# MTS-PRO2SAT: Hybrid Mutation Tabu Search Algorithm in Optimizing Probabilistic 2 Satisfiability in Discrete Hopfield Neural Network

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## Abstract

**:**

## 1. Introduction

- (a)
- To propose a modified metaheuristic algorithm, namely the hybrid mutation tabu search algorithm, which integrates a mutation operator and segment operation as neighborhood operations. Through these newly constructed neighborhood operations, the tabu search algorithm has been successfully incorporated into the systematic logic for minimizing the cost function.
- (b)
- To optimize the retrieval phase by introducing a mutation operator. The mutation operator randomly flips the literals in unsatisfied clauses, aiming to disrupt the bias in the final state of neurons and enhance the diversity of the global minimum.
- (c)
- To propose the average similarity metric to evaluate the variability and diversity of solutions in the retrieval phase by calculating the similarity between non-repeated solutions and benchmark solutions.
- (d)
- This study examined the effectiveness of the mutation tabu search algorithm through the analysis of different performance measures. The performance of the proposed approach was thoroughly evaluated against that of the leading logical rule in both the learning and retrieval phase.

## 2. Motivation

#### 2.1. Inefficient Learning Phase of DHNNs

#### 2.2. Limited Solution Diversity in Retrieval Phase of DHNN

## 3. Probabilistic 2 Satisfiability (PRO2SAT)

## 4. ${\mathit{P}}_{\mathit{P}\mathit{R}\mathit{O}\mathbf{2}\mathit{S}\mathit{A}\mathit{T}}$ in Discrete Hopfield Neural Network (DHNN)

## 5. Objective Function of PRO2SAT in Learning Phase

_{i}represents the i-th second-order clauses, and ${f}_{{P}_{PRO2SAT}}({S}_{i})$ represents the fitness value of the i-th clause of PRO2SAT, which is calculated as shown in Equation (17):

## 6. Proposed Metaheuristics

#### 6.1. Proposed Mutation Tabu Search Algorithm (MTS)

#### 6.1.1. Initialization

#### 6.1.2. Generation Strategy to Neighborhood Solution

#### 6.1.3. Generation Strategy to Candidate Solution

**Candidate solution**$N{S}_{*}$: the neighborhood solution with the maximum fitness is selected as the candidate solution, as defined in Equation (20).

**Tabu table**(TBT): it refers to the storage structure used to store the tabu operations, where two main indicators are available: tabu operations (TBS) and tabu length (L). In this paper, the mutation segment number of the neighborhood solution was used as the tabu operation to avoid the repeated search of the obtained solution, expand the search area of the algorithm, and escape the local optimum. The tabu length is usually set to a positive integer; too short a tabu length tends to fall into the local optimal solution prematurely, and too long a tabu length leads to prolonged computation. As a result, all the neighborhood solutions are tabued, and it is impossible to continue computing. Therefore, according to [25], we set the tabu length to 3.

**Update of the local optimal solution and the global optimal solution**: it determines whether the mutation segment number of the candidate solution has been stored in TBT when implementing the neighborhood operation. If not, it is written to TBT. The local optimal solution is updated and the candidate solution values are assigned to the local optimal solution. Note that, if the new local optimal solution outperforms the global optimal solution, it is necessary to update the global solution, and assign the new local optimal solution value to the global optimal solution. If the mutation segment number of the candidate solution has been recorded in TBT, in order to expand the search domain of the MTS algorithm, the current candidate solution should be discarded and the solution with the maximum fitness among the remaining solutions in the neighborhood solution should be selected as the new candidate solution $N{S}_{t+1}^{*}$. The above process is repeated until the local optimal solution and the global optimal solution are generated.

**Flouting rule**: If the mutation segments number of the candidate solution is available in TBT, but the fitness is greater than the current global optimal solution, the tabu operations can be neglected. The mutation segment number of the candidate solution is written to TBT, and the local optimal solution and the global optimal solution are updated accordingly, as defined in Equations (24) and (25).

