Multimodal Functions Research Articles

An Improved Whale Optimizer with Multiple Strategies for Intelligent Prediction of Talent Stability

Talent resources are a primary resource and an important driving force for economic and social development. At present, researchers have conducted studies on talent introduction, but there is a paucity of research work on the stability of talent introduction. This paper presents the first study on talent stability in higher education, aiming to design an intelligent prediction model for talent stability in higher education using a kernel extreme learning machine (KELM) and proposing a differential evolution crisscross whale optimization algorithm (DECCWOA) for optimizing the model parameters. By introducing the crossover operator, the exchange of information regarding individuals is facilitated and the problem of dimensional lag is improved. Differential evolution operation is performed in a certain period of time to perturb the population by using the differences in individuals to ensure the diversity of the population. Furthermore, 35 benchmark functions of 23 baseline functions and CEC2014 were selected for comparison experiments in order to demonstrate the optimization performance of the DECCWOA. It is shown that the DECCWOA can achieve high accuracy and fast convergence in solving both unimodal and multimodal functions. In addition, the DECCWOA is combined with KELM and feature selection (DECCWOA-KELM-FS) to achieve efficient talent stability intelligence prediction for universities or colleges in Wenzhou. The results show that the performance of the proposed model outperforms other comparative algorithms. This study proposes a DECCWOA optimizer and constructs an intelligent prediction of talent stability system. The designed system can be used as a reliable method of predicting talent mobility in higher education.

Explaining Exploration–Exploitation in Humans

Human as well as algorithmic searches are performed to balance exploration and exploitation. The search task in this paper is the global optimization of a 2D multimodal function, unknown to the searcher. Thus, the task presents the following features: (i) uncertainty (i.e., information about the function can be acquired only through function observations), (ii) sequentiality (i.e., the choice of the next point to observe depends on the previous ones), and (iii) limited budget (i.e., a maximum number of sequential choices allowed to the players). The data about human behavior are gathered through a gaming app whose screen represents all the possible locations the player can click on. The associated value of the unknown function is shown to the player. Experimental data are gathered from 39 subjects playing 10 different tasks each. Decisions are analyzed in a Pareto optimality setting—improvement vs. uncertainty. The experimental results show that the most significant deviations from the Pareto rationality are associated with a behavior named “exasperated exploration”, close to random search. This behavior shows a statistically significant association with stressful situations occurring when, according to their current belief, the human feels there are no chances to improve over the best value observed so far, while the remaining budget is running out. To classify between Pareto and Not-Pareto decisions, an explainable/interpretable Machine Learning model based on Decision Tree learning is developed. The resulting model is used to implement a synthetic human searcher/optimizer successively compared against Bayesian Optimization. On half of the test problems, the synthetic human results as more effective and efficient.

The hybrid average subtraction and standard deviation based optimizer

In this paper, we propose a new metaheuristic algorithm named the Hybrid Average Subtraction and Standard Deviation based Optimizer (HASSO) to solve the optimization problem. The proposed algorithm uses the amount of information gathered from averages, subtraction, standard deviation, and hybrid population members to explore the search space in order to reach the near-optimal or quasi-optimal solution. The mathematical modeling of the proposed algorithm has been presented in detail. We solve twenty-three well-known important functions containing the unimodal and multimodal functions to obtain their quasi-optimal solution. The results obtained for unimodal and multimodal functions indicate the high exploitation ability of HASSO in converging towards the global optima. The convergence plot of the algorithm shows that the use of five phases in the algorithm makes the algorithm tend towards the optimal or quasi-optimal solution in fewer iterations. In addition to this, the obtained results are compared with nine classical algorithms, and the sensitivity of the algorithm to the population size and number of iterations is also evaluated. Our algorithm is also statistically analyzed for superiority over other algorithms. From this analysis, HASSO’s superiority over the other nine classical algorithms in providing a highly accurate solution with a lesser number of iterations is clearly shown.

