Multi-Objective Multiagent Credit Assignment in NSGA-II Using Difference Evaluations

This study compared the performances of three state-of-theart evolutionary multi-objective optimization algorithms: the Non-Dominated Sorted Genetic Algorithm II (NSGA-II), the Epsilon-Dominance Non-Dominated Sorted Genetic Algorithm II (e -NSGA-II) and the Epsilon-Dominance Multi-Objective Evolutionary Algorithm ( e MOEA), on a constrained biobjective crop-mix planning problem (BCP). The BCP test case objectives were to: (i) maximize crop production and (ii) minimize planting area. The BCP problem was enumerated to provide the true Pareto-optimal solution set to facilitate rigorous testing of the evolutionary multi-objective optimization algorithms. The performances of the three algorithms were assessed and compared using three performance metrics (additive epsilon indicator, inverted generational distance and spacing). For the comparative study, a graphical comparison with respect to true Pareto front of optimal crop planning problem was provided. Results of the analyses indicated that the e MOEA greatly outperforms the NSGAII and the e -NSGA-II.

For a high-speed and high-acceleration planar parallel mechanism, a design for control (DFC) approach is proposed for the optimization of the mechanism structure. Several evaluation indexes of kinematics and dynamics performance are established for the object functions and constraints, and then a standard multi-objective optimization problem can be formed. The dimension synthesis is carried out by solving the multi-objective optimization with NSGA-II (non-dominated sorting genetic algorithm-II) algorithm. The final result shows that the mechanism with the optimized parameters can eliminate the systematic coupling and enhance the dynamic performance greatly, which provides a solid mechanism hardware platform for designing the high-speed control system.

Almost all real-world and engineering problems involve multi-objective optimization of some sort that is often constrained. To solve these constrained multi-objective optimization problems, constrained multi-objective optimization evolutionary algorithms (CMOEAs) are enlisted. These CMOEAs require specific constraint handling techniques. This study aims to address and test the most successful constraint handling techniques, seven different penalty constraint techniques, as applied to the Non-dominated Sorting Genetic Algorithm II (NSGA-II). In this paper, NSGA-II is chosen because of its high popularity amongst evolutionary algorithms. Inverted Generational Distance and Hypervolume are the main metrics that are discussed to compare the constraint handling techniques. NSGA-II is applied on 13 constrained multi-objective problems known as CF1-CF10, C1-DTLZ1, C2-DTLZ2, and C3-DTLZ4. The result of IGD and HV values are compared and the feasibility proportions of each combination on each problem are shown. The results of simulation present interesting findings that have been presented at the end of paper as discussion and conclusion.

Mathematical runtime analysis for the non-dominated sorting genetic algorithm II (NSGA-II)

The non-dominated sorting genetic algorithm II (NSGA-II) is the most intensively used multi-objective evolutionary algorithm (MOEA) in real-world applications. However, in contrast to several simple MOEAs analyzed also via mathematical means, no such study exists for the NSGA-II so far. In this work, we show that mathematical runtime analyses are feasible also for the NSGA-II. As particular results, we prove that with a population size larger than the Pareto front size by a constant factor, the NSGA-II with two classic mutation operators and three different ways to select the parents satisfies the same asymptotic runtime guarantees as the SEMO and GSEMO algorithms on the basic OneMinMax and LOTZ benchmark functions. However, if the population size is only equal to the size of the Pareto front, then the NSGA-II cannot efficiently compute the full Pareto front (for an exponential number of iterations, the population will always miss a constant fraction of the Pareto front). Our experiments confirm the above findings.