#### 6.1.4. Fitness Assessment

#### 6.1.5. Mutation

#### 6.2. Baseline Model

- (a)
- GA [11]: This study integrated the strengths of the genetic algorithm and Hopfield neural network to efficiently find solutions to the logic satisfiability problem. The genetic algorithm is a global search algorithm which usually uses a binary encoding technique for optimization problems [26]. At the time of solving, the initial solution to the actual problem is coded to form a gene string, i.e., a chromosome, which is selected, crossed, and mutated to form a new chromosome. The resulting chromosome will be retained if it is closer to the maximum fitness than the previous one.
- (b)
- EA [12]: The EA algorithm is inspired by the election of a national president. The algorithm was proposed by Emami, H [27], which combined the features of evolutionary algorithms and SIA. In this algorithm [13], positive advertisement, negative advertisement, and coalition were used to implement intelligent search in a synergy mechanism. All solutions will be considered as voters from whom candidates will be selected, and each candidate determines his own voters based on his social relationship to the voters, thus forming a political party. As the leader of this party, the candidate will positively influence his own supporters (voters) through positive advertisement and positively influence the supporters (voters) for the leaders of other parties through negative advertisement in order to expand the search space and increase their probability of being elected. The most popular candidate will ultimately receive the most votes.
- (c)
- ACO [14]: Kho introduced the ACO algorithm in HDNNs (Kho, 2021). In this work, the ant colony algorithm is used to minimize the cost function of the corresponding logic rule in DHNNs. In the ACO algorithm, the pheromone density is used to find the optimal path, thus achieving a zero-cost function without consuming more learning iterations. In this study, the potential application of the ACO algorithm was fully demonstrated in optimization problems, including propositional logic.
- (d)
- EDA [28]: This algorithm is a probability-based population evolution algorithm. By generating a new population through random sampling, the evolution of the population is achieved through iterations. The main function of MTS is to estimate and predict the distribution of the data. This algorithm is able to predict future data trends by inferring the distribution of the data through statistical analysis. The standard EDA has two important operations, namely the selection operations and the modeling of the probability distribution. The selection operation is the same as the selection strategy in GA. The probability distribution model can be a univariate marginal distribution algorithm (UMDA), which is calculated.
- (e)
- DE [29]: A novel binary differential evolution algorithm based on Taper-shaped transfer functions (T-NBDE) is proposed to address the knapsack problem in [29]. The DE and GA algorithms are both evolutionary algorithms, which are adaptive global search algorithms first proposed by Price and Storn in the 1990s to solve the real number solution optimization problems. The DE algorithm has the main features of a simple structure, easy implementation, robustness and fast convergence, etc. In addition, the DE algorithm also has memory function, which can dynamically track the search situation, and the control parameters mainly include population size, variation operator, crossover operator, and selection operator. Unlike the GA algorithm, the variation operators of DE randomly select three individuals as parents for mutation operation to form new individuals.
- (f)
- GWO [23]: The GWO algorithm has been successfully applied to RDHNNs in the work by Ba et al. [16]. Therefore, we used GWO in the learning phase of PRO2SAT for comparative analysis. It is a heuristic optimization algorithm based on the behavior of grey wolf packs in nature, which simulates the social hierarchy of grey wolves and divides the individuals within the pack into four classes: head wolf (${X}_{\alpha}$), subordinate wolf (${X}_{\beta}$), common wolf (${X}_{\delta}$), and bottom wolf (${X}_{\omega}$). In the GWO algorithm, each grey wolf represents a potential solution, and each wolf has an adaptation value. The higher the adaptation value, the better the indicated solution. Starting from any position in the solution space, the individual with the best fitness is set as the leader wolf $\alpha $, the one with the second fitness is set as the subordinate wolf $\beta $, the one with the third fitness is set as the ordinary wolf $\delta $, and the rest are the bottom wolves $\omega $. The leader wolf is responsible for guiding the behavior of the pack, the subordinate wolf assists the leader wolf in making decisions, the ordinary wolf obeys the leader wolf and the subordinate wolf, and dominates the bottom wolf to catch and hunt the target. The bottom wolf needs to obey the guidance of other wolves and follows other wolves to complete hunting, and mainly takes charge of the balance of intra-pack relationships.
- (g)
- (PSO [30]: The PSO algorithm, proposed by Eberhart and Kenndy in 1995, is a type of SIA algorithm, which is widely used in the field of combinatorial optimization, and it can be applied to PRO2SAT. Inspired by the foraging behavior of a flock of birds, this algorithm includes evolutionary theory. The idea of the PSO algorithm is that each individual searches for a better solution based on the optimal solution that has been found and compares the optimal solution currently found by the population to update its speed. The algorithm searches for the global optimal solution by constantly updating the position and velocity of the population, which is a process of movement from simple individuals to complex global solutions.
- (h)
- SA [31]: The SA algorithm is a probability-based search algorithm proposed by Kirkpatrick, Gelatt, and Vecchi in 1983 to describe the physical annealing process of an object. It can be used to solve complex optimization problems. The basic idea is the process of finding the global optimal solution of the objective function randomly in the space of all local solutions starting from an initial solution (annealing point) in an isothermal process combined with the probabilistic sudden jump property, i.e., the ability to probabilistically jump out of each local solution and finally obtain the global optimal solution. The advantage of the SA algorithm is that it can find the global optimal solution and have a relatively large search space to find a better solution.
- (i)
- ABC [13]: This study introduces the combination of the artificial bee colony algorithm with the Hopfield network to minimize or maximize the cost function of any combinatorial problem. With this literature approach, we processed the secondary values using the ABC algorithm. Each bee was assigned an initial nectar source, and the employed bee dances through the solution space to compute the new nectar source. This explores the solution space of consistent solutions during the learning phase of the Hopfield neural network and identifies potential solutions. The combination of the ABC algorithm and the Hopfield neural network demonstrates the superior performance of the artificial bee colony in solving the 2 Satisfiability problem.

## 7. Experimental Setup

#### 7.1. Simulation Design

#### 7.2. Parameters Assignment

#### 7.3. Performance Evaluation Metrics

#### 7.3.1. Learning Phase Metrics

- (a)
- Mean absolute error of clause adaptation ($MA{E}_{learn}$)

- (b)
- Mean absolute error of the adaptation of logic rules ($LRTR$)

- (c)
- Mean similarity of consistency interpretations ($GL{I}_{avg}$)

- l indicates the overall occurrence count of $({S}_{i}^{bs}=1,{S}_{i}=1)$ in ${C}_{{S}_{i}^{bs},{S}_{i}}$;
- m indicates the overall occurrence count of $({S}_{i}^{bs}=1,{S}_{i}=-1)$ in ${C}_{{S}_{i}^{bs},{S}_{i}}$;
- n indicates the overall occurrence count of $({S}_{i}^{bs}=-1,{S}_{i}=1)$ in ${C}_{{S}_{i}^{bs},{S}_{i}}$;
- o indicates the overall occurrence count of $({S}_{i}^{bs}=-1,{S}_{i}=-1)$ in ${C}_{{S}_{i}^{bs},{S}_{i}}$.

- (d)
- Computational Time ($CT$)

- (e)
- Average iterations ($AI$)

#### 7.3.2. Retrieval Phase Metrics

- (f)
- Global minimum proportion ($ZM$)

- (g)
- The average similarity of solutions ($T{V}_{similar}$)

- (h)
- Mean Absolute Error of Logic Rule Energy ($MA{E}_{test}$)

- (i)
- Friedman Statistical Analysis (${F}_{d}$)

#### 7.4. Simulation Dataset

## 8. Result and Discussions

#### 8.1. Learning Phase

- (a)
- Experimental model for $\eta =\left\{0.1,0.2,0.3,0.4,0.5\right\}$

- (b)
- Experimental model for $\eta =\left\{0.6,0.7,0.8,0.9\right\}$

_{TC}and MTS

_{TC}, respectively. For a more intuitive comparison of algorithm time complexity, we set n and w to the same value. By comparing GA

_{TC}and MTS

_{TC}, it can be observed that MTS

_{TC}has a smaller time complexity. The advantage of the MTS algorithm mainly manifests in the aspect of the fitness evaluation. This advantage stems from the feedback correction mechanism implemented in the MTS algorithm during computation, specifically the guided search for the highest-quality solution using a tabu list. In contrast, the GA algorithm generates multiple offspring chromosomes in each iteration, leading to increased computational complexity in evaluating the fitness of the next generation, thus consuming more time. Additionally, the mutation operation in the GA algorithm requires traversing every gene in all chromosomes, directly resulting in a more significant time gap compared to the MTS algorithm. Therefore, the MTS algorithm exhibits higher search efficiency and performance, especially in terms of fitness evaluation.

