The Vehicle Routing Problem: State-of-the-Art Classiﬁcation and Review

: Transportation planning has been established as a key topic in the literature and social production practices. An increasing number of researchers are studying vehicle routing problems (VRPs) and their variants considering real-life applications and scenarios. Furthermore, with the rapid growth in the processing speed and memory capacity of computers, various algorithms can be used to solve increasingly complex instances of VRPs. In this study, we analyzed recent literature published between 2019 and August of 2021 using a taxonomic framework. We reviewed recent research according to models and solutions, and divided models into three categories of customer-related, vehicle-related, and depot-related models. We classiﬁed solution algorithms into exact, heuristic, and meta-heuristic algorithms. The main contribution of our study is a classiﬁcation table that is available online as Appendix A. This classiﬁcation table should enable future researchers to ﬁnd relevant literature easily and provide readers with recent trends and solution methodologies in the ﬁeld of VRPs and some well-known variants.


Introduction
Problems related to the distribution of goods between warehouses and customers are generally considered as vehicle routing problems (VRPs).The VRP was first proposed by Dantzig and Ramser [1] in 1959 to model how a fleet of homogeneous trucks could serve the demand for oil from a number of gas stations from a central hub with a minimum travel distance.Five years later, Clarke and Wright [2] added more practical restrictions to VRPs in which the delivery of goods to each customer must occur within a set of bounds.This type of problem model became known as the VRP with time windows (VRPTW), which is one of the most widely studied topics in the field of operations research [3].
However, current VRP models differ significantly from those introduced by Dantzig and Ramser [1] and Clarke and Wright [2], because they aim to incorporate real-world complexities.Because VRPs are some of the most critical challenges faced by logistics companies, an increasing amount of research is focusing on VRPs.Several surveys and taxonomies for VRPs can be found in [3][4][5][6]  Solving VRPs is computationally expensive and categorized as NP-hard [11], because realworld problems involve complex constraints such as time windows, time-dependent travel times (reflecting traffic congestion), multiple depots, and heterogeneous fleets.These features introduce significant complexity and have dramatically evolved the VRP research landscape.
The processing speed and memory capacity of computers has grown rapidly, enabling the processing of increasingly complex instances of VRPs and widespread application of logistics distribution scenarios.The number of VRP solution methods introduced in the academic literature has grown rapidly over the past few decades.According to Eksioglu et al. [4], the VRP represents an evolving field of operations research that has been growing exponentially at a rate of 6% per year, which makes it difficult to keep track of developments in the field and obtain a clear overview of which variants and solution methods are relatively novel.
The VRP family can be considered as two combinatorial senses: (1) the number of possible solutions, which grow exponentially with computer science and algorithm design; and (2) the number of conceivable problem variants, which also grow exponentially with a variety of problem attributes [12].This survey classifies the academic literature on VRPs from the perspective of solution methodologies, as well as the detailed characteristics of VRPs.Because we base our classification on the taxonomy presented in [4], we restrict our analysis to articles published between 2019 and August of 2021.Therefore, we do not intend to provide an exhaustive overview of VRP literature.To the best of our knowledge, this article provides the first structured classification of recent VRP literature based on solution and problem attributes.
The main contribution of our paper is a classification table that is available online as Appendix A. This classification table should enable future researchers to find relevant literature easily by eliminating or selecting characteristics in the taxonomy, leaving only articles tailored to their interests.The main objective of this work is to provide readers with recent trends and solution methodologies in the field of VRPs and some well-known variants.This survey is expected to help future researchers identify a problem domain and promising topics for research.
Section 2 defines the scope of this survey and Section 3 introduces the VRP and its variants.A comprehensive survey of state-of-the-art strategies currently used for solving VRPs is presented in Section 4. Section 5 summarizes our observations and conclusions.

Scope of the Survey
We analyzed recent literature published between 2019 and August of 2021 using a taxonomic framework.Classification is followed by a survey that uses the taxonomy to evaluate trends in the field and identify which articles contribute to these trends.We restricted the reviewed literature to the following features: only relevant articles published in English-language journals were considered, meaning books, conference proceedings, and dissertations were excluded.
To extract the most relevant literature and keep the number of articles manageable, the following search strategy was applied.First, only articles containing "vehicle routing" as title words or keywords were selected.Second, the search was limited to articles that were extended by highly cited articles published in any ranked journal (Google Scholar top 20), excluding review papers.For papers published in 2021, which are too recent to have cite ranking, we selected the top five pages from Google Scholar, each of which had 10 cited articles, as well as two review papers written by Moghdani et al. [13] and Asghari and Al-e (2021).Third, the abstracts of selected articles were read to determine their relevance to the subject.
This search strategy resulted in a final set of 88 articles.Although this selection is not exhaustive, it contains the majority of recent articles on VRPs and can be considered as representative of the field.

