Teaching–LearningBased Optimization (TLBO) algorithms simulate the teaching–learning phenomenon of a classroom to solve multidimensional, linear and nonlinear problems with appreciable efficiency. In this paper, the basic TLBO algorithm is improved to enhance its exploration and exploitation capacities by introducing the concept of number of teachers, adaptive teaching factor, tutorial training and self motivated learning. Performance of the improved TLBO algorithm is assessed by implementing it on a range of standard unconstrained benchmark functions having different characteristics. The results of optimization obtained using the improved TLBO algorithm are validated by comparing them with those obtained using the basic TLBO and other optimization algorithms available in the literature.
Evolutionary algorithms ; Swarm intelligence based algorithms ; Improved teaching–learningbased optimization ; Unconstrained benchmark functions
The problem of finding the global optimum of a function with large numbers of local minima arises in many scientific applications. In typical applications, the search space is large and multidimensional. Many of these problems cannot be solved analytically, and consequently, they have to be addressed by numerical algorithms. Moreover, in many cases, global optimization problems are nondifferentiable. Hence, the gradientbased methods cannot be used for finding the global optimum of such problems. To overcome these problems, several modern heuristic algorithms have been developed for searching nearoptimum solutions to the problems. These algorithms can be classified into different groups, depending on the criteria being considered, such as populationbased, iterative based, stochastic, deterministic, etc. Depending on the nature of the phenomenon simulated by the algorithms, the populationbased heuristic algorithms have two important groups: Evolutionary Algorithms (EA) and swarm intelligence based algorithms.
Some of the recognized evolutionary algorithms are: Genetic Algorithms (GA) [1] , Differential Evolution (DE) [2] and [3] , Evolution Strategy (ES) [4] , Evolution Programming (EP) [5] , Artificial Immune Algorithm (AIA) [6] , and Bacteria Foraging Optimization (BFO) [7] etc. Some of the well known swarm intelligence based algorithms are: Particle Swarm Optimization (PSO) [8] , Ant Colony Optimization (ACO) [9] , Shuffled Frog Leaping (SFL) [10] , and Artificial Bee Colony (ABC) algorithms [11] , [12] , [13] and [14] , etc. Besides the evolutionary and swarm intelligence based algorithms, there are some other algorithms which work on the principles of different natural phenomena. Some of them are: the Harmony Search (HS) algorithm [15] , the Gravitational Search Algorithm (GSA) [16] , BiogeographyBased Optimization (BBO) [17] , the Grenade Explosion Method (GEM) [18] , the league championship algorithm [19] and the charged system search [20] and [21] .
In order to improve the performance of the abovementioned algorithms, the exploration and exploitation capacities of different algorithms are combined with each other and hybrid algorithms are produced. Several authors have hybridized different algorithms to improve the performance of individual algorithms [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] and [31] . Similarly, the performance of existing algorithms is enhanced by modifying their exploration and exploitation capacities [31] , [32] , [33] and [34] .
All evolutionary and swarm intelligence based algorithms are probabilistic algorithms and require common controlling parameters, like population size and number of generations. Besides common control parameters, different algorithms require their own algorithmspecific control parameters. For example, GA uses mutation rate and crossover rate. Similarly, PSO uses inertia weight, and social and cognitive parameters. The proper tuning of the algorithmspecific parameters is a very crucial factor affecting the performance of optimization algorithms. The improper tuning of algorithmspecific parameters either increases computational effort or yields the local optimal solution. Considering this fact, recently, Rao et al. [35] and [36] and Rao and Patel [37] introduced the TeachingLearningBased Optimization (TLBO) algorithm, which does not require any algorithmspecific parameters. The TLBO requires only common controlling parameters like population size and number of generations for its working. Common control parameters are common in running any population based optimization algorithms; algorithmspecific parameters are specific to that algorithm and different algorithms have different specific parameters to control. However, the TLBO algorithm does not have any algorithmspecific parameters to control and it requires only the control of the common control parameters. Contrary to the opinion expressed by Črepinšek et al. [38] that TLBO is not a parameterless algorithm, Rao and Patel [37] clearly explained that TLBO is an algorithmspecific parameterless algorithm. In fact, all comments made by Črepinšek et al. [38] about the TLBO algorithm were already addressed by Rao and Patel [37] .
In the present work, some improvements in the basic TLBO algorithm are introduced to enhance its exploration and exploitation capacities, and the performance of the Improved TeachingLearningBased Optimization (ITLBO) algorithm is investigated for parameter optimization of unconstrained benchmark functions available in the literature.
The next section describes the basic TLBO algorithm.
Teachinglearning is an important process where every individual tries to learn something from other individuals to improve themselves. Rao et al. [35] and [36] and Rao and Patel [37] proposed an algorithm, known as TeachingLearningBased Optimization (TLBO), which simulates the traditional teachinglearning phenomenon of a classroom. The algorithm simulates two fundamental modes of learning: (i) through the teacher (known as the teacher phase) and (ii) interacting with other learners (known as the learner phase). TLBO is a populationbased algorithm, where a group of students (i.e. learner) is considered the population and the different subjects offered to the learners are analogous with the different design variables of the optimization problem. The results of the learner are analogous to the fitness value of the optimization problem. The best solution in the entire population is considered as the teacher. The operation of the TLBO algorithm is explained below with the teacher phase and learner phase [37] .
This phase of the algorithm simulates the learning of the students (i.e. learners) through the teacher. During this phase, a teacher conveys knowledge among the learners and makes an effort to increase the mean result of the class. Suppose there are ‘ ’ number of subjects (i.e. design variables) offered to ‘ ’ number of learners (i.e. population size, ). At any sequential teachinglearning cycle, is the mean result of the learners in a particular subject ‘ ’ ( ). Since a teacher is the most experienced and knowledgeable person on a subject, the best learner in the entire population is considered a teacher in the algorithm. Let be the result of the best learner considering all the subjects who is identified as a teacher for that cycle. The teacher will put maximum effort into increasing the knowledge level of the whole class, but learners will gain knowledge according to the quality of teaching delivered by a teacher and the quality of learners present in the class. Considering this fact, the difference between the result of the teacher and the mean result of the learners in each subject is expressed as:

