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Top1. Introduction
Most of the real-world optimization problems have multiple objectives with a large number of decision variables. These problems are known as Multi-Objective Optimization Problems (MOOPs). Instead of a single solution as in Single-Objective Optimization Problems (SOOPs), MOOPs have a set of optimal solutions. MOOPs are difficult to solve than SOOPs due to attributes like conflicting objectives, dimensionality, time constraints and in-homogeneity.
MOOP differs from SOOP in two aspects. First is the number of objective functions and second is the criteria used to compare the solutions (Deb, 2014). SOOP typically has a single optimal solution and solutions are compared based on their fitness value. In contrast, MOOPs have a set of optimal solutions, which are known as Pareto front (Deb, 2001; Deb 2014). A solution from such set cannot be dominated by any other solution; that’s why these solutions are also known as non-dominated or non-inferior solutions (Coello et al., 2004). A Pareto front corresponds to a curve or an extremely complex hyper-surface in objective space. Based on the additional subjective preference information of the problem, the user can select any single optimal solution out of the Pareto front solutions.
MOOPs solving approaches are categorized into two types, namely priori and posteriori approach (Rao et al., 2016). The priori approaches solve MOOP by assigning weights of importance to the objective functions and later convert it into an SOOP. The weighted sum method, ϵ-constraint method, weighted metric method, weighted aggregation approach, value function method, Benson’s method and goal programming methods fall in the category of priori approaches (Deb, 2001). The posteriori approaches do not require any weight assignment concept. These approaches provide a Pareto front for MOOPs during the simulation.
Nature-Inspired Algorithms (NIAs) are now among the widely used meta-heuristic algorithms for solving SOOPs, due to their flexibility and efficiency (Fister Jr et al., 2013). NIAs have also shown their promising performance for MOOPs. In the past, many multi-objective variants of the Single-Objective Optimization Algorithm (SOOA) have been proposed by various researchers. These algorithms are named as Multi-Objective Optimization Algorithms (MOOAs). Most of the NIAs are based on the social behavior of the species in nature (Fister Jr et al., 2013). Some of the well-established MOOAs are Non-dominated Sorting Genetic Algorithm-II (NSGA-II) (Deb et al., 2002), Vector Evaluated Genetic Algorithm (VEGA) (Schaffer, 1985), Multi-Objective Evolutionary Algorithm (MOEA) (Coello et al., 2007), Strength Pareto Evolutionary Algorithm-2 (SPEA-2) (Zitzler et al., 2001), Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) (Zhang and Li, 2007), Multi-Objective Differential Evolution (MODE) (Xue et al., 2003), Pareto Envelope-based Selection Algorithm-II (PESA-II) (Corne et al., 2000; 2001), Multi-Objective Particle Swarm Optimization (MOPSO) (Mostaghim and Teich, 2003; Coello et al., 2004), Multi-Objective Artificial Bee Colony (MOABC) (Akbari et al., 2012), Multi-Objective Firefly Algorithm (MOFA) (Yang, 2013), Multi-Objective Teaching Learning based Optimization (MOTLBO) (Zou et al., 2013), Multi-Objective Flower Pollination Algorithm (MOFPA) (Yang et al., 2013), Multi-Objective Jaya Algorithm (Rao et al., 2016), ϵ-Constraint Heat Transfer Search (ϵ-HTS) (Tawhid and Savsani, 2018) and others. These aforementioned algorithms are the extended versions of SOOAs for solving the MOOPs. Although many MOOAs are available for solving the MOOPs, there still remains scope for a new generalized algorithm with improved robustness, efficiency and correctness (Wolpert and Macready, 1997). The MOOAs have aimed to find an approximate set of solutions as close as possible to the solutions of the Pareto front of MOOP and all solutions are diverse and uniformly distributed in Pareto front (Li et al., 2015).