Showing papers on "Simulated annealing published in 2021"

A metaheuristic optimization approach for energy efficiency in the IoT networks

Abstract: © 2020 John Wiley & Sons, Ltd. Recently Internet of Things (IoT) is being used in several fields like smart city, agriculture, weather forecasting, smart grids, waste management, etc. Even though IoT has huge potential in several applications, there are some areas for improvement. In the current work, we have concentrated on minimizing the energy consumption of sensors in the IoT network that will lead to an increase in the network lifetime. In this work, to optimize the energy consumption, most appropriate Cluster Head (CH) is chosen in the IoT network. The proposed work makes use of a hybrid metaheuristic algorithm, namely, Whale Optimization Algorithm (WOA) with Simulated Annealing (SA). To select the optimal CH in the clusters of IoT network, several performance metrics such as the number of alive nodes, load, temperature, residual energy, cost function have been used. The proposed approach is then compared with several state-of-the-art optimization algorithms like Artificial Bee Colony algorithm, Genetic Algorithm, Adaptive Gravitational Search algorithm, WOA. The results prove the superiority of the proposed hybrid approach over existing approaches.

SGOA: annealing-behaved grasshopper optimizer for global tasks

Abstract: An improved grasshopper optimization algorithm (GOA) is proposed in this paper, termed as SGOA, which combines simulated annealing (SA) mechanism with the original GOA that is a natural optimizer widely used in finance, medical and other fields, and receives more promising results based on grasshopper behavior. To compare performance of the SGOA and other algorithms, an investigation to select CEC2017 benchmark function as the test set was carried out. Also, the Friedman assessment was performed to check the significance of the proposed method against other counterparts. In comparison with ten meta-heuristic algorithms such as differential evolution (DE), the proposed SGOA can rank first in the CEC2017, and also ranks first in comparison with ten advanced algorithms. The simulation results reveal that the SA strategy notably improves the exploration and exploitation capacity of GOA. Moreover, the SGOA is also applied to engineering problems and parameter optimization of the kernel extreme learning machine (KELM). After optimizing the parameters of KELM using SGOA, the model was applied to two datasets, Cleveland Heart Dataset and Japanese Bankruptcy Dataset, and they achieved an accuracy of 79.2% and 83.5%, respectively, which were better than the KELM model obtained other algorithms. In these practical applications, it is indicated that the proposed SGOA can provide effective assistance in settling complex optimization problems with impressive results.

A hybrid Harris Hawks optimization algorithm with simulated annealing for feature selection

Abstract: The significant growth of modern technology and smart systems has left a massive production of big data. Not only are the dimensional problems that face the big data, but there are also other emerging problems such as redundancy, irrelevance, or noise of the features. Therefore, feature selection (FS) has become an urgent need to search for the optimal subset of features. This paper presents a hybrid version of the Harris Hawks Optimization algorithm based on Bitwise operations and Simulated Annealing (HHOBSA) to solve the FS problem for classification purposes using wrapper methods. Two bitwise operations (AND bitwise operation and OR bitwise operation) can randomly transfer the most informative features from the best solution to the others in the populations to raise their qualities. The Simulate Annealing (SA) boosts the performance of the HHOBSA algorithm and helps to flee from the local optima. A standard wrapper method K-nearest neighbors with Euclidean distance metric works as an evaluator for the new solutions. A comparison between HHOBSA and other state-of-the-art algorithms is presented based on 24 standard datasets and 19 artificial datasets and their dimension sizes can reach up to thousands. The artificial datasets help to study the effects of different dimensions of data, noise ratios, and the size of samples on the FS process. We employ several performance measures, including classification accuracy, fitness values, size of selected features, and computational time. We conduct two statistical significance tests of HHOBSA like paired-samples T and Wilcoxon signed ranks. The proposed algorithm presented superior results compared to other algorithms.

