Strategic Network Design for Parcel Delivery with Drones

Strategic Network Design for Parcel Delivery withDrones under Competition

Gohram Baloch, Fatma GzaraDepartment of Management Sciences, University of Waterloo, ON Canada N2L 3G1

[email protected]

[email protected]

This paper studies the economic desirability of UAV parcel delivery and its effect on e-retailer distribution

network while taking into account technological limitations, government regulations, and customer behavior.

We consider an e-retailer offering multiple same day delivery services including a fast UAV service and

develop a distribution network design formulation under service based competition where the services offered

by the e-retailer not only compete with the stores (convenience, grocery, etc.), but also with each other.

Competition is incorporated using the Multinomial Logit market share model. To solve the resulting nonlinear

mathematical formulation, we develop a novel logic-based Benders decomposition approach. We build a case

based on NYC, carry out extensive numerical testing, and perform sensitivity analyses over delivery charge,

delivery time, government regulations, technological limitations, customer behavior, and market size. The

results show that government regulations, technological limitations, and service charge decisions play a vital

role in the future of UAV delivery.

Key words : UAV; drone; market share models; facility location; logic-based benders decomposition

1. Introduction

Unmanned aerial vehicles (UAVs) or drones have been used in military applications as early as 1916

(Cook 2007). As the technology improved, their applications extended to surveillance and moni-

toring (Maza et al. 2010, Krishnamoorthy et al. 2012), weather research (Darack 2012), delivery of

medical supplies (Wang 2016, Thiels et al. 2015), and emergency response (Adams and Friedland

2011). Yet, when in 2013 Amazon revealed its plan for “Prime Air” service to deliver packages

using UAVs within 30 minutes, it was faced with significant skepticism. The idea that our skies

1

Baloch and Gzara: UAV service with competition2

would be crowded with UAVs sounded like science fiction. While being confident that UAVs will

be as common as delivery trucks in a few years, Amazon’s CEO Jeff Bezos admitted in a 2014

interview with The Telegraph (Quinn 2015) that regulations lag behind and pose a serious obstacle.

Logistics practitioners also stated technology limitations, safety, privacy, and public perception as

major issues that may hinder the use of UAV technology for parcel delivery (Lewis 2014a, Keeney

2016, Wang 2016). Despite these hurdles, Amazon’s announcement started a race among compa-

nies like Google, Walmart, DHL, and Zookal to develop the technology and the logistics strategies

to enable the use of UAVs not only in last mile parcel delivery but also in first mile delivery,

inter- and intra-facility distribution, and delivery to remote and difficult to access regions (Butter

2015, Hovrtek 2018). A remarkable application is that by DHL’s “Parcelcopter 3.0” making 130

successful parcel deliveries in remote areas of Bavaria, Germany in 2017 (Burgess 2017). Recently,

Amazon has successfully delivered its first Prime Air package containing a TV streaming stick and

a bag of popcorn to a customer in UK (Hern 2016). Other successful applications include hybrid

truck-UAV delivery by UPS in Florida, USA (Stewart 2017), and UAV package delivery to islands

by Chinese e-commerce giant, Alibaba (Xinhua 2017). Despite these promising applications, UAV

parcel delivery is not yet a full scale reality. Whether the attractiveness of the technology will

overcome the regulatory and social obstacles is yet to be determined.

Unlike trucks, autonomous UAVs fly without a human pilot, are fast as they do not use congested

road networks (Lewis 2014b, Wang 2016), and are significantly cheaper (Welch 2015, D’Andrea

2014, Hickey 2014, Keeney 2016). Hence they provide a perfect solution for the e-retail industry.

The latter captured 11.7% of the total U.S retail sales in 2016 with a growth rate of 8-12% (Statista

2016, Intelligence 2017). Similar growth is observed globally. For example, in 2016, the Chinese e-

retail market captured 15.5% of the total retail sales with a growth rate of 26.2% (ECN 2017). This

growth is largely due to millennials who embrace online shopping but are ever more sensitive to

delivery time and delivery charge (Hsu 2016). Yet, it is not clear how customer preferences and their

sensitivity to delivery charge and delivery time affects their choice of a fast UAV delivery service


versus traditional in-person shopping. On the other hand, UAVs are limited by package weight,

travel range, and landing area. For example Amazon Prime Air can carry a package weighing up

to 2.5 kg and travels up to 24 km (Keeney 2016). DHL’s Parcelcopter carries a package of up to

2 kg with a travel range of 16 km (Franco 2016). Regulations require that an UAV is monitored

by a certified operator even though UAVs like Prime Air and Parcelcopter are autonomous and

can operate without human intervention. These limitations together with customer preferences are

expected to play a crucial role in determining the future of UAV parcel delivery.

In this paper, we study the economic feasibility of UAV parcel delivery in terms of its impact

on an e-retailer’s distribution network while taking into account customer preferences, locational

decisions, and regulatory and technological limitations. These research questions are of most inter-

est to an e-retailer like Amazon, that already offers a set of delivery services such as Same day

and Prime Now delivery services, and that plans to introduce a new and expedited UAV delivery

service: Prime Air. While UAVs may be integrated in a hybrid truck-UAV delivery system, the

distinctive feature of instant delivery is compromised and a hybrid system may not yield as fast a

delivery as direct drone delivery from the warehouse to the customer location. In order to achieve

short delivery times, direct UAV delivery from the e-retailer facilities is required, which may in turn

require the redesign of the distribution network partly due to the limited flight range of UAVs. On

the other hand, analysis of the top ordered products by Prime Now service reveals that these are

mostly consumer products bought for immediate use and are otherwise available at convenience

and grocery stores (Chronicle 2015). As such, an expedited UAV delivery service does not only

compete with other services offered by the e-retailer but also with physical stores in close proximity

to the customer.

We investigate the questions that an e-retailer faces when deciding whether or not to offer a UAV

parcel delivery service. The decision depends on social, regulatory, and technological challenges

facing UAVs. We incorporate social challenges by modelling the market share captured by UAV

service as a function of customer preferences for the different online services and in-person shopping,


as well as their sensitivity to delivery time and delivery cost. We use the Multinomial Logit (MNL)

market share model (Cooper, Nakanishi, and Eliashberg 1988) where the market share captured by

a service is probabilistic and a function of the utility derived from that service relative to the other

services available in the market. We model utility with five attributes: inherent attractiveness of

the service, travel time, travel cost, delivery charge, and delivery time. If regulation requires human

monitoring of UAVs, their operating cost, and consequently the corresponding delivery charge,

would increase. Furthermore, we incorporate technological limitations through allowing different

types of packages: those that may be delivered by UAVs and those that may not. Landing area

requirements like building type are incorporated in estimating the maximum market share that

UAV service may attract. Finally, the flying range is factored into the design of the distribution

network to determine whether a customer may be offered a UAV service. Ultimately, we model the

following key decisions (1) how many facilities to open and where, (2) which services to offer at an

open facility, and (3) which services to be made available to each customer zone.

The main contributions of this paper are as follows. To the best of our knowledge, this is the first

attempt to pose the above research questions in relation to UAV parcel delivery and its impact on

the e-retail industry, and to develop a quantitative model to answer these questions. The model

is also generic in nature and several possible extensions are proposed in Section 4. We develop a

logic-based Benders decomposition (LBBD) approach to solve the nonlinear mixed integer model

to optimality and within very short time, a few seconds in most cases. The proposed algorithm

is also applicable to existing models in competitive facility location (CFL) literature. Also, our

work is the first to use a multionominal logit market share model in CFLP to locate multiple

facilities with a profit maximization objective, and present an exact solution approach for such

a model. Finally, we construct a new case study based in New York City and perform extensive

numerical testing to analyze the economic feasibility and added value of UAV delivery under

varying levels of technological limitations, regulatory requirements, and customer preferences. The

modelling and analysis presented in this paper may be used not only by e-retailers but by any


retail business to assess the added value of offering UAV delivery. For example, a business concept

under development is to offer a UAV leasing service to local businesses such as pizza restaurants,

pharmacies, convenience stores, etc., who would independently operate UAVs to deliver customer

orders (Luci 2017). It may also be used by regulating bodies to assess the impact of regulations

before putting them in effect. We would like to note that we do not exclude the possibility of

using hybrid UAV-truck delivery as that may still be used for existing e-retailer services and would

only impact the delivery cost and/or delivery charge of these services, which are parameters in our

modelling.

The outline for the paper is as follows. In Section 2, we present the problem statement, develop

the nonlinear mixed integer formulation and the market share model. Section 3 presents a novel

logic-based Benders decomposition approach, derives strong Benders cuts, and details the solution

of the subproblems. Section 4 details several model extensions that could be solved using the

proposed solution approach. In Section 5, we carry out extensive numerical testing using a new

case study based in New York City (NYC), and perform sensitivity analyses over delivery charges,

delivery time, government regulations, technological limitations, customer behavior, and market

size. In Section 6, we show the effectiveness of the proposed Benders algorithm by comparing it to

an equivalent mixed integer formulation that we develop. Finally, concluding remarks and future

research directions are presented in Section 7.

1.1. Related Work

The industry interest in UAVs sparked a similar interest in the research community and led to a

significant increase in research output. Substantial research is ongoing to address technical issues

associated with commercial UAVs such as safety issues including hijacking threats and collision

with nearby obstacles, limited endurance, and payload capacity (Mahony, Kumar, and Corke 2012,

Kahn et al. 2017, Pounds, Bersak, and Dollar 2012, Allen 2005). Studies on the economics of

UAVs are mostly limited to industry reports (Hickey 2014, Keeney 2016, Wang 2016). An excellent

economic analysis is presented by ARK Invest (Keeney 2016) and attempts to determine unit UAV


delivery cost by taking into account facility upgrade costs, operator salary, fuel cost, and UAV

and battery purchase costs. D’Andrea (2014) models UAV fuel cost using energy consumption

functions. In the Management Science/Operations Research literature, Campbell, Sweeney II, and

Zhang (2017) and Carlsson and Song (2017) analyze the economic feasibility of a hybrid truck-

UAV delivery system using continuous approximation models. Murray and Chu (2015) are the first

to address operational challenges associated with truck-UAV delivery and introduce two delivery

systems that are modelled as extensions of the traveling salesman problem (TSP). In one system,

truck deliveries form a tour and UAVs depart from and land on the truck as it makes delivery

stops. In the second system, UAVs make direct deliveries from a distribution center while the truck

makes deliveries to customers that are not within UAV maximum range. Subsequent works by Ha

et al. (2015), Agatz, Bouman, and Schmidt (2018), Poikonen, Wasil, and Golden (2018) consider a

similar problem and propose efficient heuristic approaches for TSP with drones (TSP-D). Ferrandez

et al. (2016) use k-means clustering to find the optimal location of drone launch sites from the

truck and present a genetic algorithm to solve the TSP-D. The overall delivery time and cost for

a hybrid truck-drone network are compared against stand-alone truck and drone systems. Other

researchers extended TSP-D to vehicle routing problem with multiple trucks and drones (Ulmer

and Thomas 2017, Wang, Poikonen, and Golden 2016, Gambella, Naoum-Sawaya, and Ghaddar

2018).