Parameters estimation of three-diode photovoltaic model using fractional-order Harris Hawks optimization algorithm

Effective and efficient systems that calculate the parameters of PV models to be optimal have attracted great interest from many researchers. In this study, a novel algorithm approach is proposed to improve the search performance of the Harris Hawks optimization (HHO) algorithm based on fractal maps. The random parameter that is effective in besiege of the prey in the exploration and exploitation stages of the HHO algorithm is adjusted using the fractal henon chaotic map. Unimodal, multimodal and fixed-dimension benchmark functions were used to verify the quality and efficiency of the proposed Fractional Henon Chaotic Harris Hawks Optimization (FCHHHO) algorithms and variants in providing optimal solutions. Moreover, RTC France photovoltaic cell and Photowatt-PWP 201 photovoltaic models with single, double and three-diodes have been determined accurately, stably and reliably. Comparative and statistical results obtained from the developed algorithms showed that the algorithms have superior performance with regard to accuracy and robustness in comparison with their variants and compared with other metaheuristic algorithms.

Dual-Stage Hybrid Learning Particle Swarm Optimization Algorithm for Global Optimization Problems

Particle swarm optimization (PSO) is a type of swarm intelligence algorithm that is frequently used to resolve specific global optimization problems due to its rapid convergence and ease of operation. However, PSO still has certain deficiencies, such as a poor trade-off between exploration and exploitation and premature convergence. Hence, this paper proposes a dual-stage hybrid learning particle swarm optimization (DHLPSO). In the algorithm, the iterative process is partitioned into two stages. The learning strategy used at each stage emphasizes exploration and exploitation, respectively. In the first stage, to increase population variety, a Manhattan distance based learning strategy is proposed. In this strategy, each particle chooses the furthest Manhattan distance particle and a better particle for learning. In the second stage, an excellent example learning strategy is adopted to perform local optimization operations on the population, in which each particle learns from the global optimal particle and a better particle. Utilizing the Gaussian mutation strategy, the algorithm's searchability in particular multimodal functions is significantly enhanced. On benchmark functions from CEC 2013, DHLPSO is evaluated alongside other PSO variants already in existence. The comparison results clearly demonstrate that, compared to other cutting-edge PSO variations, DHLPSO implements highly competitive performance in handling global optimization problems.

Comparison of Recent Meta-Heuristic Optimization Algorithms Using Different Benchmark Functions

Meta-heuristic optimization algorithms are used in many application areas to solve optimization problems. In recent years, meta-heuristic optimization algorithms have gained importance over deterministic search algorithms in solving optimization problems. However, none of the techniques are equally effective in solving all optimization problems. Therefore, researchers have focused on either improving current meta-heuristic optimization techniques or developing new ones. Many alternative meta-heuristic algorithms inspired by nature have been developed to solve complex optimization problems. It is important to compare the performances of the developed algorithms through statistical analysis and determine the better algorithm. This paper compares the performances of sixteen meta-heuristic optimization algorithms (AWDA, MAO, TSA, TSO, ESMA, DOA, LHHO, DSSA, LSMA, AOSMA, AGWOCS, CDDO, GEO, BES, LFD, HHO) presented in the literature between 2021 and 2022. In this context, various test functions, including single-mode, multi-mode, and fixed-size multi-mode benchmark functions, were used to evaluate the efficiency of the algorithms used.

Local search enhanced Aquila optimization algorithm ameliorated with an ensemble of Wavelet mutation strategies for complex optimization problems

Aquila Optimization Algorithm (AQUILA) is a newly emerged metaheuristic optimizer for solving global optimization problems, which is based on intrinsic hunting behaviors of the foraging aquila individuals. However, this stochastic optimization method suffers from some algorithm-specific drawbacks, such as premature convergence to the local optimum points over the search hyperspace due to the lack of solution diversity in the population. To conquer this algorithmic deficiency, an ensemble of Wavelet mutation operators has been implemented into the standard AQUILA to enhance the explorative capabilities of the algorithm by diversifying the search domain as much as possible. Furthermore, a brand-new local search scheme empowered by the synergetic interactions of elite opposition-based learning and a simple-yet-effective exploitative manipulation equation is introduced into the base AQUILA to intensify on the previously visited promising regions. The proposed learning schemes are stochastically applied to the obtained solutions from the base Aquila algorithm to refine the overall solution quality and amend the premature convergence problem. It is also aimed to investigate whether the collective application of Wavelet mutation operators with different types entails a significant improvement in the general search effectivity of the algorithm rather than their individual efforts. Numerical experiments made on a suite of unconstrained unimodal and multimodal benchmark functions reveal that this hybridization with AQUILA has improved the general solution accuracy and stability to very high standards, outperforming its contemporary counterparts in the comparative statistical analysis. Furthermore, an exhaustive benchmark analysis has been performed on fourteen constrained real-world complex engineering problems.