( 1) 
where is the result of the teacher (i.e. best learner) in subject . is the teaching factor, which decides the value of mean to be changed, and is the random number in the range . The value of can be either 1 or 2. The value of is decided randomly with equal probability as:

( 2) 
where is the random number in the range [0, 1]. is not a parameter of the TLBO algorithm. The value of is not given as an input to the algorithm and its value is randomly decided by the algorithm using Eq. (2) .
Based on the , the existing solution is updated in the teacher phase according to the following expression:

( 3) 
where is the updated value of . Accept if it gives a better function value. All the accepted function values at the end of the teacher phase are maintained, and these values become the input to the learner phase.
It may be noted that the values of and affect the performance of the TLBO algorithm. is the random number in the range [0, 1] and is the teaching factor. However, the values of and are generated randomly in the algorithm and these parameters are not supplied as input to the algorithm (unlike supplying crossover and mutation probabilities in GA, inertia weight and cognitive and social parameters in PSO, and colony size and limit in ABC, etc.). Thus, tuning of and is not required in the TLBO algorithm (unlike the tuning of crossover and mutation probabilities in GA, inertia weight and cognitive and social parameters in PSO, and colony size and limit in ABC, etc.). TLBO requires tuning of only the common control parameters, like population size and number of generations, for its working, and these common control parameters are required for the working of all population based optimization algorithms. Thus, TLBO can be called an algorithmspecific parameterless algorithm.
This phase of the algorithm simulates the learning of the students (i.e. learners) through interaction among themselves. The students can also gain knowledge by discussing and interacting with other students. A learner will learn new information if the other learners have more knowledge than him or her. The learning phenomenon of this phase is expressed below.
Randomly select two learners, and , such that , where, and are the updated values of and , respectively, at the end of the teacher phase.

( 4a) 
(The above equations are for maximization problems, the reverse is true for minimization problems.)
Accept if it gives a better function value.
In the basic TLBO algorithm, the result of the learners is improved either by a single teacher (through classroom teaching) or by interacting with other learners. However, in the traditional teachinglearning environment, the students also learn during tutorial hours by discussing with their fellow classmates or even by discussion with the teacher himself/herself. Moreover, sometime students are self motivated and try to learn by themselves. Furthermore, the teaching factor in the basic TLBO algorithm is either 2 or 1, which reflects two extreme circumstances where a learner learns either everything or nothing from the teacher. In this system, a teacher has to expend more effort to improve the results of learners. During the course of optimization, this situation results in a slower convergence rate of the optimization problem. Considering this fact, to enhance the exploration and exploitation capacities, some improvements have been introduced to the basic TLBO algorithm. Rao and Patel [39] and [40] made some modifications to the basic TLBO algorithm and applied the same to the optimization of a two stage thermoelectric cooler and heat exchangers. In the present work, the previous modifications are further enhanced and a new modification is introduced to improve the performance of the algorithm.
In the basic TLBO algorithm, there is only one teacher who teaches the learners and tries to improve the mean result of the class. In this system of teachinglearning, it might be possible that the efforts of the teacher are distributed and students also pay less attention, which will reduce the intensity of learning. Moreover, if the class contains a higher number of belowaverage students, then, the teacher has to put more effort into improving their results; even with this effort, there may not be any apparent improvement in the results. In the optimization algorithm, this fact results in a higher number of function evaluations to reach optimum solution and yields a poor convergence rate. In order to overcome this issue, the basic TLBO algorithm is improved by introducing more than one teacher for the learners. By means of this modification, the entire class is split into different groups of learners as per their level (i.e. results), and an individual teacher is assigned to an individual group of learners. Now, each teacher tries to improve the results of his or her assigned group and if the level (i.e. results) of the group reaches up to the level of the assigned teacher, then this group is assigned to a better teacher. This modification is explained in the implementation steps of the algorithm.
The concept of number of teachers is to carry out the population sorting during the course of optimization and, thereby, to avoid the premature convergence of the algorithm.
Another modification is related to the teaching factor ( ) of the basic TLBO algorithm. The teaching factor decides the value of mean to be changed. In the basic TLBO, the decision of the teaching factor is a heuristic step and it can be either 1 or 2. This practice is corresponding to a situation where learners learn nothing from the teacher or learn all the things from the teacher, respectively. But, in an actual teachinglearning phenomenon, this fraction is not always at its end state for learners but varies inbetween also. The learners may learn in any proportion from the teacher. In the optimization algorithm, the lower value of allows the fine search in small steps, but causes slow convergence. A larger value of speeds up the search, but it reduces the exploration capability. Considering this fact, the teaching factor is modified as:

( 5a) 
where is the result of any learner, , considering all the subjects at iteration, , and is the result of the teacher at the same iteration, . Thus, in the ITLBO algorithm, the teaching factor varies automatically during the search. Automatic tuning of TF improves the performance of the algorithm.
It may be noted that the adaptive teaching factor in TLBO is generated within the algorithm, based on the result of learner and teacher. Thus, the adaptive teaching factor is not supplied as an input parameter to the algorithm.
This modification is based on the fact that the students can also learn by discussing with their fellow classmates or even with the teacher during the tutorial hours while solving the problems and assignments. Since the students can increase their knowledge by discussion with other students or the teacher, we incorporate this search mechanism into the teacher phase. Mathematical expression of this modification is given in the implementation steps of the algorithm.
In the basic TLBO algorithm, the results of the students are improved either by learning from the teacher or by interacting with the other students. However, it is also possible that students are self motivated and improve their knowledge by selflearning. Thus, the selflearning aspect to improvise the knowledge is considered in the ITLBO algorithm.


