Multi-objective Moth-Flame Optimization Algorithm for Engineering Problems

MOGBO: A new Multiobjective Gradient-Based Optimizer for real-world structural optimization problems

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

Although various optimization methods for solving single-objective problems have been developed in the last few decades, these methods have lost their eligibility due to the fact that today’s engineering problems are toward multiple objective optimization problems, in real applications. For single-objective optimization problems, for example, in case of a minimization problem, this value is the decision vector giving the smallest objective that can be achieved within the specified constraints. Hence the minimum decision vector within all possible (feasible) solution vectors is the so-called optimal solution and/or optimal design. However, in multi-objective optimization problems, since a different objective value is generated against each decision vector, the superiority of the solutions over each other is determined by considering the trade-off among the objective values. Therefore, the solution of multi-objective optimization problems, unlike single-objective problems, is a set of vectors rather than a single decision vector. In multi-objective optimization problems, especially if there are intricate objectives, the computational cost of the problem increases. In other words, while synchronously trying to maximize one of the objectives and to minimize another one makes it difficult to find the global optimum design. One of the important techniques used in multi-objective optimization problems is Pareto optimality which enables to select the global optimum solution taking into account the trade-off among all objectives. In this context, using of derivative-based methods has decreased, but the use of metaheuristic methods has increased due to the rapid availability of global optimum solution. This is because the improvements in the field of optimization are progressing in proportion to technology and varying according to the needs. In this chapter, one of the recent metaheuristic optimization methods based on swarm intelligence that is so-called a Pareto-based bat algorithm inspired by the behavior of determining the direction and distance of an object using the echo of the sound called the echolocation of bats is used in order to obtain optimum solutions for multi-objective engineering design problems. In this regard, a four-bar planar truss, a real-sized welded steel beam as well as a multi-layer radar absorber are selected as multi-objective engineering design optimization problems. In case the obtained results (optimal designs) are examined, the potency and the reliability of the proposed multi-objective Pareto-based bat algorithm are demonstrated.

Due to increased search complexity in multi-objective optimization, premature convergence becomes a problem. Complex engineering problems poses high number of variables with many constraints. Hence, more difficult benchmark problems must be utilized to validate new algorithms performance. A well-known optimizer, Multi-Objective Particle Swarm Optimizer (MOPSO), has a few weakness that needs to be addressed, specifically its convergence in high dimensional problems and its constraints handling capability. For these reasons, we propose a modified MOPSO (M-MOPSO) to improve upon these aspects. M-MOPSO is compared with four other algorithms namely, MOPSO, Multi-Objective Grey Wolf Optimizer (MOGWO), Multi-Objective Evolutionary Algorithm based on Decompositions (MOEA/D) and Multi-Objective Differential Evolution (MODE). M-MOPSO emerged as the best algorithm in eight out of the ten constrained benchmark problems. It also shows promising results in bioprocess application problems and tumor treatment problems. In overall, M-MOPSO was able to solve multi-objective problems with good convergence and is suitable to be used in real world problem.

In this paper, the problems of fuel cost and emission reduction based on multi-objective optimization are studied and solved. Various multi-objective optimization algorithms, such as the multi-objective ant lion optimization (MOALO), the multi-objective multiverse optimization (MOMVO), the multi-objective particle swarm optimization (MOPSO), the multi-objective thermal exchange optimization (MOTEO), and the multi-objective grasshopper optimization algorithm (MOGOA), are used to reduce the cost and emissions of generation fuel. With equality and inequality restrictions, the suggested work is carried out in an IEEE 30 bus system. To determine the lowest system-wide minimization of generating cost and emission, the results of several optimization strategies are compared. In terms of lowering fuel costs and emissions, the MOGOA performs better than any other multi-objective algorithms.

This paper addresses a multi-objective approach for simultaneous dynamic expansion planning of conventional sub-transmission grid and Regional Virtual Power Plant (RVPP). In such a feed-in tariff electricity market, it will assist RVVP’s stakeholders in deciding whether or not they should invest in new equipment’s installation. A Multi-Objective Particle Swarm Optimization (MOPSO) algorithm as a heuristic optimization method is proposed to eliminate the conventional centralized planning, which has led to competition between Regional Electric Company (REC) and RVPP for power delivery in sub-transmission system. Two objective functions are considered for simultaneous expansion planning of these two systems. The first one takes the minimum cost of the REC, as the sub-transmission grid operator, into account while the other one considers profit maximization for RVPP. To achieve the goals, MOPSO algorithm is employed to find the best expansion of REC and best location and capacity of RVPP’s resources. Having solved the proposed multi-objective optimization problem, a Pareto front is determined to show the trade-off between REC and RVPPs' contributions in joint optimal expansion planning of conventional sub-transmission grid and the RVPP. To demonstrate the applicability and effectiveness of the proposed approach, a realistic sub-transmission system in Guilan Province, Iran is used as a test system, and the results are evaluated accordingly.

