Workload Equity in Vehicle Routing Problems: A Survey and Analysis

Over the past two decades, equity aspects have been considered in a growing number of models and methods for vehicle routing problems (VRPs). Equity concerns most often relate to fairly allocating workloads and to balancing the utilization of resources, and many practical applications have been reported in the literature. However, there has been only limited discussion about how workload equity should be modeled in VRPs, and various measures for optimizing such objectives have been proposed and implemented without a critical evaluation of their respective merits and consequences. This article addresses this gap with an analysis of classical and alternative equity functions for biobjective VRP models. In our survey, we review and categorize the existing literature on equitable VRPs. In the analysis, we identify a set of axiomatic properties that an ideal equity measure should satisfy, collect six common measures, and point out important connections between their properties and those of the resulting Pareto-optimal solutions. To gauge the extent of these implications, we also conduct a numerical study on small biobjective VRP instances solvable to optimality. Our study reveals two undesirable consequences when optimizing equity with nonmonotonic functions: Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent, i.e. composed of tours whose workloads are all equal to or longer than those of other Pareto-optimal solutions. We show that the extent of these phenomena should not be underestimated. The results of our biobjective analysis are valid also for weighted sum, constraint-based, or single-objective models. Based on this analysis, we conclude that monotonic equity functions are more appropriate for certain types of VRP models, and suggest promising avenues for further research.Comment: Accepted Manuscrip

The Vehicle Routing Problem with Service Level Constraints

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems

The Dynamic Multi-objective Multi-vehicle Covering Tour Problem

This work introduces a new routing problem called the Dynamic Multi-Objective Multi-vehicle Covering Tour Problem (DMOMCTP). The DMOMCTPs is a combinatorial optimization problem that represents the problem of routing multiple vehicles to survey an area in which unpredictable target nodes may appear during execution. The formulation includes multiple objectives that include minimizing the cost of the combined tour cost, minimizing the longest tour cost, minimizing the distance to nodes to be covered and maximizing the distance to hazardous nodes. This study adapts several existing algorithms to the problem with several operator and solution encoding variations. The efficacy of this set of solvers is measured against six problem instances created from existing Traveling Salesman Problem instances which represent several real countries. The results indicate that repair operators, variable length solution encodings and variable-length operators obtain a better approximation of the true Pareto front

An updated annotated bibliography on arc routing problems

The number of arc routing publications has increased significantly in the last decade. Such an increase justifies a second annotated bibliography, a sequel to Corberán and Prins (Networks 56 (2010), 50–69), discussing arc routing studies from 2010 onwards. These studies are grouped into three main sections: single vehicle problems, multiple vehicle problems and applications. Each main section catalogs problems according to their specifics. Section 2 is therefore composed of four subsections, namely: the Chinese Postman Problem, the Rural Postman Problem, the General Routing Problem (GRP) and Arc Routing Problems (ARPs) with profits. Section 3, devoted to the multiple vehicle case, begins with three subsections on the Capacitated Arc Routing Problem (CARP) and then delves into several variants of multiple ARPs, ending with GRPs and problems with profits. Section 4 is devoted to applications, including distribution and collection routes, outdoor activities, post-disaster operations, road cleaning and marking. As new applications emerge and existing applications continue to be used and adapted, the future of arc routing research looks promising.info:eu-repo/semantics/publishedVersio

Comparision of the walk techniques for fitness state space analysis in vehicle routing problem

The Vehicle Routing Problem (VRP) is a highly researched discrete optimization task. The first article dealing with this problem was published by Dantzig and Ramster in 1959 under the name Truck Dispatching Problem. Since then, several versions of VRP have been developed. The task is NP difficult, it can be solved only in the foreseeable future, relying on different heuristic algorithms. The geometrical property of the state space influences the efficiency of the optimization method. In this paper, we present an analysis of the following state space methods: adaptive, reverse adaptive and uphill-downhill walk. In our paper, the efficiency of four operators are analysed on a complex Vehicle Routing Problem. These operators are the 2-opt, Partially Matched Crossover, Cycle Crossover and Order Crossover. Based on the test results, the 2-opt and Partially Matched Crossover are superior to the other two methods

Adapting a Hyper-heuristic to Respond to Scalability Issues in Combinatorial Optimisation

