# Predicting the Mechanical Properties of Heat-Treated Woods Using Optimization-Algorithm-Based BPNN

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis of the Algorithm

#### 2.1. Back-Propagation (BP) Neural Network Models

#### 2.2. The Traditional DBO Algorithm

#### 2.2.1. Dung Beetle Ball Rolling

#### Obstacle-Free Mode

#### Barrier Mode

#### 2.2.2. Dung Beetle Breeding

#### 2.2.3. Dung Beetle Foraging

#### 2.2.4. Dung Beetle Stealing

## 3. Proposed Method

#### 3.1. Improved Dung Beetle Optimizer

#### 3.1.1. Piece-Wise Linear Chaotic Mapping

#### 3.1.2. Self-Adaptive Parameter Adjustment Tactics

#### 3.1.3. Dimension Learning-Enhanced Foraging Search Strategy

#### 3.2. The IDBO-BP Algorithm

## 4. Evaluate the Effectiveness of the Suggested IDBO Model

#### 4.1. Benchmark Functions

#### 4.2. Contrast Algorithm and Experimental Parameter Settings

^{®}Core™ i7-11700 processor with 2.5 GHz and 16 GB RAM using MATLAB 2019a for simulation. The optimal fitness, mean fitness and standard error of fitness for the IDBO algorithm and its comparative algorithms are presented in Table 4, where bold values indicate the best consequences. Additionally, the bottom three lines of each table show the ’w/t/l’ for the wins (w), ties (t) and losses (l) of each algorithm.

#### 4.3. Evaluation of Exploration and Exploitation

#### 4.4. Evaluation of Convergence Curves

#### 4.5. Local Optimal Circumvention Evaluation

#### 4.6. High-Dimensional Robustness Evaluation

#### 4.7. Statistical Analysis

## 5. Experimental Research

#### 5.1. Data Preprocessing

#### 5.2. Model Parameter Setting

#### 5.2.1. Selection of Activation Functions

#### 5.2.2. Determination of the Topology

#### Determination of the Number of Neurons in the Hidden Layer

#### Determination of the Number of Hidden Layers

#### 5.3. Model Assessment Standards

^{2}). MAE reflects the discrepancy between the algorithm’s optimal value and the theoretical optimal value. It indicates the algorithm’s exploration capability and convergence accuracy. MSE measures the standard error between the predicted and true values, and as the standard error lowers, the model accuracy increases. MAPE measures the relative error between the predicted and true values as a percentage. It is useful for comparing models with different scales of data. R2 measures the model fit to the data, and as it comes closer to 1, the model fit becomes better, and vice versa. As it comes closer to 0, the fit becomes worse. The equations are as follows:

