Constructive Heuristics for Periodic Electric Vehicle Routing Problem

Tayeb Oulad Kouider, Wahiba Ramdane Cherif-Khettaf and Ammar Oulamara

Universit

e de Lorraine, Lorraine Research Laboratory in Computer Science and its Applications - LORIA (UMR 7503),

Campus Scientiﬁque, 615 Rue du Jardin botanique, 54506 Vandœuvre-les-Nancy, France

Keywords:

Periodic Vehicle Routing, Electric Vehicle, Charging Station.

Abstract:

This paper introduces a new variant of the electric vehicle routing problem, named PEVRP for Periodic Elec-

tric Vehicles Routing Problem, in which the routing and charging are planned over a multi-period horizon,

subject to frequency constraints, limited ﬂeet of electric vehicles available at the depot, intermediate facilities

for charging during the trips, and partial charging. The objective of the PEVRP is to minimize the total cost

of routing and charging over the time horizon, such that each vehicle could be charged nightly at the deport,

and during the day at charging stations if refuelling is necessary. We propose two constructive heuristics.

The ﬁrst one is based on clustering technique that aims at allocating customers per vehicle and per period,

and then constructs tour for each vehicle visiting customers and charging stations for refuelling. The second

one is based on best insertion strategy, in which each customer is allocated to its best position that minimizes

charging and routing cost. Using several parameters setting, we compared and analysed the results of the two

proposed approaches on 50 new instances derived from EVRP instances of the literature.

1 INTRODUCTION

Several urban centers encounter to mobility and

goods transportation problems. As they become lar-

ger, trafﬁc congestion, energy consumption, and car-

bon emissions are increasing, imposing many adverse

consequences in terms of environment. In order to

cope with these environmental problems, public insti-

tutions have restricted access to these urban centers,

by imposing public policies. However, these public

policies have limited impact on the problems gene-

rated by transport activities. Others alternatives are

focused on the development of clean vehicles, such

as electric or hydrogen vehicles.

Thanks to technological progress, especially the

storage capacity of batteries, electric vehicle beco-

mes a promising tool to meet the challenge of de-

carbonising transport activities. Services using elec-

tric vehicles are already deployed to meet the demand

of mobility through several cities, especially for daily

commuting such as Autolib in Paris. However, there

are some factors that prevent massive use of vehicles

in all transport activities. These factors are mainly,

limited to electric vehicle driving range, long char-

ging time of electric vehicle batteries, and the avai-

lability of a charging infrastructure. However, to en-

sure a successful deployment of electric vehicles in

the short-term, it is signiﬁcant to target development

towards (i) speciﬁc usage categories where the elec-

tric vehicle is the most suitable (e.g. urban transport,

urban logistic, business ﬂeet) in terms of the driving

range, load capacity, and operating cost, and (ii) ma-

nage operations of a complex landscape of ecosystem

of EVs (vehicles - chargers - electricity grid - ﬂeet

management) with a focus on the new optimization

challenges aiming to develop efﬁcient models and de-

cision tools to manage the ecosystem of electric vehi-

cles. In fact, to plan their activities, several ﬂeet ma-

nagers use decision software tools either at tactical

(sizing of the ﬂeet, etc.) or operational (route opti-

mization, performance monitoring, tracking of vehi-

cles, etc.) level. However, the existing route plan-

ning software are not suitable for planning routing of

electric vehicles, since they do not take into account

the speciﬁcities of the ecosystem of electric vehicles

(autonomy, charging time, charging rate of batteries,

type and the availability of charging stations, capa-

city of the electricity grid, the cost of energy, etc.),

and these tools need to be upgraded in order to con-

sider the ecosystem of electric vehicle. This upgrade

requires the development of new and efﬁcient optimi-

zation models and decision tools to manage the whole

ecosystem of the electric vehicles.

In this paper, we focus our study on a speciﬁc

usage in which electric vehicles are most suitable such

as parcel or mail delivery in which agents distribute

264

Kouider, T., Cherif-Khettaf, W. and Oulamara, A.

Constructive Heuristics for Periodic Electric Vehicle Routing Problem.

