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In this paper, we have conducted a literature review on the recent developments and publications involving the vehicle routing problem and its variants, namely vehicle routing problem with time windows (VRPTW) and the capacitated vehicle routing problem (CVRP) and also their variants. The VRP is classified as an NP-hard problem. Hence, the use of exact optimization methods may be difficult to solve these problems in acceptable CPU times, when the problem involves real-world data sets that are very large. The vehicle routing problem comes under combinatorial problem. Hence, to get solutions in determining routes which are realistic and very close to the optimal solution, we use heuristics and meta-heuristics. In this paper we discuss the various exact methods and the heuristics and meta-heuristics used to solve the VRP and its variants.

The Vehicle Routing Problem (VRP) is used to design an optimal route for a fleet of vehicles to service a set of customers, given a set of constraints. The VRP is used in supply chain management in the physical delivery of goods and services. There are several variants to the VRP. These are formulated based on the nature of the transported goods, the quality of service required and the characteristics of the customers and the vehicles. The VRP is of the NP-hard type.

The vehicle routing problem (VRP) has been very extensively studied in the optimization literature. It started with the seminal papers of [1,2]. Now, VRP offers a wealth of heuristic and metaheuristic approaches, which are surveyed in the papers of [3-5]. The VRP is so widely studied because of its wide applicability and its importance in determining efficient strategies for reducing operational costs in distribution networks. Today, exact VRP methods have a size limit of 50 - 100 orders depending on the VRP variant and the time–response requirements. Consequently, current research concentrates on approximate algorithms that are capable of finding high quality solutions in limited time, in order to be applicable to reallife problem instances that are characterized by large vehicle fleets and affect significantly logistics and distribution strategies.

The VRP was first stated by [

The VRP can be defined as the problem of designing least cost delivery routes from a depot to a set of geographically dispersed locations (customers) subject to a set of constraints.

Dynamic Vehicle Routing Problems (DVRP), sometimes referred to as On-line Vehicle Routing Problems, have recently arisen due to the advances in information and communication technologies that enable information to be obtained and processed in real-time. In DVRP, some of the orders are known in advance before the start of the working day, but as the day progresses, new orders arrive and the system has to incorporate them into an evolving schedule. The existence of a communication system between the dispatcher (where the tours are calculated, e.g. headquarter of the company) and the drivers is assumed. The dispatcher can periodically communicate to the drivers about the new visits assigned to them. In this way, during the day, each driver always has knowledge about the next customers assigned to him/her

The classical VRP is defined as follows: Let G = (V, A) be a directed graph where V = {0,…,n} is the vertex set and A = {(i, j) : i, j Î V, i ≠ j} is the arc set. Vertex 0 represents the depot whereas the remaining vertices correspond to customers. A fleet of m identical vehicles of capacity Q is based at the depot. The fleet size is given a priori or is a decision variable. Each customer i has a non-negative demand q_{i.}

The heuristic methods for the VRP can be divided into construction heuristics, improvement heuristics and metaheuristics

There are different classes or variations of VRP like the capacitated VRP (CVRP), VRP with Time Windows (VRPTW). In the CVRP, a fleet of identical vehicles located at a central depot has to be optimally routed to supply a set of customers with known demands.

The capacitated VRP (CVRP) is described as the graph theoretic problem: Let G = (V, E) be a complete and undirected graph where V = {0,…,n} is the vertex set and E is the edge set. Vertex set V_{c }= {1, … , n} corresponds to n customers, whereas vertex 0 corresponds to the depot.

The objective of the VRPTW is to serve a number of customers within predefined time windows at minimum cost (in terms of distance travelled), without violating the capacity and total trip time constraints for each vehicle. Combinatorial optimisation problems of this kind are non-polynomial-hard (NP-hard) and are hence best solved by using heuristics. The most important metaheuristics used to solve the VRPTW are Tabu search (TS), genetic algorithm (GA), evolutionary algorithms (EA) and ant colony optimisation algorithm (ACO).