Improved Artificial Immune System Algorithm for Type-2 Fuzzy Flexible Job Shop Scheduling Problem

Abstract: In practical applications, particularly in flexible manufacturing systems, there is a high level of uncertainty. A type-2 fuzzy logic system (T2FS) has several parameters and an enhanced ability to handle high levels of uncertainty. This article proposes an improved artificial immune system (IAIS) algorithm to solve a special case of the flexible job shop scheduling problem (FJSP), where the processing time of each job is a nonsymmetric triangular interval T2FS (IT2FS) value. First, a novel affinity calculation method considering the IT2FS values is developed. Then, four problem-specific initialization heuristics are designed to enhance both quality and diversity. To enhance the exploitation abilities, six local search approaches are conducted for the routing and scheduling vectors, respectively. Next, a simulated annealing method is embedded to accept antibodies with low affinity, which can enhance the exploration abilities of the algorithm. Moreover, a novel population diversity heuristic is presented to eliminate antibodies with high crowding values. Five efficient algorithms are selected for a detailed comparison, and the simulation results demonstrate that the proposed IAIS algorithm is effective for IT2FS FJSPs.

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

Abstract: In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance.

Metaheuristics: a comprehensive overview and classification along with bibliometric analysis

Abstract: Research in metaheuristics for global optimization problems are currently experiencing an overload of wide range of available metaheuristic-based solution approaches. Since the commencement of the first set of classical metaheuristic algorithms namely genetic, particle swarm optimization, ant colony optimization, simulated annealing and tabu search in the early 70s to late 90s, several new advancements have been recorded with an exponential growth in the novel proposals of new generation metaheuristic algorithms. Because these algorithms are neither entirely judged based on their performance values nor according to the useful insight they may provide, but rather the attention is given to the novelty of the processes they purportedly models, these area of study will continue to periodically see the arrival of several new similar techniques in the future. However, there is an obvious reason to keep track of the progressions of these algorithms by collating their general algorithmic profiles in terms of design inspirational source, classification based on swarm or evolutionary search concept, existing variation from the original design, and application areas. In this paper, we present a relatively new taxonomic classification list of both classical and new generation sets of metaheuristic algorithms available in the literature, with the aim of providing an easily accessible collection of popular optimization tools for the global optimization research community who are at the forefront in utilizing these tools for solving complex and difficult real-world problems. Furthermore, we also examined the bibliometric analysis of this field of metaheuristic for the last 30 years.

Optimal location, selection, and operation of battery energy storage systems and renewable distributed generation in medium–low voltage distribution networks

Abstract: This paper presents a methodology for the optimal location, selection, and operation of battery energy storage systems (BESSs) and renewable distributed generators (DGs) in medium–low voltage distribution systems. A mixed-integer non-linear programming model is presented to formulate the problem, and a planning-operation decomposition methodology is proposed to solve it. The proposed methodology is separated into two problems (planning and operation problems). The planning problem is related to the location and selection of these devices, and the operation problem is responsible for finding the optimal BESS operating scheme. For solving the planning problem is used a simulated annealing algorithm with a defined neighborhood structure that uses a sensitivity analysis based on the Zbus matrix. A novel decomposition method, which guarantees near-global-optimal solutions with low computational effort, is proposed for solving the operation problem. The effectiveness and accuracy of the proposed decomposition method is validated and tested on an 11-node test system from the specialized literature, and the robustness of the proposed method is assessed and tested on a modified version of an IEEE 135-node test system. The proposed planning-operation decomposition methodology is tested on a real medium–low voltage distribution system of 230 nodes. To verify the efficiency of the proposed methodology, four cases are compared: (I) without BESS and DGs, (II) with DGs, (III) with BESS, and (IV) with BESS and DGs. The numerical results demonstrate the effectiveness and robustness of the proposed methodology.

A Hybrid Iterated Greedy Algorithm for a Crane Transportation Flexible Job Shop Problem

Abstract: In this study, we propose an efficient optimization algorithm that is a hybrid of the iterated greedy and simulated annealing algorithms (hereinafter, referred to as IGSA) to solve the flexible job shop scheduling problem with crane transportation processes (CFJSP). Two objectives are simultaneously considered, namely, the minimization of the maximum completion time and the energy consumptions during machine processing and crane transportation. Different from the methods in the literature, crane lift operations have been investigated for the first time to consider the processing time and energy consumptions involved during the crane lift process. The IGSA algorithm is then developed to solve the CFJSPs considered. In the proposed IGSA algorithm, first, each solution is represented by a 2-D vector, where one vector represents the scheduling sequence and the other vector shows the assignment of machines. Subsequently, an improved construction heuristic considering the problem features is proposed, which can decrease the number of replicated insertion positions for the destruction operations. Furthermore, to balance the exploration abilities and time complexity of the proposed algorithm, a problem-specific exploration heuristic is developed. Finally, a set of randomly generated instances based on realistic industrial processes is tested. Through comprehensive computational comparisons and statistical analyses, the highly effective performance of the proposed algorithm is favorably compared against several efficient algorithms.