(If the equality is not met, select the closest to the value calculated above)



where and,

where .
At this point, it is important to clarify that in the TLBO and ITLBO algorithms, the solution is updated in the teacher phase as well as in the learner phase. Also, in the duplicate elimination step, if duplicate solutions are present, then they are randomly modified. So, the total number of function evaluations in the TLBO algorithm is = {(2 × population size × number of generations) + (function evaluations required for duplicate elimination)}. In the entire experimental work of this paper, the above formula is used to count the number of function evaluations while conducting experiments with TLBO and ITLBO algorithms.
In this section, the ability of the ITLBO algorithm is assessed by implementing it for the parameter optimization of several unconstrained benchmark functions with different dimensions and search space. Results obtained using the ITLBO algorithm are compared with the results of the basic TLBO algorithm, as well as with other optimization algorithms available in literature. The considered benchmark functions have different characteristics, like unimodality/multimodality, separability/nonseparability and regularity/nonregularity.
This experiment is conducted to identify the ability of the ITLBO algorithm to achieve the global optimum value. In this experiment, eight different benchmark functions are tested using the TLBO and ITLBO algorithms, which were earlier solved using ABC and modified ABC by Akay and Karaboga [33] . The details of the benchmark functions are given in Table 1 . Previously, Akay and Karaboga [33] experimented all functions with 30 000 maximum function evaluations. To maintain the consistency in the comparison, TLBO and ITLBO algorithms are also experimented with the same maximum function evaluations.
No.  Function  Formulation  Search range  Initialization range  

1  Sphere  10  
2  Rosenbrock  10  
3  Ackley  10  
4  Griewank  10  
5  Weierstrass  10  
6  Rastrigin  10  
7  NCRastrigin  10  
8  Schwefel  10 
D: Dimension.
Each benchmark function is experimented 30 times with TLBO and ITLBO algorithms and comparative results, in the form of mean value and standard deviation of objective function obtained after 30 independent runs, are shown in Table 2 . Except TLBO and ITLBO algorithms, the rest of the results are taken from the previous work of Akay and Karaboga [33] . Moreover, the ITLBO algorithm is experimented with different numbers of teachers, and the effect on the obtained objective function value is reported in Table 2 .
Mean ± SD  Mean ± SD  Mean ± SD  Mean ± SD  

