/ + 𝐶0,0/q represents the delay costs (missed deliveries and lateness). Equations (7) and

(8) define the time-window constraint. The time needed by each driver or drone to deliver all

parcels in its delivery zone (travelling time plus service time) must be smaller than the aimed

time-window Tp (called “planned time-window”) fixed by the company strategy. The drones

are subject to two more constraints: a maximum range constraint 2 ∙ 𝐷i'j (9) due to the

limitation of the battery autonomy as well as a maximum payload constraint 𝑊i'j(10).

Equation (11) specifies that the sum of lengths Li of all rings must be equal to the radius R of

the global delivery region. Equations (12) and (13) prevent any holes in the system. If a ring i

is not served by either some trucks nor some drones, the next rings will not be served either.

3.4.2 Solution

For the rings served by trucks, Yin & al. found the solution to the optimization problem of the

VRP. For rings served by drones, the main results are summarized below. By testing the models

with numerical values, the results showed that the drones are quickly limited by their range and

cannot even reach the second ring of the optimal truck delivery system (model 0). Thus, to

simplify the model, it was decided to ignore the binary variables Xij and Xjk in the model.

Instead, the maximum distance the drones can reach in the delivery region is defined and the

30

trucks are set to deliver the rest of the delivery region beyond this point. As a consequence, the

objective function (6) has now only one independent decision variable, Tp.

1) For i ≥ 1, the widths of each service zone are equal. Yin & al. showed that although the central ring is approximated as a triangle and not a rectangle, it may not need to be differentiated due to its small size in most cases. Thus, the width of each zone in a ring i is:

𝜔" = 8 9:;

(16)

2) The radial distance between the different rings can be found recursively by starting by

the outer ring R of the delivery region:

𝑟#/,"HG =p9m√z;{bq·/CHN·b

:m√z;{b (17)

𝑟0/,"HG = max(𝑟"HGN , 𝑟"HG1}~�, 𝑟"HG� ) (18)

with: 𝑟0/,"HGN = N·beaH/C(√�m√z;{eabea)GH√�H√z;{eabea

(19)

𝑟0/,"HG1}~� =

1��aC√�

:H√� (20)

𝑟0/,"HG� = /C√z;�H�}~�√z;�

(21)

and: 𝑟0/,"HGN ≤ 𝑇 · 𝑉0/ (22)

𝑟0/,"HG� ≤ 1}~�:

(23)

3) Number of vehicles needed in the 𝑖#� ring (regardless of whether it is served by trucks

or drones):

𝑁" =�paC��aCq

�B

(24)

For the second part of the objective function which relates to the delay costs, the number of late

deliveries 𝑝 and the amount of lateness 𝑑 are defined based on the distribution of available time

windows. First, a time threshold (equations (25) to (30)) is defined to evaluate in which state

each ring (and thus each delivery zone) is. There are three states illustrated on the Figure 9. If

a ring cannot be started (state a), it means that all deliveries due on this ring will be late and the

31

total lateness for a zone will be equal to the number of customers in this zone times the average

lateness of a delivery. If a ring cannot be finished (state b), only a part of the deliveries can be

completed on time. To finish with, if a ring can be finished (state c), all deliveries are on time,

meaning that there is no late deliveries nor lateness.

Equations (31) to (39) below give the number of late deliveries and the amount of lateness for

each scenario.

Figure 9: The three possible states of a ring (Yin, Hansen, & Shen, 2016)

4) The time needed to finish the first ring is:

𝑇�G,#/ = �:��m:B��;·

��B�

b a� + 2𝜔G𝐿G𝛿𝜏#/ (25)

𝑇�G,0/ = :B��;bea

8EB�9F:+E G

;B�F:+ 2𝜔G𝐿G𝜏0/ (26)

5) The time needed to start the ith ring (1 ≥ i ≥ K) is:

𝑇#/,"� = 𝑇#/,"HG� + �C��b

(27)

𝑇0/,"� = ∑ E �QbeaF"HG

V�G = 𝑇0/,"HG� + �C��bea

(28)

6) The time needed to finish the ith ring (1 ≥ i ≥ K) is:

32

𝑇#/,"� = 𝑇#/,"HG� +(:�CH�C��)m:BC�C;·

��BCH:BC��C��;·

��BC��

b a+(2𝜔"𝐿"𝛿𝜏#/ − 2𝜔"HG𝐿" − 1𝛿𝜏#/) (29)

𝑇0/,"� = 𝑇0/,"HG� + :BC�C;bea

8EBC9F:+E G

;BCF:+ 2𝜔"𝐿"𝜏0/ (30)

7) The number of late packages is:

• If the ring is in state a (Tp ≤ T�� i.e. the ring cannot be started):

𝑝#/,"' = 𝑝0/,"' = 2𝐿"𝜔"𝛿 · :�E∑ �QC��

Q�� m��CF

:BC (31)

• If the ring is in state b (T�� ≤ Tp ≤ T�� i.e. the ring cannot be finished):

𝑝#/,"� = � ;(NC�HN~ )

�¡à

m¢£C�

�¡àm{à;BC

¤ · �:�·∑ (�QC��

Q�� m��C)

:BC� (32)

𝑝0/,"� =

⎝

⎜⎛ ;(NC

�HN~ )

¢¡ea

Ë£C� F�m© �

¢£Cª�m;{ea

⎠

⎟⎞· �

:�·∑ (�QC��Q�� m��C)

:BC� (33)

• If the ring is in state c (T�� ≤ Tp ≤ T�� i.e. the ring cannot be finished):

𝑝#/,"& = 𝑝0/,"& = 0 (34)

8) The amount of lateness is: • If the ring is in state a (Tp ≤ T�� i.e. the ring cannot be started):

𝑑#/,"' = 𝑝#/,"' · ©pN a,C

® HN~ mN a,C� HN~ q

:ª (35)

𝑑0/,"' = 𝑝#/,"' · ©pNea,C

® HN~ mNea,C� HN~ q

:ª (36)


𝑑#/,"� = 𝑝#/,"� · ©N a,C� HN~ :

ª (37)

𝑑0/,"� = 𝑝0/,"� · ©Nea,C� HN~ :

ª (38)

• If the ring is in state c (T�� ≤ Tp) i.e. the ring can be finished):

𝑑#/,"& = 𝑑0/,"& = 0 (39)

33

Finally, the total expected number of late deliveries and the total expected amount of lateness

can be obtained by calculating the numerical integral of equations (31) to (39) over the

probability of the available time window P(Tav).

