An Orthogonal Evolutionary Algorithm With Learning Automata for Multiobjective Optimization.

A Multiobjective Optimization Problem (MOP) requires the optimization of several objective functions simultaneously, usually in conflict with each other. One of the most efficient algorithms for solving MOPs is MOEA/D (Multiobjective Evolutionary Algorithm Based on Decomposition), which decomposes a MOP into single-objective optimization subproblems and solves them using information from neighboring subproblems. MOEA/D variants with other evolutionary operators have emerged over the years, improving their efficiency in various MOPs. Recently, the IWOA (Improved Whale Optimization Algorithm) was proposed, an optimization algorithm bioinspired by the whale hunting method hybridized with Differential Evolution, which presented excellent results in single-objective optimization problems. This work proposes the MOEA/D-IWOA algorithm, which associates characteristics of the evolutionary operators of the IWOA to MOEA/D. Computational experiments were accomplished to analyze the performance of the MOEA/D-IWOA in benchmark MOPs suites. The results were compared with those obtained by the MOEA/D, Non-dominated Sorting Genetic Algorithm II (NSGA-II), Third Evolution Step of Generalized Differential Evolution (GDE3), Improving the Strength Pareto Evolutionary Algorithm (SPEA2), and Indicator-Based Evolutionary Algorithm (IBEA) algorithms in the Hypervolume and Inverted Generational Distance Plus (IGD+) indicators. The MOEA/D-IWOA proved to be competitive, with a good performance profile, in addition to presenting the best results in some POMs.

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) is a popular algorithm for solving Multi-Objective Problems (MOPs). The main component of MOEA/D is to decompose a MOP into easier sub-problems using a set of weight vectors. The choice of the number of weight vectors significantly impacts the performance of MOEA/D. However, the right choice for this number varies, given different MOPs and search stages. We adaptively change the number of vectors by removing unnecessary vectors and adding new ones in empty areas of the objective space. Our MOEA/D variant uses the Consolidation Ratio to decide when to change the number of vectors and to decide where to add or remove these weighted vectors. We investigate the effects of this adaptive MOEA/D against MOEA/D with a poorly chosen set of vectors, a MOEA/D with fine-tuned vectors and MOEA/D with Adaptive Weight Adjustment on two commonly used benchmark functions. We analyse the algorithms in terms of hypervolume, IGD and entropy performance. Our results show that the proposed method is equivalent to MOEA/D with fine-tuned vectors and superior to MOEA/D with poorly defined vectors. Thus, our adaptive mechanism mitigates problems related to the choice of the number of weight vectors in MOEA/D, increasing the final performance of MOEA/D by filling empty areas of the objective space and avoiding premature stagnation of the search progress.

In optimization problems it is often necessary to perform an optimization based on more than one objective. The goal of the multiobjective optimization is usually to find an approximation of the Pareto front which contains solutions that represent the best possible trade-offs between the objectives. In a multiobjective scenario it is important to both improve the solutions in terms of the objectives and to find a wide variety of available options. Evolutionary algorithms are often used for multiobjective optimization because they maintain a population of individuals which directly represent a set of solutions of the optimization problem. multiobjective evolutionary algorithm based on decomposition (MOEA/D) is one of the most effective multiobjective algorithms currently used in the literature. This paper investigates several methods which increase the selective pressure to the outside of the Pareto front in the case of the MOEA/D algorithm. Experiments show that by applying greater selective pressure in the outwards direction the quality of results obtained by the algorithm can be increased. The proposed methods were tested on nine test instances with complicated Pareto sets. In the tests the new methods outperformed the original MOEA/D algorithm.

Multi objective optimization problems often involve computationally expensive objective functions, making traditional evolutionary algorithms impractical in real-world applications. Surrogate-assisted approaches have emerged as an effective solution by approximating expensive evaluations with machine learning models. In this study, we compare the performance of standard NSGA-II with a surrogate-assisted, uncertainty-aware variant against several benchmark multi-objective algorithms, including Generalized Differential Evolution (GDE), Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), Optimized Multi-Objective Particle Swarm Optimization (OMOPSO), and Strength Pareto Evolutionary Algorithm 2 (SPEA2), on the vehicle crashworthiness design problem. Performance was assessed using Inverted Generational Distance (IGD) and Hypervolume (HV) over 100 independent runs. The results show that surrogate-assisted NSGA-II significantly outperforms the baseline NSGA-II and other algorithms, achieving the lowest IGD (0.2976) and the highest HV (1.6023E+16). These findings demonstrate the potential of incorporating uncertainty-aware surrogate models to enhance the efficiency and solution quality of evolutionary optimization in computationally demanding engineering applications.