Optimization of micromixer with Cantor fractal baffle based on simulated annealing algorithm

Abstract: To maximize the mixing performance of the micromixer with the Cantor fractal baffle structure, single-objective optimization of the micromixer with different Reynolds (Res) is carried out. The three-dimensional Navier-Stokes equation is used to numerically analyze the fluid flow and mixing in the micromixer. We choose three parameters related to the geometry of the Cantor fractal baffle inside the microchannel as the best design variables. The mixing index at the outlet of the micromixer is used as the objective function. And conduct parameter studies to explore the influence of the design variables on the objective function. For the parameter study of the design space, the Latin hypercube sampling (LHS) method is used as an experimental design technique. It is used to select design points in the design space. We use surrogate modeling of response surface functions to approximate the objective function. When Re is different, the simulated annealing algorithm is used to optimize the objective of the established surrogate modeling, and finally the optimal structure configuration of the micromixer is obtained. In this article, we combine the fractal principle with the simulated annealing algorithm to improve the mixing performance of the micromixer. This is not involved in previous studies. The results show that the mixing performance of the optimized micromixer has indeed been significantly improved. When Re = 0.1, 1, 10, and 100, the mixing efficiency of the optimized micromixer is increased by 7.64%, 17.75%, 14.08%, and 0.91%, respectively, compared with the reference design.

Mind mappings: enabling efficient algorithm-accelerator mapping space search

Abstract: Modern day computing increasingly relies on specialization to satiate growing performance and efficiency requirements. A core challenge in designing such specialized hardware architectures is how to perform mapping space search, i.e., search for an optimal mapping from algorithm to hardware. Prior work shows that choosing an inefficient mapping can lead to multiplicative-factor efficiency overheads. Additionally, the search space is not only large but also non-convex and non-smooth, precluding advanced search techniques. As a result, previous works are forced to implement mapping space search using expert choices or sub-optimal search heuristics. This work proposes Mind Mappings, a novel gradient-based search method for algorithm-accelerator mapping space search. The key idea is to derive a smooth, differentiable approximation to the otherwise non-smooth, non-convex search space. With a smooth, differentiable approximation, we can leverage efficient gradient-based search algorithms to find high-quality mappings. We extensively compare Mind Mappings to black-box optimization schemes used in prior work. When tasked to find mappings for two important workloads (CNN and MTTKRP), Mind Mapping finds mappings that achieve an average 1.40×, 1.76×, and 1.29× (when run for a fixed number of steps) and 3.16×, 4.19×, and 2.90× (when run for a fixed amount of time) better energy-delay product (EDP) relative to Simulated Annealing, Genetic Algorithms and Reinforcement Learning, respectively. Meanwhile, Mind Mappings returns mappings with only 5.32× higher EDP than a possibly unachievable theoretical lower-bound, indicating proximity to the global optima.

A hybrid mine blast algorithm for feature selection problems

Abstract: Feature selection (FS) is the process of finding the least possible number of features that are able to describe a dataset in the same way as the original features. Feature selection is a crucial preprocessing step for data mining techniques as it improves the performance of the prediction process in terms of speed and accuracy and also provides a better understanding of stored data. The success of the FS process depends on achieving a balance between two important factors, namely selecting the minimal number of features and maintaining the maximum accuracy in the results. In this paper, two methods are proposed to improve the FS process. Firstly, the mine blast algorithm (MBA) is introduced to optimize the FS process in the exploration phase. Secondly, the MBA is hybridized with simulated annealing as a local search in the exploitation phase to enhance the solutions located by the MBA. The proposed approaches (MBA and MBA–SA) are tested on 18 benchmark datasets from the UCI repository, and the comprehensive experimental results indicate that MBA–SA achieved good performance when compared with five approaches in the literature.

