Minimum Makespan Multi-vehicle Dial-a-Ride

Dial a ride problems consist of a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs, where each object requires to be moved from its source to destination vertex. We consider the multi-vehicle Dial a ride problem, with each vehicle having capacity k and its own depot-vertex, where the objective is to minimize the maximum completion time (makespan) of the vehicles. We study the "preemptive" version of the problem, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log^3 n)-approximation algorithm for preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint. We also show that the approximation ratios improve by a log-factor when the underlying metric is induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles with possibly different speeds, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot

A survey of approximation algorithms for capacitated vehicle routing problems

Finding the shortest travelling tour of vehicles with capacity k from the depot to the customers is called the Capacity vehicle routing problem (CVRP). CVRP plays an essential position in logistics systems, and it is the most intensively studied problem in combinatorial optimization. In complexity, CVRP with k $\ge$ 3 is an NP-hard problem, and it is APX-hard as well. We already knew that it could not be approximated in metric space. Moreover, it is the first problem resisting Arora's famous approximation framework. So, whether there is, a polynomial-time (1+$\epsilon$)-approximation for the Euclidean CVRP for any $\epsilon>0$ is still an open problem. This paper will summarize the research progress from history to up-to-date developments. The survey will be updated periodically.Comment: First submissio

The commodity-split multi-compartment capacitated arc routing problem

The purpose of this paper is to develop a data-driven matheuristic for the Commodity-Split Multi-Compartment Capacitated Arc Routing Problem (CSMC-CARP). This problem arises in curbside waste collection, where there are different recyclable waste types called fractions. The CSMC-CARP is defined on an undirected graph with a limited heterogeneous fleet of multi-compartment vehicle types based at a depot, where each compartment's capacity can vary depending on the waste fraction assigned to it and on the compression factor of that fraction in that vehicle type. The aim is to determine a set of least-cost routes starting and ending at the depot, such that the demand of each edge for each waste fraction is collected exactly once by one vehicle, without violating the capacity of any compartment. The CSMC-CARP consists of three decision levels: selecting the number of vehicles of each type, assigning waste fractions to the compartments of each selected vehicle, and routing the vehicles. Our three-phase algorithm decomposes the problem into incomplete solution representations and heuristically solves one or more decision levels at a time. The first phase selects a subset of attractive compartment assignments from all assignments of all vehicle types. The second phase solves the CSMC-CARP with an unlimited fleet of the selected assignments. This is done by our C-split tour splitting algorithm, which can simultaneously split a giant tour of required edges into feasible routes while making decisions on the fractions that are collected by each route. The third phase selects the set of best routes servicing all fractions of all required edges without exceeding the number of vehicles available of each type. The algorithm is applied to real-life instances arising from recyclable waste collection operations in Denmark, with graph sizes up to 6,149 nodes and 3,797 required edges, the waste sorted in three to six fractions, and four to six vehicle types with one to four compartments. Computational results show that the generated solutions favor combining different fractions together in vehicles with higher numbers of compartments, and that the algorithm adapts well to the characteristics of the data, in terms of the graph, vehicle types, degree of sorting, and to skewness in demand among waste fractions.</p

The commodity-split multi-compartment capacitated arc routing problem

The purpose of this paper is to develop a data-driven matheuristic for the Commodity-Split Multi-Compartment Capacitated Arc Routing Problem (CSMC-CARP). This problem arises in curbside waste collection, where there are different recyclable waste types called fractions. The CSMC-CARP is defined on an undirected graph with a limited heterogeneous fleet of multi-compartment vehicle types based at a depot, where each compartment's capacity can vary depending on the waste fraction assigned to it and on the compression factor of that fraction in that vehicle type. The aim is to determine a set of least-cost routes starting and ending at the depot, such that the demand of each edge for each waste fraction is collected exactly once by one vehicle, without violating the capacity of any compartment. The CSMC-CARP consists of three decision levels: selecting the number of vehicles of each type, assigning waste fractions to the compartments of each selected vehicle, and routing the vehicles. Our three-phase algorithm decomposes the problem into incomplete solution representations and heuristically solves one or more decision levels at a time. The first phase selects a subset of attractive compartment assignments from all assignments of all vehicle types. The second phase solves the CSMC-CARP with an unlimited fleet of the selected assignments. This is done by our C-split tour splitting algorithm, which can simultaneously split a giant tour of required edges into feasible routes while making decisions on the fractions that are collected by each route. The third phase selects the set of best routes servicing all fractions of all required edges without exceeding the number of vehicles available of each type. The algorithm is applied to real-life instances arising from recyclable waste collection operations in Denmark, with graph sizes up to 6,149 nodes and 3,797 required edges, the waste sorted in three to six fractions, and four to six vehicle types with one to four compartments. Computational results show that the generated solutions favor combining different fractions together in vehicles with higher numbers of compartments, and that the algorithm adapts well to the characteristics of the data, in terms of the graph, vehicle types, degree of sorting, and to skewness in demand among waste fractions.</p

A hybrid algorithm combining path scanning and biased random sampling for the Arc Routing Problem

The Arc Routing Problem is a kind of NP-hard routing problems where the demand is located in some of the arcs connecting nodes and should be completely served fulfilling certain constraints. This paper presents a hybrid algorithm which combines a classical heuristic with biased random sampling, to solve the Capacitated Arc Routing Problem (CARP). This new algorithm is compared with the classical Path scanning heuristic, reaching results which outperform it. As discussed in the paper, the methodology presented is flexible, can be easily parallelised and it does not require any complex fine-tuning process. Some preliminary tests show the potential of the proposed approach as well as its limitationsPostprint (published version

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

Application of the Branch and Cut Method to the Vehicle Routing Problem

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. We investigate three classes of valid inequalities for the CVRP, multistars, pathbin inequalities and hypotours and give computational results we obtained with a Branch and Cut implementation

Efficient routing of snow removal vehicles

This research addresses the problem of finding a minimum cost set of routes for vehicles in a road network subject to some constraints. Extensions, such as multiple service requirements, and mixed networks have been considered. Variations of this problem exist in many practical applications such as snow removal, refuse collection, mail delivery, etc. An exact algorithm was developed using integer programming to solve small size problems. Since the problem is NP-hard, a heuristic algorithm needs to be developed. An algorithm was developed based on the Greedy Randomized Adaptive Search Procedure (GRASP) heuristic, in which each replication consists of applying a construction heuristic to find feasible and good quality solutions, followed by a local search heuristic. A simulated annealing heuristic was developed to improve the solutions obtained from the construction heuristic. The best overall solution was selected from the results of several replications. The heuristic was tested on four sets of problem instances (total of 115 instances) obtained from the literature. The simulated annealing heuristic was able to achieve average improvements of up to 26.36% over the construction results on these problem instances. The results obtained with the developed heuristic were compared to the results obtained with recent heuristics developed by other authors. The developed heuristic improved the best-known solution found by other authors on 18 of the 115 instances and matched the results on 89 of those instances. It worked specially better with larger problems. The average deviations to known lower bounds for all four datasets were found to range between 0.21 and 2.61%