#### 8.2. Retrieval Phase

#### 8.3. Friedman Test

#### 8.4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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$\mathit{N},\mathit{\eta}$ | $\mathit{N}\mathit{\eta}$ | ${\mathit{P}}_{\mathit{P}\mathit{R}\mathit{O}2\mathit{S}\mathit{A}\mathit{T}}$ |
---|---|---|

8, 0.1 | 1 | $(\neg {x}_{1}\vee {x}_{2})\wedge (\neg {x}_{3}\vee \neg {x}_{4})\wedge (\neg {x}_{5}\vee \neg {x}_{6})\wedge (\neg {x}_{7}\vee \neg {x}_{8})$ |

8, 0.3 | 2 | $({x}_{1}\vee \neg {x}_{2})\wedge (\neg {x}_{3}\vee \neg {x}_{4})\wedge (\neg {x}_{5}\vee {x}_{6})\wedge (\neg {x}_{7}\vee \neg {x}_{8})$ |

8, 0.5 | 4 | $({x}_{1}\vee \neg {x}_{2})\wedge (\neg {x}_{3}\vee {x}_{4})\wedge (\neg {x}_{5}\vee {x}_{6})\wedge ({x}_{7}\vee \neg {x}_{8})$ |

8, 0.7 | 6 | $({x}_{1}\vee \neg {x}_{2})\wedge ({x}_{3}\vee {x}_{4})\wedge (\neg {x}_{5}\vee {x}_{6})\wedge ({x}_{7}\vee {x}_{8})$ |

8, 0.9 | 7 | $({x}_{1}\vee {x}_{2})\wedge (\neg {x}_{3}\vee {x}_{4})\wedge ({x}_{5}\vee {x}_{6})\wedge ({x}_{7}\vee {x}_{8})$ |

Parameter | Parameter Value |
---|---|

Number of neurons ($NN$) | $10\le NN\le 120$ |

Neuron combination (a) | 100 |

Number of trials (b) | 100 |

Number of learnings (c) | 100 |

Tolerance value (Tol) | 0.001 |

Activation function | HTAF |

Synaptic weight method | Wan Abdullah method |

Initialization of neuron states | Random |

CPU computing time | 24 h |

Ratio of positive literal ($\eta $) | $\left\{0.1,0.2,\cdots ,0.9\right\}$ |

Control probability (${p}_{l}$) | {$\xi $,$\mathrm{max}(\eta ,1-\eta )$} |

Length of tabu table ($L$) | 3 |

Number of segments (w) | 8 |

Number of neighborhood operation segments | 2 |

Number of mutated neurons | 2 |

Neighborhood solution | 14 |

Selection rate | 1 |

Mutation rate | 1 |

Maximum number of iterations | 100 |

**Table 3.**The average $LRTR$ value of all proposed algorithms in the learning phase for $\eta =\left\{0.1,0.2,0.3,0.4,0.5\right\}$. The bold values indicate the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 0.000 | 0.690 | 0.000 | 0.850 | 0.000 | 1.000 | 0.000 | 0.220 | 1.000 | |

Mean | 0.263 | 1.000 | 0.262 | 0.937 | 0.620 | 0.954 | 0.609 | 1.000 | 0.536 | 0.850 | 1.000 | |

Std | 0.389 | 0.000 | 0.406 | 0.093 | 0.434 | 0.055 | 0.443 | 0.000 | 0.455 | 0.261 | 0.000 | |

Avg Rank | 9.375 | 3.667 | 9.625 | 5.333 | 6.458 | 4.958 | 6.917 | 3.667 | 7.458 | 4.875 | 3.667 | |

0.2 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 0.000 | 0.780 | 0.000 | 0.800 | 0.000 | 1.000 | 0.000 | 0.040 | 1.000 | |

Mean | 0.256 | 1.000 | 0.294 | 0.943 | 0.613 | 0.953 | 0.603 | 1.000 | 0.524 | 0.790 | 1.000 | |

Std | 0.386 | 0.000 | 0.423 | 0.069 | 0.438 | 0.062 | 0.445 | 0.000 | 0.465 | 0.341 | 0.000 | |

Avg Rank | 9.583 | 3.625 | 9.500 | 5.042 | 6.667 | 5.000 | 6.708 | 3.625 | 7.417 | 5.208 | 3.625 | |

0.3 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 0.000 | 0.780 | 0.100 | 0.870 | 0.000 | 1.000 | 0.000 | 0.050 | 1.000 | |

Mean | 0.253 | 1.000 | 0.360 | 0.953 | 0.615 | 0.971 | 0.608 | 1.000 | 0.523 | 0.774 | 1.000 | |

Std | 0.383 | 0.000 | 0.448 | 0.069 | 0.432 | 0.044 | 0.443 | 0.000 | 0.457 | 0.349 | 0.000 | |

Avg Rank | 9.792 | 3.625 | 9.000 | 5.208 | 6.250 | 5.042 | 7.000 | 3.625 | 7.635 | 5.208 | 3.625 | |

0.4 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 0.000 | 0.74 | 0.000 | 0.81 | 0.000 | 1.000 | 0.000 | 0.040 | 1.000 | |

Mean | 0.263 | 1.000 | 0.483 | 0.939 | 0.620 | 0.956 | 0.599 | 1.000 | 0.520 | 0.754 | 1.000 | |

Std | 0.394 | 0.000 | 0.459 | 0.078 | 0.435 | 0.065 | 0.445 | 0.000 | 0.462 | 0.370 | 0.000 | |

Avg Rank | 9.792 | 3.583 | 8.333 | 5.417 | 6.833 | 5.083 | 7.000 | 3.583 | 7.583 | 5.208 | 3.583 | |

0.5 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 0.820 | 0.830 | 0.000 | 0.880 | 0.000 | 1.000 | 0.000 | 0.030 | 1.000 | |

Mean | 0.266 | 1.000 | 0.960 | 0.957 | 0.612 | 0.963 | 0.614 | 1.000 | 0.525 | 0.750 | 1.000 | |

Std | 0.395 | 0.000 | 0.064 | 0.064 | 0.438 | 0.045 | 0.439 | 0.000 | 0.463 | 0.375 | 0.000 | |

Avg Rank | 9.917 | 3.833 | 5.167 | 5.167 | 7.5 | 5.583 | 7.042 | 3.833 | 8.208 | 5.917 | 3.833 |

**Table 4.**The average $LRTR$ value of all proposed algorithms in the learning phase for $\eta =\left\{0.6,0.7,0.8,0.9\right\}$. The bold values indicate the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.6 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 1.000 | 0.770 | 0.010 | 0.820 | 0.000 | 1.000 | 0.000 | 0.050 | 1.000 | |