The development of a heuristic to solve an optimisation problem in a new domain, or a specific variation of an existing problem domain, is often beyond the means of many smaller businesses. This is largely due to the task normally needing to be assigned to a human expert, and such experts tend to be scarce and expensive. One of the aims of hyper-heuristic research is to automate all or part of the heuristic development process and thereby bring the generation of new heuristics within the means of more organisations. A second aim of hyper-heuristic research is to ensure that the process by which a domain specific heuristic is developed is itself independent of the problem domain. This enables a hyper-heuristic to exist and operate above the combinatorial optimisation problem “domain barrier” and generalise across different problem domains. A common issue with heuristic development is that a heuristic is often designed or evolved using small size problem instances and then assumed to perform well on larger problem instances. The goal of this thesis is to extend current hyper-heuristic research towards answering the question: How can a hyper-heuristic efficiently and effectively adapt the selection, generation and manipulation of domain specific heuristics as you move from small size and/or narrow domain problems to larger size and/or wider domain problems? In other words, how can different hyperheuristics respond to scalability issues? Each hyper-heuristic has its own strengths and weaknesses. In the context of hyper-heuristic research, this thesis contributes towards understanding scalability issues by firstly developing a compact and effective heuristic that can be applied to other problem instances of differing sizes in a compatible problem domain. We construct a hyper-heuristic for the Capacitated Vehicle Routing Problem domain to establish whether a heuristic for a specific problem domain can be developed which is compact and easy to interpret. The results show that generation of a simple but effective heuristic is possible. Secondly we develop two different types of hyper-heuristic and compare their performance across different combinatorial optimisation problem domains. We construct and compare simplified versions of two existing hyper-heuristics (adaptive and grammar-based), and analyse how each handles the trade-off between computation speed and quality of the solution. The performance of the two hyper-heuristics are tested on seven different problem domains compatible with the HyFlex (Hyper-heuristic Flexible) framework. The results indicate that the adaptive hyper-heuristic is able to deliver solutions of a pre-defined quality in a shorter computational time than the grammar-based hyper-heuristic. Thirdly we investigate how the adaptive hyper-heuristic developed in the second stage of this thesis can respond to problem instances of the same size, but containing different features and complexity. We investigate how, with minimal knowledge about the problem domain and features of the instance being worked on, a hyper-heuristic can modify its processes to respond to problem instances containing different features and problem domains of different complexity. In this stage we allow the adaptive hyper-heuristic to select alternative vectors for the selection of problem domain operators, and acceptance criteria used to determine whether solutions should be retained or discarded. We identify a consistent difference between the best performing pairings of selection vector and acceptance criteria, and those pairings which perform poorly. This thesis shows that hyper-heuristics can respond to scalability issues, although not all do so with equal ease. The flexibility of an adaptive hyper-heuristic enables it to perform faster than the more rigid grammar-based hyper-heuristic, but at the expense of losing a reusable heuristic

A solution for a real-time stochastic capacitated vehicle routing problem with time windows

Real-time distribution planning presents major difficulties when applied to large problems. Commonly, this planning is associated to the capacitated vehicle routing problem with time windows (CVRPTW), deeply studied in the literature. In this paper we propose an optimization system developed to be integrated with an existing Enterprise Resource Planning (ERP) without causing major disruption to the current distribution process of a company. The proposed system includes: a route optimization module, a module implementing the communications within and to the outside of the system, a non-relational database to provide local storage of information relevant to the optimization procedure, and a cartographic subsystem. The proposed architecture is able to deal with dynamic problems included in the specification of the project, namely: arrival of new orders while already optimizing as well as locking and closing of routes by the system administrator. A back-office graphical interface was also implemented and some results are presented

Distance-constrained vehicle routing problem: exact and approximate solution (mathematical programming)

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The asymmetric distance-constrained vehicle routing problem (ADVRP) looks at finding vehicle tours to connect all customers with a depot, such that the total distance is minimised; each customer is visited once by one vehicle; every tour starts and ends at a depot; and the travelled distance by each vehicle is less than or equal to the given maximum value. We present three basic results in this thesis. In the first one, we present a general flow-based formulation to ADVRP. It is suitable for symmetric and asymmetric instances. It has been compared with the adapted Bus School Routing formulation and appears to solve the ADVRP faster. Comparisons are performed on random test instances with up to 200 customers. We reach a conclusion that our general formulation outperforms the adapted one. Moreover, it finds the optimal solution for small test instances quickly. For large instances, there is a high probability that an optimal solution can be found or at least improve upon the value of the best feasible solution found so far, compared to the other formulation which stops because of the time condition. This formulation is more general than Kara formulation since it does not require the distance matrix to satisfy the triangle inequality. The second result improves and modifies an old branch-and-bound method suggested by Laporte et al. in 1987. It is based on reformulating a distance-constrained vehicle routing problem into a travelling salesman problem and uses the assignment problem as a lower bounding procedure. In addition, its algorithm uses the best-first strategy and new branching rules. Since this method was fast but memory consuming, it would stop before optimality is proven. Therefore, we introduce randomness in choosing the node of the search tree in case we have more than one choice (usually we choose the smallest objective function). If an optimal solution is not found, then restart is required due to memory issues, so we restart our procedure. In that way, we get a multistart branch and bound method. Computational experiments show that we are able to exactly solve large test instances with up to 1000 customers. As far as we know, those instances are much larger than instances considered for other VRP models and exact solution approaches from recent literature. So, despite its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in literature. Moreover, this approach is general and may be used in solving other types of vehicle routing problems. In the third result, we use VNS as a heuristic to find the best feasible solution for groups of instances. We wanted to determine how far the difference is between the best feasible solution obtained by VNS and the value of optimal solution in order to use the output of VNS as an initial feasible solution (upper bound procedure) to improve our multistart method. Unfortunately, based on the search strategy (best first search), using a heuristic to find an initial feasible solution is not useful. The reason for this is because the branch and bound is able to find the first feasible solution quickly. In other words, in our method using a good initial feasible solution as an upper bound will not increase the speed of the search. However, this would be different for the depth first search. However, we found a big gap between VNS feasible solution and an optimal solution, so VNS can not be used alone unless for large test instances when other exact methods are not able to find any feasible solution because of memory or stopping conditions