Interfaces with Other Disciplines
Shortest path tour problem with time windows

https://doi.org/10.1016/j.ejor.2019.08.052Get rights and content

Highlights

  • We present a new variant of the Shortest Path Tour Problem.

  • Time Windows constraints are taken into account.

  • The theoretical properties of the problem are investigated.

  • An optimal solution approach based dynamic programming is devised.

  • Extensive computational experiments are carried out on a significant set of test problems.

Abstract

This paper aims at studying a new variant of the shortest path tour problem, where time window constraints are taken into account. This is the first work dealing with the shortest path tour problem with time windows. The problem is formally described and its theoretical properties are analyzed. We prove that it belongs to the NP-hard class of complexity by polynomial reduction from the knapsack problem. An optimal solution approach based on the dynamic programming paradigm is devised. Labelling algorithms are defined along with well-tailored pruning strategies based on cost and time. The correctness of the bounding strategies is proven and the empirical behavior is analyzed in depth. In order to evaluate the performance of the proposed approach, extensive computational experiments have been carried out on a significant set of test problems derived from benchmarks for the shortest path tour problem. Sensitivity analysis is carried out by considering both algorithmic and instance parameters.

Introduction

The Shortest Path Tour Problem (SPTP) has been firstly introduced by Bajaj (1971) as a constrained version of the Shortest Path Problem (SPP). The problem is formulated on a graph and the nodes are clustered in several ordering sets. A cost is associated with each arc of the graph. The aim is to find a path with minimum cost, such that the sets are served in the order they are defined. In particular, a set is said to be served if at least one node belonging to it appears in the path. The optimal solution can be composed by subpaths of not served nodes, that play the role of connections between two consecutive served nodes.

One of the main applications of SPTP can be found in the domain of Network Functions Virtualization (NFV). NFV decouples the network service functionality from the underlying network, computer, and storage resources, and allows communication services to be composed by stitching together functional building blocks that may not be co-located and may be offered by different providers. The orchestration is the process of arranging and coordinating multiple network services to deliver the desired functionality. Orchestration relies on a “marketplace of services”: a repository of services and network functionalities that are available to users. An NFV marketplace planner has to construct a path from source to destination that visits virtual nodes where instances of these services have been deployed (Bhat, 2017).

Although the SPTP was introduced more than 40 years ago in Bajaj (1971), it has not been studied for a long time. Its resolution is proposed as an exercise of Bertsekas’s Dynamic Programming and Optimal Control book (Bertsekas, 2005), where it is asked to formulate it as a dynamic programming problem.

To the best of our knowledge, the first consistent work on SPTP is Festa (2012), in which it is proved that SPTP belongs to P complexity class, reducing it to the SPP. Several algorithms for its solution are proposed and some correlations with the Uncapacitated Facility Location Problems (UFLP) are explored. Later, Festa, Guerriero, Laganá, and Musmanno (2013) proposed a dynamic programming based labeling algorithm, and Bhat and Rouskas (2017) developed a depth-first tour search algorithm that outperformed the state of the art.

Several different variants of the problem have been proposed. Ferone, Festa, Guerriero, and Laganá (2016b) proposed a variant of SPTP, named Constrained SPTP (CSPTP), in which the solution path P can cross each arc of the input graph at most once. They proved that CSPTP is an NP-hard problem and proposed a Branch & Bound (B&B) algorithm to optimally solve it and a GRASP meta-heuristic. Independently, de Andrade and Saraiva (2018) and Ferone, Festa, and Guerriero (2019a) proposed two similar mathematical models. The former solved the problem using CPLEX, the latter proposed an efficient B&B.

Finally, Carrabs, Cerulli, Festa, and Laureana (2017) proposed the Forward SPTP, in which it is possible to visit a node in Tk if and only if at least a node of each previous cluster, T1,,Tk1, has been already visited. They proposed a polynomial algorithm that is able to manage also the classical SPTP.

In this paper, we analyze a time-constrained version of the SPTP that includes time windows associated with each served node (SPTPTW). In this version of the problem, traversing time is associated with each arc and a feasible path must guarantee that the time needed to reach a served node does not exceed the due-date defined by the time window constraint.

Routing problems with time windows are widely studied in the scientific literature. The shortest path problem with time windows (SPPTW) was addressed for the first time in Desrosiers, Pelletier, and Soumis (1983) as a subproblem of a route construction problem. It was designed a label correcting method that generalizes the approach proposed by Gallo and Pallottino (1988) for the SPP.

