Fleet Of Capacitated Vehicles Research Articles

Deep reinforcement learning for stochastic last-mile delivery with crowdshipping

We study a setting in which a company not only has a fleet of capacitated vehicles and drivers available to make deliveries but may also use the services of occasional drivers (ODs) willing to make deliveries using their own vehicles in return for a small fee. Under such a business model, a.k.a crowdshipping, the company seeks to make all the deliveries at the minimum total cost, i.e., the cost associated with their vehicles plus the compensation paid to the ODs.We consider a stochastic and dynamic last-mile delivery environment in which customer delivery orders, as well as ODs available for deliveries, arrive randomly throughout the day, within fixed time windows.We present a novel deep reinforcement learning (DRL) approach to the problem that can deal with large problem instances. We formulate the action selection problem as a mixed-integer optimization program.The DRL approach is compared against other optimization under uncertainty approaches, namely, sample-average approximation (SAA) and distributionally robust optimization (DRO). The results show the effectiveness of the DRL approach by examining out-of-sample performance.

A Branch-and-Cut Algorithm for the Inventory Routing Problem with Product Substitution

The inventory routing problem arises in vendor managed systems, in which a supplier is responsible for replenishing inventories of products at a set of retailers and manages the logistics over a finite planning horizon with a fleet of capacitated vehicles. We consider this problem in the context of the distribution of two different quality products with one-way product substitution, where the high-quality product may be used to meet the demand for the low-quality product. The routes of vehicles and the quantities of products sent to each retailer in each period are determined in such a way that no stockouts occur and the total cost associated with inventory holding, substitution and transportation is minimized. In this study, we derive a mixed integer linear programming formulation for the problem, strengthen this formulation with valid inequalities and develop a branch-and-cut algorithm as an exact solution method. We conduct experiments using benchmark and randomly generated instances to analyze the effectiveness of our solution method and investigate the relationship between product substitution decisions and system costs under different demand, supply, vehicle capacity and substitution cost settings.

The complexity of the unit stop number problem and its implications to other related problems

The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of autonomous electric vehicles. In such a system, clients request for a ride from an origin station to a destination station, and a fleet of capacitated vehicles must satisfy all requests. The goal is to minimize the number of pick-up/drop-off operations. In this paper we focus on a constrained unitary variant, called Unit Stop Number Problem, where each client requests for a single seat in the vehicles. This variant was recently conjectured to be NP-Hard. In this regard, we show how this problem relates to other problems known in the literature in order to derive some polynomial-time solvable variants. Moreover, we provide a positive answer to the conjecture by showing that the problem is NP-Hard for any fixed capacity greater than or equal to 2, even for the case where the graph of requests is restricted to the class of planar bipartite graphs. Our proof of NP-Hardness also improves the complexity results known in the literature for the related problems identified.

Compact Formulations for Split Delivery Routing Problems

Split delivery routing problems are concerned with serving the demand of a set of customers with a fleet of capacitated vehicles at minimum cost, where a customer can be served by more than one vehicle if beneficial. They generalize traditional variants of routing problems and have applications in commercial and humanitarian logistics. Previously, formulations involving only commonly used arc-based variables have provided only relaxations for split delivery variants, as the possibility of visiting customers more than once introduces modeling challenges. The only known compact formulations are based on variables indexed by vehicle or by visit number and perform poorly when using general-purpose integer programming software. We present compact formulations that avoid the use of these types of variables and that can model split delivery routing problems with and without time windows. Computational experiments demonstrate their superior performance over existing compact formulations. We also develop a branch-and-cut algorithm that balances the efficiency derived from a relaxed formulation with the strength derived from one of the proposed formulations and demonstrate its efficacy on a large set of benchmark instances. The algorithm solves 95 instances to proven optimality for the first time and improves the best known lower and/or upper bound for many other instances.

Pickup and delivery problems with autonomous vehicles on rings

In this paper we introduce a new class of Pickup and Delivery problems on circles (or rings). These problems arise in the field of public transportation systems where autonomous (i.e. driverless) vehicles travel on circular networks. We consider a set of stations arranged in a circle and a set of transportation requests. Each request asks for the transportation of a certain quantity from a pickup station to a delivery station. A fleet of capacitated vehicles is available at the depot. In the first part of the paper we propose a classification scheme for these problems. In the second part, we address the variants in which the vehicles are allowed to move in a single direction of the circle (either clockwise or counterclockwise) and the objective is to minimize the number of tours on the ring while serving all the requests. We provide a complexity analysis for this class of problems. We develop polynomial time algorithms for the variants that are polynomially solvable and proofs of NP-hardness for the variants that are NP-hard. In addition, for the latter, we provide mathematical formulations and perform computational tests that show the effectiveness of these formulations. Finally, we compare optimal solutions with those obtained using a straightforward greedy algorithm.

