Pareto Optimization of a Half Car Passive Suspension Model Using a Novel Multiobjective Heat Transfer Search Algorithm

Hindawi Modelling and Simulation in Engineering Volume 2017, Article ID 2034907, 17 pages https://doi.org/10.1155/2017/2034907 Research Article Pareto Optimization of a Half Car Passive Suspension Model Using a Novel Multiobjective Heat Transfer Search Algorithm Vimal Savsani,1 Vivek Patel,1 Bhargav Gadhvi,2 and Mohamed Tawhid3 1 Mechanical Engineering Department, School of Technology, Pandit Deendayal Petroleum University, Gandhinagar, Gujarat 382007, India 2 Simon Fraser University, Burnaby, BC, Canada 3 Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, BC, Canada Correspondence should be addressed to Vimal Savsani; Received 16 August 2016; Revised 18 January 2017; Accepted 24 January 2017; Published 3 May 2017 Academic Editor: Mohamed B. Trabia Copyright © 2017 Vimal Savsani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Most of the modern multiobjective optimization algorithms are based on the search technique of genetic algorithms; however the search techniques of other recently developed metaheuristics are emerging topics among researchers. This paper proposes a novel multiobjective optimization algorithm named multiobjective heat transfer search (MOHTS) algorithm, which is based on the search technique of heat transfer search (HTS) algorithm. MOHTS employs the elitist nondominated sorting and crowding distance approach of an elitist based nondominated sorting genetic algorithm-II (NSGA-II) for obtaining different nondomination levels and to preserve the diversity among the optimal set of solutions, respectively. The capability in yielding a Pareto front as close as possible to the true Pareto front of MOHTS has been tested on the multiobjective optimization problem of the vehicle suspension design, which has a set of five second-order linear ordinary differential equations. Half car passive ride model with two different sets of five objectives is employed for optimizing the suspension parameters using MOHTS and NSGA-II. The optimization studies demonstrate that MOHTS achieves the better nondominated Pareto front with the widespread (diveresed) set of optimal solutions as compared to NSGA-II, and further the comparison of the extreme points of the obtained Pareto front reveals the dominance of MOHTS over NSGA-II, multiobjective uniform diversity genetic algorithm (MUGA), and combined PSO-GA based MOEA. 1. Introduction In recent time, the multiobjective evolutionary algorithms (MOEAs) have gained enormous attention in solving the engineering optimization problems with more than one objective. The multi/many objective optimization problems (MOOPs) differ from their single objective counterparts (SOOPs) in terms of both their problem definitions/statements and methods to solve such problems. MOOPs have the conflicting objectives to be optimized simultaneously and solving these yields a set of optimal or trade-off or Pareto solutions (Pareto front points), whereas SOOPs have a single objective at a given time and the solution is usually a single optimal point. The classical methods such as calculus based methods, gradient based methods, and elimination and interpolation methods and the like for solving SOOPs and methods, for instance, weighted sum method and 𝜀 constraint method for solving MOOPs are limited to the problems with simple objective functions and constraints. This is due to their nature to get stuck to a suboptimal solution, their convergence dependency on initial guess, and their unsuitability in solving a large variety of optimization problems [1]. Alternatively evolutionary algorithms (EAs), or metaheuristics, have had a remarkable success in finding the global optimum of complex problems nearly in all the disciplines of the knowledge. This success of EAs and their nature of using a population of solutions had led the researchers to employing the search techniques of EAs to optimize the MOOPs. Such algorithms to solve MOOPs are referred to as multiobjective evolutionary algorithms (MOEAs) when they use EAs as their basic search techniques and are referred to as simply multiobjective optimization algorithms (MOOAs) when they use any metaheuristics in general. Primarily all the methods to solve MOOPs have two goals to 2 attain. The first is to find the Pareto front solutions as close as possible to the optimal Pareto front and the second is to maintain diversity among the optimal set of solutions [1]. One of the oldest attempts to have employed the EA to form a population based multiobjective evolutionary algorithm was by Schaffer [2] and the method is known as vector evaluated genetic algorithm (VEGA). VEGA does the selection for each objective separately [3], but its incapability in finding all the Pareto front solutions had limited its applications to very few real world optimization problems. These demerits of VEGA were overcome by algorithms which were inspired by the work of Goldberg and Holland [4], such as multiobjective genetic algorithm (MOGA) by Knowles and Corne [5], nondominated sorting genetic algorithm (NSGA) by Srinivas and Deb [6], and niched Pareto genetic algorithm (NPGA) by Horn et al. [7]. These new algorithms introduced the concept of Pareto optimality into the selection process which awarded them a great success in application to various disciplines of engineering as thoroughly described in [8]. Consequently the researchers had started experimenting various ways in assigning fitness values to the populations and in maintaining the diversity of optimal points which led to the development of new state-of-the-art MOEAs, for example, strength Pareto evolutionary algorithm (SPEA2 and SPEA) by Zhou et al. [9] and by Zitzler and Thiele [3], Pareto archived evolution strategy (PAES) by Knowles and Corne [5], Pareto envelope-based selection algorithm (PESA and PESA-II) by Corne et al. [10] and Horn et al. [7], an elitist based nondominated sorting genetic algorithm-II (NSGAII) by Deb et al. [11], and a multiobjective evolutionary algorithm based on decomposition (MOEA/D) by Zhang and Li [12] and its improved versions described in [9]. In addition, swarm intelligence based search algorithms for multiobjective optimization [13–15] have also been developed and applied to a variety of MOOPs. Several more recently developed multiobjective algorithms [16–19] based on the search techniques other than the ones used in genetic algorithm [4] or EAs and particle swarm optimization [20] have also shown a great potential in obtaining the Pareto front closest to the true Pareto front. In this study a nonlinear MOOP of optimizing the vehicle suspension system design having conflicting objectives is optimized. The goal is to optimize the design of passive suspension system of the passenger road vehicle by using a 2-dimensional pitch-heave ride mod (...truncated)