In order to ensure high-quality and on-time delivery in logistic distribution processes, it is necessary to efficiently manage the delivery 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
The vehicle routing problem (VRP) is an NP-hard optimization problem that aims to determine a set of least-cost delivery routes from a depot to a set of geographically scattered customers, subject to side constraints [
In the past decade, the European Union (EU) has announced many new actions and regulations related to greenhouse gas (GHG) emissions in the transport sector [
Due to the limited battery capacity, the range that delivery BEVs can achieve with a fully charged battery is 160-240 km [
Here we present, to our knowledge, the most recent literature reviews on the E-VRP and related problems.
Juan et al. [
Montoya [
The most recent survey on the E-VRP is presented by Pelletier et al. [
In this paper, a survey on the E-VRP is presented, which includes approaches for solving the E-VRP and related problems that emerged with BEVs integration in the logistic processes. The focus is not on the economic and environmental challenges related to BEVs. Most of the latest literature reviews were published in 2016; hence, to the best of our knowledge, there is no published research that summarizes the state-of-the-art research in the E-VRP field. In this paper we outline the following contributions: a review of the recent energy consumption models that could be used in BEV routing models; an updated literature review and a concise table summary of already reviewed E-VRP variants such as GVRP, mixed fleet, BSSs, partial recharges, and different charging technologies; a review of the additional emerged E-VRP variants, which include hybrid vehicles, CS siting, nonlinear charging function, dynamic traffic conditions and charging schedule optimization; a comprehensive analysis of operation research procedures in the E-VRP, which includes an overview of the procedures employed for solving various E-VRP variants, highlighting state-of-the-art procedures, and a concise table summary of the applied procedures.
The remainder of this paper is organized as follows. In Section
With BEV penetration in logistic distribution processes, a problem of routing a fleet of BEVs has emerged: the E-VRP. The E-VRP aims to design least-cost BEV routes in order to serve a set of customers by taking into account often used constraints: vehicle load capacity, customer time windows, working hours, etc. [
To the best of our knowledge, the first research regarding the routing of an electric fleet was published by Gonçalves et al. [
In the VRP, it is customary for the primary objective to minimize the total number of vehicles used (
Overview of the E-VRP variants and related problems.
Reference | Problem name | Charging | Other | Objective | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
C | TW | MIX | LRP | F | L | NL | PR | DFC | BS | TD | H | ||||
Bektaş and Laporte [ | PRP | X | X | Emission model, speed limitation | Total costs: labor, fuel and emission as a function of load and speed | ||||||||||
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Conrad and Figliozzi [ | Recharging VRP | X | X | X | Vehicle number and total traveling costs: distance, service time, recharging time | ||||||||||
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Gonçalves et al. [ | VRPPD with mixed fleet | X | X | X | No CS location, time constraint | Total traveling costs: fixed and variable | |||||||||
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Demir et al. [ | PRP | X | X | Emission model | Total traveling costs: labor, fuel and emission as a function of load and speed; speed optimization | ||||||||||
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Erdoğan and Miller-Hooks [ | GVRP | X | AFVs, limited route duration | Vehicle number and total traveled distance | |||||||||||
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Omidvar and Tavakkoli-Moghaddam [ | GVRP | X | X | X | X | AFVs, limited fuel capacity and route duration, congestion management | Total costs: vehicle fixed costs, distance, time and emission | ||||||||
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Abdallah [ | PHEVRPTW | X | X | X | X | X | Electric charge cost is neglected | Routing costs: time run on the fossil oil | |||||||
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Barco et al. [ | E-VRP and charge scheduling | X | X | X | X | X | Energy consumption and battery degradation model, private and public CS, time-dependent energy rates | Energy consumption | |||||||
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Davis and Figliozzi [ | X | X | Energy consumption model with speed profiles, limit route duration and energy consumption (battery capacity), no recharging during the route | Total costs: vehicle purchase, energy, maintenance, tax incentive, battery replacement, routing | |||||||||||
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Van Duin et al. [ | FSMVRPTW, EVFSMVRPTW | X | X | X | No recharge, range constrained by battery, relaxed time window constraints, emission | Total costs: vehicle fixed costs, time and distance | |||||||||
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Adler and Mirchandani [ | Online routing of BEVs | X | Battery reservations, waiting for fully charged battery | The average vehicle delay time | |||||||||||
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Alesiani and Maslekar [ | BEV routing | X | Waiting time cost at CS is proportional to the number of EVs in CS, limiting the number of CSs in route and number of vehicles in CS, energy consumption model | Traveling, charging and energy consumption costs | |||||||||||
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Demir et al. [ | PRP | X | X | Fuel consumption and emission model | Bi-objective minimization of | ||||||||||
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Felipe et al. [ | GVRP-MTPR | X | X | X | X | Total recharging costs: fixed and variable | |||||||||
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Preis et al. [ | Energy-optimized routing of BEVs | X | X | X | Energy consumption model | Energy consumption | |||||||||
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Sassi et al. [ | HEVRP-TDMF VRP-HFCC VRP-MFHEV | X | X | X | X | X | Time-dependent charging costs, operating windows and power limitation of CS, compatibility of BEVs with chargers, electricity grid capacity [ | Vehicle number and total costs: fixed, routing, charging and waiting costs | |||||||
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Schneider et al. [ | E-VRPTW | X | X | X | Vehicle number and total traveled distance | ||||||||||
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Zündorf [ | EVRC | X | X | X | X | Battery constrained SPP, different CS types: regular, superchargers and BSS, energy consumption model | Travel time | ||||||||