Online Weighting Factor Optimization by Simplified Simulated Annealing for Finite Set Predictive Control

Abstract: Model predictive control brings many advantages and it simplifies the control scheme in power electronics. However, tuning the weighting factor is one of the important open discussions on this topic. There are online and offline methods that have been introduced to select the weighting factor. The online methods are preferred because they are more feasible. In this article, an online weighting factor optimization method based on the simulated annealing algorithm is proposed. The energy of the ripple is used as a convergence criterion. The presented method can be converged in a few steps and it does not impose cumbersome computations. Therefore, the optimal voltage will be identical for a range of the weighting factor. Furthermore, the used search algorithm is parameter independent. The proposed method is implemented for an induction motor but it is also applicable for other applications. The proposed method is validated by the experimental tests.

Mind Mappings: Enabling Efficient Algorithm-Accelerator Mapping Space Search

Abstract: Modern day computing increasingly relies on specialization to satiate growing performance and efficiency requirements. A core challenge in designing such specialized hardware architectures is how to perform mapping space search, i.e., search for an optimal mapping from algorithm to hardware. Prior work shows that choosing an inefficient mapping can lead to multiplicative-factor efficiency overheads. Additionally, the search space is not only large but also non-convex and non-smooth, precluding advanced search techniques. As a result, previous works are forced to implement mapping space search using expert choices or sub-optimal search heuristics. This work proposes Mind Mappings, a novel gradient-based search method for algorithm-accelerator mapping space search. The key idea is to derive a smooth, differentiable approximation to the otherwise non-smooth, non-convex search space. With a smooth, differentiable approximation, we can leverage efficient gradient-based search algorithms to find high-quality mappings. We extensively compare Mind Mappings to black-box optimization schemes used in prior work. When tasked to find mappings for two important workloads (CNN and MTTKRP), the proposed search finds mappings that achieve an average $1.40\times$, $1.76\times$, and $1.29\times$ (when run for a fixed number of steps) and $3.16\times$, $4.19\times$, and $2.90\times$ (when run for a fixed amount of time) better energy-delay product (EDP) relative to Simulated Annealing, Genetic Algorithms and Reinforcement Learning, respectively. Meanwhile, Mind Mappings returns mappings with only $5.32\times$ higher EDP than a possibly unachievable theoretical lower-bound, indicating proximity to the global optima.

Metaheuristic approaches to design and address multi-echelon sugarcane closed-loop supply chain network

Abstract: The sugarcane industry is technologically pioneering in the area of food production. On the other side, this industry produces a huge amount of by-products. Proper handling of these by-products has remained a challenge. An efficient multi-echelon Sugarcane Supply Chain Network (SSCN) is designed and proposed in this paper to handle the by-products produced by the sugarcane industry that can be utilized further with little modification. It helps to reduce the overall working cost of the network. Usually, the supply chain problems are complex in nature, and complexity further increases with increasing problem instances. Metaheuristics techniques are, in general, applied to handle such NP-hard problems. This work proposes three hybrid metaheuristics algorithms, namely H-GASA, a hybrid of Genetic Algorithm with Simulated Annealing, H-KASA, a hybrid of Keshtel Algorithm with Simulated Annealing, and H-RDASA, a hybrid of Red Deer Algorithm with Simulated Annealing to handle the complexity of the problem. The algorithms’ performance is probed using the Taguchi experiments, and the best combinations of parameters are identified. This hybrid algorithms’ efficacy is compared with their basic version of the algorithms, i.e. GA, KA, RDA, and SA using different criteria. A set of test problems is generated to ensure the capability of the presented model. The obtained results suggest that H-KASA significantly outperforms in small-sized problems, while the H-RDASA significantly outperforms in medium- and large-sized problem instances. In addition, the sensitivity analysis confirms that by adopting this proposed multi-echelon SSCN, decision-makers can achieve a significant cost reduction of 8.3% in terms of the total cost.

