Computational Intelligence and Neuroscience

Volume 2018, Article ID 4231647, 19 pages

https://doi.org/10.1155/2018/4231647

## A Modified Sine-Cosine Algorithm Based on Neighborhood Search and Greedy Levy Mutation

^{1}School of Information Engineering, Baise University, Baise 533000, China^{2}School of Tourism Management, Baise University, Baise 533000, China^{3}School of Business Administration, Baise University, Baise 533000, China^{4}School of Politics and Public Affair Management, Baise University, Baise 533000, China

Correspondence should be addressed to Chiwen Qu; moc.361@newihcuq

Received 26 February 2018; Revised 26 April 2018; Accepted 30 April 2018; Published 4 July 2018

Academic Editor: Paulo Moura Oliveira

Copyright © 2018 Chiwen Qu 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

For the deficiency of the basic sine-cosine algorithm in dealing with global optimization problems such as the low solution precision and the slow convergence speed, a new improved sine-cosine algorithm is proposed in this paper. The improvement involves three optimization strategies. Firstly, the method of exponential decreasing conversion parameter and linear decreasing inertia weight is adopted to balance the global exploration and local development ability of the algorithm. Secondly, it uses the random individuals near the optimal individuals to replace the optimal individuals in the primary algorithm, which allows the algorithm to easily jump out of the local optimum and increases the search range effectively. Finally, the greedy Levy mutation strategy is used for the optimal individuals to enhance the local development ability of the algorithm. The experimental results show that the proposed algorithm can effectively avoid falling into the local optimum, and it has faster convergence speed and higher optimization accuracy.

#### 1. Introduction

Many problems in the field of engineering practice and scientific research come down to the global optimization problems. The traditional methods which purely lie upon the exactly mathematical mode have unsatisfactory effect in solving such optimization problems. These problems need to be continuous and derivable when the traditional methods are used for solving such practical engineering optimization problems, and these methods do not have the ability of global optimization for the multimodal, strong-nonlinearity, and dynamic change problems [1]. Accordingly, many scholars begin to explore new solution methods. The swarm intelligence optimization algorithm is a kind of global optimization algorithm designed by simulating the mutual cooperation behavior mechanism of gregarious biology in nature. Compared with the traditional optimization methods, the swarm intelligence optimization algorithm is characterized by simple principle and fewer adjustment parameters, and the gradient information and strong global optimization algorithm of problems are not required. So it is widely used in the engineering field of function optimization [2–4], combinatorial optimization [5], neural network training [6, 7], and image processing. At present, many swarm intelligence optimization algorithms are proposed [2, 8–15] like particle swarm optimization (PSO) [8], differential evolution (DE) [9, 10], artificial bee colony algorithm (ABC) [2, 11], cuckoo search (CS) [12, 13], and flower pollination algorithm (FPA) [14, 15].

Sine-cosine algorithm (SCA) is a new swarm intelligence optimization algorithm proposed by Mirjalili in 2016 [16]. This algorithm has been concerned and studied by many scholars due to its simple implementation and less parameter setting, and its optimization search can be realized through simple variation of sine and cosine function values. It has been successfully applied to solving the parameter optimization of support vector regression [17], short-term hydrothermal scheduling [18], and other engineering fields at present. However, as with other swarm intelligence algorithms, this algorithm also has the disadvantage of low optimization precision and slow convergence speed. Many scholars have put forward various improved sine-cosine algorithms from different perspectives in order to overcome this disadvantage in last two years. Elaziz et al. [19] proposed a sine-cosine algorithm based on the opposition method, and the more accurate solutions is obtained. Nenavath et al. [20] adopted a hybrid algorithm by combining differential evolution with sine-cosine to solve the problem of global optimization and target tracking. This algorithm has faster convergence speed and ability of seeking the optimal solution compared with the basic sine-cosine algorithm and differential evolution algorithm. Reddy et al. [21] applied a new binary variant of sine-cosine algorithm to solve the PBUC (profit-based unit commitment) problem. Sindhu et al. [22] used the elitism strategy and new updating mechanism to improve the sine-cosine algorithm, which improved the accuracy of classification in the selection of features or attributes. Kumar et al. [23] proposed a new sine-cosine optimization algorithm with the hybrid Cauchy and Gaussian mutations in order to track MPP (maximum power point) quickly and efficiently. Mahdad et al. [24] presented a sine-cosine algorithm coordinated with the interactive process to improve the security of the power system aimed at loading margin stability and faults at specified important branches. Bureerat et al. [25] adopted an adaptive differential sine-cosine algorithm to solve the problem of structural damage detection. Turgut et al. [26] combined backtracking search algorithm (BSA) and sine-cosine algorithm (SCA) to obtain the optimal design for the shell and tube evaporator. Attia et al. [27] embed Levy's flight into the original sine-cosine algorithm to increase the local search ability of the algorithm and avoided the algorithm being trapped in a local optimal defect. Tawhid et al. [28] used elite nondominated sorting to obtain different nondominated grades and applied crowd distance method to maintain the diversity of optimal solution sets, putting forward a multiobjective SCA algorithm. Issa et al. [29] presented an enhanced version of SCA by embedding the particle swarm optimization algorithm in SCA(ASCA-PSO). The ASCA-PSO algorithm makes full use of developing ability of the particle swarm optimization algorithm in the search space, which is stronger than that of the SCA. In the tests of some functions, it is found that the search performance of ASCA-PSO is apparently superior to that of SCA and other recently proposed basic metaheuristic algorithms. Rizk-Allah et al. [30] proposed a multiorthogonal sine-cosine algorithm (MOSCA) based on a multiorthogonal search strategy (MOSS) to solve the problem of engineering designs. The MOSCA algorithm eliminated the disadvantages which are that the basic SCA lacked exploitability and it was easily trapped in local optimum.

