Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

Vehicle routing with dynamic travel times: A queueing approach

Transportation is an important component of supply chain competitiveness since it plays a major role in the inbound, inter-facility, and outbound logistics. In this context, assigning and scheduling vehicle routes is a crucial management problem. In this paper, a vehicle routing problem with dynamic travel times due to potential traffic congestion is considered. The approach developed introduces mainly the traffic congestion component based on queueing theory. This is an innovative modeling scheme to capture travel times. The queueing approach is compared with other approaches and its potential benefits are described and quantified. Moreover, the optimization of the starting times of a route at the distribution center is evaluated. Finally, the trade-off between solution quality and calculation time is discussed. Numerous test instances are used, both to illustrate the appropriateness of the approach as well as to show that time-independent solutions are often unrealistic within a congested traffic environment, which is usually the case on European road networks.     

Ambulance location and relocation problems with time-dependent travel times

EMERGENCY SERVICE PROVIDERS ARE FACING THE FOLLOWING PROBLEM: how and where to locate vehicles in order to cover potential future demand effectively. Ambulances are supposed to be located at designated locations such that in case of an emergency the patients can be reached in a time-efficient manner. A patient is said to be covered by a vehicle if (s)he can be reached by an ambulance within a predefined time limit. Due to variations in speed and the resulting travel times it is not sufficient to solve the static ambulance location problem once using fixed average travel times, as the coverage areas themselves change throughout the day. Hence we developed a multi-period version, taking into account time-varying coverage areas, where we allow vehicles to be repositioned in order to maintain a certain coverage standard throughout the planning horizon. We have formulated a mixed integer program for the problem at hand, which tries to optimize coverage at various points in time simultaneously. The problem is solved metaheuristically using variable neighborhood search. We show that it is essential to consider time-dependent variations in travel times and coverage respectively. When ignoring them the resulting objective will be overestimated by more than 24%. By taking into account these variations explicitly the solution on average can be improved by more than 10%.     

An iterated local search algorithm for the time-dependent vehicle routing problem with time windows

We generalize the standard vehicle routing problem with time windows by allowing both traveling times and traveling costs to be time-dependent functions. In our algorithm, we use a local search to determine routes of the vehicles. When we evaluate a neighborhood solution, we must compute an optimal time schedule for each route. We show that this subproblem can be efficiently solved by dynamic programming, which is incorporated in the local search algorithm. The neighborhood of our local search consists of slight modifications of the standard neighborhoods called 2- opt^*, cross exchange and Or-opt. We propose an algorithm that evaluates solutions in these neighborhoods more efficiently than the ones computing the dynamic programming from scratch by utilizing the information from the past dynamic programming recursion used to evaluate the current solution. We further propose a filtering method that restricts the search space in the neighborhoods to avoid many solutions having no prospect of improvement. We then develop an iterated local search algorithm that incorporates all the above ingredients. Finally we report computational results of our iterated local search algorithm compared against existing methods, and confirm the effectiveness of the restriction of the neighborhoods and the benefits of the proposed generalization.     

Gossip algorithms for heterogeneous multi-vehicle routing problems

In this paper we address a class of heterogeneous multi-vehicle task assignment and routing problems. We propose two distributed algorithms based on gossip communication: the first algorithm is based on a local exact optimization and the second is based on a local approximate greedy heuristic. We consider the case where a set of heterogeneous tasks arbitrarily distributed in a plane has to be serviced by a set of mobile robots, each with a given movement speed and task execution speed. Our goal is to minimize the maximum execution time of robots.     

Vehicle routing problems on a line-shaped network with release time constraints

This work considers the vehicle routing problem on a line with the constraint that each customer is visited after its release time. It is already known that the single-vehicle case is polynomially solvable. We present polynomial time algorithms for two variants of the multi-vehicle case.     

A branch-and-cut algorithm for vehicle routing problems

We present a branch-and-cut algorithm for the identical customer Vehicle Routing Problem. Transforming the problem into an equivalent Path-Partitioning Problem allows us to exploit its polyhedral structure and to generate strong cuts corresponding to facet-inducing inequalities. By using cuts defined by multistars, partial multistars and generalized subtour elimination constraints, we are able to consistently solve 60-city problems to proven optimality and are currently attempting to solve problems involving a hundred cities. We also present details of the computer implementation and our computational results.     

Shortest paths in piecewise continuous time-dependent networks

We consider a shortest path problem, where the travel times on the arcs may vary with time and waiting at any node is allowed. Simple adaptations of the Dijkstra algorithm may fail to solve the problem, when discontinuities exist. We propose a new Dijkstra-like algorithm that overcomes these difficulties.     