#### 5.4. Model Performance Comparison Analysis

## 6. Conclusions

- This article proposes the IDBO algorithm to address the limitations of the DBO algorithm. PWLCM mapping is employed to initialize the population and preserve versatility. An adaptive parameter adjustment strategy is introduced to enhance search range and efficiency. Additionally, a DLF strategy is implemented to equilibrium for exploration and exploitation search capabilities, increasing the likelihood of escaping local optima and improving later searchability. The performance of IDBO is evaluated against four basic meta-heuristic algorithms, including DBO, for 14 benchmark functions. The algorithms are ranked by applying the Wilcoxon signed-rank and Friedman tests. The outcomes demonstrate that IDBO outperforms other algorithms in finding solutions for both low- and high-dimensional functions with a single mode or multiple modes, which verifies the effectiveness of the improvement strategy, and it is highly competitive with other meta-heuristics.
- In this paper, five prediction models are separately developed using the IDBO-BP model to predict the LCS, TRS, TME, RH and TH of larch wood after heat treatment with temperature, duration and relative humidity as input variables. The outcomes indicate that the MAE, MSE and MAPE values of the IDBO-BP model are considerably diminished compared with the primitive BP neural network model. The results show that optimizing neural networks model with IDBO significantly improves the prediction accuracy of wood mechanical properties. In addition to comparing the original BP neural network model, this paper also compares it with the TSSA-BP, GWO-BP, IGWO-BP and DBO-BP models. The results denote that the forecast outcomes of the IDBO-BP model are closer to the true values, indicating significant optimization and improved prediction ability.
- This paper compares the optimal prediction models with different parameters and their corresponding topologies and activation functions, and it shows in Table A3 that the same model with different parameters does not necessarily have the same optimal topology. For the LCS, TRS, TME, RH and TH of heat-treated larch wood predicted in this paper, the five most accurate topologies of IDBO-BP that minimize the error are 3-2-1, 3-4-6-1, 3-4-1, 3-5-1 and 3-4-1, respectively.
- The Friedman test can only reflect the quality of the solution, not the diversity of the solution. Therefore, some algorithms may have significant differences in the diversity of solutions, but not in the quality of solutions. The Friedman test is also less robust in some extreme cases; for example, if an algorithm obtains an exceptionally good or bad solution, it may influence the rank and rank mean of other algorithms, thus obscuring the differences between other algorithms. This is illustrated in Table A5. The original BP model ranks first in MSE for both the training and test sets, which may be more susceptible to outlier data because MSE magnifies the prediction error. However, Table A5 also shows that, although the original BP model performs well for MSE, its overall ranking for both the test and training sets is inferior to that of the other five models.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Test Temperature/°C | Test Time/h | Test Humidity/% | Longitudinal Compressive Strength/MPa | Transverse Rupture Strength/MPa | Transverse Modulus of Elasticity/GPa | Radial Hardness/MPa | Tangential Hardness/MPa |
---|---|---|---|---|---|---|---|