The constraints of the Vehicle Routing Problem with Time Windows (VRPTW) consist of a set of identical vehicles, a central depot node, a set of customer nodes and a network connecting the depot and customers. There are N + 1 customers and K vehicles. The depot node is denoted as customer 0. Each arc in the network represents a connection between two nodes and also indicates the direction it travels. Each route starts from the depot. The number of routes in the network is equal to the number of vehicles used. One vehicle is dedicated to one route. A cost c_{ij} and a travel time t_{ij} are associated with each arc of the network.

In Solomon’s 56 VRPTW 100-customer instances [_{ij, }the travel time t_{ij} and the Euclidean distance between the customer nodes equal each other.

Each customer in the node can be visited only once by one of the vehicles. Every vehicle has the same capacity q_{k} and every customer has a varying demand m_{i}. q_{k} must be greater than or equal to the sum of all demands on the route travelled by the vehicle k., which means that no vehicle can be overloaded. The time window constraint is denoted by a predefined time interval, given an earliest arrival time and latest arrival time. The vehicles must arrive at the customers not later than the latest arrival time. If vehicles arrive earlier than the earliest arrival time, waiting occurs. Each customer also imposes a service time to the route, taking into consideration the loading/unloading time of goods. In Solomon’s instances, the service time is assumed to be unique regardless of the load quantity needed to be handled. Vehicles are also supposed to complete their individual routes within a total route time, which is essentially the time window of the depot.

There are three types of principal decision variables in VRPTW. The principal decision variable x_{ijk} (i, j Î {0, 1, 2,…, N}; k Î {1, 2,…, K}; i ≠ j) is 1 if vehicle k travels from customer i to customer j, and 0 otherwise. The decision variable T_{i} denotes the time when a vehicle arrives at the customer, and w_{i} denotes the waiting time at node i. The objective is to design a network that satisfies all constraints, at the same time minimizing the total travel cost. The model is mathematically formulated below:

T_{i} arrival time at node i w_{i} wait time at node i x_{ijk} Î {0,1}, 0 if there is no arc from node i to node j, and 1 otherwise,

Parameters:

K total number of vehicles N total number of customers c_{ij} cost incurred on arc from node i to j t_{ij} travel time between node i and j m_{i} demand at node i q_{k} capacity of vehicle k e_{i} earliest arrival time at node i l_{i} latest arrival time at node i f_{i} service time at node i r_{k} maximum route time allowed for vehicle k Minimize

subject to:

The objective function minimizes the total cost of travel of all the vehicles in completing their tours. Constraint set 1 guarantees that the number of tours is K by selecting at most K outgoing arcs from the depot (I = 0).

The constraint set 2 ensures that for each vehicle, there is exactly one outgoing arc from the depot is selected. Similarly, the constraint set 3 ensures that for each vehicle, there is exactly one arc entering into the node with respect to depot (i = 0). These two constraint sets (Constraint set 2 and constraint set 3) jointly ensure that a complete tour for each vehicle is ensured.

The constraint set 4 makes sure that from each node i only one arc for each vehicle emanates from it. The constraint set 5 ensures that for each node j, only one arc for each vehicle enters into it. These two constrains (Constraint set 4 and Constraint set 5) make sure that each vehicle visits each node only once.

The constraint set 6 sees that for each vehicle, the total demand (load) allocated to it is less than or equal to its capacity.

The constraint set 7 ensures that the total time of travel of the route of each vehicle is less than or equal to the maximum route time allocation to that vehicle.

The constraint set 8 sets the arrival time, waiting time and service time of each vehicle at the depot to zero. The constraint set 9 guarantees that the arrival time of each vehicle at the node j is less than the specified arrival time at that node. The constraint set 10 guarantees that the sum of the arrival time and the waiting time of each vehicle at each node i is more than equal to the earliest arrival time at that node and less than or equal to the latest arrival time at that node i, i = 1, 2, 3, …, N. Constraint sets (8) - (10) define the time windows. These formulation completely specify the feasible solutions for the VRPTW.