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Bruglieri et al. [ | E-VRPTW | X | X | X | X | Vehicle number and total travel, recharging and waiting time | |||||||||
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Goeke and Schneider [ | E-VRPTWMF | X | X | X | X | Energy consumption model: varying BEV load, road slope | Vehicle number and different objectives: (i) distance, (ii) costs: labor, driver wage, vehicle propulsion - electric energy and diesel costs, (iii) (ii) + battery replacement cost | ||||||||
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Lebeau et al. [ | FSMVRPTW-EV | X | X | X | X | Energy consumption model based on the collected data, recharge only at the depot | Total costs: vehicles fixed and operating costs, labor costs | ||||||||
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Moghaddam [ | E-VRPTWPR | X | X | X | X | Capacitated CSs | Vehicle number and number of CSs | ||||||||
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Pourazarm et al. [ | Single/Multi BEV routing | X | X | X | Homogeneous and in-homogeneous CSs | Total time | |||||||||
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Schneider et al. [ | VRPIS (EVRPRF) | X | X | Total travel and fixed vehicle costs | |||||||||||
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Yang and Sun [ | BSS-EV-LRP | X | X | X | Total routing and construction costs | ||||||||||
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Desaulniers et al. [ | E-VRPTW -SF/MF/SP/MP | X | X | X | X | Vehicle number and total routing costs | |||||||||
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Doppstadt et al. [ | HEV-TSP | X | X | Four modes of travel: combustion, electric, charging and boost mode, no CSs visits | Total costs as long as maximal route duration is not overrun | ||||||||||
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Hiermann et al. [ | E-FSMFTW | X | X | X | X | Vehicle number and total costs: vehicle fixed and routing costs | |||||||||
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Keskin and Çatay [ | E-VRPTWPR | X | X | X | X | Vehicle number and total traveled distance | |||||||||
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Koç and Karaoglan [ | GVRP | X | AFVs, limited route duration | Vehicle number and total traveled distance | |||||||||||
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Lin et al. [ | E-VRP | X | X | Energy consumption model and load effect | Total costs: battery charging, travel time and waiting costs | ||||||||||
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Masliakova [ | Routing and charging of electric buses | X | X | X | Energy consumption model, two types of buses depending on the charging event: en-route or at depot, homogeneous and in-homogeneous CSs | Investment and operations costs, a travel time of passengers | |||||||||
| |||||||||||||||
Mirmohammadi et al. [ | Periodic green VRP | X | X | X | X | Periodic routing, primary and secondary time windows, static traffic conditions within a period | Total emissions, total service time and penalties | ||||||||
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Montoya et al. [ | GVRP | X | AFVs, limited route duration | Vehicle number and total traveled distance | |||||||||||
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Roberti and Wen [ | E-TSPTW | X | X | X | Total traveled distance | ||||||||||
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Schiffer et al. [ | E-LRPTWPR | X | X | X | X | X | CSs at customers’ locations, pickup and delivery, multiple driver shifts and multiple planning periods, emissions | Total costs during the planning period: CS and BEV investment costs, fixed costs (tax, maintenance), distance dependent costs (energy) | |||||||
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Wen et al. [ | E-VSP | X | X | Timetable bus trips, multiple depots, time windows of depots and CSs | Total cost: vehicles and traveling costs | ||||||||||
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Andelmin and Bartolini [ | GVRP | X | AFVs, limited route duration | Vehicle number and traveled distance | |||||||||||
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Çatay and Keskin [ | E-VRPTWPR | X | X | X | X | X | Normal and fast charger at CSs | Vehicle number and total recharging costs | |||||||
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Froger et al. [ | E-VRP-NL-C | X | X | X | Capacitated CSs | Total travel, service, charging and waiting time | |||||||||
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Hof et al. [ | BSS-EV-LRP | X | X | X | Total routing and construction costs | ||||||||||
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Leggieri and Haouari [ | GVRP | X | AFVs, limited route duration | Vehicle number and total traveled distance | |||||||||||
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Mancini [ | HVRP | X | Full instant recharge | Total traveled distance with penalties for using internal combustion engine | |||||||||||
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Montoya et al. [ | E-VRP-NL | X | X | X | CS types: slow, moderate and rapid | Total travel and recharging time | |||||||||
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Schiffer and Walther [ | E-LRPTWPR | X | X | X | X | X | Total distance, number of vehicles and CSs used, total costs: investment costs of BEVs and CSs, and operational costs | ||||||||
| |||||||||||||||
Shao et al. [ | EVRP-CTVTT | X | X | X | X | Total costs: travel, charging, penalty, and fixed vehicle costs | |||||||||
| |||||||||||||||
Sweda et al. [ | Adaptive routing and recharging policies of EVs | X | X | Heterogeneous CSs - the probability of being available and expected waiting time, origin-destination pairs | Traveling, waiting and recharging costs | ||||||||||
| |||||||||||||||
Vincent et al. [ | HVRP | X | X | Vertex demand in CVRP is associated with travel time for each arc in HVRP | Total costs | ||||||||||
| |||||||||||||||
Amiri et al. [ | BSS location & scheduling | X | X | Multi-objective - minimization of | |||||||||||
| |||||||||||||||
Bruglieri et al. [ | E-VReP | X | X | One way car sharing service, workers with bicycles go to the EVs locations and relocate them, battery level demand request | Multi-objective: | ||||||||||
| |||||||||||||||
Joo and Lim [ | EV routing | Energy SPP, no recharging, energy consumption model | Minimize energy consumption and average speed on the path | ||||||||||||
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Keskin and Çatay [ | E-VRPTW-FC | X | X | X | X | X | Vehicle number and total recharging costs | ||||||||
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Keskin et al. [ | E-VRPTW-FC | X | X | X | X | M/M/1 queuing system at capacitated CSs, battery capacity restriction, four planning intervals in a day, partial recharge not evident in paper | Total cost: energy cost, routing, labor and penalties for late arrivals | ||||||||