VRP
In addition to the classical VRP, several variants have also been studied.Capacitated VRP (CVRP), VRPTW, VRP with heterogeneous fleets (HFVRP), time-dependent VRP (TD-VRP), and multi-depot VRP (MDVRP) are some of these variants.The classical VRP can be described as follows.Let G = (V, A) be a graph, where V = {v 0 , v 1 , v 2 , . . . ,v N }, where {v 1 , v 2 , . . . ,v N } is the node set representing customers to be served and v 0 is the depot.Each customer is characterized by a demand D i .A = v i , v j : v i , v j ∈ V is the arc set (subscript indicates sequence) linking nodes i and j with a distance d ij .Let M m = {m 1 , m 2 , m 3 , . . . ,m m } denote the vehicle set, where each vehicle has a maximum load capacity cap m , meaning the total load of vehicle m cannot exceed the maximum load capacity cap m .To reflect a real distribution scenario accurately, different features are considered according to the settings of heterogeneous models.The goal of the VRP is to derive optimal vehicle routes such that each customer is visited exactly once by one vehicle and each vehicle starts and ends its route at the depot.The following assumptions are adopted: 1.
The depot has a demand equal to zero.

2.
Each customer location is serviced by only one vehicle.
Each vehicle shall not exceed its maximum load capacity cap m . 5.
Each vehicle starts and ends its route at the depot.6.
Customer demand, distribution distances between customers, and delivery costs are known.
The notations used for problem definition are summarized as Tables 1-3.Traditional logistics models focus on minimizing the total cost of a network.This is where the concept of the VRP is best applied.We follow this concept and add the fixed cost f c m of a vehicle, which represents rent cost or operating costs, to the total cost to minimize the total number of vehicles.We also include the variable cost vc m of delivery using each type of vehicle to optimize vehicle scheduling.Additional constraints appear in the target calculation in the form of penalty functions to enforce vehicle limit constraints.The objective of minimizing the total cost is defined as follows: subject to the following constraints: Routing: Demand and capacities: The objective function in Equation ( 1) is the total cost, which includes the fixed cost and variable cost.Constraint (2) states that each vehicle should return to the depot, where the subscript is zero.Constraint (3) ensures that each node can only be visited once in a route.Constraint (4) states that, if a vehicle arrives at a node, it must leave that node, thereby ensuring route continuity.Constraints (5) and ( 6) impose restrictions on the amounts of demand and capacity.Constraint (7) defines the maximum number of available vehicles veh m .