A New Fusion of ASO with SA Algorithm and Its Applications to MLP Training and DC Motor Speed Control

Abstract: An improved version of atom search optimization (ASO) algorithm is proposed in this paper The search capability of ASO was improved by using simulated annealing (SA) algorithm as an embedded part of it The proposed hybrid algorithm was named as hASO-SA and used for optimizing nonlinear and linearized problems such as training multilayer perceptron (MLP) and proportional-integral-derivative controller design for DC motor speed regulation as well as testing benchmark functions of unimodal, multimodal, hybrid and composition types The obtained results on classical and CEC2014 benchmark functions were compared with other metaheuristic algorithms, including two other SA-based hybrid versions, which showed the greater capability of the proposed approach In addition, nonparametric statistical test was performed for further verification of the superior performance of hASO-SA In terms of MLP training, several datasets were used and the obtained results were compared with respective competitive algorithms The results clearly indicated the performance of the proposed algorithm to be better For the case of controller design, the performance evaluation was performed by comparing it with the recent studies adopting the same controller parameters and limits as well as objective function The transient, frequency and robustness analysis demonstrated the superior ability of the proposed approach In brief, the comparative analyses indicated the proposed algorithm to be successful for optimization problems with different nature

Optimization of Complex System Reliability using Hybrid Grey Wolf Optimizer

Abstract: Reliability allocation to increase the total reliability has become a successful way to increase the efficiency of the complex industrial system designs. A lot of research in the past have tackled this problem to a great extent. This is evident from the different techniques developed so far to achieve the target. Stochastic metaheuristics like simulated annealing, Tabu search (TS), Particle Swarm Optimization (PSO), Cuckoo Search Optimization (CS), Genetic Algorithm (GA), Grey wolf optimization technique (GWO) etc. have been used in recent years. This paper proposes a framework for implementing a hybrid PSO-GWO algorithm for solving some reliability allocation and optimization problems. A comparison of the results obtained is done with the results of other well-known methods like PSO, GWO, etc. The supremacy/competitiveness of the proposed framework is demonstrated from the numerical experiments. These results with regard to the time taken for the computation and quality of solution outperform the previously obtained results by the other well-known optimization methods.

A genetic programming hyper-heuristic for the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times

Abstract: In this paper, a genetic programming hyper heuristic (GP-HH) algorithm is proposed to solve the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times (DAPFSP-SDST) and the objective of makespan minimization. The main idea is to use genetic programming (GP) as the high level strategy to generate heuristic sequences from a pre-designed low-level heuristics (LLHs) set. In each generation, the heuristic sequences are evolved by GP and then successively operated on the solution space for better solutions. Additionally, simulated annealing is embedded into each LLH to improve the local search ability. An effective encoding and decoding pair is also presented for the algorithm to obtain feasible schedules. Finally, computational simulation and comparison are both carried out on a benchmark set and the results demonstrate the effectiveness of the proposed GP-HH. The best-known solutions are updated for 333 out of the 540 benchmark instances.

A new artificial ecosystem-based optimization integrated with Nelder-Mead method for PID controller design of buck converter

Abstract: Over the last decade, there has been a constant development in control techniques for DC-DC power converters which can be classified as linear and nonlinear. Researchers focus on obtaining maximum efficiency using linear control techniques to avoid complexity although nonlinear control techniques may achieve full dynamic capabilities of the converter. This paper has a similar purpose in which a novel hybrid metaheuristic optimization algorithm (AEONM) is proposed to design an optimal PID controller for DC-DC buck converter’s output voltage regulation. The AEONM employs artificial ecosystem-based optimization (AEO) algorithm with Nelder-Mead (NM) simplex method to ensure optimal PID controller parameters are efficiently tuned to control output voltage of the buck converter. Initial evaluations are performed on benchmark functions. Then, the performance of AEONM-based PID is validated through comparative results of statistical boxplot, non-parametric test, transient response, frequency response, time-domain integral-error-performance indices, disturbance rejection and robustness using AEO, particle swarm optimization and differential evolution. A comparative performance analysis of transient and frequency responses is also performed against simulated annealing, whale optimization and genetic algorithms for further performance assessment. The comparisons have shown the proposed hybrid AEONM algorithm to be superior in terms of enhancing the buck converter’s transient and frequency responses.

