Heuristics for the multi-period orienteering problem with multiple time windows

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Abstract

We present the multi-period orienteering problem with multiple time windows (MuPOPTW), a new routing problem combining objective and constraints of the orienteering problem (OP) and team orienteering problem (TOP), constraints from standard vehicle routing problems, and original constraints from a real-world application. The problem itself comes from a real industrial case. Specific route duration constraints result in a route feasibility subproblem. We propose an exact algorithm for this subproblem, and we embed it in a variable neighborhood search method to solve the whole routing problem. We then provide experimental results for this method. We compare them to a commercial solver. We also adapt our method to standard benchmark OP and TOP instances, and provide comparative tables with state-of-the-art algorithms.

Introduction

In this paper we present a solution procedure for the individual route planning of field workers and sales representatives (e.g. sales representatives of food retailers or of the pharmaceutic industry). The problem stems from a software distribution company which sells decision support systems for salesman and marketing departments. The developed algorithm will be integrated in an application for scheduling and planning future working days of sales representatives in an off-line fashion. Field workers and sales representatives can use this tool off-line to plan their trips for a certain planning horizon individually.

Field workers and sales representatives have the duty to visit their customers on a regular basis in specific time intervals. The different requirements for the planning of the customer visits and the resulting planning of the traveling salesman tour can be different for different branches. In the following we consider a variant of sales representatives which has the flexibility to schedule his/her customer visit by him-/herself on a weekly basis. This is a decentralized approach where the salesman can organize the schedule of visits—in comparison to that a centralistic approach is presented in Polacek et al. [17]. An usual planning process of the sales representative is that he/she decides at the end of the week which customers or potential customers should be visited for the next week. The mandatory customers are selected. Some of them have specific time windows when they can be visited. There are even some customers who have a different time window for every day. E.g. one customer can be visited on Monday from 9.00 to 12.00 but on Tuesday he/she can only be visited from 11.00 to 12.00 or from 13.00 to 15.00. Note that up to two different time windows per day are allowed.

As already mentioned sales representatives have to visit their customers with long-term relations on a regular basis in order to present new products. In our problem formulation these are the mandatory customers. Moreover it is also very important for the success of a company to acquire new customers. To consider this situation potential customers located nearby should be possibly integrated into the customer tours. When a salesman for tires visits regularly the garages which sells the companies’ tires—he/she should possibly also include garages which are not customers so far. These are the optional customers in our problem. The optional customer selection decision has to be integrated in the solution procedure. For several branches purchasable databases exist, providing the potential customers who can be integrated in such routes. These potential customers do not have to be visited and some of them are only included in the schedule when good combinations result with the mandatory customer.

The contribution of the paper is threefold:

  • We introduce and define a new problem for scheduling the customer visits of sales representatives. This problem is a generalization of the orienteering problem (OP), team orienteering problem (TOP), orienteering problem with time windows (OPTW) and team orienteering problem with time windows (TOPTW), and is denoted as the multi-period orienteering problem with multiple time windows (MuPOPTW).

  • We develop an exact algorithm for the route feasibility check when having multiple time windows. Multiple time windows can occur in any routing problems with time window requirements.

  • We develop an efficient variable neighborhood search algorithm for this specific problem and apply it also to standard instances.

In the TOP a set of customers needs to be selected and a set of tours organized in order to maximize the profit. Feillet et al. (cf. [8]) gather these problem types for a single tour under the name TSP with profits. The used objective functions may be the maximization of the collected total profit (OP), the minimization of the total traveling costs (prize-collecting TSP) by reaching a predefined minimum profit, or the optimization of a combination of both (profitable tour problem). Archetti et al. (cf. [1]) name the extension of the TSP with profits to multiple tours as vehicle routing problems (VRP) with profits. The extension of the OP to multiple tours, which is a special case of VRP with profits, was introduced under the name TOP by Chao et al. (cf. [5]). The OPTW and TOPTW are generalizations of OP and TOP, with time window constraints. OPTW was introduced by Righini and Salani and solved to optimality with a dynamic programming algorithm [19], [20]. Vansteenwegen introduced the TOPTW, and provided a first heuristic approach for the OPTW and TOPTW [28].