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Kullman et al. [ | E-VRP-PP | X | X | X | Public and private CSs, capacitated CS, single charging technology per CS | Expected time to visit all the customers | |||||||||
| |||||||||||||||
Li et al. [ | MBFM & recharging problem | X | X | Electric, diesel, compressed natural gas and hybrid-diesel buses | Total network benefit of replacing old vehicles with new ones within the planning horizon and budget constraints | ||||||||||
| |||||||||||||||
Lu et al. [ | MTFSP | X | X | X | X | Travel request are known a priori in time-varying origin-destination tables, service and deadheaded trips with different consumption rate, BEVs have higher priority than ICEVs | Total operating cost of a taxi company | ||||||||
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Masmoudi et al. [ | DARP-EV | X | X | X | X | Energy consumption model of Genikomsakis and Mitrentsis [ | Total routing costs (distance) | ||||||||
| |||||||||||||||
Paz et al. [ | MDEVLRPTW - BS/PR/BSPR | X | X | X | X | X | X | Three MIP models depending on the partial recharge and BSS | Total traveled distance | ||||||
| |||||||||||||||
Pelletier et al. [ | EFV-CSP | X | X | X | Preemptive charging with a limited number of chargers and charging events at the depot, time-dependent energy costs, FRD charge, grid restriction, cyclic and calendar battery degradation | Total charging costs | |||||||||
| |||||||||||||||
Poonthalir and Nadarajan [ | F-GVRP | X | Fuel consumption model, varying speed, AFVs, limited route duration | Bi-objective: | |||||||||||
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Schiffer and Walther [ | LRPIF | X | X | X | X | X | Loading and refueling facilities | Total costs: investment costs of vehicles and facilities, routing costs | |||||||
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Schiffer and Walther [ | RELRPTWPR | X | X | X | X | X | Uncertain customer pattern scenarios over working days regarding the spatial customer distribution, demand and service time windows | Total costs: investment costs of vehicles and facilities, routing costs | |||||||
| |||||||||||||||
Shao et al. [ | E-VRP | X | X | Energy consumption model: cargo load, uncertain travel speed | Total costs: travel, charging and fixed vehicle costs | ||||||||||
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Wang et al. [ | BEV routing | X | Parking fee, capacitated CSs - queuing time | Multi-objective minimization of | |||||||||||
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Zhang et al. [ | E-VRP | X | Energy consumption model, emissions, static speed, charging time unknown | Energy consumption | |||||||||||
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Basso et al. [ | 2sEVRP | X | X | X | X | Each CSs can have different charging rate, energy consumption model for road segments: speed profile, road slope, acceleration | Total energy consumption | ||||||||
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Breunig et al. [ | E2EVRP | X | Two echelons - first ICEVs and second BEVs, full recharge when visiting CSs, charging time unknown | Total routing costs | |||||||||||
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Bruglieri et al. [ | GVRP | X | AFVs, limited route duration | Vehicle number and total traveled distance | |||||||||||
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Froger et al. [ | E-VRP-NL | X | X | X | Total travel and charging time | ||||||||||
| |||||||||||||||
Hiermann et al. [ | | X | X | X | X | X | X | Propulsion mode decision | Total costs: fixed and variable | ||||||
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Jie et al. [ | 2E-EVRP-BSS | X | X | Routing in two echelons, sensitivity analysis of battery driving range and vehicle emissions | Total routing costs, the battery swapping costs and the handling costs at the satellites | ||||||||||
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Koyuncu and Yavuz [ | MGVRP | X | X | X | X | X | X | ICEVs (fixed recharge time) and AFVs, node- and -arc MILP formulation, single recharge technology per CS, additional modeling: customer demands, customer vehicle restrictions, subscription or pay-as-you-go refueling costs, completely heterogeneous fleet, last-mile delivery and closed time windows | Total traveling cost | ||||||
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Macrina et al. [ | GMFVRP-PRTW | X | X | X | X | X | X | X | Single recharge technology per CS, but different charging technologies between CSs, energy consumption model for road segments with time-dependent speeds | Cost of energy recharged during the route and at the depot, fuel costs, and cost related to traveled distance | |||||
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Macrina et al. [ | GMFVRP-PRTW | X | X | X | X | X | X | Single recharge technology per CS, but different charging technologies between CSs | Recharging, routing and activation costs, limit emissions | ||||||
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Normasari et al. [ | CGVRP | X | X | AFVs, limited route duration | Vehicle number and total traveled distance |
The major problem that BEVs in delivery processes are facing is the limited driving range. Grunditz and Thiringer [
BEVs are more likely to be used on short distances and/or in urban areas where they are more effective than ICEVs due to the low driving speed, low noise production, frequent stops, and financial incentives. In cases when the average route length is short, such as the average FedEx route length in the USA, which is 68 km [
Due to the low specific energy, energy consumption should be precisely estimated in order to achieve a BEV’s maximal driving range and to reduce the overall routing costs. The energy consumption can be estimated by simulation models but due to the complexity of the E-VRP and unknown driving cycles in advance, mostly macroscopic models with several real-world approximations are applied in the BEV routing models. In the available literature, energy consumption is often estimated using longitudinal dynamics model (LDM). Here, the LDM of Asamer et al. [
Goeke and Schneider [
Genikomsakis and Mitrentsis [
Macrina et al. [
Basso et al. [
Asamer et al. [
Preis et al. [
In real-life conditions, speeds on the roads are time-dependent and can be described as the speed profile over the observed time period. Speed profile depends on the road type, driver behavior, traffic (accidents, recurrent congestions), weather conditions, etc. [
Pelletier et al. [
Many different VRP variants were researched over the years. With BEV appearance, researchers started to adapt them to the E-VRP context. Due to the specific characteristics of BEV routing, some new problem-specific variants emerged. Here, we present some of the most relevant E-VRP variants and related problems.