Multi-objective evolutionary algorithm based on multiple neighborhoods local search for multi-objective distributed hybrid flow shop scheduling problem

Abstract: In order to be competitive in today’s rapidly changing business world, enterprises have transformed a centralized to a decentralized structure in many areas of decision. It brings a critical problem that is how to schedule the production resources efficiently among these decentralized production centers. This paper studies a multi-objective distributed hybrid flow shop scheduling problem (MDHFSP) with the objectives of minimizing makespan, total weighted earliness and tardiness, and total workload. In the MDHFSP, a set of jobs have to be assigned to several factories, and each factory contains a hybrid flow shop scheduling problem with several parallel machines in each stage. A multi-objective evolutionary algorithm based on multiple neighborhoods local search (MOEA-LS) is proposed to solve the MDHFSP. In the initialization phase, a weighting mechanism is used to decide which position is the best one for each job when constructing a new sequence. Several multiple neighborhoods local search operators based on the three objectives are designed to generate offsprings. Some worse neighboring solutions are replaced by the solutions in the achieve set with a simulated annealing probability. In order to avoid trapping into local optimum, an adaptive weight updating mechanism is utilized when the achieve set has no change. The comprehensive comparison with other classic multi-objective optimization algorithms shows the proposed algorithm is very efficient for the MDHFSP.

A Simulated Annealing Algorithm and Grid Map-Based UAV Coverage Path Planning Method for 3D Reconstruction

Abstract: With the extensive application of 3D maps, acquiring high-quality images with unmanned aerial vehicles (UAVs) for precise 3D reconstruction has become a prominent topic of study. In this research, we proposed a coverage path planning method for UAVs to achieve full coverage of a target area and to collect high-resolution images while considering the overlap ratio of the collected images and energy consumption of clustered UAVs. The overlap ratio of the collected image set is guaranteed through a map decomposition method, which can ensure that the reconstruction results will not get affected by model breaking. In consideration of the small battery capacity of common commercial quadrotor UAVs, ray-scan-based area division was adopted to segment the target area, and near-optimized paths in subareas were calculated by a simulated annealing algorithm to find near-optimized paths, which can achieve balanced task assignment for UAV formations and minimum energy consumption for each UAV. The proposed system was validated through a site experiment and achieved a reduction in path length of approximately 12.6% compared to the traditional zigzag path.

Path Planning for Autonomous Underwater Vehicles Under the Influence of Ocean Currents Based on a Fusion Heuristic Algorithm

Abstract: Recently, research on path planning for the autonomous underwater vehicles (AUVs) has developed rapidly. Heuristic algorithms have been widely used to plan a path for AUV, but most traditional heuristic algorithms are facing two problems, one is slow convergence speed, the other is premature convergence. To solve the above problems, this paper proposes a new heuristic algorithms fusion, which improves the genetic algorithm with the ant colony optimization algorithm and the simulated annealing algorithm. In addition, to accelerate convergence and expand the search space of the algorithm, some algorithms like trying to cross, path self-smoothing and probability of genetic operation adjust adaptively are proposed. The advantages of the proposed algorithm are reflected through simulated comparative experiments. Besides, this paper proposes an ocean current model and a kinematics model to solve the problem of AUV path planning under the influence of ocean currents.

A new multi-objective mathematical model for a Citrus supply chain network design: Metaheuristic algorithms

Abstract: Nowadays, the citrus supply chain has been motivated by both industrial practitioners and researchers due to several real-world applications. This study considers a four-echelon citrus supply chain, consisting of gardeners, distribution centers, citrus storage, and fruit market. A Mixed Integer Non-Linear Programming (MINLP) model is formulated, which seeks to minimize the total cost and maximize the profit of the Citrus supply chain network. Due to the complexity of the model when considering large-scale samples, two well-known meta-heuristic algorithms such as Ant Colony Optimization (ACO) and Simulated Annealing (SA) algorithms have been utilized. Additionally, a new multi-objective ACO algorithm based on a set of non-dominated solutions form the Pareto frontier developed to solve the mathematical model. An extensive comparison based on different measurements analyzed to find a performance solution for the developed problem in the three sizes (small, medium, and large-scale). Finally, the various outcomes of numerical experiments indicate that the MOACO algorithm is more reliable than other algorithms.