Water distribution networks (WDN) model optimization is an important part of smart water systems to achieve optimal strategies. WDN optimization focuses on the nonlinearity of the discharge head loss equation, the availability of discrete properties of pipe sizes, and the conservation of constraints. Multi-objective evolutionary algorithms (MOEAs) have been proposed and successfully applied in the field of WDN design optimization. Previous studies have focused on comparing the optimization effects of algorithms in water distribution networks, ignoring the problems of unbalanced pressure distribution and water hammer at the nodes of the pipe network caused by the complex terrain in mountainous areas. In this paper, a multi-objective water supply optimization model that integrated cost, reliability, and water quality was established for a mountainous WDN in real engineering. The method of traversing the nodes to solve the water age was introduced to find a more scientific and practical water age solution model, with setting the weight function to evaluate the water age of the water supply model comprehensively. Strength Pareto Evolutionary Algorithm II (SPEA-II) and Non-dominated Sorting Genetic Algorithm II (NSGA-II) were adopted to optimize the WDN design model in the mountainous complex terrain. The significance levels of the number of Pareto solutions (NOPS) and running time are 0.029 and 0.001, respectively, indicating that the two algorithms have significant differences. Compared to NSGA-II, SPEA-II has a better convergence rate and running time in multi-objective water supply optimization design. The solution set distribution of SPEA-II is more concentrated than that of NSGA-II, also the numerical value is better. The number of SPEA-II optimization schemes is larger and the scheme is more effective. Among them, the Pareto solution set of SPEA-II can obtain more desirable optimization results on cost, reliability index (RI) and water age. In summary, the study provides valuable information for decision makers in WDN with complex terrain.

Non-orthogonal multiple access (NOMA) deployment in future wireless networks has been recently considered a promising radio access technology to enhance spectral efficiency (SE). However, gain in SE comes always with the cost of energy efficiency (EE). In this paper, we investigate the SE and EE tradeoff in downlink NOMA with the consideration of quality of service (QoS) requirements based on three population-based multi-objective evolutionary algorithms (MOEAs): multi-objective particle swarm optimization (MOPSO), non-dominated sorting genetic algorithm-II (NSGA-II) and strength Pareto evolutionary algorithm-2 (SPEA2). The tradeoff is optimized and Pareto optimal solutions are obtained through MOEAs. The effectiveness of the algorithms is evaluated based on the hypervolume metric and the capability of solving multi-objective optimization problems. Simulation results reveal that SPEA2 outperforms NSGA-II and MOPSO. Furthermore, NSGA-II is the loser among all algorithms in terms of finding Pareto optimal results.

Distribution system (DS) service restoration is a very complex multi-objective and multi-constraint optimization problem, which requires high-quality Pareto fronts for helping the DS operators' work. This paper proposes a multi-objective evolutionary algorithm (MOEA) that combines the advantages of MOEAs in Tables with the properties of the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to recover Pareto fronts of the service problem. Its main differentials are its ability of recovering high-quality Pareto fronts, even in large-scale DSs, and of prioritizing switching operations in Remotely Controlled Switches (RCSs), independently of amount of RCSs. In opposite of Manually Controlled Switches (MCSs), RCS can be faster and cheapest operated from operating center. Several tests were carried out to validate and to compare the proposed MOEA to three MOEAs published in literature. Four metrics were used for assessing the quality of Pareto fronts generated and a Welch's t hypothesis test enabled the comparison of MOEAs' performance. Test results indicate the proposed MOEA outperforms those three MOEAs published in literature.

This paper proposes an evolutionary alternative to conventional two-phase planning methods for solving the integrated crew scheduling (ICS) problem. The approach models and formulates the ICS problem as a combinational optimization problem with multiple constraints and objectives. An integrated evolutionary framework is proposed for simultaneously considering crew pairing and crew rostering subproblems. To improve the efficiency of the Pareto set explorer, the solution methodology applies a novel variant of the nondominated sorting genetic algorithm II (NSGA-II), one of the most popular multiobjective optimization evolutionary algorithms. The proposed variant features problem-dependent constraint handling and a bounded front policy to reserve diverse individuals. The proposed approach is verified and validated in a case study of a real-world short-haul airline crew scheduling problem. The experimental results obtained by the proposed integrated approach are then compared with a real-world airline plan generated by the conventional sequential method. An aircraft schedule recovery problem is also studied to compare solution performance between the conventional NSGA-II method and the proposed NSGA-II variant. The comparison results confirm that the proposed variant obtains solutions that are superior in two aspects: First, the proposed NSGA-II variant obtains better convergence in the studied problems compared with the original version; second, the results explored by the variant enable decision makers to select from multiple crew schedules, which are superior to the real-world airline crew plan while considering the same objectives and constraints.

