Strategies for Electric Location-routing Problems Considering Short

and Long Term Horizons

Victor Hugo Vidigal Corr

1 a

, Andr

e Gustavo dos Santos

1 b

and Thiago Henrique Nogueira

2 c

Departamento de Inform

atica, Federal University of Vic¸osa, Av. Peter Henry Rolfs, S/n, Vic¸osa, Minas Gerais, Brazil

Departamento de Engenharia de Produc¸

ao, Federal University of Vic¸osa,

Rodovia BR 230 KM 7, Rio Parana

ıba, Minas Gerais, Brazil

Keywords:

Electric Location-routing Problem, Long-term Horizon Planning, Variable Neighborhood Search.

Abstract:

Recent climate data has risen attention to many problems related to the global warming effect caused by the

emission of greenhouse gases. The rise in the global average temperature has many consequences and it is very

close to the established threshold in which, immediate actions must be taken or the damage to our planet will be

irreversible. The transportation sector, responsible for 23% of the global CO

emissions, and the public power,

aware of the situation, have been trying to innovate and solutions such as electric vehicles are getting much

attention and growing in popularity. This work aims to help logistic companies by proposing a metaheuristic

algorithm and a novel methodology for the planning of electric vehicle infrastructures composed by battery

recharging stations and battery swap stations. Different from previous works, we consider a long-term horizon

planning by using the proposed algorithm itself to pre-process data and improve results by considering the

synergy of long-term location and short-term routing problems. Computational experiments shows that our

algorithm is able to reduce the cost of electric vehicles infrastructures compared to previous work.

1 INTRODUCTION

The concerns on the global warming effect in our

planet has risen the attention of many people around

the globe and has being an important subject to

many heads of state, even generating some con-

ﬂicts in diplomatic relationships. According to The

Guardian (Harveyn, 2020), during the Climate Ambi-

tion Summit 2020, the United Nations secretary gen-

eral, Ant

onio Guterres, urged for governments around

the world to declare climate emergency due to re-

cent analysis on climate data collected over the last

years. As stated by the World Meteorological Orga-

nization (WMO, 2019), the global average tempera-

ture in 2019 was 1.1 degree Celsius above the pre-

industrial period - the second highest since the record

began - and, such worries rises from the last Inter-

governmental Panel on Climate Change (IPCC) report

(Rogelj et al., 2018) that discuss the consequences of

an increase of 1.5 degree Celsius in the global aver-

age temperature above the pre-industrial era, in both

https://orcid.org/0000-0002-8390-7075

https://orcid.org/0000-0002-5743-3523

https://orcid.org/0000-0002-6602-8458

environmental and economic aspects. Among many

problems here we cite a few: the decrease of pollina-

tion of crops and plants due to insects’ lost of habitat,

the change in weather events patterns, making them

more dangerous to humans and other species, and the

endangerment of over 6 million people that live in

coastal areas that are vulnerable to the sea-level rise

at the given temperature increase.

The IPCC report also discusses how can humanity

mitigate the path to the 1.5 degree Celsius increase

in the context of sustainable development. One of its

highlights is the transportation sector’s contribution

in the emission of Greenhouse Gases (GHGs) that,

in 2014, was responsible for 23% of global energy-

related CO

emissions. Zhao et al. (2019), analyze

China’s GHGs emission and shows that electric ve-

hicle (EV) deployment has a better long-term de-

carbonization effect, while fuel economy regulations

shows a better result in the short term. It is concluded

that by adopting a deployment plan of electric vehi-

cles together with fuel economy regulations, by the

year 2026, China could reach its peak of gases emis-

sions, 4 years earlier then what was agreed during

the 2015 United Nations Climate Change Conference,

where China government agreed to achieve its emis-

Corrêa, V., Santos, A. and Nogueira, T.

Strategies for Electric Location-routing Problems Considering Short and Long Term Horizons.

