A Survey on the Electric Vehicle Routing Problem: Variants...

Review ArticleA Survey on the Electric Vehicle Routing Problem: Variants andSolution Approaches

Tomislav ErdeliT and TonIi CariT

Faculty of Transport and Traffic Sciences, University of Zagreb, Vukelićeva Street 4, Zagreb, Croatia

Correspondence should be addressed to Tomislav Erdelić; [email protected]

Received 14 December 2018; Revised 5 March 2019; Accepted 2 April 2019; Published 9 May 2019

Guest Editor: Eduardo Lalla-Ruiz

Copyright © 2019 Tomislav Erdelić and Tonči Carić. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In order to ensure high-quality and on-time delivery in logistic distribution processes, it is necessary to efficiently manage thedelivery fleet. Nowadays, due to the new policies and regulations related to greenhouse gas emission in the transport sector,logistic companies are paying higher penalties for each emission gram of CO2/km. With electric vehicle market penetration, manycompanies are evaluating the integration of electric vehicles in their fleet, as they do not have local greenhouse gas emissions,produce minimal noise, and are independent of the fluctuating oil price. The well-researched vehicle routing problem (VRP) isextended to the electric vehicle routing problem (E-VRP), which takes into account specific characteristics of electric vehicles.In this paper, a literature review on recent developments regarding the E-VRP is presented. The challenges that emerged with theintegration of electric vehicles in the delivery processes are described, together with electric vehicle characteristics and recent energyconsumption models. Several variants of the E-VRP and related problems are observed. To cope with the new routing challenges inE-VRP, efficient VRP heuristics and metaheuristics had to be adapted. An overview of the state-of-the-art procedures for solvingthe E-VRP and related problems is presented.

1. Introduction

The vehicle routing problem (VRP) is an NP-hard optimiza-tion problem that aims to determine a set of least-cost deliv-ery routes from a depot to a set of geographically scatteredcustomers, subject to side constraints [1]. The problem wasfirst defined by Dantzig and Ramser [2] as the Truck Dis-patching Problem. VRP is a generalization of the well-knowntraveling salesman problem (TSP), which aims to design oneleast-cost route to visit all the customers. The problem hasapplications in several real-life optimization problems, whichhas led to the definition of many problem variants over theyears: limited vehicle load capacity (capacitated VRP, CVRP),customer time windows (VRPwith time windows, VRPTW),multiple depots (multidepot VRP, MDVRP), pickup anddelivery (VRP with pickup and delivery, VRPPD), time-dependent travel time (time-dependent VRP, TD-VRP), het-erogeneous fleet (mixed fleet VRP, MFVRP), etc. [3, 4].Due to the complexity of the problem, exact procedures areonly capable of optimally solving small-sized problems: up

to 360 customers for CVRP [5] and 50-100 customers forVRPTW [6]. Over the years, a vast number of heuristics,metaheuristics, and hybrid procedures were proposed forsolving different VRP problems.

In the past decade, the European Union (EU) hasannounced many new actions and regulations related togreenhouse gas (GHG) emissions in the transport sector[7]. External factors and the rise of social and ecologicalawareness have prompted green initiatives in many com-panies. Conventional internal combustion engine vehicles(ICEVs), which are dependent on limited fossil fuels, severelypollute the environment, especially in congested urban areas.According to WEEA [8], the EU intends to decrease GHGemissions by 20% and 40% by 2020 and 2030, respectively.Sbihi and Eglese [9] introduced the research field of greenlogistics, which deals with the sustainability of deliveryprocesses by taking into account environmental and socialfactors. With the electric vehicle (EV) market penetration,many logistic companies evaluated the use of the EVs intheir vehicle fleet in order to decrease GHG emissions and,

HindawiJournal of Advanced TransportationVolume 2019, Article ID 5075671, 48 pageshttps://doi.org/10.1155/2019/5075671

http://orcid.org/0000-0002-8012-0090http://orcid.org/0000-0001-8564-4304https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5075671

2 Journal of Advanced Transportation

therefore, reduce the charges for every emission gram ofCO2/km. EVs have several advantages compared to ICEVs:(i) they do not have local GHG emissions; (ii) they produceminimal noise; (iii) they can be powered from renewableenergy sources; and (iv) they are independent of the fluctu-ating oil price [10, 11]. There are two basic configurations ofEVs: the battery electric vehicle (BEV), which is exclusivelypowered from batteries mounted inside the vehicle, and thehybrid electric vehicle (HEV), which can be powered frombatteries inside the vehicle or by other energy sources, mostcommonly internal combustion engine. The plug-in HEV(PHEV) can be recharged by connecting the plug to theelectric power source. In this paper mostly BEVs for logisticpurposes are considered, where two main problems come tothe fore: limited driving range and the need for additionalrecharging infrastructure.

Due to the limited battery capacity, the range that deliveryBEVs can achieve with a fully charged battery is 160-240 km[12], which is much lower than the 480-650 km range ofICEVs [13]. To achieve a similar driving range as ICEVs, BEVshave to visit charging stations (CSs) more frequently. Today,there is still a lack of CSs in the road network infrastructure,and their locations and energy demand should be plannedin future infrastructural plans. For an empty BEV to becomeoperable again, battery energy has to be renewed at a CS.This can be performed in two ways: (i) by swapping emptybatteries with fully charged ones at Battery Swapping Station(BSS) or (ii) by charging at CS [14, 15]. The former processcan be performed in a time comparable to the refueling timeof ICEVs. In the latter process BEVs recharge their batteries atCSs by plugging into the electric power source. The rechargetime depends on the state of charge (SoC) when entering theCS, the desired SoC level when leaving CS, and the chargingfunction.

1.1. Recent Literature Reviews and Scientific Contribution.Here we present, to our knowledge, the most recent literaturereviews on the E-VRP and related problems.

Juan et al. [16] presented a review regarding the envi-ronmental, strategic, and operational challenges of EV inte-gration in logistics and transportation activities. The authorsperformed a comprehensive analysis of environmental chal-lenges including the transportation impact on the pollutionand EVs’ possible contribution to the reduction of carbonemissions. Regarding the strategic challenges, the authorspresented issues related to CSs: battery swap technology,different charging technologies, CS location problem, limitednumber of charges, and charging network; issues related tothe EVs: mixed fleet of ICEVs and EVs, economic challengeswhen integrating EVs in a fleet, and routing constraints.The solution approaches for the E-VRP were presented ingeneral, as a class of VRP solving procedures. Similarly,Margaritis et al. [17] presented practical and research chal-lenges of EVs focusing on battery development, lack ofcharger compatibility, systematic energy management, thelack of optimization procedures that could minimize theEV routing and scheduling decisions, cooling/heating usage,financial sources, novel policies,measures for EVdeploymentin transport services, etc.

Montoya [18] researched several variants of the E-VRP:green VRP (GVRP), E-VRP with partial recharging andnonlinear charging functions, and the technician routingproblem with a mixed fleet of ICEVs and EVs. For eachproblem, effective solving procedures were proposed: multi-space sampling heuristic, iterated local search enhanced withheuristic concentration, and two-phase parallel metaheuris-tic based on solving a set of subproblems and extended set-covering formulation. The authors also formulated a fixedroute vehicle charging problem (FRVCP) with and withouttime windows to optimize the charging decisions for a routewith a fixed customer sequence. The authors did not focus asmuch on reviewing the E-VRP literature, especially not on theprocedures for solving the problem. Compared to Montoya[18], this paper did not focus as much on the FRVCP andtechnician routing problem or on detailed procedures usedto solve those problems.

The most recent survey on the E-VRP is presented byPelletier et al. [19] and it includes technical backgroundon EV types and batteries, EV market penetration, EVcompetitiveness and incentives, and an overview of theexisting research regarding EVs in transportation science.The authors provided a comprehensive review on, at the time,the latest solving procedures for the E-VRP with mixed fleetand optimal paths and covered several key papers regardingpartial recharges, hybrid vehicles, and different chargingtechnologies.

In this paper, a survey on the E-VRP is presented,which includes approaches for solving the E-VRP and relatedproblems that emerged with BEVs integration in the logisticprocesses.The focus is not on the economic and environmen-tal challenges related to BEVs. Most of the latest literaturereviews were published in 2016; hence, to the best of ourknowledge, there is no published research that summarizesthe state-of-the-art research in the E-VRP field. In this paperwe outline the following contributions:

(i) a review of the recent energy consumption modelsthat could be used in BEV routing models;

(ii) an updated literature review and a concise tablesummary of already reviewed E-VRP variants suchas GVRP, mixed fleet, BSSs, partial recharges, anddifferent charging technologies;

(iii) a review of the additional emerged E-VRP variants,which include hybrid vehicles, CS siting, nonlinearcharging function, dynamic traffic conditions andcharging schedule optimization;

(iv) a comprehensive analysis of operation research proce-dures in the E-VRP, which includes an overview of theprocedures employed for solving various E-VRP vari-ants, highlighting state-of-the-art procedures, and aconcise table summary of the applied procedures.

1.2. Organization of This Paper. The remainder of this paperis organized as follows. In Section 2, the E-VRP is describedand the literature review of the basic problem formulationis presented. In Section 3, basic characteristics and recentenergy consumption models of BEVs are described together

Journal of Advanced Transportation 3

with the BEV’s application and evaluation in the deliveryprocesses. In Section 4, variants of the E-VRP are presentedwith some related problems from the literature. In Section 5,approaches for solving E-VRP are presented, which includestate-of-the-art exact, heuristic, metaheuristic, and hybridprocedures.The conclusion and future research directions aregiven in Section 6.

