A New Method for Solving the Flow Shop Scheduling Problem on Symmetric Networks Using a Hybrid Nature-Inspired Algorithm

: Recently, symmetric networks have received much attention in various applications. They are a single route for incoming and outgoing network trafﬁc. In symmetric networks, one of the fundamental categories of wide-ranging scheduling problems with several practical applications is the FSSP. Strictly speaking, a scheduling issue is found when assigning resources to the activities to maximize goals. The difﬁculty of ﬁnding solutions in polynomial time makes the ﬂow shop scheduling problem (FSSP) NP-hard. Hence, the utilization of a hybrid optimization technique, a new approach to the ﬂow shop scheduling issue, on symmetric networks is given in the current research. In order to address this issue, each party’s strengths are maximized and their weaknesses reduced, and this study integrates the Ant Colony Algorithm with Particle Swarm Optimization (ACO-PSO). Even though these methods have been employed before, their hybrid approach improves their resilience in a variety of sectors. The ACO-PSO is put to the test by contrasting it with innovative algorithms in the literature. The search space is ﬁrst ﬁlled with a variety of solutions by the algorithm. Using pheromones in the mutual region, the ACO algorithm locally controls mobility. Moreover, the PSO-based random interaction among the solutions yields the global maximum. The PSO’s random interaction among the solutions typically results in the global maximum. The computational research demonstrates that the recommended ACO-PSO method outperforms the existing ones by a large margin. The Friedman test also shows that the average algorithm ranks for ACO and PSO are 1.79 and 2.08, respectively. The proposed method has an average rank of 2.13 as well. It indicates that the suggested algorithm’s effectiveness increased.


Introduction
Symmetry, a fundamental idea in numerous branches of physics, can be expressed explicitly using mathematical methods, namely, invariance under a particular group action. Symmetric networks are expressive enough to represent the behavior of any nonoscillating recurrent network that is based on binary threshold units. Recent scientific emphasis has been drawn to symmetry in complex network systems [1]. One of the fascinating problems

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Minimizing the delay time of FSSPs on symmetric networks using a hybrid algorithm; • Minimizing the makespan of FSSPs on symmetric networks using a hybrid algorithm.
Section 2 contains a list of related works. The problem statement, the method for the FSSP, and the fitness function are explained in Section 3. The recommended optimization algorithm is described in Section 4. Experimental results are addressed and explained in Section 5. The concluding section brings the article to a close.

Related Work
When uncertainties exist, procedures for tackling an FSSP can be categorized into two groups. The first group develops a stable mathematical model and transforms uncertainties into a deterministic problem [22,23]. Substitute steps often set defined, unique terminal intervals utilizing discrete or constant scenarios and then solve them using available algorithms [24,25]. In addition to the difficulties, the second group modifies a current timetable technique: rescheduling strategies [26]. The production time of a job can be exceedingly unpredictable in the real world [27,28]. Up to now, there have been three kinds of approaches to the FSSP to fix the ambiguity of processing time. The initial one is to model the unknown processing period utilizing a gamma distribution, with the predicted and global quest skills. Empirical findings showed that the suggested EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To accommodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing the mechanism of the probability transition matrix. However, they used Pareto Optimal (MPSO) to cope with different targets. The findings revealed that MPSO is better than PSO because there was an increase in the chance of finding the best solution from the MPSO solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. According to Table 1, several methods have been used for the FSSP, each of which has its own features. Since the FSSP is one of the main challenges of the symmetric networks, this paper proposed a method for the FSSP using ACO-PSO to decrease delay time and makespan, which is presented in the next section. Table 1. Comparing the discussed works side by side.

Article Goals Main Features
Goli, Ala [32] Optimization of non-permutation FSSP and lot-sizing simultaneously Symmetry 2023, 15, x FOR PEER REVIEW 5 of Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features.