DOI: 10.5220/0010500407950802

In Proceedings of the 23rd International Conference on Enterprise Information Systems (ICEIS 2021) - Volume 1, pages 795-802

ISBN: 978-989-758-509-8

795

sion peak of CO

before the year 2030. The use of

electric vehicle is not the sole solution for this huge

climate problem that the world is facing, other ac-

tions such the improvement in the cleanliness of en-

ergy production must be taken and will be required

to reverse the pathway to the 1.5 degrees Celsius in-

crease.

Changing the engine proﬁle in vehicle ﬂeets is not

an easy task. Due to the use of internal batteries to

store energy, EVs have some critical differences com-

pared to ICE vehicles that makes this transition much

more difﬁcult. EVs require a longer time to recharge

the batteries reducing drastically its range, while, in

contrast, ICE vehicles require no more than a few

minutes to go from empty to full gas tank, extend-

ing its range quickly. Due to the limitations in battery

technology currently used in EVs, its autonomy has

limitations that directly impact its range. While the

fastest recharge method, the super charger power out-

let, can provide up to 80% of battery capacity in 40

minutes, if one is available, the most common power

outlet can take many hours to provide a full battery

charge. This makes the vehicle’s range the second

most important factor that individuals consider when

purchasing an electric vehicle, after the price range

(Cecere et al., 2018).

Logistics companies are a lot more sensible to this

problem since their activities are very time-sensitive

and waiting for the vehicle’s battery to recharge may

cause issues in their delivery performance. Waiting

too much for the battery to recharge, can potentially

make companies need more vehicles to be able to ful-

ﬁll all the customers demand, or in some cases make

even impossible for the company to reach a deter-

mined customer in a day. A different approach to the

EVs battery recharging problem, that is not exclusive

to, but very helpful in the matter of logistic compa-

nies, is the battery swapping. This method allows the

driver to head the vehicle to a special facility, named

battery swap station (BSS), where the battery can be

quickly replaced by a fully charged one. This method

is getting much attention in China where some com-

panies already use it. A company called NIO has

141 BSSs installed across the country and claims to

have made more then 700000 battery swaps (Tianyu,

2020).

Despite the difﬁculties, companies are trying to in-

corporate cleaner transportation in their ﬂeets. Given

the climate urgency, they are either trying to ﬁt gov-

ernment laws and make use of incentive polices or

trying to innovate in order to catch the public at-

tention. Companies such as DHL have already an-

nounced back in 2014 that they would start to incor-

porate EVs in their ﬂeets (DHL, 2021) and in 2018

they even started to operate with electric vehicles in

the state of Rio de Janeiro, Brazil (DHL, 2018). To

make the adoption of EVs easier for the general pub-

lic and companies, the presence of a reliable and well

built infrastructure network that can provide the ser-

vices necessary for the vehicle’s operation is a deci-

sive step and optimization techniques can be used to

design them in a more effective way and with lower

costs.

The Electric Location-Routing Problem (ELRP) is

a combinatorial optimization problem in which a ﬂeet

of electric vehicles must have their routes deﬁned to

serve a set of customers alongside their travel and the

location to install a set of facilities must be chosen,

where the vehicles can be recharged or have its bat-

tery replaced so they can ﬁnish their delivery routes.

This is a combination of two others well known op-

timization problem, the facility location and vehicle

routing problems. Due to their classiﬁcation as a NP-

Hard problem, the ELRP is also in this category and

there is no known algorithm able to solve it in a poly-

nomial time. Many variants have been proposed, for

example: the possibility of partial charging at BRS

(Schiffer and Walther, 2017); combination of BRSs

and BSSs (Paz et al., 2018); and stations with differ-

ent charging speed (Li-Ying and Yuan-Bin, 2015).

In this work we study the Multi-Depot Electric

Location-Routing Problem with Time Windows, Bat-

tery Swapping and Partial Recharging (Paz et al.,

2018; Corr

ea and dos Santos, 2020), for the design

of EVs infrastructures that incorporate the well estab-

lished BRSs and the innovating BSSs in an uniﬁed

network for logistic companies. We extended the pre-

vious works by developing a new heuristic algorithm,

due to the ELRP’s NP-Hard classiﬁcation, to solve

the problem for large-sized instances and address the

two subproblems simultaneously, one with short-term

and the other with long-term characteristics to further

optimize electric vehicle infrastructures. It is a step

to solve an issue often found in the location-routing

problem literature, as in most of the previous works

the location component is solved in short-term hori-

zon in order to optimize the routing component, but

in real-life applications the solution of the location

component is to be used in a long-term horizon, for

several short-term routing problems.