Sphere  Rosenbrock  Ackley  Griewank  
PSO–w  7.96E−051 ± 3.56E−050  3.08E+000 ± 7.69E−001  1.58E−014 ± 1.60E−014  9.69E−002 ± 5.01E−002 
PSO–cf  9.84E−105 ± 4.21E−104  6.98E−001 ± 1.46E+000  9.18E−001 ± 1.01E+000  1.19E−001 ± 7.11E−002 
PSO–wlocal  2.13E−035 ± 6.17E−035  3.92E+000 ± 1.19E+000  6.04E−015 ± 1.67E−015  7.80E−002 ± 3.79E−002 
PSO–cflocal  1.37E−079 ± 5.60E−079  8.60E−001 ± 1.56E+000  5.78E−002 ± 2.58E−001  2.80E−002 ± 6.34E−002 
UPSO  9.84E−118 ± 3.56E−117  1.40E+000 ± 1.88E+000  1.33E+000 ± 1.48E+000  1.04E−001 ± 7.10E−002 
FDR  2.21E−090 ± 9.88E−090  8.67E−001 ± 1.63E+000  3.18E−014 ± 6.40E−014  9.24E−002 ± 5.61E−002 
FIPS  3.15E−030 ± 4.56E−030  2.78E+000 ± 2.26E−001  3.75E−015 ± 2.13E−014  1.31E−001 ± 9.32E−002 
CPSOH  4.98E−045 ± 1.00E−044  1.53E+000 ± 1.70E+000  1.49E−014 ± 6.97E−015  4.07E−002 ± 2.80E−002 
CLPSO  5.15E−029 ± 2.16E−28  2.46E+000 ± 1.70E+000  4.32E−10 ± 2.55E−014  4.56E−003 ± 4.81E−003 
ABC  7.09E−017 ± 4.11E−017  2.08E+000 ± 2.44E+000  4.58E−016 ± 1.76E−016  1.57E−002 ± 9.06E−003 
Modified ABC  7.04E−017 ± 4.55E−017  4.42E−001 ± 8.67E−001  3.32E−016 ± 1.84E−016  1.52E−002 ± 1.28E−002 
TLBO  0.00 ± 0.00  1.72E+00 ± 6.62E−01  3.55E−15 ± 8.32E−31  0.00 ± 0.00 
ITLBO  
0.00 ± 0.00  1.29E+00 ± 3.97E−01  3.11E−15 ± 4.52E−15  0.00 ± 0.00  
0.00 ± 0.00  1.13E+00 ± 4.29E−01  2.93E−15 ± 1.74E−15  0.00 ± 0.00  
0.00 ± 0.00  6.34E−01 ± 2.53E−01  2.02E−15 ± 1.51E−15  0.00 ± 0.00  
0.00 ± 0.00  2.00E−01 ± 1.42E−01  1.42E−15 ± 1.83E−15  0.00 ± 0.00  
Weierstrass  Rastrigin  NCRastrigin  Schwefel  
PSO–w  2.28E−003 ± 7.04E−003  5.82E+000 ± 2.96E+000  4.05E+000 ± 2.58E+000  3.20E+002 ± 1.85E+002 
PSO–cf  6.69E−001 ± 7.17E−001  1.25E+001 ± 5.17E+000  1.20E+001 ± 4.99E+000  9.87E+002 ± 2.76E+002 
PSO–wlocal  1.41E−006 ± 6.31E−006  3.88E+000 ± 2.30E+000  4.77E+000 ± 2.84E+000  3.26E+002 ± 1.32E+002 
PSO–cflocal  7.85E−002 ± 5.16E−002  9.05E+000 ± 3.48E+000  5.95E+000 ± 2.60E+000  8.78E+002 ± 2.93E+002 
UPSO  1.14E+000 ± 1.17E+00  1.17E+001 ± 6.11E+000  5.85E+000 ± 3.15E+000  1.08E+003 ± 2.68E+002 
FDR  3.01E−003 ± 7.20E−003  7.51E+000 ± 3.05E+000  3.35E+000 ± 2.01E+000  8.51E+002 ± 2.76E+002 
FIPS  2.02E−003 ± 6.40E−003  2.12E+000 ± 1.33E+000  4.35E+000 ± 2.80E+000  7.10E+001 ± 1.50E+002 
CPSOH  1.07E−015 ± 1.67E−015  0 ± 0  2.00E−001 ± 4.10E−001  2.13E+002 ± 1.41E+002 
CLPSO  0 ± 0  0 ± 0  0 ± 0  0 ± 0 
ABC  9.01E−006 ± 4.61E−005  1.61E−016 ± 5.20E−016  6.64E−017 ± 3.96E−017  7.91E+000 ± 2.95E+001 
Modified ABC  0.00E+000 ± 0.00E+000  1.14E−007 ± 6.16E−007  1.58E−011 ± 7.62E−011  3.96E+000 ± 2.13E+001 
TLBO  2.42E−05 ± 1.38E−20  6.77E−08 ± 3.68E−07  2.65E−08 ± 1.23E−07  2.94E+02 ± 2.68E+02 
ITLBO  
9.51E−06 ± 1.74E−05  3.62E−12 ± 7.82E−11  1.07E−08 ± 6.19E−08  2.73E+02 ± 2.04E+02  
3.17E−06 ± 2.66E−06  2.16E−15 ± 9.13E−16  5.16E−09 ± 4.43E−09  2.62E+02 ± 2.13+02  
0.00 ± 0.00  0.00 ± 0.00  7.78E−016 ± 4.19E−015  1.49E+02 ± 1.21+02  
0.00 ± 0.00  0.00 ± 0.00  0.00 ± 0.00  1.10E+02 ± 1.06E+02 
Results of algorithms except TLBO and ITLBO are taken from Ref. [33] .
It is observed from the results that the ITLBO algorithm has achieved the global optimum value for Sphere, Griewank, Weierstrass, Rastrtigin and NCrastrigin functions, within the specified number of function evaluations. For the Rosenbrock function, ITLBO performs better than the rest of the algorithms. The performance of TLBO and ITLBO algorithms is better than the rest of the considered algorithms for Sphere and Griewank functions. For Weierstrass, Rastrigin and NCrastrigin functions, the performance of ITLBO and CLPSO are identical and better than the rest of the considered algorithms. For the Ackley function, ABC and ITLBO algorithms perform equally well. For the Schwefel function, the modified ABC algorithm performs better than the rest of the considered algorithms.
It is observed from the results that the fitness value of the objective function is improved as the number of teachers is increased from 1 to 4 for the ITLBO algorithm. During the experimentation, it is observed that with a further increase in the number of teachers beyond 4, the improvement in the fitness value of the objective function is insignificant and it involves significant increment in computational effort.
To indentify the computational effort and consistency of the ITLBO algorithm, eight different benchmark functions considered by Ahrari and Atai [18] are tested in this experiment. The results obtained using the ITLBO algorithm are compared with the basic TLBO algorithm, along with other well known optimization algorithms. The details of the benchmark functions are given in Table 3 .
No.  Function  Formulation  Search range  

1  De Jong  2  
2  GoldStein–Price  2  
3  Branin  2  
4  Martin and Gaddy  2  
5  Rosenbrock  2  
6  Rosenbrock  2  
7  Rosenbrock  3  
8  Hyper Sphere  6 
D: Dimension.
To maintain the consistency in the comparison between all comparative algorithms, the execution of the TLBO and ITLBO algorithms are stopped when the difference between the fitness obtained by the algorithm and the global optimum value is less than 0.1% (in cases where the optimum value is 0, the solution is accepted if it differs from the optimum value by less than 0.001). While making this complete study, the ITLBO algorithm is examined for different numbers of teachers and its effect on the performance of the algorithm is included in the results. Each benchmark function is experimented 100 times with the TLBO and ITLBO algorithms and the comparative results in the form of mean function evaluations and success percentage is shown in Table 4 . The results of the other algorithms are taken from Ahrari and Atai [18] .
MNFE  Succ %  MNFE  Succ %  MNFE  Succ %  MNFE  Succ %  