The energy shortest path problem (E-SPP) and electric TSP (E-TSP) can be considered as two of the simplest forms of the E-VRP. In most of the VRP variants, graph arcs have positive weight values that can represent distance, travel time, cost, etc. To compute the shortest path between customers, the most commonly applied algorithms are Dijkstra, Bellman-Ford,
As VRP is the generalization of the TSP, the E-VRP is closely related to the E-TSP, in which a set of customers has to be served by only one BEV. Roberti and Wen [
In today’s vehicle fleets, mostly ICEVs are present. Transition to an almost wholly electric fleet is a very challenging economic task. Therefore, most companies are gradually integrating BEVs into their existing ICEV fleet. Routing algorithms have to be upgraded and adapted for electric fleet characteristics, as on-time priority is harder to achieve.
Fleet size and mix VRP (FSM-VRP) was first introduced by Golden et al. [
As compensation for the limited driving range of BEVs, HEVs, which have both an internal combustion engine and an electric engine, have been developed. Two main types of HEVs are present on the market: the
One of the first papers that dealt with the PHEVs routing problem was published by Abdallah [
In the beginning, in most of the E-VRP problems, full recharge was considered when a BEV visited a CS [
Several papers have analyzed strategies of partial recharging and formulated the problem as E-VRPTW with partial recharging (E-VRPTWPR). Bruglieri et al. [
Today, multiple charging technologies are present: (i) slow, 3 kW (6-8 h); (ii) fast, 7-43 kW (1-2 h); and (iii) rapid, 50-250 kW (5-30 min) [
In most of the E-VRP related literature, either linear or constant charging time is considered. Most of the BEVs have lithium-ion batteries installed, which are often charged in constant-current constant-voltage (CC-CV) phases: first by constant current until approx. 80% of the SoC value and then by a constant voltage. In the CC phase, SoC increases linearly, and in the CV phase, the current drops exponentially and SoC increases nonlinearly in time, which prolongs charging time [
Due to the currently low BEV market share, the number of CSs installed in the road infrastructure is also relatively low. Therefore, great potential lies 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 and vehicle routes supplying customers from these depots [
Schiffer et al. [
Instead of charging at CS, at specially designed BSS, empty or nearly empty batteries can be replaced with fully charged ones [
Breunig et al. [
Many companies that use BEVs prefer charging the vehicles at their own facilities in order to charge the vehicles between the delivery routes and during specific periods of the day. In such occasions, there are usually a limited number of chargers at the depot, typically fewer than the fleet size; therefore, the efficient charging schedule at the depot has to be determined.
Pelletier et al. [
Barco et al. [
Most of the E-VRP research considers static conditions on the road network. The traffic states change recurrently, depending on the time of the day, day of the week, and season, or nonrecurrently when a traffic incident occurs, such as an accident [
Many researchers are dealing with the scheduling of bus/taxi routes that are fixed or could be slightly altered. Li et al. [
Bruglieri et al. [
Masmoudi et al. [
Paz et al. [
Schiffer and Walther [
Keskin et al. [
Kullman et al. [
The green vehicle routing problem focuses on the reduction of routing pollution on the environment. The key idea is to promote the use of sustainable energy sources and minimize overall emissions. Regarding the emission of BEV, Álvarez Fernández [
Erdoğan and Miller-Hooks [
Poonthalir and Nadarajan [
Normasari et al. [
Macrina et al. [
Koyuncu and Yavuz [
As BEVs have no local emissions, the E-VRP is closely related to the minimization of GHG emissions, where a problem-specific GVRP variant called the pollution routing problem (PRP) was introduced [
It can be noted that several papers are dealing with fuel consumption and pollution emissions and not AFVs, but the term
The E-VRP can be formulated more generally as the VRP with intermediate stops (VRPIS), in which the vehicle visits intermediate or intraroute facilities to replenish/unload the goods or to refuel [
Most of the researchers are using a single-objective function that represents travel distance, total costs, energy consumption, total time, etc. The multiobjective variants are still relatively scarcely applied. Generally, the solution of the multiobjective problem is not the optimal solution for all of the objectives, but rather it is satisfactory in those terms.
Demir et al. [
The already mentioned F-GVRP of Poonthalir and Nadarajan [
Amiri et al. [
Bruglieri et al. [
Table
Since VRP is a well-researched problem, a large number of procedures for solving the problem have been proposed. Due to the NP-hardness of the problem and a large number of customers in real-life problems, most of the procedures used in real-life applications are heuristics, metaheuristics, and hybrid combinations. For small-sized problems, a great number of exact procedures have been proposed. Many of the VRP procedures in the available literature are with an adaptation applicable for solving the E-VRP. An overview of the applied procedures for solving the E-VRP and related problems is presented in Table
Overview of the applied procedures.