2. Electric Vehicle Routing Problem

With BEV penetration in logistic distribution processes, aproblem of routing a fleet of BEVs has emerged: the E-VRP. The E-VRP aims to design least-cost BEV routes inorder to serve a set of customers by taking into accountoften used constraints: vehicle load capacity, customer timewindows, working hours, etc. [3, 20]. Additionally, BEVshave the limited driving range which directly corresponds tomore frequent recharging events at CSs. CSs can be built atseparate locations as public CSs or mounted at customers’locations as private CSs. The time needed to travel to a CSand the recharging time are important aspects of fleet routing,especially if customer time windows are taken into account.

To the best of our knowledge, the first research regardingthe routing of an electric fleet was published by Gonçalveset al. [21], with the authors observing VRPPD using a mixedfleet of EVs and ICEVs. Refueling of the vehicle is performedat the location where the need for the refueling occurred andthe total refuel time is computed, based on the total distancetraveled by the EV. Conrad and Figliozzi [22] formulatedrecharging VRP in which vehicles with limited driving rangeare allowed to refuel at customers’ locations during the routeto up to 80% of the vehicle’s battery capacity. The authorsdescribed an application of the model and analyzed theimpact of different driving ranges, fixed charging times, andtime windows on EV routes and solution quality. The resultsindicated that customer time windows greatly limit routedistance when recharging time is long and vehicle rangeis constrained. Erdoğan and Miller-Hooks [23] formulatedGVRP in which a fleet of vehicles is powered by alterna-tive fuels (alternative fuel vehicle, AFV): biodiesel, ethanol,hydrogen, methanol, natural gas, electricity, etc. AFVs canrefuel at separately located stations, with fixed refuelingtime. The authors did not consider customer time windowsand vehicle load capacity constraints. New problem-specificinstances were developed and two heuristics for solving theproblem were applied. The results showed that the limitationof driving range severely increased the number of refuelingstations and the total traveled distance in the solution. Asimilar problem was researched by Omidvar and Tavakkoli-Moghaddam [24], in which the authors added vehicle loadconstraint, customer time windows, and congestion manage-ment, as the vehicle can stay at the customer’s location duringthe congestion hours. The authors minimized emission andpollution costs by applying commercial software for solvingsmall instances and two metaheuristics for solving largerinstances. Schneider et al. [25] published the first researchon routing a BEV fleet by taking into account possiblevisits to CSs and charging time dependent on the SoC levelwhen entering a CS. The problem was formulated as E-VRP

with time windows (E-VRPTW) as they took into accountload, battery, and time window constraints. The authorsformulated E-VRPTW as the mixed integer linear program(MILP) on the complete directed graph 𝐺, where customersare modeled as graph vertices and paths between customersare modeled as graph arcs. Here, the basic model formulationis given. Let 𝑉 = {1, . . . , 𝑁} be a set of geographicallyscattered customers who need to be served, and let 𝐹 be a setof CSs for BEVs. In order to allow multiple visits to the sameCS, a virtual set of CSs 𝐹 is defined. Vertices 0 and 𝑁 + 1denote the depot, and every route begins with vertex 0 andends with vertex 𝑁 + 1 (𝑉0,𝑁+1 = 𝑉 ∪ {0} ∪ {𝑁 + 1}). Graph𝐺 is defined as 𝐺 = (𝑉0,𝑁+1 ∪ 𝐹, 𝐴), where 𝐴 is set of arcs,𝐴 = {(𝑖, 𝑗) | 𝑖, 𝑗 ∈ 𝑉0,𝑁+1 ∪ 𝐹, 𝑖 ̸= 𝑗}. Depending on the real-life constraints, different (𝑖, 𝑗) arc values can be interpretedas distance 𝑑𝑖𝑗, travel time 𝑡𝑖𝑗, energy consumption 𝑒𝑖𝑗, speedV𝑖𝑗, cost 𝑐𝑖𝑗, etc. The binary variable 𝑥𝑖𝑗 = {0, 1} is equal to 1if arc (𝑖, 𝑗) is traversed in the solution, and 0 otherwise. Thewhole MILP program for the E-VRPTW with equations forload, battery, time windows, flow, and subtour constraints ispresented by Schneider et al. [25].

In the VRP, it is customary for the primary objectiveto minimize the total number of vehicles used (1) and thento minimize the total distance traveled (2) or some otherobjective functions [26]. Total vehicle number is a primaryobjective as generally greater savings can be achieved withfewer vehicles (vehicle fixed costs, labor cost, etc.). Such anobjective is contradictory as with fewer vehicles total traveleddistance increases and vice versa. By taking into account thehigh purchase cost of BEVs, such a hierarchical objectiveseems justifiable in BEV routing applications [25, 27].

min ∑𝑗∈𝑉∪𝐹

𝑥0𝑗 (1)min ∑

𝑖∈𝑉0∪𝐹,𝑗∈𝑉𝑁+1∪𝐹

,𝑖 ̸=𝑗

𝑑𝑖𝑗𝑥𝑖𝑗 (2)Objective functions can be complex with simultaneous mini-mization of vehicle number, total traveled distance [28], totaltravel times [29], total routing cost and planning horizon [11,27, 30, 31], GHG emission [32, 33], energy consumption [34–36], etc. Total routing costs of BEVs usually consist of BEVacquisition cost, circulation tax, maintenance, costs relatedto the energy consumption (electric energy price), cost ofbattery pack renewal after its lifetime, labor costs, etc. Insteadof a single-objective function, some authors use multiobjec-tive function, i.e., fuel consumption and total driving time[37], fuel consumption and route cost [38], battery swappingand charge scheduling [39], etc. An overview of differentobjectives in E-VRP is presented in Table 1 in the columnObjective.

3. Battery Electric Vehicles inDelivery Processes

The major problem that BEVs in delivery processes arefacing is the limited driving range. Grunditz and Thiringer[101] analyzed over 40 globally available BEVs, which can


Table1:Overviewof

theE

-VRP

varia

ntsa

ndrelatedprob

lems.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Bektaş

andLapo

rte

[40]

PRP

XX

Emissionmod

el,speed

limitatio

n

Totalcosts:

labo

r,fueland

emissionas

afun

ctionof

load

andspeed

Con

radandFigliozzi

[22]

Recharging

VRP

XX

XVe

hiclen

umbera

ndtotal

travelingcosts

:distance,service

time,recharging

time

Gon

çalves

etal.[21]

VRP

PDwith

mixed

fleet

XX

XNoCS

locatio

n,tim

econstraint

Totaltravelin

gcosts

:fixedand

varia

ble

Dem

iretal.[32]

PRP

XX

Emissionmod

el

Totaltravelin

gcosts

:labor,fuel

andem

issionas

afun

ctionof

load

andspeed;speed

optim

ization

Erdo

ğanand

Miller-H

ooks

[23]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Omidvara

ndTavakkoli-

Moghadd

am[24]

GVRP

XX

XX

AFV

s,lim

itedfuel

capacityandroute

duratio

n,congestio

nmanagem

ent

Totalcosts:

vehiclefi

xedcosts

,dista

nce,tim

eand

emission

Abdallah[41]

PHEV

RPTW

XX

XX

XElectriccharge

costis

neglected

Routingcosts

:tim

erun

onthe

fossiloil

Barcoetal.[34,42]

E-VRP

and

charge

schedu

ling

XX

XX

X

Energy

consum

ption

andbatte

rydegradation

mod

el,priv

atea

ndpu

blicCS

,tim

e-depend

entenergy

rates

Energy

consum

ption

DavisandFigliozzi

[10]

XX

Energy

consum

ption

mod

elwith

speed

profi

les,lim

itroute

duratio

nandenergy

consum

ption(battery

capacity),no

recharging

durin

gther

oute

Totalcosts:

vehiclep

urchase,

energy,m

aintenance,tax

incentive,batte

ryreplacem

ent,

routing

VanDuinetal.[43]

FSMVRP

TW,

EVFSMVRP

TWX

XX

Norecharge,range

constrainedby

batte

ry,

relaxedtim

ewindo

wconstraints,em

ission

Totalcosts:

vehiclefi

xedcosts

,tim

eand

dista

nce


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Adlera

ndMirc

hand

ani[44

]Onliner

outin

gof

BEVs

XBa

ttery

reservations,

waitin

gforfullycharged

batte

ryTh

eaverage

vehicled

elay

time

Alesia

niandMaslekar

[45]

BEVrouting

X

Waitin

gtim

ecostatC

Sisprop

ortio

naltothe

numbero

fEVsinCS

,lim

iting

then

umbero

fCS

sinroutea

ndnu

mber

ofvehiclesin

CS,energy

consum

ptionmod

el

Traveling,charging

andenergy

consum

ptioncosts

Dem

iretal.[37]

PRP

XX

Fuelconsum

ptionand

emissionmod

el

Bi-objectiv

eminim

izationof(1)

fuelconsum

ptionand(2)tot

aldrivingtim

e

Felip

eetal.[46]

GVRP

-MTP

RX

XX

XTo

talrechargingcosts

:fixedand

varia

ble

Preise

tal.[35]

Energy-

optim

ized

routingof

BEVs

XX

XEn

ergy

consum

ption

mod

elEn

ergy

consum

ption

Sassietal.[47]

Sassietal.[48]

Sassietal.[49]

HEV

RP-TDMF

VRP

-HFC

CVRP

-MFH

EVX

XX

XX

Time-depend

ent

charging

costs

,operatin

gwindo

wsa

ndpo

wer

limitatio

nof

CS,

compatib

ilityof

BEVs

with

chargers,electric

itygrid

capacity[47,49]