The remainder of this work is organized as fol-

lows: in Section 2 we present a brief review on the

literature related to electric vehicle infrastructures; in

Section 3 we present a formal deﬁnition of the prob-

lem; in Sections 4 and 5 we present a heuristic algo-

rithm and our methodology to improve solutions with

a preprocessing procedure; in Section 6 we present

the test instances elaborated for the experiments and

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the results obtained with them; and ﬁnally, in Section

7 we discuss the results and present some insights for

future works.

2 LITERATURE REVIEW

An electric vehicle infrastructure can be composed by

a range of elements going from battery recharging sta-

tions (BRS) to special road lanes that allow cars to

have their batteries charged wirelessly while driving.

However, most of the studies consider either the use

of battery swap stations (BSS), BRS or a combination

of both. A slightly different approach was made by

(Cui et al., 2018). They considered a special type of

BRS that is placed in the truck body, so they can move

the station to another place according to the daily traf-

ﬁc ﬂow. This vehicle is called mobile charging ve-

hicle (MCV) and it is requested by customers via a

mobile app and must be drove to the scheduled place.

The MCV can also support multiple types of charges.

The authors present a mathematical formulation of

the problem based on the single-depot ELRP. The ob-

jective is to minimize the total travelled distance and

determine the vehicles’ route, the MCV’s allocation,

and the type of charger in each station; while meeting

the time-windows, battery charge and vehicle capac-

ity constraints.

Li-Ying and Yuan-Bin (2015) elaborate a mixed

integer programming model and an algorithm for the

Multiple Charging Station Location-Routing Problem

with Time Window of Electric Vehicle (EV-MCS-

LRPTW). Besides the usual constraints and features

of the ELRP like time windows, vehicle battery and

vehicle loading capacity constraints, they also con-

sider four types of charging stations: slow charging

station (SCS), fast charging station (FCS), super-fast

charging station (SFCS), and the BSS. The model’s

objective function is to minimize the total cost in-

cluding the cost of siting stations, the cost of electric-

ity to recharge batteries and the drivers wage. The

designed algorithm is an Adaptive Variable Neigh-

borhood Search mixed with the Tabu Search algo-

rithm. It was implemented on Java with a single

core code and the model implemented on the CPLEX

solver. Experiments were conducted with instances

for the Pollution-Routing Problem (PRP) (Demir

et al., 2012), adapted for the EV-MCS-LRPTW, a to-

tal of nine sets of instances with different sizes. The

results show that the proposed algorithm can ﬁnd near

optimal solutions in the smaller sets of instances and

provide convincing results in moderate run time on

the largest sets. As future work, they pointed out to

extend the problem to multiple depots, to consider a

mixed ﬂeet of electric and internal combustion vehi-

cles and also to consider public charging stations.

Schiffer and Walther (2017) deﬁne a more com-

plete version of the ELRP in which time windows

and partial recharging are considered. Time windows

constraints are very common in practical applications

and partial recharging allow the electric vehicle to use

service time to recharge battery, increasing its auton-

omy. The authors present a novel mixed integer pro-

gramming model called Electrical Location-Routing

Problem with Time Windows and Partial Recharging

(ELRP-TWPR). They also provide 5 different objec-

tive functions to meet various requirements, e.g. min-

imizing distance traveled, number of vehicles used,

number of charging stations sited, number of vehicles

plus number of recharging stations (convex combina-

tion) and total costs. They conducted experiments

with new instances created based on instances from

the literature and compared the solutions obtained uti-

lizing all of the 5 objective functions proposed. As

suggestion for future works they highlighted the im-

plementation of heuristic solution method in order to

solve large instances for real world problems, the ap-

plication of the proposed model in practical case stud-

ies and extend the model for heterogeneous or mixed

ﬂeet.

