A multi-objective improved teaching–learning based optimization algorithm (MO-ITLBO)
Introduction
Multi-objective optimization has been applied in the fields of science, engineering, economics, logistics, etc., when optimal decision is required to be taken in presence of trade-offs between two or more conflicting objectives. Many real-world optimization problems involve multiple conflicting objectives. Therefore, multi-objective optimization (MOO) has attracted much attention of researchers and many algorithms have been developed for solving multi-objective optimization problems in the last decade. In the MOO problems a set of solutions must be found with respect to a trade-off between objectives since there is no single solution available [55].
The concept of multi-objective optimization problem (MOOP) can be expressed as minimizing (or maximizing) function F(X) where, F(X) = {f1(X), … , fn(X)} is a vector in objective space and X is a vector in a decision space. The goal of optimization is to find a set of decision vectors, which maps them to the set of best solutions in the objective space for multi-objective problems [1]. The obtained result in multi-objective optimization is a set of solutions since objective functions are competing with each other with different importance. The ideal solution set has this attribute that there is no other solution existing in other sets that has better objective function values among all objectives. This ideal set is a non-dominated set and is known as the Pareto-optimal set. These solutions form a balanced surface in the objective space which is known as Pareto front [3].
From the study on MOO so far, it is observed that the multi-objective optimization methods can be classified in different categories such as aggregative, lexicographic, sub-population, Pareto-based, and hybrid methods [3]. Among the multi-objective methods, the majority of research is concentrated on Pareto-based approaches [8]. Pareto-based methods select a part of individuals based on the Pareto dominance notion as leaders. Usually leaders are maintained in an external archive. These methods use different approaches to select non-dominated individuals as leaders.
Different types of MOO algorithms have been proposed in recent years to solve problems with multiple objectives. Evolutionary algorithms are among most popular meta-heuristics to address MOO problems. Generalized Differential Evolution 3 (GDE3) [19], Archive-based Micro Genetic Algorithm (AMGA) [46], NSGA-II [11], Local Search Based Evolutionary Multi-Objective Optimization Algorithm (NSGAIILS) [41], Differential Evolution with Self-adaptation and Local Search Algorithm (DECMOSA-SQP) [51], Multi-Objective Self-adaptive Differential Evolution Algorithm with Objective-wise Learning Strategies (OWMOSaDE) [16], Dynamical Multi-Objective Evolutionary Algorithm (DMOEADD) [25], Multi-Objective Evolutionary Programming (MOEP) [31], Multi-Objective Evolutionary Algorithm based on Decomposition (MOEAD) [53], Multiple Trajectory Search (MTS) [47], LiuLi Algorithm [24], Clustering Multi-Objective Evolutionary Algorithm (Clustering MOEA) [48], Enhancing MOEA/D with Guided Mutation and Priority Update (MOEADGM) [5] and an Orthogonal Multi-objective Evolutionary Algorithm with Lower-dimensional Crossover (OMOEAII) [13] are some of the competitive evolutionary MOO algorithms that aimed to obtain approximate Pareto front for multi-objective problems.
Similarly, several types of swarm intelligence based MOO algorithms have been presented in the literature to address MOO problem. Multi-objective Particle Swarm optimization (MOPSO) [9], Interactive Particle Swarm Optimization (IPSO) [2], Covering Pareto-optimal fronts by sub swarms in multi-objective particle swarm optimization [28], PSO-based multi-objective optimization with dynamic population size and adaptive local archives [21], particle swarm inspired evolutionary algorithm (PS-EA) for multi-objective optimization problem [42], Dynamic Multiple Swarms in Multi-Objective Particle Swarm Optimization (DSMOPSO) [50], multi-objective bee swarm optimization [4], Autonomous bee colony optimization for multi-objective function [52], Multi-objective artificial bee colony algorithm [3], [14] and a novel multi-objective optimization algorithm based on artificial bee colony [57] are some of the competitive swarm intelligence based MOO algorithms that aimed to obtain approximate Pareto front for multi-objective problems.
Recently, several authors carried out the work for the performance improvement of multi-objective algorithms. Wang et al. [49] focus on improving the accuracy and the diversity of multi-objective evolutionary algorithm through the local application of evolutionary operators to selected sub-populations. Martin et al. [26] propose a new multi-objective evolutionary model to mine a set of quantitative association rules. Chen and Zou [6] perform runtime analysis of a multi-objective evolutionary algorithm for obtaining finite approximations of Pareto fronts. Li et al. [22] presents the method for achieving balance between proximity and diversity in multi-objective evolutionary algorithm. Chen et al. [7] presents a new archive-updating strategy by combining convergence with diversity for multi-objective evolutionary algorithm. Jiao et al. [17] proposes a direction vectors based co-evolutionary multi-objective optimization algorithm that introduces the decomposition idea from MOEA/D to co-evolutionary algorithms. Tan et al. [44] proposes a uniformly designed multi-objective differential algorithm based on decomposition for optimizing multi-objective problems. Pedro and Takahashi [30] present the interactive non-dominated Sorting algorithm with Preference model for multi-objective optimization problem. Kundu et al. [20] proposed multi-objective artificial weed colony (IWC) algorithm. IWC algorithm simulates the ecological process of weeds colonization and distribution.