Other than Hong, Kuby, and Murray (2017), researchers are more focused on the operational

challenges associated with drones without addressing key network design questions. Hong, Kuby,

and Murray (2017) develop a maximal coverage location model with a given number of warehouses

and charging stations. The objective is to maximize drone coverage while minimizing the aver-

age network distance between the warehouse and charging stations. However, the location of the

warehouses is fixed and the model only decides on locating charging stations. To the best of our

knowledge, none of the research addresses the economic feasibility of UAV parcel delivery in terms

of its impact on an e-retailer’s distribution network while taking into account customer preferences,

locational decisions, and regulatory and technological limitations.


Studies within the marketing literature primarily focus on investigating customer behaviour. In

this regard, the work by Hsiao (2009) and Schmid, Schmutz, and Axhausen (2016) are relevant in

our context as they attempt to estimate customer sensitivity to travel time, travel cost, delivery

time, and delivery charges based on market surveys. Schmid, Schmutz, and Axhausen (2016) focus

on grocery and electronic goods while Hsiao (2009) focuses on books. Both use market survey

results in MNL models to determine the utility derived by customers from online shopping and

in-store shopping. We use similar market share modelling but within an optimization modelling

framework. The market share model parameters allow an analysis of different customer segments

and how the e-retailer decision to offer UAV delivery changes as a result.

Our work closely relates to competitive facility location (CFL) literature where locational deci-

sions are based on competition in the market. We present a new variant of CFL problem where

there is no competition between an e-retailer’s own facilities but rather the services offered by

an e-retailer are competing against each other and nearby stores. Farahani et al. (2014) provide

a comprehensive review of the existing literature on facility location problems under three main

types of competition: static, dynamic, and foresight. Static competition is when the new entrant

assumes that the attributes of existing competitors do not change following its entrance into the

market. Dynamic competition is when the new entrant makes decisions assuming that competitive

characteristics of existing rivals may change following its entrance into the market. Competition

with foresight is when the rivals (follower) soon join the market once the new entrant (leader)

enters the market. Table 1 classifies/lists papers on CFL according to each type of competition and

compared to our work based on model assumptions, competitive characteristics, strategic decisions,

objective function, and solution approach.

In static competition, nearly all the papers use utility based attraction models (multionomial

logit or multiplicative competitive interaction market share models), also referred to as gravity

models, to capture probabilistic customer behavior. These works consider same service being offered

and players (facilities) in the market compete on facility characteristics such as travel distance,


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travel time, etc. In such cases, the profit maximization objective is equivalent to the market share

maximization objective when facility costs are constrained by a budget. This is because profit

margins are the same for all competing players. Under market share maximization, the objective

function is a non-decreasing concave function, and easy to deal with. However, since we consider

multiple services with different profit margins, profit maximization makes more sense. A profit

maximization objective function is more difficult to deal with because it yields a non-concave

objective function and piece-wise linear approximation techniques (Aboolian, Berman, and Krass

2007a,b) used to solve conventional CFL problems do not apply. We are not aware of any work

other than Fernandez et al. (2007a) that considers a profit maximization objective function with

attraction models. Fernandez et al. (2007a) present exact and heuristic solution approaches to solve

the CFLP for profit maximization. However, the model decides only on locating a single facility

which greatly limits the applicability of the solution methodology in general context of locating

multiple facilities. On the other hand, the literature dealing with competition with foresight and

dynamic competition consider linear demand models for profit maximization problems (Rezapour

and Farahani (2014), Rhim, Ho, and Karmarkar (2003), Meng, Huang, and Cheu (2009), Nasiri

et al. (2018)). In linear models, the market share captured by a player (e.g. facility, firm, or service)

is a linear function of the attributes of all competing players. However, Cooper, Nakanishi, and

Eliashberg (1988) correctly point out that such models do not meet logical consistency requirements.

To the best of our knowledge, this makes our work the first to use a multionominal logit (MNL)

market share model in a competitive facility location problem to locate multiple facilities with a

profit maximization objective. The proposed demand model in this paper is different from the ones

used in the CFL literature in terms of the competitive characteristics. Although, some of these

characteristics are frequently studied in the marketing literature, we are not aware of any work

that incorporates delivery price, delivery time, travel time, travel cost, and package weight in the

demand model.

To solve the challenging nonlinear nonconcave optimization models, we develop an efficient logic-

based Benders decomposition (LBBD) approach and show that it is equally applicable to market


share maximization problems under budget constraint. Logic-based Benders decomposition gener-

alizes the classical Benders decomposition approach by relaxing the linear subproblem requirement

(Hooker 2000, Hooker and Ottosson 2003). In this approach, the original problem is divided into a

master problem and subproblem(s). In the master problem, some decision variables and constraints

are fixed/removed. Optimal solutions of the master problem are used in the subproblem(s) to gener-

ate Benders cuts that are added back to the master problem. This iterative process continues until

an optimal solution is found. Unlike classical Benders decomposition where standardized cuts are

added using dual information of the subproblem, the cuts in LBBD are problem-specific (Hooker

2007). Therefore, while deriving optimality or feasibility cuts, the modeler needs to ensure that the

cuts are strong and may be computed with little computational effort. Logic-based Benders decom-

position has been used in a variety of applications including scheduling (Jain and Grossmann 2001),

network design (Garg and Smith 2008), and location (Fazel-Zarandi and Beck 2012, Fazel-Zarandi,

Berman, and Beck 2013, Wheatley, Gzara, and Jewkes 2015). Within location problems, LBBD

is applied to facility location and vehicle assignment problems Fazel-Zarandi and Beck (2012),

Fazel-Zarandi, Berman, and Beck (2013), and location-inventory problems Wheatley, Gzara, and

Jewkes (2015). Unlike the literature on location problems using logic-based Benders, we define

LBBD cuts using location decision variables as opposed to assignment variables which significantly

improves the computational efficiency by reducing the number of binary decision variables in the

master problem. A novel approach is also proposed to compute stronger cut coefficients with little

computational effort using the properties of MNL model. To the best of our knowledge, our work

is the first to use LBBD in a competitive facility location (CFL) problem.

2. Problem Definition

Consider an e-retailer that offers a set of same day delivery services {1, ..., n−1} and plans to offer

a new UAV service n. Let S = {1, ..., n} be the set of all e-retailer services. Amazon, for example

has 2-hour and 12-hour same day delivery services and plans on offering 30-minute UAV delivery

service. The e-retailer competes with existing stores e.g., retail, convenience, and department,


that offer in-person shopping service n + 1. The e-retailer wants to decide on optimal network

configuration by opening facilities from a set of discrete candidate locations J offering same services

in S = {1, ..., n}. We assume that the network is designed from scratch which allows us to present

a comparative study between networks with and without UAVs to investigate how offering a UAV

service affects the network design. Later in Section 4.3, we show how a UAV service could be added

to an existing network. Opening a facility at location j ∈ J incurs a fixed cost Lj. An additional

fixed service cost Fs is incurred for offering service s ∈ S. We assume that facilities have ample

capacity to service all assigned demand regions. The capacity limitation is dealt with through

stocking and replenishment decisions.

Customer demand originates from a set of finite customer zones I with two types of packages P =

{0,1} where p= 0 denotes packages that are not deliverable by UAV and p= 1 refers to packages

that may be delivered by UAVs. A package is defined as a bundle of products a customer buys in

one order. A package cannot be delivered by a UAV due to two reasons: its weight exceeds UAV

weight limit or landing at the customer location is not possible. A binary parameter asp is calculated

apriori which indicates whether service s ∈ S can deliver package p ∈ P . As per FAA regulations,

UAV weight including the package must not exceed 55 lbs. This regulation is incorporated within

parameter asp which equals zero for the packages that exceeds the weight limit. Another binary

parameter, rijs, indicates whether delivery to customer zone i∈ I from facility j ∈ J using service

s ∈ S is possible, and is calculated based on the distance metric (Euclidean or Manhattan) being

used. The parameter rijs takes into account maximum delivery range of services as well as other

regulatory limitations such as airspace restrictions and dedicated paths. Current FAA regulations

prohibit UAVs to fly over people and require the flight path to be limited within class G airspace.

As such, UAVs are required to fly in dedicated airspace, for instance, flying over the road network.

In our modelling approach, these restrictions only impact the reachibility which is captured by rijs.

However, the regulation to keep UAV within visual line-of-sight must be relaxed by the government

for commercial use of UAVs. Until then, UAVs cannot operate. FAA regulations also require that


an UAV is monitored by a certified operator and must fly under 100 mph. Hiring certified operators

increases operator cost while flying speed affects the number of UAVs an e-retailer has to purchase.

As such, these regulations impact UAV delivery cost and are incorporated within the unit delivery

cost parameter cijs. Refer to Section 5.1 for detailed calculations.

Three sets of binary decision variables (wj, xjs, and yijs) are defined where wj takes value 1

when facility j ∈ J is open and xjs takes value 1 when service s ∈ S is offered at location j ∈ J

while yijs equals 1 if customer zone i ∈ I is assigned facility j for service s ∈ S. We also define

two sets of continuous decision variables Disp and dijsp, where Disp denotes the demand captured

by service s ∈ S for package p ∈ P in zone i ∈ I and dijsp is the portion serviced by facility j ∈ J .

The demand Disp, captured by a service depends on the other competing services available to the

customers. The competition between the services is modelled using a multinominal logit (MNL)

model detailed next.

2.1. Market share model

We use a multinomial logit (MNL) model to predict the demand captured by each service in a

competitive environment. Customer zone i ∈ I has a maximum market size Nip for package p ∈ P

which is distributed between the services in S0 = S∪{n+ 1} which compete on five distinct factors:

(1) inherent attractiveness, β0s (2) travel time, TTi, (3) travel cost, TCi, (4) delivery charge, qs, and

(5) delivery time DTs. To incorporate social resistance against UAVs, we may set β0n <β0s, ∀s ∈

S0\{n}, i.e., the inherent attractiveness of UAV service is, possibly significantly, less than that of

other services.

We assume that there is no competition between the stores and customers visit their nearest

store. It is further assumed that there is static competition between services offered by the e-retailer

and stores, i.e., the characteristics of the services offered will not change once delivery by UAV

service is made available.

The utility function of customer zone i∈ I for package p∈ P is:

Uip =∑s∈S

((∑j∈J

aspyijs

)exp(β0s−βdtDTs−βdcqs)

)+ exp(β0,n+1−βttTTi−βtcTCi) (2.1)


where βtt, βtc, βdt, and βdc are sensitivity parameters to travel time, travel cost, delivery time, and

delivery charges, respectively. The first expression on the left of (2.1) is the utility captured by

the services offered by the e-retailer. The second expression is the utility captured by stores where

customer i ∈ I visits the nearest store. The market share captured by service s ∈ S in customer

zone i∈ I for package p∈ P is:

MSisp =

(∑j∈J

aspyijs


Uip(2.2)

The standard market share model assumes that market size is perfectly inelastic which limits

the applicability of the model to capture market expansion or shrinkage. When more services are

available to the customers, the probability of lost sales decreases. As a result, the overall market

size increases. Similarly, a proportion of the market is lost since not all competing services in the

market are included in the model. Hence, we use an exponential expenditure function (Berman

and Krass 2002) to determine the proportion of the maximum market size Nip that is captured by

all services offered by the e-retailer and the store. The expenditure function is

g(Uip) = 1− exp(−λUip) (2.3)

and market size is Nip×g(Uip). Parameter λ represents the elasticity of market size with respect to

total utility Uip. When elasticity λ→∞, g(Uip)→ 1, and the maximum market size is fully captured.