De Jong  Goldstein and Price  Branin  Martin and Gaddy  
SIMPSA  –  –  –  –  –  –  –  – 
NESIMPSA  –  –  –  –  –  –  –  – 
GA  10 160  100  5662  100  7325  100  2488  100 
ANTS  6000  100  5330  100  1936  100  1688  100 
Bee Colony  868  100  999  100  1657  100  526  100 
GEM  746  100  701  100  689  100  258  100 
TLBO  1070  100  452  100  443  100  422  100 
ITLBO  
836  100  412  100  438  100  350  100  
784  100  386  100  421  100  312  100  
738  100  302  100  390  100  246  100  
722  100  288  100  367  100  233  100  
Rosenbrock ( )  Rosenbrock ( )  Rosenbrock ( )  Hyper sphere ( )  
SIMPSA  10 780  100  12 500  100  21 177  99  –  – 
NESIMPSA  4508  100  5007  100  3053  94  –  – 
GA  10 212  100  –  –  –  –  15 468  100 
ANTS  6842  100  7505  100  8471  100  22 050  100 
Bee Colony  631  100  2306  100  28 529  100  7113  100 
GEM  572  100  2289  100  82 188  100  423  100 
TLBO  669  100  1986  100  21 426  100  417  100 
ITLBO  
646  100  1356  100  20 462  100  410  100  
602  100  1268  100  20 208  100  396  100  
554  100  1024  100  18 490  100  382  100  
522  100  964  100  17 696  100  376  100 
Results of algorithms except TLBO and ITLBO are taken from Ref. [18] .
It is observed from Table 4 that except for function 7 (i.e. Rosenbrock ( )), the ITLBO algorithm requires less number of function evaluations than other algorithms to reach the global optimum value, with a very high success rate of 100%. For the Rosenbrock 4 dimension function, the ant colony system (ANTS) performs better than ITLBO with 100% success rate. Similar to previous experiments, here, also, the results are improved further as the number of teachers is increased from 1 to 4 at the cost of more computational time.
In this experiment, the performance of the ITLBO algorithm is compared with the recently developed ABC algorithm, along with its improvised versions (IABC and GABC) and hybrid version (PSABC). In this part of the work, TLBO and ITLBO are experimented on 23 unconstrained benchmark functions (as shown in Table 5 ), which was earlier attempted by Li et al. [31] . This experiment is conducted from small scale to large scale by considering the dimensions 20, 30 and 50 of all the benchmark functions.
No.  Function  Formulation  Search range  C 

1  Sphere  US  
2  Schwefel 2.22  UN  
3  Schwefel 1.2  UN  
4  Schwefel 2.21  UN  
5  Rosenbrock  UN  
6  Step  US  
7  Quartic  US  
8  Schwefel  MS  
9  Rastrigin  MS  
10  Ackley  MN  
11  Griewank  MN  
12  Penalized  MN  
13  Penalized 2  MN  
14  Foxholes  MS  
15  Kowalik  MN  
16  6 Hump camel back  MN  
17  Branin  MS  
18  GoldStein–Price  MN  
19  Hartman 3  MN  
20  Hartman 6  MN  
21  Shekel 5  MN  
22  Shekel 7  MN  
23  Shekel 10  MN 
C: Characteristic, U: Unimodal, M: Multimodal, S: Separable, N: Non  separable.
Li et al. [31] attempted all these functions using ABC, IABC, GABC and PSABC with colony size 40 and number of cycles of 400 (i.e. 40 000 maximum function evaluations). But, it is observed that in the PSABC algorithm, three different food positions are generated for each employed bee and the corresponding nectar amount is calculated for each position. Out of these three food positions, the employed bee selected the best food position based on the calculated nectar amount. Similarly, for each onlooker bee, three more food positions are generated and out of these three positions, the onlooker bees select the best position. In that way, the total number of function evaluations in the PSABC algorithm is not equal to colony size multiplied by number of cycles. In the PSABC algorithm, three fitness evaluations are required for each employed bee for selecting the best food position. Similarly, three fitness evaluations are required for each onlooker bee for selecting the best food position. So, the total number of function evaluations for the PSABC algorithm is equal to 3*colony size*number of cycles. Considering this fact, in the present work, TLBO and ITLBO are implemented with 40 000 function evaluations to compare its performance with ABC, IABC and GABC algorithms, and 120 000 function evaluations to compare its performance with the PSABC algorithm.
In this experiment, each benchmark function is experimented 30 times with TLBO and ITLBO algorithms and the results are obtained in the form of mean solution and standard deviation of objective function after 30 independent runs of the algorithms. Table 6 shows the comparative results of ABC, IABC, GABC, TLBO and ITLBO algorithms for the first 13 functions with 40 000 maximum function evaluations. Except for TLBO and ITLBO algorithms, the rest of the results are taken from the previous work of Li et al. [31] .
Table 6.
Comparative results of TLBO and ITLBO algorithms with different variants of ABC algorithm over 30 independent runs (for functions 1–13 of Table 5 with 40 000 maximum function evaluations).
ABC [31]  IABC [31]  GABC [31]  TLBO  ITLBO  