Vehiclen

umbera

ndtotalcosts:

fixed,rou

ting,charging

and

waitin

gcosts

Schn

eidere

tal.[25]

E-VRP

TWX

XX

Vehiclen

umbera

ndtotal

traveled

dista

nce

Zünd

orf[50]

EVRC

XX

XX

Batte

ryconstrainedSP

P,different

CStypes:

regu

lar,superchargers

andBS

S,energy

consum

ptionmod

el

Traveltim

e

Brug

lierietal.[29,51]

E-VRP

TWX

XX

XVe

hiclen

umbera

ndtotaltravel,

recharging

andwaitin

gtim

e

Goeke

andSchn

eider

[30]

E-VRP

TWMF

XX

XX

Energy

consum

ption

mod

el:varying

BEV

load,roadslo

pe

Vehiclen

umbera

nddifferent

objectives:(i)dista

nce,(ii)c

osts:

labo

r,driver

wage,vehicle

prop

ulsio

n-e

lectric

energy

and

dieselcosts

,(iii)(ii)

+batte

ryreplacem

entcost


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Lebeau

etal.[52]

FSMVRP

TW-

EVX

XX

X

Energy

consum

ption

mod

elbasedon

the

collected

data,recharge

onlyatthed

epot

Totalcosts:

vehiclesfixed

and

operatingcosts

,labor

costs

Moghadd

am[53]

E-VRP

TWPR

XX

XX

CapacitatedCS

sVe

hiclen

umbera

ndnu

mbero

fCS

s

Pourazarm

etal.[54]

Sing

le/M

ulti

BEVrouting

XX

XHom

ogeneous

and

in-hom

ogeneous

CSs

Totaltim

e

Schn

eidere

tal.[55]

VRP

IS(EVRP

RF)

XX

Totaltraveland

fixed

vehicle

costs

Yang

andSun[56]

BSS-EV

-LRP

XX

XTo

talrou

tingandconstructio

ncosts

Desaulniersetal.[57]

E-VRP

TW-SF/MF/SP

/MP

XX

XX

Vehiclen

umbera

ndtotalrou

ting

costs

Dop

pstadt

etal.[58]

HEV

-TSP

XX

Four

mod

esof

travel:

combu

stion

,electric

,charging

andbo

ost

mod

e,no

CSsv

isits

Totalcostsas

long

asmaxim

alrouted

urationisno

toverrun

Hierm

annetal.[31]

E-FSMFT

WX

XX

XVe

hiclen

umbera

ndtotalcosts:

vehiclefi

xedandroutingcosts

Keskin

andÇa

tay[

28]

E-VRP

TWPR

XX

XX

Vehiclen

umbera

ndtotal

traveled

dista

nce

Koça

ndKa

raoglan

[59]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Linetal.[60

]E-VRP

XX

Energy

consum

ption

mod

elandload

effect

Totalcosts:

batte

rycharging

,traveltim

eand

waitin

gcosts

Masliakova

[61]

Routingand

charging

ofelectricbu

ses

XX

X

Energy

consum

ption

mod

el,two

typeso

fbu

sesd

epending

onthe

charging

event:en-rou

teor

atdepo

t,ho

mogeneous

and

in-hom

ogeneous

CSs

Investm

entand

operations

costs

,atraveltim

eofp

assengers

Mirm

oham

madietal.

[62]

Perio

dicg

reen

VRP

XX

XX

Perio

dicr

outin

g,prim

aryandsecond

ary

timew

indo

ws,sta

tictraffi

ccon

ditio

nswith

inap

eriod

Totalemissions,totalservice

timea

ndpenalties


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Mon

toya

etal.[63]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Robertiand

Wen

[64]

E-TS

PTW

XX

XTo

taltraveleddista

nce

Schiffere

tal.[11,27]

E-LR

PTWPR

XX

XX

X

CSsa

tcustomers’

locatio

ns,picku

pand

delivery,multip

ledriver

shiftsa

ndmultip

leplanning

perio

ds,

emissions

Totalcostsdu

ringthep

lann

ing

perio

d:CS

andBE

Vinvestm

ent

costs

,fixedcosts

(tax,

maintenance),dista

nce

depend

entcosts(energy)

Wen

etal.[65]

E-VSP

XX

Timetablebu

strip

s,multip

ledepo

ts,tim

ewindo

wso

fdepotsa

ndCS

s

Totalcost:vehicles

andtraveling

costs

And

elmin

and

Bartolini[66]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtraveled

dista

nce

ÇatayandKe

skin

[67]

E-VRP

TWPR

XX

XX

XNormalandfastcharger

atCS

sVe

hiclen

umbera

ndtotal

recharging

costs

Froger

etal.[68]

E-VRP

-NL-C

XX

XCapacitatedCS

sTo

taltravel,s

ervice,charginga

ndwaitin

gtim

e

Hof

etal.[69]

BSS-EV

-LRP

XX

XTo

talrou

tingandconstructio

ncosts

LeggieriandHaouari

[70]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Mancini

[71]

HVRP

XFu

llinsta

ntrecharge

Totaltraveleddista

ncew

ithpenalties

foru

singinternal

combu

stion

engine

Mon

toya

etal.[72]

E-VRP

-NL

XX

XCS

types:slo

w,mod

erate

andrapid

Totaltraveland

recharging

time

Schiffera

ndWalther

[73]

E-LR

PTWPR

XX

XX

X

Totald

istance,num

bero

fvehiclesandCS

sused,totalcosts:

investm

entcostsof

BEVsa

ndCS

s,andop

erationalcosts

Shao

etal.[74]

EVRP

-CTV

TTX

XX

XTo

talcosts:

travel,charging,

penalty,and

fixed

vehiclec

osts


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Swedaetal.[75]

Adaptiv

eroutingand

recharging

policieso

fEVs

XX

Heterogeneous

CSs-

the

prob

ability

ofbeing

availablea

ndexpected

waitin

gtim

e,origin-destin

ationpairs

Traveling,waitin

gand

recharging

costs

Vincentetal.[76]

HVRP

XX

Vertex

demandin

CVRP

isassociated

with

travel

timefor

each

arcin

HVRP

Totalcosts

Amiri

etal.[39]

BSSlocatio

n&

schedu

ling

XX

Multi-ob

jective-

minim

ization

of(1)ba

ttery

charging

and

powe

rlossc

osts;(2)de

viation

from

nominalvoltage;and(3)

networkcapacityreleasing

Brug

lierietal.[77]

E-VRe

PX

X

One

way

carsharin

gservice,wo

rkersw

ithbicycles

goto

theE

Vs

locatio

nsandrelocate

them

,battery

level

demandrequ

est

Multi-ob

jective:(1)

minim

izationof

thew

orkers

employed;(2)m

inim

izationof

thed

urationof

thelon

gestroute;

and(3)ma

ximizationof

the

numbero

fservedrelocatio

nrequ

ests

JooandLim

[78]

EVrouting

Energy

SPP,no

recharging

,energy

consum

ptionmod

el

Minim

izee

nergyconsum

ption

andaveragespeed

onthep

ath

Keskin

andÇa

tay[

79]

E-VRP

TW-FC

XX

XX

XVe

hiclen

umbera

ndtotal

recharging

costs

Keskin

etal.[80]

E-VRP

TW-FC

XX

XX

M/M

/1qu

euingsyste

matcapacitatedCS

s,batte

rycapacity

restr

ictio

n,four

planning

intervalsina

day,partialrechargen

otevidentinpaper

Totalcost:energy

cost,

routing,

labo

rand

penalties

forlate

arriv

als

Kullm

anetal.[81]

E-VRP

-PP

XX

X

PublicandprivateC

Ss,

capacitatedCS

,single

charging

techno

logy

per

CS

Expected

timetovisit

allthe

custo

mers

Lietal.[82]

MBF

M&

recharging

prob

lem

XX

Electric,diesel,

compressednaturalgas

andhybrid-dieselbuses

Totaln

etwo

rkbenefit

ofreplacingoldvehicles

with

new

ones

with

inthep

lann

ingho

rizon

andbu

dgetconstraints


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Luetal.[83]

MTF

SPX

XX

X

Travelrequ

estarekn

own

aprio

riin

time-varying

origin-destin

ationtables,

servicea

nddeadheaded

tripsw

ithdifferent

consum

ptionrate,B

EVs

have

high

erprioritythan

ICEV

s

Totaloperatin

gcostof

ataxi

company

Masmou

dietal.[84]

DARP

-EV

XX

XX

Energy

consum

ption

mod

elof

Genikom

sakis

andMitrentsis[85]w

ithconstant

speed,

acceleratio

nandroad

slope;differentvehicle

resources:

accompanyingperson

seat,handicapp

edperson

seat,stre

tcher

and/or

awheelchair,

limiteduser

ridetim

e

Totalrou

tingcosts

(distance)

Paze

tal.[86]

MDEV

LRPT

W-B

S/PR

/BSP

RX

XX

XX

XTh

reeM

IPmod

els

depend

ingon

thep

artia

lrecharge

andBS

STo

taltraveleddista

nce

Pelletie

retal.[87]

EFV-

CSP

XX

X

Preemptivec

harging

with

alim

itednu

mbero

fchargersandcharging

eventsatthed

epot,

time-depend

entenergy

costs

,FRD

charge,grid

restr

ictio

n,cyclicand

calend

arbatte

rydegradation

Totalchargingcosts

Poon

thalirand

Nadarajan

[38]

