Discrete Optimization
Dynamic traveling salesman problem with stochastic release dates

https://doi.org/10.1016/j.ejor.2019.07.062Get rights and content

Highlights

  • We introduce a dynamic and stochastic variant of a routing problem with release dates.

  • A heuristic reoptimization approach is proposed.

  • Two models for solving the problem at each stage of the reoptimization are proposed.

  • An exhaustive computational study is presented to assess the presented approaches.

Abstract

The dynamic traveling salesman problem with stochastic release dates (DTSP-srd) is a problem in which a supplier has to deliver parcels to its customers. These parcels are delivered to its depot while the distribution is taking place. The arrival time of a parcel to the depot is called its release date. In the DTSP-srd, release dates are stochastic and dynamically updated as the distribution takes place. The objective of the problem is the minimization of the total time needed to serve all customers, given by the sum of the traveling time and the waiting time at the depot. The problem is represented as a Markov Decision Process and is solved through a reoptimization approach. Two models are proposed for the problem to be solved at each stage. The first model is stochastic and exploits the entire probabilistic information available for the release dates. The second model is deterministic and uses an estimation of the release dates. An instance generation procedure is proposed to simulate the evolution of the information to perform computational tests. The results show that a more frequent reoptimization provides better results across all tested instances and that the stochastic model performs better than the deterministic model. The main drawback of the stochastic model lies in the computational time required to evaluate a solution, which makes an iteration of the heuristic substantially more time-consuming than in the case where the deterministic model is used.

Introduction

In a last mile distribution system, the supplier often faces the uncertainty related to the delivery of goods to its depot. Information therefore plays a crucial role in an efficient distribution planning. In this setting, we study the dynamic traveling salesman problem with stochastic release dates (DTSP-srd) that is a problem in which a supplier receives from its suppliers goods to be distributed to customers. The goods to be delivered are available for the distribution only after they have arrived to the depot of the supplier. The arrival time of a parcel to the depot is called its release date. In the DTSP-srd, release dates are considered to be stochastic and dynamically updated as the distribution takes place. The DTSP-srd finds application in many real world problems. The steadily growing interest in an environment-aware supply chain management has led to a significant research for an eco-friendly delivery to city centers. In this context, the distribution is implemented by means of distribution centers, where goods are unloaded from the trucks, consolidated, and delivered to the customers by means of hybrid or electric vehicles. The supplier has to face the uncertainty of the arrival times of the trucks to plan for an efficient distribution. The goal is to define an efficient distribution plan serving all customers. The DTSP-srd also finds application in cross docking operations, where the routing of outbound vehicles is planned according to the arrival time of inbound vehicles. In addition, as discussed in Klapp, Erera, Toriello, 2018a, Klapp, Erera, Toriello, 2018b, the DTSP-srd finds application in the Same-Day Delivery (SDD) service, where the distribution of parcels, typically ordered on-line, from a distribution center to final customers, overlaps with their delivery from the suppliers to the distribution center.

While problems with release dates, defined as the earliest availabilities of jobs for processing, have been widely studied in the context of machine sequencing (see Pinedo, 2016), only in recent years the concept of release dates has been introduced for vehicle routing problems. Cattaruzza, Absi, and Feillet (2016) introduce the multi-trip vehicle routing problem with time windows and release dates (MTVRPTW-R) in which all information is assumed to be static and deterministic. The authors propose a genetic algorithm for the problem with an underlying giant tour decomposition procedure. The complexity of the traveling salesman problem with release dates (TSP-rd) is studied in Archetti, Feillet, and Speranza (2015). The authors introduce two variants of the problem, one considering a deadline for the completion of the distribution and seeking the minimization of the total travel time, referred to as TSP-rd(distance), and the other minimizing the total time required to complete the distribution, referred to as TSP-rd(time). The complexity of the problem is studied on special topologies of the graph. Reyes, Erera, and Savelsbergh (2018) extends this work to consider service guarantee, a fixed maximum delay between the release date and the delivery time. The TSP-rd(time) is studied in Archetti, Feillet, Mor, and Speranza (2018) where a mathematical programming formulation is described and an iterated local search is developed. Two versions of the heuristic are tested on instances derived from TSP instances by Solomon (1987) and from the TSPLIB (see Reinelt, 1991).

Stochasticity has been widely studied in the literature on distribution problems. We refer to Gendreau, Jabali, and Rei (2016) for a review of the recent advances and future directions of the stochastic vehicle routing literature and to Ritzinger, Puchinger, and Hartl (2016) for a survey on the dynamic and stochastic vehicle routing problems. Despite this, problems related to the study of uncertain release dates in routing have only recently received scientific interest. Uncertain arrival times are considered in Klapp, Erera, and Toriello (2018b), where the authors study the dispatch wave problem. At any stage (wave), requests are either known or potential. The stochasticity of potential requests considers both the probability of non-arrival and the probability of the arrival at a specific wave. The minimization of the expected vehicle operating costs and the penalties for unserved requests is sought on a special topology of the graph, i.e., the line. In Klapp et al. (2018a) the authors extend the work to a general network. A deterministic model is used to find an optimal a priori solution to the stochastic variant and two dynamic policies are developed. Finally, the trade-off between minimizing operational costs and maximizing the total order coverage is studied. Klapp et al. (2018a) is probably the work more closely related to the one presented in this paper. It is therefore worth highlighting the key differences with respect to the problem and the approach presented. In the problem discussed by these authors, probabilistic information is available for unknown customers to describe if and when a customer request will be placed. The distribution takes place over a finite planning horizon. Customers can be rejected or outsourced at the end of the planning horizon at the price of a penalty. The planning horizon is divided in a fixed and finite number of dispatch times, called waves, in which decisions are made on the customers to be served. In the problem discussed in this paper, in contrast, customers are known and probabilistic information is available on their release dates. Customers cannot be rejected and, while the objective is to minimize the completion time, the planning horizon is unbounded. A key difference in the approach to the problem is the fact that stages do not happen at fixed times but are dynamically set according to the information available for the release dates.

