In evolutionary algorithms the convergence rate of the algorithm is given prime importance for solving an optimization problem. The ability of the algorithm to obtain the global optima value is one aspect and the faster convergence is the other aspect. It is studied in the evolutionary techniques literature that there are few good techniques, often achieve global optima results but at the cost of the convergence speed. Those algorithms are good candidates for use in the areas where the main focus is on the quality of results rather than the convergence speed. In real world applications, the faster computation of accurate results is the ultimate aim. In recent time, a new optimization technique called Teaching learning based optimization [1] is gaining popularity [2-8] due to its ability to achieve better results in comparatively faster convergence time to techniques like Genetic Algorithms (GA) [9,10], Particle swarm Optimizations (PSO) [11-17] Differential Evolution (DE) [18-20] and some of its variants like DE with Time Varying Scale Factor (DETVSF), DE with Random Scale Factor (DERSF) [21] etc. The main reason for TLBO being faster to all other contemporary evolutionary techniques is it has no parameters to tune. However, in evolutionary computation research there has been always attempts to improve any given findings further and further. This work is an attempt to improve the convergence characteristics of TLBO further without sacrificing the accuracies obtained in TLBO and in some occasions trying to even better the accuracies.

In our proposed work, the attempt is made to include a parameter called as “weight” in the basic TLBO equations. Our proposed algorithm is known as weighted TLBO (WTLBO). The philosophy behind inclusion of this parameter is justified in Section 3 of the paper. The inclusion of this parameter is found not only bettering the convergence speed of TLBO, even providing better results for few problems. The performance of WTLBO for solving global function optimization problems is compared with basic TLBO and other evolutionary techniques. It can be revealed from the results analysis that our proposed approach outperforms all approaches investigated in this paper.

The remaining of the paper is organized as follows: in Section 2, we give a brief description of TLBO. In Section 3, we describe the proposed Weighted TeachingLearning-Based Optimizer (WTLBO). In Section 4, experimental settings and numerical results are given. The paper concludes with Section 5.

This optimization method is based on the effect of the influence of a teacher on the output of learners in a class. It is a population based method and like other population based methods it uses a population of solutions to proceed to the global solution. A group of learners constitute the population in TLBO. In any optimization algorithms there are numbers of different design variables. The different design variables in TLBO are analogous to different subjects offered to learners and the learners’ result is analogous to the “fitness”, as in other population-based optimization techniques. As the teacher is considered the most learned person in the society, the best solution so far is analogous to Teacher in TLBO. The process of TLBO is divided into two parts. The first part consists of the “Teacher Phase” and the second part consists of the “Learner Phase”. The “Teacher Phase” means learning from the teacher and the “Learner Phase” means learning through the interaction between learners. In the sub-sections below, we briefly discuss the implementation of TLBO.

Since the original TLBO is based on the principles of teaching-learning approach, we can always draw analogy with the real class room or learning scenario while designing TLBO algorithm. Although, a teacher always wishes that his/her student should achieve the knowledge equal to him in fast possible time but at times it becomes difficult for a student due to his/her forgetting characteristics. Teaching-learning process is an iterative process wherein the continuous interaction takes place for the transfer of knowledge. Every time a teacher interacts with a student he/she finds that the student is able to recall part of the lessons learnt from the last session. This is mainly due to the physiological phenomena of neurons in the brain. In this work we have considered this as our motivation to include a parameter known as “weight” in the Equations (4) and (6) of original TLBO. In contrast to the original TLBO, in our approach while computing the new learner value the part of its previous value is considered and that is decided by a weight factor w.

It is generally believed to be a good idea to encourage the individuals to sample diverse zones of the search space during the early stages of the search. During the later stages it is important to adjust the movements of trial solutions finely so that they can explore the interior of a relatively small space in which the suspected global optimum lies. To meet this objective we reduce the value of the weight factor linearly with time from a (predetermined) maximum to a (predetermined) minimum value:

where and are the maximum and minimum values of weight factor w, i iteration is the current iteration number and maxiteration is the maximum number of allowable iterations. and are selected to be 0.9 and 0.1, respectively.

