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Formulations for the orienteering problem with additional constraints

Abstract

This paper addresses a variant of the Orienteering Problem taking into account mandatory visits and exclusionary constraints (conflicts among nodes). Five mixed integer linear formulations are adapted from the Traveling Salesman Problem literature in order to provide a robust formulation for this problem. The main difference among these formulations lies in the way they deal with the subtour elimination constraints. The performance of the proposed formulations is evaluated over a large set of instances. Computational results reveal that the model that avoids subtours by means of a single-commodity flow formulation allows to solve to optimality more instances than the other formulations, within a time limit of 1 h.

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Acknowledgements

We sincerely thank to CONACYT, UANL-PAICYT 2015, and DGIIP of Universidad Técnica Federico Santa María (Grant USM 28.15.20) for their support to this work. Thanks are due to the anonymous referees for their valuable comments.

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Correspondence to M. Angélica Salazar-Aguilar.

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Appendix 1

Appendix 1

Tables 4, 5, 6, 7, 8, 9, 10, 11, and 12 display the results obtained by CPLEX for each instance of the OPMVC, by using the OPMVC-DL, OPMVC-GG, OPMVC-W, OPMVC-DFJ, and OPMVC-C formulations. Each table contains the following columns:

Table 4 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 1
Table 5 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 2
Table 6 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 3
Table 7 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 4
Table 8 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 5
  • Instance: Name of the instance

  • z: Objective function value. This cell contains the objective function value of a feasible solution (for models OPMVC-DL, OPMVC-GG, and OPMVC-W) or an upper bound (for models OPMVC-DFJ and OPMVC-C). A number followed by an \(^*\) indicates that the solver reached the optimal solution but it was not able to prove its optimality. This cell is empty if the solver did not find an integer solution or an upper bound, respectively, within the time limit.

  • \(\sum y\): Number of nodes visited in the feasible solution. The cell is empty is the solver did not report a feasible solution within the time limit.

  • Time: Running time in seconds. If the running time is smaller than the limit (3600 seconds), then the reported solution is optimal. If the solver ran out of time, this cell contains “>3600”.

Table 9 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 6
Table 10 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 7
Table 11 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 8
Table 12 Best integer solution values and upper bounds reported by CPLEX (after 1 h) for instances in Class 9

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Palomo-Martínez, P.J., Salazar-Aguilar, M.A. & Albornoz, V.M. Formulations for the orienteering problem with additional constraints. Ann Oper Res 258, 503–545 (2017). https://doi.org/10.1007/s10479-017-2408-4

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Keywords

  • Orienteering problem
  • Subtour elimination constraints
  • Exclusionary constraints
  • Selective traveling salesman problem