The most suitable problem definition for the problem of the sales representatives is the TOPTW. In fact the representatives problem combines optional customers from all orienteering problems and mandatory customers from more traditional routing problems. A multi-period horizon is also an important characteristic here, along with multiple time windows. This means that each customer may be visited on different days, on different time slots, and also that each customer may have several time windows for each given day. In practice, a customer has zero, one or two time windows per day. In our problem formulation the time windows are not allowed to overlap.

The profits of the optional customers are estimated. It is the expected probability that the potential customer can become a customer, multiplied by the expected turnover linked to this customer.

We designed a metaheuristic solution approach to solve the considered problem. Recent publications show that variable neighborhood search (VNS) algorithms fare very well when compared to other solution techniques, especially for routing problems (see e.g. [18], [16], [11]).

The VNS is an improvement metaheuristic introduced in [15], [14]. The basic idea is a systematic change of the neighborhood used within a local search. Here, several neighborhood structures are applied instead of a single one, as it is the case in a standard local search implementation. The algorithm is applied to standard instances for the TOP and also to real-world instances from a company.

The remainder of the paper is organized as follows. A detailed description of the problem is given in Section 2, as well as a mathematical model. The description of the algorithm for checking route feasibility is reported in Section 4. The metaheuristic solution procedure based on ideas of VNS is reported in Section 3. Detailed numerical results are reported in Section 5.

Section snippets

Model formulation

We consider a TOP with multiple time windows. Potential and mandatory customers have to be visited in a planning horizon. The planning horizon consists of several periods—in each period one route is planned. In our problem we introduce additional time aspects. We have a maximum route duration of the daily routes which cannot be exceeded. Furthermore, a maximum working time in the planning horizon is allowed. Consider a planning horizon of three days—the maximum route duration may not exceed 10 h

Heuristic algorithms for the MuPOPTW

We developed two heuristic algorithms for the MuPOPTW: a constructive heuristic, and a VNS. The constructive heuristic is deterministic, and provides a starting solution for the VNS, which is a stochastic local search algorithm. Both these methods rely on the route feasibility check algorithm detailed in the next section, and make intensive use of it. Therefore, all feasibility aspects relative to multiple time windows are delegated to this route feasibility check algorithm. Concerning the

Route feasibility check

In this section, we give an overview of the exact algorithm used to determine whether a given route is feasible or not with regards to time constraints, namely time windows and route duration limit. A more detailed description and a proof are provided in Appendix A.

For the sake of simplicity, we suppose that the customers in the route are now named {1,,n}. Since we are working on a route assigned to a given day of the horizon, t becomes a fixed value. Therefore we do not mention the index t in

MuPOPTW instances

We used information provided by the industrial partner to generate 60 problem instances. The industrial partner has five sales representatives and they sell software for geomarketing applications. Since the partner wants to keep the decision of assigning customers to representatives, each instance concerns only one representative, but on a horizon of three days, which implies solutions with three routes, one per day. A pool of customer locations as well as a distance matrix were given, along

Conclusion

In this paper, we introduced the multi-period orienteering problem with multiple time windows, which is a problem encountered by sales representatives when they want to schedule their working week. This problem is too hard for contemporary commercial solvers, at least for interesting size instances. We therefore developed a metaheuristic (VNS) to provide good solutions in reasonable amounts of time. The MuPOPTW also includes a feasibility subproblem, featuring multiple time windows, which is a

Acknowledgments

We would like to thank Guenter Kiechle for the project management and Andreas Wagner for implementing the information system and the user interface. We also wish to thank the two anonymous referees for their valuable comments, which helped us improving the quality of this paper. Furthermore we thank Zoltan Daroczi for providing the real-world data. Financial support from the Austrian Research Promotion Agency (FFG - project Bridge #814299) is gratefully acknowledged.

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