A recently proposed swarm inspired optimization algorithm based on the navigation approach of Moths in space entitle as Moth-Flame Optimization (MFO) algorithm, is used for solve equality and inequality constrained optimization, real challenging layout problems. The navigating strategy of moths in universe entitles transverse orientation, a well active mechanism for travel so far distance in the straight direction. In fact, artificial lights trick moths, so they follow a deadly spiral path. MFO algorithm gives the competitive results with both continuous and discrete control variables. Real Challenging Constrained Optimization is a way of optimising an objective function in presence of constraints on some control variables. MFO have an ability to solve both constraints that may be either hard constrained or soft constrained. A statical representation of MFO algorithm expresses the best objective function value with reference to accuracy and standard deviation over recently proposed and most popular optimization algorithms. Fourteen constrained benchmark function of real engineering problems have been calculated and gained solutions were compared with the solution obtained by various recognized algorithms. The results obtained through MFO algorithm represent better solutions in the field of engineering design problems among many recently developed algorithms.

Battle Royale Optimizer (BRO) is a recently proposed optimization algorithm that has added a new category named game-based optimization algorithms to the existing categorization of optimization algorithms. Both continuous and binary versions of this algorithm have already been proposed. Generally, optimization problems can be divided into single-objective and multi-objective problems. Although BRO has successfully solved single-objective optimization problems, no multi-objective version has been proposed for it yet. This gap motivated us to design and implement the multi-objective version of BRO (MOBRO). Although there are some multi-objective optimization algorithms in the literature, according to the no-free-lunch theorem, no optimization algorithm can efficiently solve all optimization problems. We applied the proposed algorithm to four benchmark datasets: CEC 2009, CEC 2018, ZDT, and DTLZ. We measured the performance of MOBRO based on three aspects: convergence, spread, and distribution, using three performance criteria: inverted generational distance, maximum spread, and spacing. We also compared its obtained results with those of three state-of-the-art optimization algorithms: the multi-objective Gray Wolf optimization algorithm (MOGWO), the multi-objective particle swarm optimization algorithm (MOPSO), the multi-objective artificial vulture's optimization algorithm (MOAVAO), the optimization algorithm for multi-objective problems (MAOA), and the multi-objective non-dominated sorting genetic algorithm III (NSGA-III). The obtained results approve that MOBRO outperforms the existing optimization algorithms in most of the benchmark suites and operates competitively with them in the others.

Industrialization and population growth have been accompanied by many problems such as waste management worldwide. Waste management and reduction have a vital role in national management. The presents study represents a multi-objective location-routing problem for hazardous wastes. The model was solved using Non dominated Sorting Genetic Algorithm-II, Multi-Objective Particle Swarm Optimization, Multi-Objective Invasive Weed Optimization, Pareto Envelope-based Selection Algorithm, Multi-Objective Evolutionary Algorithm Based on Decomposition and Multi-Objective Grey Wolf Optimizer algorithms. The findings revealed that the Multi-Objective Invasive Weed Optimization algorithm was the best and the most efficient among the algorithms used in this study. Obtaining income from the incineration of the wastes and reducing the risk of COVID-19 infection are the first innovation of the present study, which considered in the presented model. The second innovation is that uncertainty was considered for some of the crucial parameters of the model while the robust fuzzy optimization model was applied. Besides, the model was solved using several meta-heuristic algorithms such as Multi-Objective Invasive Weed Optimization, Multi-Objective Evolutionary Algorithm Based on Decomposition and Multi-Objective Grey Wolf Optimizer, which were rarely used in literature. Eventually, the most efficient algorithm was identified by comparing the considered algorithms.

