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Parallel algorithms for continuous multifacility competitive location problems

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Abstract

We consider a continuous location problem in which a firm wants to set up two or more new facilities in a competitive environment. Both the locations and the qualities of the new facilities are to be found so as to maximize the profit obtained by the firm. This hard-to-solve global optimization problem has been addressed in Redondo et al. (Evol. Comput.17(1), 21–53, 2009) using several heuristic approaches. Through a comprehensive computational study, it was shown that the evolutionary algorithm uego is the heuristic which provides the best solutions. In this work, uego is parallelized in order to reduce the computational time of the sequential version, while preserving its capability at finding the optimal solutions. The parallelization follows a coarse-grain model, where each processing element executes the uego algorithm independently of the others during most of the time. Nevertheless, some genetic information can migrate from a processor to another occasionally, according to a migratory policy. Two migration processes, named Ring-Opt and Ring-Fusion2, have been adapted to cope the multiple facilities location problem, and a superlinear speedup has been obtained.

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Abbreviations

i :

Index of demand points, i = 1, . . . , n.

j :

Index of existing facilities, j = 1, . . . , m (the first k of those m facilities belong to the chain).

l :

Index of new facilities, l = 1, . . . , p.

z l :

Location of the l-th new facility, z l  = (x l , y l ).

α l :

Quality of the l-th new facility (α l > 0).

nf l :

Variables of the l-th facility, nf l  = (z l , α l ).

nf :

Variables of the problem, nf = (nf 1, . . . , nf p ).

p i :

Location of the i-th demand point.

w i :

Demand (or buying power) at p i .

f j :

Location of the j-th existing facility.

\({d_{i}^{\rm min}}\) :

Minimum distance from p i at which the new facilities can be located.

d ij :

Distance between p i and f j .

α ij :

Quality of f j as perceived by p i .

g i (·):

A non-negative non-decreasing function.

\({{\alpha_{ij}} / {g_i(d_{ij})}}\) :

Attraction that p i feels for f j .

γ i :

Weight for the quality of the new facilities as perceived by demand point p i .

S :

Region of the plane where the new facilities can be located.

q min :

Minimum allowed quality for the new facilities.

q max :

Maximum allowed quality for the new facilities.

\({d_{iz_l}}\) :

Distance between p i and z l .

\({{\gamma_i\alpha_l} / {g_i(d_{iz_l})}}\) :

Attraction that p i feels for nf l .

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Authors and Affiliations

  1. Department of Computer Architecture and Electronics, University of Almería, Almería, Spain

    J. L. Redondo, I. García & P. M. Ortigosa

  2. Department of Statistics and Operations Research, Faculty of Mathematics, University of Murcia, 30100, Espinardo–Murcia, Spain

    J. Fernández

Authors
  1. J. L. Redondo
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  2. J. Fernández
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  3. I. García
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  4. P. M. Ortigosa
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Corresponding author

Correspondence to J. Fernández.

Additional information

This work has been funded by grants from the Spanish Ministry of Science and Innovation (TIN2008-01117, ECO2008-00667/ECON) and Junta de Andalucía (P06-TIC-01426, P08-TIC-3518), in part financed by the European Regional Development Fund (ERDF).

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Redondo, J.L., Fernández, J., García, I. et al. Parallel algorithms for continuous multifacility competitive location problems. J Glob Optim 50, 557–573 (2011). https://doi.org/10.1007/s10898-009-9455-6

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