F-GVRP

X

Fuelconsum

ption

mod

el,varying

speed,

AFV

s,lim

itedroute

duratio

n

Bi-objectiv

e:(1)rou

tingcosts

;and(2)fue

lcon

sumption

Schiffera

ndWalther

[88]

LRPIF

XX

XX

XLo

adingandrefueling

facilities

Totalcosts:

investm

entcostsof

vehiclesandfacilities,routing

costs


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

Schiffera

ndWalther

[89]

RELR

PTWPR

XX

XX

X

Uncertain

custo

mer

patte

rnscenarioso

ver

workingdays

regarding

thes

patia

lcustomer

distr

ibution,

demand

andservicetim

ewindo

ws

Totalcosts:

investm

entcostsof

vehiclesandfacilities,routing

costs

Shao

etal.[90]

E-VRP

XX

Energy

consum

ption

mod

el:cargo

load,

uncertaintravelspeed

Totalcosts:

travel,chargingand

fixed

vehiclec

osts

Wangetal.[91]

BEVrouting

XParkingfee,capacitated

CSs-

queuingtim

e

Multi-ob

jectivem

inim

izationof

(1)traveltim

e:driving,qu

euing

andcharging

time;(2)ch

arging

costs

:electric

ity,service

and

parkingfee;(3)en

ergy

consum

ption

Zhangetal.[36]

E-VRP

X

Energy

consum

ption

mod

el,emissions,static

speed,charging

time

unkn

own

Energy

consum

ption

Bassoetal.[92]

2sEV

RPX

XX

X

Each

CSsc

anhave

different

charging

rate,

energy

consum

ption

mod

elforroad

segm

ents:

speedprofi

le,

road

slope,acceleration

Totalenergyconsum

ption

Breunigetal.[93]

E2EV

RPX

Twoechelons

-first

ICEV

sand

second

BEVs,

fullrecharge

when

visitingCS

s,charging

timeu

nkno

wn

Totalrou

tingcosts

Brug

lierietal.[94]

GVRP

XAFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Froger

etal.[95]

E-VRP

-NL

XX

XTo

taltraveland

charging

time

Hierm

annetal.[96]

H2E-FT

WX

XX

XX

XProp

ulsio

nmod

edecisio

nTo

talcosts:

fixed

andvaria

ble

Jieetal.[97]

2E-EVRP

-BSS

XX

Routingin

twoechelons,

sensitivity

analysisof

batte

rydrivingrange

andvehiclee

missions

Totalrou

tingcosts

,the

batte

rysw

apping

costs

andtheh

andling

costs

atthes

atellites


Table1:Con

tinued.

Reference

Prob

lem

name

Charging

Other

Objectiv

eC

TWMIX

LRP

FL

NL

PRDFC

BSTD

H

KoyuncuandYavuz

[98]

MGVRP

XX

XX

XX

ICEV

s(fixed

recharge

time)

andAFV

s,no

de-

and-arc

MILP

form

ulation,

single

recharge

techno

logy

per

CS,add

ition

almod

eling:

custo

merdemands,

custo

mervehicle

restr

ictio

ns,sub

scrip

tion

orpay-as-you

-go

refuelingcosts

,completely

heterogeneou

sfleet,

last-

mile

deliveryand

closed

timew

indo

ws

Totaltravelin

gcost

Macrin

aetal.[99]

GMFV

RP-

PRTW

XX

XX

XX

X

Sing

lerecharge

techno

logy

perC

S,bu

tdifferent

charging

techno

logies

between

CSs,energy

consum

ptionmod

elfor

road

segm

entswith

time-depend

entspeeds

Costo

fenergyr

echarged

durin

gther

outeandatthed

epot,fuel

costs

,and

costrelatedto

traveled

dista

nce

Macrin

aetal.[33]

GMFV

RP-

PRTW

XX

XX

XX

Sing

lerecharge

techno

logy

perC

S,bu

tdifferent

charging

techno

logies

between

CSs

Recharging

,rou

tingand

activ

ationcosts

,lim

item

issions

Normasarietal.[100]

CGVRP

XX

AFV

s,lim

itedroute

duratio

nVe

hiclen

umbera

ndtotal

traveled

dista

nce

Refer

ence:referencedpaper;Problem

name:thenameof

theanalyzed

prob

lem;13columns

representin

gcharacteris

ticso

fthe

prob

lem

inthefollo

wingorder:C:

vehicleload

(cargo)c

apacity,T

W:customer

time

windo

ws,MIX

:heterogeneous

(mixed)fl

eet,LR

P:locatio

nroutingprob

lem,F:fixed(con

stant)refuel(recharge)tim

e,L:lin

earc

hargingprocess,NL:no

nlinearc

hargingprocess,PR

:partia

lrecharges

trategy,DFC

:different

charging

techno

logies,BS:batte

rysw

apstr

ategy,TD

:tim

e-depend

enttraveltim

es,H

:hybrid

vehicles,O

ther:som

especialcharacteris

ticofthep

roblem

;Objective:theo

bjectiv

efun

ctionforthe

optim

ization.


be categorized into small, medium-large, high-performing,and sports cars. All of the BEV models utilize lithium-basedbatteries, especially lithium-ion [102] with battery capacityand distance varying within the ranges 12-90 kWh and 85-528 km, respectively. An averagemedium-sized personal BEVhas a battery capacity of 30 kWh, which is enough to travel250 km. In the delivery process, mostly light vans and freightBEVs are used, which have a shorter driving range (160-240 km) compared to the driving range of ICEVs (480-650km) [13, 43, 103]. The reason is that the battery has lowerspecific energy (130 Wh/kg) than the fossil oil (1233Wh/kg),and the amount of energy that can be stored in the batteryis much lower than in the fossil oil. Batteries mounted inBEVs are mostly the main cause of high acquisition costsand technical limitations as the battery degrades over time,resulting in decreased maximal capacity. Pelletier et al. [104]concluded that the battery should be replaced after fiveto ten years or after 1,000 to 2,000 cycles with large SoCvariations. The authors also described factors that influencesuch battery degradation: overcharging, overdischarging,high and low temperatures, high SoC during storage, largedepth of discharge, etc., and they presented battery degra-dation models that can be used in goods distribution withBEVs.

3.1. BEV Application. BEVs are more likely to be used onshort distances and/or in urban areas where they are moreeffective than ICEVs due to the low driving speed, lownoise production, frequent stops, and financial incentives.In cases when the average route length is short, such as theaverage FedEx route length in the USA, which is 68 km[12], BEVs can be applied directly and recharging can beperformed on return to the depot. BEVs are already beingapplied in such occasions: DHL, UPS, FedEx, and Coca-Cola [105, 106] use BEVs mostly for last-mile deliveriesas distances are shorter and vehicle loads are lower. Manycompanies are performing case studies of integrating BEVsin their delivery fleet. Lin et al. [60] were debating the useof BEVs in time-precise deliveries as the long rechargingtime at CS causes hard completion of an on-time delivery,which then significantly increases the overall routing costs.Davis and Figliozzi [10] and van Duin et al. [43] evaluateda wide range of scenarios to compare the routing costs ofICEVs and BEVs. Van Duin et al. [43] concluded that BEVshave the ability to efficiently perform urban freight transportand meanwhile to reduce the GHG emissions and noisenuisance. Davis and Figliozzi [10] reported that BEVs are notcompetitive if the solution to the same problem results in ahigher number of BEVs than the number of ICEVs. For BEVsto be competitive, the authors pointed out a combinationof several key elements: high daily distance (as much asmaximum BEV driving range), low speeds and congestions,frequent customer stops, the reduction of a BEV’s purchasecost by tax incentives or technology development, and longplanning horizon. On the other hand, Schiffer et al. [27]conducted a case study using freight BEVs for deliveries witha planning horizon of five years and compared the resultsto the delivery done by conventional freight trucks. Severalcharacteristics of the performed case study are important:

(i) delivery radius up to 190 km from the depot; (ii) CSslocated at customers’ locations, which allows simultaneouscharging while unloading goods; (iii) already existing strongcurrent at CSs; and (iv) vehicles returning to the depot at leastonce a day. Overall, the authors concluded that there is nooperational limitation when using BEVs compared to ICEVs,as the used number of vehicles and total traveled distance arecompetitive, and at the same time, the overall costs are lowerwith almost 25% less CO2 emission. A year later, Schiffer etal. [11] repeated the case study with more realistic costs whena strong current at the CS is not available and concludedthat, with the higher CS investment costs, BEVs are nolonger competitive. To fully assess the integration of BEVs inlogistic processes, the authors pointed out three key elements:different network structures, future CO2 emission policies,and future technology development (battery capacity andcharging infrastructure).