𝑝 = ∑ ©∫ 𝑝#/,"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑝#/,"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª +"

∑ ©∫ 𝑝0/,"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑝0/,"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª" (40)

𝑑 = ∑ ©∫ 𝑑#/,"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑑#/,"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª" +

∑ ©∫ 𝑝0/,"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑝0/,"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª" (41)

3.5 Model 2: Trucks and drones launched from the trucks

In the model 2, a truck and a drone work as a pair (Figure 6c). The truck driver leaves

the depot with the drone “parked” on the roof of the truck. Once it has reached the assigned

delivery zone, the drone will take off to deliver its pre-loaded parcels. During this time, the

truck driver keeps delivering its own parcels. When the drone is done with its deliveries, it

meets again with the truck at a customer’s place. After the landing of the drone on the roof, the

truck driver can reload the drone with new parcels and change its battery if necessary. This

process goes on till all deliveries have been completed. The truck and the drone meet one last

time before going back to the distribution centre.

3.5.1 Additional assumptions

Some additional assumptions from the ones described in section 3.2 apply for model 2.

• The truck and the drone always meet at a customer’s place which was assigned to the

truck.

• The duration between the take-off of the drone from the roof of the truck and the time

it lands again on it is called a loop (Figure 10). The pair repeats the same loops (each

vehicle serves the same number of customers per loop) during the whole service of the

zone. Each loop of the drone is subject to the same three constraints as the ones in the

34

first model: the time window 𝑇2 , the maximum payload 𝑊i'j and the maximum range

𝐷i'jconstraints.

Figure 10: Illustration of one truck-drone loop in the delivery region (own representation)

3.5.2 Formulation

1) The travel time per loop in a zone of the 𝑖#� ring is:

𝑇#/,"$%%2 = #𝑐𝑢𝑠𝑡0/ ∙

Gb a∙B∙;

+ #𝑐𝑢𝑠𝑡#/ �E£�m

�£∙¢F

b a+ 𝜏#/� (42)

𝑇0/,"$%%2 = #𝑐𝑢𝑠𝑡#/ ∙

E£�m�

£∙¢F

bea+ #𝑐𝑢𝑠𝑡0/ »

8£�¼ m

�£�∙¢�

bea+ 𝜏0/½ (43)

Where #𝑐𝑢𝑠𝑡#/ = and #𝑐𝑢𝑠𝑡0/are the number of customers served respectively by the truck

and the drone during one loop.

2) The number of loops during which a drone can fly before changing the battery is:

𝑁$%%2¾,0/�'##h/¿ = 𝑖𝑛𝑡 ©Á'gÂh

bea∙ GÃ²Ä(NÅÆÆO,àH{à;NÅÆÆO,ea)

ª (44)

3) The total number of loops needed to deliver the entire assigned zone is:

𝑁$%%2¾ = 𝑖𝑛𝑡 E ÇC∙;#&È¾#àm#&È¾#ea

F (45)

The optimization problem of the VRP model is the same as the one from model 0 explained by

Yin & al. (2016) in their research, with the additional constraints (49) to (54). The goal again

is to minimize the number of vehicle pairs (e.g. the total travelled distance).

35

minNO

∑ [(𝑁#/,"∙ 𝐶,,#/ + 𝐷#/," ∙ 𝐶1,#/) + (𝑁0/," ∙ 𝐶,,0/ + 𝐷0/," ∙ 𝐶1,0/)" ] (46)

+𝐶2 ∙ (𝑝#/ + 𝑝0/) + 𝐶$ ∙ (𝑑#/ + 𝑑0/)

s.t 𝑡"$"ghH�'È$ + 𝑡"$%&'$ + 𝑆" ≤ 𝑇2 (47)

∑ 𝐿"" = 𝑅 (48)

#𝑐𝑢𝑠𝑡0/ ≤�}~��

(49)

𝑇0/,"$%%2 ≤ É²ÊËÌ

ÍÎÏ (50)

𝑇0/,"$%%2 ≤ TÐ (51)

𝑇#/,"$%%2 ≤ TÐ (52)

𝑇#/,"$%%2 − 𝜏#/ ≤

Á'gÂhb a

(53)

𝑇#/,"$%%2, 𝑇0/,"

$%%2, #𝑐𝑢𝑠𝑡0/, #𝑐𝑢𝑠𝑡#/ ≥ 0 (54)

The objective function (46) is the same as the model 1 without any binary variables as all rings

re served by both trucks and drones. The equation (49) limits the number of customers that a

drone can serve within one loop due to the maximum payload Wmax that the drone can carry.

The equations (50) to (53) specify that the duration of one loop must not exceed the planned

time window nor the maximum range for drones.

3.5.3 Solution

The solutions of the optimization problem are presented below:

1) For i ≥ 1, the widths of each service zone are equal. Yin & al. showed that although the central ring is approximated as a triangle and not a rectangle, it may not need to be differentiated due to its small size in most cases. Thus, the width of each zone of the 𝑖#� ring is:

𝜔" = 8 9:;

(55)

2) The number of customers served by each drone and each truck within one loop in a service zone is:

#𝑐𝑢𝑠𝑡0/ =Ñ��

(56)

36

with: #𝑐𝑢𝑠𝑡0/ ≤ Ò��

8£�¼ m

�£�∙¢�

m{ea∙bea (57)

#𝑐𝑢𝑠𝑡0/ ≤ ÓÔ∙bea

8£�¼ m

�£�∙¢�

m{ea∙bea (58)

#𝑐𝑢𝑠𝑡0/ ≤ TÐ ∙ 𝑉#/ ∙ 𝜔 ∙ 𝛿 (59)