DOI: 10.5220/0006630502640271

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 264-271

ISBN: 978-989-758-285-1

commodities to clients. More precisely, we consi-

der the periodic electric vehicle routing problem, in

which a set of customers require visits on one or more

days within a planning period, and there is a set of fea-

sible visit options for each customer. Customers must

be assigned to a feasible visit option. The typical ob-

jective is to minimize the total cost including charging

and traveling costs over the planning period, more de-

tails of the problem are provided in section 3. The rest

of the paper is organized as follows. Section 2 provi-

des a selective review on the studied problem. Section

3 gives more details on the constraints and charac-

teristics of our problem. Section 4 proposes solving

approaches based on constructive heuristics. Section

5 presents experimental results. Section 6 concludes

this study with a short summary and provides an out-

look on future research.

2 RELATED WORK

The design of weekly transportation plans arises in di-

verse applications in urban logistics, such as the col-

lection of recyclables or mails, the routing of healt-

hcare nurses, the transportation of elderly or disabled

persons, etc. This gives rise to a so-called pe- rio-

dic vehicle routing problems (PVRP), which has been

introduced in (Christoﬁdes and Beasley, 1984). The

objective of the PVRP is to ﬁnd a set of routes over

time horizon of h periods of days that minimizes to-

tal travel time while satisfying vehicle capacity, pre-

determined visit frequency for each client, and spa-

cing constraints. For a client, the spacing constraints

can be deﬁned either by minimum and maximum pe-

riods between consecutive visits, or by a set of allo-

wed period combinations. The periodic vehicle rou-

ting problem consists of selecting for each client a day

combinations (tactical decisions), and then of con-

structing routes for clients assigned to each day (ope-

rational decisions). Several objective functions can

be considered, such as minimizing the traveled dis-

tance, intra-days load balancing, or total transporta-

tion cost. Since the PVRP is NP-Hard problem, most

of works in literature presents heuristic and metaheu-

ristic approaches (Mancini, 2016), (Dayarian et al.,

2016), (Baldacci et al., 2011). A survey on the PVRP

can be found in (Francis et al., 2008).

Electric vehicle routing problems have attracted

close attention from researchers and business organi-

zations in recent years. Several variants of electric

vehicle routing problems have been studied in the li-

terature. (Schneider et al., 2014) introduced the Elec-

tric Vehicle Routing Problem with Time Windows and

Recharging Stations. The EVRP problem with mixed

ﬂeet is addressed in (Goeke and Schneider, 2015).

Authors consider a ﬂeet of electric and conventional

vehicles with time windows constraints. (Hiermann

et al., 2016) consider a heterogeneous ﬂeet of vehicles

that differ in their capacity, battery size and acquisi-

tion cost. In (Felipe et al., 2014), the authors pre-

sent a variation of the electric vehicle routing problem

in which different charging technologies are conside-

red and partial EV charging is allowed. (Schiffer and

Walther, 2017) proposed an electric location routing

problem with time windows and partial recharging.

The papers consider routing of electric vehicles and

siting decisions for charging stations simultaneously

in order to support strategic decisions of logistics ﬂeet

operators. In (Sassi et al., 2015b), (Sassi et al., 2015a)

a rich variant of Electric Vehicles Routing Problem

related to a real application is proposed. This vari-

ant considers a Mixed ﬂeet of conventional and he-

terogeneous electric vehicles and includes different

charging technologies, partial EV charging, compa-

tibility between vehicles. The charging stations could

propose different charging costs even if they propose

the same charging technology and they are subject to

operating time windows constraints The only study

in the literature that addressed the multi-periodic as-

pect for electric vehicles could be found in (Zhang

et al., 2017), but the routing and the charging over the

period is not considered. This study deals with the

multi-period planning of the charger location problem

for EVs considering facilities capacity and dynamic

demands. The aim is to determine the locations of

chargers as well as the number of charging modules at

each station over multiple time periods. In summary,

the current EVRP literature is limited to daily or stra-

tegic planning. Although EVs routing problems have

attracted close attention from researchers and busi-

ness organizations in recent years, the periodic ex-

tension of electric vehicles routing problem has never

been studied. In our study to be presented below, we

propose a new variant named PEVRP (Periodic Elec-

tric Vehicle Routing Problem), which deals with tacti-

cal and operational decisions level for electric vehi-

cles routing and charging. The aim is to deﬁne a rou-

ting and a charging plan for each vehicle over a plan-

ning horizon. Two constructive heuristics are propo-

sed. The ﬁrst one is based on clustering technique

that aims at allocating customers per vehicle and per

period, and then constructs tours for each vehicle vi-

siting customers and charging stations for refuelling.