Article Goals Main Features
Goli, Ala [32] Optimization of non-permutation FSSP and lot-sizing simultaneously Improving total energy consumption Minimizing time for providing Pareto solutions Alburaikan, Garg [6] Minimizing processing times of three-stage FSSPs under fuzziness Reducing the rental cost Lei, Guo [33] Proposing a multi-action deep reinforcement learning framework for flexible job shop scheduling problem Minimizing the total completion time Increasing efficiency Liu and Shi [34] Generating efficient dispatching rules for identifying complete schedules Minimizing the total completion time Producing high-quality schedules Gu, Li [35] Providing an improved HCSA to solve the MOPFSP Improving the performance and convergence speed of the algorithm Motair [36] Finding the optimal solution for solving the problem for a job sequence with up to 14 jobs Increasing efficiency Reducing delay time Zuo, Fan [8] Proposing a MPABC for EHFSP Minimizing makespan Minimizing total tardiness Improving total energy consumption Huang, Pan [37] Providing a mathematical model and an iterated greedy algorithm Minimizing the makespan Not considering release date, no-wait, or blocking Rashid and Osman [38] Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP Reducing the total tardiness Helping decision makers make a rational deci sion Improving total energy consumption Symmetry 2023, 15, x FOR PEER REVIEW 5 of Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features.

Article Goals Main Features
Goli, Ala [32] Optimization of non-permutation FSSP and lot-sizing simultaneously Improving total energy consumption Minimizing time for providing Pareto solutions Alburaikan, Garg [6] Minimizing processing times of three-stage FSSPs under fuzziness Reducing the rental cost Lei, Guo [33] Proposing a multi-action deep reinforcement learning framework for flexible job shop scheduling problem Minimizing the total completion time Increasing efficiency Liu and Shi [34] Generating efficient dispatching rules for identifying complete schedules Minimizing the total completion time Producing high-quality schedules Gu, Li [35] Providing an improved HCSA to solve the MOPFSP Improving the performance and convergence speed of the algorithm Motair [36] Finding the optimal solution for solving the problem for a job sequence with up to 14 jobs Increasing efficiency Reducing delay time Zuo, Fan [8] Proposing a MPABC for EHFSP Minimizing makespan Minimizing total tardiness Improving total energy consumption Huang, Pan [37] Providing a mathematical model and an iterated greedy algorithm Minimizing the makespan Not considering release date, no-wait, or blocking Rashid and Osman [38] Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP Reducing the total tardiness Helping decision makers make a rational deci sion Minimizing time for providing Pareto solutions Alburaikan, Garg [6] Minimizing processing times of three-stage FSSPs under fuzziness Symmetry 2023, 15, x FOR PEER REVIEW 5 of Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Providing one energy conservation strategy and a cellular grey optimizer Improving production efficiency Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features.

Article Goals Main Features
Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP Reducing the total tardiness Helping decision makers make a rational deci sion Reducing the total tardiness commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP

Reducing the total tardiness Helping decision makers make a rational deci sion
Helping decision makers make a rational decision EA-MOA algorithm is robust and efficient. Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient. Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features.  [38] Optimizing the production scheduling with the FA algorithm.
Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP Reducing the total tardiness Helping decision makers make a rational deci sion Minimizing the energy consumption were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient. Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient. Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient. Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features. Additionally, Li, Sang [41] suggested an Energy-Aware Multi-Objective Optimiz tion Algorithm (EA-MOA) to tackle the hybrid FSSP, considering the setup energy util zation. There was a suggested method for encoding and decoding. Eight categories o neighborhood architectures and an adaptive neighborhood structure choice framewor were also built in this article to develop both the capacities of exploration and exploitatio Moreover, a right-shifting approach was planned to boost the energy utilization goal o the strategy. Lastly, a hybrid deep exploitation/exploration approach was suggested t balance the local and global quest skills. Empirical findings showed that the suggeste EA-MOA algorithm is robust and efficient.
Santosa and Siswanto [42] suggested a discrete PSO tackling the hybrid FSSP. To a commodate the FSSP, they adjusted the PSO. The adjustment was achieved utilizing th mechanism of the probability transition matrix. However, they used Pareto Optim (MPSO) to cope with different targets. The findings revealed that MPSO is better than PS because there was an increase in the chance of finding the best solution from the MPS solution package. In comparison, the MPSO solution bundle was similar to the optimum solution. Table 1 compares the available methods for FSSP and shows the main features.  [38] Optimizing the production scheduling with the FA algorithm. Minimizing makespan Improving energy utilization Lu, Gao [39] Providing one energy conservation strategy and a cellular grey optimizer Developing a variable neighborhood search relying on the issue properly Improving production efficiency Improving energy consumption Decreasing noise pollution Fu, Ding [40] Designing a MO-DFWA to cope with the studied FSSP Reducing the total tardiness Helping decision makers make a rational deci sion Minimizing the makespan of FSSP on symmetric networks