#𝑐𝑢𝑠𝑡#/ = min(#𝑐𝑢𝑠𝑡#/(�µ), #𝑐𝑢𝑠𝑡#/

(�G), #𝑐𝑢𝑠𝑡#/(�:), #𝑐𝑢𝑠𝑡#/

(�9)) (60)

with: #𝑐𝑢𝑠𝑡#/(�µ) =

1}~�H#&È¾#ea�8£�¼ m

�£�∙¢�

∗m{ea∙bea�

8£�¼ m

�£�∙¢�

(61)

#𝑐𝑢𝑠𝑡#/(�G) =

NO∙beaH#&È¾#ea�8£�¼ m

�£�∙¢�

∗m{ea∙bea�

8£�¼ m

�£�∙¢�

(62)

#𝑐𝑢𝑠𝑡#/(�:) =

NO∙b aH#Ö×Øèa£∙¢

£�m

�£∙¢m{à∙b a

(63)

#𝑐𝑢𝑠𝑡#/(�9) =

#&È¾#ea

⎝

⎜⎛¨£

�¼ � �

£�∙¢�¡ea

m{eaH�

¡à∙£∙¢

⎠

⎟⎞m{à

£��

�£∙¢

¡àm{àH

¨£�¼ � �

£�∙¢�¡ea

(64)

3) The radial distance between the different rings can be found recursively by starting by

the outer ring R of the delivery region:

𝑟"HG =NOmÙH/CÚ

�£¢#Ö×Ø`à�#Ö×Øèa

»i'jEpNÅÆÆO,àH{àq,NÅÆÆO,eaFmÙmÛ·Ü}~�

¡ea·}~�©EÝÅÆÆO,à�ÞàF,ÝÅÆÆO,eaª½ß

�¡à

H �£¢#Ö×Ø`à�#Ö×Øèa

»i'jEpNÅÆÆO,àH{àq,NÅÆÆO,eaFmÙmÛ·Ü}~�

¡ea·}~�©EÝÅÆÆO,à�ÞàF,ÝÅÆÆO,eaª½

(65)

with

r� ≤ (𝑇2 + 𝛼) · 𝑉#/ (66)

4) The number of vehicles needed in the 𝑖#�ring is:

𝑁",#/ = 𝑁",0/ =�paC��aCq

�B

(67)

37

5) The time needed to finish the first ring is:

𝑇�G =:B;��

#&È¾#àm#&È¾#ea�𝑚𝑎𝑥 Ep𝑇$%%2,#/ − 𝜏#/q, 𝑇$%%2,0/F + 𝛼 +

å·1}~�

bea·i'jEpNÅÆÆO,àH{àq,NÅÆÆO,eaF� − 𝛼 (68)

6) The time needed to start the ith ring (1 ≥ i ≥ K) is:

𝑇"� = 𝑇"HG� + �C��b a

(69)

7) The time needed to finish the ith ring (1 ≥ i ≥ K) is:

𝑇"� = 𝑇"HG� + 2𝜔𝛿𝐿𝑖#𝑐𝑢𝑠𝑡𝑡𝑟+#𝑐𝑢𝑠𝑡𝑑𝑟

�𝑚𝑎𝑥 Ep𝑇𝑙𝑜𝑜𝑝,𝑡𝑟 − 𝜏𝑡𝑟q, 𝑇𝑙𝑜𝑜𝑝,𝑑𝑟F + 𝛼 +𝛽·𝐷𝑚𝑎𝑥

𝑉𝑑𝑟·𝑚𝑎𝑥Ep𝑇𝑙𝑜𝑜𝑝,𝑡𝑟−𝜏𝑡𝑟q,𝑇𝑙𝑜𝑜𝑝,𝑑𝑟F� − 𝛼 (70)

8) The number of late packages for each truck and drone serving a zone in the ith ring is:

• If the ring is in state a (Tp ≤ T�� i.e. the ring cannot be started):

𝑝"' = 𝑝0/,"' = 2𝐿"𝜔"𝛿 · :�E∑ �QC��

Q�� m��CF

:BC (71)


𝑝"� =

⎝

⎜⎛ (NC

�HN~ mÙ)·(#𝑐𝑢𝑠𝑡𝑡𝑟+#𝑐𝑢𝑠𝑡𝑑𝑟)

2𝜔𝛿𝐿𝑖#𝑐𝑢𝑠𝑡𝑡𝑟+#𝑐𝑢𝑠𝑡𝑑𝑟

»𝑚𝑎𝑥Ep𝑇𝑙𝑜𝑜𝑝,𝑡𝑟−𝜏𝑡𝑟q,𝑇𝑙𝑜𝑜𝑝,𝑑𝑟F+𝛼+𝛽·𝐷𝑚𝑎𝑥

𝑉𝑑𝑟·𝑚𝑎𝑥©E𝑇𝑙𝑜𝑜𝑝,𝑡𝑟−𝜏𝑡𝑟F,𝑇𝑙𝑜𝑜𝑝,𝑑𝑟ª½⎠

⎟⎞· �

:�·∑ (�QC��Q�� m��C)

:BC� (72)

• If the ring is in state c (T�� ≤ Tp ≤ T�� i.e. the ring cannot be finished):

𝑝"& = 0 (73)

9) The amount of lateness is: • If the ring is in state a (Tp ≤ T�� i.e. the ring cannot be started):

𝑑"' = 𝑝"' · ©pNC

®HN~ mNC�HN~ q

:ª (74)


𝑑"� = 𝑝"� · ©N a,C� HN~ :

ª (75)

38

• If the ring is in state c (T�� ≤ Tp) i.e. the ring can be finished):

𝑑"& = 0 (76)

Finally, the total expected number of late deliveries and the total expected amount of lateness

can be obtained by calculating the numerical integral of equations (71) to (76) over the

probability of the available time window P(Tav).

𝑝 = ∑ ©∫ 𝑝"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑝"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª" (77)

𝑑 = ∑ ©∫ 𝑑"' · 𝑃(𝑇'±)𝑑T²³ +∫ 𝑑"� · 𝑃(𝑇'±)𝑑T²³#�´�CN~ �N®C

N®CN~ �µ

ª" (78)

40

4 Results In this section, the empirical results found by the application of the models are presented.

The three delivery concepts are tested and compared with the estimated medium-term vision of

the technical performances of the delivery-drones.