The second one is based on best insertion strategy, in

which each customer is allocated to it best position

that minimizes charging and routing cost.

Constructive Heuristics for Periodic Electric Vehicle Routing Problem

265

3 PROBLEM DEFINITION

The Periodic Electric Vehicle Routing Problem (PE-

VRP) is deﬁned on complete directed graph G =

(V, A). V denotes the set of vertices composed of the

set C of n customers, a depot denoted by 0, and the

set B of ns external charging stations. The set of arcs

is denoted by A = {(i, j) | i, j ∈V } .

Each arc (i, j) of A is described by distance d

i, j

travel time t

i, j

, and travel cost c

i, j

. When an arc (i, j)

is travelled by an electric Vehicle (EV), it consumes

an amount of energy e

i, j

= r ×d

i, j

, where r denotes

a constant energy consumption rate. Each customer i

has a demand q

, and a service time s

. We consider

a time horizon H of np periods typically ”days”, in

which each customer i has a frequency f (i) = 1, and

a set of allowed visit days D(i) ∈ H. This means that

customer i must be serviced one time in D(i), but at

most once in the chosen day.

The set B is deﬁned by ns external charging sta-

tions that can be visited during each day of the plan-

ning horizon. In most studies of the literature on rou-

ting with electric vehicles, the charging cost depends

either on the amount of powers delivered by the char-

gers or on the total time spent for charging the vehi-

cles. In this paper, we consider a ﬁxed charging cost

Cc, that neither depends on the amount of the deli-

vered energy nor on the time needed to charge the

vehicle. This assumption is more realistic since char-

ging service providers are energy operators and law

prohibits companies that offer charging services from

reselling the energy, as they sell services, prices do

not depend on the amount of energy, but depends on

the quality of service (fast or slow charging) (Sassi

et al., 2015b) (Sassi et al., 2015a).The depot contains

charging points, allowing free charging at night and

during the day. The amount of power delivered to

each vehicle k at the night of day h is a decision varia-

ble P

h,0,k

, deﬁning the vehicle’s initial state of charge

at the beginning of the trip of the vehicle k for the day

h + 1, h ∈ 1...np (P

np,0,k

deﬁnes the charging at night

for day 1 for the vehicle k). A feasible solution to our

problem is composed of a set of routes and a charging

planning for each vehicle over the planning horizon.

A feasible route is a sequence of nodes that satisﬁes

the following set of constraints:

• Each route must start and end at the depot;

• the overall amount of goods delivered along the

route, given by the sum of the demands q

for each

visited customer, must not exceed the vehicle ca-

pacity Q;

• the total duration of each route, calculated as the

sum of all travel durations required to visit a set of

customers, the time required to charge the vehicle

during the day, the service time of each customer,

could not exceed T ;

• no more than m electric vehicles are used;

• each customer should be visited once during the

planning horizon, and the visit day must be in

D(i).

The PEVRP consists of assigning each client i to one

service day deﬁned by D(i) to minimize the total

cost of routing and charging over H. The objective

function to be minimized is f (x) = α × f

(x) +Cc ×

nbs(x) where: f

is the total distance of the solution x

over the planning period H, and nbs is the total num-

ber of visits to charging stations over the planning pe-

riod. α is a given weight representing the cost of one

unity of distance.

4 SOLVING APPROACHES

The PEVRP is obviously NP-hard because it includes

the basic (single-period) EVRP as a particular case, so

large instance can hardly be solved by exact methods.

The best way to tackle this problem is using heuristic

approaches. In this section, we investigated extension

of the best insertion heuristic, namely BIH (Best In-

sertion Heuristic). Another approach, based on Clus-

tering Analysis, namely CLH (Clustering heuristic) is

also proposed. In the following, we provide details of

each heuristic.