Cooperative path planning optimization for multiple UAVs with communication constraints

Path planning is a complicated optimization problem that is crucial for the safe flight of unmanned aerial vehicles (UAVs). Especially in the scenarios involving multiple UAVs, this problem is highly challenging due to the constraints of complex environments, various tasks and inherent UAV maneuverability. In this paper, a cooperative path planning model for multiple UAVs is presented. In addition to common limitations such as the path length minimization, UAV maneuverability limitation and collision avoidance, the communication requirements between UAVs and the impact of obstacles in the flight environment on the quality of communication are also taken into account in the presented path planning model. On this basis, the corresponding objective function is designed. Then, an improved particle swarm optimization (PSO) algorithm is proposed to solve the above path planning problem. Utilizing the ideas of the dynamic multi-swarm PSO (DMSPSO) algorithm and the comprehensive learning PSO (CLPSO) algorithm, the proposed algorithm, denoted as CL-DMSPSO, further improves the performance of both algorithms. The effectiveness and superiority of the novel CL-DMSPSO algorithm is verified on benchmark functions, especially for complex multimodal functions. Finally, we present an effective path planning method using CL-DMSPSO to generate optimized flyable paths for multiple UAVs. And simulation and comparison results on the designed scenario indicate the proposed UAV path planning method can efficiently plan high-quality paths for UAVs and demonstrate the advantages of the proposed CL-DMSPSO algorithm compared with other PSO algorithms in UAV path planning.

Fabrication and Functionality Integration Technologies for Small-Scale Soft Robots.

Small-scale soft robots are attracting increasing interest for visible and potential applications owing to their safety and tolerance resulting from their intrinsic soft bodies or compliant structures. However, it is not sufficient that the soft bodies merely provide support or system protection. More importantly, to meet the increasing demands of controllable operation and real-time feedback in unstructured/complicated scenarios, these robots are required to perform simplex and multimodal functionalities for sensing, communicating, and interacting with external environments during large or dynamic deformation with the risk of mismatch or delamination. Challenges are encountered during fabrication and integration, including the selection and fabrication of composite/materials and structures, integration of active/passive functional modules with robust interfaces, particularly with highly deformable soft/stretchable bodies. Here, methods and strategies of fabricating structural soft bodies and integrating them with functional modules for developing small-scale soft robots are investigated. Utilizing templating, 3D printing, transfer printing, and swelling, small-scale soft robots can be endowed with several perceptual capabilities corresponding to diverse stimulus, such as light, heat, magnetism, and force. The integration of sensing and functionalities effectively enhances the agility, adaptability, and universality of soft robots when applied in various fields, including smart manufacturing, medical surgery, biomimetics, and other interdisciplinary sciences.

A Hybrid Multi-Objective Optimization Method and Its Application to Electromagnetic Device Designs

Optimization algorithms play a critical role in electromagnetic device designs due to the ever-increasing technological and economical competition. Although evolutionary algorithm-based methods have successfully been applied to different design problems, these methods exhibit deficiencies when solving complex problems with multimodal and discontinuous objective functions, which is quite common in electromagnetic device optimization designs. In this paper, a hybrid multi-objective optimization algorithm based on a non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective particle swarm optimization method (MOPSO) is proposed. In order to enhance the convergence and diversity performance of the algorithm, a new population update mechanism of MOPSO is introduced. Moreover, an adaptive operator involving crossover and mutation is presented to achieve a better balance between global and local searches. The performance of the hybrid algorithm is validated using standard test functions and the multi-objective design of a superconducting magnetic energy storage (SMES) device. Numerical results demonstrate the effectiveness and superiority of the proposed method.