4.1 Input parameters

The numerical values for the inputs parameters were gathered based on the work of Yin

& al (2016), different interviews of UAV systems researchers, as well as different articles on

UAVs ((Pogue, 2016), (Ric, 2015)). They are summarized in the Table 2 below.

VEHICLE INPUTS

Drone Truck

Speed Vdr = 50 mph Vtr = 30 mph

Handling time per

delivery tdr = 0.008 h (0.5min) ttr = 0.025 h (1.5 min)

Parcel loading time a = 0.025 h (1.5 min) -

Battery replacement time b = 0.025 h (1.5 min) -

Range (back and forth) Dmax =10 mi -

Maximum payload Wmax = 10 lbs (4.5kg) -

Avg weight of a parcel 𝑤= 2.5 lbs (1.1kg) -

Launch point - R_min_dr = 0 mi

Veh-based cost 𝐶,,0/ = 2.5 $/veh/day 𝐶,,#/ = 15 $/veh/day

Mile-based cost 𝐶1,0/ = 0.03 $/veh/mi 𝐶1,#/ = 1.5 $/veh/mi

Cost per missed delivery 𝐶2 = 60 $/delivery 𝐶2 = 60 $/delivery

41

Cost per hour of lateness 𝐶$ = 6 $/h 𝐶$ = 6 $/h

SERVICE AREA INPUTS

Customer density ∂ = 0.8 customer/mi2

Radius of the service area R = 30 mi

Offset offset = 1

Table 2: Input parameters

4.1.1 Characteristics of drones

This study aims at assessing the future situation (in a 10 to 15-year vision) of delivery

drones, by estimating the potential evolution of the technical performances of drones. The

current ones were summed up in the Table 1 (section 2.1.2).

Increasing the range and the payload capacity of the drone strongly depends on the

improvements of battery capacity as it defines the amount of power available to generate the

propeller thrust required to counter the force of gravity and move forward (Ric, 2015). The

longer range and heavier payload capacity are wanted, the more powerful and therefore bigger

and heavier the battery must be. To carry this heavier total weight the battery must be more

powerful and so on in a vicious circle. The challenge here is to build batteries that are at the

same time small and powerful. There is for instance some active research on hydrogen fuel

cells, which could offer a cheaper, greener and less noisy alternative to internal combustion

vehicles and a lighter and more autonomous one to batteries (Plaza, 2017). However, the

research must still overcome some challenges such as the inefficiency of producing,

transporting and storing hydrogen and the high costs of the technology (Romm, 2014). Based

on this information, it was decided that in a 15-year-vision, the range could possibly increase

from 6 to 10 mi (round-trip), the payload capacity from 5 to 10 lbs, and the speed from 40 to

50 mph. Moreover, it was assumed that the delivery-UAV will by then be able to carry and

deliver several packages in one single flight. Concerning the weight of the parcels, Amazon’s

vice president Paul Misener claimed that 86% of Amazon’s packages weigh less than 5 lbs

42

(Pogue, 2016). Among those packages, an average weight per parcel of 2.5 lbs was estimated.

This value is quite uncertain and will be tested with a sensitivity analysis in section 5. However,

the important parameter is not really the weight of a parcel or the maximum payload a drone

can carry but rather their ratio, i.e how many parcels a drone can carry simultaneously. For the

first tests of the models, this number is equal to four parcels.

4.1.2 Costs

The delay costs Cp and Cl as well as the operational costs for trucks are the same used by Yin

& al. in their study (2016). The vehicle-based costs include a base salary for the driver, the

initial investment costs for the acquisition of the vehicle as well as the maintenance and

insurance costs. A initial investment cost of 40’000$ with a life-span of 10 years were

estimated. The mile-based costs include the fuel costs and the driver’s salary per mile.

Concerning the drones, the vehicle-based costs were defined based on an estimation made by

Zhiwei & al. (2014). They assumed an initial investment (maintenance included) of 4’000$ for

a life-span of 5 years and no driver costs. The mile-based costs were calculated based on the

work of D’Andrea (2014), in which he calculated the costs related to energy and power for a

potential delivery drone.

4.1.3 Flight delay and time window distribution

To define the type of distribution for the available time window Tav, Yin & al. (2016) gathered

some delay data from the FAA ASPM (Federal Aviation Administration, Aviation Systems

Performance Metrics) individual flight database. They analyzed the flight delay of two leading

cargo operators, FedEx and UPS at three important airports (Atlanta, Newark and Orlando) and

collected the 2012 FedEx overnight flights and the 2013 UPS flights at those airports. From

Figure 11, it is observed that most flights do not experience any delay. For the delayed flights,

their number decreases with the amount of delay. To simplify the translation of the flight delay

distribution to the available time window distribution, Yin & al. assumed a one-to-one

correspondence between the flight delay and the time window change for the last-mile delivery

segment. This assumption was maintained for the new models. They choose to approximate the

trend by a triangular distribution with a mode of 0. They accepted the fact that this

approximation works well for all flight delays expect the cases with a maximum time window

43

(no flight delay). This leads to time window distributions such as depicted in Figure 12. Each

distribution is defined by its minimum available time window and the maximum available time

window is fixed and equal to three hours. When the available time window is equal to three

hours (the maximum), it means that there is no delay in the system. On the contrary, if the

available time window is equal to zero hours, it implies that the flight delay is so important that

the time left for trucks to deliver the parcels on time is zero: all packages will be late. Based on

this information, it means that the black distribution in Figure 12 (with a minimum available

time window of zero) is the one suffering the most delay whereas the blue distribution (with a

minimum available time window of two) is the one suffering the least delay.

Figure 11: Flight delay distribution at destination airports (FFA ASPM)

Figure 12: Triangular distribution of available time windows (Yin, Hansen, & Shen, 2016)

44

4.2 Numerical results

In this section, the numerical results of the simulations with the input parameters

described in section 4.1 are presented.

4.2.1 Optimal planned time window

Figure 13 illustrates the evolution of total costs depending on the planned time window (i.e. the

strategic size of the fleet) under different available time window distributions. As it was

explained in section 4.1, the distributions are defined by their minimum available time window.

