Journal of Applied Mathematics

Volume 2015, Article ID 165601, 12 pages

http://dx.doi.org/10.1155/2015/165601

## Multiobjective Optimization Method Based on Adaptive Parameter Harmony Search Algorithm

^{1}K.L.N College of Engineering, Pottapalayam 630611, India^{2}Sri Krishna College of Technology, Coimbatore 641 042, India

Received 17 October 2014; Revised 8 January 2015; Accepted 8 January 2015

Academic Editor: Zong Woo Geem

Copyright © 2015 P. Sabarinath 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.

#### Abstract

The present trend in industries is to improve the techniques currently used in design and manufacture of products in order to meet the challenges of the competitive market. The crucial task nowadays is to find the optimal design and machining parameters so as to minimize the production costs. Design optimization involves more numbers of design variables with multiple and conflicting objectives, subjected to complex nonlinear constraints. The complexity of optimal design of machine elements creates the requirement for increasingly effective algorithms. Solving a nonlinear multiobjective optimization problem requires significant computing effort. From the literature it is evident that metaheuristic algorithms are performing better in dealing with multiobjective optimization. In this paper, we extend the recently developed parameter adaptive harmony search algorithm to solve multiobjective design optimization problems using the weighted sum approach. To determine the best weightage set for this analysis, a performance index based on least average error is used to determine the index of each weightage set. The proposed approach is applied to solve a biobjective design optimization of disc brake problem and a newly formulated biobjective design optimization of helical spring problem. The results reveal that the proposed approach is performing better than other algorithms.

#### 1. Introduction

Engineering design optimization has recently received lot of attention from designers so as to produce better designs. It saves time, cost, and energy involved. Engineering design optimization often deals with many design objectives under nonlinear and complex constraints. Also the design problem is subjected to constraints limited by cost, weight, and material properties like strength, design specifications, and availability of resources. The design objectives are often conflicting in nature and hence finding the true Pareto optimal front is difficult. Conventionally, numerous mathematical methods such as linear, nonlinear, dynamic, and geometric programming have been developed in the past to solve engineering design optimization problems. But these methods have many drawbacks and no single method is found suitable for solving all types of engineering design optimization problems.

Even for a single objective design optimization problem involving large number of nonlinear design variables, arriving at a global best solution is not an easy task. The shortcoming of mathematical methods gives way for metaheuristics algorithm to solve engineering design optimization problems. Metaheuristics algorithms have proven to be powerful in solving this kind of optimization since they combine rules and randomness to mimic natural phenomena such as biological systems (Genetic Algorithm), animal behavior (ant algorithm, tabu search), and physical annealing process (Simulated Annealing). GA is a popular metaheuristics algorithm which has been broadly applied to solve various design optimization problems and proven to be successful in finding the global best than traditional methods. These include the design optimization of machine elements such as gears [1], gearbox [2], journal bearing [3], magnetic thrust bearing [4], rolling element bearing [5], and automotive wheel bearing unit [6]. Swarm intelligence is another important concept in many recent metaheuristics algorithms such as particle swarm optimization, firefly algorithm, artificial bee colony algorithm, and cuckoo search.

Particle swarm optimization mimics the group behavior of birds and fish schools [7]. Sabarinath et al. [8] have applied PSO algorithm for solving optimal design of belt pulley system. Yang [9] developed firefly algorithm that imitates the flashing behavior of tropic firefly swarms. Yang and Deb [10] also proposed cuckoo search algorithm based on the behavior of cuckoo birds. Recently Geem et al. [11] proposed a new harmony search (HS) that draws its motivation not from a biological or physical process like most other metaheuristic optimization algorithms, but from an arty one, that is, the improvisation process of musicians in the hunt for a magnificent harmony. Geem et al. [11] explained the similarity between musical performance and optimization that explains the strength and language of HS. The attempt to find the harmony in music is similar to finding the optimality in an optimization process and the musician’s improvisations are similar to local and global search schemes in optimization techniques. HS algorithm uses random search based on harmony memory considering rate (HMCR) and the pitch adjusting rate (PAR) instead of gradient search. Even though the basic algorithm is successful in solving variety of problems still there is a growing diversity of modified HS algorithms that try to find improved performance. Recently Kumar et al. [12] had proposed a new version of HS algorithm, namely, parameter adaptive harmony search (PAHS) to solve standard benchmark functions in single objective optimization successfully. This PAHS algorithm is extended to solve multiobjective design optimization problems by weighted sum approach. In this weighted sum PAHS approach, a performance index based on least average error [13] is used to evaluate the performance of each weightage set.

#### 2. Literature Review

##### 2.1. Previous Work on Disc Brake Optimization

The proposed multiobjective disc brake optimization problem was introduced by Osyczka and Kundu [21]. The authors used the modified distance method in genetic algorithm to solve the disc brake problem and compared their results with that of a plain stochastic method. Ray and Liew [22] used a swarm metaphor approach in which a new optimization algorithm based on behavioural concepts similar to real swarm was proposed to solve the same problem. Yıldız et al. [23] used hybrid robust genetic algorithm combining Taguchi’s method and genetic algorithm. The combination of genetic algorithm with robust parameter design through a smaller population of individuals resulted in a solution that lead to better parameter values for design optimization problems. L16 orthogonal arrays were considered to design the experiments for this problem. The optimal levels of the design parameters were found using ANOVA with respect to the effects of parameters on the objectives and constraints. Yıldız [24] used hybrid method combining immune algorithm with a hill climbing local search algorithm for solving complex real-world optimization problems. The results of the proposed hybrid approach for multiobjective disc brake problem were compared with the previous solutions reported in literature.