120 | 0.5 | 0 | 41.9 | 67.4 | 9.093 | 14.12 | 15.56 |

120 | 0.5 | 40 | 39.8 | 65.3 | 9.038 | 13.02 | 14.69 |

120 | 0.5 | 60 | 39.7 | 69.7 | 9.1 | 14.67 | 15.08 |

120 | 0.5 | 100 | 39.5 | 67.2 | 8.845 | 14.65 | 15.45 |

120 | 1 | 0 | 39.5 | 67.8 | 8.649 | 13.98 | 14.36 |

120 | 1 | 40 | 39.5 | 66.4 | 8.752 | 12.98 | 15.59 |

120 | 1 | 60 | 39.4 | 67.8 | 9.245 | 13.78 | 15.32 |

120 | 1 | 100 | 39.2 | 63.1 | 7.895 | 14.55 | 14.23 |

120 | 2 | 0 | 39.2 | 66.9 | 9.074 | 13.33 | 14.23 |

120 | 2 | 40 | 39.1 | 68.2 | 8.945 | 12.55 | 14.58 |

120 | 2 | 60 | 39.1 | 65.2 | 8.854 | 13.25 | 14.89 |

120 | 2 | 100 | 39.1 | 63.2 | 8.933 | 13.36 | 14.56 |

120 | 3 | 0 | 39.1 | 66.5 | 8.9 | 13.56 | 14.78 |

120 | 3 | 40 | 39.1 | 67.6 | 8.963 | 13.45 | 14.45 |

120 | 3 | 60 | 38.9 | 66.6 | 8.745 | 13.01 | 14.69 |

120 | 3 | 100 | 38.9 | 64.2 | 8.745 | 12.45 | 14.78 |

140 | 0.5 | 0 | 38.9 | 66.7 | 8.978 | 14.69 | 15.56 |

140 | 0.5 | 40 | 38.9 | 67.5 | 8.845 | 13.06 | 15.02 |

140 | 0.5 | 60 | 38.7 | 66.8 | 9.155 | 14.02 | 14.23 |

140 | 0.5 | 100 | 38.7 | 65.3 | 8.877 | 15.02 | 15.01 |

140 | 1 | 0 | 38.6 | 66.5 | 9.179 | 14.16 | 15.68 |

140 | 1 | 40 | 38.6 | 64.5 | 9.137 | 13.05 | 15.01 |

140 | 1 | 60 | 38.5 | 67.2 | 9.024 | 13.49 | 15.17 |

140 | 1 | 100 | 38.5 | 63.1 | 8.823 | 13.45 | 15.48 |

140 | 2 | 0 | 38.4 | 66.3 | 8.823 | 13.54 | 14.69 |

140 | 2 | 40 | 38.4 | 65.7 | 8.852 | 14.69 | 14.58 |

140 | 2 | 60 | 38.2 | 67.1 | 8.799 | 13.99 | 14.74 |

140 | 2 | 100 | 38.2 | 62.7 | 8.9 | 14.28 | 15.63 |

140 | 3 | 0 | 38.2 | 65.4 | 8.811 | 14.39 | 14.23 |

140 | 3 | 40 | 38.2 | 64.6 | 8.934 | 13.23 | 14.56 |

140 | 3 | 60 | 38.1 | 65.5 | 8.654 | 14.23 | 13.65 |

140 | 3 | 100 | 38.1 | 62.1 | 8.798 | 13.56 | 14.02 |

160 | 0.5 | 0 | 38.1 | 66.3 | 8.788 | 14.89 | 14.99 |

160 | 0.5 | 40 | 38 | 66.9 | 9.011 | 14.87 | 14.36 |

160 | 0.5 | 60 | 37.9 | 66.3 | 8.745 | 14.58 | 14.78 |

160 | 0.5 | 100 | 37.8 | 65.8 | 8.712 | 14.69 | 15.69 |

160 | 1 | 0 | 37.8 | 62.4 | 8.679 | 13.42 | 14.56 |

160 | 1 | 40 | 37.6 | 61.4 | 8.645 | 14.09 | 15.3 |

160 | 1 | 60 | 37.6 | 62.2 | 8.798 | 14.69 | 15.9 |

160 | 1 | 100 | 37.6 | 62.8 | 8.679 | 13.58 | 15.63 |

160 | 2 | 0 | 37.6 | 62.2 | 8.727 | 14.63 | 13.92 |

160 | 2 | 40 | 37.6 | 62.1 | 8.557 | 14.02 | 14.17 |

160 | 2 | 60 | 37.5 | 63.1 | 8.687 | 15.17 | 14.28 |

160 | 2 | 100 | 37.5 | 60.9 | 8.611 | 14.65 | 15.09 |

160 | 3 | 0 | 37.5 | 61.9 | 8.611 | 13.65 | 14.36 |

160 | 3 | 40 | 37.4 | 61.5 | 8.534 | 13.47 | 14.56 |

160 | 3 | 60 | 37.3 | 60.8 | 8.601 | 13.58 | 13.89 |

160 | 3 | 100 | 37.2 | 60.5 | 8.552 | 13.69 | 14.36 |