When λ is low, the market size is small. Basuroy and Nguyen (1998) suggest a conceptually similar

expenditure function to estimate market size as MSZ0(Uip)θ, where MSZ0 is the base market size

and 0 ≤ θ < 1 reflects the size of market expansion with respect to utility. This function is also

applicable to our modelling approach and solution methodology. The demand captured by service

s∈ S in customer zone i for package p is Disp =Nip× g(Uip)×MSisp, or

Disp =

Nip (1− exp(1−λUip))

(∑j∈J

aspyijs


∑s∈S

((∑j∈J

aspyijs


)+ exp(β0,n+1−βttTTi−βtcTCi)

. (2.4)


Network RepresentationS set of e-retailer services S = {1, . . . , n} where n is the new UAV serviceS0 set of services offered in the market, S0 = S ∪{n+ 1} where n+ 1 is instore serviceJ set of candidate LocationsI set of customer zonesP set of Packages, P = {0,1}asp equals 1 if service s∈ S can deliver package p∈ Prijs equals 1 if zone i∈ I is within the maximum range of service s∈ S from facility j ∈ J

Cost ParametersLj cost of opening facility j ∈ JFs cost of offering service s∈ S at a facilityα profit margin (in percentage)πp package valuecijs delivery cost per unit to zone i∈ I from facility j ∈ J using service s∈ Sqs delivery charge for service s∈ S

Market share model ParametersNip maximum market size (in units) in zone i∈ I for package p∈ PDTs delivery time for service s∈ Sqs delivery charge for service s∈ STTi travel time for customers in zone i∈ I to the nearest storeTCi travel Cost for customers in zone i∈ I to the nearest storeβ0s inherent attractiveness of service s∈ Sβdt delivery time (in hours) senstivity parameterβdc delivery charge (in dollars) senstivity parameterβtt travel time (in hours) senstivity parameterβtc travel cost (in dollars) senstivity parameterg(Uip) expenditure functionλ elasticity of market size w.r.t to Uip

Decision variableswj equals 1 if facility j ∈ J is open.xjs equals 1 if service s∈ S is offered at facility j ∈ J .yijs equals 1 if zone i∈ I is assigned to facility j ∈ J for service s∈ SUip utility function of zone i∈ I for package p∈ PDisp Demand captured by service s∈ S for package p∈ P in zone i∈ Idijsp Portion of Disp serviced by facility j ∈ J

Table 2 Model parameters and decision variables

The expressions of Disp = f(yi11, ..., yijn+1) and Uip = u(yi11, ..., yijn+1) are functions of the deci-

sion variable yijs. In fact, Disp and Uip are decision variables, and equations (2.1) and (2.4) are

constraints to e-retailer’s problem detailed next.

2.2. Mathematical formulation

In this section, we formally define the complete mathematical formulation for e-retailer’s problem

[NP] using the modelling parameters and decision variables listed in Table 2. Model [NP] is

[NP]: max∑i∈I

∑j∈J

∑s∈S

∑p∈P

(απp + qs− cijs)dijsp−∑j∈J

∑s∈S

Fsxjs−∑j∈J

Ljwj (2.5)


s.t.∑j∈J

yijs ≤ 1 i∈ I, s∈ S, (2.6)

yijs ≤ rijsxjs i∈ I, j ∈ J, s∈ S, (2.7)

xjs ≤wj j ∈ J, s∈ S (2.8)

dijsp ≤Myijs i∈ I, j ∈ J, s∈ S,p∈ P, (2.9)∑j∈J

dijsp ≤Disp i∈ I, s∈ S,p∈ P, (2.10)

(2.1), (2.4)

yijs ∈ {0,1} i∈ I, j ∈ J, s∈ S, (2.11)

wj ∈ {0,1} j ∈ J, (2.12)

xjs ∈ {0,1} j ∈ J, s∈ S, (2.13)

dijsp, Disp, Uip ≥ 0 i∈ I, j ∈ J, s∈ S,p∈ P. (2.14)

The objective function (2.5) maximizes the overall profitability of the e-retailer expressed as the

difference between revenues, delivery costs, fixed facility costs, and fixed service costs. Constraint

(2.6) ensures that customer zone i∈ I is served using service s∈ S by only one facility. Service s∈ S

may be offered from facility j ∈ J if delivery to the customer zone i ∈ I is possible i.e., rijs = 1,

and it is available i.e., xjs = 1, as indicated by constraint (2.10). Constraint (2.8) ensures that a

facility j ∈ J can offer services only if it is open. Constraint (2.9) ensures that demand serviced

dijsp for package p ∈ P using service s ∈ S can only be satisfied by facility j ∈ J if the customer

zone i∈ I is assigned to it, where M is a large number to ensure that the constraint is nonbinding

for yijs=1. Constraint (2.10) limits the demand serviced by all facilities∑j∈J

dijsp to customer zone

i∈ I for package p∈ P using service s∈ S to the demand captured by that service Disp. The latter

depends on the utility that the customer zone i∈ I derives from that service relative to the utility

derived from other services and is completely defined when equations (2.1), and (2.4) of the market

share model equations are added as constraints. Constraints (2.11), (2.12), and (2.13) are binary

requirements for variables wj, xjs, and yijs respectively. Constraints (2.14) are the nonnegativity

requirements for variables dijsp,Disp,Uip.

Note that the capacity for UAV deliveries in a given time period is factored in delivery costs, see

Table B3 for detailed calculations of unit UAV delivery cost. At facility j, the yearly demand for

UAV service n is Y Dj =∑i∈I

∑p∈P

dijnp, where dijnp is the solution to model [NP]. The average hourly


demand Hj for UAV service at facility j is calculated as Hj =Y Dj

365×14. Based on average hourly

demand, the number of UAVs required at facility j ∈ J is calculated as NFj =Hj×ss

Φ, where ss is a

safety factor set by the e-retailer to have sufficient additional UAVs, and Φ is the minimum number

of deliveries a UAV can make within one hour. Safety factor ss takes into account fluctuations in

hourly demand and the time associated with different overhead activities such as battery charging,

battery swap, and delivery time under different weather conditions.

Although the safety factor takes into account fluctuations in hourly demand, the demand during

certain days may be much higher. To consider this variability in demand, Hjt is defined as the

demand at facility j on day t ∈ Y. Let Πt be the proportion of the demand on day t ∈ Y. The

average hourly demand Hjt for UAV service at facility j on day t is then calculated as Hjt =Y Dj×Πt

14.

Similarly, the number of UAVs required at facility j ∈ J on day t is calculated as NFjt =Hjt×ss

Φ.

As such, the total number of UAVs required at facility j equals maxt∈Y{NFjt}.

3. A Logic-Based Benders Decomposition Approach

When the services offered at the facilities are known, the problem reduces to assigning each cus-

tomer zone to the nearest open facility offering a given service s ∈ S, and to decide whether this

service s is offered or not. We exploit this feature to decompose [NP] into a location-service mas-

ter problem (LSMP) that makes locational decisions, and a set of |I| customer service-assignment

subproblems [SPi] where the assignment decisions are made. We develop a Location-Assignment

Benders (LAB) algorithm that iterates between the master problem and subproblems to solve the

nonlinear formulation to optimality. First, the master problem LSMP is solved to make locational

decisions:

[LSMP]: max∑i∈I

Zi−∑j∈J

∑s∈S

Fsxjs−∑j∈J

Ljwj (3.1)

s.t.∑j∈J

∑s∈S

∑p∈P

(απp + qs− cijs)dijsp−Zi = 0 i∈ I, (3.2)∑j∈J

dijsp ≤Dmaxisp i∈ I, s∈ S,p∈ P, (3.3)

dijsp ≤ rijsDmaxisp xjs i∈ I, j ∈ J, s∈ S,p∈ P, (3.4)

Baloch and Gzara: UAV service with competition17∑

j∈J

∑s∈S

dijsp ≤ TDmaxip i∈ I, p∈ P, (3.5)

cuts, (3.6)

(2.8), (2.13), (2.12), (2.14),

Zi ≥ 0 i∈ I (3.7)

where decision variable Zi captures the total revenue minus the delivery cost associated with

serving customer zone i ∈ I, as defined by constraint (3.2). The objective function maximizes

the total profit, which is the same as (2.5). Constraints (3.6) are Benders optimality cuts that

are generated each time the subproblem is solved. Deriving optimality cuts from the subproblem

solution is explained in detail in Section 3.2. Constraints (3.3), (3.4) and (3.5) are valid constraints,

and are added to tighten the relaxation. Constraint (3.3) ensures that the demand serviced by all

facilities,∑j∈J

dijsp, does not exceed the maximum demand that can be captured by service s ∈ S,

Dmaxisp . Constraint (3.5) ensures that for package p ∈ P in customer zone i ∈ I, demand serviced∑

j∈J

∑s∈S

dijsp, must not exceed the maximum total demand TDmaxip that e-retailer can capture. By

Lemma 1, Dmaxisp is achieved when only that service is made available to the customer zone i∈ I for

package p∈ P and TDmaxip is achieved when all services in S are offered. These results are proven in

Appendix A and follows from properties of the MNL models (Cooper, Nakanishi, and Eliashberg

1988).

Lemma 1 For customer zone i∈ I and package p∈ P ,

1. the maximum demand that service s ∈ S may capture, denoted by Dmaxisp , is achieved when

only that service is made available; and

2. the maximum total demand that the e-retailer may capture, denoted by TDmaxip , is achieved

when all services s∈ S are made available.

3.1. Customer Service-Assignment Subproblems

One of the advantages of logic-based Benders decomposition approach is that the subproblem can

take any form including an optimization problem or a feasibility problem (Jain and Grossmann

2001, Hooker 2005, Fazel-Zarandi and Beck 2012, Fazel-Zarandi, Berman, and Beck 2013, Wheatley,


Gzara, and Jewkes 2015, Roshanaei et al. 2017). In contrast to the methodologies in literature,

the subproblem in our case is a nonlinear optimization problem. When the location and service

decisions are known, [NP] reduces to |I| customer-service assignment subproblems [SPi]:

[SPi] : max∑j∈J

∑s∈S

(απp + qs− cijs)dijsp (3.8)

s.t. yijs ≤ rijsxjs, j ∈ J, s∈ S, (3.9)

(2.7)− (2.9), (2.11), (2.14), (2.1), (2.4)

where xjs is [LSMP] solution. Subproblem [SPi] finds an optimal assignment of customer zone i∈ I

to offered services at open facilities. Such an assignment depends on the demand values determined

by constraints (2.1), and (2.4). Since the latter are nonlinear, [SPi] remains challenging to solve.