Proposed Method
The wide range of problems, such as permutation flow shop, job shop, and open shop scheduling problems, have been extensively investigated using exact or heuristic methods [43,44]. However, the methods introduced for scheduling tasks contain some drawbacks: long scheduling process, not considering delay, makespan, and in some cases, not considering efficient and effective use of resources. This section introduces a new method for task scheduling to address these issues. First, a problem statement is defined, and then a proposed method is designed and described based on it.

Symmetric Networks
All nodes in a network have the same degree. By symmetry, the medial centrality of all nodes is constant. If C B (w) = c then [45]: For example, we call a symmetric topology a bi-directional ring (of sizek = |P T |) if for every i ∈ {0|P T |−1} we have:

Problem Statement
The issue is best described as follows: At one of the f flow shop businesses with m machines, n jobs must be arranged. Any jobs can be processed by these machines. A task cannot be moved to another firm after it has been allocated to one; it must be completed by that firm. Each job j on each machine i has a processing time indicated by the symbol p ij . The issue establishes the sequence π f , made up of n f jobs, that must be scheduled in each f. Hence, the sequence π = π 1 . . . . .π f . . . . .π F results in a solution π. Imagine C f i.j is the completion time related to j in i (allocated to f ), and C f max = C max π f makes up the makespan of f. In addition, C max = C max (π) means the global makespan. Moreover, π f [ ] is utilized to indicate the element of f in position i. By using f max , the global makespan can also be written as C f max max [47].

Designing a Method
FSSP concerns mapping a set of tasks to resources to use resources effectively and efficiently to process and produce results [48]. FSSP is an NP-hard issue, where m machines do n tasks. Each of these machines must process each task in this set to have less implementation time. From an economic perspective, proper use of tasks and task scheduling solutions among machines reduces processing costs and increases the efficient use of resources.

Ant Colony Optimization (ACO)
ACO utilizes swarm-inspired methods to solve complex issues [49]. The main idea of ACO is to employ ant search behavior in finding food [50]. When an ant is trying to find food, it uses a particular chemical called a pheromone to communicate with other ants [51]. At the beginning of the search, the ant starts the search randomly, and when it finds a food path, it leaves the pheromone in that path. Other ants can access the food source using the leftover pheromone [52,53]. When this process is repeated several times, the ants are gradually attracted to shorter paths and paths with a large amount of pheromones [54]. This algorithm and its advantages can also search and obtain an optimal response in task scheduling. When a new request enters the scheduling cycle, the probability of ants choosing paths from existing hosts is calculated using Equation (3). In other words, this relation indicates the possibility of sending request j to host i. τ i,j represents the concentration of pheromones, and the heuristic function η i,j is calculated by Equations (4)-(6) in each iteration. κ and β are the factors affecting the pheromone and the heuristic function, respectively, both of which take a value of one in the proposed method. us is a set of remaining requests that are not currently scheduled, and r is a request. Equations (5)-(7) indicate the fitness of host i for the three sources used in the proposed method. ηM i indicates the quality of the amount of memory remaining in host i, and similarly, ηC i, and ηD i indicate the quality of the amount of processor and storage remaining in host i, respectively. ω 1 , ω 2 , and ω 3 are the weight of memory, processor, and storage as three utilized sources. They have been used with constant and determined amounts. Max m , Max c , and Max d represent the maximum amount of memory, the maximum amount of processor consumption, and the maximum storage capacity each host can have. MR i represents the amount of memory remaining in host i, and UC i and UD i represent the amount of processor consumption and storage of host i, respectively [55]. In the ACO algorithm, the heuristic function belongs to local points [56]. Each ant has a personal opinion about choosing the next path [57]. As a result, according to the relations above, each ant chooses the next path based on the remaining resources of each path. Therefore, from a memory perspective, the longer the path memory, the better. However, from a processor and storage perspective, less use means more residual resources. Thus, the average of the remaining resources from these three resources is calculated using Equation (4) and based on the weight.
The general protocol of the ACO algorithm handles the four-stage scheduling [58]: Step 1: Initialization. The parameters of the ACO and the ants' initial locations are initialized.
Step 2: Solution construction. According to a probabilistic state transformation principle, each ant builds a full solution to the issue. The principle of state transformation relies largely on the pheromone state and the visibility of ants.
Step 3: Pheromone updating rule. The pheromone trails' strength on each edge is modified by the pheromone-updating principle when each ant has built a solution.
Step 4: Terminating criterion. Steps 2-3 are iterated before the condition for termination is set.

