Computational Intelligence and Neuroscience

Volume 2015, Article ID 292576, 15 pages

http://dx.doi.org/10.1155/2015/292576

## Nonlinear Inertia Weighted Teaching-Learning-Based Optimization for Solving Global Optimization Problem

^{1}School of Mechanical and Precision Instrumental Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China^{2}School of Information Engineering, Tibet University for Nationalities, Xianyang, Shaanxi 712082, China

Received 28 June 2015; Revised 11 August 2015; Accepted 17 August 2015

Academic Editor: Michael Schmuker

Copyright © 2015 Zong-Sheng Wu 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

Teaching-learning-based optimization (TLBO) algorithm is proposed in recent years that simulates the teaching-learning phenomenon of a classroom to effectively solve global optimization of multidimensional, linear, and nonlinear problems over continuous spaces. In this paper, an improved teaching-learning-based optimization algorithm is presented, which is called nonlinear inertia weighted teaching-learning-based optimization (NIWTLBO) algorithm. This algorithm introduces a nonlinear inertia weighted factor into the basic TLBO to control the memory rate of learners and uses a dynamic inertia weighted factor to replace the original random number in teacher phase and learner phase. The proposed algorithm is tested on a number of benchmark functions, and its performance comparisons are provided against the basic TLBO and some other well-known optimization algorithms. The experiment results show that the proposed algorithm has a faster convergence rate and better performance than the basic TLBO and some other algorithms as well.

#### 1. Introduction

Most of the swarm intelligent optimization studies and applications have been focused on nature-inspired algorithms. Numerous population-based and nature-inspired optimization algorithms have been presented, such as the Ant Colony Optimization (ACO), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), and Differential Evolution (DE). These optimization algorithms are based on different natural phenomena. ACO works based on the behavior of ant colony searching foods from the source to a destination [1, 2]. GA applies the theory of Darwin based on the survival of the fittest to the optimization problems [3, 4]. PSO emulates the collaborative behavior of birds flocking and fish schooling in searching for foods [5–7]. ABC uses the foraging behavior of a honey bee [8–10]. DE derived from the Genetic Algorithm, which is an efficient global optimizer in the continuous search domain [11, 12]. These algorithms have been applied to many engineering optimization problems and proven effective in solving specific types of problems. However, various algorithms have their own advantages and disadvantages in solving diverse problems. Generally, a good optimization algorithm should possess the three essential conditions. First, the algorithm has the ability of obtaining the true global optima value. Second, the convergence speed of the algorithms should be fast. Third, the program should have a minimum of control parameters so that it will be easy to use. If an optimization algorithm meets the above three conditions at the same time, it would be a great algorithm. Some optimization techniques often achieve global optima results but at the cost of the convergence speed. Those algorithms tend to focus on the quality of computational results rather than the convergence speed. However, the higher calculation accuracy and faster convergence speed are the ultimate aim in the practical applications.

Recently, Rao et al. [13, 14] proposed a teaching-learning-based optimization (TLBO) algorithm, inspired by the phenomenon of teaching and learning in a class. The TLBO requires only the common control parameters like population size and numbers of generation and that does not require any algorithm-specific control parameters; that is, it is a parameter-less algorithm [15]. Thus, there is no burden of tuning control parameters in the TLBO algorithm. Hence, the TLBO algorithm is simpler and more effective and involves relatively less computational cost. What it is more important is that the TLBO algorithm has the ability to achieve better results at comparatively faster convergence speed to other algorithms mentioned above. Therefore, the TLBO algorithm has been successfully applied in diverse optimization fields such as mechanical engineering, task scheduling, production planning and control, and vehicle-routing problems in transportation [16–20]. Similar to other swarm intelligent optimization algorithms, the basic TLBO can be improved further and further. In order to improve the performance of TLBO, several variants of the TLBO have been proposed. Rao and Patel presented an elitist TLBO (ETLBO) algorithm [15] to solve complex constrained optimization problems and used a modified version of TLBO algorithm [17] to solve the multiobjective optimization problem of heat exchangers. Sultana and Roy [19] proposed a quasioppositional teaching-learning-based optimization (QOTLBO) methodology in order to find the optimal location of the distributed generator to simultaneously optimize power loss, voltage stability index, and voltage deviation of radial distribution network. Ghasemi et al. [20] used Lévy mutation strategy based on TLBO for optimal settings of optimal power flow problem control variables. Furthermore, some improved TLBO algorithms have been proposed to solve the global function optimization problem [21–24] and the multiobjective optimization problem [17, 25, 26].