A variant of the ELRP called battery swap station

location-routing problem with stochastic demands

(BSS-EV-LRPSD) is introduced by (Zhang et al.,

2019). The main contribution is the addition of

stochastic demands which turn the proposed model

more applicable in real life situations. For this prob-

lem they propose an algorithm called Hybrid Vari-

able Neighborhood Search (HVNS) which incorpo-

rates the Binary Particle Swarm Optimization (BPSO)

and a Variable Neighborhood Search (VNS) algo-

rithms. The basic concept behind it is that both BPSO

and VNS are used iteratively to solve the BSS location

problem and routing planning. They present as well,

a Pareto optimality for the BSS location stage. The

HVNS is compared with ﬁve others algorithm from

the literature and its performance is evaluated using

adapted test instances from other authors. In general,

the HVNS was able to ﬁnd better solutions then the

other algorithms and showed good stability and con-

vergence. As future works that can be done to im-

prove their model, time windows constrains and BSS

capacity can be considered to give more applicabil-

ity to the model. Their work does not consider the

possibility of recharging station and the energy con-

sumption and traveling times are constants.

A mixed integer programming model for the

ELRP with simultaneous BSS and BRS is presented

by (Paz et al., 2018). The authors presents three mod-

Strategies for Electric Location-routing Problems Considering Short and Long Term Horizons

797

els of which two are for the BSS and BRS individually

and a third one for BSS and BRS mixed. The model’s

objective function is to minimize the total traveled

distance. The models were solved with CPLEX 12.5

and the experiments limited to a maximum run-time

of 8 hours. It is worth to mention the amount of big M

constraints in the model, a total of 12 out of 21 con-

straints have big M, which makes the model less ef-

ﬁcient. Some preprocessing was done to improve the

computational time. In their methodology they use a

set of dummy nodes representing duplicated stations,

so they can be visited multiple times, being one ex-

tra visit per dummy node set. They suggested as fu-

ture works, the design of solution strategies for large-

scale instances and application on real case data. A

step in this direction was made by Corr

ea and dos

Santos (2020), who proposed a hybrid heuristic for

the same problem: permutations of the customers

are generated by a Simulated Annealing (SA) heuris-

tic; a Greedy Randomized Adaptive Search Proce-

dure (GRASP) maps each permutation into an array

of routes inserting BSS stations by a greedy choice,

later improved by a local search conducted by a VNS.

The results were compared to the MILP model solved

via Gurobi limited to two hours. While the solver

was not able to ﬁnd solution in most instances, the

SA found solution in every case and was able to ﬁnd

some global optimal solutions.

3 PROBLEM DEFINITION

The problem studied here can be divided in two.

First, a daily delivery optimization is considered as

a variant of the ELRP called the Multi-Depot Elec-

tric Location-Routing Problem with Time Windows,

Battery Swapping and Partial Recharging (Paz et al.,

2018; Corr

ea and dos Santos, 2020) and is deﬁned as

follow. A set of customers, each one with time win-

dow and demand parameters, must have their demand

supplied by an electric vehicle within its time win-

dow. A set of vehicles must depart from the depots,

perform a route delivering goods and return to the

same initial depot. The vehicle has a limited auton-

omy based on its battery capacity and to address this

problem two measures can be taken: (i) use the ser-

vice time to perform a recharge in the customer’s BRS

or (ii) go to a BSS and have its battery swapped for a

fully charged one. When at the customer, the vehicle

can use the service time to give its battery a charge

but should not stay longer than that. Additionally, the

locations of the stations to be sited and the amount of

depots must be decided. The goal is to site depots,

site battery swap stations, deﬁne the route of the ve-

hicles for goods delivering and determine a recharge

plan by deﬁning when and where the vehicles have

to recharge or swap the battery; while minimizing the

total cost composed by the BSSs installation cost, the

vehicle cost, the drivers wage, depot siting cost and

energy cost in BSSs and BRSs.

