Enhanced parallel cat swarm optimization based on the Taguchi method

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Abstract

In this paper, we present an enhanced parallel cat swarm optimization (EPCSO) method for solving numerical optimization problems. The parallel cat swarm optimization (PCSO) method is an optimization algorithm designed to solve numerical optimization problems under the conditions of a small population size and a few iteration numbers. The Taguchi method is widely used in the industry for optimizing the product and the process conditions. By adopting the Taguchi method into the tracing mode process of the PCSO method, we propose the EPCSO method with better accuracy and less computational time. In this paper, five test functions are used to evaluate the accuracy of the proposed EPCSO method. The experimental results show that the proposed EPCSO method gets higher accuracies than the existing PSO-based methods and requires less computational time than the PCSO method. We also apply the proposed method to solve the aircraft schedule recovery problem. The experimental results show that the proposed EPCSO method can provide the optimum recovered aircraft schedule in a very short time. The proposed EPCSO method gets the same recovery schedule having the same total delay time, the same delayed flight numbers and the same number of long delay flights as the Liu, Chen, and Chou method (2009). The optimal solutions can be found by the proposed EPCSO method in a very short time.

Highlights

► We present the EPCSO method for solving numerical optimization problems. ► Five test functions are used to evaluate the accuracy of the proposed EPCSO method. ► The proposed EPCSO method gets higher accuracies than the existing PSO-based methods. ► It also can provide the optimum recovered aircraft schedule in a very short time.

Introduction

In recent years, some artificial intelligence (AI) methods have been presented to solve optimization problems. Chu et al., 2006, Chu and Tsai, 2007 presented the cat swarm optimization (CSO) method for solving optimization problems by retaining the natural behaviors of cats. Tsai, Pan, Chen, Liao, and Hao (2008) presented a parallel cat swarm optimization (PCSO) method based on the framework of parallelizing the structure of the CSO method. Genetic algorithms (GA) have successfully been used in the internet service (Elliott & Krzymien, 2009) and impedance measurements (Janeiro & Ramos, 2009); particle swarm optimization (PSO) techniques have successfully been used to design antennas (Wu et al., 2009) and to construct parameters in neural network systems (Lin, Chen, & Lin, 2009); ant colony optimization (ACO) techniques have successfully been used to solve the traveling salesman problem (TSP) (Dorigo & Gambardella, 1997) and the routing problem of networks (Pinto, Nägele, Dejori, Runkler, & Sousa, 2009); artificial bee colony (ABC) techniques have successfully been used to solve the lot-streaming flow shop scheduling problem (Pan, Tasgetiren, Suganthan, & Chua, 2011); cat swarm optimization (CSO) techniques have successfully been used to adjust the parameters of the SVM (Lin et al., 2009). Moreover, in the industry, the Taguchi method (Taguchi, Chowdhury, & Taguchi, 2000) has successfully been used for optimizing the product-line design and the process conditions. Tsai, Liu, and Chou (2004) successfully adopted the Taguchi method into the GA’s crossover process and presented the hybrid Taguchi-genetic algorithm (HTGA).

Although the parallel cat swarm optimization (PCSO) method presented in Tsai et al. (2008) has the ability to find the near best solution under more strict conditions, its computational speed is not efficient. It is obvious that to reduce the computational time of the PCSO method and to keep high accuracy results with a small population size simultaneously are the desired goals of the PCSO method. Therefore, in this paper, we propose the enhanced parallel cat swarm optimization (EPCSO) method by adopting the orthogonal array of the Taguchi method into the tracing mode process of the PCSO method. The proposed EPCSO method can successfully been used for solving optimization problems.

Section snippets

Parallel cat swarm optimization

Tsai et al. (2008) have proposed the parallel cat swarm optimization (PCSO) method for solving optimization problems. The basic idea of the PCSO method utilizes the major structure of the cat swarm optimization (CSO) method proposed by Chu et al. (2006). The CSO method has two modes, i.e., the seeking mode and the tracing mode, for simulating the behaviors of cats to move the individuals in the solution space. By adjusting the parameter MR, the ratio of individuals moved by the seeking process

The Taguchi method

The Taguchi method (Taguchi et al., 2000) is an important tool for robust design. It is widely used for optimizing the product-line design and the process conditions due to the fact that it can provide high quality products with low development costs (Tsai et al., 2004). One of the major tools in the Taguchi method is called the orthogonal array, which is adopted by the proposed EPCSO method. In the EPCSO method, only the two-level orthogonal array is used to take part in the process. In the

The proposed enhanced parallel cat swarm optimization (EPCSO) method

In Tsai et al. (2008), we have proposed the PCSO method by organizing the artificial agents into a predefined number of groups. When the agent moves in the process of the parallel tracing mode, it collects the local best information found so far by its own group and use the information to update its velocity. Although it presents higher accuracies and faster convergence than the CSO method (Chu et al., 2006) under the conditions of a few number of artificial agents and a short limited

Experimental results

In order to evaluate the accuracy of the proposed EPCSO method, a series of experiments are taken with five familiar benchmark functions, shown as follows:f1(X)=d=1Dxd2Initial Range:[-100,100]D,f2(X)=d=1Dxd24000-d=1Dcosxdd+1Initial Range:[0,600]D,f3(X)=-20exp-0.21Dd=1Dxd2-exp1Dd=1Dcos(2πxd)+20+exp(1)Initial Range:[-32,32]D,f4(X)=d=1Dxd2-10cos(2πxd)+10Initial Range:[-5,5]D,f5(X)=d=1Db=1dxb2Initial Range:[-100,100]D.All the benchmark functions are evaluated according to the conditions

Apply the proposed EPCSO method to solve the aircraft schedule recovery problem

In this section, we apply the proposed EPCSO method to solve the aircraft schedule recovery problem and compare the experimental results of the proposed method with the HTGA method (Liu, Chen, & Chou, 2009). The aircraft schedule recovery is necessary when the established aircraft schedule has to be changed due to some inevitable reasons, e.g., the weather or mechanical problems. It can be dealt with by swapping aircraft, delaying the flights, dispatching other aircrafts to support or taking

Conclusions

In this paper, we have presented the enhanced parallel cat swarm optimization (EPCSO) method for solving optimization problems. In this paper, five test functions are used to evaluate the accuracy of the proposed EPCSO method. The experimental results show that the proposed EPCSO method gets higher accuracies with less computational time than the existing methods. We also have applied the proposed method to solve the aircraft schedule recovery problem. Aircraft schedule recovery contains many

Acknowledgement

The authors thank Mrs. Szu-Ping Hao, Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, for her help during this work.

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