Comparative Analysis of Genetic Crossover Operators in Knapsack Problem

: The Genetic Algorithm (GA) is an evolutionary algorithms and technique based on natural selections of individuals called chromosomes. In this paper, a method for solving Knapsack problem via GA (Genetic Algorithm) is presented. We compared six different crossovers: Crossover single point, Crossover Two point, Crossover Scattered, Crossover Heuristic, Crossover Arithmetic and Crossover Intermediate. Three different dimensions of knapsack problems are used to test the convergence of knapsack problem. Based on our experimental results, two point crossovers (TP) emerged the best result to solve knapsack problem. ©JASEM

The knapsack problem (KP) has been used in many real life problem such as investment decision making (Peeta, 2010), project selection (Mavrotas, 2008) and (Hartvigsen, 2006) applied it in vote-trading problem.The Knapsack problem can be defined as a set of items, each with a weight(w) and a profit(p), determine the number(n) of each item to include in a collection(j) so that the total weight is less than or equal to a given limit and the total profit(p) is as large as possible.Mathematically it can be represented as follows: The difficulty of the problem is caused by the integrality requirement of equation (3

MATERIALS AND METHODS
In GA Crossover operators is used to divide a pair of selected chromosomes into two or more parts.It consists of combining the chromosomes of two parents to produce a new offspring (child).The reason behind using crossover is that the new chromosomes being formed (child) may be better than both of the parents, if it takes the best chromosomes from both parents.For the purpose of this work, the following Crossover will be use:

Single point Crossover (SP)
A single point crossover involves the two mating chromosomes (parent) are cut once at corresponding points and the selection after the cuts exchanged.Fig. 1 below shows the single point crossover (SP).The shaded area is the crossover point. Parent Heuristic crossover (HE), produces an offspring of the two parents which lies a small distance away from the parent with better fitness value in the direction away from the parent with the worse fitness value. ℎ Arithmetic Crossover (AM): In Arithmetic crossover (AC), it produces an offspring (child) that are weighted arithmetic mean of two parents,is random value between [0,1].If parent 1 and parent 2 are the Parents, and parent 1 has the better fitness value, the function returns a child (offspring) Crossover Scattered (SC): Crossover scattered (SC) creates a random binary chromosomes and selects the genes where the chromosome is 1 from the first parent, and the genes where the chromosome is 0 from the second parent and later combines the genes to form a child.Figure 3 below shows the scattered crossover.

RESULTS AND DISCUSION
In this study, we shall be using three different dimensions (5,10,15) of Knapsack problem that were used by (kaushik Kumar, 2014).All parameters used in this study are given Table 1 below:

Table I :
Parameters of Genetic Algorithm

Table 2 :
From Table II below heuristic crossover (HE), arithmetic crossover (AM) and intermediate crossover always stuck on local maximums in most cases especially in all the three dimensions used in this paper.Moreover, two point crossover, single point crossover and scattered crossover never stuck on local maximums and they all reach the global maximum point in all the three dimensions.Overall, two point crossover (TP) ranks the best among all other crossover in terms of the averages of mean and standard deviation, followed by scattered (SC) and single point (SP).Results of SP, TP, HE,SC, IT and AM HAKIMI, D; OYEWOLA, DO; YAHAYA, Y; BOLARIN, G