The second problem is the short and long-term

conﬂict between the facility location and the vehicle

routing problems. Considering that a logistic com-

pany will serve certain region for a long period of

time, the infrastructure optimized by the ELRP con-

sidering a single day might not be optimal or not even

be usable for other days of delivery. Considering

the given scenario the second problem is the long-

term optimization of electric vehicle infrastructures

according to the one described as the ﬁrst problem,

but routes are still to be decided for each day.

4 VARIABLE NEIGHBORHOOD

The Variable Neighborhood Search (VNS) (Hansen

and Mladenovi

c, 1997) is a metaheuristic widely used

to solve optimization problems and was used in this

work. It work as follows: starting from an initial so-

lution, the VNS cycles between a solution shake step

and a local search procedure until it reaches a stop cri-

teria. It is essential to deﬁne a solution representation,

a polynomial algorithm capable of generating an ini-

tial solution and a set of neighborhood structures to

be used to perform movements in the solution.

The solution is represented as an array of routes,

in which every route starts in a depot represented by

the ﬁrst element and ﬁnishes in the same depot repre-

sented by an arrival node in the last position. Figure 1

illustrates the solution representation. Vertex 0 is the

depot, vertexes 1 and 2 are BSSs - BSS 2 is sited and

1 is not - and the rest are customers.

Figure 1: The solution represented as an array of routes.

The VNS requires to be provided with an initial so-

lution, so a greed algorithm is purposed. The greed

algorithm starts siting a depot in the most populous

depot-able city and creating an initial route with this

depot and no customer nor BSS. This initial route is

then, added to the yet empty set of routes. The algo-

rithm then, search in the last route added in the set

for the closest customer from the last visited place on

the route. When found, the customer is added in the

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

798

route and the procedure addStations proposed by

Corr

ea and dos Santos (2020) veriﬁes if the route is

feasible. addStations is a procedure that generates

a recharge plan for a route, i.e., determine where the

vehicle must recharge its battery and where to swap

the battery, so it can complete the route. If a recharge

plan is possible, the customer is added into the route,

and is removed from the customer set. When no cus-

tomer is found, i.e., the vehicle has not enough battery

to keep driving, the route is closed and a new empty

route with the current depot is created and the algo-

rithm repeats. When the last created route remains

empty after checking every customer, the remaining

customers cannot be reached from any route starting

from the current depot, so the last created route is

deleted and the process is repeated by opening a de-

pot in the next depot-able city in population size. In

the case of the last created route remained empty af-

ter checking every customers and the current depot is

sited in the last depot-able city, the given instance is

infeasible. The initial solution algorithm is detailed in

Algorithm 1.

Algorithm 1: Initial solution algorithm.

Input: D = depots, S = stations, C = customers

Output: routes

sort(D); // by population size

routes ← set(); // create the route set

for d in D do

while depotOk = true do

depotOk ← false;

addEmptyRoute(routes, D);

route ← last(routes);

while size(C) > 0 do

c ← searchCustomer(route);

r ← route; // backup route

addCustomer(c, route);

addStations(route);

if isFeasible(r) = true then

C.remove(c);

depotOk ← true;

else

/* skip customer */

route ← r ; // restore route

aux.add(c); C.remove(c);

end

C ← aux;

route ← last(routes);

if isEmpty(route) = false then

routes.remove(route);

depotOk ← false;

else

addEmptyRoute(routes, D);

end

In total, 5 neighborhood structures are used,

namely: Union Route, Shift Customer, BSS Replace-

ment, Change Depot and 2OPT. None of them are

able to directly modify the solution’s recharge plan.

They modify characteristics such as BSSs available,

depots sited or customers order of visit. However,

they all use the addStations procedure to recon-

struct the recharge plan of the modiﬁed routes, so they

indirectly modify this characteristic. Union Route

consists of a neighborhood structure where 2 routes

are chosen and a new set of routes is constructed using

the customers from those routes. The idea is to solve

a subproblem consisting of only customers contained

in the two selected routes and use the same logic of

the greed initial solution algorithm to reconstruct the

routes. Using this structure, 2 routes may generate

a larger one or a set of 2 or more different routes.