A capacitated multi pickup online food delivery problem with time windows: a branch-and-cut algorithm.

Online food delivery companies, nowadays, allow a customer to place an order for a combination of dishes from one or more restaurants. To meet a customer’s demand, the company then arranges for the pickups from different restaurants, perhaps located distantly, before delivering them to the customer. The company caters to a set of such customers, through a fleet of capacitated vehicles. In this paper, we propose an augmented two-index formulation to find least cost vehicle routes to satisfy the set of customer requests under the constraints of time windows of the pickup and delivery locations along with the capacity constraints of the fleet of vehicles. We use branch-and-cut algorithm to solve the benchmark problem instances. The proposed formulation and the problem specific valid inequalities lead to more efficient solution performance as against the existing formulations. The computational experiments indicate that the proposed approach can solve a majority of benchmark problem instances with 25, 35 and 50 nodes, many of which were reported unsolved using exact solution approaches in the extant literature.

A Large Neighborhood Search for the Pickup and Delivery Problem with Time Windows, Split Loads and Transshipments

In the pickup and delivery problem (PDP) a fleet of capacitated vehicles must be routed in order to satisfy a set of customer requests. In this article, we address a variant of the one-to-one PDP where each customer location can be visited several times and load may be transshipped from one vehicle to another at specific locations. The goal is to minimize the sum of travel costs and transshipment costs. The corresponding problem is called the PDP with time windows, split loads and transshipments. We present an arc-based mixed-integer formulation for the problem and propose a large neighbourhood search metaheuristic for solving it. Extensive computational experiments using both benchmark instances from the literature and instances based on real-world data illustrate the performance of the solution method and give insights regarding the benefits of combining split loads and transshipments.

The split-delivery mixed capacitated arc-routing problem: Applications and a forest-based tabu search approach

Motivated by practical applications such as the water sprinkling services of urban administration bureaus in big cities, we study a split-delivery mixed capacitated arc-routing problem. In this problem, a fleet of capacitated vehicles need to serve a set of arcs/edges in a split delivery manner by traversing a mixed graph. We provide a mathematical formulation and analyze some properties of the problem that could be utilized to accelerate the search process of a meta-heuristic algorithm. A forest-based tabu search algorithm is proposed to efficiently solve the problem. Numerical experiments show that our algorithm can quickly produce high quality solutions compared with the state-of-the-art approaches. In addition, we analyze how different patterns of a mixed graph affect the quality of solutions.

Exact solution methods for the multi-period vehicle routing problem with due dates

In this paper we study the multi-period vehicle routing problem with due dates. A supplier has to determine a distribution plan to visit a set of customers over a given planning horizon. Each customer is associated with a release date and a due date, that is, the date at which the goods required by the customer become available at the supplier’s depot, and the date by which the customer has to be visited, respectively. A fleet of capacitated vehicles is available at the depot to perform the distribution and the objective is to minimize the distribution costs and the costs related to delayed deliveries. New families of valid inequalities are presented that allow us to improve a branch-and-bound algorithm proposed in the literature. The new branch-and-bound algorithm reduces to 5.1% the optimality gap which is 12.1% for the known branch-and-bound on instances with one vehicle. A variable MIP neighborhood descent (VMND) algorithm is also presented, which speeds up the search for high quality solutions through a local search heuristic embedded in the branch-and-bound algorithm. Computational tests are performed to assess the quality of the VMND algorithm against the new branch-and-bound algorithm. The computational results show that the VMND algorithm improves 35 out of 80 solutions on instances with one vehicle, reducing the average optimality gap from 5.1% to 3.6%. It further matches the performance of the new branch-and-bound algorithm on another 35 instances. On instances with two and three vehicles the average optimality gap obtained by the VMND algorithm is 5.5% and 5.6%, respectively.

An Adaptive Iterated Local Search for the Mixed Capacitated General Routing Problem

We study the mixed capacitated general routing problem (MCGRP) in which a fleet of capacitated vehicles has to serve a set of requests by traversing a mixed weighted graph. The requests may be located on nodes, edges, and arcs. The problem has theoretical interest because it is a generalization of the capacitated vehicle routing problem (CVRP), the capacitated arc routing problem (CARP), and the general routing problem. It is also of great practical interest since it is often a more accurate model for real-world cases than its widely studied specializations, particularly for so-called street routing applications. Examples are urban waste collection, snow removal, and newspaper delivery. We propose a new iterated local search metaheuristic for the problem that also includes vital mechanisms from adaptive large neighborhood search combined with further intensification through local search. The method utilizes selected, tailored, and novel local search and large neighborhood search operators, as well as a new local search strategy. Computational experiments show that the proposed metaheuristic is highly effective on five published benchmarks for the MCGRP. The metaheuristic yields excellent results also on seven standard CARP data sets, and good results on four well-known CVRP benchmarks, including improvement of the best known upper bound for one instance.