4.1 Best Insertion Heuristic (BIH)

The Best Insertion Heuristic (BIH) directly builds in

parallel the tours for each day. Roughly speaking, m

tours, initially empty, are deﬁned for each day. At

each iteration and for each day, we consider the inser-

tion in all available non-empty tours and in one new

empty tour without exceeding m tours by day. For

each customer i and for any possible day d ∈ D(i),

the algorithm simulates the insertion of i in all pos-

sible positions of all considered tours of period d. If

the residual vehicle energy is not enough to add i in

a given position k of a considered tour tr, the algo-

rithm simulates the insertion of i and a charging sta-

tion b ∈ B ∪{0} simultaneously. For the insertion of

b, we choose the less costly charging station that al-

low satisfying the energy constraint. The total cost

variation of f (x) is computed for each insertion si-

mulation. The customer (and eventually the charging

station) with the minimum insertion cost is inserted

at the end of each iteration at its best position. The

best position is given by a day h ∈D(i), the tour tr in

the considered tours of h, and a position k ∈ tr for i

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

266

(and if needed a position k

∈tr for b ). The heuristic

stops if all clients are inserted or when no insertion is

possible.

4.2 Clustering Heuristic (CLH)

The proposed CLH algorithm proceeds in four steps.

The ﬁrst step aims at creating m initial clusters in each

day, one for each available vehicle. In the second step,

for each day d, the algorithm CLH tries to dispatch the

maximum number of remaining customers in the m

available clusters of the day d, without inserting any

charging station, and using as criterion the smoothing

of Distance(cl

) overs all clusters of the day d. In step

3, the customers not inserted in step 2 due to energy

constraint, will be considered. In this step, the in-

sertion objective is the minimization of the additional

energy consumption for each cluster. Finally, in the

fourth step, a best insertion TSP heuristic is used to

ﬁnd a feasible route in each cluster for each day. More

precisely, for each cluster cl

, we compute an estima-

tion of 1) the distance Distance(cl

), 2) the energy

consumption Energy(cl

), and 3) the Time Time(cl

)

needed to perform an electric TSP tour, starting from

depot, visiting once all nodes in cl

, and charging sta-

tions if necessary, and returning to the depot. We also

compute the Load(cl

) that represents the total quan-

tity to be served in the cluster cl

. Details of the CLH

steps are given bellow.

Distance(cl

) is an estimation of the length of the

tour starting at the depot, visiting all customers in cl

and ends at the depot. It is computed according to (1)

(2) (3).

Distance(cl

) = min(Distance

(cl

), Distance

(cl

))

(1)

Distance

(cl

) is an overestimation computed ac-

cording to the costly arc in cl

Distance

(cl

) uses an approximation based on

the distance of each customers to the center of the

cluster.

Distance

(cl

) = |cl

−1|× max

i, j∈cl

i, j

+ 2 ×max

i∈cl

i,0

(2)

Distance

(cl

) = 2×

∑

i∈cl

center(cl

),i

+2×d

center(cl

),0

(3)

Energy(cl

) = r ×Distance(cl

)

Time(cl

) = min(Time

(cl

), Time

(cl

))

Time

(cl

) (respectively Time

(cl

)) is computed as

Distance

(cl

)(respectively Distance

(cl

)) by repla-

cing d

i, j

by t

i, j

Load(cl

) =

∑

i∈cl

Check − Energy − Feasibility(cl

): this function

check the energy feasibility of the solution that will

be obtained in cl

and return true if Energy(cl

) ≤ E.

Check −Constraints −Feasibility(cl

): this function

check the feasibility of the solution that will be obtai-

ned in cl

and return true If Load(cl

) ≤ Q and

Time(cl

) ≤ T .

Step 1. Cluster Initialization

The cluster initialization starts by assigning all cus-

tomers having one allowed day visit (all i ∈V , with

|D(i)|= 1), because these clients will be the most dif-

ﬁcult to insert in the tours. Let L

be the list of custo-

mers of the day d, and ListE

the list of the exclusive

clients i for day d, such that D(i) = {d}. The cluster

initialization algorithms is given bellow :

1. d := 1

2. ListE

= {i ∈ V, |D(i)| = 1 and D(i) = {d}},

Nbcluster

=|ListE

3. If Nbcluster

≥ m go to step 4, Else go to step 7

4. Initialization: for each given client i ∈ListE

, one

cluster cl

d,a

= {i} is created

5. While Nbcluster

≥ m do search for the closest

clusters a and b, such that Check −Constraints −

Feasibility(a ∪b) = true, merge a and b in one

new cluster a and delete b.