Hence, in the teacher phase the new set of improved learners can be

and a set of improved learners in learner phase as

We have divided our experimental works into four sections. In Section 4.1, we have done performance comparison of WTLBO against basic algorithms like PSO, DE and TLBO to establish that our proposed approach performs better than above algorithms for investigated problems. We used 20 benchmark problems in order to test the performance of the PSO, DE, TLBO and the WTLBO algorithms. This set is large enough to include many different kinds of problems such as unimodal, multimodal, regular, irregular, separable, non-separable and multidimensional. For experiments in section 4.2 to Section 4.4 few functions from these 20 functions are used and those are mentioned in the comparison tables in respective sections of the paper. Initial range, formulation, characteristics and the dimensions of these problems are listed in Table 1.

In Section 4.2 of our experiments, attempts are made to compare our proposed approach with the recent variants of PSO as per [22,23]. The results of these variants are directly taken from [22,23] and compared with WTLBO. In Section 4.3, the performance comparisons are made with the recent variants of DE as per [22]. The Section 4.4 of our experiments devote to the performance comparison of WTLBO with Artificial Bee Colony (ABC) variants as in [24-27]. Readers may be intimated here that in all such above mentioned comparisons we have simulated WTLBO and basic PSO, DE and TLBO of our own but gained results of other algorithms directly from the referred papers.

For comparing the speed of the algorithms, the first thing we require is a fair time measurement. The number of iterations or generations cannot be accepted as a time measure since the algorithms perform different amount of works in their inner loops, and they have different population sizes. Hence, we choose the number of fitness function evaluations (FEs) as a measure of computation time instead of generations or iterations. Since the algorithms are stochastic in nature, the results of two successive runs usually do not match. Hence, we have taken different independent runs (with different seeds of the random number generator) of each algorithm. Numbers of FEs for different algorithms which are compared with WTLBO are taken as in [22-27]. However, for WTLBO we have chosen 2.0 × 10⁴ as maximum number of FEs. The exact numbers of FEs in which we get optimal results with WTLBO are given in the Table 2.

Finally, we would like to point out that all the experiment codes are implemented in MATLAB. The experiments are conducted on a Pentium 4, 1 GB memory desktop in Windows XP 2002 environment

In this paper an improved version of basic Teachinglearning based optimization (TLBO) technique is suggested and the performance of this compared with existing TLBO and other evolutionary computation techniTable 3. Performance comparisons of PSO, DE, TLBO, WTLBO in term of fitness value (mean and standard deviation over 30 independent runs) after each algorithm was terminated after running for 20,000 FEs or when it reached the global minimum value before completely running for 20,000 FEs.

Table 4. t value, significant at a 0.05 level of significance by two tailed test using Table 3.

Table 5. Performance comparisons WTLBO, OEA, HPSO-TVAC, CLPSO and APSO in term of fitness value (mean and standard deviation over 30 independent runs) after OEA running of 3.0 × 10⁵ FEs, HPSO-TVAC, CLPSO and APSO use running 2.0 × 10⁵ FEs, and WTLBO running for 2.0 × 10⁴.

Table 6. Performance comparisons WTLBO, IADE, jDE and SaDE in term of fitness value (mean and standard deviation over 30 independent runs) after each algorithm was terminated after running by given FEs.

Table 7. Performance comparisons of WTLBO, CABC, GABC, RABC and IABC in term of fitness value (mean and standard deviation over 30 independent runs) after each algorithm was terminated after running by given FEs.

ques like PSO, DE, ABC and its variants. The proposed approach, known as Weighted-TLBO (WTLBO) is based on the natural phenomena of human brain (a learner’s brain) in forgetting the lessons learnt in last session. The paper suggested an inclusion of a parameter known as “weight” to address this phenomenon while using the learning equation in teaching and learning phases of basic TLBO algorithm. Although, inclusion of a parameter such as weight might seem to increase the complexity of the basic TLBO algorithm while tuning the parameter, the suggested approach in our work in setting up the weight parameter eases the task and able to provide better results compared to basic TLBO and all other investigated algorithms in this work for several benchmark functions. Our proposed WTLBO not only able to find global optima results but also does in faster computation time. This fact is verified from the number of function evaluations for WTLBO in each case from the results shown in our paper.

As a further research, we need to verify how this approach behaves with many other benchmark functions and some real world problems in data clustering.