Powell and Chen (1998) presented a label-correcting algorithm extending for the SPPTW the rule applied to the SPP by Glover, Glover, and Klingman (1984). Later, Desrochers and Soumis (1988) presented a label-setting algorithm that exhibits a pseudopolynomial complexity. The SPPTW arises as subproblem when the vehicle routing problem with time windows is solved via column-generation approaches (Liberatore, Righini, Salani, 2011, Qureshi, Taniguchi, Yamada, 2009, Tagmouti, Gendreau, Potvin, 2007). For more details on the SPPTW and the related solution strategies, the reader is referred to Di Puglia Pugliese and Guerriero (2013a).

The SPTPTW shows some similarities with the Generalized Vehicle Routing Problem with Time Windows (GVRPTW) (Moccia, Cordeau, & Laporte, 2012), an extension of the Generalized Vehicle Routing Problem (GVRP) (Ghiani & Improta, 2000) since a clustered substructure for the nodes is considered. However, in the GVRPTW, the set of nodes is partitioned in sets of customers, and each set must be visited (served) exactly once. Moreover, since the sets are not ordered, the decisions involved in the problem are node selection (which node in each set is visited) and node sequencing (the sequence in which customer sets are served).

In the case of the SPTPTW, even though each subset must be served exactly once, it is possible to traverse each subset more than once (more details are given in Section 2). In addition, the start and end location of the vehicle can be different and, finally, the most important difference is that the service order of the subsets is given as input in the SPTPTW. For this reason, the decision about the node sequencing must not be taken.

To the best of our knowledge, this is the first time that tour constraints are included in the well-studied SPPTW. The SPTPTW has several applications. For instance, in tourism path planning, a tourist wants to visit a set of interesting areas, such as parks, museums, historical building, and squares. She/he can group these areas in several sets giving an order of visits to those sets. In each set, alternative places are collecting. The aim is to provide a tour, where one place belonging to each set is visited following the order imposed by the sets. In this context, one has to take into account the opening of the closing time of the interesting areas and the time needed to reach the target place in order to guarantee the maximum satisfaction from the planned tour.

Another example is cargo delivery. A vehicle must deliver orders to N regions in specific time windows, and the orders can be ranked respect to a priority. In this case, the regions can be represented as node subsets, and the priority is given by the order of the subsets.

The paper is organized as follows. In Section 2, the problem is formally described and theoretical properties are analyzed. Section 3 describes an optimal solution approach based on dynamic programming strategy along with tailored bounding techniques. Section 4 shows the computational results carried out considering several network topologies. The behavior of the proposed solution approach is analyzed in depth by reporting sensitivity analysis varying both algorithmic and network parameters. Section 5 concludes the paper providing some directions for future research.

Section snippets

Problem definition

Let G(V, A) be a directed graph, where V is the set of n nodes and A={(i,j):i,jV} is the set of m arcs. The set V contains two special nodes: the source node s and the destination node d. A non-negative cost cij and a non-negative transit time tij is associated with each arc (i, j) ∈ A. Let N denote a certain number of node subsets T1,,TN, such that ThTk=, for all h,k=1,,N, h ≠ k. Without loss of generality, we assume that T1={s} and TN={d}. Let T=h=1NTh. Moreover, a non-negative service

Solution approach

In this section, we describe a dynamic programming algorithm (Labeling) to solve the SPTPTW. It is worth observing that the proposed algorithm is an extension of the dynamic programming strategy developed in Festa et al. (2013).

Let yi=(ci,τi,ri,πsi) be a label associated with a path πsi. The label yi stores the information related to the path, i.e., the cost ci=c(πsi), the resource τi, indicating the arrival time to node i plus the service time needed to serve i, and the resource indicating the

Experimental results

In this Section we evaluate the behavior of the proposed algorithm. We consider two versions, named DP that implements Algorithm 2, and DPB in which the bounding strategies are taken into account. In addition, several label selection policies are tested. Both DP and DPB have been coded in C++ and compiled with gcc 5.4.0 with the flag -std=c++14. All tests were run on a Intel® Core™ i7-4510U CPU @ 2.00GHz  ×  4, with 8GB of RAM, under the Linux (Ubuntu 16.04) operating system. All the tests have

Conclusions and future work

In this paper, we described and studied the shortest path tour problem with time windows (SPTPTW).

We presented a dynamic programming algorithm with some bounding strategies to improve its performance. The experimental results show how the use of the bounding strategies effectively improves the computational times of the algorithm. Furthermore, an extensive analysis of the experimental results has been performed, in order to understand how the different instance parameters impact the performance

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