3.2. Energy Consumption. Due to the low specific energy,energy consumption should be precisely estimated in orderto achieve a BEV’s maximal driving range and to reducethe overall routing costs. The energy consumption can beestimated by simulation models but due to the complexity ofthe E-VRP and unknown driving cycles in advance, mostlymacroscopic models with several real-world approximationsare applied in the BEV routing models. In the availableliterature, energy consumption is often estimated using lon-gitudinal dynamics model (LDM). Here, the LDM of Asameret al. [107] is presented. Force 𝐹 needed to accelerate and toovercome resistances (grade, rolling, and air) is given by (3),where 𝑚 is vehicle mass (mostly empty vehicle), 𝑎 accelera-tion, V vehicle speed, 𝑔 gravitational constant, 𝑓 the inertiaforce of vehicle rotating parts (up to 5% of the total vehiclemass), 𝛼 road slope, 𝑐𝑟 rolling friction coefficient, 𝑐𝑑 air dragcoefficient, 𝜌 air density, and 𝐴 vehicle frontal air surface. If𝐹 ≥ 0, the vehicle is accelerating and power is needed forthe movement of BEV (motor mode); otherwise, if 𝐹 < 0,deceleration (braking) or driving downhill is occurring andenergy is returned into theBEV’s battery as the electric enginehas the ability to return the energy (recuperating mode).By process of recuperation, up to 15% of totally consumedenergy can be returned [51, 108]. Electric power that comesfrom the battery is divided into the auxiliary power 𝑃0 andmechanical power 𝑃𝑚 = 𝐹V. Auxiliary power is spent onthe electronic devices in the vehicle: heating, ventilation,light, etc., which can shorten the BEV’s range up to 30%[109]. Battery power 𝑃𝑏 can be computed by (4), where 𝜇𝑚 isthe transmission coefficient between the electric motor anddrivetrain, 𝜇𝑒 is the conversion ratio from chemical energyin the battery to electric energy, and 𝜇𝑔 is the conversionratio from mechanical energy on wheels to chemical energystored in the battery. Energy is returned into the batteryonly if the force 𝐹 is lower than zero and speed is higherthan the experimentally determined value V𝑚𝑖𝑛 [107]. Energyconsumption can be computed by the time integration of(4).

𝐹 = 𝑚𝑔 sin 𝛼⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Grade

+ 𝑐𝑟𝑚𝑔 cos 𝛼⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Rolling

+ 0.5𝑐𝑑𝜌𝐴V2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Air

+ 𝑓𝑚𝑎⏟⏟⏟⏟⏟⏟⏟Acc.

(3)


𝑃𝑏 = {{{{{{{{{𝜇𝑒 (𝜇𝑚𝐹V + 𝑃0) , if 𝐹 ≥ 0{{{0, if V ≤ V𝑚𝑖𝑛𝜇𝑔𝐹V + 𝑃0, else, if 𝐹 < 0 (4)

Goeke and Schneider [30] extended the energy consump-tion model by taking into account variable vehicle load masswhen delivering goods without acceleration and brakingprocesses. In the CVRP variants, as customers are beingserved, vehicle load is decreasing. The authors concluded thatactual load strongly improves the solution quality, as a largenumber of solutions generated without taking into accountload distribution tend to be infeasible due to the violation ofbattery capacity or time windows.

Genikomsakis and Mitrentsis [85] presented a morerealistic electric engine model: the load-efficiency curve isapproximated by piecewise function and the normalizationfactor is added to take into account the motor size. Theauthors used the recuperation energy factor dependent onthe vehicle speed as follows: below V𝑚𝑖𝑛 there is no energyrecuperation, beyond V𝑚𝑎𝑥 maximum energy is recuperated,and in between linear interpolation of energy recuperationis assumed. Therefore, regarding the energy recuperation theauthors observed two cases: (i) when recuperated energyexceeds the consumption of the auxiliary devices and theexcess of energy is stored into the battery; (ii) when recu-perated energy is not sufficient to cover the consumptionof the auxiliary devices and thus the energy is drawn fromthe battery. The authors compared their model on nine char-acteristic driving cycles (short/long, congested/uncongested,highway, etc.) to the FASTSim simulation tool [110], and thisresulted in a relative energy consumption error of up to 4%.The developed energy consumption model was used to createa database of coefficients for several key characteristics:motortype, motor power, battery type, road type, road slope, roadspeed limit, etc.

Macrina et al. [99] developed an energy model formixed GVRP with partial recharging and time windows.The authors modeled vehicle speed on arc (𝑖, 𝑗) with threephases ℎ: acceleration (ℎ = 1), constant speed (ℎ = 2),and deceleration (ℎ = 3), resulting with either triangularfunction (1 → 3) or trapezoidal function (1 → 2 →3). The fuel consumption of ICEV is based on the similarLDM model, presented by (5) and (6), where 𝑢𝑖 is the load(cargo) weight, 𝜁 fuel-to-air mass ratio, 𝜅 heating value oftypical diesel fuel, 𝜓 conversion factor, 𝑘 engine frictioncoefficient, 𝑁𝑒 radial engine speed, 𝐷𝑒 engine displacement,𝜇𝑑 diesel engine efficiency, 𝜇𝑑𝑡 drivetrain efficiency, and 𝑡ℎtravel spent in phase ℎ [40, 111]. For the consumption of BEV,the authors proposed a similar model to that presented by(7), where 𝜂+ℎ and 𝜂−ℎ are the efficiency of the electric enginein motor and recuperating mode. The authors compared theproposed energy consumption model to the basic model,where energy consumed is proportional to the distancetraveled, and the energy consumption model of Goeke andSchneider [30], where acceleration and braking processes arenot taken into account. The results showed that, comparedto the proposed energy consumption model, basic energyconsumption model and the model of Goeke and Schneider

[30] produce an average relative error of 70% and 4%,respectively.𝐹𝑖𝑗ℎ (𝑢𝑖)= (𝑔 sin 𝛼 + 𝑐𝑟𝑔 cos 𝛼 + 𝑎 (𝑡𝑖𝑗ℎ)) (𝑚 + 𝑢𝑖) + 0.5𝑐𝑑𝜌𝐴V (𝑡𝑖𝑗ℎ)2 (5)𝐹𝑀𝑖𝑗 (𝑢𝑖) = ∑

ℎ=1,2,3

( 𝜁𝜅𝜓)(𝑘𝑁𝑒𝐷𝑒 + 𝐹𝑖𝑗ℎ (𝑢𝑖) V (𝑡𝑖𝑗ℎ)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑃𝑀𝑖𝑗ℎ

(𝑢𝑖)

/𝜇𝑑𝜇𝑑𝑡)𝑡ℎ (6)𝐸𝐸𝑖𝑗 (𝑢𝑖) = ∑

ℎ=1,2,3

(𝐹𝑖𝑗ℎ (𝑢𝑖) V (𝑡𝑖𝑗ℎ) /𝜂ℎ) 𝑡ℎ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐸𝐸𝑖𝑗ℎ

(𝑢𝑖)

,𝜂ℎ= {{{

𝜂+ℎ ≤ 1, 𝐸𝐸𝑖𝑗ℎ (𝑢𝑖) > 0 & 0 ≤ 𝑃𝑀𝑖𝑗ℎ (𝑢𝑖) ≤ 100 kW (ℎ = 1, 2)𝜂−ℎ ≥ 1, 𝐸𝐸𝑖𝑗ℎ (𝑢𝑖) < 0 & − 100 ≤ 𝑃𝑀𝑖𝑗ℎ (𝑢𝑖) ≤ 0 kW (ℎ = 3)(7)

Basso et al. [92] developed an energy consumption modelfor two-stage E-VRP (2sEVRP) that includes detailed topog-raphy and speed profiles. The similar LDMmodel of Asameret al. [107] is applied for computing the energy consumptionon a road segment (link), but with time-dependent speed,varying mass, constant slope, and link distance. For each link,similar to Macrina et al. [99], the three or two characteristicphases of speed curve can be distinguished. In such a way,acceleration and breaking processes before and after theintersection are taken into account. As mass changes duringthe delivery, the energy consumption is expressed as a linearfunction of mass. Then, the shortest energy paths betweenall vertices pairs (customers, depot, CSs) without chargingconstraint are determined by applying the Bellman-Fordalgorithm [112]. A nominal mass value between the mass offull and empty vehicle is used in the computation. The exper-iments showed an average energy estimation error of 2.28%and improved energy feasibility compared to some of theprevious consumptionmodels.The authors also reported thatthe use of average speed consumed at least 24% less energy.

Asamer et al. [107] analyzed the data collected from theBEV trips in order to determine the dependency betweenBEV energy share and average trip speed. Results showed thaton lower speeds V ≤ 30 km/h, most of the energy is spenton the acceleration and auxiliary devices, while on the higherspeeds V ≥ 80 km/h, approx. 70% of total energy is spent onovercoming the air drag force.The slope of the terrain in totalenergy consumption has the lowest share, and its value is evendecreasing with the increase of average trip speed.The rollingresistance energy consumption share does not depend on theaverage trip speed.

Preis et al. [35] analyzed the energy optimal routing ofBEV, where the routes were designed in different terrainslopes and battery capacity scenarios. The authors concludedthat energy savings grow linearly with the maximum altitudedifference and that decreasing the battery capacity of BEVsdoes not affect the recharging schedule but increases thetotal energy consumption. Fiori et al. [113] compared thepower-based consumption model of BEV and ICEV [114].The authors concluded that BEVs and ICEVs have differentfuel/energy-optimized assignment. The faster routes increase


the BEV’s energy consumption while the congested and low-speed arterial routes consume less energy. Masmoudi etal. [84] compared the realistic energy consumption modelof Genikomsakis and Mitrentsis [85] to the constant con-sumption model (237.5 W/km) on instances for the dial-a-ride problem with EVs. The authors concluded that usingthe realistic model is more efficient with 0.14% differencebetween realistic and constant energy consumption from thebest-known solutions (BKS). To emphasize the importanceof the energy consumption model and energy minimiza-tion, Zhang et al. [36] compared the distance and energyminimization and concluded that the distance-minimizingobjective consumes 16.44% more energy than the energy-minimizing objective.

