The one‐commodity pickup and delivery travelling salesman problem on a path or a tree

Abstract

Optimization algorithms for both path and tree topology classes of the one-commodity pickup and delivery travelling salesman problem (1-PDTSP) are proposed in this article, which focus on minimizing the route distance to transport products among pickup and delivery customers by a single vehicle with a limited capacity of k . Each pickup customer provides one unit volume of the product while each delivery customer requires one unit volume of the product. For the path case, we propose an O(n 2 / min (k , n)) algorithm for any arbitrary k , and two O(n) algorithms for k = 1 and k = ∞. For the tree case, O(n 2 ) and O(n) algorithms are proposed for k = 1 and k = ∞, respectively. Moreover, when k is arbitrary, the problem becomes NP-hard in the strong sense.

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

“…Note that since we require that the vehicle leave and return to the depot without any bike, our problem is different to the problem considered in previous study (Hernandez-Perez and Salazar-Gonzalez 2002, 2004a, 2007Hernandez-Perez et al 2009;Louveaux and Salazar-Gonzalez 2008;Wang et al 2006;Zhao et al 2009), therefore our numerical results are not comparable to the results provided therein.…”

“…The goal is to minimize the total cost of the trip. The depot serves as a "buffer", i.e., the vehicle is allowed to leave the depot with an amount of load and/or to return to the depot with an amount of remains, see Hernandez-Perez and Salazar-Gonzalez (2002, 2004a, 2007, Hernandez-Perez et al (2009), Louveaux andSalazar-Gonzalez (2008), Wang et al (2006) and Zhao et al (2009). There have been several studies focusing on this problem since it was proposed.…”

“…In Hernandez-Perez (2002) and Salazar-Gonzalez (2004a), the authors presented an integer programming formulation with exponential number of constrains and a branch-and-cut algorithm for the 1-PDTSP. In Wang et al (2006), the 1-PDTSP on special graphs with path or tree topology is studied. In Hernandez-Perez and Salazar-Gonzalez (2007), by observing the close connection between the 1-PDTSP and the capacitated vehicle routing problem, the authors presented new valid inequalities for the 1-PDTSP and incorporated these inequalities to their branch-and-cut algorithm.…”

Static pickup and delivery problems: a classification scheme and survey

“…The Q-DTSP on a line and on a tree has been studied by Wang et al (2006). For the line case, the authors have presented an O(n 2 /min{Q, n}) algorithm and have shown how to solve the unit capacity version (Q = 1) and the uncapacitated version (Q = ∞) in linear time.…”

“…They have also presented a heuristic with a worst-case performance ratio of 1.5, and they have shown that in the single commodity case the problem can be solved in polynomial time. A linear time algorithm for the single-commodity, a non-preemptive version of the SP on a tree, is presented by Wang et al (2006) is a many-to-many, single-commodity problem. The set of customer locations is partitioned into pickup and delivery customers to be served by a single vehicle based at the depot.…”