Balanced multi-objective optimization algorithm using improvement based reference points approach

Highlights

• We suggest an enhancement for exploration and exploitation factors of the Equilibrium Optimizer (EO) to balance between exploration and exploitation operators in the proposed multi-objective equilibrium optimizer.

• We propose an improvement-based reference points method increasing the diversity between the population members in multi-objective optimization problems.

• The CEC 2020, CEC 2009, DTLZ, and ZDT test problems are solved using a multi-objective equilibrium optimizer algorithm.

• Our proposed algorithm outperforms the other algorithms in terms of the spread and inverted generational distance measures for different test problems that satisfy all characteristics for optimization problems.

Abstract

In this work, we explore a novel multi-objective optimization algorithm to identify a set of solutions that could be optimal for more than one task. The proposed approach is used to generate a set of solutions that balance the tradeoff between convergence and diversity in multi-objective optimization problems. Equilibrium Optimizer (EO) algorithm is a novel developed meta-heuristic algorithm inspired by the physics laws. In this paper, we propose a Multi-objective Equilibrium Optimizer Algorithm (MEOA) for tackling multi-objective optimization problems. We suggest an enhancement for exploration and exploitation factors of the EO algorithm to randomize the values of these factors with decreasing the initial value of the exploration factor with the iteration and increasing the exploitation factor to accelerate the convergence toward the best solution. To achieve good convergence and well-distributed solutions, the proposed algorithm is integrated with the Improvement-Based Reference Points Method (IBRPM). The proposed approach is applied to the CEC 2020, CEC 2009, DTLZ, and ZDT test functions. Also, the inverted generational and spread spacing metrics are used to compare the proposed algorithm with the most recent evolutionary algorithms. It's obvious from the results that the proposed algorithm is better in both convergence and diversity.

Introduction

Multi-objective optimization problems (MOPs) have gained their prominence in many real-world problems such as task scheduling [1], Water Distribution Networks (WDNs) [2], wind speed forecasting [3], protein structure [4], and traveling salesman problem [5]. When addressing MOP, decision-makers become frustrated because they need to make the optimal decision regarding two or more conflicting objectives. For example, there are two objectives in WDNs; one is to minimize the capital cost, while the other is to maximize network reliability. MOPs seek to optimize several and competitive objective functions. There are three approaches followed in tackling MOPs [6], [7], [8]. The first approach is priori in which all the objectives aggregated into a single objective using a weight that specifies its significance. Thus, we handle MOP as a single-objective problem. The second approach is posteriori, which tries to find set solutions known as Pareto-optimal solutions. After that, the decision-makers come to select one solution that satisfies their obligations [9]. Unlike the priori approach in giving weights for the objectives, the posteriori approach is more informed and justified [10]. Finally, the interactive approach comes to involve the human inside the optimization process and his effect in the algorithm search space.

Nobody can conceal the impressive success of meta-heuristic algorithms in solving many optimization problems [11], [12], [13], [14], [15], [16] due to their robustness and easy implementation. So, we will review some of the meta-heuristic dedicated to solve MOPs. The Evolutionary Algorithms (EAs) have attracted many researchers for tackling MOPs [17], [18], [19], [20]. Generally, the aim of Multi-objective evolutionary algorithms (MOEAs) is to obtain a set of Pareto-optimal solutions in a single run. Evolutionary algorithms can be classified into three groups: decomposition-based algorithms (e.g., MOEA/D [22] and MOEA/D-AWA [58]), dominance-based algorithms (e.g., NSGA-II [59], B-NSGA-III [60] and U-NSGA-III [21]), and indicator-based algorithms (e.g., IBEA [60] and HypE [61]). Some of the recent multi-objective evolutionary algorithms will be reviewed within the next subsections.