Mean | 0.300 | 1.000 | 1.000 | 0.935 | 0.265 | 0.920 | 0.225 | 1.000 | 0.150 | 0.420 | 1.000 | |

Std | 0.391 | 0.000 | 0.000 | 0.084 | 0.436 | 0.057 | 0.445 | 0.000 | 0.458 | 0.367 | 0.000 | |

Avg Rank | 10.417 | 4.042 | 4.042 | 5.625 | 6.917 | 5.542 | 7.250 | 4.042 | 8.167 | 5.917 | 4.042 | |

0.7 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 1.000 | 0.740 | 0.000 | 0.83 | 0.000 | 1.000 | 0.000 | 0.070 | 1.000 | |

Mean | 0.510 | 1.000 | 1.000 | 0.950 | 0.140 | 0.965 | 0.150 | 1.000 | 0.255 | 0.475 | 1.000 | |

Std | 0.393 | 0.000 | 0.000 | 0.080 | 0.434 | 0.061 | 0.439 | 0.000 | 0.463 | 0.355 | 0.000 | |

Avg Rank | 10.375 | 4.083 | 4.083 | 5.375 | 7.000 | 5.708 | 7.167 | 4.083 | 8.208 | 5.833 | 4.083 | |

0.8 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 1.000 | 0.740 | 0.000 | 0.800 | 0.000 | 1.000 | 0.000 | 0.160 | 1.000 | |

Mean | 0.150 | 1.000 | 1.000 | 0.930 | 0.100 | 0.930 | 0.350 | 1.000 | 0.575 | 0.705 | 1.000 | |

Std | 0.395 | 0.000 | 0.000 | 0.090 | 0.443 | 0.071 | 0.447 | 0.000 | 0.457 | 0.298 | 0.000 | |

Avg Rank | 9.917 | 4.000 | 4.000 | 5.625 | 6.833 | 5.625 | 7.625 | 4.000 | 8.667 | 5.708 | 4.000 | |

0.9 | Max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Min | 0.000 | 1.000 | 1.000 | 0.740 | 0.000 | 0.700 | 0.000 | 1.000 | 0.000 | 0.000 | 1.000 | |

Mean | 0.170 | 1.000 | 1.000 | 0.895 | 0.300 | 0.945 | 0.200 | 1.000 | 0.225 | 0.400 | 1.000 | |

Std | 0.388 | 0.000 | 0.000 | 0.086 | 0.430 | 0.082 | 0.445 | 0.000 | 0.462 | 0.341 | 0.000 | |

Avg Rank | 9.875 | 4.167 | 4.167 | 5.500 | 7.333 | 5.792 | 7.208 | 4.167 | 8.167 | 5.458 | 4.167 |

**Table 5.**Matrix diagram of $GL{I}_{avg}$ for MTS and EA models. The bold value indicates the superior result among the mentioned metrics.

η | Algorithm | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | MTS | 0.457 | 0.946 | 1.426 | 1.949 | 2.462 | 3.150 | 3.626 | 4.212 | 4.785 | 5.161 | 5.776 | 6.242 |

EA | 0.466 | 0.919 | 1.422 | 1.873 | 2.498 | 3.076 | 3.588 | 4.259 | 4.641 | 5.278 | 5.855 | 6.505 | |

0.2 | MTS | 0.800 | 1.652 | 2.586 | 3.610 | 4.564 | 5.324 | 6.510 | 7.409 | 8.278 | 9.232 | 9.991 | 11.074 |

EA | 0.814 | 1.629 | 2.514 | 3.465 | 4.257 | 5.418 | 6.439 | 7.497 | 8.339 | 9.481 | 10.072 | 11.181 | |

0.3 | MTS | 1.072 | 2.146 | 3.464 | 4.713 | 5.894 | 7.462 | 8.457 | 9.725 | 11.091 | 12.319 | 13.426 | 14.755 |

EA | 1.062 | 2.175 | 3.352 | 4.556 | 5.648 | 7.089 | 8.397 | 9.698 | 10.752 | 12.402 | 13.340 | 14.484 | |

0.4 | MTS | 1.198 | 2.447 | 3.876 | 5.284 | 6.736 | 8.285 | 9.791 | 11.122 | 12.610 | 13.891 | 15.477 | 16.744 |

EA | 1.197 | 2.512 | 3.834 | 5.201 | 6.688 | 8.112 | 9.697 | 10.942 | 12.663 | 13.986 | 15.338 | 16.892 | |

0.5 | MTS | 1.252 | 2.590 | 4.020 | 5.611 | 7.098 | 8.682 | 10.061 | 11.571 | 13.080 | 14.564 | 15.870 | 17.312 |

EA | 1.241 | 2.561 | 3.902 | 5.211 | 6.944 | 8.421 | 9.791 | 11.501 | 12.936 | 14.555 | 16.169 | 17.662 | |

0.6 | MTS | 1.198 | 2.506 | 3.857 | 5.316 | 6.754 | 8.175 | 9.725 | 11.160 | 12.761 | 13.992 | 15.323 | 16.634 |

EA | 1.184 | 2.443 | 3.841 | 5.249 | 6.667 | 8.058 | 9.690 | 11.122 | 12.723 | 13.981 | 15.630 | 16.839 | |

0.7 | MTS | 1.062 | 2.191 | 3.400 | 4.520 | 6.032 | 7.248 | 8.420 | 9.775 | 11.084 | 12.265 | 13.502 | 14.629 |

EA | 1.056 | 2.163 | 3.328 | 4.463 | 5.692 | 7.057 | 8.533 | 9.633 | 11.107 | 12.203 | 13.448 | 14.723 | |

0.8 | MTS | 0.816 | 1.691 | 2.586 | 3.518 | 4.432 | 5.670 | 6.519 | 7.476 | 8.221 | 9.145 | 10.194 | 11.138 |

EA | 0.808 | 1.686 | 2.471 | 3.496 | 4.451 | 5.379 | 6.319 | 7.407 | 8.237 | 9.250 | 10.140 | 11.228 | |

0.9 | MTS | 0.458 | 0.918 | 1.390 | 2.068 | 2.721 | 3.325 | 3.759 | 4.228 | 4.708 | 5.339 | 5.661 | 6.138 |

EA | 0.455 | 0.957 | 1.423 | 1.862 | 2.401 | 2.926 | 3.649 | 4.218 | 4.554 | 5.244 | 5.814 | 6.249 |

**Table 6.**Matrix representing the frequency of the maximum $GL{I}_{avg}$. The bold value indicates the superior result among the mentioned metrics.