Particle Swarm Optimization (PSO)
PSO is a powerful optimization method. As with fish and bird flocks, the cooperative conduct of insect colonies is inspired by this technique [59]. After all, it is interesting how these animals pass in one direction, often breaking into 2 groups to escape a predator and reforming the initial group [52]. Each one utilizes local information it can obtain to determine its relocation of its nearest neighbors. Simple rules such as 'keep reasonably close to other individuals', 'walk in the same direction', and 'go at the same pace' are adequate to preserve community cohesion and allow dynamic and adapted collective attitude. The PSO is planned to optimize nonlinear continuous functions [60]. Each individual is a particle reflecting a possible solution to optimize the issue. Let f the fitness function be reduced. Each particle k can be illustrated at iteration t by the characteristics below [61]: 1. Each particle k is demonstrated in n-dimension search space by its position vector X t k = X t k1 . X t k2 . . . . . X t kn and its velocity vector V t k = V t k1 . V t k2 . . . . . V t kn at iteration t, where X t kd : the position of the k at the dimension d = 1. n and V t kd : the velocity of the k at the dimension d = 1. n.
2. Each particle k remembers its best location attained until iteration t is denoted by Equation (8).
3. P t g is the best position in the swarm until iteration t and defined by Equation (9).
4. The current velocity of the dth is updated based on Equation (10).
ω controls the impact of the previous velocity on the current one, c 1 . c 2 are two positive constants employed to decide whether the particles prefer moving toward to P t k or P t g position, and r 1 .r 2 are random distributed real numbers in [0, 1]. 5. The position vector of the particle k at iteration t is given by Equation (11) [62].
7. The global best position found so far is obtained using Equation (13).
Here, the objective f is to reduce the makespan C max . According to their series in the vector location, the method must first schedule the workers on the towing machines utilizing a list algorithm to determine this value. Afterward, in the second unit, it calculates the average completion time. The pseudo-code of the PSO algorithm can be summarized [63]: Step 1: Initialization. Initialize an array of particles based on random locations.
Step 2: Local best updating. Assess the fitness function of the particles and update the P k i using the best current value.
Step 3: Global best updating. Calculate the current best global value among the existing positions and update P k g .
Step 4: Solution construction. Each particle is moved to a new spot by altering the velocities.

Particle Representation
Representing a particle phase tries to translate a solution to the FSSP into a particle (or ant). In this problem, a position in a particle represents a job number, and the number of positions (i.e., particle size) corresponds to the number of jobs, as shown in Figure 1. For example, consider particle 2 (J 4 → J 1 → J 2 → J 3 ). The J 4 is scheduled on the corresponding stage and assigned to a machine that can complete J 4 at the earliest completion time. Then the J 1 is scheduled, and so on. Here, the objective f is to reduce the makespan Cmax. According to their vector location, the method must first schedule the workers on the towing m lizing a list algorithm to determine this value. Afterward, in the second unit the average completion time. The pseudo-code of the PSO algorithm can be [63]: Step 1: Initialization. Initialize an array of particles based on random lo Step 2: Local best updating. Assess the fitness function of the particle the P i k using the best current value.
Step 3: Global best updating. Calculate the current best global value am ing positions and update P g k .
Step 4: Solution construction. Each particle is moved to a new spot by velocities.