Mean  SD  Mean  SD  Mean  SD  Mean  SD  Mean  SD  
Sphere  20  6.18E−16  2.11E−16  0.00  0.00  3.19E−16  7.39E−17  0.00  0.00  0.00  0.00 
30  3.62E−09  5.85E−09  0.00  0.00  6.26E−16  1.08E−16  0.00  0.00  0.00  0.00  
50  1.11E−05  1.25E−05  0.00  0.00  1.25E−05  6.05E−09  0.00  0.00  0.00  0.00  
Schwefel 2.22  20  1.35E−10  7.15E−11  0.00  0.00  9.36E−16  1.33E−16  0.00  0.00  0.00  0.00 
30  5.11E−06  2.23E−06  0.00  0.00  1.31E−10  4.69E−11  0.00  0.00  0.00  0.00  
50  2.92E−03  9.05E−04  0.00  0.00  2.37E−05  6.19E−06  0.00  0.00  0.00  0.00  
Schwefel 1.2  20  3.13E+03  1.19E+03  4.54E+03  2.69E+03  2.69E+03  1.46E+03  3.29E−38  1.20E−37  0.00  0.00 
30  1.24E+04  3.01E+03  1.43E+04  2.73E+03  1.09E+04  2.57E+03  3.25E−27  8.21E−27  0.00  0.00  
50  4.57E+04  6.46E+03  4.69E+04  7.36E+03  4.12E+04  5.83E+03  1.38E−21  4.00E−21  0.00  0.00  
Schwefel 2.21  20  3.9602  1.37E+00  0.00  0.00  0.3325  1.08E+00  7.19E−278  6.90E−278  0.00  0.00 
30  24.5694  5.66E+00  1.21E−197  0.00  12.6211  2.66E+00  3.96E−253  4.24E−253  4.7E−324  0.00  
50  56.3380  4.84E+00  25.5055  5.67E+00  45.3075  4.32E+00  4.77E−234  5.11E−234  4.9E−324  0.00  
Rosenbrock  20  1.1114  1.80E+00  15.7165  1.40E+00  1.6769  2.90E+00  16.0706  3.68E−01  11.0955  8.71E−01 
30  4.5509  4.88E+00  26.4282  1.40E+00  7.4796  1.91E+01  26.6567  2.94E−01  22.7934  5.82E−01  
50  48.03  4.67E+01  47.0280  8.60E−01  25.7164  3.18E+01  47.0162  3.56E−01  43.9786  4.55E−01  
Step  20  5.55E−16  1.69E−16  6.31E−16  2.13E−16  3.34E−16  1.02E−16  1.99E−20  5.03E−20  6.16E−33  4.11E−33 
30  2.49E−09  3.68E−09  3.84E−10  2.32E−10  6.45E−16  1.11E−16  2.74E−09  5.36E−09  1.17E−26  3.55E−26  
50  1.36E−05  1.75E−05  1.84E−05  1.74E−05  5.65E−09  3.69E−09  6.26E−04  6.33E−04  1.39E−11  1.61E−11  
1.01E−02  6.31E−03  6.45E−03  Quartic  20  6.51E−02  2.03E−02  8.71E−03  3.24E−03  3.31E−02  7.93E−03  1.71E−02 
30  1.56E−01  4.65E−02  1.96E−02  9.34E−03  8.48E−02  2.79E−02  1.71E−02  8.95E−03  8.29E−03  4.30E−03  
50  4.88E−01  1.07E−01  8.83E−02  2.55E−02  2.46E−01  4.72E−02  1.59E−02  8.11E−03  9.68E−03  3.88E−03  
Schwefel  20  −8327.49  6.63E+01  −8323.770  7.40E+01  −8355.92  7.23E+01  −8105.47  1.74E+02  −8202.98  1.27E+02 
30  −12 130.31  1.59E+02  −12 251.030  1.67E+02  −12 407.29  1.06E+02  −12 311.72  2.21E+02  −12 351.4  1.35E+02  
50  −19 326.50  2.66E+02  −19 313.490  2.77E+02  −19 975.29  2.31E+02  −20 437.84  1.48E+02  −20 533.71  2.46E+02  
Rastrigin  20  1.41E−11  4.05E−11  0.00  0.00  0.00  0.00  1.95E−13  2.32E−13  0.00  0.00 
30  0.4531  5.15E−01  0.00  0.00  0.0331  1.81E−01  1.87E−12  6.66E−12  0.00  0.00  
50  8.4433  2.70E+00  0.00  0.00  2.1733  1.07E+00  2.03E−12  5.46E−12  0.00  0.00  
Ackley  20  2.83E−09  2.58E−09  8.88E−16  0.00  2.75E−14  3.58E−15  3.55E−15  8.32E−31  7.11E−16  1.50E−15 
30  2.75E−05  2.13E−05  8.88E−16  0.00  7.78E−10  2.98E−10  3.55E−15  8.32E−31  1.42E−15  1.83E−15  
50  4.71E−02  3.40E−02  8.88E−16  0.00  1.11E−04  3.88E−05  3.55E−15  8.32E−31  1.42E−15  1.83E−15  
Griewank  20  3.71E−03  6.61E−03  0.00  0.00  6.02E−04  2.23E−03  0.00  0.00  0.00  0.00 
30  3.81E−03  8.45E−03  0.00  0.00  6.96E−04  2.26E−03  0.00  0.00  0.00  0.00  
50  1.19E−02  1.97E−02  0.00  0.00  1.04E−03  2.74E−03  0.00  0.00  0.00  0.00  
Penalized  20  4.06E−16  9.42E−17  4.17E−16  1.09E−16  3.26E−16  6.67E−17  1.13E−06  1.15E−06  4.00E−08  9.72E−15 
30  1.18E−10  2.56E−10  7.10E−12  5.25E−12  5.86E−16  1.13E−16  6.16E−03  2.34E−02  2.67E−08  1.15E−13  
50  8.95E−06  3.21E−05  5.42E−07  2.98E−07  9.30E−11  7.96E−11  6.01E−02  6.71E−02  5.72E−08  2.81E−08  
Penalized 2  20  6.93E−08  2.92E−07  1.75E−16  4.54E−16  6.55E−08  2.44E−07  1.13E−06  1.15E−06  2.54E−08  3.77E−11 
30  2.27E−07  4.12E−07  4.78E−08  2.04E−07  2.17E−07  5.66E−07  6.16E−03  2.34E−02  2.55E−08  4.89E−11  
50  1.35E−05  2.78E−05  2.41E−05  4.35E−05  8.87E−07  1.53E−06  6.01E−02  6.71E−02  1.82E−06  1.08E−06 
It is observed from the results that the ITLBO algorithm outperforms the rest of the considered algorithms for Schwefel1.2, Step and Quartic functions for all the dimensions. For the Schwefel 2.21 function, the ITLBO outperforms the other algorithms for dimensions 30 and 50, while the performances of ITLBO and IABC are identical for dimension 10. For the Schwefel function, the performance of the ITLBO is better than rest of the algorithms for dimension 50, while performances of GABC and ABC, IABC and GABC are better than the ITLBO for dimensions 30 and 20, respectively. GABC outperforms the other algorithms for the penalized 1 function. For the penalized 2 function, performances of IABC, ITLBO and GABC are better than other algorithms for dimensions 20, 30 and 50, respectively. For the Rosenbrock function, the performance of basic ABC and GABC is better than the other algorithms for dimensions 20 and 30, while GABC is better than the other algorithms for dimension 50. For Sphere, Schwefel 2.22 and Griewank functions, TLBO, ITLBO and IABC perform equally well for all the dimensions. Similarly, for the Rastrigin function, performances of IABC and ITLBO are identical and better than the other considered algorithms. For the Ackley function, performances of IABC, GABC, TLBO and ITLBO algorithms are more or less identical.
Table 7 shows the comparative results of PSABC, TLBO and ITLBO algorithms for the first 13 functions with 120 000 maximum function evaluations. It is observed from the results that ITLBO outperforms the basic TLBO and PSABC algorithms for Step and Quartic functions (for all the dimensions) and the Schwefel 2.21 function (for dimensions 30 and 50). The PSABC outperforms the TLBO and ITLBO for Rosenbrock and Schwefel functions. For the Schwefel 1.2 function, the performance of TLBO and ITLBO is identical and better than the PSABC algorithm. Performance of PSABC and ITLBO is identical for the Rastrigin function, while performance of all three algorithms is identical for Sphere, Schwefel 2.22 and Griewank functions. For Ackley, penalized 1 and penalized 2 functions, performances of PSABC and ITLBO are more or less similar.
PSABC [31]  TLBO  ITLBO  