Yang and Deb [25] used multiobjective cuckoo search (MOCS) algorithm for solving this disc brake problem by considering two objective functions. The authors extended the original cuckoo search for single objective optimization by Yang and Deb for multiobjective optimization by modifying the first and third rule of three idealized rules of original cuckoo search. The same author [26] used multiobjective firefly algorithm (MOFA) for solving this disc brake problem. By extending the basic ideas of FA, Yang et al. developed multiobjective firefly algorithm (MOFA). Yang et al. [27] successfully extended a flower algorithm for single objective optimization to solve multiobjective design problems. The author solved the biobjective disc brake problem using multiobjective flower pollination algorithm (MOFPA). Reynoso-Meza et al. [28] used the evaluation of design concepts and the analysis of multiple Pareto fronts in multicriteria decision-making using level diagrams. They addressed multiobjective design optimization problem of disc brake by considering the friction surfaces as 4 and 6 to obtain Pareto fronts.

##### 2.2. Previous Work on Helical Compression Spring Optimization

In single objective spring design optimization problem, two different cases were proposed with objective function of volume minimization as case I and weight minimization as case II. This single objective design problem had been solved using many optimization algorithms as two different cases. In this section we will look into a few reviews on case I of a compression spring design problem since the proposed multiobjective helical compression spring optimization problem is the conversion of well-known standard benchmark problem of single objective design optimization, that is, volume minimization of helical compression spring (case I). For both the cases, the design variables are common. The three design variables are the wire diameter , the mean coil diameter , and the number of active coils . But the data type of design variables, objective function, and constraints of these two cases are different. Sandgren [15] used integer programming to solve this problem. Chen and Tsao [16] used simple genetic algorithm to minimize the volume of the helical compression spring. Wu and Chow [17] used metagenetic parameter in GA to solve the same problem. Another improved version of GA called geneAS was used by Deb and Goyal [29] to address this problem. The same problem was solved by discrete version of PSO by Kennedy and Eberhart [20] and by using differential evolution algorithm by Lampinen and Zelinka [19]. Guo et al. [18] used swarm intelligence to optimize the design of helical compression spring. Datta and Figueira [30] used real-integer-discrete-coded PSO for solving this case I problem. He et al. [31] used an improved version of PSO to solve this case I spring problem. Deb et al. [32] proposed NSGA II algorithm for solving some biobjective design optimization of mechanical components. He used case I spring problem and converted it to a biobjective problem by adding one more objective of minimizing stress induced in the spring. In this work, we have converted the case I spring problem into a biobjective problem by adding one more objective of maximizing the strain energy stored in the spring. In all works related with case I spring problem, the previous researchers used FPS unit system. But in this work, all data are converted into SI units and the results of this proposed multiobjective spring problem are presented in SI units.

The main objectives of this work are (i) multiobjective design optimization of the disc brake and compression spring using the weighted sum approach of parameter adaptive harmony search algorithm and (ii) to demonstrate the effectiveness of this proposed algorithm for multiobjective optimization of machine components. So far, the weighted sum approach of parameter adaptive harmony search algorithm has not been tried or experimented for the multiobjective optimization of machine components. In this paper, ability of the algorithm is demonstrated using a performance index based on least average error. The optimization results obtained by using this proposed algorithm are compared with those obtained by previous researchers using other methods.

##### 2.3. Previous Works on HS Algorithm

HS algorithm is widely used in solving various optimization problems such as pipe network design [33], design of coffer dam drainage pipes [34], water distribution networks [35], satellite heat pipe design [36], design of steel frame [37], power economic load dispatch [38], economic power dispatch [39], power flow problem [40], vehicle routing [41], orienteering [42], robot application [43], and data clustering [44]. Mahdavi et al. [45] proposed improved HS algorithm by changing the parameters of HS algorithm dynamically. HS algorithm has been implemented by many researchers for the past few years to solve optimization problems in many fields of engineering, science, and technology. To increase the performance of HS algorithm, several improvements were done periodically in the past. From the literature, it is observed that mainly two improvements had been considered. First improvement is in terms of tuning the parameters of HS algorithm and the other one is in terms of hybridizing the components of HS algorithm with other metaheuristic algorithms. In this work, we are concentrating only with the first improvement.

Mahdavi et al. [45] first attempted an improvement in HS algorithm by proposing a new variant of HS known as improved HS (IHS) algorithm. They changed the variants such as pitch adjusting rate (PAR) and bandwidth (BW) dynamically with generations. Linear increase in PAR and exponential decrease in BW were allowed between prespecified minimum and maximum range.

Kong et al. [46] proposed an adaptive HS (AHS) algorithm to adjust PAR and BW. In this approach, PAR was changed dynamically in response to objective function values while BW was tuned for each variable. Omran and Mahdavi [47] proposed a global best HS algorithm inspired by PSO algorithm to improve the performance of HS algorithm. The difficulty in finding the lower and upper bounds of BW was solved by using global best particle which is the fittest particle in the swarm in terms of objective function values than other particles. A self-adaptive mechanism for selecting BW was proposed by Das et al. [48], known as explorative HS algorithm. BW was recomputed for each iteration while the other parameters were kept fixed. The other improvements in HS variants were reported in the literature [14, 41, 44, 48–55]. In all the above cases, only PAR and BW values were changed whereas HMCR was kept fixed. This fixed value of HMCR is the key factor which prevents to get global optimal solution. In order to overcome this difficulty, a parameter adaptive harmony search (PAHS) algorithm was recently proposed by Kumar et al. [12]. A detailed description of the proposed parameter adaptive harmony search (PAHS) algorithm can be seen from [12]. However, for the sake of easiness a brief introduction and a step by step computational procedure for implementing this algorithm are given in the following section. A flow chart showing the Steps 1 to 5 of the proposed PAHS approach is given in Figure 1.