Particle Representation
Representing a particle phase tries to translate a solution to the FSSP i (or ant). In this problem, a position in a particle represents a job number, an of positions (i.e., particle size) corresponds to the number of jobs, as shown For example, consider particle 2 (J4→J1→J2→J3). The J4 is scheduled on the c stage and assigned to a machine that can complete J4 at the earliest completio the J1 is scheduled, and so on.

Proposed Method for Flow Shop Scheduling
Most current feature selection algorithms struggle with the issue of loca nation [65][66][67]. ACO, GA, and PSO randomize the search space and use hist get around this problem. However, evolution operators such as crossover a are absent from ACO and PSO. However, both have memory, which is nece ating useful algorithms. In contrast to GA, where individuals compete with PSO and ACO are more effective since agents cooperate (particles or ants) [6 some other methods that are proposed for scheduling problems, such as t proach, have some other disadvantages. One disadvantage is that the Pareto not very suitable for combining with local search since several local moveme affect how well a solution ranks [69]. Thus, in this research, a hybrid method to solve the flow shop scheduling issue, which has a higher speed. The prop uses two heuristic and fitness functions to achieve makespan and optimal following, its integration with the PSO algorithm in achieving convergen

Proposed Method for Flow Shop Scheduling
Most current feature selection algorithms struggle with the issue of local optima stagnation [65][66][67]. ACO, GA, and PSO randomize the search space and use historical data to get around this problem. However, evolution operators such as crossover and mutation are absent from ACO and PSO. However, both have memory, which is necessary for creating useful algorithms. In contrast to GA, where individuals compete with one another, PSO and ACO are more effective since agents cooperate (particles or ants) [68]. Moreover, some other methods that are proposed for scheduling problems, such as the Pareto approach, have some other disadvantages. One disadvantage is that the Pareto technique is not very suitable for combining with local search since several local movements might not affect how well a solution ranks [69]. Thus, in this research, a hybrid method is employed to solve the flow shop scheduling issue, which has a higher speed. The proposed method uses two heuristic and fitness functions to achieve makespan and optimal delay. In the following, its integration with the PSO algorithm in achieving convergence solves the problem of scheduling tasks with optimal makespan and minimum delay of tasks. In the proposed method, tasks are sent to processing hosts as a set of requests, including the number of requested resources and the amount of time required to use these resources. Resources in this method include all three processors, memory, and storage. The proposed method uses two heuristic and fitness functions to achieve makespan and optimal delay.

