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A novel GRASP solution approach for the Orienteering Problem

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Abstract

The Orienteering Problem (OP) is a well-known variant of the Traveling Salesman Problem. In this paper, a novel Greedy Randomized Adaptive Search Procedure (GRASP) solution is proposed to solve the OP. The proposed method is shown to outperform state-of-the-art heuristics for the OP in producing high quality solutions. In comparison with the best known solutions of standard benchmark instances, the method can find the optimal or the best known solution of about 70 % of the instances in a reasonable time, which is about 17 % better than the best known approach in the literature. Moreover, a significant improvement is achieved on the solution of two standard benchmark instances.

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Notes

  1. Greedy Randomized Adaptive Search Procedure (Feo and Resende 1995).

  2. If the travel distance time matrix satisfies the triangular inequality, this can be done in time complexity O(l). The algorithm is similar to the linear algorithm for solving the well-known “the smallest sub-array with sum greater than a given value” problem. Using the same algorithm, one can find j values for all positions of i (Lines 21–30). This O(l) algorithm is applied even if the triangular inequality condition does not hold, sacrificing some quality.

  3. The values for Sec., #iter, and size are reported for the best solution among the 5,000 generated solutions, while the other outputs are reported according to the best result of each replication.

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Acknowledgments

The authors would like to thank the authors of Fischetti et al. (1998) and Campos et al. (2014) for sharing their results.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran

    Morteza Keshtkaran & Koorush Ziarati

Authors
  1. Morteza Keshtkaran
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  2. Koorush Ziarati
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Correspondence to Morteza Keshtkaran.

Appendices

Appendix 1: GRASP-SR parameter selection

In the construction phase of GRASP-SR (Sect. 3.1), the candidate list was restricted to the paths having an improvement within the fraction \(\alpha = 0.2\) of the profit gained through the most profitable path in the candidate list. Campos et al. (2014) presented some experiments to show that the value of 0.2 is a good choice for their work. In this appendix, we show that this value is also a good choice in this work.

We considered the 48 TSP-based problem instances of Fischetti et al. (1998) having no more than 100 vertices (16 instances in each Generation). GRASP-SR was run for different values of \(\alpha \) (0, 0.2, 0.4, 0.6, 0.8, 1.0) 100 times on each instance and the best solution of these 100 runs were kept for each instance. All the results were obtained in less than 2 min. For each value of the parameter \(\alpha \), Table 6 shows the average deviation of the solutions with respect to the optimal solutions (Dev.) and the number of optimal solutions that GRASP-SR with the given parameter has been able to find (Optimal).

The results show that when \(\alpha \) is equal to 0 or 0.2, a larger number of instances can be solved to optimality. Additionally, when \(\alpha \) is set to 0.2, the smallest average deviation of the results from the optimal solutions is obtained.

Table 6 Effect of different \(\alpha \) values on the results of GRASP-SR
Full size table

Appendix 2: Detailed results on the orienteering problem

In this section, detailed results for the TSP-based benchmark instances are presented (Tables 7,  8,  9,  10,  11 and  12). The description of the tables provided in this section was presented in Sect. 4. In addition, the values in bold indicate the best solution among the solutions of the reported exact and heuristic approaches.

Some additional notes should be considered. As Silberholz and Golden (2010) mentioned, herein the value of \(T_{max}\) has been corrected for problem gr229 to 67,301, which was incorrectly listed as 1765 in Fischetti et al. (1998). The correct value of \(T_{max}\) for problem lin318 is 21,015, but since the value of 21,045 has been used in other works, we also used this value for our experimental results. Moreover, the profit values produced by Campos et al. (2014) for the Generation 3 instances does not contain the profit value of 100 for the instances rat99, kroc100, kroe100, pr124, and krob150. Due to floating-point precision errors, for the farthest vertex from the origin of these instances, the value of 99 has been produced instead of 100. It influenced our results for instances rat99 and kroe100, resulting in a value one more than the optimal solutions reported by Campos et al. (2014). Therefore, the same profit values were used as used in Campos et al. (2014) for a fair comparison.

Table 7 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), TS (Gendreau et al. 1998b), and 2-PIA (Silberholz and Golden 2010) approaches on Generation 1 of the TSP-based benchmark instances
Full size table
Table 8 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), TS (Gendreau et al. 1998b), and 2-PIA (Silberholz and Golden 2010) approaches on Generation 2 of the TSP-based benchmark instances
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Table 9 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), TS (Gendreau et al. 1998b), and 2-PIA (Silberholz and Golden 2010) approaches on Generation 3 of the TSP-based benchmark instances
Full size table
Table 10 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), GRASP and GRASP-PR(Campos et al. 2014) approaches on Generation 1 of the TSP-based benchmark instances
Full size table
Table 11 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), GRASP and GRASP-PR(Campos et al. 2014) approaches on Generation 2 of the TSP-based benchmark instances
Full size table
Table 12 Detailed results of GRASP-SR in comparison with the exact Branch-and-Cut (BC) (Fischetti et al. 1998), GRASP and GRASP-PR(Campos et al. 2014) approaches on Generation 3 of the TSP-based benchmark instances
Full size table

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Keshtkaran, M., Ziarati, K. A novel GRASP solution approach for the Orienteering Problem. J Heuristics 22, 699–726 (2016). https://doi.org/10.1007/s10732-016-9316-7

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