Dynamic pickup and delivery problems

In the last decade, there has been an increasing body of research in dynamic vehicle routing problems. This article surveys the subclass of those problems called dynamic pickup and delivery problems, in which objects or people have to be collected and delivered in real-time. It discusses some general issues as well as solution strategies.     

The pickup and delivery problem with transfers: Formulation and a branch-and-cut solution method

In this paper, a strict formulation of a generalization of the classical pickup and delivery problem is presented. Here, we add the flexibility of providing the option for passengers to transfer from one vehicle to another at specific locations. As part of the mathematical formulation, we include transfer nodes where vehicles may interact interchanging passengers. Additional variables to keep track of customers along their route are considered. The formulation has been proven to work correctly, and by means of a simple example instance, we conclude that there exist some configurations in which a scheme allowing transfers results in better quality optimal solutions. Finally, a solution method based on Benders decomposition is addressed. We compare the computational effort of this application with a straight branch and bound strategy; we also provide insights to develop more efficient set partitioning formulations and associated algorithms for solving real-size problems.     

Solving the pickup and delivery problem with three-dimensional loading constraints and reloading ban

In this paper, we extend the classical Pickup and Delivery Problem (PDP) to an integrated routing and three-dimensional loading problem, called PDP with three-dimensional loading constraints (3L-PDP). We are given a set of requests and a homogeneous fleet of vehicles. A set of routes of minimum total length has to be determined such that each request is transported from a loading site to the corresponding unloading site. In the 3L-PDP, each request is given as set of rectangular boxes and the vehicle capacity is replaced by a 3D loading space.This paper is the second one in a series of articles on 3L-PDP. As in the first paper we are dealing with constraints which guarantee that no reloading effort will occur. Here the focus is laid on the reloading ban, a packing constraint that ensures identical placements of same boxes in different packing plans. The reloading ban allows for better solutions in terms of travel distance than a routing constraint that was used in the first paper to preclude any reloading effort. To implement this packing constraint a new type of packing procedure is needed that is capable to generate a series of interrelated packing plans per route. This packing procedure, designed as tree search algorithm, and the corresponding concept of packing checks is the main contribution of the paper at hand. The packing procedure and a large neighborhood search procedure for routing form a hybrid algorithm for the 3L-PDP. Computational experiments were performed using 54 3L-PDP benchmark instances.     

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.     

Static pickup and delivery problems: a classification scheme and survey

Pickup and delivery problems constitute an important class of vehicle routing problems in which objects or people have to be collected and distributed. This paper introduces a general framework to model a large collection of pickup and delivery problems, as well as a three-field classification scheme for these problems. It surveys the methods used for solving them. This invited paper is discussed in the comments available at: , , , , .     

A column generation approach for location-routing problems with pickup and delivery

In this paper we formulate an integer programming model for the Location and Routing Problem with Pickup and Delivery. We propose a column generation scheme and implement, for the subproblem, a label-setting algorithm for the shortest path with pickup and delivery and time windows problem. We also propose a set of heuristics to speed up this process. To validate the model, we implement the column generation scheme and test it on different instances developed in this paper. We also provide an analysis of how the costs of opening depots and the fixed cost of routes affect the optimal solution.     

Branch-and-cut with lazy separation for the vehicle routing problem with simultaneous pickup and delivery

We propose a branch-and-cut algorithm for the VRPSPD where the constraints that ensure that the capacities are not exceeded in the middle of a route are applied in a lazy fashion. The algorithm was tested in 87 instances with 50–200 customers, finding improved lower bounds and several new optimal solutions.     

An empirical study on the benefit of split loads with the pickup and delivery problem

Splitting loads such that the delivery of certain loads is completed in multiple trips rather than one trip has been shown to have benefit for both the classic Vehicle Routing Problem (VRP) and the Pickup and Delivery Problem (PDP). However, the magnitude of the benefit may be affected by various problem characteristics. In this paper, we characterize those real world environments in which split loads are most likely to be beneficial. Based on practitioner interest, we determine how the benefit is affected by the mean load size and variance, number of origins relative to the number of destinations, the percentage of origin–destination pairs with a load requiring service, and the clustering of origin and destination locations. We find that the magnitude of benefit is greatest for load sizes just over one half vehicle capacity as these loads can not be combined without splitting, while they are the easiest to combine on a vehicle with splitting; increases as the number of loads sharing an origin or destination increases because there are more potential load combinations to split at each stop; and increases as the average distance from an origin to a destination increases because splitting loads reduces the trips from origins to destinations.     