6. d = d + 1, if d = np + 1 stop else go to 2

7. For each given customer i ∈ListE

, a cluster cl

d,a

= {i} is created. As we need to have m initial clus-

ters, we must add m −Nbcluster

clusters. Let

be L(z) a list of the customers not considered in

the available clusters (d ∈ D(i) and |D(i)| = z,

z ∈2..np). Choose randomly m−Nbcluster

cus-

tomers such as each customer form a new clus-

ter considering at ﬁrst L(2), then L(3), and so

L(z + 1) until m clusters are formed.

Step 2. Customer’s Insertion without Additional

Energy

Let list

be the list of available clusters over all the

planning horizon, and L be the list of customers i

not assigned to any cluster, sorted in increasing or-

der according to the value |D(i)|. The algorithm

repeats the following two steps until L = ∅ . In

the ﬁrst step, the algorithm selects i from the head

of L, and scans all feasible insertions in each day

d ∈ D(i) and in each cluster of day d, using Check −

Constraints − Feasibility(cl

∪{i}). In the second

step, the cluster a (see formula (4)) that minimizes

the distance increases over all clusters is selected (if

Check −Constraints − Feasibility(cl

∪ {i})= false

Constructive Heuristics for Periodic Electric Vehicle Routing Problem

267

for each d then put a := ∅). If a 6= ∅, and if

Check −Energy −Feasibility(cl

∪{i}) = true, insert

i in a and delete i from L, else insert i in L

. If a = ∅,

delete i from L and insert i in L

a = arg min

∀i∈L

∀cl

∈list

{Distance(cl

∪{i})−Distance(cl

)}

(4)

At the end of this algorithm, the list L

will contain all

customers ejected due to the violation of capacity and

time constraint. The list L

will contain the clients

ejected due to the violation of the energy constraint.

All customers in L

will be introduced in the next step.

Step 3. Customer’s Insertion with Additional

Energy

This step aims to insert all customers from L

while

minimizing the increase of the energy consumption.

We know that all customers in L

verify the capa-

city and the time constraints for at least one clus-

ter. At ﬁrst, L

is sorted according to the value

of |D(i)|, then we select the cluster a that veriﬁes

Check −Constraints −Feasibility(a ∪{i}) and mini-

mizes the energy increase according to the following

formula.

a = arg min

∀i∈L

∀cl

∈list

{Energy(cl

∪{i}) −Energy(cl

)}

(5)

Step 4. Route Construction

In this step, each cluster will be considered to con-

struct a tour using the best insertion method. This

method builds a tour, starting by an empty tour com-

posed by a loop in the depot, and extends the tour

customer per customer. At each iteration, for each

customer i, the algorithm simulates the insertion of i

in each position in the tour. If the residual vehicle

energy is not enough to add i, the algorithm simulates

simultaneously the insertion of i and a charging sta-

tion b ∈ B ∪{0}. The total cost variation of f (x) is

computed for each insertion simulation. The custo-

mer i with the minimum insertion cost is inserted at

it best position at the end of each iteration.The cost

of inserting a customer includes the cost of inserting

charging stations if necessary.

5 COMPUTATIONAL

EXPERIMENTS

5.1 Data Sets

Our methods are implemented using C++. All com-

putations are carried out on an Intel Core (TM) i7-

5600U CPU, 2.60 GHz processor, with 8GB RAM

memory. In this paper, we proposed a new PEVRP

instances inspired by the data instances provided by

(Felipe et al., 2014). These instances are divided on

two types: in instances of type A, the depot is cen-

trally located and there are nine charging stations, and

in instances of type B, the depot is at a corner and

there are only ﬁve charging stations (including char-

ging at the depot).

We considered a limited homogeneous ﬂeet of

vehicles and we adjust parameters of cited above ar-

ticle to our problem. We consider the following set-

tings:

• Number of customers: n=100 and n=200

• Number of vehicles: in the interval [

√

]

• Battery capacity: 20 kWh (equivalent to a range

of 160 km).