For each distribution, an optimal planned time window Tp* can be identified (coloured dots in

the Figure 13), as the one minimizing the total delivery costs. The Tp*s and their associated total

costs are gathered in Table 3.

Quite logically, the optimal planned time window increases and the costs decrease with the rise

of the minimum available time window. The less delay the system has to absorb, the larger

delivery time window (i.e. the smallest fleet) it can aims for. For the scenario with the smallest

minimum time window (i.e. highest maximum delay, represented in blue in Figure 13), the

optimal planned time window represents 56% of the no-delay optimal time window for the

models 0 and 1 with an increase factor for total costs of about 3. For model 2, the minimum

optimal planned time window is 52% of the maximum one (no-delay scenario) for a

multiplication of total costs by 3.3.

Figure 13: Total cost vs planned time windows under different available time window

distributions (own representation)

45

Min available time

Model window 0.0 0.75 1.5 2.25 3.00

Model 0 1.67

(2.39)

2.00

(1.16)

2.33

(1.11)

2.75

(0.87)

3.0

(0.77)

Model 1 1.67

(2.22)

2.00

(1.43)

2.42

(1.04)

2.83

(0.87)

3.0

(0.80)

Model 2 1.58

(1.52)

1.83

(0.88)

2.08

(0.65)

2.58

(0.53)

3.0

(0.46)

Table 3: Optimal time window Tp and (associated total costs) under different available time

window distributions (own results)

4.2.2 Optimization of the delivery system

Figure 14 shows the optimal solutions for each distribution of available time window. Whereas

the optimal time window increases with the minimum available time window (Figure 14.a) and

their relationship is rather linear, the total costs, the number of vehicles, the number of packages

and the amount of lateness decrease with the minimum available time window and their

relationship is convex (Figure 14.b to .f).

Figure 14: Optimal solutions of programing and simulation models (own results)

46

In terms of total costs, models 0 and 1 perform similarly whereas the model 2 optimizes the

system drastically, reducing the total costs of about 3’000 to 7’500 $/day under every

distribution of available time window.

To understand this difference, it is interesting to look at Figure 15, which shows the distribution

of costs between trucks and drones. With the second concept, the drones are responsible for up

to 30% of total costs under the maximum delay scenario (i.e. minimum available time window

equal to 0). This contribution decreases with the delay to reach a share of about 5% with the

no-delay scenario. With concept 1, the share of drones in total costs is close to zero. This

difference between models 1 and 2 is not surprising looking at Figure 14.c and .d. The number

of drones for the model 1 is limited to 12 vehicles and do not change with the distributions of

available time window, whereas in the model 2, the number of drones are between 40 and 110

vehicles. This means that the concept 1 with drones launched from the depot is very unefficient.

The drones do not bring any added value to the system and the number of trucks is roughly the

same as in model 0. The causes of this inefficiency will be investigated with a sensibility

analysis in section 0.

A last interesting information can be observed in Figure 16. The proportion of operational costs

for the models 0 and 1 is much smaller than the one for the model 2. Indeed, the unit operational

costs for trucks are much bigger than the one for drones (15$/veh/day vs. 2,5$/veh/day for the

vehicle-based cost and 1.5$/mi vs. 0,03$/mi for the mile-based costs). The much smaller size

Figure 15: Proportion of costs [%] associated to trucks and drones (own results)

47

of the truck fleet with the model 2 leads to a serious decrease in the opertional costs compared

to the two other models.

Figure 16: Proportion of operational and delay costs (own representation)

48

5 Sensitivity Analysis

This section aims at analyzing the reason why the performance of models 1 and 2 is so

different. Whereas the insertion of drones launched from the trucks (model 2) considerably

decreases the daily delivery costs, the drones launched form the depot do not bring any added-

value to the traditional truck delivery system. The potential impact on total costs of the technical

performances of drones (number of parcels carried simultaneously and range) are tested with a

sensibility analysis. An average delay of 0,5 hours (i.e. a minimum available time window of

1.5h) has been chosen.

5.1 Number of parcels carried simultaneously

More than the maximum payload, the interesting element for drone deliveries is the

number of packages that each vehicle can carry simultaneously, defining the number of

customers it can serve within one loop. In the results obtained in the previous section (section

4) the number was set to a maximum number of 4 parcels per loop (an average payload of 2.5lbs

per parcel for a maximum payload of 10lbs per drone).

The Figure 17 shows that for both models, the total costs do not evolve beyond a maximum

number of carried parcels of five. The same behaviour is observed for the number of trucks and

drones. This means that no matter how many parcels a drone can carry simultaneously, it faces

another limitation that prevents the improvement of the system.

49

Figure 17: Sensibility analysis for the number of parcels carried simultaneously by a drone

within one loop (own results)

5.2 Range of the drone

The other technical component limiting the performance of the drone is probably its

range. In the section 4, the estimated range for drones was set to 10 miles. A potential increase

of the range and its consequences on the total costs and on the size of the fleet are illustrated in

the Figure 18.

Unlike the number of carried parcels, the variation of the range has big consequences on the

total delivery costs, particularly for the model 1. Till a range of 45 miles, the costs decrease of

about 2’500$/day compared to the scenario with a range of 10 miles, before increasing again

and reaching a plateau at 9’400$/day (i.e. a decrease of 1’000$/day). Concerning the trucks,

with a range of 35 miles their number decreases also of about 40% compared to the model 0

50

before reching the plateau of a 25% difference. Concerning the drones, their number increases

as a high rate with the range, reaching a number of 600 drones for the highest ranges.

The model 2 is much less affected by the range improvement. This is due to that fact that the

performance of the truck-drone pair in this model does not depend so much on the range alone

but on the combination of the range oand the number of carried parcels. This assumption is

verified with the results on the Figure 19.

