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Capacitated ring arborescence problems with profits

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Abstract

In this work, we introduce profit-oriented capacitated ring arborescence problems and present exact and heuristic algorithms. These combinatorial network design problems ask for optimized bi-level networks taking into account arc costs and node profits. Solutions combine circuits on the inner level with arborescences on the outer level of the networks. We consider the prize-collecting, the budget-constrained and the target-profit models and develop corresponding exact branch-and-cut algorithms based on mixed-integer formulations and valid inequalities. Iterated local search heuristics based on the exploration of problem-specific neighborhoods are elaborated to strengthen the upper bounds. For a set of hard literature derived instances with up to 51 nodes, we provide computational results which give evidence for the efficiency of the proposed approaches. Furthermore, we extensively analyze the performance of our methods, the obtained solution networks and the impact of the cutting planes on the obtained lower bounds.

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Notes

  1. The instances and the solutions can be requested from the corresponding author.

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Acknowledgements

We are indebted to the anonymous referees who helped to notably improve this paper through their valuable comments.

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Authors and Affiliations

  1. Department of Industrial and Manufacturing Engineering, California Polytechnic State University, 1 Grand Avenue, San Luis Obispo, CA, 93407, USA

    Alessandro Hill

  2. Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi”, University of Bologna, Via Venezia 52, 47521, Cesena, Italy

    Roberto Baldacci

  3. Faculty of Computing, University of Mato Grosso do Sul, Campo Grande, MS, Brazil

    Edna Ayako Hoshino

Authors
  1. Alessandro Hill
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  2. Roberto Baldacci
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  3. Edna Ayako Hoshino
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Corresponding author

Correspondence to Alessandro Hill.

Appendix A

Appendix A

In the following, we present the detailed results obtained by our algorithms for the used set of test instances. Each line of Table 3 is associated with a PC-RAP instance, whose number is given in column P. The numbers of given nodes (\(|V'|\),\(|U_2|\), \(|U_1|\), |W|) and the capacity bounds (q, m) are given in the corresponding columns. The bounds found by the branch-and-cut method within a one hour time limit are given in columns lb and ub. The initial objective value that was found by our heuristic is given in heu. Note that for the maximization problems (PC-RAP, BC-RAP), the start solution, obtained heuristically, gives a lower bound, whereas for TP-RAP, we obtain an upper bound. Column \(\delta \) contains the optimality gap (\(100(ub-lb)/lb\)) and the collected profit is given in a dedicated column. Whenever \(\delta =0\) the corresponding instance has been solved to optimality. The number of branching nodes and run time in seconds is given in column \(\#\) and column t, respectively. Results for BC-RAP and TP-RAP are listed in Table 4. In addition to the information described above, the budget (B), target profit (T), and the sum of the arc costs in the best integer feasible solution (costs) are provided.

Table 3 Instance properties, and results for PC-RAP
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Table 4 Results for BC-RAP and TP-RAP instances
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Hill, A., Baldacci, R. & Hoshino, E.A. Capacitated ring arborescence problems with profits. OR Spectrum 41, 357–389 (2019). https://doi.org/10.1007/s00291-018-0539-x

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