Although a number of multiobjective evolutionary algorithms have been proposed over the last two decades, not much effort has been made to deal with variable linkages in multiobjective optimization. Recently, we have suggested a general framework of multiobjective evolutionary algorithms based on decomposition (MOEA/D) [1]. MOEA/D decomposes a MOP into a number of scalar optimization subproblems by a conventional decomposition method. The optimal solution to each of these problems is a Pareto optimal solution to the MOP under consideration. An appropriate decomposition could make these individual Pareto solutions evenly distribute along the Pareto optimal front. MOEA/D aims at solving these scalar optimization subproblems simultaneously. In this paper, we propose, under the framework of MOEA/D, a multiobjective differential evolution based decomposition (MODE/D) for tackling variable linkages. Our experimental results show that MODE/D outperforms several other MOEAs on several test problems with variable linkages.

Hybrid flow shop scheduling (HFS) is an on sought problem modelling for production manufacturing. Due to its impact on productivity, researchers from different backgrounds have been attracted to solve its optimum solution. The HFS is a complex dilemma and provides ample solutions, thus inviting researchers to propose niche optimization methods for the problem. Recently, researchers have moved on to multi-objective solutions. In real-world situations, HFS is known for multi-objective problems, and consequently, the need for optimum solutions in multi-objective HFS is a necessity. Regarding sustainability topic, energy utilization is mainly considered as one of the objectives, including the common makespan criteria. This paper presents the existing multi-objective approach for solving energy utilization and makespan problems in HFS scheduling using Non-Dominated Sorting Genetic Algorithm-III (NSGA-III), and a comparison to other optimization models was subjected for analysis. The model was compared with the most sought algorithm and latest multi-objective algorithms, Strength Pareto Evolutionary Algorithm 2 (SPEA -II), Multi-Objective Algorithm Particle Swarm Optimization (MOPSO), Pareto Envelope-based Selection Algorithm II (PESA-II) and Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D). The research interest starts with problem modelling, followed by a computational experiment with an existing multi-objective approach conducted using twelve HFS benchmark problems. Then, a case study problem is presented to assess all models. The numerical results showed that the NSGA-III obtained 50% best overall for distribution performance metrics and 42% best in convergence performance metrics for HFS benchmark problems. In addition, the case study results show that NSGA-III obtained the best overall convergence and distribution performance metrics. The results show that NSGA-III can search for the best fitness solution without compromising makespan and total energy utilization. In the future, these multi-objective algorithms’ potential can be further investigated for hybrid flow shop scheduling problems.

Multiobjective evolutionary algorithms based on decomposition (MOEA/D) decompose a multiobjective optimization problem (MOP) into a group of subproblems and optimizes them at the same time. The reproduction method in MOEA/D, which generates offspring solutions, has crucial effect on the performance of algorithm. As the difficulties of MOPs increases, it requires much higher efficiency for the reproduction methods in MOEA/D. However, for the complex optimization problems whose PS shape is complicated, the original reproduction method used in MOEA/D might not be suitable to generate excellent offspring solutions. In order to improve the property of the reproduction method for MOEA/D, this paper proposes an external archive matching strategy which selects solutions’ most matching archive solutions as parent solutions. The offspring solutions generated by this strategy can maintain a good convergence ability. To balance convergence and diversity, a perturbed learning scheme is used to extend the search space of the solutions. The experimental results on three groups of test problems reveal that the solutions obtained by MOEA/D-EAM have better convergence and diversity than the other four state-of-the-art algorithms.

Wireless sensor networks design requires high quality location assignment and energy efficient power assignment for maximizing the network coverage and lifetime. Classical deployment and power assignment approaches optimize these two objectives individually or by combining them together in a single objective or by constraining one and optimizing the other. In this article a multi-objective deployment and power assignment problem (DPAP) is formulated and a multi-objective evolutionary algorithm based on decomposition (MOEA/D) is specialized. Following the MOEA/D's framework the above multiobjective optimization problem (MOP) is decomposed into many scalar single objective problems. The sub-problems are solved simultaneously by using neighborhood information. Additionally, unique problem-specific, parameter-rising, genetic operators and local search heuristics were designed specifically for the DPAP. In addition, a new encoding scheme is designed to represent a WSN based on the DPAP's design variables. Simulation results show that MOEA/D provides a high quality set of alternative solutions without any prior knowledge on the objectives preference.

Given that the definition of the multi-objective optimization problem is raised when number of objectives is increased in number at the optimization problem, where not only the number of objectives but also the computational resources which are needed to solve the problem, is also more desired. Therefore, novel approaches had required to solve multi-objective optimization problem in a reasonable time. One of this novel approach is utilization of the decomposition method with the evolutionary algorithm/operator. This algorithm was called multi-objective evolutionary algorithm based on decomposition (MOEA/D). Later on, variants have been proposed to improve the performance of the MOEA/D algorithm. However, a general comparison between these variants has needed for demonstrate the performance of these algorithm. For this reason, in this research the variants of MOEA/D algorithms have implemented on benchmark problems (DTLZ and MaF) and the performances has compared with each other. Two metrics had selected to evaluate/compare the performances of the variants. The metrics are IGD and Spread metrics. The results at the end of the implementations suggest that adaptive weighting idea is the most promising idea to increase the performance of the MOEA/D algorithm.

