# Bi-objective location problem with balanced allocation of customers and Bernoulli demands: two solution approaches

## Abstract

A bi-objective stochastic capacitated multi-facility location–allocation problem is presented where the customer demands have Bernoulli distributions. The capacity of a facility for accepting customers is limited so that if the number of allocated customers to the facility is more than its capacity, a shortage will occur. The problem is formulated as a bi-objective mathematical programming model. The first objective is to find optimal locations of facilities among potential locations and optimal allocations of stochastic customers to the facilities so that the total sum of fixed costs of establishment of the facilities and the expected values of servicing and shortage costs is minimized. The second objective is to balance the number of allocated customers to the facilities. To solve small problems, the augmented \(\varepsilon \)-constraint method is used. Also, two metaheuristic solution approaches, non-dominated sorting genetic algorithm II (*NSGA-II*) and controlled elitist non-dominated sorting genetic algorithm II (*CNSGA-II*), are presented for solving large problems. Several sample problems are generated and with various criteria are tested to show the performance of the proposed model and the solution approaches.

## Keywords

Capacitated location–allocation problem Bi-objective model Stochastic demands Augmented \(\varepsilon \)-constraint method Metaheuristic algorithm## Notes

### Acknowledgements

The first author acknowledges University of Garmsar and the second author thanks Sharif University of Technology for supporting this work.

### Compliance with ethical standards

### Conflict of interest

All authors declares that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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