• Energy consumption: 0.125 kWh/km.

• Average speed: 80 km/h.

• Vehicle capacity: 20000 Kg.

• Maximum route duration: 8 h.

• Recharging power: 10kWh and a ﬁxed cost of

charging service of 2.5 e

Furthermore, we randomly generated visit days

for each client respecting the following rule: in each

day, the exclusive clients are selected randomly, then

for the rest of clients we randomly generate the visi-

ting days.

As the generation of visiting days being inﬂuen-

tial on the toughness of the problem, we generate 10

instances for each setting parameters by varying the

visiting days for each customer.

5.2 Comparative Analysis

In this section, we analyse the performance of the pro-

posed methods on the PEVRP generated instances.

Our ﬁrst computation are on instances with 100

clients and 2 vehicles and we allow only two visiting

days. Table 1 provides results of heuristics CLH and

BIH on ﬁve setting parameters for each type-instance

(A and B). For each setting we generate 10 random in-

stances.Thus, we have 100 tested instances in table 1.

The average CPU time in seconds (CPU), the average

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

268

Table 1: Instances with 100 clients and 2 vehicles.

CLH BIH

CPU f (x) N.V.C CPU f (x) N.V.C

A1 0,44 90,47 0,00 % 4,10 87,56 0,00%

A2 0,56 104,69 0,00 % 4,42 88,01 0,00%

A3 0,61 102,25 0,00 % 3,85 83,73 0,00%

A4 0,56 106,49 0,00 % 4,14 91,61 0,00%

A5 0,65 105,31 0,40% 4,51 97,98 0,00%

B1 0,46 61,13 0,00% 4,71 45,09 0,00%

B2 0,63 65,45 0,00% 5,37 48,36 0,00%

B3 0,65 65,91 0,00% 5,24 47,68 0,00%

B4 0,59 68,34 0,00% 5,11 50,47 0,00%

B5 0,62 66,74 0,00% 4,86 47,87 0,00%

Average 0,58 83,68 0,04% 4,63 68,84 0,00%

cost of the solution (f(x)) and the average number of

non serviced clients (NVC) over all 10 instances are

reported in table 1 as results of computation. The

CPU time is indicated in the ﬁrst column, the cost of

solution and the number of non serviced clients are re-

ported in the second and third columns, respectively,

for each solving method.

Regarding results of table 1, clearly CLH heuris-

tic is faster than BIH heuristic, however, CLH fails to

visit all clients, whereas heuristic BIH is able to vi-

sit all clients in all instances. BIH heuristic performs

Figure 1: Evaluation of the cost variation.

Figure 2: Evaluation of the CPU variation.

Table 2: Instances with 200 clients and 2 days.

v=2

CLH BIH

Inst CPU N.V.C f (x) CPU N.V.C f (x)

A1 2,96 0,30% 104,03 e 28,41 0,00% 107,47 e

A2 3,18 0,00% 105,12 e 27,70 0,00% 112,27 e

A3 3,38 0,70% 108,31 e 28,70 0,20% 111,27 e

A4 2,84 0,00% 109,56 e 28,84 0,00% 110,40 e

A5 3,22 0,90% 110,19 e 26,33 0,60% 111,31 e

B1 3,85 0,00% 93,90 e 41,83 0,00% 71,30 e

B2 4,63 0,00% 98,06 e 38,35 0,00% 74,84 e

B3 4,56 0,00% 97,08 e 36,12 0,00% 73,72 e

B4 4,31 0,00% 98,86 e 35,98 0,00% 74,01 e

B5 4,29 0,00% 98,01 e 35,00 0,00% 74,82 e

Average 3,72 0,19% 102,31 e 32,73 0,08% 92,14 e

v=3

CLH BIH

Inst CPU N.V.C f (x) CPU N.V.C f (x)

A1 1,78 0,30% 140,22 e 26,13 0,00% 138,49 e

A2 2,03 0,20% 137,27 e 24,99 0,00% 128,64 e

A3 1,83 0,60% 142,81 e 25,73 0,00% 138,62 e

A4 1,77 0,00% 144,24 e 25,39 0,00% 134,73 e

A5 1,89 0,60% 138,42 e 25,07 0,10% 135,77 e

B1 1,81 0,10% 97,09 e 30,26 0,00% 68,64 e

B2 2,48 0,00% 104,55 e 30,00 0,00% 70,40 e

B3 2,38 0,00% 104,44 e 27,83 0,00% 69,47 e

B4 2,45 0,00% 105,91 e 31,37 0,00% 73,64 e

B5 2,45 0,00% 104,40 e 29,02 0,00% 71,48 e

Average 2,09 0,18% 121,93 e 27,58 0,01% 102,99 e

better than CLH even if it’s slower. Furthermore, the

heuristic CLH is more suitable for instances of type B

than instances of type A.