We explain two main characteristics of [SPi] and develop a fast enumeration based algorithm

to solve industry-scale instances. We also present special cases where the resulting optimization

problem has a non-decreasing concave objective function that can be solved without enumerating

over all scenarios. When xjs and wj are known, customer zone i ∈ I is assigned to the nearest

open facility offering that service to minimize delivery costs, cijs. If rijsxjs = 0 ∀j ∈ J , customer

zone i ∈ I is assigned to a dummy facility with sufficiently large penalty to ensure that service

s ∈ S is not offered. When all customer zones are assigned, we need to decide on the optimal set

of services to be made available to each zone which is a binary non-convex nonlinear optimization

problem. We use an enumeration based approach to solve the problem. Note that the decomposition

reduces the original problem to a point where only service decisions need to be optimized for each

customer zone separately. Moreover, the services offered by an e-retailer are limited in practice.

For instance, Amazon offers a total of six main delivery services (Prime Now (2-hour), Same-Day

Delivery, One-Day Delivery, Release-Date Delivery, Free Shipping, and Amazon Locker) and not

all of these services are offered at all customer locations (Amazon 2018). As such, the enumeration

based approach performs extremely well for industry-scale instances.

Let P(S) be the power set of S. For zone i ∈ I and package p ∈ P , the demand captured by

service s∈ S when the set of services e∈P(S) is offered is precalculated as

Dispe =Nip(1− exp(−λpU ipe))MSispe ∀ i∈ I, s∈ S,p∈ P,e∈P(S) (3.10)


where U ipe is the utility of customer zone i∈ I for package p∈ P when the set of services e∈P(S)

is available. MSispe calculates the market share of package p∈ P from customer zone i∈ I captured

by service s∈ S when the set of services available is e∈P(S). The optimal solution of [SPi] is then

SP i = maxe∈P(S)

{∑s∈S

∑p∈P

(απp + qs−MCis)Dispe} ∀i∈ I (3.11)

where MCis denotes minimum cost to deliver a package to zone i ∈ I using service s ∈ S. Note

that Dispe is computed only once at the start of the LAB algorithm and the resulting subproblem

(3.11) is solved in |P(S)| iterations to find the set of services e ∈P(S) to be offered to customer

zone i∈ I which maximizes operating profits SP i.

Under a special case, when the objective function coefficients (απp + qs−MCis) are same for all

services in S and products in P , the resulting subproblem objective is a non-decreasing concave

function and is therefore solvable in polynominal time. If ∃j ∈ J, s∈ S: rijs×xjs = 1, then by Lemma

1∑

j∈J yijs = 1 i.e., service s must be offered to customer zone i by some facility to maximize the

total demand captured. As such, enumerating over all possible set of services is not required and

the optimal solution is to offer each service when ∃j ∈ J, s∈ S: rijs×xjs = 1.

3.2. Logic-based Benders cuts

This paper presents a novel way to define cuts using location-service decision variables and inde-

pendent of the assignment variables. This allows to remove assignment variables from the master

problem, making it extremely efficient to solve repeatedly. The number of binary decision variables

in the master problem reduces from |J |(|S|(|I|+ 1)) to |J |(|S|+ 1) when location-service based

Benders cuts are used instead of assignment-based cuts. The cut coefficients are also tailored to

the location-service decisions variables and are calculated with little computational effort. Previous

work in the facility location literature (Fazel-Zarandi and Beck 2012, Fazel-Zarandi, Berman, and

Beck 2013, Wheatley, Gzara, and Jewkes 2015), however, use assignment variables in defining the

cuts.

Consider a solution (xjs,wj,Zi) obtained from [LSMP]. [LSMP] provides an upper bound to

the original problem, and a lower bound is calculated as∑i∈I

SP i−∑j∈J

∑s∈S

Fsxjs−∑j∈J

Ljwj. Since


market share constraints are dropped in the master problem [LSMP] and are replaced by an upper

bound on Disp, at any given iteration, Zi ≥ SP i. When the operating profits Zi, in [LSMP] equal

the operating profits calculated in subproblem SP i, ∀ i ∈ I, an optimal solution is reached. At a

given iteration k, let Ok = {j ∈ J, s ∈ S : xjs = 1}. If ∃i ∈ I : Zi > SP i, we add Benders optimality

cuts to the master problem [LSMP]. A valid Benders cut is defined by Chu and Xia (2004) as any

logical expression that eliminates the current master solution (x,w,Z) if it is not feasible to the

original problem [NP], and it must not eliminate any solution that is feasible to the original problem

[NP]. At each iteration, either optimality or feasibility cuts are added to the master problem. When

the subproblem is feasible, optimality cuts are added to improve the lower bound (Roshanaei et al.

2017). If the subproblem is infeasible, feasibility cuts are added to the master problem (Jain and

Grossmann 2001, Hooker 2005, Fazel-Zarandi and Beck 2012, Wheatley, Gzara, and Jewkes 2015,

Fazel-Zarandi, Berman, and Beck 2013). In our problem, the subproblem is always feasible and

feasibility cuts are therefore not required. We define optimality cut

Zi ≤ SP i +∑j∈Ok

∑s∈Ok

γijs(1−xjs) +∑j /∈Ok

∑s/∈Ok

δijsxjs i∈ I (3.12)

where γijs and δijs are cut coefficients. The effectiveness of cut (3.12) depends on the cut coefficients.

A large value is likely to result in total enumeration. A significant contribution of the paper is in

calculating effective cut coefficients with little computational effort. We present a novel approach

that calculates right cut coefficients in (3.12) without eliminating any solution that is feasible to

the original problem and is computationally efficient.

The cut coefficients are designed to capture the change in operating profits where γijs captures

the minimum possible decrease in SP i when an available service s∈ S at facility j ∈ J (i.e. xjs = 1)

is closed. For a given service s at a facility j, minimum decrease in SP i is achieved when only

xjs = 0. We define γijs =Mmaxijs −Mmax

i where Mmaxijs = SP i when only xjs = 0, and Mmax

i = SP i

when xjs = 1 ∀ j ∈ J, s∈ S. Similarly, δijs captures the maximum possible increase in SP i when a

service s ∈ S that is not offered at facility j ∈ J (i.e. xjs = 0) is opened. δijs =Mijs−Mmini where

Mijs = SP i when only xjs = 1, and Mmini = 0 when xjs = 0 ∀ j ∈ J, s ∈ S. As such, γijs is the


minimum decrease in SP i when only service s at facility j is closed and δijs is the maximum increase

in SP i when only service s at facility j is open. In the same fashion, valid Benders optimality cuts

can also be computed for the market share maximization problem.

If the same set of services are offered in subsequent iterations,∑j∈Ok

∑s∈Ok

(1−xjs) and∑j /∈Ok

∑s/∈Ok

xjs

equal 0, reducing the cut to Zi ≤ SP i which is violated if Zi takes a value greater than SP i. As

such, cut (3.12) ensures that either the current solution is changed or the operating profit is reduced

to SP i, i ∈ I, i.e., it eliminates the current solution if it is infeasible. This proves that the cut

satisfies the first condition. Since the maximum change is considered to calculate cut coefficients,

(3.12) does not remove any feasible solution and is a valid Benders cut.

LAB algorithm is an iterative process that alternates between [LSMP] and [SPi]. At a given

iteration k, an optimal solution (x,w,Z) to [LSMP] is used to solve subproblems [SPi]. If ∀i ∈ I :

Zi = SP i, the solution (w,x,Z) is optimal. This rarely happens in early iterations since Disp in

[LSMP] is overestimated. If ∃i ∈ I :Zi 6= SP i, |I| optimality cuts (3.12), are calculated and added

to [LSMP]. The algorithm stops when Zi = SP i, i ∈ I. The overall iterative algorithm (LAB) is

shown in Figure 1.


Initialize k= 1

Compute

Dispe, γijs, δijs

Solve LSMP

[SP1] [SP2] [SP|I|]

xjs optimal

∀ i∈ I :Zi = SP i?

SP i

Zi

Stop, optimal

solution found

yes

Add Opti-

mality cuts to

LSMP, k++

no

Figure 1 Location-Assignment Benders Algorithm (LAB)


Algorithm 1 Pseudo code for LAB Algorithm

Require: Benders cut coefficients γijs, δijs and demand values Dispe

Initialization1: k ← 02: Z ← ∞3: SP ← 0

Main Loop4: while

∑i∈I

Zi 6=∑i∈I

SP i do

5: Solve [LSMP] . obtain solution (x,Z)6: Z ← Z7: x ← x8: for customer zone i∈ I do9: for service s∈ S do

10: MCis ← ∞11: for facility j ∈ J do12: if rijs×xjs = 1 & cijs <MCis then13: MCis ← cijs . Assigns zone to the nearest open facility offering service s14: end if15: end for16: end for17: Solve [SPi], . obtain (SP)18: SP i ← SPi19: Derive the optimality cut and add to [LSMP]20: end for21: k ← k+ 122: end while


4. Model Extensions

In this section, we discuss extensions to account for market share maximization objective, facility

costs with economies of scale, and redesign of an existing network to add UAV service. We spec-

ify the changes in the modelling and explain when and how the solution method is modified to

accommodate the extensions.

4.1. Market Share Maximization Problem

In competitive facility location problems (CFLP), the problem is often modelled as market share

maximization under budget constraint. The e-retailer’s market share maximization problem [MS]

can easily be modelled as

[MS]: max∑i∈I

∑j∈J

∑s∈S

∑p∈P

dijsp (4.1)

s.t. (2.1), (2.4), (2.6)− (2.14),∑j∈J

∑s∈S

Fsxjs +∑j∈J

Ljwj ≤B, (4.2)

where (4.2) is the budget constraint ensuring that total fixed facility and service costs do not

exceed the available budget B. Model [LSMP] is modified by defining Zi =∑j∈J

∑s∈S

∑p∈P

dijsp as the

total demand captured from customer zone i and the objective is to maximize∑i∈I

Zi under budget

constraint (4.2). For the market share maximization problem [MS], if ∃j ∈ J, s ∈ S: rijs × xjs =

1, then by Lemma 1∑

j∈J yijs = 1 i.e., service s must be offered to customer zone i by some

facility to maximize the total demand captured. Disp is then simply calculated using Equation

(2.4) and subproblem solution is SP i =∑s∈S

∑p∈P

Disp. Note that under market share maximization,

the subproblem is solvable in polynomial time without a need for enumeration.

4.2. Facility costs with economies of scale

Model [NP] assumes a fixed cost Lj for opening a facility, and a fixed cost Fs for offering service s.

This assumes that the fixed costs are independent of the number of services offered. Often times,

substantial cost savings might be achieved through economies of scale by offering multiple services

at the same location. Economies of scale may be captured by defining fixed facility cost as a piece-

wise linear function of the number of services offered. We define a new decision variable ηj that


counts the number of services offered at facility j. The facility fixed cost Lj = fj(ηj) is then a

nonlinear concave function. Let T = {0,1, ..., |S|} be the set of possible values that ηj can take. The

nonlinear function may be linearized by defining SOS1 variables σtj ∀ t ∈ T, j ∈ J . A parameter

btj = t ∀t ∈ T , is also defined representing breakpoints of the number of services. The modified

model with economies of scale [NPE] is as follows.