In real-life conditions, speeds on the roads are time-dependent and can be described as the speed profile over theobserved timeperiod. Speed profile depends on the road type,driver behavior, traffic (accidents, recurrent congestions),weather conditions, etc. [115, 116]. A large number of param-eters make it hard to predict a BEV’s energy consumption indifferent traffic scenarios. Therefore, some researchers applydata-driven approaches to predict the energy consumptionof a BEV. De Cauwer et al. [117] developed a model for BEVenergy consumption by applying multiple linear regression toreal-world measured BEV data. The proposed multiple linearregression model (MLR) has seven features: distance, speed,energy consumed by auxiliary devices, positive elevation,negative elevation, temperature, and kinetic energy changeper unit distance. Results showed that the prediction errorof trip energy consumption is within 25%. De Cauwer etal. [115] applied a similar MLR model for predicting energyconsumption on road segments, suited for BEV routing. Theauthors used a neural network based on the road-trafficand weather-related features to predict the speed profile ofa road segment. The error prediction of the trip’s energyconsumption is 12-14%. Lebeau et al. [52] used real-lifedata to model the energy consumed and recuperated onthe trip through least square analysis. The used functiondepends on the trip duration, measured energy consumption,temperature, and correction parameter.The𝑅2 for the energyconsumption model is 0.93 and 0.77 for the recuperatedenergymodel.Wu et al. [118] presented an empirical and ana-lytical method that can estimate a BEV’s instantaneous powerin real time and overall trip energy consumption, with theaverage prediction error up to 15.6%. Fiori andMarzano [119]proposed a backward microscopic power-based LDM energyconsumption model based on the known driving cycle. Theacceleration, speed, and regenerative braking efficiency weremodeled as functions of time. Based on the real driving BEVdata, for each vehicle, six parameters were optimized: threeparameters regarding the drivetrain and battery efficiency,the parameter for determining when the regenerative brakingis occurring, and two traction power thresholds. The authorsconcluded that the proposed model is flexible enough to beapplied to any kind of driving cycle and BEV type with givenkinematic profile and characteristics.

Pelletier et al. [104] presented battery degradation modelsthat could be used in BEV routing applications, as batteriesare a crucial part of the economic BEV routing. Several

lithium-ion battery degradation mechanisms with storageand operating conditions that affect battery lifespan weredescribed. Barco et al. [42] applied a battery degradationmodel based on the temperature, SoC, and depth of dischargein their airport shuttle service with BEVs. It was observed thatthe consideration of the battery degradation model affects thecharging patterns.

4. Variants of the Electric VehicleRouting Problem

Many different VRP variants were researched over the years.With BEV appearance, researchers started to adapt them tothe E-VRP context. Due to the specific characteristics of BEVrouting, some new problem-specific variants emerged. Here,we present some of the most relevant E-VRP variants andrelated problems.

4.1. Energy Shortest Path Problem and the Electric TravelingSalesman Problem. The energy shortest path problem (E-SPP) and electric TSP (E-TSP) can be considered as two ofthe simplest forms of the E-VRP. Inmost of the VRP variants,graph arcs have positive weight values that can representdistance, travel time, cost, etc. To compute the shortest pathbetween customers, the most commonly applied algorithmsare Dijkstra, Bellman-Ford, A∗, contraction hierarchies, etc.[108, 120]. By taking into account the recuperated energy ofBEV, some arc weights could have a negative value, whichmakes most of the shortest path algorithms inapplicable.To overcome this problem, Artmeier et al. [108] formalizedenergy-efficient routing with rechargeable batteries as aspecial case of the constrained SPP (CSPP) in which thebattery charge is limited with its capacity, and there is norecharging at CS. The authors adapted the Bellman-Fordshortest path algorithm with time complexity O(𝑛3) in orderto solve the energy CSPP. To overcome negative graph edges,Eisner et al. [121] used Johnson’s shifting technique [122]to transform negative edge cost functions into nonnegativeones and applied Dijsktra’s algorithm with time complexityof O(𝑛 log (𝑛) + 𝑚) [123]. Storandt [124] introduced CSPPfor EVs with battery swapping, while Sweda and Klabjan[125] added recharging events at CSs with a differentiablecharging function. Zündorf [50] solved the CSPP for BEVrouting with battery constraints, different types of CSs, andnonlinear charging process.The author developed a chargingfunction propagating algorithm in order to minimize traveltime and applied CH andA∗ algorithms to solve the problem.Liao et al. [103] considered the BEV’s shortest travel pathproblem and presented a dynamic programming algorithmfor solving the problem that runs in O(𝑘𝑛2), where 𝑘 is theupper bound on the number of BSSs. More recently, Strehleret al. [126] observed recharging events in the graph verticesand arcs for the energy-efficient shortest routes of BEVs andHEVs, while Zhang et al. [127] dealt with the electric vehicleroute planning with recharging problem (EVRC), whichminimizes the overall travel and charging time and takes intoaccount CSs’ locations, partial recharging, nonlinear charg-ing functions, service time duration, and service frequency atCS.


As VRP is the generalization of the TSP, the E-VRP isclosely related to the E-TSP, in which a set of customershas to be served by only one BEV. Roberti and Wen [64]formulated the E-TSP with time windows (E-TSPTW) as acompact MILP program with binary variables for rechargingpaths with intermediate stops. The authors observed full andpartial recharge policies and applied three-phase heuristic forsolving the problem. The authors solved multiple instances,among them the 13 E-VRPTW instances of Schneider et al.[25], which were solved optimally with only one vehicle.Doppstadt et al. [58] formulated the HEV-TSP with fourworking modes of HEV and provided new test instances.Liao et al. [103] introduced the EV touring problem, as thegeneralization of TSP, with the minimization of the totalrequired time. Two scenarios with battery swap scheme wereobserved: the on-site station model, in which each city hasa BSS; the off-site station model, in which BSS is located atan acceptable distance from the city. The authors proposedefficient polynomial time algorithms for the problem.

4.2. Heterogeneous or Mixed Vehicle Fleet. In today’s vehiclefleets, mostly ICEVs are present. Transition to an almostwholly electric fleet is a very challenging economic task.Therefore, most companies are gradually integrating BEVsinto their existing ICEV fleet. Routing algorithms have to beupgraded and adapted for electric fleet characteristics, as on-time priority is harder to achieve.

Fleet size and mix VRP (FSM-VRP) was first introducedby Golden et al. [128], where the authors considered routinga fleet of vehicles with different acquisition costs and therouting cost is dependent on the vehicle type. Goeke andSchneider [30] formulated E-VRPTW and mixed fleet (E-VRPTWMF) with equal vehicle load capacities of BEVsand ICEVs, and equal BEV battery capacities. The authorscompared the percentage share of BEVs to the total trav-eled distance obtained with three objective functions: totaltraveled distance, total costs with battery costs, and totalcosts without battery costs. The traveled distance on onebattery pack was assumed to be 241,350 km (150,000 miles)with the payment of an additional $600 per kWh in caseof replacement. The results showed that the BEV’s share intotal traveled distance increased significantly when batterycosts are not considered, while in the other two scenariosICEVs performed most of the deliveries. Hiermann et al.[31] analyzed a similar problem, the electric fleet size andmix VRPTW and recharging stations (E-FSMFTW), withdifferent vehicle load and battery capacities. The authorsshowed the overall positive influence of the heterogeneousvehicle fleet on the generalized total cost function. In mostof the instances, three to four vehicle types are used in thefinal solution. A similar analysis on small-sized instanceswith different BEVs, ICEVs, and HEVs was undertaken byLebeau et al. [52].The authors defined seven groups of vehicletypes that could be used for the delivery, from small vans andquadricycles through diesel-only or electric-only groups to agroup of all vehicle types. The results showed the followingaspects of routing: (i) the fleet with different vehicle typesreduced the total routing costs; (ii) in the large van group,ICEVs outperformed BEVs; and (iii) HEVs showed a great

application in deliveries solelymade by trucks. Sassi et al. [47–49] also observed a heterogeneous fleet of ICEVs and BEVswith different load and battery capacities, different operatingcosts, time-dependent charges, and the compatibility of BEVsand chargers at CSs. The authors focused on the proceduresfor solving the problem, so they did not offer any compar-ison between the ICEV and BEV solutions. Hiermann etal. [96] introduced the Hybrid heterogeneous electric fleetrouting problem with time windows and recharging stations(H2E-FTW), in which the fleet consists of ICEVs, BEVs,and PHEVs. The authors compared the overall costs of thesolutions obtained with a homogeneous fleet of ICEVs, BEVs,and PHEVs to the optimized solution with a mixed fleet. Thegap for the ICEV fleet is the largest, with an average value of60% in the extreme case when the fuel cost is the highest. Incontrast, when the electric cost is the highest, the BEVfleet onaverage produces a 40% gap, while the PHEV fleet shows thelowest gap value, up to 25%. The authors pointed out that, inmixed solutions, BEVs are preferred for clustered instances,ICEVs for randomly distributed instances, and PHEVs forrandomly clustered and distributed instances. Overall, theresults showed that operational costs can be 7% lower in amixed case compared to the homogeneous case.

4.3. Hybrid Vehicles. As compensation for the limited drivingrange of BEVs, HEVs, which have both an internal com-bustion engine and an electric engine, have been developed.Two main types of HEVs are present on the market: theseries hybrid, in which only the electric motor drives the trainand the internal combustion engine is used to recharge thebattery pack, and the parallel hybrid, which uses both internalcombustion engine and electric engine to drive the train,where the electric engine is more efficient in stop-and-goactivities and the internal combustion engine ismore efficientat high speeds.We focus on the PHEVas a version of a parallelhybrid in which batteries can be recharged by connecting aplug to the electric power source. The PHEVs have an optionto decide during the route to run on either electric energyor fossil oil. This enables the visiting of customers far fromthe depot with almost no refuel/recharge during the route.As PHEVshave two engines, their load is heavier compared tothe BEVs and ICEVs, and therefore they have a higher energyconsumption rate.The time spent on the deliveries is shorter,whichmakes it easier to achieve time-precise deliveries and toreduce the costs of recharging, at the expense of higher costsdue to fossil oil consumption.