A survey of recently developed metaheuristics and their comparative analysis

The aim of this study was to gather, discuss, and compare recently developed metaheuristics to understand the pace of development in the field of metaheuristics and make some recommendations for the research community and practitioners. By thoroughly and comprehensively searching the literature and narrowing the search results, we created with a list of 57 novel metaheuristic algorithms. Based on the availability of the source code, we reviewed and analysed the optimization capability of 26 of these algorithms through a series of experiments. We also evaluated the exploitation and exploration capabilities of these metaheuristics by using 50 unimodal functions and 50 multimodal functions, respectively. In addition, we assessed the capability of these algorithms to balance exploration and exploitation by using 29 shifted, rotated, composite, and hybrid CEC-BC-2017 benchmark functions. Moreover, we evaluated the applicability of these metaheuristics on four real-world constrained engineering optimization problems. To rank the algorithms, we performed a nonparametric statistical test, the Friedman mean rank test. Based on the statistical results for the unimodal and multimodal functions, we declared that the GBO, PO, and MRFO algorithms have better exploration and exploitation capabilities. Based on the results for the CEC-BC-2017 benchmark functions, we found the MPA, FBI, and HBO algorithms to be the most balanced. Finally, based on the results for the constrained engineering optimization problems, we declared that the HBO, GBO, and MA algorithms are the most suitable. Collectively, we confidently recommend the GBO, MPA, PO, and HBO algorithms for real-world optimization problems.

Ultra-fast electro-reduction and activation of graphene for high energy density wearable supercapacitor asymmetrically designed with MXene

Controlled perforation of graphene is vital to surpass the performance of supercapacitors that rely on their pristine form. However, their practical utilization has been halted by energy-inefficient and lengthy processing. Here, we are reporting a pulse Joule heating strategy for on-site reduction and activation to realize a multimodal porous framework made of perforated graphene using millisecond current pulses. The multimodal porosity and surface functionalities of graphene were regulated at an ultrafast rate by passing a transient current. As-developed ready-to-use electrode composed of nano-to-macro multimodal porosity displays high areal capacitance of 380.2 mF cm−2 in symmetric two-electrode configuration, which is nearly 1.6 times higher than the non-electro activated counterpart. Furthermore, a high-performance wearable asymmetric supercapacitor with an areal energy density of 107.8 μWh cm−2 was realized using this multimodal porous graphene in combination with suitable negative electrodes made of MXene. High energy density, together with stable and repeatable performance of the wearable device for 10000 cycles of charge-discharge and 5000 cycles of bending, signifies the importance of the as-developed device for practical wearable applications. Direct, simple processing of electrodes and orders of magnitude lower cost–and–processing–time can make the process appealing for practical wearable and other energy storage applications.

An Improved Particle-Swarm-Optimization Algorithm for a Prediction Model of Steel Slab Temperature

Aiming at the problem of the low accuracy of temperature prediction, a mathematical model for predicting the temperature of a steel billet is developed. For the process of temperature prediction, an improved particle-swarm-optimization algorithm (called XPSO) is developed. XPSO was designed based on a multiple swarm scheme to improve the global search capability and robustness; thus, it can improve the low accuracy of prediction and overcome the problem of easy entrapment into local optima. In the XPSO, the multiple swarm scheme comprises four modified components: (1) the strategy of improving the positional initialization; (2) the mutation strategy for particle swarms; (3) the adjustment strategy of inertia weights; (4) the strategy of jumping out local optima. Based on widely used unimodal, multimodal and composite benchmark functions, the effectiveness of the XPSO algorithm was verified by comparing it with some popular variant PSO algorithms (PSO, IPSO, IPSO2, HPSO, CPSO). Then, the XPSO was applied to predict the temperatures of steel billets based on simulation data sets and measured data sets. Finally, the obtained results show that the XPSO is more accurate than other PSO algorithms and other optimization approaches (WOA, IA, GWO, DE, ABC) for temperature prediction of steel billets.

Comparison of shear viscosity and normal stress measurements by rotational and on-line slit rheometers with tube model predictions

In-extruder measurements of shear viscosity and normal stresses are important as these measurement techniques allow determining the rheological state of the polymer melt at processing conditions up to high shear rates. However, validation of viscosity and normal stress data obtained by in-line slit rheometers at high shear rates is difficult due to a lack of overlap of the in-line data and the off-line measurements by rotational rheometers limited to lower shear rates. Here, shear viscosity and normal stress data measured in-line at large shear rates during extrusion and off-line at low shear rates are compared to predictions of the Doi-Edwards model and the Hierarchical Multi-Mode Molecular Stress Function (HMMSF) model using linear-viscoelastic off-line small amplitude oscillating shear data of two polystyrenes and a low-density polyethylene as input parameters. For polystyrene, the results of this investigation do not only validate the experimental data obtained by rotational as well as slit-die rheometry, but also demonstrate the agreement between experiments and models up to very high shear rates, which were not experimentally accessible earlier. The low-density polyethylene shows a more complex behaviour, which follows the HMMSF model at low shear rates, but approaches the Doi-Edwards model at high shear rates.

