Open vehicle routing problem with driver nodes and time deadlines

In this paper, we consider a variant of the open vehicle routing problem in which vehicles depart from the depot, visit a set of customers, and end their routes at special nodes called driver nodes. A driver node can be the home of the driver or a parking lot where the vehicle will stay overnight. The resulting problem is referred to as the open vehicle routing problem with driver nodes (OVRP-d). We consider three classes of OVRP-d: with no time constraints, with a maximum route duration, and with both a maximum route duration as well as time deadlines for visiting customers. For the solution of these problems, which are not addressed previously in the literature, we develop a new tabu search heuristic. Computational results on randomly generated instances indicate that the new heuristic exhibits a good performance both in terms of the solution quality and computation time.     

The open vehicle routing problem with time windows

In this paper, we consider the open vehicle routing problem with time windows (OVRPTW). The OVRPTW seeks to find a set of non-depot returning vehicle routes, for a fleet of capacitated vehicles, to satisfy customers’ requirements, within fixed time intervals that represent the earliest and latest times during the day that customers’ service can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns. The model is solved using a greedy look-ahead route construction heuristic algorithm, which utilizes time windows related information via composite customer selection and route-insertion criteria. These criteria exploit the interrelationships between customers, introduced by time windows, that dictate the sequence in which vehicles must visit customers. Computational results on a set of benchmark problems from the literature provide very good results and indicate the applicability of the methodology in real-life routing applications.     

Integer linear programming models for a cement delivery problem

We consider a cement delivery problem with an heterogeneous fleet of vehicles and several depots. The demands of the customers are typically larger than the capacity of the vehicles which means that most customers are visited several times. This is a split delivery vehicle routing problem with additional constraints. We first propose a two phase solution method that assigns deliveries to the vehicles, and then builds vehicle routes. Both subproblems are formulated as integer linear programming problems. We then show how to combine the two phases in a single integer linear program. Experiments on real life instances are performed to compare the performance of the two solution methods.     

Multi-ant colony system (MACS) for a vehicle routing problem with backhauls

The vehicle routing problem with backhaul (VRPB) is an extension of the capacitated vehicle routing problem (CVRP). In VRPB, there are linehaul as well as backhaul customers. The number of vehicles is considered to be fixed and deliveries for linehaul customers must be made before any pickups from backhaul customers. The objective is to design routes for the vehicles so that the total distance traveled is minimized. We use multi-ant colony system (MACS) to solve VRPB which is a combinatorial optimization problem. Ant colony system (ACS) is an algorithmic approach inspired by foraging behavior of real ants. Artificial ants are used to construct a solution by using pheromone information from previously generated solutions. The proposed MACS algorithm uses a new construction rule as well as two multi-route local search schemes. An extensive numerical experiment is performed on benchmark problems available in the literature.     

Vehicle routing with a sparse feasibility graph

In this paper we introduce the concept of a feasibility graph for vehicle routing problems, a graph where two customers are linked if and only if it is possible for them to be successive (adjacent) customers on some feasible vehicle route. We consider the problem of designing vehicle routes when the underlying feasibility graph is sparse, i.e. when any customer has only a few other customers to which they can be adjacent on a vehicle route. This problem arose during a consultancy study that involved the design of fixed vehicle routes delivering to contiguous (adjacent) postal districts. A heuristic algorithm for the problem is presented and computational results given for a number of test problems involving up to 856 customers.     

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.     

A multi-level composite heuristic for the multi-depot vehicle fleet mix problem

The problem of simultaneously allocating customers to depots, finding the delivery routes and determining the vehicle fleet composition is addressed. A multi-level composite heuristic is proposed and two reduction tests are designed to enhance its efficiency. The proposed heuristic is tested on benchmark problems involving up to 360 customers, 2 to 9 depots and 5 different vehicle capacities. When tested on the special case, the multi-depot vehicle routing, our heuristic yields solutions almost as good as those found by the best known heuristics but using only 5 to 10% of their computing time. Encouraging results were also obtained for the case where the vehicles have different capacities.     

A heuristic method for the open vehicle routing problem

The open vehicle routing problem (OVRP) differs from the classic vehicle routing problem (VRP) because the vehicles either are not required to return to the depot, or they have to return by revisiting the customers assigned to them in the reverse order. Therefore, the vehicle routes are not closed paths but open ones. A heuristic method for solving this new problem, based on a minimum spanning tree with penalties procedure, is presented. Computational results are provided.     

