Improved sine cosine algorithm with crossover scheme for global optimization
Introduction
Optimization is the process of selecting or determining the best solution from the available set of possible solutions. Nowadays optimization is unavoidable in engineering, science, finance and various other fields. The general form of single objective optimization problem can be defined as: where, and are real valued functions and are known as objective function, inequality constraints and equality constraints. These functions may be linear or non-linear. is dimensional decision vector, and are the lower and upper limits for the th component of a decision vector. and represents the total number of inequality and equality constraints.
In the literature, a large number of traditional techniques and exact methods are available which have been applied to solve the optimization problems of various fields such as multiproduct economic production quantity problem [1], feature selection [2], machine learning [3], supply chain [4], [5], [6], [7], [8], [9], [10], selective maintenance scheduling [11], Single machine scheduling [12], vehicle routing problem [13], and many others [14], [15]. But, these traditional methods cannot be applied everywhere. The real-life optimization problems where the continuity, differentiability, convexity of the objective function and/or constraints are absent, the traditional techniques fail to start the process of determination of optima. Nature Inspired Optimization techniques works as a boon in all such cases because in these techniques any information regarding the problem is not needed. From some past decades, the Nature Inspired Optimization has received enormous attention. Generally, these techniques mimic the intelligent behavior of various phenomenon which occurs in nature. These techniques treated the problem as a black box and try to explore all the search regions of the search space which are promising. Based on the number of search agents, nature inspired optimization techniques/algorithms can be grouped into two categories — single solution based algorithms and population-based algorithms. In single solution based algorithms, the search process is followed by a single search agent. Hill Climbing [16], Simulated Annealing [17] are some algorithms which are single solution based. In population based algorithms, multiple search agents proceed the search process. The advantage of population-based algorithms is the sharing of useful information between the search agents and this increases the exploration ability of search agents as compared to single solution based algorithms. Also, the multiple agents help each other to avoid the local solutions. Genetic Algorithm (GA) [18], Particle Swarm Optimization (PSO) [19], Differential Evolution (DE) [20], Ant Colony Optimization (ACO) [21], Artificial Bee Colony (ABC) algorithm [22], Grey Wolf Optimizer (GWO) [23], Ant Lion Optimizer (ALO) [24], Moth-flame Optimization (MFO) [25], Whale Optimization Algorithm (WOA) [26] Invasive weed optimization algorithm [27], Flower Pollination Algorithm [28], [29] are some examples of population based algorithms.
Sine cosine algorithm (SCA) [30] is a recently developedpopulation-based approach to solve global optimization problems. SCA uses the characteristics of sine and cosine trigonometric functions in the search process. In [30], SCA has shown its efficiency in terms of exploration and exploitation. Due to its exploration ability it has been applied to solve many real-life applications. Hafez et al. [31] have used classical SCA for the feature selection problem. SCA was used to train a feed-forward neural network [32]. In [33], SCA was also used for Handwritten Arabic Manuscript Image Binarization. In [34], the application of SCA is demonstrated for training artificial neural network in the problem of load forecasting. In [35], SCA is used for data clustering problem. In [36], the parameters of support vector regression have been optimized by SCA. In [37], the performance of SCA has been used for the optimization of a vehicle engine connecting rod. In [38], the loading margin stability is optimized using SCA to improve the power system security. In [39], the solution of optimal power flow problem is obtained using SCA. In [40], unified power quality conditioner allocation in the distribution system is addressed by SCA. In [41], SCA is used to optimize the complementary metal-oxide semiconductor analog circuits.
Although the literature shows that the SCA has enough ability to explore the search space but like other algorithms, it faces some difficulties like local optima stagnation, slow convergence and skipping of true solutions while solving the real-life problems. To alienate these issues from classical SCA, some attempts have been done in the literature. For example — In [42], an improved version of SCA is proposed based on orthogonal parallel information for global optimization tasks. In [43], the levy flight strategy is integrated into SCA to avoid the stagnation at local solutions. In [44], SCA has been hybridized with opposition based learning to train the feed-forward neural network. In [45], an improved version of SCA, by incorporating the opposition based learning, is introduced to solve the global optimization problems. In [46], an improved version of SCA is used for unit commitment problem. In [47], a novel weighted update mechanism is used to improve the performance of SCA. In [48], the search equations of classical SCA are modified to enhance the search ability in promising directions and to avoid the problem of stagnation at local solutions. In [49], the modified SCA has been combined with an extreme learning machine for pathological brain detection. In [50] SCA and Water Wave Optimization algorithm is hybridized to propose an efficient optimizer. In [51], the SCA has been hybridized with DE and employed on structural damage detection problems. In [52], the SCA and DE are hybridized and applied for visual tracking. In [53], SCA is hybridized with multi-orthogonal search strategy for manufacturing optimization problems. In [54], hybrid Q-learning sine–cosine-based strategy is proposed for addressing the combinatorial test problems. The multiobjective version of SCA is used to develop a novel forecasting system for wind speed forecasting [55] and for engineering design problems [56].
Although, there are many attempts have been done to improve the performance of SCA, but from the analysis of their results on benchmark test problems, it can be observed that in some cases SCA still suffers from the problem of overflow of diversity and stagnation at local optima. Therefore in the present work, the search mechanism of SCA has been improved by modifying its search equations, so that the above mentioned issues of SCA can be avoided.
