The multi-commodity one-to-one pickup-and-delivery traveling salesman problem (original) (raw)

The one-commodity pickup-and-delivery traveling salesman problem: Inequalities and algorithms

Networks, 2007

This article concerns the "One-commodity Pickup-and-Delivery Traveling Salesman Problem" (1-PDTSP), in which a single vehicle of fixed capacity must either pick up or deliver known amounts of a single commodity to a given list of customers. It is assumed that the product collected from the pickup customers can be supplied to the delivery customers, and that the initial load of the vehicle leaving the depot can be any quantity. The problem is to find a minimum-cost sequence of the customers in such a way that the vehicle's capacity is never exceeded. This article points out a close connection between the 1-PDTSP and the classical "Capacitated Vehicle Routing Problem" (CVRP), and it presents new inequalities for the 1-PDTSP adapted from recent inequalities for the CVRP. These inequalities have been implemented in a branch-and-cut framework to solve to optimality the 1-PDTSP that outperforms a previous algorithm . Larger instances (with up to 100 customers) are now solved to optimality. The classical "Traveling Salesman Problem with Pickups and Deliveries" (TSPPD) is a particular case of the 1-PDTSP, and this observation gives an additional motivation for this article. The here-proposed algorithm for the 1-PDTSP was able to solve to optimality TSPPD instances with up to 260 customers.

Heuristics procedures to solve the multi-commodity Pickup-and-Delivery Traveling Salesman Problem

The "multi-commodity Pickup-and-Delivery Traveling Salesman Problem" (m-PDTSP) is a generalization of the well-known "Traveling Salesman Problem" in which cities correspond to customers providing or requiring known amounts of m different products, and the vehicle has a known capacity. Each customer must be visited exactly once by the vehicle serving the demands of the different products while minimizing the total travel distance. It is assumed that a unit of a product collected from a customer can be supplied to any other customer that requires this product. We discuss heuristic algorithms for the m-PDTSP. First, we introduce a heuristic that finds a solution of the m-PDTSP. After, we present several procedures which improve a solution (even when this solution is infeasible). These improvement procedures include a branch-and-algorithm where some variables are fixed. Computational experiments on randomly generated instances.

Heuristics for the One-Commodity Pickup-and-Delivery Traveling Salesman Problem

Transportation Science, 2004

T his paper deals with a generalisation of the well-known traveling salesman problem (TSP) in which cities correspond to customers providing or requiring known amounts of a product, and the vehicle has a given upper limit capacity. Each customer must be visited exactly once by the vehicle serving the demands while minimising the total travel distance. It is assumed that any unit of product collected from a pickup customer can be delivered to any delivery customer. This problem is called one-commodity pickup-and-delivery TSP (1-PDTSP). We propose two heuristic approaches for the problem: (1) is based on a greedy algorithm and improved with a k-optimality criterion and is based on a branch-and-cut procedure for finding an optimal local solution. The proposal can easily be used to solve the classical "TSP with pickup-and-delivery," a version studied in literature and involving two commodities. The approaches have been applied to solve hard instances with up to 500 customers.

A branch-and-cut algorithm for a traveling salesman problem with pickup and delivery

Discrete Applied Mathematics, 2004

We study a generalization of the well-known traveling salesman problem (TSP) where each customer provides or requires a given non-zero amount of product, and the vehicle in a depot has a given capacity. Each customer and the depot must be visited exactly once by the vehicle supplying the demand while minimizing the total travel distance. We assume that the product collected from pickup customers can be delivered to delivery customers. We introduce a 0-1 integer linear model for this problem and describe a branch-and-cut procedure for ÿnding an optimal solution. The model and the algorithm are adapted to solve instances of TSP with pickup and delivery. Some computational results are presented to analyze the performance of our proposal.

An Algorithm for the One Commodity Pickup and Delivery Traveling Salesman Problem with Restricted Depot

Networks and Spatial Economics, 2015

In this paper we study the one commodity pickup-and-delivery traveling salesman problem with restricted depot (1-PDTSP-RD), which is a generalization of the classical traveling salesman problem (TSP). We first introduce a polynomial size integer programming formulation for the problem and then study the feasibility issue which is shown to be N P-complete by itself. In particular, we prove sufficient conditions for the feasibility of the problem and provide a polynomial algorithm to find a feasible solution. We also develop a bound on the cost of the 1-PDTSP-RD solution in terms of the cost of the TSP solution. Based on this bound, we provide a heuristic algorithm to solve the 1PDTSP-RD. Extensive numerical experiments are performed to evaluate the efficiency of both the exact and approximation algorithms.

An exact algorithm for the Traveling Salesman Problem with Deliveries and Collections

Networks, 2003

In this paper, we describe a new integer programming formulation for the Traveling Salesman Problem with mixed Deliveries and Collections (TSPDC) based on a two-commodity network flow approach. We present new lower bounds that are derived from the linear relaxation of the new formulation by adding valid inequalities, in a cutting-plane fashion. The resulting lower bounds are embedded in a branch-and-cut algorithm for the optimal solution of the TSPDC. Computational results on different classes of test problems taken from the literature indicate the effectiveness of the proposed method.

A hybrid heuristic approach for the multi-commodity pickup-and-delivery traveling salesman problem

European Journal of Operational Research, 2016

We address in this article the multi-commodity pickup-and-delivery traveling salesman problem, which is a routing problem for a capacitated vehicle that has to serve a set of customers that provide or require certain amounts of m different products. Each customer must be visited exactly once by the vehicle, and it is assumed that a unit of a product collected from a customer can be supplied to any other customer that requires that product. Each product is allowed to have several sources and several destinations. The objective is to minimize the total travel distance. We propose a hybrid three-stage heuristic approach that combines a procedure to generate initial solutions with several local search operators and shaking procedures, one of them based on solving an integer programming model. Extensive computational experiments on randomly generated instances with up to 400 locations and 5 products show the effectiveness of the approach.

The traveling salesman problem with pickup, delivery, and ride-time constraints

Networks, 2015

In the traveling salesman problem with pickup, delivery, and ride-time constraints (TSPPD-RT), a vehicle located at a depot is required to service a number of requests where the requests are known before the route is formed. Each request consists of (i) a pickup location (origin), (ii) a delivery location (destination), and (iii) a maximum allowable travel time from the origin to the destination (maximum ride-time). The problem is to design a tour for the vehicle that (i) starts and ends at the depot, (ii) services all requests, (iii) ensures that each request's ride-time does not exceed its maximum ride-time, and (iv) minimizes the total travel time required by the vehicle to service all requests (objective function). A capacity constraint that may be present is that the weight or volume of the undelivered requests on the vehicle must always be no greater than the vehicle's capacity. In this article, we concurrently analyze the TSPPD-RT with capacity constraints and without capacity constraints. We describe two mathematical formulations of the problem. These formulations are used to derive new lower bounds on the solution to the problem. Then, we provide two exact methods for finding the optimal route that minimizes the total travel cost. Our extensive computational analysis on both versions of the TSPPD-RT shows that the proposed algorithms are capable of solving to optimality instances involving up to 50 requests.

Integer Models for the Asymmetric Traveling Salesman Problem with Pickup and Delivery

2018

We propose a new Mixed Integer Programming formulation for the Asymmetric Traveling Salesman Problem with Pickup and Delivery, along with valid inequalities for the Sarin-Sherali-Bhootra formulation. We study these models in their complete forms, relax complicating constraints of these models, and compare their performance. Finally, we present computational results showing the promise of these formulations when applied to pickup and delivery problems.