Combining Penalty Function with Modified Chicken Swarm Optimization for Constrained Optimization

In many mechanical designs, such as airborne electro-optical platform, optical lenses, mechanical containers, speed reducer, and so on, lightweight design has always been our goal. Under various constraints, obtaining the minimum of some parameter is the optimization problem we often encounter in the engineering works. Chicken Swarm Optimization (CSO), a new bioinspired algorithm, is namely applied to deal with these kinds of problems. This paper firstly describes the origin and the basic model of the CSO and shows the result of applying the CSO to the algorithm test functions and a fair statistical comparison of the CSO with Bat Algorithm (BA) and modified Bat Algorithm based on Differential Evolution (DEBA) on the same test functions. Then, the CSO algorithm is modified. After that, the modified CSO is used to do the test on the previous test functions in order to be compared with the basic CSO, BA and DEBA. Finally, the modified CSO is combined with a dynamic penalty function to solve nonlinear constrained optimization problems and compared with other algorithms. From the results of all the tests, we can see that the CSO outperforms many other algorithms or their modified ones in terms of both optimization accuracy and stability. However, the modified CSO gets better performances than the CSO. As well, the modified CSO combined with penalty function is better than the CSO and many other optimization algorithms for constrained optimization problems. Introduction CSO algorithm (Meng, X.B. et al. 2014) was proposed on the Sixth International Conference on Swarm Intelligence in 2014. Through the observation of the individual and the entire flock of chickens, the researchers found that chicken has mature cognitive ability, communication skill and learning ability, and in the flock there exists an almost strict hierarchical order, similar to our teams. Flock activity exhibits complex and efficient swarm intelligence, which can be associated with the objective problem to be optimized. For example, in the optimization design for inner frame of airborne electro-optical platform, in order to improve the dynamic performance of airborne electrooptical platform and reduce the adverse effect of vibration environment on image quality, there needs to minimize the combined compliance index with the constraint that the fundamental frequency is below a certain value (Wang, P. et al. 2014). Through ideally modeling the flock structure, identity of chicken, chickens’ relationships and foraging law, researchers got this new bio-inspired algorithm. Mimicking flock activity pattern and foraging law, the CSO gathers the whole wisdom of the flock. The CSO, owning the characteristics of simplicity and good scalability, is a naturally adaptive multiple swarm algorithm. Like many swarm intelligence algorithms, the CSO is a kind of stochastic optimization algorithm, using an iterative approach to solve the objective problem. Thus it just needs low mathematical analyticity and doesn’t require that the target function is derivable. It can handle not only continuous problems but also discrete problems. Besides, its parameter configuration is simple. International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 2015) © 2015. The authors Published by Atlantis Press 1899 Mathematical Model The mathematical model of the CSO can be understood in the following way: firstly, confirm the flock structure, namely the number of the roosters, the hens, the chicks and the mother hens; secondly, set fixed identities for all of the chickens; thirdly, establish the mathematical model by chickens’ identities and their foraging laws; and fourthly, set a certain interval to update the relationship of chickens regularly. Flock Structure Suppose there are N chickens in total in the flock. The proportion of roosters and hens is rp and hp, respectively. In hens, the proportion of mother hens is mp. Assume RN, HN, CN and MN indicate the number of the roosters, the hens, the chicks and the mother hens, respectively. Then the relationship among them can be expressed as follows.