The Proposed Method
This algorithm improves the search results after the ACO's search step is designed to optimize the particle swarm. The reason for this fact is the high convergence speed of this algorithm, which makes the proposed method complete the scheduling operation as soon as convergence is achieved. The main use of PSO is to obtain the best individual and overall solution in each iteration to solve the scheduling problem for new requests [70]. Equation (10) is used to obtain the velocity vector in the proposed method. S pby represents the best individual solution of ant y to iteration t. S gb is the best overall answer. Values C 1 and C 2 are two random weighting factors to choose either individual or general solutions. The difference is that the best individual solution signifies the best answer of the ant y in the iteration of t. In the main PSO algorithm, the addition operator and weighting factors are used to move the particle to the best individual solution and a certain amount toward the best general solution. However, the addition operator has attained another meaning for simplification and scheduling in the proposed method. In this paper, the A+B relation means selecting the A and B solutions in the population swarm optimization algorithm.
The process is such that to generate the velocity vector after the random production of weighting factors, according to these weighting factors, the solutions are selected from one of the sets of best individual answers or best overall answers using the Roulette Wheel selection method. If the scheduling is selected from a solution already in the list, that scheduling will be rejected, and another path will be selected again from another solution. The same process is repeated to obtain a new position, and the roulette wheel selection method is employed. The only difference is that the selection coefficient for selecting the velocity vector or the previous position is the same. Ultimately, the probability of selection between both of them is considered equal.
After each ant has produced its solution, Equation (14) evaluates each ant's score from the existing scheduling. The best solution for the current iteration is searched using this equation, and the best overall answer value is updated. Equation (15) uses the fuzzy set theory to sum the answers related to the makespan and the delay. In this equation, σ 1 and σ 2 are weighting factors. In Equations (16)- (18), the values Υ 1 , Υ 2 , Υ 3 , b 1 , and b 2 are weighting factors. A value of V is a characteristic of resource utilization distribution, and a value of D is known as the property of maximum use difference. Equation (15) represents makespan in hosts. The lower the value of this equation, the better the makespan. In Equation (16), the values CV mr , CV c , and CV d , respectively, determine the residual coefficient of variation for memory, processor, and storage among all hosts. Since this method uses three different resources, their values need to be normalized. Using the coefficient of variation can solve this issue. For example, if the value of CV mr is small, it means that the difference between the remaining values of memory is small among all hosts, and as a result, one minus this value results in a larger number. The values MR max and MR min express the maximum and minimum amount of remaining memory, respectively. The values UC max and UC min indicate the maximum and minimum CPU consumption, respectively. The values UD max and UD min are the maximum and minimum values of storage utilization, respectively.
In the proposed method, only the total pheromone values are updated. In other words, increasing the pheromone is possible if the selected path in the current iteration belongs to the best solution. Otherwise, in Equation (19), the value of f cb equals zero, resulting in increased evaporation of the pheromone evaporation process.
where p is the evaporation rate, and f cb is the best solution value for the current iteration. In the proposed method, two conditions are considered for completing the iterations of the hybrid algorithm and obtaining the scheduling. The first situation is when the algorithm has reached the maximum value possible for repetition. The second situation is one in which the function of the solution's solutions does not change for some time. The pseudo-code of the proposed hybrid method is outlined in Algorithm 1. The best solution for the flow shop scheduling Generate n particle agents randomly and distribute them uniformly over the search space while (itr < Countmax) For i = 1 to n Update the pheromone of the particle i, For j = 1 to m Update the pheromone of all attributes trail by increasing the pheromone in the paths followed by i.
End for End for For i = 1 to n Update the pheromone If the fitness of the current position < their best neighbors' fitness value, Then move to the neighbor's pattern If the fitness value of the last position is higher than the new position, Then replace that lower fitness value with the higher-fitness-value If the newly generated particle has more fitness than the earlier fitness, Then update the pheromone End for End.

O
Steps in Hybrid ACO-PSO Optimization Algorithm The steps of the proposed method are described in the following [71]: Step 1: Receiving the inputs.
Set the servers with their resource capacity. Set the VMs with their resource requirements.
Initialize population size, number of ants, and maximum number of iterations.
Step 3: Generate the initial population by VM placement solutions based on each server's power consumption and resource wastage.
Step 4: Update the particle-pheromone table with solutions obtained from ACO.
Step 6: Calculate the fitness of all particles and find their local best position.
Step 8: Update the particle's position.
Step 9: Update the particle-pheromone table with solutions obtained from PSO.
Step 10: Discover the global-best solution.
Step 11: Go to step 3 if the iteration value is less than maximum number of iterations, or go to step 12.
Step 12: Output the global-best solution for VM placement. Figure 2 depicts the flowchart of the proposed method.
Symmetry 2023, 15, x FOR PEER REVIEW 12 of 23 Step 10: Discover the global-best solution.
Step 11: Go to step 3 if the iteration value is less than maximum number of iterations, or go to step 12.
Step 12: Output the global-best solution for VM placement. Figure 2 depicts the flowchart of the proposed method.

Fitness Function
A fitness function determines the appropriateness of the solution. To calculate fitness values, the relative error is employed. By using synchronous master-slave parallelization, the fitness is achieved from Equations (20)

Fitness Function
A fitness function determines the appropriateness of the solution. To calculate fitness values, the relative error is employed. By using synchronous master-slave parallelization,

Experimental Results
These two goals (makespan and delay) are the most important, so they are considered in this paper. If the other goals, such as the flow time, tardy jobs, tardiness, earliness, etc., are added, it is likely to affect the results of these two goals. In this section, the simulation process is explained first. The algorithm is then examined with two other algorithms in three different scenarios, and the results are presented. At the end of this section, the statistical test results are presented.