[NPE]: max∑i∈I

∑j∈J

∑s∈S

∑p∈P

(απp + qs− cijs)dijsp−∑j∈J

∑s∈S

Fsxjs−∑j∈J

∑t∈T

fj(btj)σtj (4.3)

s.t. (2.1), (2.4), (2.5)− (2.7), (2.9)− (2.13), (2.14),

ηj =∑s∈S

xjs j ∈ J, (4.4)

ηj =∑t∈T

btjσtj j ∈ J, (4.5)∑t∈T

σtj = 1 j ∈ J, (4.6)

σtj ≥ 0, σtj→ SOS1 t∈ T, j ∈ J, (4.7)

where Constraint (4.4) counts the number of services offered at facility j ∈ J . Constraints (4.5)

and (4.6) defines ηj as convex combination of two consecutive breakpoints of the number of services

and∑

t∈T fj(btj)σkj is a convex combination of two breakpoint service costs.

4.3. Facility location and relocation Problem

So far, we consider the design of a network from scratch. However, when a network already exists

the question may be whether a redesign is necessary. We now show how to modify model [NP]

to allow for possibly closing existing facilities and opening new ones. We partition the candidate

locations into two sets J = JE ∪JN , where JE be the set of existing facilities, JN be the set of new

candidate facilities. Let Sj be the set of services that are already offered at the existing facility

j ∈ JE. The objective function is then modified as

max∑i∈I

∑j∈J

∑s∈S

∑p∈P

(απp + qs− cijs)dijsp−∑j∈JN

∑s∈S

FNs xjs−

∑j∈JN

LNj wj (4.8)

−∑j∈JE

LEj wj −∑j∈JE

CLj (1−wj)−∑j∈JE

∑s∈S

FEjsxjs−

∑j∈JE

∑s∈Sj

CSEs (1−xjs)

where FNs is the fixed cost of offering service s ∈ S at a new facility, LNj cost of opening the new

facility j ∈ JN , LEj is the fixed cost of operating an existing facility, CLj is the cost of closing an


existing facility j ∈ JE, FEjs is the fixed service cost (could be a new or an existing service) at the

existing facility j ∈ JE, and CSEs are the costs associated with closing service s∈ Sj at an existing

facility. The above objective function takes into account two important factors while deciding on

relocation of the facilities, first, it considers the cost of closing a facility or a service, and secondly,

if the facility is not closed, it would incur fixed cost of LEj to operate it and is expected to be less

than LNj since there are no initial setup costs. Note that fixed service costs FEjs depends on whether

the service is already being offered at the facility j ∈ JE or it has to be added.

5. The case of NYC

An actual network, shown in Figure 2, is constructed for New York City (NYC), the most populated

city in the USA with a population of around 8.5 million, and is spread over a land area of 789

km2 (NYC 2016). There are several reasons that motivate selecting NYC for the case study. First,

the world’s largest e-retail company, Amazon.com, first started its 2-hour delivery operations in

NYC. UAV service shares similar characteristics with 2-hour delivery service to make instantaneous

deliveries. Selecting a city where such a service is already offered allows for investigating the effects

of introducing UAV delivery service. Second, NYC’s land area allows the possibility of multiple

facilities to open. NYC also poses challenges because of high rises. There is ongoing research as

how apartment buildings should be designed or updated to accommodate UAV deliveries. One

such concept is “DragonFly” by UK-based industrial design studio where packages are dropped on

landing pads installed on the sides of buildings allowing direct delivery to customer apartments

(PriestmanGoode 2018). Our study helps answer the question of whether advancing the technology

to allow delivery to tall buildings is worth it and how this impacts the overall design of the network.

Finally, availability of data online was a major reason in using NYC and Amazon in the case study.

Data available on Amazon.com is used to estimate market share model parameters and other cost

figures. Section 5.1 details the data used.

5.1. Data used

Physical network and facility costs NYC consists of five boroughs that are divided into

Neighborhood Tabulation Areas (NTAs) as shown in Figure 2 (NTA 2015). The centroid of each


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Man

hattan

Bronx

Stat

en Is

land

Quee

ns

Brooklyn

Legend

●

Facility

Stores

25000

50000

75000

100000

125000Population

Figure 2 New York City Network

NTA is used as a customer zone. To locate competitive stores (retail, grocery and departmental),

we use Google Earth software to find their exact locations. Moreover, 20 candidate facility locations

are selected such that they are evenly distributed over the network. In practice, there already exists

a network with open facilities. For the sake of simplicity, we assume that the network is designed

from scratch and no such facilities are open. However, we can easily design a UAV network given

an existing network by forcing wj = 1 for already open facilities in the model. Each facility is

assumed to be 50,000 sq.ft based on the fact that Amazon has 50,000 sq.ft distribution center

in Manhattan to offer same day delivery services in NYC. To estimate yearly facility cost Lj,

we consider warehouse lease rates, labor and miscellaneous costs. Yearly labor and miscellaneous

costs for a 50,000 sq.ft facility are estimated to be $1.5 million based on the study conducted

by Boyd Company (Boyd 2014). To calculate the lease rate of a facility j ∈ J located in a given

borough b, we use Jll (2015) report and online leasing website LoopNet (LoopNet 2017) to estimate

minimum (minb) and maximum (maxb) lease rates ($/sq.ft) in each borough b. Yearly facility cost


Lj = 50,000 ∗U(minb,maxb) + 1,500,000, where U(minb,maxb) randomly generates a lease rate

for a facility in a given borough b. Each facility can offer three types of services S = {1,2,3}. The

maximum range of 2-hour (s= 1), 12-hour (s= 2), and UAV (s= 3) services are assumed to be

20 km, 40 km, and 10 km, respectively. Amazon’s Prime Air travels up to 24 km while DHL’s

Parcelcopter has a maximum flying range of 16 km which averages to 20 km. Since the UAV must

return back to the facility, its maximum range is set to 202

= 10 km. For UAV service, we set rij3 = 0

for all customer zones that are within short distance to the airport and is shaded in grey in Figure

2. For 2-hour and 12-hour services, the maximum ranges are estimated by calculating the distance

between Amazon’s current distribution center in Manhattan and the farthest location (based on zip

code) where each service is available. It turns out that 12-hour service is available to all boroughs

in New York City while 2-hour service is not offered in Staten Island. The distances between the

nodes are calculated using geosphere package in R (Hijmans 2017). The yearly additional cost of

offering service s= 1 or s= 2 at a facility, Fs = $250,000. Yearly cost of offering UAV service s= 3

is estimated to be $1,000,000 based on the ARK Invest industry report (Keeney 2016).

Package delivery costs and service charges Unit delivery costs cijs for each service s ∈ S

are presented in Table 3. To estimate UAV delivery cost cij3, we use a similar methodology as

presented by ARK Invest (Keeney 2016). Detailed calculations are shown in Table C3, where the

amortized UAV and battery costs are used along with operator salary cost to estimate cost per

delivery. To do so, we estimate the number of UAVs and batteries an e-retailer has to purchase to

meet its yearly demand YD. The maximum hourly demand Hmax is the average hourly demand

multiplied by a safety factor ss. The latter accounts for fluctuations in hourly demand, the time

associated with overhead activities such as battery charging and battery swap, and fluctuations in

delivery time because of weather conditions. The value of the safety factor is estimated based on

two industry reports. In a market survey by Statista (2015), out of four time windows, 48% of the

respondents shop online during the peak-time window. Based on these figures, the ratio of peak

time demand to average demand is estimated as ss = 48%14

= 1.92. Using the data from Albright


(2015) who reports the percentage of hourly online retail sales, we estimate ss= 1.6. We round up

the estimated values and set the safety factor to ss= 2.0 to take into account other operational

fluctuations.

Assuming that a UAV flies at a speed of 40 km/h and has a range of 10 km, it can make atleast

two deliveries per hour and as such, we set the number of UAVs required to Hmax2

. We further

assume that the number of extra batteries required equals the number of UAVs to ensure zero

charging time. Using the number of UAVs and batteries, the purchasing cost PC, is calculated

and amortized over a period of five years (rate, r = 20%) to estimate yearly amortized cost, AC =

r × PC. Several reports suggest that FAA new regulations for delivery by UAVs would require

certified operators to monitor the UAV activity (Wang 2016, Keeney 2016). However, the number

of UAVs ND, an operator can manage simultaneously are highly speculated ranging from 1 to 30.

For the base case scenario, we assume that ND = 10, i.e., 10 UAVs per operator are allowed. To

calculate the number of UAV operators required, we assume that the delivery process is automated

and operators’ role is to monitor flight operations from a central control room in the city. In case of

an emergency, however, an operator flies the UAV manually. An operator can make two deliveries

per hour and as such, the number of operators required per hour then equals Hmax2×ND and is used to

estimate yearly operator salary cost, OC. The total yearly cost, TC =AC+OC, and unit delivery

cost, cij0 = TCYD

+ 0.10 = $2.62, where $0.10 is the battery charging cost per delivery.

For 2-hour service s= 1, Amazon uses its Flex Program where independent drivers are paid $20

per hour to make deliveries (Chuang 2016). We estimate cij1 = $10 if a driver makes 4 deliveries in

a two hour window. For 12-hour service s= 2, cij2 = $6 based on the analysis presented by Wohlsen

(2013). These cost figures are well aligned with Amazon’s delivery charges. For each service s∈ S,

delivery charges qs are assumed to be equal to the unit delivery cost cijs as e-retailers usually do

not earn profits from delivery charges. In fact, Amazon reports revenue earned from delivery to be

less than delivery costs incurred (Statista 2018).


Lj : 50000×U(minb,maxb) + 1,500,000 yearly facility costs

π0 = π1 = 20, α= 0.30 package price and profit margin

f1 = $250,000 2-hour delivery - yearly facility costs

f2 = $250,000 same-day delivery - yearly facility costs

f3 = $1,000,000 30-minute delivery by UAV - yearly facility costs

cij1 = q2 = $10 delivery cost and charges per package using 2-hour service

cij2 = q3 = $6 delivery cost and charges per package using same-day service

cij3 = q1 = $2.62 delivery cost and charges per package using UAV service

r1 : 20km Range of two-hour delivery at a facility

r2 : 40km Range of same day delivery at a facility

r3 : 10km Range of delivery-by-UAV

β04 = 0.00, β01 =−2.00, β02 =−2.00, β03 =−2.22, Inherent attractiveness of the services available.

βtt = 1.4 Travel time (in hours) sensitivity parameter.

βtc = 0.035 Travel cost sensitivity parameter.

βdt = 0.092 Delivery time (in hours) sensitivity parameter.

βdc = 0.34 Delivery charges (in dollars) sensitivity parameter.

λ= 0.5 Demand elasticity parameter, elastic.