180 | 0.5 | 0 | 37.2 | 65.9 | 8.601 | 15.21 | 14.03 |

180 | 0.5 | 40 | 37.1 | 65.3 | 8.689 | 15.98 | 14.56 |

180 | 0.5 | 60 | 37.1 | 66.1 | 8.645 | 16.01 | 13.97 |

180 | 0.5 | 100 | 36.9 | 65.7 | 8.599 | 14.32 | 14.33 |

180 | 1 | 0 | 36.9 | 65.4 | 8.623 | 15.09 | 13.79 |

180 | 1 | 40 | 36.9 | 64.9 | 8.645 | 14.98 | 14.25 |

180 | 1 | 60 | 36.7 | 66.3 | 8.579 | 15.45 | 14.08 |

180 | 1 | 100 | 36.7 | 64.8 | 8.545 | 14.33 | 13.64 |

180 | 2 | 0 | 36.6 | 65.1 | 8.574 | 14.65 | 13.69 |

180 | 2 | 40 | 36.5 | 65.8 | 8.6 | 14.13 | 13.59 |

180 | 2 | 60 | 36.5 | 64.5 | 8.532 | 13.99 | 14.49 |

180 | 2 | 100 | 36.1 | 64.2 | 8.544 | 15.1 | 13.54 |

180 | 3 | 0 | 36 | 64.1 | 8.6 | 14.21 | 14.06 |

180 | 3 | 40 | 35.9 | 64.2 | 8.541 | 13.99 | 14.21 |

180 | 3 | 60 | 35.8 | 64.8 | 8.456 | 14.58 | 13.98 |

180 | 3 | 100 | 35.8 | 63.8 | 8.499 | 14.99 | 13.69 |

200 | 0.5 | 0 | 35.8 | 62.1 | 8.483 | 12 | 13.6 |

200 | 0.5 | 40 | 35.5 | 60.6 | 8.475 | 11.96 | 12.99 |

200 | 0.5 | 60 | 35.4 | 59.9 | 8.399 | 11.45 | 13.21 |

200 | 1 | 0 | 35.4 | 61.9 | 8.422 | 11.69 | 12.98 |

200 | 1 | 40 | 35.1 | 60.8 | 8.489 | 11.46 | 12.64 |

200 | 1 | 60 | 34.6 | 61.2 | 8.321 | 11.54 | 12.35 |

200 | 2 | 0 | 34.5 | 61.2 | 8.369 | 11.99 | 13.02 |

200 | 2 | 40 | 34.5 | 60.8 | 8.354 | 11.15 | 12.69 |

200 | 2 | 60 | 34.2 | 60.5 | 8.211 | 10.65 | 12.49 |

200 | 3 | 0 | 34.1 | 60.9 | 8.249 | 10.68 | 12.73 |

200 | 3 | 40 | 34.1 | 59.8 | 8.231 | 11.05 | 12.57 |

200 | 3 | 60 | 34.1 | 58.2 | 8.011 | 10.22 | 12.37 |

210 | 0.5 | 0 | 33.9 | 50.1 | 7.856 | 10.23 | 10.98 |

210 | 0.5 | 40 | 33.8 | 50.8 | 7.789 | 10.59 | 9.98 |

210 | 0.5 | 60 | 33.2 | 49.9 | 7.865 | 10.55 | 10.23 |

210 | 1 | 0 | 32.9 | 50.6 | 7.765 | 10.21 | 10.65 |

210 | 1 | 40 | 32.9 | 49.8 | 7.712 | 9.98 | 10.21 |

210 | 1 | 60 | 32.8 | 48.9 | 7.498 | 10.01 | 10.65 |

210 | 2 | 0 | 32.5 | 49.1 | 7.689 | 9.98 | 9.64 |

210 | 2 | 40 | 32.1 | 49.5 | 7.712 | 9.65 | 9.35 |

210 | 2 | 60 | 31.8 | 49.6 | 7.623 | 10.03 | 9.67 |

210 | 3 | 0 | 31.5 | 47.8 | 7.5 | 9.21 | 8.91 |

210 | 3 | 40 | 30.8 | 46.5 | 7.412 | 9.1 | 8.21 |

210 | 3 | 60 | 30.5 | 45.1 | 7.321 | 9.03 | 8.99 |

Alg. | D | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | Avg. Rank | Overall Rank |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

30 | 3 | 3 | 5 | 4.67 | 3.67 | 2.5 | 3 | 3 | 1.83 | 2.33 | 3 | 3 | 4.33 | 3.33 | 3.26 | 3 | |

WOA | 50 | 3 | 2.33 | 5 | 4.67 | 3.67 | 2.5 | 3 | 3 | 2.17 | 3.67 | 3 | 3 | 4.33 | 3.33 | 3.33 | 3 |

100 | 3 | 3 | 5 | 4.67 | 3.67 | 2.5 | 3 | 3 | 2.17 | 3.67 | 3 | 3 | 4.33 | 3.33 | 3.38 | 4 | |