Model | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | Probability |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MTS | 7 | 6 | 8 | 9 | 7 | 8 | 8 | 7 | 5 | 4 | 4 | 1 | 69% |

EA | 2 | 3 | 1 | 0 | 2 | 1 | 1 | 2 | 4 | 5 | 5 | 8 | 31% |

Index | GA | Time Complexity | MTS | Time Complexity |
---|---|---|---|---|

1 | Selection operation | O(clogc) | Initializing the initial solutions | O(clogc) |

2 | Crossover operation | O(m^{2} + nlogn) | Generation strategy to neighborhood solution | O(w × NN/2 + wlogw + w) |

3 | Mutation operation | O(n × NN) | Generation strategy to candidate solution | O(2) |

4 | Fitness evaluation | O(n × NN) | Fitness evaluation | O(NN/2) |

collect | GA_{TC} = O(clogc + w × NN/2 + wlogw +m^{2} +w × NN) | MTS_{TC} = O(clogc + w × NN/2 + wlogw + w+2 + NN/2) |

**Table 8.**The average $ZM$ value of all proposed algorithms during the retrieval phase at $\eta =\left\{0.1,0.2,0.3,0.4,0.5\right\}$. The bold value indicates the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | Max | 1.000 /20 | 1.000 | 1.000 /20 | 1.000 /70 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /70 | 1.000 | 0.000 /80 | 0.690 /120 | 0.000 /120 | 0.780 /120 | 0.000 /120 | 1.000 | 0.000 /110 | 0.180 /120 | 1.000 | |

Mean | 0.262 | 1.000 | 0.262 | 0.937 | 0.620 | 0.954 | 0.609 | 1.000 | 0.536 | 0.850 | 1.000 | |

Std | 0.405 | 0.000 | 0.424 | 0.097 | 0.445 | 0.075 | 0.467 | 0.000 | 0.479 | 0.282 | 0.000 | |

Avg Rank | 9.540 | 3.670 | 9.620 | 5.330 | 6.370 | 4.830 | 6.540 | 3.670 | 7.580 | 5.170 | 3.670 | |

0.2 | Max | 1.000 /20 | 1.000 | 1.000 /20 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /80 | 1.000 | 0.000 /80 | 0.780 /120 | 0.000 /120 | 0.800 /120 | 0.000 /110 | 1.000 | 0.000 /120 | 0.040 /120 | 1.000 | |

Mean | 0.256 | 1.000 | 0.294 | 0.943 | 0.613 | 0.953 | 0.603 | 1.000 | 0.524 | 0.790 | 1.000 | |

Std | 0.403 | 0.000 | 0.442 | 0.087 | 0.458 | 0.065 | 0.465 | 0.000 | 0.485 | 0.356 | 0.000 | |

Avg Rank | 9.580 | 3.620 | 9.500 | 5.000 | 6.710 | 5.000 | 6.750 | 3.620 | 7.460 | 5.120 | 3.620 | |

0.3 | Max | 1.000 /20 | 1.000 | 1.000 /30 | 1.000 /60 | 1.000 /60 | 1.000 /50 | 1.000 /50 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /90 | 1.000 | 0.000 /100 | 0.780 /120 | 0.010 /120 | 0.870 /120 | 0.000 /110 | 1.000 | 0.000 /110 | 0.050 /120 | 1.000 | |

Mean | 0.255 | 1.000 | 0.360 | 0.953 | 0.615 | 0.971 | 0.608 | 1.000 | 0.523 | 0.774 | 1.000 | |

Std | 0.401 | 0.000 | 0.468 | 0.719 | 0.451 | 0.046 | 0.462 | 0.000 | 0.477 | 0.365 | 0.000 | |

Avg Rank | 9.750 | 3.620 | 9.040 | 5.210 | 6.250 | 5.040 | 7.000 | 3.620 | 7.620 | 5.210 | 3.620 | |

0.4 | Max | 1.000 /20 | 1.000 | 1.000 /40 | 1.000 /50 | 1.000 /50 | 1.000 /60 | 1.000 /50 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /80 | 1.000 | 0.000 /120 | 0.740 /120 | 0.000 /110 | 0.810 /120 | 0.000 /110 | 1.000 | 0.000 /110 | 0.040 /120 | 1.000 | |

Mean | 0.264 | 1.000 | 0.483 | 0.939 | 0.538 | 0.956 | 0.517 | 1.000 | 0.520 | 0.754 | 1.000 | |

Std | 0.412 | 0.000 | 0.479 | 0.082 | 0.471 | 0.068 | 0.477 | 0.000 | 0.483 | 0.386 | 0.000 | |

Avg Rank | 9.710 | 3.580 | 8.170 | 5.330 | 7.080 | 5.080 | 7.250 | 3.580 | 7.420 | 5.210 | 3.580 | |

0.5 | Max | 1.000 /20 | 1.000 | 1.000 /70 | 1.000 /70 | 1.000 /50 | 1.000 /50 | 1.000 /60 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /70 | 1.000 | 0.820 /120 | 0.830 /120 | 0.000 /110 | 0.890 /120 | 0.000 /120 | 1.000 | 0.000 /100 | 0.030 /120 | 1.000 | |

Mean | 0.267 | 1.000 | 0.960 | 0.957 | 0.612 | 0.963 | 0.614 | 1.000 | 0.525 | 0.750 | 1.000 | |

Std | 0.413 | 0.000 | 0.665 | 0.669 | 0.458 | 0.047 | 0.459 | 0.000 | 0.483 | 0.392 | 0.000 | |

Avg Rank | 9.920 | 3.830 | 5.170 | 5.170 | 7.500 | 5.580 | 7.040 | 3.830 | 8.210 | 5.920 | 3.830 |

**Table 9.**The average $ZM$ value of all proposed algorithms during the retrieval phase at $\eta =\left\{0.6,0.7,0.8,0.9\right\}$. The bold value indicates the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.6 | Max | 1.000 /10 | 1.000 | 1.000 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 | 1.000 /50 | 1.000 /50 | 1.000 |

Min | 0.000 /70 | 1.000 | 1.000 | 0.770 /120 | 0.010 /120 | 0.820 /120 | 0.000 /100 | 1.000 | 0.000 /110 | 0.050 /120 | 1.000 | |

Mean | 0.258 | 1.000 | 1.000 | 0.942 | 0.611 | 0.955 | 0.607 | 1.000 | 0.529 | 0.746 | 1.000 | |