Decision diagrams for solving traveling salesman problems with pickup and delivery in real time

The Traveling Salesman Problem with Pickup and Delivery seeks a minimum cost path with pickups preceding deliveries. It is important in on-demand last-mile logistics, such as ride sharing and meal delivery. We examine the use of low-width Decision Diagrams in a branch-and-bound with and without Assignment Problem inference duals as a primal heuristic for finding good solutions within strict time budgets. We show these diagrams can be more effective than similarly structured hybrid Constraint Programming techniques for real-time decision making.     

A multi-commodity, capacitated pickup and delivery problem: The single and two-vehicle cases

We explore dynamic programming solutions for a multi-commodity, capacitated pickup and delivery problem. Cargo flows are given by an origin/destination matrix which is not necessarily symmetric. This problem is a generalization of several known pickup and delivery problems, as regards both problem structure and objective function. Solution approaches are developed for the single-vehicle and two-vehicle cases. The fact that for each cargo that goes from a node i to another node j there may be a cargo going in the opposite direction provides the motivation for the two-vehicle case, because one may conceivably consider solutions where no cargoes that travel in opposite directions between node pairs are carried by the same vehicle. Yet, it is shown that such scenarios are generally sub-optimal. As expected, the computational effort of the single vehicle algorithm is exponential in the number of cargoes. For the two-vehicle case, said effort is of an order of magnitude that is not higher than that of the single-vehicle case. Some rudimentary examples are presented or both the single-vehicle and two-vehicle cases so as to better illustrate the method.     

A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.     

Real-time vehicle rerouting problems with time windows

This paper introduces and studies real-time vehicle rerouting problems with time windows, applicable to delivery and/or pickup services that undergo service disruptions due to vehicle breakdowns. In such problems, one or more vehicles need to be rerouted, in real-time, to perform uninitiated services, with the objective to minimize a weighted sum of operating, service cancellation and route disruption costs. A Lagrangian relaxation based-heuristic is developed, which includes an insertion based-algorithm to obtain a feasible solution for the primal problem. A dynamic programming based algorithm solves heuristically the shortest path problems with resource constraints that result from the Lagrangian relaxation. Computational experiments show that the developed Lagrangian heuristic performs very well.     

Multiple pickup and delivery traveling salesman problem with last-in-first-out loading and distance constraints

We extend the traveling salesman problem with pickup and delivery and LIFO loading (TSPPDL) by considering two additional factors, namely the use of multiple vehicles and a limitation on the total distance that a vehicle can travel; both of these factors occur commonly in practice. We call the resultant problem the multiple pickup and delivery traveling salesman problem with LIFO loading and distance constraints (MTSPPD-LD). This paper presents a thorough preliminary investigation of the MTSPPD-LD. We propose six new neighborhood operators for the problem that can be used in search heuristics or meta-heuristics. We also devise a two-stage approach for solving the problem, where the first stage focuses on minimizing the number of vehicles required and the second stage minimizes the total travel distance. We consider two possible approaches for the first stage (simulated annealing and ejection pool) and two for the second stage (variable neighborhood search and probabilistic tabu search). Our computational results serve as benchmarks for future researchers on the problem.     

Maritime crude oil transportation - A split pickup and split delivery problem

The maritime oil tanker routing and scheduling problem is known to the literature since before 1950. In the presented problem, oil tankers transport crude oil from supply points to demand locations around the globe. The objective is to find ship routes, load sizes, as well as port arrival and departure times, in a way that minimizes transportation costs. We introduce a path flow model where paths are ship routes. Continuous variables distribute the cargo between the different routes. Multiple products are transported by a heterogeneous fleet of tankers. Pickup and delivery requirements are not paired to cargos beforehand and arbitrary split of amounts is allowed. Small realistic test instances can be solved with route pre-generation for this model. The results indicate possible simplifications and stimulate further research.     

Lasso solution strategies for the vehicle routing problem with pickups and deliveries

This paper considers the vehicle routing problem with pickups and deliveries (VRPPD) where the same customer may require both a delivery and a pickup. This is the case, for instance, of breweries that deliver beer or mineral water bottles to a set of customers and collect empty bottles from the same customers. It is possible to relax the customary practice of performing a pickup when delivering at a customer, and postpone the pickup until the vehicle has sufficient free capacity. In the case of breweries, these solutions will often consist of routes in which bottles are first delivered until the vehicle is partly unloaded, then both pickup and delivery are performed at the remaining customers, and finally empty bottles are picked up from the first visited customers. These customers are revisited in reverse order, thus giving rise to lasso shaped solutions. Another possibility is to relax the traditional problem even more and allow customers to be visited twice either in two different routes or at different times on the same route, giving rise to a general solution. This article develops a tabu search algorithm capable of producing lasso solutions. A general solution can be reached by first duplicating each customer and generating a Hamiltonian solution on the extended set of customers. Test results show that while general solutions outperform other solution shapes in term of cost, their computation can be time consuming. The best lasso solution generated within a given time limit is generally better than the best general solution produced with the same computing effort.     

