Elsevier

Applied Mathematics and Computation

Volume 265, 15 August 2015, Pages 533-556
Applied Mathematics and Computation

Teaching-learning based optimization with global crossover for global optimization problems

https://doi.org/10.1016/j.amc.2015.05.012Get rights and content

Abstract

Teaching learning based optimization (TLBO) is a newly developed population-based meta-heuristic algorithm. It has better global searching capability but it also easily got stuck on local optima when solving global optimization problems. This paper develops a new variant of TLBO, called teaching learning based optimization with global crossover (TLBO-GC), for improving the performance of TLBO. In teaching phase, a perturbed scheme is proposed to prevent the current best solution from getting trapped in local minima. And a new global crossover strategy is incorporated into the learning phase, which aims at balancing local and global searching effectively. The performance of TLBO-GC is assessed by solving global optimization functions with different characteristics. Compared to the TLBO, several modified TLBOs and other promising heuristic methods, numerical results reveal that the TLBO-GC has better optimization performance.

Introduction

Optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. To the best of our knowledge, since Fermat and Lagrange found calculus-based formulas for identifying optima, a wide variety of numerical optimization methods have been proposed and reported for solving global optimization problems. However, traditional methods (such as linear programming, nonlinear programming, simplex, steepest descent, Newton's method, etc.) are limited in that they require substantial gradient information and depend strongly on the starting point of the search [1]. Heuristic or meta-heuristic algorithms (such as genetic algorithm (GA) [2] and differential evolution (DE) [3], particle swarm optimization (PSO) [4] and artificial bee colony (ABC) [5], harmony search (HS) [6], etc.) are more effective in general optimization problems as they do not require the objective functions satisfy specific conditions or have favorable mathematical properties. Own to their impressive advantages, these heuristic or meta-heuristic algorithms and their improved variants have been paid more and more attention and have been widely used in various engineering optimization problems.

Teaching learning based optimization (TLBO) algorithm has been developed by Rao et al. in 2011 [21]. Like other nature-inspired algorithm, TLBO is a population based meta-heuristic algorithm which imitates the natural phenomena of knowledge dissemination. One highlight of the TLBO algorithm is that it does not require any algorithm-specific parameters, which was stressed by Rao and Patel [8] and Črepinšek [9]. It seems to be a rising star among a number of meta-heuristics with relatively competitive performance [9]. TLBO has been paid much more attention and has been successfully applied to a wide range of real-world optimization problems [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45].

Since TLBO algorithm is introduced in 2011, several variants have been proposed by researchers for providing better performance or expanding application fields. These variants can be broadly classified under two categories, such as; (1) algorithm performance improvement; (2) algorithm application. A brief overview of the above categories of the TLBO algorithms is highlighted below.

Algorithm performance improvement: Owing to TLBO algorithm has several impressive advantages such as simple structure, easy implementation, quick convergence and free algorithm-specific parameters, it gained a growing concern in recent years. Rao and Patel developed an elitist TLBO algorithm, which integrated with an elitism concept to solve a series of constrained optimization problems [8]. Soon afterward, Rao and Patel proposed an improved TLBO algorithm, named I-TLBO [10]. The I-TLBO algorithm introduced the concept of number of teachers, adaptive teaching factor, tutorial training and self-motivated learning to enhance exploration and exploitation capacities. A weighted teaching–learning-based optimization (WTLBO) was introduced by Satapathy et al., which attempts to use a weight parameter to increase the convergence rate of TLBO algorithm [11]. Meanwhile, Satapathy et al. presented a TLBO based on orthogonal design [12], which combined TLBO algorithm with orthogonal design to make TLBO algorithm converge faster. A new variant of TLBO called modified TLBO (mTLBO) introduced by Satapathy et al, which used the concept of tutorial class to improve the performance of TLBO algorithm [13]. Zou et al. employed dynamic group strategy, random learning strategy and quantum-behaved learning strategy to maintain TLBO algorithm population diversity and discourage premature convergence, and then presented an improved version of TLBO called DGSTLBO [14]. Experiment results demonstrated that the DGSTLBO algorithm is an effective method for global optimization problems. Besides, bare-bones teaching–learning based optimization also proposed by Zou et al. in 2014 [15]. Chen et al. developed a TLBO algorithm with producer–scrounger model (PSTLBO) for global optimization. Numerical results indicate that the PSTLBO algorithm is generally better than some EAs in terms of the convergence accuracy, speed and reliability [16]. Besides, a new improved TLBO algorithm is also proposed by Chen et. al for global optimization algorithm [17]. In order to further ameliorate the performance of TLBO, some other improved or ameliorated multi-objective TLBO algorithms are reported in recent years, such as Patel and Savsani presented an efficient multi-objective improved teaching learning based optimization (MO-ITLBO) algorithm [18], grid-based approach and Pareto dominance are incorporated into the MO-ITLBO algorithm for solving multi-objective optimization problems. Zou et al. proposed a new version of TLBO algorithm for multi-objective optimization problems, which adopt the nondominated sorting concept and the mechanism of crowding distance computation [19]. Rao and Waghmare made a comparative study for teaching learning based optimization algorithm on multi-objective unconstrained and constrained functions [20].