For a better performance study of the two heuris-

tics, we considered several settings of parameters. In

the following we will use the notation (n, d, v) to des-

cribe a setting parameters, n being the number of cus-

tomers, d the number of available days and v the num-

ber of vehicles. As before, each type-instance has 5

setting parameters, and for each setting parameters,

we generate 10 instances. Thus, for each setting we

have 50 instances.

Firstly, we set the number of customers to 100,

the number of available days to 2, and the number

of vehicles to 3 and 4. Then we increase the num-

ber of available days to 3, while setting v to 2. Fi-

nally, we took the 200 customers instances, and va-

rying d and v between 2 and 3. The results given in

tables 1 to 5 show that BIH remains better than CLH

in terms of solution cost for all instances. The BIH

method successfully inserted all customers except for

two instances.The average percentage of non-inserted

clients remains very low for BIH, whereas CLH can-

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Table 3: Instances with 100clients and 2 days.

v=2

CLH BIH

Instance CPU N.V.C f (x) CPU N.V.C f (x)

A1 0,44 0,00% 90,47 e 4,10 0,00% 87,56 e

A2 0,56 0,00% 104,69 e 4,42 0,00% 88,01 e

A3 0,61 0,00% 102,25 e 3,85 0,00% 83,73 e

A4 0,56 0,00% 106,49 e 4,14 0,00% 91,61 e

A5 0,65 0,40% 105,31 e 4,51 0,00% 97,98 e

B1 0,46 0,00% 61,13 e 4,71 0,00% 45,09 e

B2 0,63 0,00% 65,45 e 5,37 0,00% 48,36 e

B3 0,65 0,00% 65,91 e 5,24 0,00% 47,68 e

B4 0,59 0,00% 68,34 e 5,11 0,00% 50,47 e

B5 0,62 0,00% 66,74 e 4,86 0,00% 47,87 e

Average 0,58 0,04% 83,68 e 4,63 0,00% 68,84 e

v=3

CLH BIH

Instance CPU N.V.C f (x) CPU N.V.C f (x)

A1 0,38 0,00% 107,63 e 3,92 0,00% 80,95 e

A2 0,46 0,00% 108,85 e 4,77 0,00% 81,97 e

A3 0,49 0,00% 104,18 e 3,83 0,00% 83,40 e

A4 0,33 0,00% 113,88 e 3,47 0,00% 89,08 e

A5 0,36 0,30% 111,09 e 3,59 0,00% 91,80 e

B1 0,45 0,00% 65,99 e 5,80 0,00% 45,84 e

B2 0,42 0,00% 72,25 e 5,38 0,00% 48,55 e

B3 0,50 0,00% 72,32 e 5,61 0,00% 47,36 e

B4 0,36 0,00% 74,01 e 4,52 0,00% 48,75 e

B5 0,42 0,00% 72,78 e 4,15 0,00% 48,32 e

Average 0,42 0,03% 90,30 e 4,50 0,00% 66,60 e

v=4

CLH BIH

Instance CPU N.V.C f (x) CPU N.V.C f (x)

A1 0,31 0,00% 106,07 e 3,56 0,00% 82,41 e

A2 0,32 0,00% 109,17 e 3,52 0,00% 82,82 e

A3 0,41 0,00% 108,48 e 3,45 0,00% 80,83 e

A4 0,33 0,20% 118,76 e 3,73 0,00% 86,98 e

A5 0,35 0,30% 113,57 e 3,72 0,00% 88,83 e

B1 0,33 0,00% 70,62 e 4,80 0,00% 45,84 e

B2 0,31 0,00% 74,37 e 4,39 0,00% 48,55 e

B3 0,36 0,00% 75,86 e 4,16 0,00% 47,36 e

B4 0,33 0,00% 76,72 e 4,31 0,00% 48,75 e

B5 0,34 0,00% 77,20 e 3,67 0,00% 48,32 e

Average 0,34 0,05% 93,08 e 3,93 0,00% 66,07 e

not manage to insert all clients in most instances. The

computing time of BIH remains higher than the com-

puting time of CLH, but this computing time of BIH

is always reasonable (it reaches at maximum 32,73

seconds).