Simulation Process
The simulations were performed on a PC with an Intel 6600 processor with 4 GB of memory and a Windows 8 operational system. To evaluate the algorithm's efficiency, MATLAB version 2019 was employed, and the system environment with several different scenarios was considered. The method's performance was compared to other approaches. The analysis of the results was reviewed and evaluated based on various criteria. Three different scenarios (with different sources) were considered to analyze the simulation results, and simulations were performed at different system scales.

Results
The issue of the FSSP is one of the main issues when trying to determine optimal scheduling for executing tasks and allocating optimal resources. The main focus of this article is minimizing delay and reducing makespan in symmetric networks. To compare the efficiency of the proposed method, the Ant Lion Optimizer (ALO) algorithm, Grey Wolf Optimization (GWO), ACO [72], PSO [42], and Motair [36] were taken into account. The values used as simulation parameters in MATLAB were considered according to Table 2. The method was implemented with tasks ranging from 2000 to 8000 in three different scenarios. The distribution was considered uniform. The efficiency of the approach was compared to the ACO-PSO algorithm in terms of delay time and makespan parameters. A comparison of evaluation parameters is discussed below.  Comparing the relative performance for different iteration counts is possible by computing the objective function, as shown in Figure 3. The fitness amount ranges from 0 to 1, as shown in Figure 3. The fitness value declines as the number of generations rises. As illustrated, the fitness value between the 140th and 240th generations fells to its lowest point of 0.26. The stability findings for the integers between 0.20 and 0.25 were achieved for 60 iterations, as shown in Figure 4. Therefore, the best and most frequent answer was 0.24.

❖ Delay Time
In this part, we compare the performance of the suggested method to the A strategy in three scenarios regarding latency.
• Scenario 1: The number of resources is 70 Figure 5 shows the amount of delay presented over 70 scheduled resources. ing to Figure 5, when the number of tasks is 2000, the delay is 5865 s in the Mota O Delay Time In this part, we compare the performance of the suggested method to the ACO-PSO strategy in three scenarios regarding latency. According to Figure 5, the delay increases with an increase in the number of tasks, which is less in the proposed method than in the compared methods except for Motaier (2021). It is a little better than the proposed method.    Figure 7, the delay increases with an increase in the number of tasks, which is less in the proposed method than in the compared methods except for Motaier (2021). It is a little better than the proposed method.
in the PSO algorithm. Similarly, when the number of tasks is 8000, the del s in the Motaier (2021), 8300 s in the proposed method, 8456 s in the GWO ALO, 9613 s in the ACO algorithm, and 8580 s in the PSO algorithm. The d with an increase in the number of tasks, which is less in the proposed meth compared methods except for Motaier (2021). It is a little better than the pro