The above mentioned algorithms are efficient and have competitive performance over the MOO problems. However, these algorithms have deficiencies in optimizing some of the multi-objective problems. This is mainly due to the complexity associated with the tuning of the control parameters of the algorithms [10]. Thus, it is required to design the new MOO algorithms for obtaining better solutions. Teaching–learning based optimization algorithm [33], [34], [38], [39], [40] is among the recently introduced meta-heuristics. The main difference between the TLBO and other optimization algorithms is: TLBO algorithm does not required the tuning of the control parameter (except population size and number of generation) for its working. This characteristic of the TLBO algorithm made it more efficient and effective than other optimization algorithms. In order to enhance the performance of the basic TLBO algorithm, modifications in the basic TLBO algorithm are reported in the literature [35], [36], [37] with its applications to the thermal engineering design problems.
Previously, some authors presented multi-objective variants of basic TLBO algorithm. Zou et al. [56] proposed multi-objective version of basic TLBO algorithm by adopting the non-dominated sorting and crowding distance sorting concepts. Medina et al. [27] proposed multi-objective teaching–learning algorithm based on decomposition where multi-objective optimization problem decomposed into different scalar optimization sub problem and then the algorithm solved these sub problems simultaneously. Niknam et al. [29] proposed θ-multi-objective teaching–learning based optimization where the optimization process is done on the phase angles which are allocated to the design variables rather than design variables themselves. The θ-MOTLBO uses an external archive named repository to store non-dominated solutions. Krishnanand et al. [18] proposed MO-TLBO algorithm utilizing the concept of non-dominated sorting to solve the environmental/economic dispatch problem.
In this work, a multi-objective variant of the improved teaching–learning based optimization (ITLBO) is presented. The ITLBO algorithm incorporates modifications in the basic TLBO algorithm and results in well balanced search mechanisms to balance exploration and exploitation in an effective manner. Moreover, the performance of the MO-ITLBO algorithm is also compared to the basic MO-TLBO algorithm to assess the individual effectiveness of both.
The remainder of this paper is organized as follows. Section 2 briefly describes the basic TLBO algorithm. Section 3 presents ITLBO algorithm. Section 4 explains the proposed MO-ITLBO algorithm. Section 5 presents experimentation on a set of standard test problems proposed for the CEC 2009. Section 6 presents statistical analysis of the results and the conclusion is presented in Section 7.
Section snippets
Teaching–learning-based optimization (TLBO) algorithm
Teaching–learning is an important motivated process where any individual tries to learn something from the other. Traditional class room teaching–learning environment is one sort of motivated process where students learn from the teacher to improve his/her knowledge. Based on this fact, Rao et al. [33], [34], [38], [39], [40] has proposed an algorithm known as teaching–learning based optimization (TLBO) which simulates the traditional teaching–learning phenomenon of class room. The algorithm
Improved TLBO (ITLBO) algorithm
The ITLBO algorithm is the improved version of the basic TLBO algorithm. Earlier the basic TLBO algorithm is modified by introducing the concept of “Number of teachers” and “Adaptive teaching factor” [35], [36], [37] as well as “Self motivated learning” [35], [37] to improvise its performance and applied it to optimization of thermal system. In the present work, the previous modifications are clubbed together and new modification in the form of tutorial training is introduced to enhance the
Multi-objective Improved TLBO (MO-ITLBO) algorithm
The MO-ITLBO is a Pareto based algorithm with an external archive to store non-dominated solutions. Note that two solutions are non-dominated when in a minimization problem such that no one has less value in all objectives than the other one or vice versa. To include the latest solutions found in the archive and check if they dominate other solutions or other archive members or versa, MO-ITLBO algorithm need an updating method. The MO-ITLBO algorithm uses ε-dominance method. In ε-dominance, the
Experimental investigation
Congress on Evolutionary Computation (CEC) provides set of benchmark functions for assessing the performance of optimization algorithms. CEC 2005, 2006 and 2010 [43], [23], [45] focused on optimization computation of single objective functions while CEC 2009 [54] focus on optimization computation of multi-objective functions.
In this section, the ability of the MO-ITLBO algorithm is assessed by applying it for the optimization of 20 well defined benchmark functions of CEC 2009 [54]. Out of 20
Statistical analysis
One of the most frequent situations for the use of statistical tests is in the multi data analysis, where the experimental results achieved by various algorithms in computational intelligence are required to be tested. Statistical tests can perform two classes of analysis: pair wise comparisons and multiple comparisons. Pair wise statistical procedures perform individual comparisons between two algorithms. Therefore, in order to carry out a comparison which involves more than two algorithms,
Conclusions
In this work, the performance of multi-objective version of improved TLBO algorithm is investigated and compared its performance against state-of-the-art algorithms presented in the literature. The comparison between the multi-objective versions of the basic and improved TLBO is also presented in this work. The MO-ITLBO algorithm considered the mathematical concept of multi teacher, tutorial training and self motivated learning in addition to traditional class room teaching. The MO-ITLBO
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