30 | 4 | 4 | 3 | 3 | 3.33 | 2.5 | 4 | 4 | 3.17 | 4 | 4 | 3.33 | 2.67 | 3.33 | 3.45 | 4 | |

GWO | 50 | 4 | 4 | 3 | 3 | 3.33 | 2.5 | 4 | 4 | 3.83 | 2.67 | 4 | 2.67 | 2.67 | 3.67 | 3.38 | 4 |

100 | 4 | 4 | 3 | 3 | 3.33 | 2.5 | 4 | 4 | 2.83 | 2.67 | 4 | 2.67 | 2.67 | 3.33 | 3.29 | 3 | |

30 | 5 | 5 | 4 | 4.33 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 2.67 | 5 | 4.71 | 5 | |

PSO | 50 | 5 | 5 | 4 | 4.33 | 5 | 5 | 5 | 5 | 4.67 | 5 | 5 | 5 | 2.67 | 5 | 4.69 | 5 |

100 | 5 | 5 | 4 | 4.33 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 2.67 | 5 | 4.71 | 5 | |

30 | 2 | 2 | 2 | 2 | 1.67 | 2.5 | 2 | 2 | 3.17 | 2.67 | 1.67 | 2.67 | 2.67 | 2.33 | 2.24 | 2 | |

DBO | 50 | 2 | 2.67 | 2 | 2 | 1.67 | 2.5 | 2 | 2 | 2.83 | 2.67 | 1.67 | 3.33 | 2.67 | 1.67 | 2.26 | 2 |

100 | 2 | 2 | 2 | 2 | 1.67 | 2.5 | 2 | 2 | 3.5 | 2.67 | 1.67 | 3.33 | 2.67 | 2.33 | 2.31 | 2 | |

30 | 1 | 1 | 1 | 1 | 1.33 | 2.5 | 1 | 1 | 1.83 | 1 | 1.33 | 1 | 2.67 | 1 | 1.33 | 1 | |

IDBO | 50 | 1 | 1 | 1 | 1 | 1.33 | 2.5 | 1 | 1 | 1.5 | 1 | 1.33 | 1 | 2.67 | 1.33 | 1.33 | 1 |

100 | 1 | 1 | 1 | 1 | 1.33 | 2.5 | 1 | 1 | 1.5 | 1 | 1.33 | 1 | 2.67 | 1 | 1.31 | 1 |

**Table A3.**Optimal prediction models with different parameters and the corresponding topologies and activation functions.

Parms | Model | Neuron Configuration | Topology | Hidden and Output Activations | Train | Test | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MSE | MAPE | R² | MAE | MSE | MAPE | R² | |||||

LCS | BP | 2 | 3-2-1 | LOGSIG-PURELIN | 0.2206 | 0.0020 | 0.9335% | 0.9770 | 0.1945 | 0.0011 | 0.5332% | 0.9868 |

GWO-BP | 3 | 3-3-1 | LOGSIG-PURELIN | 0.1956 | 0.1054 | 0.5263% | 0.9785 | 0.1372 | 0.0724 | 0.4919% | 0.9940 | |

TSSA-BP | 4 | 3-4-1 | LOGSIG-TANSIG | 0.1592 | 0.1282 | 0.4342% | 0.9759 | 0.1481 | 0.0642 | 0.4270% | 0.9887 | |

DBO-BP | (3, 7) | 3-3-7-1 | LOGSIG-PURELIN | 0.1412 | 0.1052 | 0.3786% | 0.9830 | 0.1503 | 0.0566 | 0.4260% | 0.9879 | |

IGWO-BP | (2, 2) | 3-2-2-1 | PURELIN-LOGSIG | 0.1652 | 0.1200 | 0.4449% | 0.9805 | 0.1291 | 0.0345 | 0.3591% | 0.9907 | |