The BSS Replacement neighborhood structure aims

to reduce the solution cost by reducing the number

of BSSs on it. To do so, it removes any occurrence

of a chosen BSS from the solution and the affected

routes are reconstructed with the addStations pro-

cedure. Change Depot works in a similar way as the

BSS Replacement, a depot is chosen and all its oc-

currences are replaced by another depot. The affected

routes are then, reconstructed with the addStations

procedure. The 2OPT structure swaps two arcs in a

route and the Shift Customer shifts forward or back-

ward a given customer by a certain number of po-

sitions in the route. In our solution representation

both structures are achieved similarly. The 2OPT is

achieved by removing the route’s recharge plan, in-

verting part of the customer order and reconstructing

the recharge plan with the addStations algorithm.

Analogous, the Shift Customer is achieved by remov-

ing the route’s recharge plan, shifting backward or

forward in an amount of positions a given customer

and reconstructing the recharge plan.

The local search is conducted by a best-

improvement hill-descent metaheuristic and the stop-

ping criterion adopted is the maximum amount of

time running and number of iterations without im-

provement.

5 PRE-PROCESSING METHOD

FOR THE LONG-TERM

LOCATION

Most of the works dealing with location-routing prob-

lem considers an amortization of the costs of in-

stalling facilities, as they are to be used for a long

period of time, not only for the particular set of cus-

Strategies for Electric Location-routing Problems Considering Short and Long Term Horizons

799

tomers considered in the routing component of the

problem. In addition to the usual amortization ap-

proach used to address this short-term and long-term

duality in Location-Routing Problems, we present

a novel methodology to further optimize problems

based on this model. Considering that logistics com-

panies will install a set of depots and BSSs that will be

ﬁxed during a very long period of time, and the cus-

tomers itself and its demand may vary quickly over

the days, our approach consists in the execution of

an algorithm to solve the ELRP multiple times con-

sidering different days of delivery, extract informa-

tion about the most frequently used facilities and re-

execute the algorithm with a subset of facilities cre-

ated with the previous solution results. The BSS lo-

cation is then decided once for the whole planning

horizon, while the routing is decide day-by-day using

the pre-selected BSSs. For this method, we consider a

set of instances consisting in many delivery days and

with the same parameters such as vehicle range, num-

ber of depot candidates, number of BSSs candidates,

etc. The difference among the instances is the cus-

tomer set, representing the different days of delivery.

We execute the VNS to solve each day individ-

ually and then, compute the frequency in which the

BSSs were used. Next, a subset of BSSs must be de-

termined to be passed to the VNS again. This is done

by sorting the BSSs by frequency of use and doing

the following steps. An initial percentage x of BSSs

is determined and the top x most frequent BSSs are

chosen. To determine whether those BSSs will be

enough to at least generate initial solution in each day

so the VNS can run, the initial solution algorithm is

executed for every day. If a feasible solution is gener-

ated for all days, the process stop. If any execution is

unable to provide feasible initial solution in any day,

the value x is increased by certain amount y and the

process is repeated. The worst case scenario happens

when x = 100, meaning that the method could not re-

move any BSS, and the ones chosen previously by the

VNS are the minimum required to generate feasible

initial solutions.

To evaluate this method, the cost considering all

instances must be computed. Instead of simply sum-

ming up every solution cost, a special calculation

must be done, otherwise, by summing up the solution

objective function values, duplicated costs are going

to be accounted. To do so, the cost is divided in two

parts, the ﬁxed and the variable. The ﬁxed cost is

composed by the costs that will be shared across the

days, such as acquiring vehicles and siting BSSs and

depots. The variable cost is composed by the daily

costs incorporating the energy cost in the BSSs and

BRSs, the ﬁxed cost of operating a BSS to swap the

battery and the drivers wage. For every solution from

the VNS daily execution, the four variable costs are

summed up and the ﬁxed cost must be processed and

then also summed up. The vehicle acquiring cost is

determined by the solution with the highest number

of routes, the depot cost is determined by the set of

depots used in every solution, as well as BSS siting

cost that is determined by the set of BSSs used in the

daily solutions.

