A Culture-Based Particle Swarm Optimization Framework for Dynamic, Constrained Multi-Objective Optimization

Introduction

Biological groups of living beings exhibit natural tendencies to flock for mutual benefits. Individuals of such flocks also accrue knowledge over time. This knowledge is often shared and exchanged as they interact. Researchers have been devoted to develop a framework that mimics the interactions and cultures in a society, namely the way individuals strive to reach their best performance through inspiration from leaders and the way beliefs arise and are strengthened based on social acceptance of good and bad (Daneshyari & Yen, 2011). The beliefs may be to follow acknowledged leaders, a list of accepted social norms, collective information on trends in society, historical information of failures, successes and etc. These beliefs affect the day-to-day activities of an individual and decide the paths taken. Such beliefs metamorphose with time. Along with this, there is always an inclination of individuals to go out of the ordinary path when faced adversity.

The aim here is effective utilization of all such knowledge for solving real-world problems that are dynamic, constrained, and multi-objective in nature. The idea of what is optimum may vary with time and under different environments- hence arises the dynamism in the problem formulation. The problem is often modeled as a minimization problem without loss of generality (as opposed to a maximization problem):min(1) where x=[x₁,x₂,…,x_n]ϵRⁿ and subject to inequality and equality constraints respectively as

The bounds on the decision space are:

The aforementioned problem is the simultaneous minimization of r objective functions within an n dimensional decision space. The parameter specifies the dynamic optimization problems where the fitness functions may vary sporadically or periodically due to environmental changes. It is worth noting that the magnitude of changes need not be the same for all objective functions. It is restricted by a total of p constraints, of which m are inequality constraints and p − m equality ones. The bounds define the upper and lower limits of the decision space. Different x values may result in feasible or infeasible solutions. A feasible individual is one which satisfies all constraints while an infeasible individual on the contrary is one that violates at least one constraint. Both these x values defined belong to the Search Space, S. The aim is to find values of within S at which are feasible as well as optimal (in the context of Pareto optimality).

The proposed hybrid approach is referred to as Culture based Constrained Dynamic Multiple Objective Particle Swarm Optimization (Culture-CDMOPSO). Culture-CDMOPSO exploits heuristics to minimize the function based on the ‘fitness’ with the uncertainties and constraints, using a variant of Cultural Algorithms (CA) (Reynolds & Peng, 2005). CA provides the overall organization of knowledge via a systematic and consolidated set of heuristic information gained through searching the hyperspace using a varied set of data structures maintained on the lines of the working philosophy of cultures and societal interactions. Particle Swarm Optimization (PSO) (Kennedy & Eberhart, 1995) provides the searching heuristics, mimicking the interactions amongst a swarm of birds flying together to reduce the aerodynamic efforts of flight and the search time of the common destination. Parts of the stated problem have been addressed individually. Yet, no literature exists to solve the stated problem in its entirety.