Dynamic Programming Strategies for the Traveling Salesman Problem with Time Window and Precedence Constraints

Efficiently handling last-mile deliveries becomes more and more important nowadays. Using drones to support classical vehicles allows improving delivery schedules as long as efficient solution methods to plan last-mile deliveries with drones are available. We study exact solution approaches for some variants of the traveling salesman problem with drone (TSP-D) in which a truck and a drone are teamed up to serve a set of customers. This combination of truck and drone can exploit the benefits of both vehicle types: the truck has a large capacity but usually low travel speed in urban areas; the drone is faster and not restricted to street networks, but its range and carrying capacity are limited. We propose a compact mixed-integer linear program (MILP) for several TSP-D variants that is based on timely synchronizing truck and drone flows; such an MILP is easy to implement but nevertheless leads to competitive results compared with the state-of-the-art MILPs. Furthermore, we introduce dynamic programming recursions to model several TSP-D variants. We show how these dynamic programming recursions can be exploited in an exact branch-and-price approach based on a set partitioning formulation using ng-route relaxation and a three-level hierarchical branching. The proposed branch-and-price can solve instances with up to 39 customers to optimality outperforming the state-of-the-art by more than doubling the manageable instance size. Finally, we analyze different scenarios and show that even a single drone can significantly reduce a route’s completion time when the drone is sufficiently fast.

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Exact and heuristic dynamic programming algorithms for the traveling salesman problem with flexible time windows

Optimization Letters ◽

10.1007/s11590-018-1342-y ◽

2018 ◽

Vol 14 (3) ◽

pp. 579-609

Author(s):

Ramon Faganello Fachini ◽

Vinícius Amaral Armentano

Keyword(s):

Dynamic Programming ◽

Traveling Salesman Problem ◽

Time Windows ◽

Traveling Salesman ◽

Heuristic Dynamic Programming ◽

The Traveling Salesman Problem ◽

Programming Algorithms

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Exact and Heuristic Algorithms for Routing AGV on Path with Precedence Constraints

Mathematical Problems in Engineering ◽

10.1155/2016/5040513 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Liang Xu ◽

Yao Wang ◽

Lin Liu ◽

Jiaxing Wang

Keyword(s):

Dynamic Programming ◽

Traveling Salesman Problem ◽

Polynomial Time ◽

Heuristic Algorithms ◽

Exact Algorithm ◽

Precedence Constraints ◽

Traveling Salesman ◽

Automated Guided Vehicle ◽

Np Complete

A new problem arises when an automated guided vehicle (AGV) is dispatched to visit a set of customers, which are usually located along a fixed wire transmitting signal to navigate the AGV. An optimal visiting sequence is desired with the objective of minimizing the total travelling distance (or time). When precedence constraints are restricted on customers, the problem is referred to as traveling salesman problem on path with precedence constraints (TSPP-PC). Whether or not it is NP-complete has no answer in the literature. In this paper, we design dynamic programming for the TSPP-PC, which is the first polynomial-time exact algorithm when the number of precedence constraints is a constant. For the problem with number of precedence constraints, part of the input can be arbitrarily large, so we provide an efficient heuristic based on the exact algorithm.

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