A unified solution framework for multi-attribute vehicle routing problems

Vehicle routing attributes are extra characteristics and decisions that complement the academic problem formulations and aim to properly account for real-life application needs. Hundreds of methods have been introduced in recent years for specific attributes, but the development of a single, general-purpose algorithm, which is both efficient and applicable to a wide family of variants remains a considerable challenge. Yet, such a development is critical for understanding the proper impact of attributes on resolution approaches, and to answer the needs of actual applications. This paper contributes towards addressing these challenges with a component-based design for heuristics, targeting multi-attribute vehicle routing problems, and an efficient general-purpose solver. The proposed Unified Hybrid Genetic Search metaheuristic relies on problem-independent unified local search, genetic operators, and advanced diversity management methods. Problem specifics are confined to a limited part of the method and are addressed by means of assignment, sequencing, and route-evaluation components, which are automatically selected and adapted and provide the fundamental operators to manage attribute specificities. Extensive computational experiments on 29 prominent vehicle routing variants, 42 benchmark instance sets and overall 1099 instances, demonstrate the remarkable performance of the method which matches or outperforms the current state-of-the-art problem-tailored algorithms. Thus, generality does not necessarily go against efficiency for these problem classes.     

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.     

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported.     

An improved ant colony optimization for vehicle routing problem

The vehicle routing problem (VRP), a well-known combinatorial optimization problem, holds a central place in logistics management. This paper proposes an improved ant colony optimization (IACO), which possesses a new strategy to update the increased pheromone, called ant-weight strategy, and a mutation operation, to solve VRP. The computational results for fourteen benchmark problems are reported and compared to those of other metaheuristic approaches.     

The time-dependent capacitated profitable tour problem with time windows and precedence constraints

We introduce the time-dependent capacitated profitable tour problem with time windows and precedence constraints. This problem concerns determining a tour and its departure time at the depot that maximizes the collected profit minus the total travel cost (measured by total travel time). To deal with road congestion, travel times are considered to be time-dependent. We develop a tailored labeling algorithm to find the optimal tour. Furthermore, we introduce dominance criteria to discard unpromising labels. Our computational results demonstrate that the algorithm is capable of solving instances with up to 150 locations (75 pickup and delivery requests) to optimality. Additionally, we present a restricted dynamic programing heuristic to improve the computation time. This heuristic does not guarantee optimality, but is able to find the optimal solution for 32 instances out of the 34 instances.     

A post-improvement procedure for the mixed load school bus routing problem

This paper aims to develop a mixed load algorithm for the school bus routing problem (SBRP) and measure its effects on the number of required vehicles. SBRP seeks to find optimal routes for a fleet of vehicles, where each vehicle transports students from their homes and to their schools while satisfying various constraints. When mixed load is allowed, students of different schools can get on the same bus at the same time. Although many of real world SBRP allow mixed load, only a few studies have considered these cases. In this paper, we present a new mixed load improvement algorithm and compare it with the only existing algorithm from the literature. Benchmark problems are proposed to compare the performances of algorithms and to stimulate other researchers’ further study. The proposed algorithm outperforms the existing algorithm on the benchmark problem instances. It has also been successfully applied to some of real-world SBRP and could reduce the required number of vehicles compared with the current practice.     

Metaheuristics for vehicle routing problems with three-dimensional loading constraints

This paper addresses an important combination of three-dimensional loading and vehicle routing, known as the Three-Dimensional Loading Capacitated Vehicle Routing Problem. The problem calls for the combined optimization of the loading of freight into vehicles and the routing of vehicles along a road network, with the aim of serving customers with minimum traveling cost. Despite its clear practical relevance in freight distribution, the literature on this problem is very limited. This is because of its high combinatorial complexity.     

Arc routing problems with time-dependent service costs

This paper studies an arc routing problem with capacity constraints and time-dependent service costs. This problem is motivated by winter gritting applications where the “timing” of each intervention is crucial. The exact problem-solving approach reported here first transforms the arc routing problem into an equivalent node routing problem. Then, a column generation scheme is used to solve the latter. The master problem is a classical set covering problem, while the subproblems are time-dependent shortest path problems with resource constraints. These subproblems are solved using an extension of a previously developed algorithm. Computational results are reported on problems derived from a set of classical instances of the vehicle routing problem with time windows.     

The school bus routing problem: A review

This paper aims to provide a comprehensive review of the school bus routing problem (SBRP). SBRP seeks to plan an efficient schedule for a fleet of school buses where each bus picks up students from various bus stops and delivers them to their designated schools while satisfying various constraints such as the maximum capacity of a bus, the maximum riding time of a student in a bus, and the time window of a school. This class of problem consists of different sub-problems involving data preparation, bus stop selection, bus route generation, school bell time adjustment, and bus scheduling. In this paper, the various assumptions, constraints, and solution methods used in the literature on SBRP are summarized. A list of issues requiring further research is also presented.     

A unified exact method for solving different classes of vehicle routing problems

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.      

Heuristics for multi-attribute vehicle routing problems: A survey and synthesis

The attributes of vehicle routing problems are additional characteristics or constraints that aim to better take into account the specificities of real applications. The variants thus formed are supported by a well-developed literature, including a large variety of heuristics. This article first reviews the main classes of attributes, providing a survey of heuristics and meta-heuristics for Multi-Attribute Vehicle Routing Problems (MAVRP). It then takes a closer look at the concepts of 64 remarkable meta-heuristics, selected objectively for their outstanding performance on 15 classic MAVRP with different attributes. This cross-analysis leads to the identification of “winning strategies” in designing effective heuristics for MAVRP. This is an important step in the development of general and efficient solution methods for dealing with the large range of vehicle routing variants.