Hybrid multi-objective Harris Hawk optimization algorithm based on elite non-dominated sorting and grid index mechanism

The Production Routing Problem (PRP) has been extensively studied in logistics optimization, yet research that explicitly integrates environmental and social sustainability dimensions remains limited. This research proposes a sustainable production routing problem (SU-PRP) as a multi-objective optimization model, minimising the total operating cost, CO₂ emissions cost, and social cost simultaneously. The complex model is addressed using Multi-Objective Particle Swarm Optimization (MOPSO), which has been modified to handle discrete decision variables in a supply chain environment. The algorithm’s performance is benchmarked against NSGA-II using three quality indicators: Spacing Metric (SM), Inverted Generational Distance (IGD), and Hypervolume (HV). Numerical testing was performed using a moderate-sized problem instance comprising two plants, five customers, two products, and a two-period planning horizon, with 663 binary variables and 546 constraints. In the results, it can be observed that MOPSO has reached a larger HV, indicating better coverage of the objective space. Nevertheless, NSGA-II outperforms MOPSO in terms of solution uniformity and convergence, as evidenced by smaller SM and IGD values. The results suggest that MOPSO is more effective in operational planning scenarios where fast computation and solution diversity are crucial, such as real-time production and routing adjustments. In contrast, NSGA-II is more appropriate than MOPSO for strategic planning tasks to obtain more accurate and evenly distributed Pareto solutions.

Renewable Energy Resources (RES) can supply a clean and a smart solution to satisfy the rapidly growing load demand. The Photovoltaic (PV) and wind turbines are considered as resources of Distributed Generation (DG). The integration of PV - DG units into distribution systems has a significant effect on reducing the system active power losses. In this paper, a new technique based on the Moth Flame Optimization (MFO) algorithm, is used to optimally allocate the PV -DG units in Radial Distribution Systems (RDSs) and determine the optimal size of each unit. The objective of the technique is to minimize the total active power losses of the system, while satisfying several operating constraints. The proposed optimization algorithm is tested using three different IEEE standard RDSs of various sizes, which are 33, 69 and 118-bus systems. In order to demonstrate the applicability and the performance of the MFO algorithm, numerical simulations are obtained when applying the MFO algorithm, and compared with different existing optimization techniques. This comparison shows the validity of the proposed MFO algorithm.

In the last decade, multiobjective Particle Swarm Optimization (MOPSO) has been observed as one of the most powerful optimization algorithms in solving Multiobjective Optimization Problems (MOPs). Nowadays, it is becoming increasingly clear that MOPSO can handle complex MOPs based on the competitive-cooperative framework. The goal of this paper is to provide a comprehensive review of MOPSO from the basic principles to hybrid evolutionary strategies. To offer the readers insights on the prominent developments of MOPSO, the key parameters on the convergence and diversity performance of MOPSO were analyzed to reflect the influence on the searching performance of particles. Then, the main advanced MOPSO methods were discussed, as well as the theoretical analysis of multiobjective optimization performance metrics. Even though some hybrid MOPSO methods show promising multiobjective optimization performance, there is much room left for researchers to improve further, in particular in terms of engineering applications. As a result, further in-depth studies are required. This paper should motivate evolutionary computation researchers to pay more attention to this practical yet challenging area.

This paper proposes a modified particle swarm optimization algorithm for engineering optimization problems with constraints, in which the penalty function is employed to the traditional PSO algorithm, and at the same time adjusts the personal optimum and global optimum to make PSO being able to solve the non-linear programming problems, then the multi-objective problem can be converted into single objective problem. Moreover, the constraint term played its role in the process of generating particles, those pariticles which don't meet the constraint condition are eliminated. The actual engineering design optimization problem is tested and the results show that the multi-objective particle swarm optimization algorithm can be used to solve the multi-objective constrained optimization problem. Comparison with Genetic Algorithm confirms that the proposed algorithm can find better solutions, and converge quickly.

This paper formulates a bi-criteria fuzzy blocking flow shop scheduling problem with fuzzy processing time and fuzzy due date. An improved multi-objective grey wolf optimization (MOGWO) algorithm is proposed to solve this combinational optimization problem. The proposed MOGWO utilizes the ranked-order-value (ROV) rule for solution representation, employs a dynamic maintenance strategy to maintain archive, and develops a thorough mechanism for leader selection. In addition, to improve the performance of the neighborhood search, a VNS structure with three randomly ranked neighborhood search operators is introduced and implemented on the members of archive that may become the selected leaders. The proposed MOGWO is tested on a fuzzy blocking flow shop scheduling problem of panel block construction, and is compared with general MOGWO and multi-objective particle swarm optimization (MOPSO). Computational results suggest that the proposed MOGWO is superior to the compared algorithms in terms of the convergence, spread and coverage of the optimal solutions. This demonstrates the feasibility and effectiveness of the proposed MOGWO.