Using Pareto-based multi-objective Evolution algorithms in decision structure to transfer the hazardous materials to safety storage centre

Multi-objective and multi-parameter optimization of two-stage thermoelectric generator in electrically series and parallel configurations through NSGA-II

In order to obtain higher quality silicon single crystal, a hybrid strategy for modeling of Czochralski silicon crystal growth and optimizing of process parameters is presented in the paper. The hybrid strategy includes the computational fluid dynamics (CFD) method, neural network of group method of data handing (GMDH), and improved nondominated sorting genetic algorithm II (NSGA‐II). The shape variable of solid‐liquid interface h and the defect evaluation criteria V/G are set to objective functions according to engineering experience and process requirement. The polynomial of the objective functions is produced by GMDH and CFD. Ultimately, an improved elitist strategy and crowding distance NSGA‐II is proposed in the paper to obtain the Pareto optimum solution by the objective functions identified by the GMDH. Compared with other optimization algorithms, the improved NSGA‐II can increase the lateral diversity and the uniform distribution of the nondominated solutions. Engineering validation proved that the proposed hybrid strategy can effectively solve the complex uncertain multiobjective optimization problem of system and provides a new method for obtaining high‐quality crystal growth process parameters.

The non-dominated sorting genetic algorithm II (NSGA-II) is applied to the concurrent generation scheduling and load management of the isolated microgrid: the joint generation — voyage scheduling of an all-electric ship (AES). The AES uses integrated power generators and an energy storage system (ESS) to match its propulsion and service loads, thus forming an isolated microgrid. The propulsion load is the variable load that determines th ship voyage; hence, the schedulings of the generator, the propulsion load and the ESS have to be processed concurrently, for the optimal operation of the AES. NSGA-II outperforms the existing methods for the AES scheduling problem by modelling the schedulings in a one-stage optimisation. Another advantage of NSGA-II applied here is its multi-objective optimisation. Existing work on ship scheduling has hitherto set the minimisation of the AES operational cost as the single optimisation objective, with the reduction of the greenhouse gas (GHG) emission merely treated as a constraint of the optimisation. The proposed work emphasises the environmental concerns and makes GHG mitigation a separate objective, thus expanding the optimisation into multi-objective. The simulation demonstrate that optimising the ship voyage jointly with the generation scheduling results in the reduction of both operational cost and GHG emission, compared to the fixed-voyage generation scheduling. Integrating the ESS dispatch into the generation scheduling further enhances the benefits.

Minimizing carbon emissions in power system operations has become a critical challenge as the energy sector transitions toward sustainability. This paper presents a high-fidelity simulation framework for CO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-optimized energy dispatch, employing an enhanced Non-Dominated Sorting Genetic Algorithm II (NSGA-II) tailored for emission-oriented optimization. Unlike traditional economic dispatch approaches that primarily focus on cost, the proposed framework explicitly minimizes total CO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> emissions while ensuring demand–supply balance, operational limits, and ramp-rate feasibility. Key enhancements to the baseline NSGA-II include adaptive crossover–mutation operators, hybrid population initialization, and constraint-repair mechanisms, which improve convergence speed, robustness, and solution diversity. The framework is validated through multiple case studies incorporating stochastic load demand and renewable variability. Results show that the enhanced NSGA-II achieves nearly 20% emission reduction compared to baseline economic dispatch, with faster convergence and a broader Pareto-optimal solution set than standard NSGA-II. Furthermore, dispatch outcomes demonstrate increased reliance on renewable and storage units, reducing the share of coal and gas generation. These findings highlight the potential of the proposed approach to support low-carbon, policy-aligned energy system operations and provide a foundation for future extensions in multi-objective optimization, real-time scheduling, and carbon pricing integration.

Multi- objective modeling and optimization of dissimilar laser welding by integrating an artificial intelligence predictive model with NSGA-II algorithm