An exact algorithm for a single-vehicle routing problem with time windows and multiple routes

This paper describes an exact algorithm for solving a problem where the same vehicle performs several routes to serve a set of customers with time windows. The motivation comes from the home delivery of perishable goods, where vehicle routes are short and must be combined to form a working day. A method based on an elementary shortest path algorithm with resource constraints is proposed to solve this problem. The method is divided into two phases: in the first phase, all non-dominated feasible routes are generated; in the second phase, some routes are selected and sequenced to form the vehicle workday. Computational results are reported on Euclidean problems derived from benchmark instances of the classical vehicle routing problem with time windows.     

A survey of recent research on location-routing problems

The design of distribution systems raises hard combinatorial optimization problems. For instance, facility location problems must be solved at the strategic decision level to place factories and warehouses, while vehicle routes must be built at the tactical or operational levels to supply customers. In fact, location and routing decisions are interdependent and studies have shown that the overall system cost may be excessive if they are tackled separately. The location-routing problem (LRP) integrates the two kinds of decisions. Given a set of potential depots with opening costs, a fleet of identical vehicles and a set of customers with known demands, the classical LRP consists in opening a subset of depots, assigning customers to them and determining vehicle routes, to minimize a total cost including the cost of open depots, the fixed costs of vehicles used, and the total cost of the routes. Since the last comprehensive survey on the LRP, published by Nagy and Salhi (2007), the number of articles devoted to this problem has grown quickly, calling a review of new research works. This paper analyzes the recent literature (72 articles) on the standard LRP and new extensions such as several distribution echelons, multiple objectives or uncertain data. Results of state-of-the-art metaheuristics are also compared on standard sets of instances for the classical LRP, the two-echelon LRP and the truck and trailer problem.     

Using clustering analysis in a capacitated location-routing problem

The location routing problem (LRP) appears as a combination of two difficult problems: the facility location problem (FLP) and the vehicle routing problem (VRP). In this work, we consider a discrete LRP with two levels: a set of potential capacitated distribution centres (DC) and a set of ordered customers. In our problem we intend to determine the set of installed DCs as well as the distribution routes (starting and ending at the DC). The problem is also constrained with capacities on the vehicles. Moreover, there is a homogeneous fleet of vehicles, carrying a single product and each customer is visited just once. As an objective we intend to minimize the routing and location costs.     

A computational comparison of algorithms for the inventory routing problem

The inventory routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Each day a fleet of trucks is dispatched over a set of routes to resupply a subset of the customers. In this paper, we describe and compare algorithms for this problem defined over a short planning period, e.g. one week. These algorithms define the set of customers to be serviced each day and produce routes for a fleet of vehicles to service those customers. Two algorithms are compared in detail, one which first allocates deliveries to days and then solves a vehicle routing problem and a second which treats the multi-day problem as a modified vehicle routing problem. The comparison is based on a set of real data obtained from a propane distribution firm in Pennsylvania. The solutions obtained by both procedures compare quite favorably with those in use by the firm.Part of this work was performed while this author was visiting the University of Waterloo.     

The multi-trip vehicle routing problem

The basic vehicle routing problem is concerned with the design of a set of routes to serve a given number of customers, minimising the total distance travelled. In that problem, each vehicle is assumed to be used only once during a planning period, which is typically a day, and therefore is unrepresentative of many practical situations, where a vehicle makes several journeys during a day. The present authors have previously published an algorithm which outperformed an experienced load planner working on the complex, real-life problems of Burton's Biscuits, where vehicles make more than one trip each day. This present paper uses a simplified version of that general algorithm, in order to compare it with a recently published heuristic specially designed for the theoretical multi-trip vehicle routing problem.     

The min-max multi-depot vehicle routing problem: heuristics and computational results

In the multi-depot vehicle routing problem (MDVRP), there are several depots where vehicles can start and end their routes. The objective is to minimize the total distance travelled by all vehicles across all depots. The min-max multi-depot vehicle routing problem (Min-Max MDVRP) is a variant of the standard MDVRP. The primary objective is to minimize the length of the longest route. We develop a heuristic (denoted by MD) for the Min-Max MDVRP that has three stages: (1) simplify the multi-depot problem into a single depot problem and solve the simplified problem; (2) improve the maximal route; (3) improve all routes by exchanging customers between routes. MD is compared with two alternative heuristics that we also develop and an existing method from the literature on a set of 20 test instances. MD produces 15 best solutions and is the top performer. Additional computational experiments on instances with uniform and non-uniform distributions of customers and varying customer-to-vehicle ratios and with real-world data further demonstrate MD’s effectiveness in producing high-quality results.     