An Improved Nonlinear Tuna Swarm Optimization Algorithm Based on Circle Chaos Map and Levy Flight Operator

The tuna swarm optimization algorithm (TSO) is a new heuristic algorithm proposed by observing the foraging behavior of tuna populations. The advantages of TSO are a simple structure and fewer parameters. Although TSO converges faster than some classical meta-heuristics algorithms, it can still be further accelerated. When TSO solves complex and challenging problems, it often easily falls into local optima. To overcome the above issue, this article proposed an improved nonlinear tuna swarm optimization algorithm based on Circle chaos map and levy flight operator (CLTSO). In order to compare it with some advanced heuristic algorithms, the performance of CLTSO is tested with unimodal functions, multimodal functions, and some CEC2014 benchmark functions. The test results of these benchmark functions are statistically analyzed using Wilcoxon, Friedman test, and MAE analysis. The experimental results and statistical analysis results indicate that CLTSO is more competitive than other advanced algorithms. Finally, this paper uses CLTSO to optimize a BP neural network in the field of artificial intelligence. A CLTSO-BP neural network model is proposed. Three popular datasets from the UCI Machine Learning and Intelligent System Center are selected to test the classification performance of the new model. The comparison result indicates that the new model has higher classification accuracy than the original BP model.

IFA-EO: An improved firefly algorithm hybridized with extremal optimization for continuous unconstrained optimization problems

As one of the evolutionary algorithms, firefly algorithm (FA) has been widely used to solve various complex optimization problems. However, FA has significant drawbacks in slow convergence rate and is easily trapped into local optimum. To tackle these defects, this paper proposes an improved FA combined with extremal optimization (EO), named IFA-EO, where three strategies are incorporated. First, to balance the tradeoff between exploration ability and exploitation ability, we adopt a new attraction model for FA operation, which combines the full attraction model and the single attraction model through the probability choice strategy. In the single attraction model, small probability accepts the worse solution to improve the diversity of the offspring. Second, the adaptive step size is proposed based on the number of iterations to dynamically adjust the attention to the exploration model or exploitation model. Third, we combine an EO algorithm with powerful ability in local-search into FA. Experiments are tested on two group popular benchmarks including complex unimodal and multimodal functions. Our experimental results demonstrate that the proposed IFA-EO algorithm can deal with various complex optimization problems and has similar or better performance than the other eight FA variants, three EO-based algorithms, and one advanced differential evolution variant in terms of accuracy and statistical results.

Grey Wolf Optimization algorithm based on Cauchy-Gaussian mutation and improved search strategy

The traditional Grey Wolf Optimization algorithm (GWO) has received widespread attention due to features of strong convergence performance, few parameters, and easy implementation. However, in actual optimization projects, there are problems of slow convergence speed and easy to fall into local optimal solution. The paper proposed a Grey Wolf Optimization algorithm based on Cauchy-Gaussian mutation and improved search strategy (CG-GWO) in response to the above problems. The Cauchy-Gaussian mutation operator is introduced to increase the population diversity of the leader wolves and improve the global search ability of the algorithm. This work retains outstanding grey wolf individuals through the greedy selection mechanism to ensure the convergence speed of the algorithm. An improved search strategy was proposed to expand the optimization space of the algorithm and improve the convergence accuracy. Experiments are performed with 16 benchmark functions covering unimodal functions, multimodal functions, and fixed-dimension multimodal functions to verify the effectiveness of the algorithm. Experimental results show that compared with four classic optimization algorithms, particle swarm optimization algorithm (PSO), whale optimization algorithm (WOA), sparrow optimization algorithm (SSA), and farmland fertility algorithm (FFA), the CG-GWO algorithm shows better convergence accuracy, convergence speed, and global search ability. The proposed algorithm shows the same better performance compared with a series of improved algorithms such as the improved grey wolf algorithm (IGWO), modified Grey Wolf Optimization algorithm (mGWO), and the Grey Wolf Optimization algorithm inspired by enhanced leadership (GLF-GWO).