IDBO-BP | 2 | 3-2-1 | LOGSIG-PURELIN | 0.1315 | 0.0996 | 0.3513% | 0.9834 | 0.0856 | 0.0122 | 0.2286% | 0.9978 | |

TRS | BP | 2 | 3-2-1 | LOGSIG-PURELIN | 1.2157 | 0.0379 | 2.4030% | 0.9363 | 1.1994 | 0.0753 | 2.4487% | 0.9275 |

TSSA-BP | 2 | 3-2-1 | LOGSIG-PURELIN | 1.2191 | 2.2895 | 1.9534% | 0.9236 | 1.2609 | 2.2655 | 2.0291% | 0.9430 | |

GWO-BP | (3, 7) | 3-3-7-1 | LOGSIG-PURELIN | 1.1097 | 1.8158 | 1.7920% | 0.9433 | 0.9837 | 1.9907 | 1.5331% | 0.9272 | |

IGWO-BP | 3 | 3-3-1 | LOGSIG-TANSIG | 1.1020 | 2.2158 | 1.8002% | 0.9310 | 0.9519 | 1.5641 | 1.5302% | 0.9557 | |

DBO-BP | 7 | 3-7-1 | POSLIN-PURELIN | 1.0349 | 1.7647 | 1.6610% | 0.9510 | 0.9345 | 1.4363 | 1.4881% | 0.9484 | |

IDBO-BP | (4, 6) | 3-4-6-1 | TANSIG-PURELIN | 0.9032 | 1.4304 | 1.4750% | 0.9601 | 0.8218 | 1.1362 | 1.3392% | 0.9683 | |

TME | BP | 3 | 3-3-1 | LOGSIG-PURELIN | 0.1710 | 0.0006 | 2.2422% | 0.8260 | 0.0928 | 0.0005 | 1.4955% | 0.8929 |

GWO-BP | (3, 7) | 3-3-7-1 | LOGSIG-PURELIN | 0.0925 | 0.0306 | 1.0968% | 0.8638 | 0.1010 | 0.0164 | 1.1576% | 0.9002 | |

TSSA-BP | (2, 2) | 3-2-2-1 | LOGSIG-PURELIN | 0.1207 | 0.0500 | 2.0647% | 0.7992 | 0.0977 | 0.0158 | 1.1322% | 0.7225 | |

IGWO-BP | 3 | 3-3-1 | LOGSIG-PURELIN | 0.1144 | 0.0324 | 1.3611% | 0.8424 | 0.0923 | 0.0155 | 1.0797% | 0.9018 | |

DBO-BP | (3, 7) | 3-3-7-1 | LOGSIG-PURELIN | 0.1024 | 0.0273 | 1.2126% | 0.8658 | 0.0879 | 0.0127 | 1.0182% | 0.9053 | |

IDBO-BP | 4 | 3-4-1 | POSLIN-PURELIN | 0.0849 | 0.0266 | 1.0132% | 0.8743 | 0.0824 | 0.0115 | 0.9340% | 0.9156 | |

RH | BP | (5, 6) | 3-5-6-1 | POSLIN-PURELIN | 0.4763 | 0.0049 | 3.5198% | 0.9249 | 0.4963 | 0.0078 | 3.8911% | 0.8986 |

TSSA-BP | (3, 7) | 3-3-7-1 | LOGSIG-PURELIN | 0.4205 | 0.3300 | 4.1028% | 0.8901 | 0.3919 | 0.3644 | 3.8402% | 0.9023 | |

GWO-BP | 5 | 3-5-1 | POSLIN-PURELIN | 0.4009 | 0.2375 | 2.9805% | 0.9233 | 0.5008 | 0.3834 | 3.8098% | 0.8785 | |

IGWO-BP | (4, 6) | 3-4-6-1 | TANSIG-PURELIN | 0.4256 | 0.2896 | 3.2035% | 0.9107 | 0.4095 | 0.2953 | 3.0822% | 0.8975 | |