Reference | Problem name | NF | Initial | Hybrid Metaheuristics Heuristics | A | Exact & Software | Instances | Description |
---|---|---|---|---|---|---|---|---|
Bektaş and Laporte [ | PRP | CPLEX - small | Generated | |||||
| ||||||||
Conrad and Figliozzi [ | Recharging VRP | Route construction | Iterative route construction and improvement | Solomon - CVRP-TW | | |||
| ||||||||
Gonçalves et al. [ | VRPPD with mixed fleet | CPLEX - small | ||||||
| ||||||||
Demir et al. [ | PRP | CWS | ALNS + speed opt. | PRP | ALNS destroy: Random, WorstDistance, WorstTime, Route, Shaw, ProximityBased, TimeBased, DemandBased, HistoricalKnowledge, Neighborhood, Zone, NodeNeighborhood | |||
| ||||||||
Erdoğan and Miller-Hooks [ | GVRP | | LS | CPLEX - small | GVRP | LS operator: Exchange | ||
| ||||||||
Omidvar and Tavakkoli-Moghaddam [ | GVRP | | SA | CPLEX - small | Generated - Solomon procedure | No description of the implementation | ||
| ||||||||
Abdallah [ | PHEVRPTW | Push forward insertion heuristic + LP optimization of charges | Heuristics for feasible upper bound, TS | SA | MATLAB, CPLEX - small, DP | Adapted Solomon | Compare three procedures for small instances: | |
| ||||||||
Barco et al. [ | E-VRP and charge scheduling | Random | Differential evolution, GA | Real | ( | |||
| ||||||||
Van Duin et al. [ | FSMVRPTW, EVFSMVRPTW | Sequential insertion heuristics | LS | Shortrec software | ||||
| ||||||||
Adler and Mirchandani [ | Online routing of BEVs | Approximate DP | Real | Markov chance-decision process | ||||
| ||||||||
Alesiani and Maslekar [ | BEV routing | X | Random | GA | Generated | Solution: a two-dimensional binary array of routes and CSs | ||
| ||||||||
Demir et al. [ | PRP | ALNS + speed opt. + heuristic | Generated | ALNS of Demir et al. [ | ||||
| ||||||||
Felipe et al. [ | GVRP-MTPR | Constructive- | LS(48A) | SA | GVRP-MTPR, GVRP, E-VRPTW | LS (48A): RechargeRelocation, 2-Opt, reinsertion with four options: first or best and yes/no update of savings | ||
| ||||||||
Preis et al. [ | Energy-optimized routing of BEVs | X | CWS | TS | MIP solver - small | Generated | Neighborhood operator: Relocate | |
| ||||||||
Sassi et al. [ | HEVRP-TDMF | CRH | ILS+LNS | Real | CRH - | |||
| ||||||||
Schneider et al. [ | E-VRPTW | X | Sweep | VNS+TS | SA | CPLEX - small | E-VRPTW, MDVRPI, GVRP | VNS - 15 neighborhood structures based on the cyclic exchange move |
| ||||||||
Zündorf [ | EVRC | Heuristics to speed up exact algorithms | Charging function propagating algorithm, CH,A∗ | Real | Battery CSPP | |||
| ||||||||
Bruglieri et al. [ | E-VRPTW | VNS with branching | CPLEX | E-VRPTW - small | VNS with Hamming distance of three: removing CS, inserting CS, merging | |||
| ||||||||
Goeke and Schneider [ | E-VRPTWMF | X | Modified insertion heuristics | ALNS+LS | SA | E-VRPTWMF, VRPTW, E-VRPTW | Initial - long BEV routes are converted to ICEV routes | |
| ||||||||
Lebeau et al. [ | FSMVRPTW-EV | CWS + tree branching | FSMVRPTW - EV | The tree is used for different combinations of vehicles - in each branch, one vehicle is selected and vertices are inserted | ||||
| ||||||||
Moghaddam [ | E-VRPTWPR | Gurobi | Generated | |||||
| ||||||||
Pourazarm et al. [ | Single/Multi BEV routing | DP | Generated | Single - electric vehicle CSPP | ||||
| ||||||||
Sassi et al. [ | VRP-MFHEV | CRH | ITS+LNS | RR | Generalized E-VRPTW | CRH - without charge scheduling at the depot | ||
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Sassi et al. [ | VRP-HFCC | CRH | ILS+LNS | RR | Real, generalized E-VRPTW | LNS destroy: Random | ||
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Schneider et al. [ | VRPIS (EVRPRF) | X | MCWS | AVNS+LS | SA | CPLEX - small | EVRPRF, GVRP, VRPIRF | AVNS shaking phase - |
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Yang and Sun [ | BSS-EV-LRP | Radius covering + MCWS | TS-MCWS+LS | CPLEX - small | BSS-EV-LRP | LS: Relocate, Exchange | ||
X | Modified sweep + greedy | SIGALNS | SA | Remove all BSSs and then apply the iterated greedy procedure to insert best BSSs | ||||
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Desaulniers et al. [ | E-VRPTW -SF/MF/SP/MP | Branch-price-and-cut GENCOL + CPLEX | E-VRPTW with modif. for SF/SP | |||||
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Doppstadt et al. [ | HEV-TSP | | ITS | CPLEX | HEV-TSP | Initial route is driven in combustion mode + LS - 2-Opt hill-climbing 4:1 ratio - four arcs have to be driven in charging mode in order to drive the last one in electric mode | ||
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Hiermann et al. [ | E-FSMFTW | X | Iterative route construction | ALNS+LS | SA | Branch-and-price+DP | E-FSMFTW, E-VRPTW | Labeling algorithm for CSs and new extension functions |
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Keskin and Çatay [ | E-VRPTWPR | Insertion heuristics | ALNS | SA | CPLEX - small | E-VRPTW | ALNS destroy options: remove customer only or with preceding/succeeding CS | |
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Koç and Karaoglan [ | GVRP | MCWS | SA | SA | Branch-and- cut | GVRP | Improved MIP formulation of GVRP | |
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Lin et al. [ | E-VRP | MATLAB | Real | The linearized model formulation including energy costs and battery charging time | ||||
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Masliakova [ | Routing and charging of electric buses | Random CSs positions + route construction | GA+AC | Real | GA: random 2-point crossover, mutation: move the CS to the nearest vertex | |||
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Mirmohammadi et al. [ | Periodic green VRP | CPLEX | Generated | |||||
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Montoya et al. [ | GVRP | ( | Modified multi-space sampling heuristic + set partitioning | Gurobi - set partitioning | GVRP | | ||
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Roberti and Wen [ | E-TSPTW | X | Random | Three-phase heuristic: VND + LS + DP | CPLEX, DP | E-TSPTW, TSPTW, E-VRPTW - single vehicle | ( | |
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Schiffer et al. [ | E-LRPTWPR | X | CWS | ALNS+DP+LS | DP | Real | ALNS destroy: large (Add, Drop, SwapPerfect, SwapPerfectOut) - change CS configuration; small (Worst, Related, Route, Shaw, StationVicinity) - change route configuration | |
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Wen et al. [ | E-VSP | Greedy constructive | ALNS | SA | CPLEX - small and post opt. | E-VSP | ALNS destroy: Random, TimeRelated, NeighboringSchedule (problem specific) | |
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Andelmin and Bartolini [ | GVRP | Exact procedure | GVRP, generated | Multigraph - arc represents possible paths between customers with inserted CS Set partitioning formulation + subset row inequalities and | ||||
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Çatay and Keskin [ | E-VRPTWPR | CPLEX - small | E-VRPTW | |||||
