Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

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

In the UAV-based urban delivery setting, the battery capacity serves as one of the key factors affecting the delivery range of drones. For a typical multi-rotor unmanned aerial vehicle (UAV), the maximum effective working range of a lithium battery for a single flight leg is approximately 20 km. In practice, the lithium battery is replaced each time the UAV completes a delivery, even though the remaining power of the replaced lithium battery is still capable of supporting the following flight legs. Therefore, the one-order-one-swapping strategy significantly brings down the battery utilisation as well as the efficiency of the delivery system. In the present paper, the UAV-based delivery problem with battery swapping decision is considered, where multiple en route pickup and deliveries are allowed. To obtain optimal decisions in terms of flight route, time and location of each battery swapping, a branch-and-price-and-cut algorithm is introduced. First we formulate this problem as a mixed integer linear programming (MILP) model, and then decompose it into a master problem and a pricing sub-problem, where the master problem is considered as a set covering model, and the sub-problem as an elementary shortest path problem with resource constraints (ESPPRC). To speed up the process of column generation, efficient heuristics are also integrated to provide basic feasible solutions. In the pricing sub-problem, a combination of heuristic algorithm and dynamic programming is implemented to improve the computational efficiency. In the numerical experiments, the proposed algorithm is tested on various scales of UAV-based urban delivery instances, the performance of which is analysed and compared with the off-the-shelf commercial solver.

Addressed in this study is a vehicle routing problem with two‐dimensional loading constraints and time windows (2L‐CVRPTW), aiming to minimize the transportation cost while satisfying the two‐dimensional loading and routing constraints with time windows. To solve this problem, for the first time a mixed‐integer linear programming model is formulated with considering practical last‐in‐first‐out loading constraints, and a branch‐and‐price‐based (BP‐based) heuristic is proposed based on a set partitioning formulation. In the heuristic, a modified labeling algorithm is proposed for the complex pricing problem, which is a relaxation of the elementary shortest path problem with resource constraints and two‐dimensional loading constraints. Therein, an effective Tabu‐maximum open space packing heuristic is proposed to verify the feasibility of the two‐dimensional packing problem of each route generated by the labeling algorithm. In addition, effective accelerating and branching strategies are introduced to improve the solving efficiency of the heuristic. To evaluate the effectiveness and the advantages of the proposed heuristic, extensive computational experiments are performed based on the generated instances. The computational results show that the proposed BP‐based heuristic can effectively solve the 2L‐CVRPTW, in which the optimal solutions can be achieved much faster than CPLEX in small‐scale problems. Relationships between the transportation cost and the characteristics of the instances are analyzed. The stability of the algorithm and the effectiveness of the accelerating strategies are verified and discussed.

This paper addresses the split-delivery vehicle routing problem with time windows (SDVRPTW) that consists of determining least-cost vehicle routes to service a set of customer demands while respecting vehicle capacity and customer time windows. The demand of each customer can be fulfilled by several vehicles. For solving this problem, we propose a new exact branch-and-price-and-cut method, where the column generation subproblem is a resource-constrained elementary shortest-path problem combined with the linear relaxation of a bounded knapsack problem. Each generated column is associated with a feasible route and a compatible delivery pattern. As opposed to existing branch-and-price methods for the SDVRPTW or its variant without time windows, integrality requirements in the integer master problem are not imposed on the variables generated dynamically, but rather on additional variables. An ad hoc label-setting algorithm is developed for solving the subproblem. Computational results show the effectiveness of the proposed method.

Solving Multi-objective Vehicle Routing Problem with Time Windows by FAGA

This paper considers a vehicle routing problem with pickup and delivery, time windows and location congestion. Locations provide a number of cumulative resources that are utilized by vehicles either during service (e.g., forklifts) or for the entirety of their visit (e.g., parking bays). Locations can become congested if insufficient resources are available, upon which vehicles must wait until a resource becomes available before proceeding. The problem is challenging from a computational standpoint since it incorporates the vehicle routing problem and the resource-constrained project scheduling problem. The main contribution of this paper is a branch-and-price-and-check model that uses a branch-and-price algorithm that solves the underlying vehicle routing problem, and a constraint programming subproblem that checks the feasibility of the location resource constraints, and then adds combinatorial nogood cuts to the master problem if the resource constraints are violated. Experimental results show the benefits of the branch-and-price-and-check approach.

Logic-based Benders decomposition for the heterogeneous fixed fleet vehicle routing problem with time windows

This paper discusses a vehicle routing problem with multiple trips, time windows, and simultaneous delivery-pickup (VRPMTTWSDP). This problem is a variant of the basic vehicle routing problem (VRP) including the following characteristics: multiple trips, time windows, and simultaneous delivery-pickup. In this paper, a solution approach based on tabu search (TS) is proposed. In the proposed TS, the sequential insertion (SI) algorithm is used to construct an initial solution. A neighbor structure is generated by applying an operator order consisting of eleven operators of relocation, exchange, and crossover operators. A tabu solution code (TSC) method is applied as a tabu restriction mechanism. Computational experiments are carried out to examine the performance of the proposed TS using hypothetical instances. The performance of the proposed TS is compared to the local search (LS) and the genetic algorithm (GA). The comparison shows that the proposed TS is better in terms of the objective function value.