In this paper, we propose a novel improved TLBO, which is called nonlinear inertia weighted TLBO (NIWTLBO). A nonlinear inertia weighted factor is introduced into the basic TLBO to control the memory rate of learners, and another dynamic inertia weighted factor is used to replace the original random number in teacher phase and learner phase. So, as a result, the NIWTLBO has faster convergence speed and higher calculation accuracy for most of these optimization problems than the basic TLBO. The performance of NIWTLBO for solving global function optimization problems is compared with basic TLBO and other optimization algorithms. The analysis results show that the proposed algorithm outperforms most of the other algorithms investigated in this paper.

The rest of this paper is organized as follows. Section 2 describes the basic TLBO algorithm in detail. In Section 3, the proposed NIWTLBO algorithm will be introduced. And Section 4 provides numerical experiments and results demonstrating the performance of NIWTLBO in comparison with other optimization algorithms. Finally, our conclusions are mentioned in Section 5.

#### 2. Teaching-Learning-Based Optimization

The basic TLBO algorithm mainly consists of two parts, namely, the teacher phase and the learner phase. In teacher phase, the students can learn from the teacher to make their knowledge level closer to the teacher’s. In learner phase, the students can learn from the interaction of other individuals to increase their knowledge. In the TLBO algorithm, a group of learners is considered as a population. Each learner is analogous to an individual of the evolutionary algorithm. The different subjects offered to the learners are considered as design variables of the optimization problem. A learner’s result is analogous to the fitness value of the objective function for optimization problems. The best learner (i.e., the best solution in the entire population) is considered as the teacher. The best solution is the best value of the objective function of the given optimization problem. The design variables are the input parameters of the objective function.

The process of basic TLBO algorithm is described below.

##### 2.1. Initialization

The notations used in TLBO are described as follows: is number of learners in a class (i.e., population size). is number of subjects offered to the learners (i.e., dimensions of design variables). MAXITER is maximum number of allowable iterations. denotes a learner in class (i.e., the individual in the population) at any iterator . denotes the result of th subject offered to th learner at th iterator. represents the teacher, that is, the best learner in a class at th iterator.

The population is randomly initialized by a search space bounded by matrix. The values of are assigned randomly using the equationwhere and . The rand represents a uniformly distributed random variable within the range . represents the lower bound of design variable. represents the upper bound of design variable.

##### 2.2. Teacher Phase

In this phase, the algorithm simulates the students learning from teachers. A good teacher can bring his or her learners up to his or her level in terms of knowledge. Hence, the mean result of a class may increase from a low level to the teacher’s level. But, in fact, it is impossible that the mean result of a class reaches the teacher’s level. Because of the individual differences and the forgetfulness of memory, the learners cannot gain all the knowledge of the teacher. A teacher can increase the mean result of a class to a certain value which depends on the capability of the whole class.

Let be the mean result of the learners on a particular subject “” () and let be the teacher at any iteration . will try to move mean towards its own level which is the new mean. is the difference between the existing mean result of each subject and the corresponding result of the teacher for each subject at the iteration . The solution is updated according to the difference between the existing and the new means given bywhere is the result of the teacher in subject at the iteration . is a random number in the range , and is the teaching factor, which decides the value of mean to be changed. can be either 1 or 2. The values of and are generated randomly in the algorithm and both of these parameters are not supplied as input to the algorithm.

