Vehicle routing problem with a heterogeneous fleet and time windows
Introduction
This paper is motivated by the last mile urban transportation and unmanned airborne vehicle (UAV) path planning applications. The last mile problem is valued in regions where land-use patterns have moved tremendous jobs and people to low-density suburbs. UAVs are widely utilized in military and civilian applications, such as surveillance and firefighting. The routing problems in these real-world applications can be classified into different problem variants of the vehicle routing problem (VRP), which has wide applicability in the fields of transportation, distribution and logistics (Toth & Vigo, 2001). VRP calls for the determination of the optimal set of routes to be taken up by a fleet of vehicles serving customers. VRP can be categorized into problem variants, such as VRP with a limited number of vehicles, VRP with time windows, and VRP with multiple depots. Many algorithms have been proposed for each problem variant in the literature. However, in system implementations, a real problem could be a generalization of various problem variants. Thus, it is difficult to evaluate the performance of algorithms for specific problem variants on real-world problems.
In this paper, a problem variant, the vehicle routing problem with a heterogeneous fleet and time windows (VRPHETW), is defined to generalize the vehicle routing problem with time windows and a limited number of vehicles (m-VRPTW), fleet size and mix vehicle routing problem with time windows (FSMVRPTW) and vehicle routing problem with a heterogeneous fleet (VRPHE). A tabu-search algorithm, which extends an existing tabu search specially designed for the m-VRPTW (Lau, Sim, & Teo, 2003), is proposed to solve the VRPHETW.
It is also observed that researchers frequently encounter problems when conducting experiments and reporting computational results. Firstly, the size of the parameter space expands in an exponential manner for an algorithm with a large number of parameters, which impedes the search for a good set of parameters. Secondly, fair comparisons are not easy to make (Silberholz & Golden, 2010). In particular, it is difficult to have access to the results and the executable routines. Thus, researchers are not able to run the algorithms and find out the details of the solving process. Thirdly, using only the best solutions, as is often done in the literature, may create a false picture on the real performance of an algorithm (Bräysy & Gendreau, 2005).
In order to overcome these problems, our proposed method is designed to contain as few parameters as possible. Meanwhile, the test cases used and executable routines developed can be provided to facilitate comparisons. The experiments are conducted on a large set of test cases to evaluate the algorithm’s performance. In the literature, the VRPHETW is less studied than other problem variants. Also, most of the papers focus on the real applications and so the test cases and results are not comparable. As such, we have evaluated our proposed algorithm on the FSMVRPTW instances, which have heterogeneous fleet, time windows and unlimited number of vehicles, as well as instances from several other problem variants of the VRP with a heterogeneous fleet. The aim of illustrating the ability of the proposed algorithm to solve numerous problem variants is to assess its potential in solving real-world problems as a real-world problem could be a generalization of various problem variants in practice.
Our contribution is in proposing an effective method to solve the VRPHETW which is able to provide reasonably good results for numerous problem variants. This is an advancement over existing work where most of the relevant algorithms would tend to solve a specific VRP variant effectively but may not perform as well for other problem variants.
This paper is organized as follows. Section 2 gives a detailed literature survey on the VRP with a heterogeneous fleet. In Section 3, the VRPHETW is described and formulated as a mathematical programming model. Subsequently in Section 4, a tabu-search algorithm is proposed to solve the VRPHETW. In Section 5, available sets of standard test cases are introduced and specially generated test cases are also proposed. Then in Section 6, experiments are conducted and the results are compared with the best-known published solutions for each problem variant. Finally, this paper concludes in Section 7 with a summary and possibilities for future research work.
Section snippets
Literature review
In the literature, several problem variants of the VRP consider a heterogeneous fleet, where vehicles have various capacities, fixed costs, variable costs, number of vehicles for each vehicle type, or latest returning times for vehicles to return to the depot. The fleet size and mix vehicle routing problem (FSMVRP) is one of the earlier problem variants with a heterogeneous fleet. In the FSMVRP, vehicles have differing vehicle capacities for different vehicle types, and the number of vehicles
Notation and problem formulation
The VRPHETW can be defined on a graph G(V, A) where V = {0, 1, …, n} is the node set, is the arc set. The depot is represented by node 0, and customers are represented by N = {1, …, n}. The arc set A comprises of connections between nodes, and there is a travel time or distance tij associated with each arc (i, j).
Each customer i(i ∊ N) has a fixed demand di and a time window [ei, li] that represents the earliest and latest times for service to customer i to start. The vehicle must stay
The tabu-search algorithm
A two-phase tabu-search algorithm is proposed to solve the VRPHETW. In Phase 1, the tabu search procedure in Lau et al. (2003) is extended to handle the scenario with a heterogeneous fleet. In Phase 2, a post-processing procedure is developed to improve the solution.
Test case suite
To evaluate the proposed algorithm, the test case suite presented here is composed of six sets of test cases in the literature and one set of test cases generated for the VRPHETW.
Proposed by Solomon (1987), the VRPTW test cases constitute the first set, which has 56 benchmark problems of six classes (C1, C2, R1, R2, RC1, RC2): classes C1 and C2 have clustered customers; classes R1 and R2 have customer locations randomly generated uniformly over a square; classes RC1 and RC2 have clustered
Computational results
Before showing the computational results, a brief discussion on the reporting of the results is provided. There are usually two types of problem backgrounds in the application of heuristics: strategic planning and operational planning. For strategic planning, computing time is not a constraint. For example, in a facility location problem, the management can have several months or even years to make a decision. Several heuristics could be executed and compared, and each heuristic could be
Conclusions
In this paper, the VRPHETW is defined to consider the problem of vehicle routing with a heterogeneous fleet, a limited number of vehicles and time windows. A two-phase solution procedure is developed based on tabu search, which is derived from an existing tabu search algorithm for the VRP with a limited number of vehicles. Due to the limited research works and results for the VRPHETW, we have evaluated our proposed algorithm on the FSMVRPTW instances, as well as other instances of the problem
Acknowledgment
This work was supported in part by the Temasek Defence Systems Institute under Grant R266-000-054-232, R266-000-054-422, and R266-000-054-592.
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