The Vehicle Routing Problem with Occasional Drivers

We consider a setting in which a company not only has a fleet of capacitated vehicles and drivers available to make deliveries, but may also use the services of occasional drivers who are willing to make a single delivery using their own vehicle in return for a small compensation if the delivery location is not too far from their own destination. The company seeks to make all the deliveries at minimum total cost, i.e., the cost associated with its own vehicles and drivers plus the compensation paid to the occasional drivers. The option to use occasional drivers to make deliveries gives rise to a new and interesting variant of the classical capacitated vehicle routing problem. We design and implement a multi-start heuristic which produces solutions with small errors when compared with optimal solutions obtained by solving an integer programming formulation with a commercial solver. A comprehensive computational study provides valuable insight into the potential of using occasional drivers to reduce delivery costs, focusing primarily on the number and flexibility of occasional drivers and the compensation scheme employed.

Multi-period Vehicle Routing Problem with Due dates

In this paper we study the Multi-period Vehicle Routing Problem with Due dates (MVRPD), where customers have to be served between a release and a due date. Customers with due dates exceeding the planning period may be postponed at a cost. A fleet of capacitated vehicles is available to perform the distribution in each day of the planning period. The objective of the problem is to find vehicle routes for each day such that the overall cost of the distribution, including transportation costs, inventory costs and penalty costs for postponed service, is minimized. We present alternative formulations for the MVRPD and enhance the formulations with valid inequalities. The formulations are solved with a branch-and-cut algorithm and computationally compared. Furthermore, we present a computational analysis aimed at highlighting managerial insights. We study the potential benefit that can be achieved by incorporating flexibility in the due dates and the number of vehicles. Finally, we highlight the effect of reducing vehicle capacity.

Optimal Shipment Policies for Distribution Systems With a Limited Fleet of Capacitated Vehicles

This paper deals with a distribution system with one origin and one destination in which the shipments are realized with a limited fleet of capacitated vehicles. An optimization problem is defined to find the optimal shipment policies in order to satisfy the external demand, by minimizing holding costs of the customer and transportation costs. Note to Practitioners - A distribution system with one origin and one destination characterized by a Vendor-Managed Inventory policy is considered. In such a system the supplier plans the deliveries to the customer in order to minimize the customer's inventory and the transportation costs. Besides the definition of some solution methods based on mathematical programming techniques, an algorithmic approach for solving the considered optimization problem is also derived in this paper. Even though this approach is suboptimal, it provides very effective solutions and its major strength stands in the possibility of easily implementing it without the use of mathematical programming solvers.

The bike sharing rebalancing problem: Mathematical formulations and benchmark instances

Bike sharing systems offer a mobility service whereby public bicycles, located at different stations across an urban area, are available for shared use. These systems contribute towards obtaining a more sustainable mobility and decreasing traffic and pollution caused by car transportation. Since the first bike sharing system was installed in Amsterdam in 1965, the number of such applications has increased remarkably so that hundreds of systems are now operating all over the world.In a bike sharing system, users can take a bicycle from a station, use it to perform a journey and then leave it at a station, not necessarily the same one of departure. This behavior typically leads to a situation in which some stations become full and others are empty. Hence, a balanced system requires the redistribution of bicycles among stations.In this paper, we address the Bike sharing Rebalancing Problem (BRP), in which a fleet of capacitated vehicles is employed in order to re-distribute the bikes with the objective of minimizing total cost. This can be viewed as a special one-commodity pickup-and-delivery capacitated vehicle routing problem. We present four mixed integer linear programming formulations of this problem. It is worth noting that the proposed formulations include an exponential number of constraints, hence, tailor-made branch-and-cut algorithms are developed in order to solve them.The mathematical formulations of the BRP were first computationally tested using data obtained for the city of Reggio Emilia, Italy. Our computational study was then extended to include bike sharing systems from other parts of the world. The information derived from the study was used to build a set of benchmark instances for the BRP which we made publicly available on the web. Extensive experimentation of the branch-and-cut algorithms presented in this paper was carried out and an interesting computational comparison of the proposed mathematical formulations is reported. Finally, several insights on the computational difficulty of the problem are highlighted.