Mean  SD  Mean  SD  Mean  SD  
Sphere  20  0.00  0.00  0.00  0.00  0.00  0.00 
30  0.00  0.00  0.00  0.00  0.00  0.00  
50  0.00  0.00  0.00  0.00  0.00  0.00  
Schwefel 2.22  20  0.00  0.00  0.00  0.00  0.00  0.00 
30  0.00  0.00  0.00  0.00  0.00  0.00  
50  0.00  0.00  0.00  0.00  0.00  0.00  
Schwefel 1.2  20  1.04E+03  6.11E+02  0.00  0.00  0.00  0.00 
30  6.11E+03  1.69E+03  0.00  0.00  0.00  0.00  
50  3.01E+04  4.11E+03  0.00  0.00  0.00  0.00  
Schwefel 2.21  20  0.00  0.00  0.00  0.00  0.00  0.00 
30  8.59E−115  4.71E−114  4.9E−324  0.00  0.00  0.00  
50  19.6683  6.31E+00  9.9E−324  0.00  0.00  0.00  
Rosenbrock  20  0.5190  1.08E+00  15.0536  2.28E−01  1.3785  8.49E−01 
30  1.5922  4.41E+00  25.4036  3.50E−01  15.032  1.2E+00  
50  34.4913  3.03E+01  45.8955  2.89E−01  38.7294  7.57E−01  
Step  20  2.61E−16  3.86E−17  9.24E−33  4.36E−33  0.00  0.00 
30  5.71E−16  8.25E−17  1.94E−29  1.88E−29  0.00  0.00  
50  1.16E−15  1.41E−16  3.26E−13  5.11E−13  1.51E−32  8.89E−33  
Quartic  20  6.52E−03  2.25E−03  1.07E−02  5.16E−03  5.16E−03  4.64E−03 
30  2.15E−02  6.88E−03  1.15E−02  3.71E−03  5.36E−03  3.72E−03  
50  6.53E−02  1.77E−02  1.17E−02  5.00E−03  5.60E−03  3.40E−03  
Schwefel  20  −8379.66  4.72E−12  −8210.23  1.66E+02  −8263.84  1.16E+02 
30  −12 564.23  2.55E+01  −12 428.60  1.53E+02  −12 519.92  1.16E+02  
50  −20 887.98  8.04E+01  −20 620.72  1.89E+02  −20 700.70  1.64E+02  
Rastrigin  20  0.00  0.00  6.41E−14  6.16E−14  0.00  0.00 
30  0.00  0.00  6.95E−13  1.64E−12  0.00  0.00  
50  0.00  0.00  7.90E−13  1.89E−12  0.00  0.00  
Ackley  20  8.88E−16  0.00  3.55E−15  8.32E−31  7.11E−16  0.00 
30  8.88E−16  0.00  3.55E−15  8.32E−31  7.11E−16  0.00  
50  8.88E−16  0.00  3.55E−15  8.32E−31  7.11E−16  0.00  
Griewank  20  0.00  0.00  0.00  0.00  0.00  0.00 
30  0.00  0.00  0.00  0.00  0.00  0.00  
50  0.00  0.00  0.00  0.00  0.00  0.00  
Penalized  20  2.55E−16  4.97E−17  4.00E−08  6.85E−24  2.42E−16  1.09E−16 
30  5.53E−16  8.68E−17  2.67E−08  6.79E−12  4.98E−16  2.14E−16  
50  1.02E−15  1.58E−16  5.18E−05  1.92E−04  9.19E−16  5.38E−16  
Penalized 2  20  2.34E−18  2.20E−18  2.34E−08  6.85E−24  1.93E−18  1.12E−18 
30  6.06E−18  5.60E−18  2.37E−08  4.91E−10  5.92E−18  4.74E−18  
50  5.05E−17  1.53E−16  1.52E−03  5.29E−03  4.87E−17  4.26E−17 
Table 8 shows the comparative results of the considered algorithms for 14 to 23 functions. Here, the results of ABC, IABC, GABC, TLBO and ITLBO algorithms are obtained with 40 000 maximum function evaluations, while the result of the PSABC algorithm is obtained with 120 000 maximum function evaluations. It is observed from the results that all the algorithms perform identically for functions 14, 16, 18, 19 and 21–23. The performance of the ITLBO is better than rest of the algorithms for the Kowalik function, while performances of different variants of ABC are better than the TLBO for the Hartman 6 function.
ABC  IABC  GABC  PSABC  TLBO  ITLBO  

