Q2 NSABC: Non-dominated sorting based multi-objective artificial bee colony algorithm and its application in data clustering Avadh Kishor n , Pramod Kumar Singh, Jay Prakash Q1 Computational Intelligence and Data Mining Research Laboratory, ABV-IIITM, Gwalior 474015, India article info Article history: Received 20 June 2015 Received in revised form 30 May 2016 Accepted 3 August 2016 Communicated by Zidong Wang Keywords: Multi-objective optimization Artificial bee colony algorithm Non-dominated sorting Crowding distance Augmented population abstract This paper presents a non-dominated sorting based multi-objective artificial bee colony algorithm NSABC to solve multi-objective optimization problems. It is an extension of the artificial bee colony algorithm ABC, which is a single objective optimization algorithm, to the multi-objective optimization domain. It uses a novel approach in the employee bee phase to steer the solutions to simultaneously achieve both the orthogonal goals in the multi-objective optimization – convergence and diversity. The onlooker bee phase is similar to the ABC except for the fitness computation to exploit the promising solutions whereas there is no change in the scout bee phase, which is used to get rid of bad solutions and add diversity in the swarm by introducing random solutions. Along with a novel way of exploring new solutions, it uses non-dominated sorting and crowding distance, inspired by the NSGA-II, to maintain the best and diverse solutions in the swarm. It is tested on the 10 two-objective and three-objective unconstrained bench- mark problems of varying nature and complexities from the CEC09 suite of test problems and is found better than or commensurable to thirteen state-of-the-art significant multi-objective optimization al- gorithms as well as other multi-objective variants of the ABC. Further, it is tested on the nine real-life data clustering problems considered from the UCI machine learning repository and proved itself better in comparison to the NSGA-II, MOVGA, and a recent multi-objective variant of the ABC named MOABC. Thus, it is observed that the NSABC is comparatively a simple, light, and powerful algorithm to solve multi-objective problems. & 2016 Elsevier B.V. All rights reserved. 1. Introduction The optimization problems, which involve a sole optimization function, are known as single objective optimization problems. The aim in such problems is to find a single optimum value of the objective function. Except for the multimodal problems, these problems are simple to solve and many conventional as well as evolutionary and swarm intelligence based algorithms are avail- able in the literature to solve such problems. However, most of the real-world problems in the engineering and other domains involve multiple objective functions, which are required to be optimized simultaneously. Such problems are known as multi-objective op- timization problems and are much harder to solve as most of the problems involve conflicting objectives. In this case, it is not pos- sible to obtain a single solution, which is optimum to all the conflicting objectives. Instead the solver obtains a set of solutions, where each solution is a trade-off of the objectives and no solution is inferior to the other solutions in this set. This set is known as the Pareto-optimal set. Generally, for the real-world problems, the number of solutions in the Pareto-optimal set is extremely large or may be even infinite as it grows exponentially with the problem size. Thus, these problems are NP-hard problems [1]. The conventional methods available in the literature to solve multi-objective optimization problems are perplexed with various problems, e.g., they get stuck at a suboptimal solution, they are not efficient in handling problems with discontinuous and/or discrete search space, a method efficient in solving one problem may not be efficient in solving other problem [2]. Here, the evolutionary and swarm based optimization algorithms come to rescue. They are specially suited to solve such problems as they obtain multiple solutions in a single run and use synergy of the solutions in the population/swarm to steer them to find the Pareto-optimal set. Moreover, they are not problem specific. Instead, they are applied to array of problems. As early as 1957, Box [3] realized a necessity to consider more than one objective function in a design to increase industrial productivity. Thereafter, in 1966 Fogel et al. [4] argued a weighted approach to handle multiple goals in two distinct scenarios. Fi- nally, Schaffer in 1984 [5,6] presented and implemented first true Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 http://dx.doi.org/10.1016/j.neucom.2016.08.003 0925-2312/& 2016 Elsevier B.V. All rights reserved. n Corresponding author. E-mail addresses: avadhkishor133@gmail.com (A. Kishor), pksingh@iiitm.ac.in (P.K. Singh), jayprakash.iiitm@gmail.com (J. Prakash). Please cite this article as: A. Kishor, et al., NSABC: Non-dominated sorting based multi-objective artificial bee colony algorithm and its application in data clustering, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.08.003i Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