An exact method for the double TSP with multiple stacks

Networks, 2013

This article studies the double traveling salesman problem with two stacks. A number of requests have to be served where each request consists in the pickup and delivery of an item. All the pickup operations have to be performed before any delivery can take place. A single vehicle is available that starts from a depot, performs all the pickup operations and returns to the depot. Then, it performs all the delivery operations and returns to the depot. The items are loaded in two stacks, each served independently from the other with a last-in-first-out policy. The objective is the minimization of the total cost of the pickup and delivery tours. We propose a branchand-bound approach to solve the problem. The algorithm uses properties of the problem both to tighten the lower bounds and to avoid the exploration of redundant subtrees. Computational results performed on benchmark instances reveal that the algorithm outperforms the other exact approaches for this problem.

INFORMS Journal on Computing, 2013

The double traveling salesman problem with multiple stacks is a variant of the pickup and delivery traveling salesman problem in which all pickups must be completed before any delivery. In addition, items can be loaded on multiple stacks in the vehicle, and each stack must obey the last-in-first-out policy. The problem consists of finding the shortest Hamiltonian cycles covering all pickup and delivery locations while ensuring the feasibility of the loading plan. We formulate the problem as two traveling salesman problems linked by infeasible path constraints. We also introduce several strengthenings of these constraints, which are used within a branch-and-cut algorithm. Computational results performed on instances from the literature show that the algorithm outperforms existing exact algorithms. Instances with up to 28 requests (58 nodes) have been solved to optimality.

Networks, 2012

This article studies a single vehicle pickup and delivery problem with loading constraints. In this problem, the vehicle contains a number of (horizontal) stacks of finite capacity for loading items from the rear of the vehicle. Each stack must satisfy a last-in-first-out constraint where any new item must be loaded on top of a stack and any unloaded item must be on top of its stack. A large neighborhood search is proposed for solving this problem. Computational results are reported on different types of randomly generated instances. Results are also reported on benchmark instances for two special cases of our problem and a comparison is provided with state-of-the-art methods.

Computers & Operations Research, 2009

The rectangular packing problem is to pack a number of rectangles into a single large rectangular sheet so as to maximize the total area covered by the rectangles packed. The paper first presents a least wasted first strategy which evaluates the positions used by the rectangles. Then a random local search is introduced to improve the results and a least wasted first heuristic algorithm (LWF) is further developed to find a desirable solution. Twenty-one rectangular-packing instances are tested by the algorithm developed, the experimental results show that the presented algorithm can achieve an optimal solution within reasonable time and is fairly efficient for dealing the rectangular packing problem. LWF still performs well when it is extended to solve zero-waste and non-zero-waste strip packing instances.

Computers & Operations Research

The Double Traveling Salesman Problem with Multiple Stacks is a pickup-and-delivery single-vehicle routing problem which performs all pickup operations before the deliveries. The vehicle has a loading space divided into stacks of a fixed height that follows a Last-In-First-Out policy. It has to collect products following a Hamiltonian tour in a pickup region, and then deliver them following a Hamiltonian tour in a delivery region. The aim is to minimize the total routing cost while satisfying the vehicle loading constraints.A generalization of this problem considers that each product is offered in several pickup locations at different prices, that is, the pickup locations are markets. That means that the pickup tour may not be Hamiltonian, and therefore the set of locations to be visited during the pickup tour is unknown in advance. The delivery locations represent customers, each requiring a product, and all of them must be visited by the vehicle. Thus, this problem has to select a...

2009

This paper addresses a variation of the traveling salesman problem with pickup and delivery in which loading and unloading operations have to be executed in a first-in-first-out fashion. It provides an integer programming formulation of the problem. It also describes five operators for improving a feasible solution, and two heuristics that utilize these operators: a probabilistic tabu search algorithm, and an iterated local search algorithm. The heuristics are evaluated on data adapted from TSPLIB instances.

This paper introduces an additive branch-and-bound algorithm for a variant of the pickup and delivery traveling salesman problem in which loading and unloading operations have to be performed in a Last-In-First-Out (LIFO) order. Two relaxations are used within the additive approach: the assignment problem and the shortest spanning rarborescence problem. The quality of the lower bounds is further improved by a set of elimination rules applied at each node of the search tree to remove from the problem arcs that cannot belong to feasible solutions because of precedence relationships. The performance of the algorithm and the effectiveness of the elimination rules are assessed on instances from the literature.

2018

The Traveling Salesman Problem with Pickup and Delivery (TSPPD) describes the problem of finding a minimum cost path in which pickups precede their associated deliveries. The TSPPD is particularly important in the growing field of Dynamic Pickup and Delivery Problems (DPDP). These include the manyto-many Dial-A-Ride Problems (DARP) of companies such as Uber and Lyft, and meal delivery services provided by Grubhub. We examine exact methods for solving TSPPDs where orders from different pickup locations can be carried simultaneously in real-time applications, in which finding high quality solutions quickly (often in less than a second) is often more important that proving the optimality of such solutions. We consider enumeration, Constraint Programming (CP), Mixed Integer Programming (MIP), and hybrid methods combining CP and MIP. Our CP formulations examine multiple techniques for ensuring pickup and delivery precedence relationships. Finally, we attempt to provide guidance about whi...

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.

The multiple traveling salesman problem (mTSP) is a generalization of the well-known traveling salesman problem (TSP), where more than one salesman is allowed to be used in the solution. Moreover, the characteristics of the mTSP seem more appropriate for real-life applications, and it is also possible to extend the problem to a wide variety of vehicle routing problems (VRPs) by incorporating some additional side constraints. Although there exists a wide body of the literature for the TSP and the VRP, the mTSP has not received the same amount of attention. The purpose of this survey is to review the problem and its practical applications, to highlight some formulations and to describe exact and heuristic solution procedures proposed for this problem.