O Makespan
This section examines the proposed method's performance in terms of makespan, with the ACO-PSO approach in three different scenarios.
• Scenario 1: The number of resources is 70 Figure 8 shows the amount of makespan that is scheduled for 70 resources. According to Figure 8, when the number of jobs is 1000, the makespan is about 500 in the proposed method, 520 in the GWO, 535 in the ALO, 595 in the Motaier, 555 in the PSO algorithm, and 575 in the ACO algorithm. Similarly, when the number of tasks is 8000, the makespan is about 1411 in the proposed method, 1490 in the GWO, 1509 in the ALO, 1600 in the Motaier, 1520 in the PSO algorithm, and 1550 in the ACO algorithm.  Figure 9 shows the amount of makespan presented on 120 s cording to Figure 9, when the number of tasks is 2000, the make proposed method, 208 in the Motaier, 210 in the GWO algorithm the PSO, and 235 in the ACO algorithm. Similarly, when the nu makespan is about 819 in the proposed method, 823 in the Mota in the ALO, 876 in the PSO algorithm, and 840 in the ACO algorit 9, as the number of jobs rises, the increase in the makespan in the than in other algorithms. • Scenario 2: The number of resources is 200 Figure 9 shows the amount of makespan presented on 120 scheduled resources. According to Figure 9, when the number of tasks is 2000, the makespan is about 200 in the proposed method, 208 in the Motaier, 210 in the GWO algorithm, 240 in the ALO, 245 in the PSO, and 235 in the ACO algorithm. Similarly, when the number of jobs is 8000, the makespan is about 819 in the proposed method, 823 in the Motaier, 835 in the GWO, 850 in the ALO, 876 in the PSO algorithm, and 840 in the ACO algorithm. According to Figure 9, as the number of jobs rises, the increase in the makespan in the proposed method is less than in other algorithms.
proposed method, 208 in the Motaier, 210 in the GWO algorithm, 240 in the ALO, 2 the PSO, and 235 in the ACO algorithm. Similarly, when the number of jobs is 8000 makespan is about 819 in the proposed method, 823 in the Motaier, 835 in the GWO in the ALO, 876 in the PSO algorithm, and 840 in the ACO algorithm. According to Fi 9, as the number of jobs rises, the increase in the makespan in the proposed method is than in other algorithms.

Statistical Analysis
The proposed method was examined analytically and compared to the ACO and PSO algorithms. Friedman has been used for many comparative studies. To provide a statisti

Statistical Analysis
The proposed method was examined analytically and compared to the ACO and PSO algorithms. Friedman has been used for many comparative studies. To provide a statistical study of the algorithms built to measure the experiment categories, we obtained the required post hoc procedures, and estimated the controlled p-values. Algorithms were statistically evaluated for the generalized implementation graphs generated at random.

H1.
The proposed method's efficiency seems to be different from that of the other methods.
H0. The proposed method's efficiency does not seem to differ from that of the other methods. Tables 3 and 4 show Friedman's results for the average rankings attained by the algorithms. To evaluate the Friedman test results and determine if the difference in the average success methods is significant, the results of Table 3 (Test Statistics) must be used.   Table 4 determines which proposed method has the least performance and which has the most. The table shows that the mean ranks of the GWO, ALO, ACO, and PSO algorithms were 2.19, 2.06, 1.99, and 1.86, respectively. However, the average of the proposed method was 2.25, indicating that the proposed method was now more efficient.

Conclusions and Future Work
In symmetric networks, scheduling deals with allocating resources over a given time frame to minimize one or more targets. Scheduling plays an essential part as a decision-making method for manufacturing and processing systems, transport, distribution structures, and even certain kinds of service. Investigators in this field have made a major contribution, from the classical issues of the literature to experimental applications. Because of its varied economic and industrial uses, several scholars have investigated the FSSP widely. The standard FSSP includes workers on a collection of machines with the same output flow. The aim is to find the job sequence that satisfies one or a set of criteria. An ACO-PSO algorithm is proposed to lessen the makespan and delay time in symmetric networks. To forecast the workload of novel input demands, ACO-PSO utilizes historical information. The proposed method selects the best host from the available hosts based on heuristic and fitness functions considering the two factors, makespan and delay, by searching all hosts over several iterations and assigning it to the host based on the requested resources. Combining the ACO and PSO algorithms can significantly reduce the delay and makespan. The proposed technique has higher effectiveness than the previous approaches. The results prove that the delay time was improved compared with the PSO and ACO algorithms; therefore, the proposed method is suitable for real-time applications. Moreover, the makespan of the suggested approach was enhanced when compared to PSO and ACO. In future work, the ACO-PSO algorithm can be extended to the FSSP considering other metrics, such as energy consumption, to realize green systems. In addition, utilizing the provided method in other distributed environments, such as the NSGA-III algorithm [73], distributionally robust optimization (DRO) [74], and Deep Belief Networks [75], is a very interesting line for future research. Furthermore, employing learning analytics and epistemic network analysis [76], grey wolf optimization [77], and Kalman filtering [78,79] for solving the FSSP can be investigated in the future. Finally, considering the dynamics approach [80,81] and human resources [82] in the symmetric networks can be investigated in future research.