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Froger et al. [ | E-VRP-NL-C | NNH-TSP | Two-stage heuristic | Gurobi - MILP models | Modified E-VRP-NL | First stage without CS capacity constraint: ILS - giant TSP route + perturbation (random double bridge) + split procedure on acyclic graph with heuristic for CSs + LS: 2-Opt, Relocate and MILP model for charging decisions; local optimum solutions are stored in a pool of solutions | ||
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Hof et al. [ | BSS-EV-LRP | X | MCWS | AVNS+LS | SA | CPLEX - small | BSS-EV-LRP - Yang and Sun [ | AVNS - Schneider et al. [ |
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Leggieri and Haouari [ | GVRP | CPLEX | GVRP | |||||
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Mancini [ | HVRP | Run MILP for | LNS | Xpress | HVRP, GVRP | Two routes are destroyed and the MILP mathematical model is run again for | ||
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Montoya et al. [ | E-VRP-NL | NNH-TSP + split | ILS (VND)+ heuristic concentration | FRVCP-Gurobi | E-VRP-NL | Sequencing first, charging (FRVCP) second | ||
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Schiffer and Walther [ | E-LRPTWPR | Gurobi | E-VRPTW - small | |||||
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Shao et al. [ | EVRP-CTVTT | Encoded mode | GA | Real | Dynamic Dijkstra algorithm for time-dependent SPP, single string encoding, each individual after the crossover or mutation needs to satisfy all the constraints - if not run the procedure again, elitism | |||
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Sweda et al. [ | Adaptive routing and recharging policies of EVs | Two specially designed heuristics | Generated | Origin-destination, | ||||
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Vincent et al. [ | HVRP | X | Modified NNH | SA | SA | Various CVRP | Neighborhood operators: Insert, Swap, Reverse | |
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Amiri et al. [ | BSS location & scheduling | GA | Generated | Non-dominated sorting GA | ||||
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Bruglieri et al. [ | E-VReP | Construction heuristics | Two-phase heuristic | CPLEX | Real | | ||
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Joo and Lim [ | EV routing | AC | Generated | Energy SPP | ||||
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Keskin and Çatay [ | E-VRPTW-FC | | ALNS + opt. | SA | CPLEX - small and opt. | GVRP-MTPR, E-VRPTW | ALNS of Keskin and Çatay [ | |
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Keskin et al. [ | E-VRPTW with waiting times | ALNS + opt. | SA | CPLEX - small and opt. | E-VRPTW | ALNS of Keskin and Çatay [ | ||
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Kullman et al. [ | E-VRP-PP | Routing policies | Exact, FRVCP | Generated | Markov decision process - approximated DP solution | |||
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Li et al. [ | MBFM & recharging problem | CPLEX | Real | Case study, new life additional benefit costs method for the replacement of vehicles (CPLEX); two routing methods: single/multi-period approach for solving the recharging problem | ||||
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Lu et al. [ | MTFSP | Network decomposition by vehicles and time | CPLEX - sub-models | Real | Multi-layer taxi-flow time-space network | |||
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Masmoudi et al. [ | DARP-EV | X | Modified insertion heuristics | EVO-VNS | CA, RR | CPLEX - small | HDARP, DARP-EV | |
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Paz et al. [ | MDEVLRPTW - BS/PR/BSPR | CPLEX | Modified E-LRPTW - small | |||||
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Pelletier et al. [ | EFV-CSP | CPLEX | Generated | The focus is on the deriving meaningful insights and not the computation prowess | ||||
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Poonthalir and Nadarajan [ | F-GVRP | NNH + random | TVa-PSOGMO | GVRP | Particle encoding: customer sequence only which is converted to solution by inserting depots and refueling stations | |||
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Schiffer and Walther [ | LRPIF | X | MCWS | ALNS+LS+DP | DP | E-LRPTWPR (old and new), BSS-EV-LRP, E-VRPTW, E-VRPTWPR | Penalty terms evaluation - corridor-based approach - a range of possible refueling times + concatenation operators | |
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Schiffer and Walther [ | RELRPTWPR | X | MCWS | ALNS+LS+DP | DP, Gurobi - small | Real | Parallelized ALNS of Schiffer and Walther [ | |
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Shao et al. [ | E-VRP | GA+LS | Real | General GA + elitism | ||||
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Wang et al. [ | BEV routing | GA + fuzzy prog. | Generated | Fuzzy programming and fuzzy preference relations are applied to transform the objective function into a single objective function | ||||
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Zhang et al. [ | E-VRP | X | Construction + insertion heuristics | AC+ILS, ALNS | Generated | Initial: | ||
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Basso et al. [ | 2sEVRP | CPLEX | Generated | Bellman-Ford algorithm for SPP | ||||
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Breunig et al. [ | E2EVRP | Greedy insertion | LNS+LS+DP | Exact - route and candidate solution enumerations | E2EVRP | LNS destroy: RelatedRemoval, Random, CloseSatellite, OpenSatellite, RemoveSingleCustomerRoutes | ||
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Bruglieri et al. [ | GVRP | Exact + Xpress | GVRP | Path-based approach - the route is a composition of paths for a subset of customers without station visits: | ||||
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Froger et al. [ | E-VRP-NL | Heuristic for FRVCP | Exact labeling - FRVCP, GUROBI - small | E-VRP-NL | Heuristic for FRVCP: at each CS, EV is charged minimum energy required to reach next CS or depot | |||
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Hiermann et al. [ | | X | GA+LS+LNS+DP | CPLEX - set partitioning | | The sequence of customers and vehicle types without CS | ||
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Jie et al. [ | 2E-EVRP-BSS | X | Extended sweep | CG-ALNS | SA | CPLEX - small | 2E-EVRP-BSS, 2E-VRP | |
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Koyuncu and Yavuz [ | MGVRP | CPLEX | GVRP | Upper and lower bound for energy state (labeling algorithms) and arrival times Lower bound on the number of vehicles | ||||
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Macrina et al. [ | GMFVRP-PRTW | CPLEX | Hybrid LNS | GMFVRP-PRTW | Initial: CPLEX to find an initial feasible solution | |||
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Macrina et al. [ | GMFVRP-PRTW | Sequential insertion heuristics | ILS | CPLEX - small | E-VRPTW | Initial: two clusters of customers for EV and ICEV routes - | ||
