An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

An adaptive insertion algorithm for the single-vehicle dial-a-ride problem with narrow time windows

The dial-a-ride problem (DARP) is a widely studied theoretical challenge related to dispatching vehicles in demand-responsive transport services, in which customers contact a vehicle operator requesting to be carried from specified origins to specified destinations. An important subproblem arising in dynamic dial-a-ride services can be identified as the single-vehicle DARP, in which the goal is to determine the optimal route for a single vehicle with respect to a generalized objective function. The main result of this work is an adaptive insertion algorithm capable of producing optimal solutions for a time constrained version of this problem, which was first studied by Psaraftis in the early 1980s. The complexity of the algorithm is analyzed and evaluated by means of computational experiments, implying that a significant advantage of the proposed method can be identified as the possibility of controlling computational work smoothly, making the algorithm applicable to any problem size.     

Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil

In this paper, we consider a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries that occurs in a major Brazilian retail group. A single depot attends 519 stores of the group distributed in 11 Brazilian states. To find good solutions to this problem, we propose heuristics as initial solutions and a scatter search (SS) approach. Next, the produced solutions are compared with the routes actually covered by the company. Our results show that the total distribution cost can be reduced significantly when such methods are used. Experimental testing with benchmark instances is used to assess the merit of our proposed procedure.     

Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model

In this paper, we address a variant of the vehicle routing problem called the vehicle routing problem with time windows and multiple routes. It considers that a given vehicle can be assigned to more than one route per planning period. We propose a new exact algorithm for this problem. Our algorithm is iterative and it relies on a pseudo-polynomial network flow model whose nodes represent time instants, and whose arcs represent feasible vehicle routes. This algorithm was tested on a set of benchmark instances from the literature. The computational results show that our method is able to solve more instances than the only other exact method described so far in the literature, and it clearly outperforms this method in terms of computing time.     

A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows

This paper presents a novel three-phase heuristic/algorithmic approach for the multi-depot routing problem with time windows and heterogeneous vehicles. It has been derived from embedding a heuristic-based clustering algorithm within a VRPTW optimization framework. To this purpose, a rigorous MILP mathematical model for the VRPTW problem is first introduced. Likewise other optimization approaches, the new formulation can efficiently solve case studies involving at most 25 nodes to optimality. To overcome this limitation, a preprocessing stage clustering nodes together is initially performed to yield a more compact cluster-based MILP problem formulation. In this way, a hierarchical hybrid procedure involving one heuristic and two algorithmic phases was developed. Phase I aims to identifying a set of cost-effective feasible clusters while Phase II assigns clusters to vehicles and sequences them on each tour by using the cluster-based MILP formulation. Ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small MILP model is finally performed at Phase III. Numerous benchmark problems featuring different sizes, clustered/random customer locations and time window distributions have been solved at acceptable CPU times.     

Simulated annealing metaheuristics for the vehicle routing problem with time windows

This paper develops simulated annealing metaheuristics for the vehicle routing and scheduling problem with time window constraints. Two different neighborhood structures, the λ-interchange mechanism of Osman and thek-node interchange process of Christofides and Beasley, are implemented. The enhancement of the annealing process with a short-term memory function via a tabu list is examined as a basis for improving the metaheuristic approach. Computational results on test problems from the literature as well as large-scale real-world problem are reported. The metaheuristics achieve solutions that compare favorably with previously reported results.     

The vehicle routing problem with coupled time windows

In this article we introduce the vehicle routing problem with coupled time windows (VRPCTW), which is an extension of the vehicle routing problem with time windows (VRPTW), where additional coupling constraints on the time windows are imposed. VRPCTW is applied to model a real-world planning problem concerning the integrated optimization of school starting times and public bus services. A mixed-integer programming formulation for the VRPCTW within this context is given. It is solved using a new meta-heuristic that combines classical construction aspects with mixed-integer preprocessing techniques, and improving hit-and-run, a randomized search strategy from global optimization. Solutions for several randomly generated and real-world instances are presented.     

A powerful route minimization heuristic for the vehicle routing problem with time windows

We suggest an efficient route minimization heuristic for the vehicle routing problem with time windows. The heuristic is based on the ejection pool, powerful insertion and guided local search strategies. Experimental results on the Gehring and Homberger’s benchmarks demonstrate that our algorithm outperforms previous approaches and found 18 new best-known solutions.     

Scatter search for the fleet size and mix vehicle routing problem with time windows

This work proposes a scatter search (SS) approach to solve the fleet size and mix vehicle routing problem with time windows (FSMVRPTW). In the FSMVRPTW the customers need to be serviced in their time windows at minimal costs by a heterogeneous fleet. Computational results on 168 benchmark problems are reported. Computational testing revealed that our algorithm presented better results compared to other methods published in the literature.     

An iterative three-component heuristic for the team orienteering problem with time windows

This paper studies the team orienteering problem with time windows, the aim of which is to maximize the total profit collected by visiting a set of customers with a limited number of vehicles. Each customer has a profit, a service time and a time window. A service provided to any customer must begin in his or her time window. We propose an iterative framework incorporating three components to solve this problem. The first two components are a local search procedure and a simulated annealing procedure. They explore the solution space and discover a set of routes. The third component recombines the routes to identify high quality solutions. Our computational results indicate that this heuristic outperforms the existing approaches in the literature in average performance by at least 0.41%. In addition, 35 new best solutions are found.     