Figure 18: Sensibility analysis for the range of the drones (own results)

The Figure 19 illustrates the minimum delivery costs that can be obtained under each available

time window distribution and with different combinations of range and number of carried

parcels. The comparison between the figures a) and b) and a) and c) shows that an increase of

the range only (doubling to 20 miles) does not improve the system (i.e. decrease the costs) with

the concept 2. If a drone can only deliver 4 parcels at a time, no matter how long it can fly, its

loop will be over after the fourth delivery and it will be constraint to come back to the truck to

be reloaded. On the other hand, a simultaneous increase in the range and the number of carried

parcels (doubling to 20 miles and 10 parcels) does decrease the total costs. For a maximum

51

delay scenario (i.e. a minimum available time window equal to 0), these technical

improvements allow to decrease the total costs of 50% compared to the traditional truck

delivery case against 40% with a range of 10 miles and a number of parcels of 4 items. The

same improvements are observed for a scenario with an average delay of 0.5 hours (i.e. a

minimum available time window equal to 1.5).

As expected for the first model, the technical improvements have a clear impact on the

performance of the system. It finally becomes more profitable than the traditional truck delivery

system (model 0). It is particularly the case when the range is tripled to 30 miles. The daily

delivery costs can be reduced by up to 2’500 $/day. Despite these improvements, the second

model with trucks launched from the trucks always stays more profitable than the first one with

drones launched form the distribution center.

Figure 19: Optimal solutions for the toal costs under different technical performances of drones

(own results)

52

6 Conclusion

This study has shown that the insertion of drones in last-mile delivery systems can decrease

the total delivery costs. However, the optimization of the system depends on several parameters.

First, the delivery concept is of great importance. A combination of trucks and drones with

drones launched from the trucks (concept 2) is easily more profitable than a combination with

drones launched from the distribution center (concept 1). The second system can decrease the

daily delivery costs by up to 40% under specific distributions of the available time window

compared to the traditional delivery system. One of the reason is the high sensitivity to the

technical performances of drones. This is the second important element for the optimization of

the traditional delivery system with trucks. The current technical performances of drones, and

particularly the maximum range, limit drastically the performance of the system with drones

launched form the distribution center. For it to start becoming profitable, the range of the drones

should reach a minimum of 20 miles. Only then, the costs can be decreased by up to 10% under

specific distributions of the available time window. For the second concept with drones

launched from the trucks, only a simultaneous increase in the range and the number of carried

parcels by the drone can lead to an additional improvement of the system.

It is important to keep in mind that this study has several limitations due to its simplifying

assumptions. The most important one is the constant customer density within the delivery

region, which do not represent the real situation. However, the goal of this study was above all

to build the models to estimate the operational and delay costs due to flight delay with a truck-

drone delivery system. A variable customer density should be tested in further research.

Another element that was not taken into account for the feasibility of the truck-drone delivery

system is its dependency to weather conditions. It would be interesting to add a distribution

over time of bad weather in the model to consider the times of critical rain and wind conditions

preventing the use of drones.

53

To finish with, the concept of combined truck-drone deliveries will not happen before the

Federal Aviation Administration changes the regulations concerning the operation of

commercial drones in the United States. Only solid evidence on the security of drones for the

population will unlock the negotiations between the FAA and the cargo companies.

55

7 Reference list Asencio, M., Gros, P., & Patry, J.-J. (2010, August). Les drones tactiques à voilure tournante

dans les engagements contemporains. Recherches & documents.

Associated Press. (2016, April 4). The Guardian. Retrieved from Proposed drone regulation could clear the way for widespread US services: https://www.theguardian.com/technology/2016/apr/04/drone-faa-regulation-delivery-services

Defense Industry Daily. (2007, October). Textron buys UAV makers AAI & Aerosonde. Retrieved from Defense Industry Daily: http://www.defenseindustrydaily.com/textron-buys-uav-makers-aai-aerosonde-03968/

Harris, M. (2016, March). Near misses between drones and airplanes on the rise in US, says FAA. Retrieved from The Guardian: https://www.theguardian.com/technology/2016/mar/25/near-misses-between-drones-and-airplanes-on-the-rise-in-us-finds-faa

Kastrenakes, J. (2017, February). UPS has a delivery truck that can launch drones. Retrieved from The Verge: https://www.theverge.com/2017/2/21/14691062/ups-drone-delivery-truck-test-completed-video

Langevin, A., & Soumis, F. (1989). Design of multiple-vehicle delivery tours satisfying time constraints. Transpn. Res. 23B, 123-138.

Leswing, K. (2015, July). Amazon wants to reserve airspace for a drone highway . Retrieved from Futurism: https://futurism.com/amazon-wants-to-reserve-airspace-for-a-drone-highway/

Macdonald, C. (2017, February 21). UPS tests 'drive by' drone deliveries in Florida to drop packages at doors Read more: http://www.dailymail.co.uk/sciencetech/article-4245826/UPS-tests-drone-deliveries-Florida-eye-cost-cuts.html#ixzz53zm2sRpZ Follow us: @MailOnline on Twitter | DailyMail on Facebook . Retrieved from Daily Mail: http://www.dailymail.co.uk/sciencetech/article-4245826/UPS-tests-drone-deliveries-Florida-eye-cost-cuts.html

Mendoza, M. (2014, September 26). DHL tests drone Parcelcopter to deliver medicine in German island. Retrieved from TECH TIMES: http://www.techtimes.com/articles/16521/20140926/dhl-tests-drone-parcelcopter-to-deliver-medicine-in-german-island.htm

56

Newell Gordon, C. D. (1986a). Design of multiple-vehicle delivery tours-I: a ring radial network. Transpn. Res. 20B, 345-363.