The authors in [22] suggested an Evolutionary Algorithm based on hypervolume measure employing the regularity property to handle the MOPs with complicated search space and the objective function. Also, Liu and Wang [23] proposed the EA for solving a particular type of constrained MOPs called Decision and Objective Spaces (DOS). It is the first time to apply the constraints for both the objective and decision spaces simultaneously. Chen et al. [24] integrated EA with reference vector adaptation and the diversity ranking method to manage the diversity of different types of Pareto fronts for MOPs and many-objective problems. Based on the decomposition strategy, the authors in [25] proposed the EA that divides the problems into several sub-problems. Then each sub-problem is solved based on the information taken from the neighboring sub-problems. Besides, the authors [26] addressed the different shapes of Pareto fronts using a multi-objective EA based on an improved inverted generational distance indicator.

Moreover, The comprehensive EA introduced by Seifollahi-Aghmiuni and Haddad [27] optimized the single-objective problems and MOPs by applying a unique structure. Most of its parameters didn't require more sensitivity analysis, but it consumes much time. Pedroso et al. [28] developed a parallel version of EA using genetic and differential evolution operators. The hypervolume as a performance measure to guide the search is computationally expensive with the increase of the number of objectives. So, Gómez et al. [29] provided a parallel EA based on an asynchronous island model with micro-populations to reduce the time execution. In [30], the author proposed a local search method, namely Pareto Archived Evolution Strategy (PAES), to maintain an archive of the previous Pareto optimal solutions for identifying the dominance between the current solutions and the archived one. Yen and Lu [31], an evolutionary approach based on the dynamic population size has been proposed for tackling the multi-objective problems. Also, the author in [32] tried to reduce the running time of the evolutionary algorithms by introducing a distributed cooperative EA, that share the computational workloads among a group of computers over the network.

In [33], a multi-objective evolutionary algorithm based decomposition improved using adaptive weight vector adjustment (MOEA/D-AWA) has been proposed to remedy the limitations of MOED/D on the problems with a complex Pareto front. In addition, B-NSGA-III [34] has been proposed to tackle the MOPs to produce much better convergence and diversity preservation in addition to stressing extreme objective-wise solutions. An indicator-based evolutionary algorithm (IBEA) [35] was developed to overcome the discrete and continuous benchmark problems by adapting search according to the arbitrary performance measures. Hypervolume estimation algorithm (Hype) [36] has been developed to overcome the computational effort required by Hypervolume indicator in addition to limitations of only solving problems with a few objectives.

Chen [37] proposed a dynamic constrained multi-objective evolutionary algorithm based on a non-dominated solution selection operator, a population selection operator, a change detection operator, and a mating selection strategy, and a change response strategy. The non-dominated solution selection operator gives the algorithm an ability to obtain a non-dominated population with diversity when the environment changes. To handle the infeasible solutions, the mating selection strategy and population selection operator was used. Additionally, the change response strategy was used to combine reinitializing the population randomly within the search space of the problem with reusing some parts of the previous solutions. A multi-objective brain storm optimization algorithm [38] for flow shop scheduling problem has been proposed to minimize two objectives together: the makespan, or also defined as the maximum completion time, and the total energy consumption. This algorithm was compared with NSGA-II and MOEA/D on a set of test problems and the experimental results show the superiority of this algorithm against the others when dealing with this problem.

It's time to expose some of the works done using the swarm algorithms for tackling the MOPs. A parallel Particle Swarm Optimization (PSO) [39] is incorporated by a cooperative coevolutionary and a strategy for limiting the velocity of the particles to avoid erratic movements. Also, Mokarram and Banan [40] supported PSO to handle MOPs based on the selection of the leader to guide the particles in the search space for diversity and fast convergence. Moreover, PSO [41] used an alternative repository of particles to help the other particles in their flight. Mousa et al. integrated PSO [42] by the PAES as a local search method. Furthermore, in [43], the authors proposed the niching particle swarm optimization integrated by a local search method for producing more solutions around the personal best. Zhang et al. [44] suggested a novel cluster-based on PSO with ring topology and an updating strategy for the leader to solve the multimodal MOPs.