In every iteration, is the updated value of . Because the optimization problem is a minimization problem, our goal is to find the minimum of . If the new value gives a better function value, then the old value is updated with the new value. The updated formula is given aswhere and represent the new and old total result of th student at the iteration , respectively. All the accepted new values at the end of the teacher phase become the input to the learner phase.

##### 2.3. Learner Phase

In learner phase, the algorithm simulates the learning of the learners through interaction among themselves. A learner interacts randomly with other learners to increase his or her knowledge. If a learner has more knowledge than others, the other learners can quickly achieve new knowledge by learning from him or her to increase their level. In this learning process, two learners are randomly selected. One is and another is , . The updated formula is given aswhere is a random number in the range . and represent the total result of th student and th student at the iteration , respectively. Accept the new value if it improves the value of the objective function. Similarly, use (5) to update the learner.

In each iteration of the TLBO, it is necessary to detect the repeated solution to the entire population. If there is a repeated solution, it needs to remove the repeated solution and generate a new individual randomly. Hence, it will expand the diversity of populations and avoid premature convergence of the algorithm. After a number of generations, the knowledge level of the entire class is smoothly approximated to a point that is considered the teacher, and the algorithm converges to a solution.

##### 2.4. Algorithm Termination

The algorithm is terminated after MAXITER iterations. The details of TLBO algorithm can be referred to in literature [13, 14].

#### 3. Nonlinear Inertia Weighted Teaching-Learning-Based Optimization

The basic TLBO algorithm is based on teaching-learning phenomenon of a classroom. In the teacher phase, the teacher tries to shift the mean of the learners towards himself or herself by teaching. In the learner phase, learners improve their knowledge by interaction among themselves. In the process of the teaching-learning, learners improve their level by accumulating knowledge. In other words, they learn new knowledge based on existing knowledge. In the real world, the teacher tends to wish that his or her students should achieve the knowledge equal to him in fast possible time. But it is impossible for a student because of his or her forgetting characteristics. In fact, a student usually forgets a part of existing knowledge due to the physiological phenomena of the brain. With increasing the iteration numbers of learning, more and more existing knowledge will be remembered. As the learning curve presented by Ebbinghaus, it describes how fast learning knowledge is in learning process. The sharpest increase occurs after the first try and then gradually evens out, meaning that less and less new knowledge is retained after each repetition. Like the forgetting curve, the learning curve is exponential. So it is necessary to add a memory weight to the existing knowledge of the student for simulating this learning scenario. According to this phenomenon, a nonlinear inertia weighted factor is introduced into (4) and (6) in the basic TLBO, and this factor is considered as memory weighted factor which controls the memory rate of learners. This nonlinear inertia weighted factor will scale the existing knowledge of the learner for computing the new value. In contrast to the basic TLBO, in our algorithm the part of previous value of the learner is decided by a weighted factor while computing the new learner value.

Accordingly, to meet the characteristic of memory to conform to the learning curve, the* nonlinear inertia weighted factor * (i.e., memory rate) is nonlinearly increased from to 1.0 over time, whose value is given aswhere iter is the current iteration number, MAXITER is the maximum number of allowable iterations, and is the minimum value of* nonlinear inertia weighted factor *. The value should be above 0.5 (here it is selected 0.6), or the individuals are worse due to remembering too little existing knowledge at first. Hence, if the value is too small, the algorithm could not converge to the true global optimal solution. curve (i.e., memory rate curve) is shown as Figure 1. The* nonlinear inertia weighted factor * is applied to the new equations shown as (10) and (11). In this modified TLBO, the individuals try to sample diverse zones of the search space during the early stages of the search. During the later stages, the individuals adjust the movements of trial solutions finely so that they can explore the interior of a relative small space.