6 COMPUTATIONAL

EXPERIMENTS

In this section, the results of the computational exper-

iment are shown and also, the test instances and the

method used to generate them. The experiments were

executed in a machine equipped with a processor In-

tel(R) Core(TM) i7-7700HQ and 16GB RAM. The al-

gorithm was implemented in C++ using the Microsoft

Visual Studio compiler. The VNS was set to stop at

25 iterations without improvement or 3600 seconds

of runtime, whichever happens ﬁrst, and its neigh-

borhood exploration order was set to random, i.e., in

each iteration, after a worsening solution is found, a

random neighborhood is chosen independently from

the previous one. The parameters for the Select BSS

subset Algorithm were set to x = 5 and y = 5, i.e.,

for the long-term location problem, at least 5% of the

BSSs used in the ﬁrst VNS round are selected, and

5% more is added until a feasible solution is found by

the greedy algorithm for all instances of the set.

6.1 Test Instances

Instances set from previous work are limited and are

not ﬁt to test our new methodology, we require in-

stances that represents a long-horizon period, with

multiple days of delivery to get a more real world rep-

resentation. Thus, we created a new instance set based

on the state of Minas Gerais, Brazil, to contemplate

every characteristic needed to represent our problem.

The new set has multiple-day instances, i.e., instances

representing multiple days of goods’ demand in the

same geographical space, with each day containing

a different set of customers but the same general pa-

rameters. The State of Minas Gerais is divided in ten

administrative regions, and 7 are used in this work,

namely: Alto Parana

ıba, Central, Centro-Oeste, Mata,

Rio Doce, Sul and Tri

angulo. The remaining 3 re-

gions were not used because they are very sparse and

were not ﬁt to our instance generation method. Each

instance set contains customers from one administra-

tive region.

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Three real world information were required: the

cities geographical coordinate, the distances between

each pair of cities, and the cities’ population. We get

the coordinates by the Open Street Map (OSM), us-

ing the Nominatim Python API. The distance matrix

was obtained with the Open Source Routing Machine

(OSRM) and the cities’ population were gathered

from the Brazilian Institute of Geography and Statis-

tics (IBGE) information retrieval system SIDRA

considering the last census done in the country in the

year 2010.

For each instance set, we had to determine one set

of cities to be candidates to receive a depot and an-

other set of cities to be a candidate to receive BSSs.

The depot-able set is determined by its city popula-

tion size. We sort the cities by this criteria and choose

the ones with highest population to be part of the set.

However, we did not choose near cities and a mini-

mum distance between the cities in this set is estab-

lished as the half the maximum vehicle range.

The BSS-able set is chosen by solving a facility

location model. The intention is to obtain a good dis-

tribution of BSSs candidates, hence minimizing the

probability of generating infeasible instances, due to

cities being very far from the depot with no BSS for

a vehicle to reach it. The proposed model is based on

the p-Center facility location problem, in which a set

of facilities must be assigned to a set of customers in

order to minimize the maximum distance of a facil-

ity to a customer. In our case, we want to deﬁne p

cities as candidates to sit BSSs in order to minimize

the maximum distance of a city to the nearest BSS

candidate.

We generated 3 instances set for each region with

a total of 30 days of customer’s demand. They differ

by the percentage of cities chosen to compose each

day. For example, the instance set Central region

has a total of 157 cities. The set central 20 contains

30 daily customers (20% of the regions size) while

the set central 80, 121. The parameter p was set as

20% of the number of cities.

6.2 Results

In order to evaluate our preprocessing methodology,

we run 10 times the VNS algorithm followed by the

VNS with the pre-processing method for each in-

stance set and compare the VNS execution average

cost with the VNS limiting the BSSs. Table 1 report

the results. Column ‘Initial’ displays the average so-

lution cost in the ﬁrst algorithm step (VNS). Column

‘%BSS’ displays the percentage reduction of BSSs in

the second step, i.e., 100 - x. Column ‘Improv.’ the

https://sidra.ibge.gov.br/home/ipca15/brasil

percentage of cost reduction in the second VNS exe-

cution with the BSSs limited. Column ‘Time’ shows

the average execution time in seconds.