In the following two ﬁgures, we compare the

average CPU and the average cost of the solutions

obtained by the two heuristics in the different settings

of instances.

Table 4: Instances with 100 clients and 3 days.

v=2

CLH BIH

Inst CPU N.V.C f (x) CPU N.V.C f (x)

A1 0,25 0,00% 123,55 e 3,40 0,00% 95,86 e

A2 0,25 0,00% 134,72 e 3,40 0,00% 94,98 e

A3 0,25 0,00% 127,15 e 3,20 0,00% 92,56 e

A4 0,25 0,00% 135,92 e 3,13 0,00% 98,83 e

A5 0,26 0,00% 130,60 e 3,36 0,00% 98,90 e

B1 0,18 0,00% 123,55 e 3,56 0,00% 45,84 e

B2 0,18 0,00% 134,72 e 3,38 0,00% 48,55 e

B3 0,18 0,00% 127,15 e 3,17 0,00% 47,36 e

B4 0,18 0,00% 135,92 e 3,46 0,00% 48,75 e

B5 0,19 0,00% 130,60 e 3,26 0,00% 48,32 e

Average 0,22 0,00% 130,39 e 3,33 0,00% 71,99 e

Table 5: Instances with n=200 and day=3.

v=2

CLH BIH

Inst CPU N.V.C f (x) CPU N.V.C f (x)

A1 1,63 0,10% 153,76 e 20,83 0,00% 148,40 e

A2 1,47 0,00% 162,50 e 20,28 0,00% 136,95 e

A3 1,50 0,20% 162,69 e 21,01 0,00% 146,32 e

A4 1,69 0,00% 162,47 e 19,93 0,00% 149,88 e

A5 1,70 0,40% 164,34 e 21,87 0,00% 147,28 e

B1 1,81 0,10% 97,09 e 25,21 0,00% 74,81 e

B2 2,48 0,00% 104,55 e 25,57 0,00% 77,17 e

B3 2,38 0,00% 104,44 e 24,31 0,00% 79,56 e

B4 2,45 0,00% 105,91 e 28,36 0,00% 77,54 e

B5 2,45 0,00% 104,40 e 28,01 0,00% 78,89 e

Average 1,96 0,08% 132,21 e 23,54 0,00% 111,68 e

If we compare the two heuristics for the 100 custo-

mers and 2 days instances in terms of cost, we could

see that the BIH heuristic performance is improved

by the increase of v, which is explained by the more

possibilities given to insert the customers and the less

needs to visit charging stations. In the other hand, the

cost of solutions given by the CLH heuristic is increa-

sing, which is due to the fact that the heuristic use in-

evitably all the vehicles every day. We predicate that

there is an optimal number of vehicles for the CLH

and that a smaller or even bigger ﬂeet increases the

cost. BIH being not constrained to use all the vehi-

cles.

6 CONCLUSION

This paper addresses a new extension of the EVRP,

named PEVRP (Periodic Electric VRP), in which the

routing and charging are planned over a multi-period

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270

horizon, subject to frequency constraints, limited ﬂeet

of electric vehicles available at the depot, intermedi-

ate facilities for charging during the trips, and partial

charging. We have proposed two constructive heu-

ristics, Best Insertion Heuristic (BIH) and Clustering

Method (CLH). These methods are tested on instan-

ces derived from EVRP instances with up to 200 cus-

tomers. Computational experiments show the effecti-

veness of the BIH approach. In further researches, we

aim to improve the CLH method by a post optimisa-

tion step, and to propose a metaheuristic approach.It

will also be interesting to develop more efﬁcient lo-

wer bounds. Another perspective is to generalize the

PEVRP model considering that f (i) ≥ 1.

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