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Normasari et al. [ | CGVRP | NNH | SA+LS | SA | CPLEX | CGVRP, GVRP | Random neighborhood structures: Swap, Insertion, station Insert/Delete |
When creating or modifying a VRP solution, two types of search procedures regarding the feasibility of the solution can be observed: allowing only feasible solutions or acceptance of infeasible solutions. The infeasible solution means that some customers are served without satisfying all of the problem constraints. Related, feasible procedures are searching in the space of feasible solutions, while infeasible procedures allow searching the space of infeasible solutions, which broadens the search. In the E-VRP, the feasibility mostly refers to vehicle load, customer time window, and battery (energy) constraints. In infeasible procedures, objective function is often defined with penalization coefficients, which are updated during the search process. At the beginning of the search, infeasible solutions are allowed in order to search a larger solution space. As the search process comes to an end, penalties for infeasible solutions increase. An example of such an objective function is given by (
Exact procedures are able to find an optimal solution for a smaller number of customers: up to 360 customers for CVRP and up to 100 customers for VRPTW [
Heuristic procedures seek to solve the problem based on the specific knowledge of the problem, usually suboptimal or close enough to a satisfactory solution. In the research field of VRP, heuristic procedures can be split into constructive and improvement heuristics.
Constructive heuristics are often used to generate an initial solution by either serial or parallel route construction. Solutions are constructed in a greedy way, which often produces solutions of the VRP that are 10-15% far from an optimal solution [
The
The
The
Many researchers are adopting some form of the insertion heuristics, which have some characteristics of already reviewed heuristics. The general idea of the insertion heuristics is to iteratively insert or add customers in available routes. In each step of the algorithm, an unserved customer together with a route and its position in the route are determined to least increase the objective function. This selection can be deterministic but is often stochastic to select at random one from
Improvement heuristics or local search (LS) explores the neighborhood of the incumbent solution, searching for a better solution. The neighborhood is explored by applying perturbation moves based on the composite neighborhood operators. The local search stops when no improving solution can be found in the neighborhood of the incumbent solution, which is then called local optima. A great variety of researchers have performed LS procedures to intensify the search, coupled together with perturbation moves to escape the local optima. Often, the perturbation moves are similar to the neighborhood operators used in LS phase. Most of the classical VRP neighborhood operators [ 2
The order of the operators affects both the solution quality and the execution time. Often, there is a question whether to make a
Neighborhood operators explore only the space of the immediate vicinity of the current solution, which often leads to local optima. In order to search a larger solution space, Shaw [
Operator evaluation is an important part that highly affects execution time and has to be efficiently implemented. For example, in VRPTW, for most of the basic neighborhood operators, the load capacity check and time window check can be computed in constant time
Many researchers employ metaheuristics to continue the exploration after the first local optima occurrence. Metaheuristics can be defined as heuristics guiding other heuristics and can be divided into neighborhood-oriented metaheuristics and population metaheuristics. Most often, multiple metaheuristics and heuristics are combined together and adapted to the problem; therefore, the term hybrid is used in such occasions.
Neighborhood-oriented heuristics iteratively explore the neighborhood of the incumbent solution. Here, we present the most used neighborhood heuristics in the E-VRP literature.
SA is one of the most used metaheuristics for acceptance criteria, in order to escape the local optima, but some others can also be applied. Tiwari et al. [
Population metaheuristics are based on the natural selection to evolve a population and survival of the fittest. They have been widely applied in the VRP field: genetic algorithm [
Barco et al. [
The overview of the applied procedures in the E-VRP field is presented in Table
We can note that a vast number of researchers use hybrid heuristics procedures with only a few focusing on the exact procedures [
The ALNS is one of the most applied procedures and it is giving the BKSs on various problem variants. Among them, we can point out the following: ALNS of Hiermann et al. [
Most of the population-based metaheuristics are not efficiently implemented so they are not giving high-quality solutions. But, the HGA of Hiermann et al. [
With the increase in popularity of electric-powered vehicles, many logistics companies have started to integrate electric vehicles into their delivery fleets. Routing a fleet of electric vehicles for delivering goods was formulated as the electric vehicle routing problem. Besides the often used load (cargo) capacity and time window constraints, E-VRP routing models have to account for the limited driving range of BEVs, which directly corresponds to the more frequent recharging needs at CS.
In this paper, a literature review on recent developments in the E-VRP field is presented. We consider BEVs to be vehicles powered only from batteries mounted inside the vehicle. A general overview of the BEV’s characteristics for goods delivery includes the driving range, battery capacity, application, and case studies. As energy consumption estimation is an important part of BEV routing, we summarized the recent research on theoretical and data-driven energy consumption models.
Due to the BEVs specific characteristics, new E-VRP variants have emerged: a mixed fleet of electric and conventional vehicles, partial recharging, simultaneous CS siting and BEV routing, nonlinear charging, different charging technologies, battery swap technology, hybrid vehicles, green routing, etc. The development of efficient heuristics was necessary to find optimal or near-optimal solutions to the new routing problems. We reviewed the state-of-the-art exact, heuristic, and hybrid procedures applied for solving various E-VRP variants. The adaptive large neighborhood search [
From the literature review, it can be noted that the electric vehicle routing research community has grown rapidly in the last few years and many problem variants have already been explored. Nevertheless, we can highlight possible future directions as follows.