Table 3 NYC example input parameters

Market share model parameters As pointed out earlier, for same day delivery, products that

are readily available at convenience and retail stores are frequently ordered. We therefore use US

grocery sales 2015 (Bender 2016) to estimate maximum market size Nip in customer zone i∈ I for

package p∈ P . Grocery sales (in dollars) are converted into units by assuming that average package

value πp = 20, p = {0,1}. The total grocery sales (in units) in a given customer zone are then

estimated based on its population relative to US population. We assume that UAV delivery is not

possible to apartment buildings (i.e., the number of units in the building µ≤ 9, for UAV delivery),

and only 86% of the packages meet UAV weight capacity based on an interview of Amazon’s CEO

Jeff Bezos in 2014 (Quinn 2015). Building size data is retrieved from American Community survey

data (ACS 2015) which presents housing characteristics for all customer zones (NTAs). As such,

in a customer zone i ∈ I, total grocery demand is divided between packages p = 0 and p = 1 to

estimate maximum market size Nip.

Sensitivity parameters of the market share model are usually estimated based on market surveys

or POS data (refer to Cooper, Nakanishi, and Eliashberg (1988)) which does not fall within the

scope of this work. We therefore use the sensitivity parameters as estimated in (Schmid, Schmutz,

and Axhausen 2016). Schmid, Schmutz, and Axhausen (2016) study consumer choice behavior


[1,2]

Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

Legend

Facility

Stores

250005000075000100000125000

Population

(a) Without UAVs

[3]

[1,3]

[1,2,3]Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

(b) With UAVs

Figure 3 NYC optimal network configuration

for online grocery shopping versus in-store. The inherent attractiveness for online shopping β0s =

−2.00 relative to in-store shopping (β04 = 0.0). β0s < 0 indicates negative attraction towards online

shopping. For UAV service, we assume β03 =−2.2 to account for social resistance. Schmid, Schmutz,

and Axhausen (2016) calculate average value of the travel time V OTT = βttβtc

= 40.0$/hr and travel

cost sensitivity βtc = 0.035. As such, travel time sensitivity βtt = V OTT ×βtc = 40.0× 0.035 = 1.4.

The study estimates the value of delivery time V ODT = βdtβdc

= $6.5/day = $0.27/hr. We set βdt =

0.27βdc in the utility function (2.1) and calculate βdc such that the percentage of the grocery market

that is captured by e-retailers equals US online grocery market share. Based on our calculations,

βdc = 0.34 and βdt = 0.092. We use demand elasticity λ= 0.5 so that on average, the expenditure

function (2.3) equals Amazon’s online grocery market share (41%). When estimating βdc and λ

from utility and expenditure functions, we do not include UAV service. This is because the market

share figures and the study conducted by Schmid, Schmutz, and Axhausen (2016) are based on

same day delivery without UAVs.

5.2. Analysis of the base case

To study the effect of UAVs on network design, we solve the NYC instance under two scenarios. In

scenario 1, UAV service is dropped from the model and the resulting optimal network is shown in


●

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● ● ●

10

20

30

40

50

0 2 4 6Nb. of UAV facilities

Rev

enue/

Cos

ts (

in m

illion

s)

●

Net Profits

Operating Profits

Facility costs

(a) Operating Profits vs facility costs

●

●

●●

●● ● ●

0%

25%

50%

75%

100%

0 2 4 6Nb. of UAV facilities

UA

V c

over

age

(%)

●

Area coverage

Population coverage

(b) UAV coverage

Figure 4 Trade-offs between net revenue, costs and coverage

Figure 3a. A single facility is open and it offers both 2-hour and 12-hour services. While 12-hour

service is available to all customer zones, only 95% of the customer zones can use the 2-hour service

as shown by the region in the red circle in Figure 3a. In scenario 2, the model is solved with all three

services. In the optimal design, three facilities are open, each offering UAV service to the customers

within its 10 km radius as depicted by the blue circles in Figure 3b. The facility opened in scenario

1 is no longer optimal for scenario 2. This shows that incorporating a UAV service may require

the relocation of existing facilities. We also observe that under scenario 2, 2-hour service (s= 1) is

made available at two facilities which improves its coverage from 95% to 98%. UAV service may

also improve the coverage of other services. In this example, the coverage of 2-hour service s= 1

improves from 95% to 98%. Offering UAV service increases facility costs by $11.2 million while it

increases operating profit by $27.7 million from $18 million to $45.7 million. Higher facility costs

are compensated by increased operating profits due to the increase in market share captured by

the e-retailer from 3.0 million to 7.6 million packages.

To further investigate the effect of the number of facilities open on the operating profits and

costs and on UAV coverage, we add a constraint to the model to fix the number of UAV facilities


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0%

25%

50%

75%

µ ≤ 2 µ ≤ 4 µ ≤ 9 µ ≤ 20 µ ≤ ∞

UAV building size µ constraint

UA

V p

opula

tion c

over

age

Figure 5 UAV population coverage vs building size µ constraint

open to a specific number and vary it between 0 and 7. Figure 4a plots the operating profits and

costs in function of the number of UAV facilities open and Figure 4b shows UAV coverage. “Area

coverage” denotes the percentage of customer zones that are within 10 km radius of an opened

facility offering UAV service. However, not all customers within 10 km can be served by UAVs due

to technological limitations including building size and package weight. Population coverage takes

these limitations into account and gives the percentage of the population for which UAV delivery is

possible. As expected, facility cost increases linearly with the number of open facilities. Operating

profits, on the other hand, increase at a decreasing rate. The net profit is maximized with three

open UAV facilities covering 75% of the NYC area and 34% of the population. Opening a fourth

facility increases area and population coverage to 84% and 38% respectively, but the increase in

operating profit is not sufficient to cover additional facility costs.

5.3. Technological limitations and network design

Currently, UAV technology is in the development phase and it is hard to predict technological

constraints associated with it. In this paper, we take into account two types of technological

limitations that a UAV delivery will face: parcel weight limit, and building size. Many reports

suggest that a UAV cannot deliver parcels weighing more than 5 lbs (Wang 2016, Rezapour and


Farahani 2014). Consequently, population coverage cannot exceed 86% due to the weight limit.

However, the type of buildings where UAV delivery would be possible is not clear yet. In our

analysis, we assumed that UAV delivery is only possible to buildings with units µ ≤ 9. In this

section, we relax the assumption. Figure 5 is a box-plot of UAV population coverage in all customer

zones under different building size requirements and shows that as the building requirement is

relaxed, the coverage increases. Coverage is maximized when delivery is possible to all building

types and reaches 86%. As the building requirement is relaxed, more facilities are open as illustrated

in Figure 6. When building requirement is µ≤ 4, a densely populated area like Manhattan is not

offered UAV service because 95% of the population live in buildings with 5 or more units. In Staten

Island, 100% of the population live in buildings with 4 units or less. However, Staten Island is not

offered UAV service due to low demand. Brooklyn and Queens are the most favorable boroughs

for UAV delivery due to high population living in buildings with fewer number of units. The above

analysis shows that advancement in technology will play a vital role in shaping the future of UAV

delivery.

5.4. Government regulations

Government regulations are expected to play a vital role in determining the future of UAVs in

last-mile delivery. These regulations may require firms to hire certified operators to monitor UAV

activity leading to higher delivery costs and delivery charges which would in turn reduce customer

attraction towards the UAV service.

5.4.1. Analysis of UAV’s target market The number of operators a firm needs to hire

depends on the number of UAVs, ND, an operator is allowed to monitor simultaneously. In the

base case scenario in Section 5.2, we set ND= 10. However, this value is highly speculative as some

reports suggest only 1 to 2 UAVs per operator (Lewis 2014a) will be allowed while others believe

this value may be as high as 30 (Keeney 2016). To determine the effect of government regulations

on UAV delivery, we solve the model under different values of ND.

Based on our calculations in Table C3, Figure 7a illustrates the relationship between ND and

UAV delivery cost cij3 which decreases at a decreasing rate as ND increases and varies significantly


[1,2]

Man

hatta

n

Bronx

Staten Is

landQ

ueen

sBrooklyn

25000

50000

75000

100000

125000Population

(a) Without UAVs

[1,3]

[1,2,3]

Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

10000

20000

30000

40000

50000

UAVCoverage

(b) µ≤ 2

[1,3]

[1,2,3]

Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

20000

40000

60000

UAVCoverage

(c) µ≤ 4

[3]

[1,3]

[1,2,3]Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

20000

40000

60000

UAVCoverage

(d) µ≤ 9

[3]

[1,3]

[1,2,3]Man

hatta

n

Bronx

Staten Is

landQ

ueen

s

Brooklyn

20000

40000

60000

UAVCoverage

(e) µ≤ 20

[3]

[1,3]

[1,2,3]

[3]

Man

hatta

n

Bronx

Staten Is

land

Que

ens

Brooklyn

30000

60000

90000

UAVCoverage

(f) µ≤∞Figure 6 UAV building size µ constraint

from $24 for ND = 1 to $1 for ND = 30. Since UAV delivery charge q3 is determined by the

delivery cost, ND would impact the demand captured by the UAV service. As such, government

regulations will have significant impact on value-added by UAVs. To study this, we solve the model

by varying market size using elasticity parameter λ between 0.01 to 1.3. under four different values

of ND: 2, 5, 10, and 30. Currently, FAA regulations allow only one UAV per operator which is

too restrictive to make UAVs economically feasible. For ND= 1, UAV service is not offered by the

e-retailer due to low demand and high service costs. However, these regulations are expected to

be relaxed in the future and we therefore vary ND between 2 to 30. Figure 7b plots e-retailer’s

profits against the market size under different government regulations where “No UAV” denotes

e-retailer’s profit in the absence of UAV service. As the market size increases, profits increase due to


0

5

10

15

20

25

0 4 8 12 16 20 24 28Nb. of UAVs per operator, ND

Unit

del

iver

y c

ost

, c

ij3

(a) UAV delivery cost as a function of Nb. of

UAVs per operator

●●●●●●●●●●●●●●●

●●●●●●

● ●● ●

● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0

30

60

90

120

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Market size, λ

Pro

fits

(in

million

s)

●

No UAV

ND = 2

ND = 5

ND = 10

ND = 30

(b) UAVs per operator effect under different

market sizes

Figure 7 Effect of regulations for UAVs on same day delivery market βtt

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

0.000

0.025

0.050

0.075

0.100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Market size, λ

PV

AD

(%)

(a) ND=2

●

●

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●

●

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●●

●●

●●

●●

●●

●●

35.0

37.5

40.0

42.5

45.0

47.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(b) ND=5

●

●

●

●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

100

200

300

400

500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(c) ND=10

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

200

400

600

800

1000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(d) ND=30

Figure 8 Effect of government regulations on PVAD

higher demand captured and profit curve shifts upwards as ND increases. To investigate the value-

added by UAVs, we plot the percentage value-added (PVAD) against market size under different

government regulations as shown in Figure 8. PVAD is calculated as the percentage increase in

profits between scenarios 1(without UAVs) and 2(with UAV).

For ND = 2, PVAD is maximized when the market size is very large as shown in Figure 8a.