The split delivery capacitated team orienteering problem

In this article, we study the capacitated team orienteering problem where split deliveries are allowed. A set of potential customers is given, each associated with a demand and a profit. The set of customers to be served by a fleet of capacitated vehicles has to be identified in such a way that the profit collected is maximized, while satisfying constraints on the maximum time duration of each route and the vehicle capacity constraints. When split deliveries are allowed, each customer may be served by more than one vehicle. We show that the profit collected by allowing split deliveries may be as large as twice the profit collected under the constraint that each customer has to be served by one vehicle at most. We then present a branch-and-price exact algorithm and a hybrid heuristic. We show the effectiveness of the proposed approaches on benchmark instances and on a new set of instances that allow us to computationally evaluate the impact of split deliveries. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(1), 16–33 2014

A column generation approach for the split delivery vehicle routing problem

AbstractIn this article we present a branch‐and‐price‐and‐cut method for the solution of the split delivery vehicle routing problem (SDVRP). The SDVRP is the problem to serve customers with a fleet of capacitated vehicles at minimum traveling cost. With respect to the classical vehicle routing problem, where each customer is visited exactly once, in the SDVRP a customer may be visited any number of times. The exact method we propose is based on a decomposition of the problem where the possible routes, with the delivery quantities, are generated in the subproblem. The generated routes are also used to find a heuristic solution to the problem. We consider both the case where the fleet of vehicles is unlimited and the case where the fleet is limited to the minimum possible number of vehicles. We solve to optimality instances with larger size with respect to previous approaches, find new best solutions to several benchmark instances and reduce the optimality gap on most of the benchmark instances. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(4), 241–254 2011

The undirected capacitated arc routing problem with profits

A profit and a demand are associated with each edge of a set of profitable edges of a given graph. A travel time is associated with each edge of the graph. A fleet of capacitated vehicles is given to serve the profitable edges. A maximum duration of the route of each vehicle is also given. The profit of an edge can be collected by one vehicle only that also serves the demand of the edge. The objective of this problem, which is called the undirected capacitated arc routing problem with profits (UCARPP), is to find a set of routes that satisfy the constraints on the duration of the route and on the capacity of the vehicle and maximize the total collected profit. We propose a branch-and-price algorithm and several heuristics. We can solve exactly instances with up to 97 profitable edges. The best heuristics find the optimal solution on most of instances where it is available.

An Optimization-Based Heuristic for the Split Delivery Vehicle Routing Problem

The split delivery vehicle routing problem is concerned with serving the demand of a set of customers with a fleet of capacitated vehicles at minimum cost. Contrary to what is assumed in the classical vehicle routing problem, a customer can be served by more than one vehicle, if convenient. We present a solution approach that integrates heuristic search with optimization by using an integer program to explore promising parts of the search space identified by a tabu search heuristic. Computational results show that the method improves the solution of the tabu search in all but one instance of a large test set.

A probabilistic analysis of a fixed partition policy for the inventory‐routing problem

AbstractWe consider the Inventory‐Routing Problem (IRP) where n geographically dispersed retailers must be supplied by a central facility. The retailers experience demand for the product at a deterministic rate, and incur holding costs for keeping inventory. Distribution is performed by a fleet of capacitated vehicles. The objective is to minimize the average transportation and inventory costs per unit time over the infinite horizon. We focus on the set of Fixed Partition Policies (FPP). In an FPP, the retailers are partitioned into disjoint and collectively exhaustive sets. Each set of retailers is served independently of the others and at its optimal replenishment rate. Previous research has measured the effectiveness of an FPP solution relative to a lower bound over all policies. We propose an additional measure that is relative to the optimal FPP. In this paper we construct a polynomial‐time partitioning scheme that is shown to yield an FPP whose cost is asymptotically within 1.5% + ϵ of the cost of an optimal FPP, for arbitrary ϵ > 0. In addition, in some cases, our polynomial‐time scheme yields an FPP whose cost is asymptotically within 1.5% + ϵ of the minimal policy's cost (over all feasible policies). © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004

An asymptotic 98.5%-effective lower bound on fixed partition policies for the inventory-routing problem

We consider the Inventory-Routing Problem where n geographically dispersed retailers must be supplied by a central facility. The retailers experience demand for a product at a deterministic rate and incur holding costs for keeping inventory. Distribution is performed by a fleet of capacitated vehicles. The objective is to minimize the average transportation and inventory costs per unit time over the infinite horizon. In this paper, we focus on the set of fixed partition policies. In a fixed partition policy, the retailers are partitioned into disjoint and collectively exhaustive sets. Each set of retailers is served independently of the others and at its optimal replenishment rate. We derive a deterministic (O( n)) lower bound on the cost of the optimal fixed partition policy. A probabilistic analysis of the performance of this bound demonstrates that it is asymptotically 98.5%-effective. That is, as the number of retailers increases, the lower bound is very close to the cost of the optimal fixed partition policy.