By our observation, there is a lack of papers regarding case studies and application cases, where actual E-VRP models could be evaluated, and some meaningful insights could be drawn. Several energy consumption models were reviewed, but only few are predicting realistic energy consumption at the road segment level in the road network. Only recently, have researchers started to integrate a nonlinear charging process, CS location problem, and hybrid electric vehicles into the E-VRP models. A few papers addressed the problem of CS capacity, as a limited number of BEVs can charge simultaneously at a CS. Several variations were observed: waiting times [
For future research regarding the solution procedures, we can highlight the following. Although there have been a couple of exact procedures developed for GVRP, only a few exact procedures have been proposed for the E-VRP and its extensions. A great number of researchers applied a population metaheuristic to solve the problem, but only a handful of them produced high-quality solutions in reasonable computation time. To improve the computation time, parallelized procedures could be used [
Here, we present an overview of the used destroy and repair operators in ALNS metaheuristic for solving EVRP variants and related problems. We divide the destroy operators in terms of whether they include customer removal (C), station/facility removal (S), or route removal (R).
Destroy operators:
Repair operators:
Two-echelon capacitated electric vehicle routing problem with battery swapping stations
Two-stage electric vehicle routing problem
Ant colony algorithm
Alternative fuel vehicle
Adaptive large neighborhood search
Battery electric vehicle
Best-known solution
Battery swap station
Electric vehicles battery swap stations location routing problem
Cauchy function
Capacitated green vehicle routing problem
Charge routing heuristic
Charging station
Constrained shortest path problem
Capacitated vehicle routing problem
Dial-a-ride problem with electric vehicles and battery swapping stations
Density based clustering algorithm
Dynamic programming
Electric fleet size and mix vehicle routing problem with time windows and recharging stations
Electric location routing problem
Electric location routing problem with time windows and partial recharges
Energy shortest path problem
Electric traveling salesman problem
Electric traveling salesman problem with time windows
Electric vehicle relocation problem
Electric vehicle routing problem
Electric vehicle routing problem with nonlinear charging functions
Electric vehicle routing problem with nonlinear charging functions and capacitated CSs
Electric vehicle routing problem with public-private recharging strategy
Electric vehicle routing problem with time windows
Electric vehicle routing problem with time windows and fast charging
Electric vehicle routing problem with time windows, S (single recharge), M (multiple recharges), F (full recharge), P (partial recharge)
Electric vehicle routing problem with time windows and mixed fleet
Electric vehicle routing problem with time windows with partial recharging
Electric vehicle scheduling problem
Electric two-echelon vehicle routing problem
Electric freight vehicles charge scheduling problem
Electric vehicle
Electric vehicle fleet size and mix vehicle routing problem with time windows
Evolutionary variable neighborhood search
Electric vehicle route planning with recharging
Electric vehicle routing problem with charging time and variable travel time
Electric vehicle routing problem with recharging facilities
Fuel efficient green vehicle routing problem
Facility-related demand
Fixed route vehicle charging problem
Fleet size and mix vehicle routing problem
Fleet size and mix vehicle routing problem with time windows
Fleet size and mix vehicle routing problem with time windows for electric vehicles
Genetic algorithm
Greenhouse gas
Green mixed fleet vehicle routing problem with partial battery recharging and time windows
Greedy mutation operator
Greedy randomized adaptive search procedure
Green vehicle routing problem
Green vehicle routing problem with multiple technologies and partial recharges
Hybrid heterogeneous electric fleet routing problem with time windows and recharging stations
Heterogeneous dial-a-ride problem
Hybrid electric vehicle
Hybrid electric vehicle traveling salesman problem
Heterogenous electric vehicle routing problem with time-dependent charging costs and a mixed fleet
Hybrid genetic algorithm
Hybrid vehicle routing problem
Internal combustion engine vehicle
Iterated local search
Iterated tabu search
Longitudinal dynamics model
Location routing problem
Location routing problem with intraroute facilities
Local search
Mixed bus fleet management problem
Modified Clark and Wright savings method
Multidepot electric vehicle location routing problem with time windows (battery swapping/partial recharging)
Multidepot vehicle routing problem
Multidepot vehicle routing problem with inter-depot routes
Mixed fleet green vehicle routing problem
Mixed integer linear program
Mixed integer program
Multiple linear regression
Mixed taxi fleet scheduling problem
Nearest neighbor heuristic
Plug-in hybrid electric vehicles
Plug-in hybrid electric vehicle routing problem with time windows
Push forward heuristics
Pollution routing problem
Particle swarm optimization
Resource constrained shortest path algorithm
Robust electric location routing problem with time windows and partial recharging
Record-to-record procedure
Simulated annealing
State of charge
Time-dependent vehicle routing problem
Tabu search
Traveling salesman problem
Traveling salesman problem with time windows
Particle swarm optimization with greedy mutation operator and time-varying acceleration coefficient
Variable neighborhood decent
Variable neighborhood search
Vehicle routing problem
Vehicle routing problem with a mixed fleet of conventional and heterogenous electric vehicles including new constraints
Vehicle routing problem with mixed fleet of conventional and heterogenous electric vehicles
Vehicle routing problem with intermediate replenishment facilities
Vehicle routing problem with intermediate stops
Vehicle routing problem with pickup and delivery
Vehicle routing problem with time windows.
The authors declare that there are no conflicts of interest regarding the publication of this article.
The research has been supported by the European Regional Development Fund under grant KK.01.2.1.01.0120 and has also been partially supported by other two sources: Croatian Science Foundation under project IP-2018-01-8323 and project KK.01.1.1.01.0009 (DATACROSS) within the activities of the Centre of Research Excellence for Data Science and Cooperative Systems supported by the Ministry of Science and Education of the Republic of Croatia.