An iterated local search algorithm for the single-vehicle cyclic inventory routing problem

The Single-Vehicle Cyclic Inventory Routing Problem (SV-CIRP) belongs to the class of Inventory Routing Problems (IRP) in which the supplier optimises both the distribution costs and the inventory costs at the customers. The goal of the SV-CIRP is to minimise both kinds of costs and to maximise the collected rewards, by selecting a subset of customers from a given set and determining the quantity to be delivered to each customer and the vehicle routes, while avoiding stockouts. A cyclic distribution plan should be developed for a single vehicle.     

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.     

Heuristic algorithms for a vehicle routing problem with simultaneous delivery and pickup and time windows in home health care

This paper addresses a vehicle scheduling problem encountered in home health care logistics. It concerns the delivery of drugs and medical devices from the home care company’s pharmacy to patients’ homes, delivery of special drugs from a hospital to patients, pickup of bio samples and unused drugs and medical devices from patients. The problem can be considered as a special vehicle routing problem with simultaneous delivery and pickup and time windows, with four types of demands: delivery from depot to patient, delivery from a hospital to patient, pickup from a patient to depot and pickup from a patient to a medical lab. Each patient is visited by one vehicle and each vehicle visits each node at most once. Patients are associated with time windows and vehicles with capacity. Two mixed-integer programming models are proposed. We then propose a Genetic Algorithm (GA) and a Tabu Search (TS) method. The GA is based on a permutation chromosome, a split procedure and local search. The TS is based on route assignment attributes of patients, an augmented cost function, route re-optimization, and attribute-based aspiration levels. These approaches are tested on test instances derived from existing VRPTW benchmarks.     

A branch-price-and-cut algorithm for the vehicle routing problem with time windows and multiple deliverymen

We address a variant of the vehicle routing problem with time windows that includes the decision of how many deliverymen should be assigned to each vehicle. In this variant, the service time at each customer depends on the size of the respective demand and on the number of deliverymen assigned to visit this customer. In addition, the objective function consists of minimizing a weighted sum of the total number of routes, number of deliverymen and traveled distance. These characteristics make this variant very challenging for exact methods. To date, only heuristic approaches have been proposed for this problem, and even the most efficient optimization solvers cannot find optimal solutions in a reasonable amount of time for instances of moderate size when using the available mathematical formulations. We propose a branch-price-and-cut method based on a new set partitioning formulation of the problem. To accelerate the convergence of the method, we rely on an interior-point column and cut generation process, a strong branching strategy and a mixed-integer programming-based primal heuristic. Additionally, a hierarchical branching strategy is used to take into account the different components of the objective function. The computational results indicate the benefits of using the proposed exact solution approach. We closed several instances of the problem and obtained upper bounds that were previously unknown in the literature.     

Integrated machine scheduling and vehicle routing with time windows

This paper integrates production and outbound distribution scheduling in order to minimize total tardiness. The overall problem consists of two subproblems. The first addresses scheduling a set of jobs on parallel machines with machine-dependent ready times. The second focusses on the delivery of completed jobs with a fleet of vehicles which may differ in their loading capacities and ready times. Job-dependent processing times, delivery time windows, service times, and destinations are taken into account. A genetic algorithm approach is introduced to solve the integrated problem as a whole. Two main questions are examined. Are the results of integrating machine scheduling and vehicle routing significantly better than those of classic decomposition approaches which break down the overall problem, solve the two subproblems successively, and merge the subsolutions to form a solution to the overall problem? And if so, is it possible to capitalize on these potentials despite the complexity of the integrated problem? Both questions are tackled by means of a numerical study. The genetic algorithm outperforms the classic decomposition approaches in case of small-size instances and is able to generate relatively good solutions for instances with up to 50 jobs, 5 machines, and 10 vehicles.     

An exact algorithm for the fixed-charge multiple knapsack problem

We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528–541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.     

A simulated annealing heuristic for the team orienteering problem with time windows

This paper presents a simulated annealing based heuristic approach for the team orienteering problem with time windows (TOPTW). Given a set of known locations, each with a score, a service time, and a time window, the TOPTW finds a set of vehicle tours that maximizes the total collected scores. Each tour is limited in length and a visit to a location must start within the location’s service time window. The proposed heuristic is applied to benchmark instances. Computational results indicate that the proposed heuristic is competitive with other solution approaches in the literature.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

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 metaheuristic based on a pool of routes for the vehicle routing problem with multiple trips and time windows

This paper studies the vehicle routing problem with multiple trips and time windows, in which vehicles are allowed to perform multiple trips during a scheduling period and each customer must be served within a given time interval. The problem is of particular importance for planning fleets of hired vehicles in common practices, such as e-grocery distributions, but this problem has received little attention in the literature. As a result of the multi-layered structure characteristic of the problem solution, we propose a pool-based metaheuristic in which various routes are first constructed to fill a pool, following which some of the routes are selected and combined to form vehicle working schedules. Finally, we conduct a series of experiments over a set of benchmark instances to evaluate and demonstrate the effectiveness of the proposed metaheuristic.