# The incorporation of fixed cost and multilevel capacities into the discrete and continuous single source capacitated facility location problem

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## Abstract

In this study we investigate the single source location problem with the presence of several possible capacities and the opening (fixed) cost of a facility that is depended on the capacity used and the area where the facility is located. Mathematical models of the problem for both the discrete and the continuous cases using the Rectilinear and Euclidean distances are produced. Our aim is to find the optimal number of open facilities, their corresponding locations, and their respective capacities alongside the assignment of the customers to the open facilities in order to minimise the total fixed and transportation costs. For relatively large problems, two solution methods are proposed namely an iterative matheuristic approach and VNS-based matheuristic technique. Dataset from the literature is adapted to assess our proposed methods. To assess the performance of the proposed solution methods, the exact method is first applied to small size instances where optimal solutions can be identified or lower and upper bounds can be recorded. Results obtained by the proposed solution methods are also reported for the larger instances.

## Keywords

Discrete and continuous capacitated location Opening fixed cost Mathematical formulation Matheuristic VNS## Notes

### Acknowledgements

We would like to thank the referees for their interesting suggestions that improved both the content as well as the presentation of the paper. The third author is supported in part by the Spanish Ministry of Economy and Competitiveness Research Project MTM2015-70260-P.

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