A simulated annealing heuristic for the multiconstraint team orienteering problem with multiple time windows
Graphical abstract
Introduction
Souffriau, Vansteenwegen, Van den Berghe and Van Oudheusden [1] proposed the multiconstraint team orienteering problem with multiple time windows (MC-TOP-MTW) as a sub-problem in the development of a personal tourist guide. The personal tourist guide helps tourists plan their visits to points of interest (POIs). Usually, when visiting a city, tourists gather relevant information and select POIs to visit because they have limited tour time. The tourists then plan their tour based on available time and POI opening hours. The planning process is often extremely complex and time-consuming. The personal tourist guide can consider all the complications and promptly suggest best itinerary for tourists. The problem of selecting POIs and planning associated tours is called the tourist trip design problem [2].
The tourist trip design problem is closely related to the well-known orienteering problem (OP) introduced by Tsiligirides [3]. The OP can be defined using a set of vertices, each associated with a score. Travel time between each pair of vertices is known. The goal of the OP is to determine a tour that maximizes the total score collected at the visited vertices. The score of each vertex can be collected at most once. The OP scores represent the strength of tourist preference for or interest in the POIs. The available tour time in the tourist trip design problem resembles the time budget in the OP. Therefore, the tourist trip design problem can be resolved by solving an OP.
Because of practical considerations or restrictions, there are several variants of the OP in the literature, including the team orienteering problem (TOP), orienteering problem with time windows (OPTW), and team orienteering problem with time windows (TOPTW). Vansteenwegen, Souffriau and Van Oudheusden [4] provide a comprehensive survey about the OP and some of its extensions.
The MC-TOP-MTW discussed in this study is an extension of the TOPTW. In the TOPTW, each location is associated with a single time window. However, in real world tourist trip design, a location may have multiple time windows. Additionally, different trips may have different time windows for certain locations. The MC-TOP-MTW considers these situations and includes additional knapsack constraints that deal with the budget limitations and “max-n type constraints” [1].
During tourist trip design, the underlying MC-TOP-MTW needs to be rapidly solved. On the other hand, given sufficient planning time, high quality solutions are desirable. For example, a tourist may have searched for information about the destination and come up with a list of attractions that s/he wants to visit before the journey starts. In this case, the tourist is willing to spend a longer time for a high-quality trip design. The main contribution of this study is developing an efficient simulated annealing with restart strategy (SA_RS) heuristic that can produce high quality MC-TOP-MTW solutions within a reasonable time. Two versions of SA_RS, SA_RSBF and SA_RSCF, were developed. The main difference between the two versions is in the acceptance probability of worse solutions used in SA. The SA_RSBF uses Boltzmann function to determine the acceptance probability of a worse solution, while the SA_RSCF accepts a worse solution based on the Cauchy function. Both SA_RSBF and SA_RSCF are compared with GRILS [1] in the computational study. The GRILS hybridizes an iterated local search (ILS) approach with a greedy randomized adaptive search procedure (GRASP).
The remainder of this paper is organized as follows. Section 2 defines the problem and reviews the literature. Section 3 then describes the proposed simulated annealing with restart strategy heuristic for the MC-TOP-MTW. Next, Section 4 details the computational experiment and discusses the computational results. Finally, Section 5 draws conclusions.
Section snippets
Problem definition and literature review
The MC-TOP-MTW is an extension of the TOPTW, which originates from the famous orienteering problem. Tsiligirides [3] defined the OP based on the sport of orienteering. Orienteering involves participants beginning from a control point, attempting to collect as many points as possible from checkpoints, and finally returning to the control point before a predetermined time. The goal is to maximize the total score collected at checkpoints. The OP is also known as the selective traveling salesperson
Simulated annealing with restart strategy heuristic for the MC-TOP-MTW
This study is motivated by the fact that the literature proposes only one efficient heuristic approach for the MC-TOP-MTW. Particularly, this study develops an SA-based heuristic for the MC-TOP-MTW. Simulated annealing is a local search-based meta-heuristic capable of escaping from local optimum by accepting, with small probability, worse solutions during the search process. SA has been successfully applied to many hard combinatorial optimization problems [21], [22], [23], [24], [25], [26], [27]
Computational study
The proposed SA_RS heuristic was coded using C language and run on a personal computer with an Intel Core 2 2.5 GHz CPU, which is comparable to the computational environment employed by Souffriau, Vansteenwegen, Van den Berghe and Van Oudheusden [1]. The performance of the proposed SA_RS heuristic using the Cauchy function (SA_RSCF) was compared with that using the Boltzmann function (SA_RSBF) and the GRILS heuristic proposed by Souffriau, Vansteenwegen, Van den Berghe and Van Oudheusden [1].
Conclusions
This study proposes a simulated annealing with restart strategy heuristic for MC-TOP-MTW. The proposed algorithm uses a solution representation scheme that is suitable for MC-TOP-MTW. It also features effective control of parameters, a good hybrid of different search strategies, and a balance between computational time and solution quality. Computational study indicates that both SA_RSBF and SA_RSCF produce high quality MC-TOP-MTW solutions. Furthermore, SA_RSCF outperforms SA_RSBF for
Acknowledgements
The authors thank Professor W. Souffriau for providing us the benchmark problems and solutions. The first author is grateful to the Ministry of Science and Technology of the Republic of China (Taiwan) and the Linkou Chang Gung Memorial Hospital for financially supporting this research under grants MOST 103-2410-H-182-006 and CARPD3B0012, respectively. The work of the second author was partially supported by the Ministry of Science and Technology of the Republic of China (Taiwan) under grant NSC
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