Vehicle routing problem with fuzzy time windows

We present an optimisation-based solution approach for a real ship planning problem,which is a combination of a variant of the multi-vehicle pickup and delivery problemwith time windows (m-PDPTW), and a multi-inventory model. This problem involves thedesign of a set of minimum cost routes for a fleet of heterogeneous ships servicing a set ofproduction and consumption harbours with a single product (ammonia). The production andinventory information at each harbour, together with the ship capacities and the location ofthe harbours, determine the number of possible arrivals at each harbour during the planningperiod, the time windows for start of service and the load quantity intervals at each arrival.We call this problem the inventory pickup and delivery problem with time windows -IPDPTW. In the mathematical programming model, we duplicate some of the variables anduse a Dantzig - Wolfe decomposition approach. Then the IPDPTW decomposes into a sub-problemfor each harbour and each ship. By synchronising the solutions from both types ofsubproblems, we get extra constraints in the master problem as compared to the masterproblem for the m-PDPTW discussed in the literature. The LP-relaxation of the masterproblem is solved by column generation, where the columns represent ship routes or harbourvisit sequences. Finally, this iterative solution process is embedded in a branch-and-boundsearch to make the solution integer optimal. Our computational results indicate that theproposed method works for the real planning problem.

The ongoing rise in e-commerce comes along with an increasing number of first-time delivery failures due to the absence of the customer at the delivery location. Failed deliveries result in rework which in turn has a large impact on the carriers’ delivery cost. In the classical vehicle routing problem (VRP) with time windows, each customer request has only one location and one time window describing where and when shipments need to be delivered. In contrast, we introduce and analyze the vehicle routing problem with delivery options (VRPDO), in which some requests can be shipped to alternative locations with possibly different time windows. Furthermore, customers may prefer some delivery options. The carrier must then select, for each request, one delivery option such that the carriers’ overall cost is minimized and a given service level regarding customer preferences is achieved. Moreover, when delivery options share a common location, e.g., a locker, capacities must be respected when assigning shipments. To solve the VRPDO exactly, we present a new branch-price-and-cut algorithm. The associated pricing subproblem is a shortest-path problem with resource constraints that we solve with a bidirectional labeling algorithm on an auxiliary network. We focus on the comparison of two alternative modeling approaches for the auxiliary network and present optimal solutions for instances with up to 100 delivery options. Moreover, we provide 17 new optimal solutions for the benchmark set for the VRP with roaming delivery locations.

In response to the rapid increase in e-commerce, distribution services, and smart technology, businesses need to solve the vehicle routing problem (VRP) and its variants to improve service quality and cut operating costs. In this paper, we introduce the VRP with time windows and time costs (VRPTWTC), which is a new but practical VRP variant that adds time windows of customers and time costs of service constraints into the optimization of the traditional VRP problem. Furthermore, we propose a masking algorithm and combine it into our proposed end-to-end and data-driven deep reinforcement learning framework to solve the VRPTWTC. Experimental results show a shorter tour length and reasonable processing time than benchmarks both on a simulated dataset and a real enterprise-level dataset. Besides, violation of time window and total time cost, which is one of the most important criterions in our problem formulation, never occurred in our experiment under the real road connection conditions when using a real map API. To the best of our knowledge, our framework is the first to effectively and efficiently solve a VRP variant problem with practical time windows and time costs constraints with a real map API to calculate routes of VRP considering real road connection conditions on an enterprise-level dataset.

A Vehicle Routing Problem with Flexible Time Windows

With the change of energy structure and the development of energy internet technology, the penetration rate of distributed generation is gradually increasing, and the location of distributed generation has an important impact on the distribution network line losses. In this paper, a mixed integer quadratic programming model with separability is considered to optimize the location of distributed generation and to reduce line losses in distribution networks, which measures line losses in distribution networks in terms of the quadratic term of current. In order to solve the problem of large-scale distributed power facility location problems with separability, a Benders decomposition algorithm incorporating matheuristics is proposed, which first decomposes the original problem into a mixed-integer linear programming master problem and a convex quadratic subproblem with separability, and then separates the generalized Benders cuts form solving the subproblem at the root node relaxed solution of the master problem. A simple in-out Stabilization technique is applied in order to efficiently raise the lower bound of the master problem at the root node. In order to avoid the use of branch-and-cut algorithms to solve the master problem to obtain optimal feasible solutions, two matheuristic repair strategies are proposed for the root node relaxation solution to obtain high-quality feasible solutions. Experimental results on 90 large-scale power facility location problems show that matheuristic repair strategies for the root node relaxation solution can find the global optimal solution.