The aim of the present paper is to introduce the improved version of classical SCA which is more efficient to search for optima during the intermediate iterations. Therefore, in the present work, an enhanced version of SCA is proposed which is named as Improved Sine Cosine Algorithm (ISCA) in the paper. ISCA, uses the personal best memory in the search equation in place global state of the population, to decide the search area. In the search equation of ISCA global search direction is also integrated to utilize the useful information of a current best solution of the problem. To maintain the best features of a solution, and to explore the search regions around the best memory of search agents, a crossover is performed between the current solution and its personal best memory. To prevent the overflow of diversity, a greedy selection has been applied between the previous and current population of solutions. The proposed modifications in the search strategy of SCA reduces the difficulties of classical SCA. The proposed algorithm considers the personal best as well as global information to explore the search space as in PSO [19], but in the proposed algorithm the crossover operator allows the search around the individual’s best memory and the sine cosine functions maintain the diversity and exploitation near the current solution using algorithm parameters. The proposed algorithm can also be called as memetic algorithm as the crossover operator has been merged with the search strategy of SCA. Crossover is an evolutionary algorithms-based strategy to evolve the search agents. The term ‘memetic algorithm’ was first presented by Moscato in [57] with local improvement strategy to search the solution. The memetic algorithm provides a local search in establishing the exploitation of the search space in an algorithm. Memetic algorithms are hybrid methods which are based on population-based search strategy [58], [59] and neighborhood-based local search framework [60].
To prove the efficacy and reliability of the proposed algorithm on real-life optimization problems, first, it has been tested on classical benchmark set as well as on standard IEEE CEC 2014 [61] and latest CEC 2017 [62] benchmark set of problems. Secondly, the proposed algorithm has been used to solve benchmark engineering optimization problems. In the last, the proposed algorithm is also applied for multilevel thresholding problem. Comparison with classical SCA indicates that the proposed algorithm is much better than classical SCA in terms of accuracy and efficacy.
In the literature, various evolutionary algorithms have been used for optimal thresholding. For example, Ayala et al. [63] have proposed a novel beta DE approach for thresholding. Maitra and Chatterjee [64] in 2008, have used a cooperative comprehensive learning based PSO for image segmentation. Ali et al. [65] have determined the optimal thresholds using synergetic DE. Sarkar et al. [66] have applied thresholding for colored images based on DE and minimum cross entropy. Hammouche et al. [67] have studied the effects of various meta-heuristic techniques on image thresholding and concluded the better performance ability of DE and rapid convergence of PSO.
In some previous recent years, various new and efficient approaches have been applied to determine the optimal thresholds for image segmentation. Aziz et al. [68] have applied the Whale Optimization Algorithm (WOA) [26] and Moth-flame Optimization [25] for multiple thresholding. In [69], Cuckoo Search (CS) and Wind Driven Optimization have been studied on image thresholding. In [70], the determination of multiple thresholds has been done using Grey Wolf Optimizer. In [71], Elephant Herding Optimization Algorithm is used to determine the optimal thresholds for image segmentation. In [72], an improved version of Bacterial Foraging Optimization is used for multilevel thresholding. In [73], Modified Firefly Algorithm is used for color image multilevel thresholding.
The remaining of the paper is organized as follows — Section 2 provides a brief description of Sine Cosine Algorithm. In Section 3, the improved version of Sine Cosine Algorithm named as ISCA is discussed in detail. In Section 4, evaluation and analysis of the proposed algorithm have been done based on 23 classical, 30 standard IEEE CEC 2014 and 30 latest IEEE CEC 2017 benchmark problems. In Section 5, the applications of ISCA is discussed on five engineering optimization problems. In Section 6, the proposed algorithm is applied to solve multilevel thresholding and finally, Section 7 concludes the work of the paper and suggests some future ideas.
Section snippets
Sine Cosine Algorithm (SCA)
The Sine Cosine Algorithm (SCA) is a recently developed meta-heuristic algorithm based on the mathematical characteristics of sine and cosine trigonometric functions. This algorithm was designed by Mirjalili in 2015 [30]. Like other population-based meta-heuristic optimization algorithms, SCA also starts with a set of randomly distributed solutions, then each solution updates their position with the help of following equations —
The proposed ISCA algorithm:
In this section, the proposed algorithm ISCA has been described in detail. The stepwise description and analysis of proposed ISCA is as follows:
Validation of proposed ISCA
The proposed improved Sine Cosine Algorithm (ISCA) can be considered more efficient optimizer as compared to classical SCA in terms of exploiting the promising regions of a search space and in keeping the personal best features within the population of solutions. To validate the strength and advantages of integrated strategies in proposed ISCA, three set of benchmark problems — classical set of 23 well-known benchmark test problems, standard IEEE CEC 2014 benchmark set and a latest set of
Applications of ISCA on engineering test problems
Validation of the proposed algorithm ISCA on an extensive set of benchmark test problems ensures the significant improvement in the search ability of solutions. The performance of ISCA on the benchmark test problems examines the reliability and efficiency of the proposed algorithm. Moreover, in this section, proposed ISCA is used to determine the solutions of engineering optimization problems. The two set of engineering test applications have been taken in this section. First contains the
Application of ISCA in multilevel thresholding
In this section, the proposed algorithm (ISCA) has been used to for multilevel thresholding in image segmentation.
Conclusions and future scope
The paper introduces an improved version of Sine Cosine Algorithm through the crossover with personal best state of solutions. In the proposed algorithm, the search equations are modified by integrating the personal best state in place of the global best state to decide the region of search space around the personal best state of a solution and to prevent from the problem of stagnation at local optima. In the search equation, global best or social direction is also integrated with random steps
Acknowledgment
The first author gratefully acknowledges to the Ministry of Human Resource and Development (MHRD), Government of India for their financial support. Grant No. MHR-02-41-113-429.
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