The modified sine-cosine algorithm (MSCA) based on neighborhood search and the greedy Levy mutation has been proposed in order to better balance the global exploration ability and local exploitation ability. The improved algorithm makes improvements in the following three aspects. Firstly, both the linear decreasing inertia weight and exponential declining conversion parameters are used to balance the global exploration and local exploitation ability, which achieves the smooth transition of algorithm from global exploration to local development. Secondly, the guidance of random individuals near the optimal solution is fully used to allow the algorithm easily jump out of the local optimum, which effectively prevents the algorithm premature convergence and increases the diversity of population. Thirdly, the greedy Levy mutation strategy is used for the optimal individuals to enhance the local development ability of the algorithm. Compared with other swarm intelligence algorithms, the improved sine-cosine algorithm has better performance in terms of searching precision, convergence speed, and stability.

#### 2. Basic Sine-Cosine Algorithm

In the basic sine-cosine algorithm, the simple variation of sine and cosine function values is used to achieve the optimization search. In this paper, the population size is* n*. The dimension of search space is* d*, and the* i*th individual in the population is . In each iteration, the update mode of can be obtained by the following equation:where* t* is the current iteration, is the* j*th dimension value of the optimal individual at iteration* t*, and is the* j*th dimension value of the individual* i* at iteration* t*. , , , and are the random numbers. and obey uniform distribution between 0 and 2. obey uniform distribution between 0 and 2, and obey uniform distribution between 0 and 1.

In (1), or jointly lead the global exploration and local development ability of the algorithm. When the value of or is greater than 1 or less than -1, the algorithm conducts a global exploration search. When the value of or is within the range of , the algorithm conducts a local development search. The value of or is within the range of . So the control parameter plays a crucial role in the global exploration, which controls the transition of the algorithm from global exploration to local development. In the basic algorithm, the control parameter adopts the linear decreasing method of (2) to guide the algorithm transit from the global exploration to the local development.where is a constant, is the current iteration, and is the maximum number of iterations.

#### 3. Modified Sine-Cosine Algorithm

##### 3.1. Exponential Decreasing Conversion Parameter

The parameter setting is crucial to the search performance in the basic sine-cosine algorithm, in which the control parameter controls the transition of algorithm from global exploration to local development. The larger value can improve the global searching ability of the algorithm, and the smaller value can enhance the local development ability of the algorithm. Therefore, is designed as the linear decreasing method of (2) in the basic algorithm to balance the global exploration and local development ability of the algorithm. In the literature [31], experimental contrast analysis is made on the linear decreasing method, parabola decreasing method, and exponential decreasing method in the basic algorithm. It is found that the exponential decreasing method is superior to the other two methods in the search performance. At the same time, the inertia weight remains unchanged in the iterative process of the basic algorithm, which may easily cause the population individuals to oscillate in the later period of search. In this paper, both the linear decreasing inertia weight and exponential decreasing conversion parameter strategy are used on the basis of (1), which can better balance the global exploration and local development ability of the algorithm. The update mode of individuals is as follows:where* t* is the current iteration, is the maximum number of iterations, is the* j*th dimension value of the optimal individual at iteration* t*, is the* j*th dimension value of the individual* i* of current iteration, and and are the maximum and minimum inertia weight, respectively.

It can be seen from (3) that the population individuals work together through the inertia weight and conversion parameter . The value of and is large in the early iterations, which is conducive to the global exploration of the algorithm. The values of and are small in later iterations, which are conducive to the local development of the algorithm so as to improve the searching precision and convergence speed of the algorithm.

##### 3.2. The Neighborhood Search of the Optimal Individual

In the basic sine-cosine algorithm, the search directions of the new individuals simply are updating process by optimal individuals in the population. Once the global optimal individuals fall into the local optimum, the whole algorithm easily gets into premature convergence. Therefore, in order to reduce the possibility of algorithm getting into the local optimum, the guiding role of the better individuals possibly existing near the optimal solution should be used. In this paper, the random individuals near the optimal solution are used to replace the current optimal individuals to guide the algorithm search, so as to improve the possibility of algorithm jumping out of the local optimum. The sine-cosine algorithm strategy for the neighborhood search of the optimal individual iswhere is the uniform distribution number within (-1, 1), and is the disturbance coefficient. Other parameters are in line with (3).

In the neighborhood search of the optimal individual, the current optimal individual is taken as the center and as the step size, and the algorithm searches between the section and . It effectively expands the search orientation and increases the probability of algorithm jumping out of the local optimum.

##### 3.3. Greedy Levy Mutation

In the basic sine-cosine algorithm, the optimal individuals lead the search direction of the whole population. But the optimal individuals lack experiential knowledge and self-learning ability. So they may hardly get effective improvement and thus get into the domain of local optimum. In order to further prevent the basic sine-cosine algorithm from getting into the local optimum and eliminate the defect of low efficiency in later period, a strategy based on greedy Levy mutation is proposed for the optimum individuals. Thus, the population individuals can jump out of the position of optimal value searched previously through the mutation operation, which retains the diversity of population. The mutation method is as follows:where is the random number that obeys the Levy distribution, is the coefficient of self-adapting variation, and is the* j*th dimension value of the optimal individual at iteration* t* (Algorithm 1).