Foxholes  0.9980  0.9980  0.9980  0.9980  0.9980  0.9980 
Kowalik  6.74E−04  3.76E−04  5.54E−04  4.14E−04  3.08E−04  3.08E−04 
6 Hump camel back  −1.0316  −1.0316  −1.0316  −1.0316  −1.0316  −1.0316 
Branin  0.7012  0.3978  0.6212  0.6300  0.3978  0.3978 
GoldsteinPrice  3.0010  3.0000  3.0000  3.0000  3.0000  3.0000 
Hartman 3  −3.8628  −3.8628  −3.8628  −3.8628  −3.8628  −3.8628 
Hartman 6  −3.3220  −3.3220  −3.3220  −3.3220  −3.2866  −3.2948 
Shekel 5  −10.1532  −10.1532  −10.1532  −10.1532  −10.1532  −10.1532 
Shekel 7  −10.4029  −10.4029  −10.4029  −10.4029  −10.4029  −10.4029 
Shekel 10  −10.5364  −10.5364  −10.5364  −10.5364  −10.5364  −10.5364 
Results of algorithms except TLBO and ITLBO are taken from Ref. [31] .
In order to identify the convergence of TLBO and ITLBO, a unimodal (Step) and a multimodal (Rastrigin) function are considered for the experiment with dimensions 20, 30 and 50. Maximum function evaluations are set as 40 000 and a graph is plotted between the function value (on logarithmic scale) and function evaluations. The function value is taken as the average of the function value for 10 different independent runs. Figure 1 and Figure 2 show the convergence graphs of unimodal and multimodal functions, respectively. It is observed from the graphs that the convergence rate of the ITLBO is faster than the basic TLBO algorithm for both unimodal and multimodal functions for all the dimensions. Similarly, Table 9 shows the computational effort of TLBO and ITLBO algorithms for the lower dimension problem (functions 14–23) in the form of the mean number of function evaluations required to achieve a global optimum value with a gap of 10^{−3} . Here, the mean number of function evaluations is obtained through 30 independent runs on each function. Here, also, the ITLBO algorithm requires less number of function evaluations than the basic TLBO algorithm to achieve the global optimum value. Moreover, as the number of teachers is increased from 1 to 4, the convergence rate of the ITLBO algorithm is also improved.

Figure 1. Convergence of TLBO and ITLBO algorithms for a unimodal function (step).


Figure 2. Convergence of TLBO and ITLBO algorithms for a multimodal function (Rastrigin).

TLBO  ITLBO  

Foxholes  524  472  431  344  278 
Kowalik  2488  2464  2412  2344  2252 
6 Hump camel back  447  426  408  339  276 
Branin  443  438  421  390  367 
GoldsteinPrice  582  570  553  511  473 
Hartman 3  547  524  492  378  310 
Hartman 6  24 847  18 998  18 542  17 326  16 696 
Shekel 5  1245  1218  1212  1124  1046 
Shekel 7  1272  1246  1228  1136  1053 
Shekel 10  1270  1251  1233  1150  1062 
An improved TLBO algorithm has been proposed for unconstrained optimization problems. Two new search mechanisms are introduced in the proposed approach in the form of tutorial training and self motivated learning. Moreover, the teaching factor of the basic TLBO algorithm is modified and an adaptive teaching factor is introduced. Furthermore, more than one teacher is introduced for the learners. The presented modifications enhance the exploration and exploitation capacities of the basic TLBO algorithm. The performance of the ITLBO algorithm is evaluated by conducting small scale to large scale experiments on various unconstrained benchmark functions and the performance is compared with that of the other stateoftheart algorithms available in the literature. Furthermore, the comparison between the basic TLBO and ITLBO is also reported. The experimental results have shown the satisfactory performance of the ITLBO algorithm for unconstrained optimization problems. The proposed algorithm can be easily customized to suit the optimization of any system involving large numbers of variables and objectives.
A possible direction for future research work is extending the ITLBO algorithm to handle single objective and multiobjective constrained optimization problems and explore its effectiveness. Analyzing the effect of the number of teachers on the fitness value of the objective function and experimentation on very large dimension problems (i.e. 100 and 500) are also possible future research directions.
Published on 06/10/16
Licence: Other
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