Chaotic Rough Particle Swarm Optimization Algorithms

The problem of finding appropriate representations for various is a subject of continued research in the field of artificial intelligence and related fields. In some practical situations, mathematical and computational tools for faithfully modeling or representing systems with uncertainties, inaccuracies or variability in computation should be provided; and it is preferable to develop models that use ranges as values. A need to provide tolerance ranges and inability to record accurate values of the variables are examples of such a situation where ranges of values must be used (Lingras, 1996). Representations with ranges improve data integrity for non-integral numerical attributes in data storage and would be preferable due to no lose of information. Rough patterns proposed by Lingras are based on an upper and a lower bound that form a rough value that can be used to effectively represent a range or set of values for variables such as daily weather, stock price ranges, fault signal, hourly traffic volume, and daily financial indicators (Lingras, 1996; Lingras & Davies, 2001). The problems involving, on input/output or somewhere at the intermediate stages, interval or, more generally, bounded and set-membership uncertainties and ambiguities may be overcome by the use of rough patterns. Further developments in rough set theory have shown that the general concept of upper and lower bounds provide a wider framework that may be useful for different types of applications (Lingras & Davies, 2001). Generating random sequences with a long period and good uniformity is very important for easily simulating complex phenomena, sampling, numerical analysis, decision making and especially in heuristic optimization. Its quality determines the reduction of storage and computation time to achieve a desired accuracy. Chaos is a deterministic, random-like process found in non-linear, dynamical system, which is non-period, non-converging and bounded. Moreover, it has a very sensitive dependence upon its initial condition and parameter (Schuster, 1998). The nature of chaos is apparently random and unpredictable and it also possesses an element of regularity. Mathematically, chaos is randomness of a simple deterministic dynamical system and chaotic system may be considered as sources of randomness. Chaotic sequences have been proven easy and fast to generate and store, there is no need for storage of long sequences (Heidari-Bateni & McGillem, 1994). Merely a few functions (chaotic maps) and few parameters (initial conditions) are needed even for very long sequences. In addition, an enormous number of different sequences can be generated simply