DBO-BP | 2 | 3-2-1 | TANSIG-PURELIN | 0.3928 | 0.2466 | 2.9423% | 0.8985 | 0.3478 | 0.2053 | 2.5616% | 0.9285 | |

IDBO-BP | 5 | 3-5-1 | POSLIN-PURELIN | 0.3353 | 0.2122 | 2.5198% | 0.9372 | 0.3236 | 0.1889 | 2.4020% | 0.9458 | |

TH | BP | (7, 5) | 3-7-5-1 | LOGSIG-LOGSIG | 0.3850 | 0.0024 | 2.8660% | 0.9412 | 0.4308 | 0.0073 | 3.4057% | 0.8996 |

GWO-BP | 2 | 3-2-1 | LOGSIG-PURELIN | 0.2857 | 0.2283 | 2.8154% | 0.9615 | 0.3720 | 0.2421 | 3.0862% | 0.8625 | |

TSSA-BP | (4, 6) | 3-4-6-1 | TANSIG-PURELIN | 0.3743 | 0.2175 | 2.8032% | 0.9354 | 0.3839 | 0.2201 | 2.9423% | 0.8896 | |

IGWO-BP | 4 | 3-4-1 | TANSIG-PURELIN | 0.3156 | 0.1671 | 2.3664% | 0.9566 | 0.3824 | 0.2373 | 2.9295% | 0.8899 | |

DBO-BP | (7, 5) | 3-7-5-1 | LOGSIG-LOGSIG | 0.3335 | 0.1700 | 2.4652% | 0.9532 | 0.3604 | 0.2092 | 2.6622% | 0.8996 | |

IDBO-BP | 4 | 3-4-1 | TANSIG-PURELIN | 0.2611 | 0.1249 | 1.9515% | 0.9676 | 0.2962 | 0.1544 | 2.1062% | 0.9399 |

Parms | Sample1-Sample 2 | Train | Test | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | Adj. Sig.^{a} | MSE | Adj. Sig.^{a} | MAPE | Adj. Sig.^{a} | R² | Adj. Sig.^{a} | MAE | Adj. Sig.^{a} | MSE | Adj. Sig.^{a} | MAPE | Adj. Sig.^{a} | R² | Adj. Sig.^{a} | ||

LCS | IDBO-BP-DBO-BP | 6.85% | 0.743 | 5.32% | 1 | 7.22% | 0.003 | −0.05% | 0.051 | 43.07% | 0.436 | 78.44% | 0.743 | 46.35% | 0.007 | −1.00% | 0.743 |

IDBO-BP-IGWO-BP | 20.40% | 1 | 17.05% | 1 | 21.05% | 0.011 | −0.30% | 0.269 | 33.71% | 0.954 | 64.60% | 0.954 | 36.35% | 0.007 | −0.71% | 0.954 | |

IDBO-BP-GWO-BP | 32.76% | 0.572 | 5.52% | 1 | 33.25% | 0.001 | −0.50% | 1 | 37.65% | 0.068 | 83.14% | 0.132 | 53.53% | 0.001 | −0.38% | 0.011 | |

IDBO-BP-TSSA-BP | 17.38% | 1 | 22.32% | 1 | 19.10% | 0.096 | −0.77% | 0.945 | 42.21% | 1 | 80.98% | 1 | 46.47% | 0.016 | −0.92% | 0.572 | |

IDBO-BP-BP | 40.38% | 0.005 | −4858.66% | 0.068 | 62.37% | 0 | −0.66% | 0.103 | 56.00% | 0.002 | −1016.77% | 0.329 | 57.13% | 0 | −1.11% | 0.002 | |

TRS | IDBO-BP-DBO-BP | 12.73% | 0.44 | 18.94% | 1 | 11.20% | 1 | −0.96% | 0.556 | 12.06% | 0.428 | 20.89% | 1 | 10.01% | 1 | −2.10% | 0.185 |

IDBO-BP-IGWO-BP | 18.04% | 0.269 | 35.45% | 1 | 18.06% | 1 | −3.13% | 0.269 | 13.67% | 0.945 | 27.36% | 1 | 12.48% | 1 | −1.32% | 0.638 | |