One of the first papers that dealt with the PHEVs routingproblem was published by Abdallah [41], where the authordefined the problem as the plug-in hybrid electric VRPTW(PHEVRPTW). The objective of the proposed problem isto minimize the routing costs on the internal combustionengine while satisfying the demand and time window con-straints. At each customer, the driver can either rechargethe vehicle battery or go to the next open time windowusing an internal combustion enginewhen the electric energyhas been depleted. Doppstadt et al. [58] defined the HEV-TSP in which the authors observed HEVs that are not plug-in and can only be charged while driving. Four workingmodes were observed, combustion-only mode, electric-only


mode, charging mode, and boost mode, when the combinedinternal combustion engine and electric engine are used.As the authors assumed that there was not a charge leftfrom the day before, the initial charging of the battery wasset to zero. The authors presented the positive effects ofusing HEVs: (i) compared to the ICEV routes, the overallcosts for the HEV routes in the tested instances reduced upto 13% and driving time increased up to 11%; (ii) savingsdepend highly on the depot location, and not on the limitedbattery capacity, which was an a priori assumption. Mancini[71] introduced hybrid VRP (HVRP), in which PHEVs canchange propulsion mode at any time and the electric engine,rather than the internal combustion engine, is promoted.Vincent et al. [76] introduced a similar HVRP problem withPHEVs and mild-HEVs, which do not have an electric-onlymode of propulsion. The results showed that PHEVs havea lower average cost per mile in all scenarios compared tothe mild-HEVs and ICEVs. Hiermann et al. [96], in theirH2E-FTW, used the electric motor of the PHEV as prioritymode choice. When a time window of a PHEV route isviolated, the mode could be changed at any time during theroute by exchanging recharging time for the correspondingamount of fuel at no additional costs.The authors showed thepositive impact of a PHEV fleet compared to ICEV and BEVfleets, especially for randomly clustered instances. PHEVs areusually represented with only 20% of the overall number ofvehicles used in the solution due to their higher consumptionand utility costs but, still, they constitute an important part ofthe fleet configuration due to their flexibility.

4.4. Partial Recharging. In the beginning, in most of the E-VRP problems, full recharge was considered when a BEVvisited a CS [25]. This can be time-consuming because,depending on the SoC level, available charging technology,and battery capacity, the vehicle can charge from five min-utes to eight hours [15]. Therefore, in real-life applications,partial recharging should be taken into account. The batteryshould be charged enough to complete the whole route orto surpass a fear that vehicle range will not be enough toperform designated tasks, the so-called range anxiety [75].This particularly has an effect on customers with narrowtime windows, where efficient charge scheduling can enablethe feasibility of the route. On the economic side, significantsavings can be achieved by applying partial recharging as aminimal amount of energy could be recharged during theday when electricity cost and energy network load are higher,and the rest of the energy could be replenished during thenight [46, 47]. In some cases, it is natural to maintain theenergy reserve. This can be done by SoC range limitation,i.e., [20, 95] [42, 47, 80]. Having an energy reserve seems evenmore important if energy consumption and range anxiety aretaken into account because up to 30%of the consumed energycan be spent on BEV’s auxiliary devices [109]. Limiting SoCvalue also helps to preserve the battery as battery capacitydecreases by overcharging and overdischarging [104].

Several papers have analyzed strategies of partial recharg-ing and formulated the problem as E-VRPTW with partialrecharging (E-VRPTWPR). Bruglieri et al. [29, 51] andMoghaddam [53] modeled the concept of partial charging in

E-VRP. Felipe et al. [46], in GRVP with multiple technologiesand partial recharges (GVRP-MTPR), and Keskin and Çatay[28], in E-VRPTWPR, ensured that after every change inthe route configuration, at the previous CS, the BEV ischarged only with the amount of energy sufficient to finishthe segment of the route until the next CS or the depot.This results in the removal of certain CSs in the route,compared to full recharge strategy, and the arrival of a vehicleat the depot with an empty battery. The authors presentedthe positive impact of partial recharges on the total costsand energy savings. A similar procedure was applied bySassi et al. [47–49] but with multiple charging technologies,different charging periods, and BEV chargers compatibilitychecks. Schiffer and Walther [73] compared full and partialrecharge by solving small E-VRPTWPR test instances usingcommercial software. The authors concluded that, in somecases, a partial recharging strategy reduces the total traveleddistance and number of visits to CSs. Montoya [18] andMontoya et al. [72] formulated the FRVCP for BEVs in which,for a fixed sequence of customers in the route, CS positionand charging amount are optimized.The results on the newlyderived test instances showed that good solutions tend toexploit partial recharges. A similar procedure was applied byKeskin and Çatay [79], Hiermann et al. [96], Froger et al. [68],and Schiffer and Walther [88, 89] to enhance the incumbentbest solution by optimizing the charging decisions along theBEV route with a fixed customer sequence. Desaulniers et al.[57] presented the effects of partial recharging by optimallysolving E-VRPTW instances containing up to 100 customers.In a case with a single recharge per route, the partial rechargereduced the routing costs by 0.97% and the number ofvehicles by 2.25%, while in a case with multiple recharges perroute these values are 1.91% and 3.80%, respectively, with asignificant increase in the average number of recharges perroute.

4.5. Different Charging Technologies. Today, multiple charg-ing technologies are present: (i) slow, 3 kW (6-8 h); (ii)fast, 7-43 kW (1-2 h); and (iii) rapid, 50-250 kW (5-30min) [15, 129]. To better control charging time in the E-VRP context, the selection of possible charging technologycould also be optimized. This could make some customerswho have narrow time windows more accessible by fastcharging at previous CSs, or if the time windows are long,an economically better approach could be slow charging.Such a problem could be extended by taking into account CSworking hours, time-dependent charging costs, the numberof available chargers and their compatibility with BEVs, thepower grid load, the charger power, etc. [46–49, 68, 80, 81, 87].Felipe et al. [46] analyzed the effect of different chargingtechnologies on the recharge cost.The authors concluded thatnone of the technologies dominated the others. The rapidand fast charging options provided slightly better results thanthe slow charging, but the best results were obtained whenjoint technologies were used as the most appropriate onecould be chosen in each case. Çatay and Keskin [67] solvedsmall-sized instances to present the insights of the quickcharge option. The results showed that quick charging mightreduce the fleet size and decrease the cost of energy needed to


operate BEVs. Keskin and Çatay [79] formulated E-VRPTWand fast charging (E-VRPTW-FC) with partial recharges andthree different charging technologies. The authors providedMILP formulation and minimized total recharging costswhile operating a minimum number of vehicles. The authorscompared two mixed integer programming (MIP) models:(i) model 1, in which binary variables are used as decisionvariables to choose which charging technology to use; (ii)model 2, based on the model of Schneider et al. [25] andKeskin and Çatay [28], in which the authors copied eachstation vertex three times depending on the charging tech-nology.The results showed that in all cases model 1 producedbetter results with shorter computation time. Testing onlarger instances showed that the fast charging option is morebeneficial when customers have narrow time windows andthat multiple charging technologies improved the overallresults (in 28 of 29 instances), while in the instances withlarger time windows the average improvement was 0.17%.

4.6. Nonlinear Charging Function. In most of the E-VRPrelated literature, either linear or constant charging timeis considered. Most of the BEVs have lithium-ion batter-ies installed, which are often charged in constant-currentconstant-voltage (CC-CV) phases: first by constant currentuntil approx. 80% of the SoC value and then by a constantvoltage. In the CC phase, SoC increases linearly, and in theCV phase, the current drops exponentially and SoC increasesnonlinearly in time, which prolongs charging time [102,104]. In only a few reviewed papers the author considerednonlinear charging process and solved the E-VRP either bylinearization per segments or by estimating charging timeby a data-driven approach [50, 68, 72, 87, 95]. Zündorf[50] developed a charging function propagation algorithmfor determining an EV battery-constrained route by takinginto account piecewise linear and concave functions of therecharging process. Montoya et al. [72] formulated the E-VRP with nonlinear charging functions (E-VRP-NL) andpresented theMILP formulation. The authors fitted piecewiselinear functions for 11, 22, and 44 kWCSs to the realmeasureddata. The results showed that neglecting the nonlinear chargecan lead to infeasible or overly expensive solutions: 12% ofthe routes in good solutions recharged the battery in thenonlinear part, after 80% of the SoC value. Froger et al. [95]proposed two new formulations for the E-VRP-NL: (i) theimproved MILP version of Montoya et al. [72] by arc-basedtracking of the SoC value; (ii) a path-based model withoutCS replication, where a sequence of vertices (a path) betweena pair of customers or depots is determined. The secondone yielded much better results with shorter computingtime. Froger et al. [68] explored similar formulations forthe E-VRP-NL with capacitated CSs (E-VRP-NL-C): (i) theCS replication-based formulation with flow and event-basedformulations forCS capacity constraints; (ii) recharging path-based formulation.

4.7. BEV Routing and CS Location. Due to the currently lowBEV market share, the number of CSs installed in the roadinfrastructure is also relatively low.Therefore, great potentiallies in the simultaneous decision-making of CS locations

and BEV routes. The classic location routing problem (LRP)consists of determining the locations of the depots andvehicle routes supplying customers from these depots [130].A modification of the LRP that deals with CS facilities isformulated as electric LRP (E-LRP) [11, 27, 73, 88].