Young’s double-slit experiment optimizer : A novel metaheuristic optimization algorithm for global and constraint optimization problems

Due to the global progress, the optimization problems are becoming more and more complex. Hence, deterministic and heuristic approaches are no longer adequate for dealing with such sophisticated problems. subsequently, metaheuristics have recently emerged as an effective alternative for addressing the optimization problems. This paper proposes a novel metaheuristic called Young’s Double-Slit Experiment (YDSE) optimizer, derived from a physical backdrop. The YDSE optimizer is inspired by Young’s double-slit experiment, which is regarded as one of the most well-known classical physics experiments, revealing the wave nature of light. In YDSE optimizer, each fringe represents a possible solution in the population. Many concepts are modeled from the experiment, such as monochromatic light waves, Huygens’ principle, constructive and destructive interference, wave intensity, amplitude, and path difference. The YDSE optimizer strikes a balance between exploration and exploitation by selecting either a constructive interference or a destructive interference based on the order number of the fringe. During the optimization process, the solution moves in search space based on its order number. If the solution has an odd number, it moves in the dark regions towards the central bright region, which is expected to contain the optimal solution. The algorithm exploits the promising areas in the bright fringe areas, which are assumed to contain the optimum. The performance of the YDSE optimizer is compared with another twelve metaheuristics using CEC 2014, CEC 2017, and CEC 2022. The benchmarks cover different unimodal, multimodal, hybrid, and composite test functions. Also, we consider ten constrained and unconstrained engineering optimization design problems. YDSE proved its superiority over the CEC 2014 and CEC 2017 winners, such as L-SHADE, LSHADE-cnEpSin, and LSHADE-SPACMA. The results and the statistical analysis demonstrated the outperformance of the proposed YDSE optimizer at a 95% confidence interval.

Research on single cell membrane algorithm and engineering application based on membrane computing theory

Membrane computing is a new computing paradigm with great significance in the field of computer science. The Multi-membrane search algorithm (MSA) is proposed based on the membrane computational population optimization theory. It showed excellent performance in the test. This paper further studies the performance characteristics of a single individual (Single Cell Membrane Algorithm, SCA) of MSA. SCA can generate adaptive solution sets for problems of different dimensions. Through transcription and reprocessing rules, new weakly correlated feasible solutions are formed for global search and local exploration. This paper is based on the unimodal Sphere function and the multimodal Rastrigr function, at dim=3, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 300, 500, 1000 and Q=1.00, 0.75, 0.50, 0.40, 0.30, 0.20, 0.10, 0.005, 0.025, 0.010, the SCA was optimized for 1000 iterations. Analyze the impact of the key parameter Q of SCA on the search performance of the algorithm in problems of different dimensions. The results show that under the set conditions, SCA has better performance when Q is 0.010 and 0.025 in the unimodal function test. In the multimodal function test, SCA has better performance when dim≤100 and Q≤0.200, and when dim>100 and Q≥0.200. In addition, this paper employs one engineering problem: I-beams to perform engineering tests on SCA and obtain results superior to other algorithms participating in the comparison. The test and comparison results show that SCA can also be used as a derivative algorithm of MSA, and has good performance.

Optimization of Robotic Task Sequencing Problems by Crowding Evolutionary Algorithms

This study solved the robotic task sequence planning problem of scheduling joint-space tours so that each task point visit is made according to expected manufacturing criteria. Multiple solutions for robotic inverse kinematic (RIK) problems were obtained for two task sequencing problems: 1) preset manufacturing sequence planning (Preset-MSP) and 2) optimal manufacturing sequence planning (Opt-MSP). First, a real-coded twin-space crowding evolutionary algorithm (TC-EA) was developed and used to explore multiple RIK joint configurations for each task point. Then, a heuristic bidirectional reference mechanism (BRM) was developed and used for efficiently solving Preset-MSP problems. By integrating BRM, the proposed discrete-coded TC-EA also efficiently solved Opt-MSP problems. To validate the proposed approach, multiple multimodal benchmark functions and task sequencing test cases were used to compare the solving capability of the proposed TC-EA and other evolutionary multimodal solvers. The experimental results showed that the proposed methods obtained better or at least comparable solutions for all test problems. For decision makers, the proposed methods have practical applications for exploring and comparing multiple robotic manufacturing plans.