A modified PBI approach for multi-objective optimization with complex Pareto fronts

A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows

Simultaneous optimization of two or more objectives is an instance of multi-objective optimization (MOO). However, for most of the Multi-Objective Problems (MOPs), no single solution can optimize all the objective functions simultaneously. Evolutionary algorithms are a type of optimization algorithm used for constructing a well-distributed optimal front more quickly than a much more efficient approach. One of the most commonly used algorithms in this regard is differential evolution (DE). To solve the problem of non-dominated sorting for Multi-Objective DE (MODE), an approach is proposed that reduces the time complexity. DE is commonly recognized as one of the best methods for solving the MOPs. Several versions of DEs have been proposed for MOPs. This article proposes a novel mutation technique, named Environmental Parameter-based Multi-Objective Differential Evolution (EP-MODE) that acquires two additional environmental parameters to preserve diversity and accelerate the convergence. The proposed approach also reduces the time complexity of non-dominated sorting for Multi-Objective DE (MODE). The proposed mutation's performance has been evaluated on DTLZ series and ZDT series benchmark test functions on the Pareto-optimal front and compared with existing multi-objective algorithms (MODE and MODE-RMO). The solution comparatively verifies that the proposed EP-MODE outperforms most of the MODEs for lower dimension functions and higher functions. HIGHLIGHTS New environmental parameters-based mutation has been introduced in a differential evolution algorithm to solve multi-objective optimization problems (EP-mode) The addition of two additional environmental parameters to EP-mode has been found to be appropriate for maintaining diversity while also accelerating the convergence rate EP-MODE evaluated on bi-objective (DTLZ series) and tri-objective (ZDT series) benchmark test functions on the Pareto-optimal front and found EP-MODE's efficiency is higher than that of the MODE algorithm and the MODE algorithm with a ranking-based mutation operator GRAPHICAL ABSTRACT

There are a large number of multi-objective optimization problems in real-world applications, like in games, that need to be solved in real time. In order to meet this pressing need, we suggests a method of parallelizing the multi-objective evolutionary algorithm based on decomposition (MOEA/D). Furthermore, a novel task decomposition strategy and scalarizing method without the ideal point are proposed for meeting the requirements of real-time and precision of the game. By combining the novel scalarizing function and GPU-based CUDA technology with the MOEA/D, a parallel MOEA/D for real-time multi-objective optimization problems is developed, namely P-MOEA/D. Experimental studies on ZDT and DTLZ benchmark problems suggest that the P-MOEA/D algorithm is efficient and fast.

Multi-objective evolutionary algorithms based on decomposition (MOEA/Ds) convert a multi-objective optimization problem (MOP) into a set of scalar sub-problems, which are then optimized collaboratively. This paper designs a multi-objective pigeon-inspired optimization algorithm based on decomposition (MPIO/D) to improve the search efficiency by using the mechanism behind the remarkable navigation capacity of homing pigeons. The map and compass operator can record the direction of descent (rising) to generate good offspring. The landmark operator is used to accelerate the convergence of sub-problems with poor convergence. Compared to six competitive MOEA/Ds, MPIO/D has shown the advantages in tacking two benchmark sets of MOPs.

To obtain propellant formulations with superior comprehensive and robustness performance, the study establishes a multi-objective optimization model that accounts for uncertainties. The model adopts a bi-layer structure. The inner layer computes performance bounds to construct uncertainty intervals, which are subsequently transformed into deterministic performance via interval order relations. The outer layer optimizes component mass fractions using MOEA/D (Multi-objective Evolutionary Algorithm Based on Decomposition) to maximize the deterministic performance. The study leverages Large Language Models (LLMs) as pre-trained optimizers to automate the operator design of MOEA/D. Designers can identify formulations that satisfy the performance requirements and robustness criteria by adjusting uncertainty levels and MOEA/D weight coefficients. The results on ZDTs and UFs demonstrate that MOEA/D-LLM achieves approximately a 4.0% improvement in hypervolume values compared to MOEA/D. Additionally, the NEPE propellant optimization case shows that MOEA/D-LLM improves the computational speed by about 13.05% and enhances hypervolume values by around 2.7% compared to MOEA/D. The specific impulse increases by 1.11%, the generation of aluminum oxide and hydrogen chloride decreases by approximately 18.43% and 16.40%, respectively, and the impact sensitivity is reduced by about 1.67%.