**Performance Study of Cultural Algorithms Based on Genetic Algorithm with Single and Multi Population for the MKP**

Deam James Azevedo da Silva1, Otávio Noura Teixeira2 and Roberto Célio Limão de Oliveira1 *1Universidade Federal do Pará (UFPA), 2Centro Universitário do Estado do Pará (CESUPA) Brazil* 

### **1. Introduction**

Evolutionary Computation (EC) is inspired from by evolution that explores the solution space by gene inheritance, mutation, and selection of the fittest candidate solutions. Since their inception in the 1960s, Evolutionary Computation has been used in various hard and complex optimization problems in search and optimization such as: combinatorial optimization, functions optimization with and without constraints, engineering problems and others (Adeyemo, 2011). This success is in part due to the unbiased nature of their operations, which can still perform well in situations with little or no domain knowledge (Reynolds, 1999). The basic EC framework consists of fairly simple steps like definition of encoding scheme, population generation method, objective function, selection strategy, crossover and mutation (Ahmed & Younas, 2011). In addition, the same procedures utilized by EC can be applied to diverse problems with relatively little reprogramming.

Cultural Algorithms (CAs), as well as Genetic Algorithm (GA), are evolutionary models that are frequently employed in optimization problems. Cultural Algorithms (CAs) are based on knowledge of an evolutionary system and were introduced by Reynolds as a means of simulating cultural evolution (Reynolds, 1994). CAs algorithms implements a dual mechanism of inheritance where are inherited characteristics of both the level of the population as well as the level of the area of belief space (culture). Algorithms that use social learning are higher than those using individual learning, because they present a better and faster convergence in the search for solutions (Reynolds, 1994). In CAs the characteristics and behaviors of individuals are represented in the Population Space. This representation can support any population-based computational model such as Genetic Algorithms, Evolutionary Programming, Genetic Programming, Differential Evolution, Immune Systems, among others (Jin & Reynolds, 1999).

Multidimensional Knapsack Problem (MKP) is a well-known nondeterministic-polynomial time-hard combinatorial optimization problem, with a wide range of applications, such as cargo loading, cutting stock problems, resource allocation in computer systems, and

Performance Study of Cultural Algorithms Based on

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MOEAs.

Genetic Algorithm with Single and Multi Population for the MKP 387

cost (Ahmed & Younas, 2011). Spillman's experiment, which applies the GA to the knapsack problem, shows that the GA does not have a good performance in relatively small size problem, but works quite well in problems that include a huge number of elements (Spillman, 1995). There are many packing problems where evolutionary methods have been applied. The simplest optimization problem and one of the most studied is the onedimensional (zero–one or 0-1) knapsack problem (Ahmed & Younas, 2011), which given a knapsack of a certain capacity, and a set of items, each one having a particular size and value, nds the set of items with maximum value which can be accommodated in the knapsack. Various real-world problems are of this type: for example, the allocation of communication channels to customers who are charged at different rates (Back et al.,

During a study of 0-1 knapsack, a number of extensions and variants have been developed such as (Ahmed & Younas, 2011): Multiple Knapsack Problems (MKP), Multidimensional Knapsack Problems (MDKP), Multi Choice Knapsack Problems (MCKP) and Multiple Choice Multidimensional Knapsack Problems (MMKP). It is also important to consider other extensions such as (Chu & Beasley, 1998): Multiconstraint Knapsack Problem, and also the term "Multidimensional Zero-one Knapsack Problem". Using alternative names for the same problem is potentially confusing, but since, historically, the designation **MKP** has been the most widely used (Chu & Beasley, 1998). Consequently, Multidimensional Knapsack Problem (MKP) is the designation selected for this work. In our previous research it was introduced a Multi Population Model on the cultural structure identified as "Multi Population Cultural Genetic Algorithm" (MCGA) (Silva & Oliveira, 2009). In MCGA model several sub-populations are connected with as ring structure, where the migration of individuals occurs after a generation interval (according to the migration based on parameter interval) with best-worst migration policy implementation. The results were satisfactory in relation to other algorithms in the literature. In another research two versions of Distributed GA (DGA) are presented as follows: standard Distributed GA (DGA) and an improved DGA (DGA-SRM), which two genetic operators are applied in parallel mode to create offspring. The term SRM represents "Self-Reproduction with Mutation", that is applied to various 0/1 multiple knapsack problems so as to improve the search performance (Aguire et al., 2000). Hybridization of memetic algorithms with Branch-and-Bound techniques (BnB) is also utilized for solving combinatorial optimization problems (Gallardo et al., 2007). BnB techniques use an implicit enumeration scheme for exploring the search space in an "intelligent" way. Yet another research utilizes adaptive GA for 0/1 Knapsack problems where special consideration is given to the penalty function where constant and self-adaptive penalty functions are adopted (Zoheir, 2002). Fitness landscape analysis techniques are used to better understand the properties of different representations that are commonly adopted when evolutionary algorithms are applied to MKP (Tavares et al., 2008). Other investigation utilizes multiple representations in a GA for the MKP (Representation-Switching GA) know as RSGA (Fukunaga, 2009). Other recent works consider two heuristics and utilize them for making comparisons to the well-known multiobjective evolutionary algorithms (MOEAs) (Kumar & Singh, 2010). While comparing MOEAs with the two heuristics, it was observed that the solutions obtained by the heuristics are far superior for larger problem instances than those obtained by

economics (Tavares et al., 2008). MKP has received wide attention from the operations research community, because it embraces many practical problems. In addition, the MKP can be seen as a general model for any kind of binary problems with positive coefficients (Glover & Kochenberger, 1996).

Many researchers have proposed the high potential of the hybrid-model for the solution of problems (Gallardo et al., 2007). The algorithms presented in this work to solve MKP are a combination of CAs with a Multi Population model. The Multi Population model is the division of a population into several smaller ones, usually called the island model. Each sub-population runs a standard sequential evolution proceeds, as if it were isolated from the rest, with occasional migration of individuals between sub-populations (Tomassini, 2005).

In order to conduct an investigation to discover improvements for MKP, this work is centered in the knowledge produced from CAs through the evolutionary process that utilizes a population-based Genetic Algorithm model, using various MKP benchmarks found in the literature. In addition, there is an interest in investigating how to deal with the Cultural Algorithms considering a population-based in Genetic Algorithms.

So as to compare test results, we implemented the follows algorithms: the standard cultural algorithm with Single Population (also known as standard CA or CA-S) and Cultural Algorithm with Multi Population defined as CA-IM with two versions: CA-IM\_1 which has fixed values for genetic operators (recombination and mutation) and CA-IM\_2 which does not have fixed values for genetic operators because these values are generated randomly. In order to evaluate the performance of the CA-IM algorithms, some comparison testing will be conducted with other two algorithms based on Distributed GA, called DGA and DGA-SRM (Aguirre et al., 2000).

The outline of the paper is as follows: in Section 2, a description with formal denition of the MKP problem and an overview of Cultural Algorithms are presented. Section 3 shows an alternative approach that explores the multi population model with Cultural Algorithms and explores how the interaction process occurs among various sub-populations. Our experimental results are shown in Section 4 and finally we show some conclusions in Section 5.