VRP Variants
Practical requirements and new challenges require extensive definitions and formulations of the VRP.For example, distance, driver working hours, time windows, traffic conditions, and so on can all arise in real-world VRPs and enrich the definition and applications of VRPs.This chapter provides an overview of recent research on different models for vehicle routing.The main goal of this chapter is to present an overview of vehicle routing and scheduling areas while discussing several real-world applications.
Some features of VRPs are summarized in Table 4 based on the research by Eksioglu et al. [4].Other variants have also been studied beyond the classical VRP.These variants include the influence of time factors, time windows of customers, maximum operating time of vehicles, differing delivery times caused by varying traffic conditions, varying characteristics of vehicles, varying capacities, varying speeds, and new types of electric vehicles.By referring to the taxonomy of [4], we divided models into three main categories: customer-related, vehicle-related, and depot-related models, which is the most important issue to represent the difference in real delivery problems.These categories have representative model features that are sorted in the tables below according to the year as shown in Tables 5-7.The objectives of VRPs can also be diversified according to different stakeholder requirements.The traditional objective of the standard VRP is to minimize a cost function, which is considered to be the total distance traveled by all vehicles.However, recent studies have focused on various negative externalities of transportation, including carbon emissions and duration.For an objective discussion, we classified single and multiple objectives according to the diversity of objectives and then listed the objectives used in different studies.The papers with the same numbers as those in Tables 2-4 are listed in Tables 8-10.Additionally, we discussed the test instances used in different studies.minimize the total cost P-n-k" instances are from [111], "RY" instances are from the "RY ATT48" in [127], and "Tai" instances are from [128] with the number of nodes ranging from 75 to 150.
150 test instances adopted from [135] 320 the number of such duplications is not known a priori and the size of the problem increases As shown in the tables above, there are different main model objectives for different years.The results are summarized in Figure 1.One can see that the time window still occupies a large proportion of model objectives and is the mainstream of current research on the VRP and its variants.This trend is closely related to the concept of the "to C" distribution, where customers focus on service satisfaction.There have been various extensions of the VRP, including the VRPTW and time-dependent problems such as those discussed in [17,24,41,75,89].Additional research has focused on heterogeneous vehicle problems that are closely related to real-life vehicle applications.With the increasing focus on environmental protection, electric vehicle distribution has also gradually become a mainstream research topic.Relevant research can be found in [49,72,81,89,116].Single-objective models still occupy a certain research space, where the objective value setting is still largely based on cost metrics (e.g., cost, distance, and  2 emission).However, unlike cost metrics in past research, the costs in the current single-objective problem research tend to be compound costs representing actual delivery costs.One can see that the time window still occupies a large proportion of model objectives and is the mainstream of current research on the VRP and its variants.This trend is closely related to the concept of the "to C" distribution, where customers focus on service satisfaction.There have been various extensions of the VRP, including the VRPTW and time-dependent problems such as those discussed in [17,24,41,75,89].Additional research has focused on heterogeneous vehicle problems that are closely related to real-life vehicle applications.With the increasing focus on environmental protection, electric vehicle distribution has also gradually become a mainstream research topic.Relevant research can be found in [49,72,81,89,116].Single-objective models still occupy a certain research space, where the objective value setting is still largely based on cost metrics (e.g., cost, distance, and CO 2 emission).However, unlike cost metrics in past research, the costs in the current single-objective problem research tend to be compound costs representing actual delivery costs.

Solutions for VRPs
Because real-world problems involve complex constraints, advanced algorithms are required to solve VRPs in complicated and constantly changing environments.The number of customers and vehicle types is increasing and the use of optimization algorithms is a key component of effective customer service and efficient operations.A large variety of VRP solution strategies have been presented in the literature.These strategies range from exact methods to heuristics and meta-heuristics.Exact methods provide optimal solutions, whereas heuristics and meta-heuristics generally yield near-optimal solutions.Exact methods are typically only suitable for small-scale problems (up to 200 customers).Because the VRP and its variants are known to have NP-hard complexity, solving larger instances optimally is very time-consuming.However, there are no bounds on problem size when solving problems using heuristics and meta-heuristics that can efficiently handle large numbers of constraints and still output near-optimal solutions.Figure 2 presents various approaches to solving the VRPs and was adapted from content in [6,149].
Appl.Sci.2021, 11, x FOR PEER REVIEW 17 of 3 exact methods to heuristics and meta-heuristics.Exact methods provide optimal solu tions, whereas heuristics and meta-heuristics generally yield near-optimal solutions.Ex act methods are typically only suitable for small-scale problems (up to 200 customers) Because the VRP and its variants are known to have NP-hard complexity, solving large instances optimally is very time-consuming.However, there are no bounds on problem size when solving problems using heuristics and meta-heuristics that can efficiently han dle large numbers of constraints and still output near-optimal solutions.Figure 2 presents various approaches to solving the VRPs and was adapted from content in [6,149].Exact methods include a variety of approaches, mainly branch and X (X: cut, bound price, and so on) approaches, as well as dynamic programming and column generation methods.In recent years, significant advances in the exact solution of VRPs have been achieved.A major milestone was the branch-and-price algorithm proposed by Pecin et al [150].The branch-and-bound (BB) method was developed to explore solution spaces im plicitly.Because the performance of BB algorithms depends on the quality of bounds ob tained throughout a tree, BB algorithms can be combined with the generation of cutting planes, forming so-called branch-and-cut algorithms, or with column generation, result ing in BAP algorithms [151].Branch and X remain the dominant VRP approaches [150,152].While branch and X approaches treat VRPs as integer linear programming (ILP or mixed ILP (MILP), dynamic programming breaks complex problems into a number o simpler sub-problems.Constraint programming is a model that interrelates different var iables using constraints.When the search space is reduced, relatively simple problems can be solved by various search algorithms [149].
Approximate methods called heuristics are designed to solve specific problems.Heu ristics focus on systematically finding an acceptable solution within a limited number o iterations.A heuristic yields solutions faster than an exact method.A meta-heuristic may be referred to as an intelligent strategy combining subordinate heuristics for exploration and exploitation.
For solution discussion, we classified exact methods, heuristic algorithms, and meta Exact methods include a variety of approaches, mainly branch and X (X: cut, bound, price, and so on) approaches, as well as dynamic programming and column generation methods.In recent years, significant advances in the exact solution of VRPs have been achieved.A major milestone was the branch-and-price algorithm proposed by Pecin et al. [150].The branch-and-bound (BB) method was developed to explore solution spaces implicitly.Because the performance of BB algorithms depends on the quality of bounds obtained throughout a tree, BB algorithms can be combined with the generation of cutting planes, forming so-called branch-and-cut algorithms, or with column generation, resulting in BAP algorithms [151].Branch and X remain the dominant VRP approaches [150,152].While branch and X approaches treat VRPs as integer linear programming (ILP) or mixed ILP (MILP), dynamic programming breaks complex problems into a number of simpler sub-problems.Constraint programming is a model that interrelates different variables using constraints.When the search space is reduced, relatively simple problems can be solved by various search algorithms [149].
Approximate methods called heuristics are designed to solve specific problems.Heuristics focus on systematically finding an acceptable solution within a limited number of iterations.A heuristic yields solutions faster than an exact method.A meta-heuristic may be referred to as an intelligent strategy combining subordinate heuristics for exploration and exploitation.
For solution discussion, we classified exact methods, heuristic algorithms, and metaheuristic algorithms in papers with the same numbers as those in Tables 11-13.The results are presented in the tables below.Regarding datasets, different studies make various adjustments to data and many use generated datasets in addition to real data, which makes it difficult to compare algorithms using a unified standard.There is still scope for significant further work in the field.Therefore, researchers should be motivated to develop publicly available datasets, and effective and efficient methods for dealing with VRPs.The gaps in the available literature mentioned above may motivate further work in these directions for researchers in this field.
For solving algorithms, with the development of the processing speed and memory capacity of computers, using the exact way such as branch and X to solve VRPs is rapid growth.However, heuristic algorithms and meta-heuristic algorithms are still the mainstream solution methods, such as SA [14], GA [35,41,45], NSG [28,47], SSO [153], and so on.It is hoped that more exact algorithms can be applied to solve VRPs in the future, and the number of nodes in the dataset that can be solved can be increased as much as possible.
Our research protocol was well defined because it aims at an efficient and thorough review of multiple VRP variants.The main goal of this study was to identify the trends of VRP variants and the algorithms applied to solve them.Additionally, papers that are considered to represent pioneering efforts from the research community were presented.The papers with the most citations were considered to be the most significant and they were discussed in detail in this review.