A constraint is called hard if it must be satisfied, while it is called soft if it can be violated. The violation of soft constraints is usually penalized and added to the objective function. The VRP with hard (resp., soft) time window constraints is abbreviated as VRPHTW (resp., VRPSTW).

An iterative route construction and improvement algorithm for the vehicle routing problem with soft time windows was proposed [

There are several heuristics proposed to solve the CVRP and its variants in the literature. Authors [

The VRPTW belongs to the class of the NP-hard combinatorial optimization problems [

Two groups of meta-heuristics have been used for solving the VRPTW Homberger and Gehring (2005) [

(1) Meta-heuristics controlling local search processes, such as tabu search [12,13], simulated annealing [

Author [

Both exact algorithms and heuristics have been used to solve the different classes of VRPs. We discuss some of these below.

Exact algorithms to solve VRP especially the capacitated VRP (CVRP) include the branch-and-bound, the branchand-cut and the branch-and-price algorithms, which are briefly described below.

A branch-cut-and-price exact algorithm was suggested by [

Authors [

Column generation or Dantzig-Wolfe decomposition provides a flexible framework that can accommodate complex constraints and time-dependent costs.

An exact solution approach was developed by [

The authors [

The Set Partitioning (SP) formulation of the CVRP was originally proposed by [

Let R denote a set of routes in which r denotes a specific route. Let a_{ir} be a binary coefficient equal to 1 if and only if vertex i Î V\{0} belongs to route r, let be the optimal cost of the route r, and let y_{k} be a binary variable equal to 1 if and only if route r is used in the optimal solution. The problem is then as given below.

(SP) minimize

subject to

A full column generation algorithm was developed by [

According [

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We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications.

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The cluster-first, route-second heuristic [

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An efficient route minimization heuristic for the vehicle routing problem with time windows was suggested by [

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Many of the most successful meta-heuristics for the large VRPTW instances are based on some form of parallel computation. In [

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A new and effective meta-heuristic algorithm, active guided evolution strategies, for the VRPTW is presented by [

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A multi-objective evolutionary algorithm (EA) for solving the VRPTW was suggested by [

Evolutionary algorithms (EAs) are optimizers based on Darwin’s theory of evolution, where only the fittest individuals survive and produce offspring to populate the next generation.

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Hybrid methods use a combination of exact, heuristic procedure or meta-heuristics to solve the VRP.

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Vehicle routing problem forms an integral part of supply chain management, which plays a significant role for productivity improvement in organizations through efficient and effective delivery of goods/ services to customers. In this paper, an attempt has been made to survey the recent developments in the vehicle routing problem (VRP) and its variants. The literature is classified into exact methods, heuristics approaches, meta-heuristics, and hybrid methods. At the beginning, a complete working mathematical model for the vehicle routing problem with time windows (VRPTW) is given. Under exact methods, a descriptive set partitioning formulation for the vehicle routing problem with time windows has been presented. Further, the contributions of different researchers under this category have been discussed.

Since, the VRPTW is a combinatorial problem, development of efficient heuristic is inevitable to find near/ global optimal solution. So, in this paper, in the next three sections, the contributions of researchers in three categories, viz. heuristics, meta-heuristics and hybrid methods, respectively, have been presented.

Under meta-heuristic, the contributions of the researchers on simulated annealing algorithms, tabu search, genetic algorithm, ant-colony optimization and GRASP applied to vehicle routing problems are presented.

Under hybrid methods, a limited number of researches which combine the meta-heuristics with exact methods are presented.

From the literature, it is clear that very few researches have been carried under hybrid methods applied to vehicle routing problem. Hence, future researchers may focus on developing efficient hybrid approaches by combining two or more of the exact methods, heuristics and metaheuristics.