**7. References**


402 Bio-Inspired Computational Algorithms and Their Applications

Table 7 shows the results from standard Cultural Algorithm (CA-S) that utilizes single population. According to results, the CA-S reaches optimum average for 100 runs only for Sento1 and Weing7. However, the results from CA-S for Petersen6, Pertersen7 and Sento2

This work presented a Cultural Algorithm (CA) with single population (CA-S) and multi population (CA-IM) in order to improve the search performance on MKP. It was observed that CA-S improves the convergence reliability and search speed. However, CA-S is not enough to reach global optimum for most problems presented. Our cultural algorithm implementation with island model (CA-IM\_1 and CA-IM\_2) allows the migration among islands sub-populations and main population through belief space structures that represent

The results have shown that the CA-IM\_1 is better than CA-IM\_2 for the benchmarks selected. The results have also shown that the CA-IM\_1 and CA-IM\_2 perform the optimum search and reach optimum values equally or above the ones reached by algorithms DGA and DGA-SRM that were chosen for comparison. The positive results obtained, give support the idea that this is a desirable approach for tackling highly constrained NP-complete problems such as the MKP. In addition, it is possible that the hybridization of cultural algorithms based on population of GA with local search techniques improves the results obtained by standard CAs. In a future work, a study will be done about the behavior of the sub-populations that are eliminated and recreated randomly. In addition a local search will be implemented to CAs as much for standard CA (single population) as for CA-IM (multi

This research was supported by the CAPES (Coordenação de Aperfeiçoamento do Pessoal de Ensino Superior, Brazil) and by FAPESPA (Fundação de Amparo à Pesquisa do Estado

Adeyemo, J.A. (2011). Reservoir Operation using Multi-objective Evolutionary Algorithms-

Agapie, A., Fagarasan, F. & Stanciulescu, B. (1997). A Genetic Algorithm for a Fitting

Aguirre, H. E. & Tanaka, K. (2006). A Model for Parallel Operators in Genetic Algorithms,

Aguirre, H. E., Tanaka, K., Sugimara, T. & Oshita, S. (2000). Improved Distributed Genetic

A Review, In: *Asian Journal of Scientific Research*, Vol.*4*, No. 1, pp.16-27, February

Problem, In: *Nuclear Instruments & Methods in Physics Research Section A*, Vol. 389,

In: *Springer Book Series Studies in Computational Intelligence, Parallel Evolutionary Computations* , Nedjah, N., Alba, E. & Macedo M., L., pp.3-31, Springer, ISBN

Algorithm with Cooperative-Competitive Genetic Operators, In: *Proc. IEEE Int.* 

outperform the results presented by DGA-SRM.

the cultural knowledge available in Cultural Algorithms.

population) so as to verify improvements on these algorithms.

No. 1-2, April 1997, pp. 288-292, ISSN 0168-9002.

9783540328391, Berlin Heidelberg.

**5. Conclusion** 

**6. Acknowledgments** 

2011, ISSN 19921454.

do Pará, Brazil).

**7. References** 

*Conf. on Systems, Man, and Cybernetics*, Vol.5, ISBN 0-7803-6583-6, pp. 3816-3822, Nashville, TN, USA, October 8-11 2000.


**21** 

**Using a Genetic Algorithm to Solve** 

The capacitated plant location problem (CPL) consists of locating a set of potential plants with capacities, and assigning a set of customers to these plants. The objective is to minimize the total fixed and shipping costs while at the same time demand of all the customers can be satisfied without violating the capacity restrictions of the plants. The CPL is a well-known combinatorial optimization problem and a number of decision problems can be obtained as special cases of CPL. There are substantial numbers of heuristic solution algorithms proposed in the literature (See Rolland et al., 1996; Holmberg & Ling, 1997; Delmaire et al., 1999; Kratica et al., 2001; He et al., 2003; Uno et al., 2005). As well, exact solution methods have been studied by many authors. These include branch-and-bound procedures, typically with linear programming relaxation (Van Roy & Erlenkotter, 1982; Geoffrion & Graves, 1974) or Lagrangiran relaxation (Cortinhal & Captivo, 2003). Van Roy (1986) used the Cross decomposition which is a hybrid of primal and dual decomposition algorithm, and Geoffrion & Graves (1974) considered Benders' decomposition to solve CPL problem. Unlike many other mixed-integer linear programming applications, however, Benders decomposition algorithm was not successful in this problem domain because of the difficulty of solving the master system. In mixed-integer linear programming problems, where Benders' algorithm is most often applied, the master problem selects values for the integer variables (the more difficult decisions) and the subproblem is a linear programming problem which selects values for the continuous variables (the easier decisions). If the constraints are explicit only in the subproblem, then the master problem is free of explicit constraints, making it more amenable to solution by genetic algorithm (GA). The fitness function of the GA is, in this case, evaluated quickly and simply by evaluating a set of linear functions. In this chapter, therefore, we discuss about a hybrid algorithm (Lai et al., 2010) and its implementation to overcome the difficulty of Benders' decomposition. The hybrid algorithm is based on the solution framework of Benders' decomposition algorithm, together with the use of GA to effectively reduce the computational difficulty. The rest of

**1. Introduction** 

 \*

Corresponding Author

**the Benders' Master Problem** 

Ming-Che Lai1 and Han-suk Sohn2,\*

*1Yu Da University,* 

*1Taiwan 2USA* 

*2New Mexico State University* 

**for Capacitated Plant Location** 