Schiffer et al. [11, 27] presented a case study for solvingELRPTWPR. It was assumed that CSs can be located atcustomers’ locations and that service time can be usedfor recharging. Overall, results indicated the viability ofcombining CSs siting and BEV routing for specific caseswhen a delivery range is not far from the depot. Schifferand Walther [73] compared ELRPTWPR and E-VRPTWPRsolutions on down-scaled test instances and concluded thatthe ELRPTWPR gives a better solution in all instances. Thereason comes from the fact that BEV can be charged whileserving, which reduces the overall route time, as there isno travel time wasted on traveling to and from CS. Hofet al. [69] and Yang and Sun [56] addressed the problemof E-LRP but with BSSs. Results indicated that decreasingthe construction cost of BSS led to the expected increasein their number in the solution. Schiffer and Walther [88]formulated the LRP with intraroute facilities (LRPIF) as amore general problem, in which intraroute facilities can beused for refueling, loading, or unloading goods. Sun et al.[131] proposed a location model for CSs without routingbased on the travel demands of urban residents. For short-distance travelers, slow CSs are utilized, while for long-distance travelers, the rapid CSs are considered. The CSs’locations are determined to maximize coverage and flowaccording to the concept of vehicle refueling. In the casestudy, the authors pointed out two critical aspects: the BEVdriving range and the budget constraint.

4.8. Battery Swap. Instead of charging at CS, at speciallydesigned BSS, empty or nearly empty batteries can bereplaced with fully charged ones [14]. The main advantagesof such procedure are the time in which it can be performedand the ability to recharge when energy network load andelectricity costs are lower, i.e., during the night. A wholereplacement procedure could last less than ten minutes,which is competitive to the refueling time of ICEVs andmuchfaster than one of the fastest charging BEV technologies.The drawbacks of such procedure are the nonstandardizedbatteries and their installation in BEVs, which makes ithard to swap empty batteries with fully charged ones. Adlerand Mirchandani [44] observed the E-VRP with swappablebatteries, already determined BSS’s locations, a fixed numberof batteries per BSS, full recharge of four hours, and afixed swapping time of two minutes. Full battery rechargeis considered so it is possible that when the vehicle arrivesat the station, there is no fully charged battery availableand the vehicle has to wait. First, the total routing cost isminimized and then the battery reservations aremade, so thatthe vehicle could avoid BSSs without available batteries. Yangand Sun [56] and Hof et al. [69] simultaneously determinedthe locations of BSSs and the vehicle routing plan andprovided different metaheuristics for solving the problem.Yang and Sun [56] formulated the problem as BSS-EV-LRPand compared the solutions of the BSS-EV-LRP instances


to the corresponding best-known CVRP solutions. The totalrouting costs for 150 km and 300 km driving ranges increasedto 7% and 3.5%, respectively, and in some cases, there wasno gap between BSS-EV-LRP and CVRP solutions. Hof et al.[69] on the same problem reported that if construction costsof BSSs are equal to zero, the BSSs would be constructed in83% of total available candidate sites.

4.9. Two-Echelon Routing Problem. Breunig et al. [93] pro-posed the electric two-echelon VRP (E2EVRP), in whichgoods are transported in two echelons: (i) in the first echelon,goods are transported by conventional freight vehicles fromthe depot to the satellite facilities; (ii) in the second echelon,goods are transported from the satellite facilities to thecustomers by light BEVs. Two vehicle types are observed inthe problem: ICEVs with higher load capacity located at thedepot and BEVs with lower load capacity located at satellitefacilities. BEVs are used for the last-mile deliveries due totheir lower pollution, noise, and size.The authors formulatedthe problem on the multigraph and applied exact proce-dures on smaller instances to solve the problem optimallyand a metaheuristic procedure on larger newly developedinstances. The results showed that the increase in CS density𝜌 decreased the detour costs following the expression 1/𝜌1.24.Also, the vehicle range has a great impact on the solutionquality: a range below 70 km produced infeasible solutionswhile a range higher than 150 km decreased recharging visitsto almost zero. Jie et al. [97] analyzed a similar problem, thetwo-echelon capacitated E-VRP with BSS (2E-EVRP-BSS), inwhich, in both echelons, BEVs are used.The BEVs in the firstechelonhave higher load and battery capacities than theBEVsin the second echelon. The authors reported that the batterydriving range is themost important aspect of routing and thatit slightly depends on the number of BSSs used.

4.10. Charging Schedule. Many companies that use BEVsprefer charging the vehicles at their own facilities in order tocharge the vehicles between the delivery routes and duringspecific periods of the day. In such occasions, there are usuallya limited number of chargers at the depot, typically fewer thanthe fleet size; therefore, the efficient charging schedule at thedepot has to be determined.

Pelletier et al. [87] considered the electric freight vehiclescharge scheduling problem (EFV-CSP) at the depot for theBEVs that deliver goods to a set of customers over multipledays. The authors included multiple charging technologies atthe depot, realistic charging process (piecewise linearizationof nonlinear charging function), time-dependent chargingcosts, grid power restriction, battery degradation costs (cyclicand calendar aging), and facility-related demand (FRD)charges representing the maximal demand registered overthe billing period. The fixed vehicle routes are known inadvance and the number of charging events in the depot islimited to avoid impractical solutions of constantly moving avehicle from one charger to another. The authors presentedthe following aspects of charge scheduling tests conductedin summer (higher electricity costs) and winter (lowerelectricity costs) periods: (i) model tries to keep the SoClower when battery degradation costs are included; (ii) in

summer, vehicles are rarely charged in peak hours, whichresults in more vehicle charging simultaneously or the useof fast chargers that retrieve more power from the grid innonpeak hours but incur higher FRD charges; (iii) to avoidcycling the battery in high SoC, it is preferable to split the longroutes into smaller ones; (iv) fast chargers are heavily usedin a high BEV utilization context; (v) grid power restrictionincreases overall energy costs, especially in summer months,and leads to infeasible solutions, limiting the number ofvehicles simultaneously charging; and (vi) total costs arealways lower with larger batteries, as smaller batteries requirelarger discharge cycles.

Barco et al. [34, 42] also analyzed charge scheduling forassigned routes in an airport shuttle scenario. The authorsdefined the set of charging actions (charge profile) over thedetermined programming horizon and minimized overalloperating costs. Sassi et al. [47, 49] also dealt with determin-ing the charging schedule of BEVs at the depot and includedtime-dependent charging costs and chargers compatibilitychecks with BEVs. Adler and Mirchandani [44] provided anonline routing model of BEVs by taking into account batteryreservations to minimize the average delay of all vehiclesby occasionally detouring them. Wen et al. [65] observedthe service time of CSs, meaning that a CS can be visitedonly in some specific time period, usually working hours.The authors proposed a battery reservation model in orderto charge the reserved battery at the defined time period.Sweda et al. [75] dealt with the possibility that a CS mightnot be available at some point in time and rerouting should beperformed.Theproblem is formulated as the adaptive routingand recharging policies for EVs. First, optimal a priori routingand recharging policies were determined and then heuristicprocedures were applied for the adaptive routing.

4.11. Dynamic Traffic Conditions. Most of the E-VRP researchconsiders static conditions on the road network. The trafficstates change recurrently, depending on the time of the day,day of the week, and season, or nonrecurrently when a trafficincident occurs, such as an accident [132–134]. TD-VRProutes a fleet of vehicles by taking into account variable traveltime on the road network [135, 136]. Shao et al. [74] observedBEVs in such time-dependent context in their E-VRP withcharging time and variable travel time (EVRP-CTVTT).The recharging time is fixed to 30 minutes and batteriesare always charged to full capacity. The authors discretizedone day into two-minute intervals and applied the dynamicDijkstra algorithm to find the shortest travel time path whenweights in the graph are not constant. The authors presenteda real-life problem and solved it by applying a geneticalgorithm with a running time of three hours. Omidvar andTavakkoli-Moghaddam [24] presented a model for routingAFVs that aims to avoid routing during congestion hours,when the pollution costs are high, by waiting at a customer’slocations. These costs were computed based on the roadspeed profile. Mirmohammadi et al. [62] presented MILPformulation for the periodic green heterogeneous VRP withtime-dependent urban traffic and time windows, which isgreen in terms of the emission minimization and not theutilization of AFVs. The planning horizon is divided into


several periods, duringwhich static traffic conditions are con-sidered.

4.12. Related Problems

4.12.1. Route Scheduling and Other Electric Variants. Manyresearchers are dealing with the scheduling of bus/taxi routesthat are fixed or could be slightly altered. Li et al. [82]addressed themixed bus fleetmanagement problem (MBFM)composed of electric, diesel, compressed natural gas andhybrid-diesel buses. The authors maximized the total benefitof replacement of old vehicles with new ones under budgetconstraints while optimizing the route assignment for eachbus during the planning period. Two routing procedures weredeveloped to solve the recharging problem: single-periodrouting and routing across multiple periods of a day. Analysisof the results on the case study of Hong-Kong showed a cost-effective scheme of fleet configuration in the following order:mixed fleet, electric, natural gas, diesel, and then hybrid-diesel buses. Wen et al. [65] also dealt with the schedulingof electric buses (electric vehicle scheduling problem, E-VSP)in order to efficiently serve a set of timetabled bus trips. Thelinear partial and full recharge were considered, with theminimization of bus numbers and the total traveled distance.Lu et al. [83] addressed the problem of optimal scheduling ofa taxi fleet with mixed BEVs and ICEVs to service advancereservations. The authors presented a multilayer taxi-flowtime-space network in order to minimize the total operatingfleet cost.

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