Table 1: Average results for each instance set.

inst Initial %BSSs Improv. Time

alto paranaiba 20 219147 0 - 3

alto paranaiba 50 229406 0 - 8

alto paranaiba 80 236943 0 - 22

central 20 308337 5 1,25 55

central 50 343652 10 1,79 963

central 80 373142 10 1,19 5537

centro oeste 20 419186 0 - 5

centro oeste 50 244848 35 3,46 38

centro oeste 80 252815 25 2,43 134

mata 20 283462 50 9,72 34

mata 50 305084 50 8,68 733

mata 80 325686 50 6,62 3312

rio doce 20 263211 5 0,93 23

rio doce 50 286969 50 6,60 274

rio doce 80 300725 45 5,00 1326

sul de minas 20 486770 30 3,57 49

sul de minas 50 335420 50 8,87 762

sul de minas 80 359989 50 8,20 3953

triangulo 20 226701 10 0,47 3

triangulo 50 248408 10 2,27 12

triangulo 80 257977 10 1,68 33

The results demonstrate that our algorithm was ca-

pable to further optimize the cost after the BSS can-

didate list reduction. The instance set in which the

reduction percentage was a 0% did not show cost re-

duction, because no BSS could be removed for the

second VNS execution, thus both executions were the

same. In most of the instances the pre-processing

method could reduce the number of BSSs installed,

thus reducing the overall cost in the long-term hori-

zon, although the transportation and energy cost on

each day may increase because of the reduced num-

ber of BSS. We can observe 29% of average reduction

percentage in the BSS candidate list and an average

of 4,28% in the cost reduction. The 4,28% of average

economy can sound a low economy and not worth,

however, considering the long run and that BSSs are

the second most expensive component in the consid-

ered electric vehicle infrastructure, this economy will

potentially grow. Take for example the instance set

sul

de minas 50. Considering the cost of half a mil-

lion dollars per BSS (Tianyu, 2020) and the reduc-

tion of 50% in the BSS allocation, over time, the sited

BSSs will be used more often and, the economy will

scale as we consider a bigger period of time.

Strategies for Electric Location-routing Problems Considering Short and Long Term Horizons

801

7 CONCLUSIONS

In this paper, we present an algorithm for the NP-

Hard problem called Multi-Depot Electric Location-

Routing Problem with Time Windows, Battery Swap-

ping and Partial Recharging and a novel methodology

to address the long and short-term conﬂict derived

from Location-Routing Problems. The algorithm is

based on the Variable Neighborhood Search and uses

a set of 5 neighborhood structures to explore the solu-

tion space with a constructive greed algorithm to pro-

vide an initial solution. In addition, we presented a

novel method to address the long and short-term prob-

lem mentioned. This method consists in the use of

the presented VNS to optimize individually each day

in the instance set and then, select a subset from the

used BSS set to reexecute the VNS limiting the BSSs

it can use.

We evaluate the proposed preprocessing method-

ology in its capabilities to further reduce electric vehi-

cle infrastructure costs. The results demonstrate a av-

erage cost reduction of 4.28% considering instances

set of 30 days of delivery, however the cost reduction

will scale as more delivery days are considered due to

the reuse of BSSs over time.

As future works, we suggest the exploration of

other ways to decide the subset of BSSs that will be

used in the second VNS execution and the creation

of instances representing the three regions not used in

this work and instances combining different regions

to execute the proposed methodology on even bigger

instances.

ACKNOWLEDGEMENTS

The authors thank Coordenac¸

ao de Aperfeic¸oamento

de Pessoal de N

ıvel Superior (CAPES) and Fundac¸

de Amparo

a Pesquisa do Estado de Minas Gerais

(FAPEMIG) for the ﬁnancial support of this project.

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