Std | 0.409 | 0.000 | 0.000 | 0.874 | 0.456 | 0.060 | 0.465 | 0.000 | 0.479 | 0.383 | 0.000 | |

Avg Rank | 10.420 | 4.040 | 4.040 | 5.620 | 6.920 | 5.540 | 7.080 | 4.040 | 8.170 | 5.920 | 4.040 | |

0.7 | Max | 1.000 /10 | 1.000 | 1.000 | 1.000 /70 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 | 1.000 /50 | 1.000 /70 | 1.000 |

Min | 0.000 /60 | 1.000 | 1.000 | 0.740 /120 | 0.020 /120 | 0.830 /120 | 0.000 /120 | 1.000 | 0.000 /100 | 0.070 /100 | 1.000 | |

Mean | 0.254 | 1.000 | 1.000 | 0.957 | 0.619 | 0.959 | 0.608 | 1.000 | 0.536 | 0.761 | 1.000 | |

Std | 0.411 | 0.000 | 0.000 | 0.084 | 0.453 | 0.064 | 0.458 | 0.000 | 0.483 | 0.371 | 0.000 | |

Avg Rank | 10.370 | 4.080 | 4.080 | 5.370 | 7.000 | 5.710 | 7.170 | 4.080 | 8.210 | 5.830 | 4.080 | |

0.8 | Max | 1.000 /20 | 1.000 | 1.000 | 1.000 /60 | 1.000 /60 | 1.000 /60 | 1.000 /50 | 1.000 | 1.000 /40 | 1.000 /70 | 1.000 |

Min | 0.000 /70 | 1.000 | 1.000 | 0.740 /120 | 0.000 /120 | 0.800 /120 | 0.000 /120 | 1.000 | 0.000 /90 | 0.160 /120 | 1.000 | |

Mean | 0.259 | 1.000 | 1.000 | 0.937 | 0.618 | 0.948 | 0.603 | 1.000 | 0.520 | 0.818 | 1.000 | |

Std | 0.413 | 0.000 | 0.000 | 0.094 | 0.463 | 0.074 | 0.467 | 0.000 | 0.478 | 0.311 | 0.000 | |

Avg Rank | 9.920 | 4.000 | 4.000 | 5.620 | 6.830 | 5.620 | 7.620 | 4.000 | 8.670 | 5.710 | 4.000 | |

0.9 | Max | 1.000 /20 | 1.000 | 1.000 | 1.000 /70 | 1.000 /50 | 1.000 /60 | 1.000 /50 | 1.000 | 1.000 /50 | 1.000 /90 | 1.000 |

Min | 0.000 /70 | 1.000 | 1.000 | 0.740 /120 | 0.000 /120 | 0.700 /120 | 0.000 /110 | 1.000 | 0.000 /100 | 0.000 /120 | 1.000 | |

Mean | 0.257 | 1.000 | 1.000 | 0.946 | 0.617 | 0.952 | 0.599 | 1.000 | 0.530 | 0.817 | 1.000 | |

Std | 0.405 | 0.000 | 0.000 | 0.090 | 0.450 | 0.086 | 0.463 | 0.000 | 0.482 | 0.356 | 0.000 | |

Avg Rank | 9.870 | 4.120 | 4.120 | 5.460 | 7.250 | 5.750 | 7.580 | 4.120 | 8.170 | 5.420 | 4.120 |

**Table 10.**The average $MA{E}_{test}$ value of all proposed algorithms during the retrieval phase at $\eta =\left\{0.1,0.2,0.3,0.4,0.5\right\}$. The bold values indicate the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | Max | 14.604 /120 | 0.000 | 16.077 /120 | 2.668 /120 | 15.498 /120 | 3.160 /120 | 14.857 /120 | 0.000 | 14.944 /120 | 12.505 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 /20 | 0.000 /50 | 0.000 /60 | 0.000 /60 | 0.000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.388 | 0.000 | 7.452 | 0.659 | 4.925 | 0.589 | 4.955 | 0.000 | 5.587 | 2.112 | 0.000 | |

Std | 5.287 | 0.000 | 5.854 | 1.013 | 6.200 | 1.021 | 6.230 | 0.000 | 6.148 | 4.151 | 0.000 | |

Avg Rank | 9.333 | 3.625 | 9.500 | 5.417 | 6.750 | 4.917 | 6.667 | 3.625 | 7.417 | 5.125 | 3.625 | |

0.2 | Max | 15.559 /120 | 0.000 | 17.358 /120 | 3.559 /120 | 14.524 /120 | 2.077 /120 | 15.431 /120 | 0.000 | 14.747 /120 | 13.961 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 /20 | 0.000 /60 | 0.000 /50 | 0.000 /60 | 0.000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.409 | 0.000 | 7.228 | 0.762 | 4.760 | 0.489 | 5.106 | 0.000 | 5.541 | 2.775 | 0.000 | |

Std | 5.422 | 0.000 | 5.740 | 1.252 | 6.049 | 0.756 | 6.286 | 0.000 | 6.050 | 4.963 | 0.000 | |

Avg Rank | 9.667 | 3.625 | 9.417 | 5.167 | 6.750 | 4.833 | 6.833 | 3.625 | 7.333 | 5.125 | 3.625 | |

0.3 | Max | 15.045 /120 | 0.000 | 14.590 /120 | 3.271 /120 | 14.255 /120 | 2.920 /120 | 14.752 /120 | 0.000 | 15.040 /120 | 14.117 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 /30 | 0.000 /60 | 0.000 /60 | 0.000 /50 | 0.000 /50 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.374 | 0.000 | 6.935 | 0.739 | 4.843 | 0.604 | 4.881 | 0.000 | 5.633 | 3.190 | 0.000 | |

Std | 5.346 | 0.000 | 5.493 | 1.197 | 6.089 | 1.019 | 6.082 | 0.000 | 6.143 | 5.226 | 0.000 | |

Avg Rank | 9.917 | 3.625 | 8.792 | 5.167 | 6.583 | 4.958 | 6.792 | 3.625 | 7.625 | 5.292 | 3.625 | |

0.4 | Max | 15.272 /120 | 0.000 | 17.675 /120 | 5.091 /120 | 15.229 /120 | 3.289 /120 | 14.988 /120 | 0.000 | 14.791 /120 | 14.167 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 /40 | 0.000 /50 | 0.000 /60 | 0.000 /60 | 0.000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.420 | 0.000 | 5.929 | 0.834 | 4.863 | 0.607 | 5.021 | 0.000 | 5.631 | 3.399 | 0.000 | |