A tabu search algorithm for the periodic vehicle routing problem with multiple vehicle trips and accessibility restrictions

In this paper, we consider a periodic vehicle routing problem that includes, in addition to the classical constraints, the possibility of a vehicle doing more than one route per day, as long as the maximum daily operation time for the vehicle is not exceeded. In addition, some constraints relating to accessibility of the vehicles to the customers, in the sense that not every vehicle can visit every customer, must be observed. We refer to the problem we consider here as the site-dependent multi-trip periodic vehicle routing problem. An algorithm based on tabu search is presented for the problem and computational results presented on randomly generated test problems that are made publicly available. Our algorithm is also tested on a number of routing problems from the literature that constitute particular cases of the proposed problem. Specifically we consider the periodic vehicle routing problem; the site-dependent vehicle routing problem; the multi-trip vehicle routing problem; and the classical vehicle routing problem. Computational results for our tabu search algorithm on test problems taken from the literature for all of these problems are presented.     

A tabu search algorithm for the multi-trip vehicle routing and scheduling problem

This paper describes a novel tabu search heuristic for the multi-trip vehicle routing and scheduling problem (MTVRSP). The method was developed to tackle real distribution problems, taking into account most of the constraints that appear in practice. In the MTVRSP, besides the constraints that are common to the basic vehicle routing problem, the following ones are present: during each day a vehicle can make more than one trip; the customers impose delivery time windows; the vehicles have different capacities considered in terms of both volume and weight; the access to some customers is restricted to some vehicles; the drivers' schedules must respect the maximum legal driving time per day and the legal time breaks; the unloading times are considered.     

An exact algorithm for team orienteering problems

Optimising routing of vehicles constitutes a major logistic stake in many industrial contexts. We are interested here in the optimal resolution of special cases of vehicle routing problems, known as team orienteering problems. In these problems, vehicles are guided by a reward that can be collected from customers, while the length of routes is limited. The main difference with classical vehicle routing problems is that not all customers have to be visited. The solution method we propose here is based on a Branch & Price algorithm. It is, as far as we know, the first exact method proposed for such problems, except for a preliminary work from Gueguen (Methodes de résolution exacte pour problémes de tournées de véhicules. Thése de doctorat, école Centrale Paris 1999) and a work from Butt and Ryan (Comput Oper Res 26(4):427–441 1999). It permits to solve instances with up to 100 customers.      

A heuristic for vehicle fleet mix problem using tabu search and set partitioning

The vehicle fleet mix problem is a special case of the vehicle routing problem where customers are served by a heterogeneous fleet of vehicles with various capacities. An efficient heuristic for determining the composition of a vehicle fleet and travelling routes was developed using tabu search and by solving set partitioning problems. Two kinds of problems have appeared in the literature, concerning fixed cost and variable cost, and these were tested for evaluation. Initial solutions were found using the modified sweeping method. Whenever a new solution in an iteration of the tabu search was obtained, optimal vehicle allocation was performed for the set of routes, which are constructed from the current solution by making a giant tour. Experiments were performed for the benchmark problems that appeared in the literature and new best-known solutions were found.     

Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation

This paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer. The solution to this vehicle routing problem with stochastic demand (VRPSD) involves the optimization of complete routing schedules with minimum travel distance, driver remuneration, and number of vehicles, subject to a number of constraints such as time windows and vehicle capacity. To solve such a multiobjective and multi-modal combinatorial optimization problem, this paper presents a multiobjective evolutionary algorithm that incorporates two VRPSD-specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions. A new way of assessing the quality of solutions to the VRPSD on top of comparing their expected costs is also proposed. It is shown that the algorithm is capable of finding useful tradeoff solutions for the VRPSD and the solutions are robust to the stochastic nature of the problem. The developed algorithm is further validated on a few VRPSD instances adapted from Solomon’s vehicle routing problem with time windows (VRPTW) benchmark problems.     

A new tabu search heuristic for the open vehicle routing problem

In this paper, another version of the vehicle routing problem (VRP)—the open vehicle routing problem (OVRP) is studied, in which the vehicles are not required to return to the depot, but if they do, it must be by revisiting the customers assigned to them in the reverse order. By exploiting the special structure of this type of problem, we present a new tabu search heuristic for finding the routes that minimize two objectives while satisfying three constraints. The computational results are provided and compared with two other methods in the literature.