A related problem considering a fleet of vehicles is discussed in Voccia, Campbell, and Thomas (2017), where the same-day delivery problem is studied. Customer requests are described by time windows or delivery deadlines. The arrival rate and geographical distribution of customer requests are known. The authors compute the time a vehicle can wait at the depot without decreasing the ability to serve customer requests and propose a consensus function to identify when it is beneficial to wait at the depot. An approximate forward dynamic programming approach is presented. The same-day delivery problem is also discussed in Ulmer, Thomas, and Mattfeld (2016). The authors investigate the case where preemptive returns are allowed, that is, the vehicles can return to the depot before their planned route is completed to load the parcels of new customers. A study on the dispatching problem in urban consolidation centers is presented in van Heeswijk, Mes, and Schutten (2017), where the delivery dispatch problem with time windows is discussed. The authors consider the case where orders are stochastic and dynamically revealed to the operator, either at the time of arrival at the distribution center or in advance, over a finite planning horizon. At each time the consolidation center must choose a subset of orders to be dispatched. Each customer request is characterized by a time window for the time of dispatch from the distribution center. The aim of the center is to find the cost-minimizing consolidation policy. As the focus of the paper is on the dispatching decisions, routing decisions are not considered as part of the optimization problem and routing costs are estimated by means of a cost function.

The contributions of the current work are the following:

  • 1.

    We introduce the DTSP-srd as a dynamic and stochastic variant of the TSP-rd(time) where the information on the release dates is assumed to be stochastic and dynamically updated as the distribution takes place. A Markov Decision Process (MDP) is presented to model the dynamic and stochastic aspects of the problem.

  • 2.

    A heuristic reoptimization approach is proposed. In particular, three strategies with increasing reoptimization frequency are introduced.

  • 3.

    Two models for solving the problem at each stage of the reoptimization are proposed. The first model is stochastic and exploits the entire probabilistic information available for the release dates. The second model is deterministic and uses a point estimation of the release dates. Both models are solved with an iterated local search.

  • 4.

    A myopic solution approach is proposed, serving customers as soon as their parcels arrive to the depot, without considering the information available about future release dates.

  • 5.

    An instance generation procedure is defined simulating the evolution of the information about the release dates by mimicking the arrival of the vehicle to the depot.

  • 6.

    An exhaustive computational study is presented to show the value of a higher reoptimization frequency and the performance of the deterministic and the stochastic model. The results, obtained on instances with 50 customers, show that a more frequent reoptimization provides better results across all tested instances. The deterministic and stochastic model have an average percentage gap from the best solution found across the three dynamic strategies of 2.92% and 1.16%, respectively. While the stochastic model performs better, its main drawback lies in the computational time required to evaluate any of the solution explored, which makes an iteration of the algorithm substantially more time-consuming. The myopic approach is shown to perform more than 12% worse than the best solution found by the two models.

The paper is organized as follows. In Section 2 the DTSP-srd is defined and modeled as an MDP. The reoptimization approach proposed for the solution of the DTSP-srd is introduced in Section 3 as well as the myopic solution approach. In Section 4 the stochastic and deterministic optimization models, to be solved at each reoptimization epoch, are introduced, an illustrative example of the two models is presented and the heuristic is described. The instance generation procedure is described in Section 5 and the computational experiments are reported in Section 6. Finally, conclusions are drawn in Section 7.

Section snippets

The dynamic traveling salesman problem with stochastic release dates

The DTSP-srd is defined as follows. Let G=(V,A) be a complete graph. A traveling time and a cost are associated with each arc (i, j) ∈ A. These two values are assumed identical and denoted by dij. It is also assumed that the triangle inequality is satisfied. The set of vertices V is composed by vertex 0, which identifies the depot, and the set N of customers, with |N|=n. Each customer is characterized by the arrival time of its parcel to the depot. We call this value its release date. A single

Solution approaches

In this section the solution approaches devised for the problem are described. First, the heuristic reoptimization approach based on the solution of the MDP described above. Then, the myopic algorithm based on the greedy strategy of sending out the vehicle as soon as there is a parcel ready to be distributed.

Optimization models

In this section the models used to select an action from the action space at each reoptimization epoch are presented. The stochastic model is introduced in Section 4.1. Then, in Section 4.2, the deterministic model is described. To illustrate the differences between the two models an example is presented in Section 4.3. Finally, the solution method proposed for the two models is described in Section 4.4.

Instance generation

The following sections describe the instance generation procedure, firstly reporting the instances from which the DTSP-srd instances are derived, and secondly defining how the release dates of each customer are generated and updated over time. In a brief overview, the release dates of customers are generated by simulating the traveling of the vehicles delivering the parcels to the depot. The dynamic release dates are updated as the vehicles travel to the depot. When the vehicle reaches the

Computational experiments

In this section, we describe the computational experiments that have been carried out to assess the performance of the two models and the three reoptimization strategies. The aim is to understand whether, in the presented reoptimization and stochastic setting, it is better to consider a point estimation or the entire stochastic information for the release dates and if increasing the number of reoptimization epochs improves the quality of the solution. We proceed as follows. In Section 6.1, the

Conclusions

In this paper the dynamic traveling salesman with stochastic release dates (DTSP-srd) is introduced. A solution approach is proposed based on reoptimization. Three strategies are introduced to define the reoptimization epochs, with increasing frequency of reoptimization, and two models are proposed for the solution of the problem at each epoch. The first one considers a point estimation of the release dates and the second one makes use of the entire probabilistic information available. The

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