Plaza, J. (2017, June). Will Hydrogen Fuel Cells helps drones stay in the air? Retrieved from Commercial UAV News: https://www.expouav.com/news/latest/hydrogen-fuel-cells-drones/

Pogue, D. (2016, January 18). Exclusive: Amazon reveals details about its crazy drone delivery program. Retrieved from Yahoo Finance: https://finance.yahoo.com/news/exclusive-amazon-reveals-details-about-1343951725436982.html

Romm, J. (2014, August). Tesla Trump Toyota Part II: the big problem with hydrogen fuel cell vehicles. Retrieved from Think Progress: https://thinkprogress.org/tesla-trumps-toyota-part-ii-the-big-problem-with-hydrogen-fuel-cell-vehicles-5917fec22274/

Schroth, F. (2017, February). UPS is testing residential delivery via drone & trucks. Retrieved from Drone Life: https://dronelife.com/2017/02/21/ups-is-testing-residential-delivery-via-drone-trucks/

Stern, J. (2013, December 1). Amazon Prime Air: Delivery by Drones Could Arrive As Early as 2015. Retrieved from abcNEWS: http://abcnews.go.com/Technology/amazon-prime-air-delivery-drones-arrive-early-2015/story?id=21064960

Wikipedia. (2017, September). Fuel cell. Retrieved from https://en.wikipedia.org/wiki/Fuel_cell#Efficiency_of_leading_fuel_cell_types

Yin, M., Hansen, M., & Shen, Z.-J. (2016, July). Estimating the impact of Flight Delay on Cargo Carriers' Ground Distribution Costs. World Conference on Transport Research (WCTR) 2016 Shanghai. Elsevier.

Yin, M., Liu, Y., & Hansen, M. (2014). Evaluating the impact of flight delay on cargo and overnight package delivery firms.

57

8 Appendix

8.1 Code - Python

58

8.1.1 Model 0

Summary

Name file (.py) Inputs Outputs Files used

BASE MODEL

1 Geometry_tr T

Vtr

density

handling

R

offset

R_min

print_on = False

Float

Float

Float

Float

Float

Float

Float

-

nbrings

summary_geom

table_geom

rings

radius

length

radius_minus

radius_avg

Area

density_rings

w_rings

Float

Matrix

Table

Array

Array

Array

Array

Array

Array

Array

Array

2 Nb_veh T

Vtr

density

handling

R

offset

R_min

print_on = False

Float

Float

Float

Float

Float

Float

Float

-

Nk

tot_nb_trucks

Array

Float

Geometry_tr_

3 Total_distance T

Vtr

density

handling

R

offset

R_min

print_on = False

Float

Float

Float

Float

Float

Float

Float

-

distance

tot_distance

Array

Float

Geometry_tr

Nb_veh

4 Op_costs T

Vtr

density

handling

R

Float

Float

Float

Float

Float

op_costs Float Nb_veh

Total_distance

59

offset

costperveh

costpermile

R_min

print_on = False

Float

Float

Float

Float

-

5 Main_op_costs

= SUMMARY of

the main results

from (1) to (4) =

VRP

T

Vtr

density

handling

R

offset

costperveh

costpermile

R_min

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

op_costs

summary_all

table_all

nbrings

tot_nb_trucks

tot_distance

Float

Matrix

Table

Float

Float

Float

Geometry_tr

Nb_veh

Total_distance

Op_costs

6 Nb_late_packages T

Vtr

density

handling

R

offset

R_min

Tav

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

-

TF

TS

late_p

tot_late_packages

Array

Array

Array

Float

Geometry_tr

Nb_veh

7 Lateness T

Vtr

density

handling

R

offset

R_min

Tav

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

-

TF

TS

laten

tot_lateness

Array

Array

Array

Float

Geometry_tr

Nb_veh

8 Delay_costs T

Vtr

density

handling

R

Float

Float

Float

Float

Float

Total_delayed_packages

Total_lateness

delay_costs

total_lateness

total_delayed_packages

Float

Float

Float

Array

Array

Nb_late_packag

es

Lateness

60

offset

R_min

minTime

maxTime

costpermissdelivery

costperminlateness

print_on = False

Float

Float

Float

Float

Float

Float

-

T_av Array

9 Main_base_model

= SUMMARY of

the main results

from (1) to (8) =

VRP

T

Vtr

density

handling

R

offset

R_min

minTime

maxTime

costperveh

costpermile

costpermissdelivery

costperminlateness

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

op_costs

summary_all

table_all

nbrings

tot_nb_trucks

tot_distance


Total_lateness

delay_costs total_lateness

total_delayed_packages

T_av

Float

Matrix

Table

Float

Float

Float

Float

Float

Float

Array

Array

Array

Geometry_tr

Nb_veh

Total_distance

Op_costs

Delay_costs

61

Code

• Geometry_tr.py

62

• Nb_veh.py

• Tot_distance.py

63

• Op_costs.py

• Main_op_costs.py

64

• Nb_late_packages.py

65

• Lateness.py

66

• Delay_costs.py

• Plot_part1_TW.py

70

8.1.2 Model 1

Summary


MODEL 1: TRUCKS + DRONES LAUNCHED FROM DEPOT

11 Geometry_dr T

Vdr

density

handling_dr

R

offset

R_min_dr

Range

payload_avg

payload_max

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

nbrings

summary_geom_dr

table_geom_dr

rings

radius

length

radius_minus

radius_avg

Area

density_rings

w_rings

Float

Matrix

Table

Array

Array

Array

Array

Array

Array

Array

Array

12 Nb_drones T

Vdr

density

handling_dr

R

offset

R_min_dr

Range

payload_avg

payload_max

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

Nk_dr

tot_nb_drones

Array

Float

Geometry_dr_

13 Total_distance_

dr

T

Vdr

density

handling_dr

R

offset

R_min_dr

Range

payload_avg

payload_max

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

distance_dr

tot_distance_dr

Array

Float

Geometry_dr

Nb_drones

71

print_on = False -

14 Op_costs_dr_tr T

Vtr

Vdr

density

handling_tr

handling_dr

R,offset

R_min_dr

Range

payload_avg

payload_max

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

tot_op_costs

op_costs_tr

op_costs_dr

Float

Float

Float

From Model 1 :

Geometry_dr

Nb_drones

Total_distance_dr

From Model 0:

Op_costs

15 Nb_late_packag

es_dr_tr

T

Vtr

Vdr

density

handling_tr

handling_dr

R,offset

R_min_dr

Range

payload_avg

payload_max

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

TF_dr

TS_dr

late_p_dr

tot_late_packages_dr

TF_tr

TS_tr

late_p_tr

tot_late_packages_tr

tot_late_packages

Array

Array

Float

Float

Array

Array

Float

Float

Float

From Model 1 :