Crow Search Algorithm (CSA) [45] is another metaheuristic employed for tackling MOPs. Nobahari and Bighashdel [46] provided a multi-objective CSA with a fitness function depending on the weight vectors. The algorithm used a chasing operator for faster convergence. Hybridization among algorithms is a trend to benefit from their strengths. In this context, Ramgouda and Chandraprakash [47] combined CSA with the fruitfly algorithm. Also, the authors in [48] incorporated the CSA with the sine cosine algorithm, which helps to make a trade-off between the exploration and the exploitation capabilities. Furthermore, The sine cosine algorithm [49] preserved the diversity of the optimal set of solutions using the elitist non-dominated sorting and crowding distance approach.

Furthermore, Differential evolution (DE) [50] has been proposed for solving the multimodal MOPs. Also, the authors [51] enhanced the performance of the Slap Swarm Algorithm (SSA) with DE for tackling the multi-objective big-data optimization. In the same context, the SSA has been proposed in [52] integrated by the PAES for tackling the MOPs. Many multi-objective algorithms are designed for dealing MOPs such as the flower pollination algorithm [53], bat algorithm [54], whale algorithm [55], grey wolf optimizer [56], grasshopper algorithm [57,58], artificial sheep algorithm [59], immune algorithm [60], spotted hyena optimizer [60], Shuffled frog leaping algorithm [61], moth flame algorithm [62,63], symbiotic organism search [64], DE solving efficiently the multi-objective optimization problems in presence of measurement noise [65], feature selection and weighting method based on MOEA/D [66], ant lion optimizer [67], Multiobjective Support Vector Machines [68], hyperspectral satellite image segmentation using Rényi entropy based on multi-level thresholding aided with DE [69], multi-objective Bi-Population Evolutionary Algorithm for flow shop scheduling problem [70], and U-NSGA-III [21].

According to the literature, most of the algorithms that have a significant effect on the MOPs belong to the evolutionary algorithms due to their ability in getting out of local minima and reaching regions the swarm algorithms couldn't reach within the search space of the problem. As a result, in this research, we motivate us to propose a new strategy to help the meta-heuristic algorithms in reaching better results by exploring other regions within the search space of the problem in the hope of finding better non-dominated solutions. Specifically, this strategy will remove unbeneficial solutions within the population and set to their positions other solutions generated in an effective manner working on promoting the exploration capability of the algorithm until reaching other regions for promoting the diversity within the whole optimization process. This strategy is integrated with an equilibrium optimizer to tackle the MOPs. Our improvement methodology within this research is as:

• An improvement in the original equilibrium optimizer has been proposed to increase the diversity among the population members and getting out of local minima. In this improvement, the particle that doesn't improve the local best solution within a number of iterations will be replaced with another solution that is generated using the genetic algorithm based on the reference points approach to enhance the diversity.

• Additionally, the exploration and exploitation factors of MOEO is randomly generated within a specific range until covering significantly the search space of the problem and accelerating the convergence toward the best solutions. This specific range will be changed with the iteration where the exploration factor will decrease while the exploitation will decrease to simulate the nature of meta-heuristic algorithms.

The main contributions of this paper include:

• The CEC 2020 test problems are solved using a multi-objective equilibrium optimizer algorithm.

• We propose an improvement-based reference points method increasing the diversity between the population members.

• We suggest an enhancement for exploration and exploitation factors of the EO to accelerate the convergence toward the best solution.

• Our proposed algorithm outperforms the other algorithms in terms of the spread and inverted generational distance measures for the CEC 2020 test problems.

We organize this paper as follows. Section 2 summarizes the equilibrium optimizer algorithm. Section 3 presents in the general description of the multi-objective optimization problems. Section 4 describes the proposed algorithm for tackling MOPs. Section 5 provides the discussion and the experimental results of the proposed algorithm using the CEC 2020, CEC 2009, ZDT, and DTLZ test functions. Section 6 demonstrates some conclusions about the proposed approach and future work.