Figure 1 .
Figure 1.Different model objectives in different years.

Figure 1 .
Figure 1.Different model objectives in different years.

Table 1 .
Sets and indices of VRP.
VNode set, where v 0 is the depot and {v 1 , v 2 , ..., v N } are customers i,j Subscripts of the customer nodes, i, j = 1, 2, ...N A A = v i , v j : v i , v j ∈ V is arcs set linking nodes i and j M mThe set of vehicles with m types
m Variable cost of vehicle type m Dm ij,mAmount carried using vehicle type m from i to j

Table 3 .
Decision variable of VRP.
X ij,mValue of one if vehicle type m travels from node i to j.Otherwise, value of zero

Table 5 .
Model categories of VRPs published in 2021.

Table 6 .
Model categories of VRPs published in 2020.

Table 7 .
Model categories of VRPs published in 2019.

Table 8 .
Model objectives of VRPs published in 2021.

Table 9 .
Model objectives of VRPs published in 2020.

Table 10 .
Model objectives of VRPs published in 2019.

Table 10 .
Cont.As shown in the tables above, there are different main model objectives for different years.The results are summarized in Figure1.

Table 11 .
Solutions to VRPs published in 2021.

Table 12 .
Solutions to VRPs published in 2020.

Table 12 .
Cont.Coded in C++ and run on a Linux cluster system with an AMD Opteron 4184 CPU (2.8 GHz and 2 GB RAM) running Ubuntu 12.04.For the general CPLEX solver, the latest version of 12.6 was used

Table 13 .
Solutions to VRPs published in 2019.Coded in Fortran 77 and run on a single thread of a 3.6 GHz Intel i7-4790 CPU with 32 GB RAM.Relies on CPLEX 12.5.1 for the resolution of linear programs and some integer sub-problems.Metaheuristic was coded in Java (JRE 1.8.0-151), and run on a single thread of a 3.4 GHz Intel i7-3770 CPU

Table A2 .
List of abbreviations for solution of VRP and its variants.