Std | 5.392 | 0.000 | 6.541 | 1.465 | 6.149 | 1.000 | 6.207 | 0.000 | 6.185 | 5.306 | 0.000 | |

Avg Rank | 9.833 | 3.667 | 8.167 | 5.458 | 6.542 | 5.250 | 6.792 | 3.667 | 7.625 | 5.333 | 3.667 | |

0.5 | Max | 15.067 /120 | 0.000 | 5.601 /120 | 3.354 /120 | 14.711 /120 | 2.880 /120 | 14.581 /120 | 0.000 | 14.902 /120 | 14.661 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 /70 | 0.000 /70 | 0.000 /50 | 0.000 /50 | 0.000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.396 | 0.000 | 0.990 | 0.666 | 5.094 | 0.498 | 4.993 | 0.000 | 5.641 | 3.392 | 0.000 | |

Std | 5.332 | 0.000 | 1.715 | 1.120 | 6.140 | 0.869 | 6.092 | 0.000 | 6.142 | 5.371 | 0.000 | |

Avg Rank | 9.833 | 3.833 | 5.333 | 5.083 | 7.542 | 5.542 | 6.875 | 3.833 | 8.292 | 6.000 | 3.833 |

**Table 11.**The average $MA{E}_{test}$ value of all proposed algorithms during the retrieval phase at $\eta =\left\{0.6,0.7,0.8,0.9\right\}$. The bold values indicate the superior result among the mentioned metrics.

η | Measure | ES | GA | ABC | EDA | PSO | ACO | DE | EA | SA | GWO | MTS |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.6 | Max | 14.723 /120 | 0.000 | 0.000 | 3.678 /120 | 15.109 /120 | 1.692 /120 | 14.908 /120 | 0.000 | 14.852 /120 | 14.030 /120 | 0.000 |

Min | 0.000 /10 | 0.000 | 0.000 | 0.000 /60 | 0.000 /60 | 0.000 /60 | 0.0000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.383 | 0.000 | 0.000 | 0.696 | 4.803 | 0.474 | 4.853 | 0.000 | 5.555 | 3.399 | 0.000 | |

Std | 5.311 | 0.000 | 0.000 | 1.142 | 6.063 | 0.695 | 6.068 | 0.000 | 6.091 | 5.386 | 0.000 | |

Avg Rank | 10.083 | 4.042 | 4.042 | 5.667 | 7.250 | 5.500 | 7.500 | 4.042 | 7.917 | 5.917 | 4.042 | |

0.7 | Max | 14.888 /120 | 0.000 | 0.000 | 2.596 /120 | 14.221 /120 | 1.691 /120 | 14.912 /120 | 0.000 | 14.865 /120 | 14.975 /120 | 0.000 |

Min | 0.000 /10 | 0.000 | 0.000 | 0.000 /70 | 0.000 /60 | 0.000 /60 | 0.000 /60 | 0.000 | 0.000 /50 | 0.000 /70 | 0.000 | |

Mean | 7.384 | 0.000 | 0.000 | 0.625 | 4.694 | 0.457 | 4.947 | 0.000 | 5.677 | 3.085 | 0.000 | |

Std | 5.357 | 0.000 | 0.000 | 0.944 | 6.009 | 0.704 | 6.190 | 0.000 | 6.209 | 5.346 | 0.000 | |

Avg Rank | 10.083 | 4.083 | 4.083 | 5.458 | 6.833 | 5.500 | 7.583 | 4.083 | 7.917 | 6.292 | 4.083 | |

0.8 | Max | 14.930 /120 | 0.000 | 0.000 | 2.892 /120 | 15.01 /120 | 2.513 /120 | 14.982 /120 | 0.000 | 14.813 /120 | 12.733 /120 | 0.000/120 |

Min | 0.000 /20 | 0.000 | 0.000 | 0.000 /60 | 0.000 /60 | 0.000 /60 | 0.000 /50 | 0.000 | 0.000 /40 | 0.000 /70 | 0.000/10 | |

Mean | 7.551 | 0.000 | 0.000 | 0.633 | 4.949 | 0.539 | 4.909 | 0.000 | 5.620 | 2.597 | 0.000 | |

Std | 5.421 | 0.000 | 0.000 | 0.962 | 6.114 | 0.932 | 6.063 | 0.000 | 6.093 | 4.531 | 0.000 | |

Avg Rank | 9.917 | 4.000 | 4.000 | 5.708 | 7.375 | 5.458 | 7.500 | 4.000 | 8.250 | 5.792 | 4.000 | |

0.9 | Max | 14.821 /120 | 0.000 | 0.000 | 2.955 /120 | 14.389 /120 | 2.222 /120 | 15.276 /120 | 0.000 | 15.069 /120 | 11.345 /120 | 0.000 |

Min | 0.000 /20 | 0.000 | 0.000 | 0.000 /70 | 0.000 /50 | 0.000 /60 | 0.000 /50 | 0.000 | 0.000 /50 | 0.000 /90 | 0.000 | |

Mean | 8.841 | 0.000 | 0.000 | 0.997 | 8.143 | 0.515 | 5.186 | 0.000 | 9.681 | 4.500 | 0.000 | |

Std | 5.304 | 0.000 | 0.000 | 0.979 | 5.835 | 0.822 | 6.216 | 0.000 | 6.197 | 3.805 | 0.000 | |

Avg Rank | 9.667 | 4.125 | 4.125 | 5.583 | 7.125 | 5.625 | 7.958 | 4.125 | 8.292 | 5.250 | 4.125 |

Model | η | Chi-Square Value, χ^{2} | p-Value | Accept(A)/Reject(R), H_{0} | Model | Chi-Square Value, χ^{2} | p-Value | Accept/Reject, H_{0} |
---|---|---|---|---|---|---|---|---|