Geometry_dr

Nb_drones

From Model 0 :

Nb_late_packages

16 Lateness_tr_dr T

Vtr

Vdr

density

handling_tr

Float

Float

Float

Float

Float

TF_dr

TS_dr

laten_dr

tot_lateness_dr

TF_tr

Array

Array

Float

Float

Array

From Model 1 :

Geometry_dr

Nb_drones

From Model 0 :

72

handling_dr

R,offset

R_min_dr

Range

payload_avg

payload_max

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

TS_tr

laten_tr

tot_lateness_tr tot_lateness

Array

Float

Float

Float

Lateness

17 Delay_costs

= SUMMARY

of the main

results from (15)

and (16)

T

Vtr

Vdr

density

handling_tr

handling_dr

R,offset

R_min_dr

minTime

maxTime

Range

payload_avg

payload_max

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

costpermissdelivery

costperminlateness

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

Total_delayed_packages_

dr

Total_lateness_dr

delay_costs_dr

total_lateness_dr

total_delayed_packages_d

r

Total_delayed_packages_t

r

Total_lateness_tr

delay_costs_tr

total_lateness_tr

total_delayed_packages_tr


Total_lateness delay_costs

T_av

Float

Float

Array

Array

Array

Float

Float

Array

Array

Array

Float

Float

Array

From Model 1 :

Geometry_dr

Lateness_tr_dr

Nb_late_packages_

dr_tr

From Model 0 :

Delay_costs

18 Main_model_1

= SUMMARY

of the main

results from (1)

to (17)

T

Vtr

Vdr

density

handling_tr

handling_dr

R,offset

Float

Float

Float

Float

Float

Float

Float

tot_op_costs

op_costs_dr

op_costs_tr

tot_nb_drones

tot_distance_dr

tot_nb_trucks

tot_distance

Float

Float

Float

Float

Float

Float

Float

From Model 1 :

Geometry_dr

Nb_drones

Total_distance_dr

Op_costs_dr_tr

Delay_costs_dr_tr

From Model 0 :

73

R_min_dr

minTime

maxTime

Range

payload_avg

payload_max

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

costpermissdelivery

costperminlateness

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

delay_costs_dr

delay_costs_tr

delay_costs

Float

Float

Float

Delay_costs

Main_op_costs

10 Plot_model_1

Code

• Geometry_dr.py

75

• Nb_drones.py

• Total_distance_dr.py

76

• Operational_costs_dr_tr.py

77

• Nb_late_packages_dr_tr.py

78

• Lateness_tr_dr.py

79

• Delay_costs_tr_dr.py

81

• Main_model_1.py

83

8.1.3 Model 2

Summary


MODEL 2: TRUCKS + DRONES LAUNCHED FROM TRUCKS

11 Geometry_tr_dr

_2

T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

nbrings

summary_geom

table_geom

rings

radius

length

radius_minus

radius_avg

Area

density_rings

w_rings

T_loop_dr

c_loop_dr

T_loop_tr

c_loop_tr

N_loops

Float

Matrix

Table

Array

Array

Array

Array

Array

Array

Array

Array

Float

Float

Float

Float

Float

12 Nb_veh_2 T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

Nk

tot_nb_trucks

tot_nb_drones

Array

Float

Float

Geometry_tr_dr_2

13 Total_distance_ T Float distance_tr Array Geometry_tr_dr_2

84

2 Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

print_on = False

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

distance_dr

tot_distance_tr,

tot_distance_dr

Array

Float

Float

Nb_veh_2

14 Op_costs_2 T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

tot_op_costs

op_costs_tr

op_costs_dr

Float

Float

Float

Nb_veh_2

Total_distance_2

15 Nb_late_packag

es_2

T

Vtr

Vdr

density

handling_tr

Float

Float

Float

Float

Float

TF_dr

TS_dr

late_p_dr

tot_late_packages_dr TF_tr

TS_tr

Array

Array

Float

Float

Array

Geometry_tr_dr_2

Nb_veh_2

85

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

late_p_tr

tot_late_packages_tr

tot_late_packages

Array

Float

Float

Float

16 Lateness_2 T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

Range

payload_avg

payload_max

battery

loading

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

TF

TS

laten

tot_lateness

tot_lateness_tr

tot_lateness_dr

Array

Array

Array

Float

Float

Float

Geometry_tr_dr_2

Nb_veh_2

17 Delay_costs_2

= SUMMARY

of the main

results from (15)

and (16)

T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

minTime

maxTime

Range

payload_avg

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float


Total_delayed_packages_tr

Total_delayed_packages_dr

Total_lateness

Total_lateness_tr

Total_lateness_dr

T_av

Float

Float

Float

Float

Float

Float

Array

Lateness_2

Nb_late_packages

_2

86

payload_max

battery

loading

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

costpermissdelivery

costperminlateness

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

18 Main_model_2

= SUMMARY

of the main

results from (1)

to (17)

T

Vtr

Vdr

density

handling_tr

handling_dr

R

offset

R_min

minTime

maxTime

Range

payload_avg

payload_max

battery

loading

costperveh_tr

costpermile_tr

costperveh_dr

costpermile_dr

costpermissdelivery

costperminlateness

print_on=Flase

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

-

tot_op_costs

tot_op_costs_tr

tot_op_costs_dr

tot_op_costs

op_costs_tr

op_costs_dr

summary_all

table_all

nbrings

tot_nb_trucks

tot_nb_drones

tot_distance_tr

tot_distance_dr


Total_delayed_packages_tr

Total_delayed_packages_dr

Total_lateness

Total_lateness_tr

Total_lateness_dr

delay_costs

delay_costs_tr

delay_costs_dr

T_av

Float

Float

Float

Float

Float

Float

Matrix

Table

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Float

Array

Geometry_tr_dr_2

Nb_veh_2

Total_distance_2

Op_costs_2

Delay_costs_2

87

Code

• Geometry_tr_dr_2.py

89

• Nb_veh_2.py

90

• Total_distance_2.py

91

• Operational_costs_2.py

92

• Nb_late_packages_2.py

93

• Lateness_2.py

95

• Delay_costs_2.py

96

• Main_model_2.py

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