An approximation algorithm for the pickup and delivery vehicle routing problem on trees

This paper presents an approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot where there are two types of demands, pickup demand and delivery demand. Customers are located on nodes of the tree, and each customer has a positive demand of pickup and/or delivery.Demands of customers are served by a fleet of identical vehicles with unit capacity. Each vehicle can serve pickup and delivery demands. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. In each tour, a vehicle begins at the depot with certain amount of goods for delivery, visits a subset of the customers in order to deliver and pick up goods and returns to the depot. At any time during the tour, a vehicle must always satisfy the capacity constraint, i.e., at any time the sum of goods to be delivered and that of goods that have been picked up is not allowed to exceed the vehicle capacity. We propose a 2-approximation algorithm for the problem.     

The tree representation for the pickup and delivery traveling salesman problem with LIFO loading

The feasible solutions of the traveling salesman problem with pickup and delivery (TSPPD) are commonly represented by vertex lists. However, when the TSPPD is required to follow a policy that loading and unloading operations must be performed in a last-in-first-out (LIFO) manner, we show that its feasible solutions can be represented by trees. Consequently, we develop a novel variable neighborhood search (VNS) heuristic for the TSPPD with last-in-first-out loading (TSPPDL) involving several search operators based on the tree data structure. Extensive experiments suggest that our VNS heuristic is superior to the current best heuristics for the TSPPDL in terms of solution quality, while requiring no more computing time as the size of the problem increases.     

Fleet routing position-based model for inventory pickup under production shutdown

This paper addresses the problem of collecting inventory of production at various plants having limited storage capacity, violation of which forces plant shutdowns. The production at plants is continuous (with known rates) and a fleet of vehicles need to be scheduled to transport the commodity from plants to a central storage or depot, possibly making multiple pickups at a given plant to avoid shutdown. One operational objective is to achieve the highest possible rate of product retrieval at the depot, relative to the total travel time of the fleet. This problem is a variant (and generalization) of the inventory routing problem. The motivating application for this paper is barge scheduling for oil pickup from off-shore oil-producing platforms with limited holding capacity, where shutdowns are prohibitively expensive. We develop a new model that is fundamentally different from standard node-arc or path formulations in the literature. The proposed model is based on assigning a unique position to each vehicle visit at a node in a chronological sequence of vehicle-nodal visits. This approach leads to substantial flexibility in modeling multiple visits to a node using multiple vehicles, while controlling the number of binary decision variables. Consequently, our position-based model solves larger model instances significantly more efficiently than the node-arc counterpart. Computational experience of the proposed model with the off-shore barge scheduling application is reported.     

The traveling salesman problem with pickup and delivery: polyhedral results and a branch-and-cut algorithm

The Traveling Salesman Problem with Pickup and Delivery (TSPPD) is defined on a graph containing pickup and delivery vertices between which there exists a one-to-one relationship. The problem consists of determining a minimum cost tour such that each pickup vertex is visited before its corresponding delivery vertex. In this paper, the TSPPD is modeled as an integer linear program and its polyhedral structure is analyzed. In particular, the dimension of the TSPPD polytope is determined and several valid inequalities, some of which are facet defining, are introduced. Separation procedures and a branch-and-cut algorithm are developed. Computational results show that the algorithm is capable of solving to optimality instances involving up to 35 pickup and delivery requests, thus more than doubling the previous record of 15.      

A heuristic algorithm for a dial-a-ride problem with time windows,multiple capacities,and multiple objectives

The paper describes a system for the solution of a static dial-a-ride routing and scheduling problem with time windows (DARPTW). The problem statement and initialization of the development project was made by the Copenhagen Fire-Fighting Service (CFFS). The CFFS needed a new system for scheduling elderly and disabled persons, involving about 50.000 requests per year. The problem is characterized by, among other things, multiple capacities and multiple objectives. The capacities refer to the fact that a vehicle may be equipped with e.g. normal seats, children seats or wheel chair places. The objectives relate to a number of concerns such as e.g. short driving time, high vehicle utilization or low costs. A solution algorithm REBUS based on an insertion heuristics was developed. The algorithm permits in a flexible way weighting of the various goals such that the solution reflects the user's preferences. The algorithm is implemented in a dynamic environment intended for on-line scheduling. Thus, a new request for service is treated in less than 1 second, permitting an interactive user interface.