Algorithm application: In recent years, TLBO algorithm and its improved variants has been successfully applied to various engineering application area. Main fields of application include mechanical engineering, combinatorial optimization, feature selection, electrical power system and other optimization fields. In mechanical engineering, the TLBO variants have been used for mechanical design optimization [21], heat exchangers [22], two-stage thermoelectric cooler [23], parameters optimization of selected casting processes [24], and plate-fin heat sink [25]. For the combinatorial optimization problems such as flexible job-shop scheduling problem [26], flow shop and job shop scheduling [27], permutation flow shop scheduling problem [28], realistic flow shop rescheduling problems [29], the TLBO algorithms performed much better than other meta-heuristic algorithms and provided some competitive solutions. The TLBO algorithms are also applied to the feature selection [30], [31]. In electrical power system, there are many researchers who have made much effort for using TLBO algorithms and its modified variants to solve electrical power problems such as dynamic economic emission dispatch [32], multi-objective optimal power flow [33], multi-area economic dispatch [34], optimal coordination of directional over-current relays [35], economic emission load dispatch problem [36], reserve constrained dynamic economic dispatch [37], optimal distributed generation location and size [38], source and transmission line maintenance outage scheduling [39]. Besides, the TLBO algorithms can also be used in other fields such as disassembly sequence planning [40], multi-level production planning in a petrochemical industry [41], design of planar steel frames [42], truss structures [43], polymerase chain reaction primer selection [44], and optimal location of distributed generator in radial distribution systems [45]. In sum, the understanding and application of TLBO algorithm above has also demonstrated its attraction and vitality in recent years.

Although the aforementioned TLBO variants have shown a better performance than the original TLBO, their convergence performance is still necessary to be further improved. Therefore, a new variant of the TLBO algorithm with global crossover strategy is proposed for solving global optimization problems. The proposed algorithm, named TLBO-GC, shows significant improvement in optimization performance for a great many classical functions with various properties. Compared to TLBO, some recently reported TLBO variants and other promising heuristic methods, TLBO-GC has relatively competitive advantages.

The remainder of this paper is organized as follows. The original TLBO algorithm and other modified variants are introduced in Section 2. In Section 3, the proposed new version of TLBO algorithm with crossover operation is described. Section 4 discusses the impact of parameter PopSize on the behavior of TLBO-GC and the effects of the perturbed scheme and global crossover on the performance of TLBO-GC, and reports the numerical results comparing the proposed algorithm with TLBO algorithm and its variants. Comparison between the proposed algorithm and some other promising meta-heuristics is also given in this section. Section 5 presents the conclusions.

Section snippets

Teaching-learning-based optimization and its variants

In this section, the original TLBO algorithm and its several improved algorithms are introduced.

Teaching learning based optimization with global crossover

TLBO is good at discovering better search space near the current good solution by applying the known information (such as the current best solution, mean solution and stochastic selected better solution), but poor at exploring a new domain of the large fixed search space quickly and absorbing new information to offspring population. That is, there exists an imbalance between the exploration and the exploitation of TLBO.

When designing optimization methods, exploration and exploitation, as two

Experimental results and analyses

In this section, the effect of the parameter PopSize, perturbed scheme and global crossover opeartions on the performance of TLBO-GC is investigated first. Then a large number of classical functions with different characteristics are used to evaluate the global capability of the proposed algorithm TLBO-GC, the original TLBO, A-TLBO, WTLBO, and OTLBO. Furthermore, in order to demonstrate the significant performance of the proposed algorithm more clearly, we compared it with several well-known

Conclusions

This paper proposes a new version of teaching–learning-based optimizations, namely TLBO-GC algorithms, for solving unconstrained optimization problems. A new global crossover method is incorporated in the proposed algorithms. Its purpose is to amend the local and global search capabilities of the original TLBO algorithm. The effects of varying control parameters on the performance of these proposed algorithms are also analyzed in detail. The performance of the proposed algorithm is tested by a

Acknowledgments

The authors are grateful to the editor and the anonymous referees for their constructive comments and recommendations, which have improved this paper significantly. The authors would also like to express their sincere thanks to Wang Yong for the useful information about DE variants on his home webpages. This work is supported by National Natural Science Foundation of China (grant nos. 61403174 and 61273155).

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