For λ ≤ 1.2, PVAD = 0% i.e., UAV service is not even offered due to low demand and high

delivery costs. At λ= 1.3, PVAD is only 0.10%. However, as government regulations are relaxed


0

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

Nb. of

open

faci

liti

es Without UAV serviceNb. UAV facilities

(a) ND=2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(b) ND=5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(c) ND=10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Market size, λ

(d) ND=30

Figure 9 Effect of government regulations on the number of UAVs facilities

i.e., ND = 5,10,30, PVAD increases significantly for all markets as shown in Figures 8b, 8c, 8d.

PVAD is maximized when the market size is small. At λ= 0.08, PVAD = 47%, 504%, and 1094%

for ND= 5, 10, and 30, respectively. To understand this phenomenon, we compare the number of

UAV facilities opened with the number of facilities without UAV service at different market sizes

as shown in Figure 9. Without UAV service, only one facility is open when λ≥ 0.08. With UAV

service, the number of facilities depend on ND, the number of UAVs per operator and on market

size. For ND = 2, UAV service is offered only for large market size at a single facility offering

other services. As ND increases to 5, 10, and 30, more facilities are open to offer UAV service for

smaller market sizes. Hence, PVAD is maximized when the market size is small as UAV service

allows e-retailers to enter these markets which would otherwise not have been possible due to small

market size. The analysis shows that UAV service may allow e-retailers to extend same day delivery

services and offer UAV service in regions with small market size when the regulations are less

restrictive. On the other hand, if regulations are more restrictive, UAV service might be restricted

to densely populated areas where technological limitations and high delivery charges may limit the

added value of UAVs.

5.4.2. Analysis of UAV delivery charges In the base case scenario, we assume that delivery

cost cijs equals service charge qs. However, service charge plays a crucial role in optimizing e-

retailer’s profits and may not always be equal to delivery cost. We vary q3 between −$5 to $18,

customer delivery charge sensitivity βdc from 0.05 to 1.00, and set ND = 2, 5, 10, and 30. Negative


0

100

200

300

−6 −3 0 3 6 9 12 15 18UAV delivery charge, q3

Pro

fits

(in

mil

lion

s)

(a) ND=2

−6 −3 0 3 6 9 12 15 18

UAV delivery charge, q3

0.25

0.50

0.75

1.00

βdc

(b) ND=5

0

100

200

300

−6 −3 0 3 6 9 12 15 18UAV delivery charge, q3

Pro

fits

(in

mil

lion

s)

(c) ND=10

−6 −3 0 3 6 9 12 15 18

UAV delivery charge, q3

0.25

0.50

0.75

1.00

βdc

(d) ND=30

Figure 10 UAV delivery charge analysis under government regulations and customer delivery charge sensitivity

delivery charge, q3 < 0 may be interpreted as a discount offered by the e-retailer for using UAV

service. Figure 10 plots profits against UAV delivery charge q3 at different levels of delivery charge

sensitivity βdc. As βdc increases, the optimal delivery charge q∗3 decreases. When βdc = 0.05, q∗3 =

$18, irrespective of government regulations. On the other hand, if customers are more sensitive,

for instance when βdc = 1.00, government regulations play a crucial role in pricing decision and

UAV’s added value. Recall the profit margin per unit = απp = 0.30× 20.0 = $6.0. When ND = 2,

cij3 = $12.2. For profits to be positive, q∗3 > 12.2− 6.0 = $6.2. Since customers are highly sensitive

to delivery charge (βdc = 1.00), even at lowest possible q3 = $7, demand is not sufficient to cover

fixed costs and as such, e-retailers do not offer UAV service as shown in Figure 10a. Similarly, for


●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ●0

100

200

300

0.00 0.25 0.50 0.75 1.00βdc

Pro

fits

(in m

illion

s) ● Without UAVsWith UAVs

(a) ND=2

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.00 0.25 0.50 0.75 1.00

βdc

(b) ND=5

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.00 0.25 0.50 0.75 1.00

βdc

(c) ND=10

●

●

●

●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.00 0.25 0.50 0.75 1.00

βdc

(d) ND=30

Figure 11 Profits under optimal UAV delivery charge q∗3

●

●

●

●

●

● ● ● ● ● ●0

5

10

15

20

25

0.0 0.1 0.2 0.3 0.4 0.5βdc

PV

AD

(%)

(a) ND=2

●● ● ● ● ● ●

●

●

●

●

200

400

600

0.0 0.1 0.2 0.3 0.4 0.5

βdc

(b) ND=5

● ● ● ● ● ● ●●

●

●

●

0

1000

2000

3000

0.0 0.1 0.2 0.3 0.4 0.5

βdc

(c) ND=10

● ● ● ● ● ● ●●

●

●

●

0

3000

6000

9000

0.0 0.1 0.2 0.3 0.4 0.5

βdc

(d) ND=30

Figure 12 Customer price sensitvity βdc and PVAD

ND= 5, cij3 = $5.0, and q∗3 >−$1.0. In this case, q∗3 = $0.0 and profits equal $1.9 million as shown

in Figure 10b. For ND = 10 and 30, the optimal delivery charge q∗3 = -$2.0 and -$3.0 respectively.

This shows that government regulations may allow an e-retailer to either offer discounts or force

it to set high UAV delivery charges.

Figure 11 plots e-retailer’s profits at optimal UAV delivery charge q∗3 against βdc for different

values of ND. It is interesting to note that when the e-retailer cannot offer discount over the

retail price (i.e., negative delivery charge), its profits decrease as βdc increases as shown in Figures

11a and 11b. However, for ND = 10 and 30, the profit function is U-curved as shown in Figures

11c and 11d. In fact, e-retailer’s profits are maximized when ND = 30 and customers are most

sensitive to delivery charge. Figure 12a plots PVAD against βdc for ND= 2 and shows that under

strict government regulations, PVAD is maximized when customers are less sensitive to delivery

charges allowing the e-retailer to charge higher prices. UAV service is therefore considered as a


●

●●

●●

● ●● ● ● ● ● ● ●

10

20

30

40

0 25 50 75 100Travel time sensitivity, βtt

Pro

fits

(in

million

s)

●

Without UAV service

With UAV service

(a) Profits

110%

115%

120%

0 25 50 75 100Travel time sensitivity, βtt

PV

AD

(b) PVAD

Figure 13 Effect of Travel time sensitivity βtt

premium service available to customers who are willing to pay higher delivery charges for a 30-

minute delivery. On the other hand, when government regulations are relaxed, PVAD is maximized

when customers are most sensitive to delivery charges as shown in Figures 12b, 12c, 12d. Relaxed

regulations allow the e-retailer to set lower delivery charges or offer discounts over the retail price

to maximize its profits. As such, UAV service will be accessible to a wide variety of customers.

5.5. Effects of competitive stores

We vary customer utility for in-store shopping by varying travel time sensitivity βtt between 0.0

to 100.0. Figure 13a plots the profits in function of βtt. As βtt increases, the e-retailer’s profit

increases at a decreasing rate. Recall the utility function (2.1), as βtt increases, customer utility

for in-store shopping decreases and the store loses market share. A proportion of the store’s lost

market share is captured by the competing services offered by the e-retailer and the rest is lost

due to the expenditure function (2.3) that allows market shrinkage when overall customer utility

decreases. As βtt increases, the store’s lost sales increase leading to increased market share captured

by the e-retailer which translates into an increase in profits. But as βtt increases, its marginal effect

on the e-retailer’s profits decreases. This is further shown in Figure 13b which plots the percentage


●

●

●

●●

● ●● ● ● ● ● ● ● ● ● ● ● ● ●

0

10

20

30

40

50

60

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Delivery time sensitivity, βdt

Pro

fits

(in

million

s) ●

Without UAV service

With UAV service

(a) Profits

0

500

1000

1500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Delivery time sensitivity, βdt

PV

AD

(%)

(b) PVAD

Figure 14 Effect of customer delivery time sensitivity βdt

value added by UAVs, PVAD, in function of βtt. Figure 13 suggests that the competitive advantage

of UAV delivery is maximized in regions where customers do not have access to stores at a close

proximity and to time sensitive customers.

5.6. Effects of customer delivery time sensitivity

Since UAVs fly over congested road networks and a single delivery per trip is made, delivery time

is expected to reduce significantly. To study this attractive feature of UAVs, we vary customer

sensitivity to delivery time, βdt and analyze its effects on value-added by UAVs. βdt is varied from

0.0 to 1.9 and the model is solved under scenarios 1 and 2. Figure 14a illustrates the effect of βdt on

e-retailer’s profits under each scenario. As βdt increases, customer utility towards online shopping

decreases exponentially which leads to reduced profits as shown in Figure 14a. Under scenario 1,

when βdt > 0.4, profits equal $0. This is because delivery time sensitive customers prefer in-store

shopping where there is zero delivery time and as such, the e-retailer does not have enough demand

to cover its facility costs. Under scenario 2, the e-retailer is able to earn profits even when βdt = 1.8.

Due to the instant delivery feature of UAVs (30-minutes), time sensitive customers place order

online and opt for UAV service. As shown in Figure 14b, PVAD is maximized when customers are

most sensitive to delivery time. When βdt = 0.0, PVAD is 46% and increases to 1686% at βdt = 0.4.


This shows that UAV service captures demand from time sensitive customers, which in turn leads

to increased e-retailer profits. It may also allow the e-retailer to enter new markets where demand

was previously low due to time sensitive customers.

6. Analysis of solution algorithm: LAB

We conducted several experiments to study the efficiency of LAB algorithm in solving profit maxi-

mization problems [NP ] by randomly generating nodes over a 1600 km2 square region. The number

of customer demand points |I|= {50,100,150,200,250}. The number of candidate facility locations

|J | = {10,20,30,40,50}. To compute the maximum demand (Nip), each customer zone i ∈ I and

package p∈ P is assigned a weight from a uniform distribution [0,1]. Randomly generated weights

are then used to proportion NYC grocery sales into different customer zones and packages to cal-

culate Nip. We assume λ= 10.0. The facility costs are randomly generated between $1 million to $2

million. High λ and low facility costs are selected to challenge the algorithm with higher demand

values and allow multiple facilities to open. Each instance is solved over two different values of

the number of stores: 10 and 100. All other parameters are the same as used for the base case in

Section 5.1. For a given |I|, |J |, and stores, 10 random instances are generated resulting in a total

of 500 instances.