0.1 | 67.727 | 1.215 × 10^{−10} | R H_{0} | 79.697 | 5.7561 × 10^{−13} | R H_{0} | ||

0.2 | 70.182 | 4.089 × 10^{−11} | R H_{0} | 80.273 | 4.4381 × 10^{−13} | R H_{0} | ||

0.3 | 69.546 | 5.426 × 10^{−11} | R H_{0} | 77.994 | 1.2399 × 10^{−12} | R H_{0} | ||

0.4 | 64.303 | 5.509 × 10^{−10} | R H_{0} | 73.484 | 9.3768 × 10^{−12} | R H_{0} | ||

0.5 | 75.046 | 4.662 × 10^{−12} | R H_{0} | 71.689 | 2.0903 × 10^{−11} | R H_{0} | ||

0.6 | 77.455 | 1.581 × 10^{−12} | R H_{0} | 78.046 | 1.2112 × 10^{−12} | R H_{0} | ||

0.7 | 72.167 | 1.689 × 10^{−12} | R H_{0} | 77.552 | 1.5129 × 10^{−12} | R H_{0} | ||

0.8 | 81.742 | 2.285 × 10^{−13} | R H_{0} | 76.690 | 2.229 × 10^{−12} | R H_{0} | ||

0.9 | 71.621 | 2.154 × 10^{−11} | R H_{0} | 71.525 | 2.249 × 10^{−11} | R H_{0} | ||

GLI_{avg} | 0.1 | 41.028 | 1.116 × 10^{−5} | R H_{0} | CT | 98.527 | 1.074 × 10^{−16} | R H_{0} |

0.2 | 40.920 | 1.166 × 10^{−5} | R H_{0} | 87.320 | 1.818 × 10^{−14} | R H_{0} | ||

0.3 | 54.616 | 3.725 × 10^{−8} | R H_{0} | 97.046 | 2.122 × 10^{−16} | R H_{0} | ||

0.4 | 53.175 | 6.909 × 10^{−8} | R H_{0} | 96.696 | 2.493 × 10^{−16} | R H_{0} | ||

0.5 | 54.191 | 4.472 × 10^{−8} | R H_{0} | 93.146 | 1.271 × 10^{−15} | R H_{0} | ||

0.6 | 58.906 | 5.835 × 10^{−9} | R H_{0} | 100.993 | 3.448 × 10^{−17} | R H_{0} | ||

0.7 | 55.253 | 2.833 × 10^{−8} | R H_{0} | 90.348 | 4.569 × 10^{−15} | R H_{0} | ||

0.8 | 63.445 | 8.032 × 10^{−10} | R H_{0} | 93.743 | 9.667 × 10^{−16} | R H_{0} | ||

0.9 | 57.113 | 1.270 × 10^{−8} | R H_{0} | 93.733 | 9.711 × 10^{−16} | R H_{0} | ||

AI | 0.1 | 76.419 | 2.517 × 10^{−12} | R H_{0} | ZM | 79.650 | 5.878 × 10^{−13} | R H_{0} |

0.2 | 72.226 | 1.645 × 10^{−11} | R H_{0} | 80.273 | 4.438 × 10^{−13} | R H_{0} | ||

0.3 | 76.282 | 2.677 × 10^{−12} | R H_{0} | 77.843 | 1.327 × 10^{−12} | R H_{0} | ||

0.4 | 74.333 | 6.416 × 10^{−12} | R H_{0} | 72.248 | 1.629 × 10^{−11} | R H_{0} | ||

0.5 | 75.121 | 4.507 × 10^{−12} | R H_{0} | 71.689 | 2.090 × 10^{−11} | R H_{0} | ||

0.6 | 81.908 | 2.120 × 10^{−13} | R H_{0} | 78.046 | 1.211 × 10^{−12} | R H_{0} | ||

0.7 | 71.021 | 2.816 × 10^{−11} | R H_{0} | 77.552 | 1.513 × 10^{−12} | R H_{0} | ||

0.8 | 76.477 | 2.452 × 10^{−12} | R H_{0} | 76.690 | 2.229 × 10^{−12} | R H_{0} | ||

0.9 | 74.019 | 7.383 × 10^{−12} | R H_{0} | 73.164 | 1.082 × 10^{−11} | R H_{0} | ||

TV_{similar} | 0.1 | 31.566 | 4.729 × 10^{−4} | R H_{0} | MAE_{test} | 74.835 | 5.124 × 10^{−12} | R H_{0} |

0.2 | 28.559 | 1.467 × 10^{−3} | R H_{0} | 78.643 | 9.257 × 10^{−13} | R H_{0} | ||

0.3 | 40.200 | 1.563 × 10^{−5} | R H_{0} | 76.486 | 2.443 × 10^{−11} | R H_{0} | ||

0.4 | 40.246 | 1.53 × 10^{−5} | R H_{0} | 68.753 | 7.714 × 10^{−11} | R H_{0} | ||

0.5 | 27.707 | 2.011 × 10^{−3} | R H_{0} | 69.862 | 4.714 × 10^{−11} | R H_{0} | ||

0.6 | 40.627 | 1.313 × 10^{−3} | R H_{0} | 73.134 | 1.097 × 10^{−11} | R H_{0} | ||

0.7 | 43.478 | 4.084 × 10^{−6} | R H_{0} | 71.994 | 1.825 × 10^{−12} | R H_{0} | ||

0.8 | 40.727 | 1.261 × 10^{−5} | R H_{0} | 73.922 | 7.790 × 10^{−12} | R H_{0} | ||

0.9 | 52.180 | 1.057 × 10^{−7} | R H_{0} | 72.254 | 1.625 × 10^{−11} | R H_{0} |

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## Share and Cite

**MDPI and ACS Style**

Chen, J.; Gao, Y.; Kasihmuddin, M.S.M.; Zheng, C.; Romli, N.A.; Mansor, M.A.; Zamri, N.E.; When, C.
MTS-PRO2SAT: Hybrid Mutation Tabu Search Algorithm in Optimizing Probabilistic 2 Satisfiability in Discrete Hopfield Neural Network. *Mathematics* **2024**, *12*, 721.
https://doi.org/10.3390/math12050721

**AMA Style**

Chen J, Gao Y, Kasihmuddin MSM, Zheng C, Romli NA, Mansor MA, Zamri NE, When C.
MTS-PRO2SAT: Hybrid Mutation Tabu Search Algorithm in Optimizing Probabilistic 2 Satisfiability in Discrete Hopfield Neural Network. *Mathematics*. 2024; 12(5):721.
https://doi.org/10.3390/math12050721

**Chicago/Turabian Style**

Chen, Ju, Yuan Gao, Mohd Shareduwan Mohd Kasihmuddin, Chengfeng Zheng, Nurul Atiqah Romli, Mohd. Asyraf Mansor, Nur Ezlin Zamri, and Chuanbiao When.
2024. "MTS-PRO2SAT: Hybrid Mutation Tabu Search Algorithm in Optimizing Probabilistic 2 Satisfiability in Discrete Hopfield Neural Network" *Mathematics* 12, no. 5: 721.
https://doi.org/10.3390/math12050721