To validate the effectiveness of the LAB algorithm, we compare its CPU time with the mixed

integer linear reformulation given in Appendix B. LAB is coded in C++ Visual Studio 2013 and all

optimization problems are solved using CPLEX version 12.6.1 on a 64-bit Windows 10 with Intel(R)

core i7-4790 3.60GHz processors and 8.00 GB RAM. Each instance is executed to an optimality

gap of 1e-09 or up to 3600 seconds in CPU time. The results are summarized in Table 4 where

values are reported as the average of all instances for each combination of |I| and |J |. The number

of open facilities is denoted by∑j∈J

Wj. “Iter” denotes the number of iterations carried out by the

LAB algorithm. “Gap” refers to the optimality gap: (UB −LB)/LB. CPU times are reported in

seconds and the ratio of CPU time of IP model and LAB algorithm is denoted by “Time ratio”. Our

algorithm performs significantly better than direct solution of the linear reformulation. On average,


|J | |I|∑j∈J

Wj LAB algorithm IPTime ratio

Iter Gap CPU time (s) Gap CPU time (s)

10 50 4 4 0.00 0.44 0.00 39.21 8910 100 3 4 0.00 1.46 0.00 210.89 14410 150 3 3 0.00 2.90 0.00 412.82 14210 200 4 4 0.00 5.53 0.00 357.11 6510 250 3 3 0.00 8.79 0.00 673.12 7720 50 4 5 0.00 2.00 0.00 261.44 13120 100 4 5 0.00 4.87 0.00 1174.63 24120 150 4 6 0.00 11.55 0.06 2433.31 21120 200 4 4 0.00 19.37 0.27 3450.15 17820 250 4 4 0.00 27.81 0.54 3602.16 13030 50 4 6 0.00 5.06 0.00 867.75 17130 100 5 4 0.00 11.02 0.26 3467.26 31530 150 4 5 0.00 20.78 0.40 3603.50 17330 200 4 4 0.00 31.26 n/a 3600.84 11530 250 5 4 0.00 41.07 n/a 3600.35 8840 50 4 5 0.00 6.36 0.01 1735.12 27340 100 4 5 0.00 13.60 0.34 3602.16 26540 150 5 3 0.00 23.76 n/a 3600.59 15240 200 5 4 0.00 39.49 n/a 3600.21 9140 250 5 4 0.00 60.15 n/a 3600.20 6050 50 5 5 0.00 8.20 0.10 3174.78 38750 100 5 5 0.00 21.54 n/a 3603.10 16750 150 5 6 0.00 41.23 n/a 3600.14 8750 200 5 5 0.00 59.16 n/a 3600.21 6150 250 5 5 0.00 85.14 n/a 3600.25 42

Average 5 5 0.00 22.10 0.12 2458.85 154

Table 4 Summary of the computational experiments

LAB is 154 times faster than Cplex. Cplex fails to close the gap in 59% of the instances while LAB

solves all instances to optimality within 92 seconds. “n/a” denote instances where Cplex fails to

find a feasible solution in 3600s. Cplex fails to find a feasible solution in 28% of the instances. This

signifies the need for the proposed LAB algorithm to solve large scale instances.

Nb. of servicesIterations

CPU time (s) [SP ] timetotal time

Gap|S| total [MP] [SP]

3 6 50.44 50.19 0.25 0.49% 0.00%

4 4 155.33 154.93 0.40 0.25% 0.00%

5 21 805.46 802.70 2.77 0.34% 0.00%

6 102 2009.78 1990.86 18.92 0.94% 0.00%

Table 5 Effect of the number of services on subproblem’s computational efficiency

To justify the use of enumeration based approach in [SP], we solve industry scale problems by

varying the number of services offered |S|. Note that the world’s largest e-retail company, Amazon

offers a maximum of 6 different delivery services in a single city. For |I|= 200 and |J |= 50, we vary


the number of services offered |S|, by the e-retailer between 3 and 6. Ten random instances are

generated and results are summarized in Table 5 where average values for each |S| are reported.

[SP] takes less than 1% of the total CPU time as shown in [SP ]time

total timecolumn. Although the CPU

time in [SP] increases with increasing number of services, the effect is not significant. Increasing

|S| results in higher number of iterations and as such, [SP] is solved several times. We note that

it takes less than a second, to solve [SP] at each iteration. The results show that the proposed

enumeration based approach in LAB algorithm works quite well for industry scale instances where

the number of services are generally limited.

7. Conclusions

We used MNL market share model and optimization modelling to study the impact of UAV delivery

on e-retailing. A novel logic-based Benders algorithm is proposed that not only solves the non-

linear model efficiently but allows several possible extensions. We analyzed the tradeoffs between

distribution costs and revenues under varying social resistance to UAVs, customer preferences,

and regulatory and technological limitations. Our results show that these challenges significantly

impact optimal distribution network configuration and UAV target markets. For example, under

the current UAV landing capabilities, a UAV delivery service may not be possible in a densely

populated area like Manhattan where demand for such a service is expected to be high. We found

that under the right technological capabilities and regulations, e-retailers are able to reach smaller

markets and more price sensitive customers possibly by offering discounts on UAV delivered orders.

The modelling and analysis presented in this paper may be used not only by e-retailers but by any

retail business and other stakeholders including regulatory bodies. For instance, regulatory bodies

may use our modelling approach to test regulations on UAV deliveries. One of the weak points of

our work is that customer sensitivity parameters are not explicitly based on customer preference

for UAV delivery, but rather on the literature that explores customer preference for online shopping

versus in-store shopping. Estimating these parameters based on a market survey is important but

is out of the scope of this work. Another extension of our work might be to incorporate charging

stations to extend the range of UAVs at the expense of higher delivery times.


Acknowledgments


Appendix A: Proof of Lemma 1

1. Recall equation (2.1) and (2.4). Equation (2.1) says that the utility Uip increases as the number ofservices offered increases. Taking the derivative of Disp with respect to Uip:

∂Disp

∂Uip=−

Nip(1− (exp(−λUip)(λUip + 1)))× (∑j∈J

aspyijs) exp(β0s−βdtDTs−βdcqs)

U2ip

< 0 (A.1)

as λ> 0, Uip > 0, and exp(−λUip)× (λUip+ 1)> 1. Hence, as the number of services offered increases, utilityUip increases which results in decreasing Disp. Therefore, maximum demand that may be captured by services∈ S, Dmax

isp is achieved when only service s∈ S is available i.e., yijs = 1 and yijs′ = 0 ∀ s′ 6= s.2. Using equation (2.4), total demand captured TDip, by the e-retailer is expressed as:

TDip =Nip(1− exp(−λUip))(Uip−USip)

Uip, (A.2)

where USip = exp(β0,n+1 − βttTTi − βtcTCi) and (Uip − USip) ≥ 0, is the utility derived by customer zonei∈ I for package p∈ P given the services offered by the e-retailer. Taking the derivative of TDip with respectto total utility Uip:

∂TDip

∂Uip=Nipe

−λUip(USip(eλUip − 1) +λUip(Uip−USip))

U2ip

> 0 (A.3)

as λ> 0, Uip > 0 , (eλUip−1)> 0, and λUip(Uip−USip)≥ 0. Hence, as Uip increases, TDip increases. Therefore,maximum total demand TDmax

ip , is achieved when the e-retailer offers all services in S to customer zone i∈ Ifor package p∈ P . �

Appendix B: Linear formulation of model NP

Three sets of binary decision variables and one set of continuous nonnegative decision variables are definedas:

tie =

{1, if customer zone i∈ I is offered set of services e∈P(S)

0, otherwise

wj =

{1, if candidate facility j ∈ J is opened

0, otherwise

xjs =

{1, if service s∈ S is offered at facility j ∈ J0, otherwise

dijspe =demand captured by facility j ∈ J using service s∈ S for package p∈ P incustomer zone i∈ I when set of services offered is e∈P(S)

The model NP is transformed into mixed integer program [IP] as:

[IP]: max∑i∈I

∑j∈J

∑s∈S

∑p∈P

∑e∈P(S)

(απp + qs− cijs)dijspe−∑j∈J

∑s∈S

Fsxjs−∑j∈J

Ljwj (B.1)

s.t.∑

e∈P(S)

tie = 1 i∈ I, (B.2)

xjs ≤wj j ∈ J, s∈ S (B.3)

dijspe ≤Mtie i∈ I, j ∈ J, s∈ S,p∈ P, e∈P(S) (B.4)

dijspe ≤Mrijsxijs i∈ I, j ∈ J, s∈ S,p∈ P, e∈P(S) (B.5)∑j∈J

dijspe ≤Dispe i∈ I, s∈ S,p∈ P, e∈P(S) (B.6)

tie ∈ {0,1} i∈ I, e∈P(S) (B.7)

xjs ∈ {0,1} j ∈ J, s∈ S (B.8)

wj ∈ {0,1} j ∈ J (B.9)

dijspe ≥ 0 i∈ I, j ∈ J, s∈ S,p∈ P, e∈P(S) (B.10)

Appendix C: Data used


Borough Minimum MaximumManhattan 70 100Staten Island 14 21Brooklyn 65 75Queens 14 25Bronx 30 40

Table C1 Yearly Lease Rate ($) per sq. ft in Boroughs

ID Borough Name NTA code Yearly Cost ($)1 Brooklyn BK82 5,000,0002 Queens QN49 2,450,0003 Queens QN01 3,000,0004 Queens QN18 2,700,0005 Staten Island SI11 2,750,0006 Staten Island SI24 2,600,0007 Brooklyn BK28 5,300,0008 Bronx BX13 3,600,0009 Staten Island SI45 2,650,00010 Bronx BX06 3,550,00011 Staten Island SI37 2,450,00012 Manhattan MN24 5,550,00013 Queens QN41 2,700,00014 Brooklyn BK31 5,000,00015 Queens QN53 3,000,00016 Manhattan MN11 6,100,00017 Brooklyn BK81 5,300,00018 Brooklyn BK42 5,250,00019 Brooklyn BK72 5,150,00020 Queens QN70 2,650,000

Average 3,837,500Table C2 Yearly facility Costs


Parameters Estimated ValueYearly demand of delivery by UAVs Y DAvg hourly demand, H Y D

365×14

maximum hourly demand, Hmax =H × 2.0 Y D365×14

× 2.0

Nb. Of UAVs required, NbUAV s= Hmax

2Y D

365×14× 2× 1

2

Nb.of batteries required, NbBatteries=NbUAV s Y D365×14

× 2× 12

Nb. Of UAVs per operator, ND 10Nb.of operators required per hour NbOperators= Hmax

2×NDYD

365×14× 2× 1

2× 1

10

Nb.of hours an operator works 8Total Nb. of operators required, TotalNbOperators=NbOperators× 14

8Y D

365×14× 2× 1

2× 1

10× 14

8

Costs calculationsCost per UAV 3000Cost per battery 200Total battery cost, BC = 200×NbBatteries Y D

365×14× 2× 1

2× 200

Total UAV cost, DC = 3000×NbUAV s Y D365×14

× 2× 12× 3000

Total purchasing cost,PC =BC +DC (3000 + 200)× Y D365×14

× 2× 12

Amortization rate, r 20%Yearly amortized cost of UAVs & Batteries,AC = r×PC (3000 + 200)× Y D

365×14× 2× 1

2× 0.20

Yearly operators salary 70,000Total yearly operators salary cost,OC = 70000×NbOperators Y D

365×14× 2× 1

2× 1

10× 14

8× 70000

Total yearly costs, TC =AC +OC (3000 + 200)× Y D365×14

× 2× 12× 0.20 + Y D

365×14× 2× 1

2× 1

10× 14

8× 70000

Battery charging cost per delivery(in dollars) 0.10

Cost per delivery, TCYD

((3000+200)× Y D

365×14×2× 12×0.20+ Y D

365×14×2× 12×

110×

148 ×70000

Y D

)+ 0.10

= 0.1252 + 2.397 + 0.10≈ 2.62

Table C3 Detailed calculations for UAV package delivery cost cijn


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Strategic Network Design for Parcel Delivery with Drones