IDBO-BP-GWO-BP | 18.61% | 0.945 | 21.22% | 1 | 17.69% | 1 | −1.79% | 1 | 16.46% | 0.061 | 42.92% | 1 | 12.65% | 1 | −4.43% | 0.02 | |

IDBO-BP-TSSA-BP | 25.92% | 0.035 | 37.52% | 1 | 24.49% | 1 | −3.96% | 0.945 | 34.83% | 0.035 | 49.85% | 1 | 34.00% | 0.572 | −2.68% | 1 | |

IDBO-BP-BP | 25.71% | 0.001 | −3676.83% | 0.001 | 38.62% | 0.096 | −2.55% | 0.103 | 31.48% | 0.006 | −1409.50% | 0 | 45.31% | 0.002 | −4.41% | 0.061 | |

TME | IDBO-BP-DBO-BP | 17.08% | 0.794 | 2.37% | 1 | 16.44% | 0.164 | −0.98% | 0.393 | 6.20% | 0.132 | 9.71% | 0.572 | 8.27% | 0.005 | −1.13% | 1 |

IDBO-BP-IGWO-BP | 25.75% | 0.269 | 17.97% | 1 | 25.56% | 0.42 | −3.79% | 0.269 | 10.66% | 0.572 | 25.86% | 1 | 13.49% | 0.954 | −1.53% | 1 | |

IDBO-BP-GWO-BP | 8.13% | 0.945 | 13.07% | 1 | 7.62% | 0.42 | −1.22% | 1 | 18.36% | 0.002 | 29.85% | 0.034 | 19.31% | 0.181 | −1.71% | 0.007 | |

IDBO-BP-TSSA-BP | 29.62% | 0.035 | 46.76% | 1 | 50.93% | 0.035 | −9.40% | 0.945 | 15.60% | 0.096 | 27.14% | 0.246 | 17.50% | 0.181 | −26.72% | 0.181 | |

IDBO-BP-BP | 50.33% | 0.001 | −4103.20% | 0.023 | 54.81% | 0.001 | −5.86% | 0.103 | 11.15% | 0.181 | −2041.22% | 0.436 | 37.55% | 0 | −2.54% | 0.068 | |

RH | IDBO-BP-DBO-BP | 14.63% | 0.361 | 13.98% | 1 | 14.36% | 1 | −4.30% | 0.185 | 6.97% | 1.000 | 8.00% | 1 | 6.23% | 1 | −1.86% | 0.119 |

IDBO-BP-IGWO-BP | 21.21% | 0.269 | 26.72% | 1 | 21.34% | 1 | −2.90% | 0.269 | 20.99% | 1 | 36.03% | 1 | 22.07% | 1 | −5.38% | 0.638 | |

IDBO-BP-GWO-BP | 16.36% | 0.945 | 10.65% | 1 | 15.46% | 1 | −1.50% | 1 | 35.40% | 1 | 50.74% | 1 | 36.95% | 1 | −7.67% | 0.02 | |

IDBO-BP-TSSA-BP | 20.25% | 0.035 | 35.70% | 0.572 | 38.58% | 1 | −5.29% | 0.945 | 17.44% | 0.096 | 48.18% | 0.954 | 37.45% | 0.572 | −4.82% | 1 | |

IDBO-BP-BP | 29.60% | 0.001 | −4200.16% | 0.016 | 28.41% | 0.048 | −1.32% | 0.103 | 34.80% | 1.000 | −2308.90% | 0.005 | 38.27% | 0.011 | −5.26% | 0.061 | |

TH | IDBO-BP-DBO-BP | 21.71% | 0.7 | 26.50% | 1 | 20.84% | 0.087 | −1.50% | 0.556 | 17.83% | 0.366 | 26.17% | 1 | 20.88% | 0.151 | −4.48% | 0.113 |

IDBO-BP-IGWO-BP | 17.27% | 0.269 | 25.22% | 1 | 17.53% | 0.42 | −1.14% | 0.269 | 22.56% | 0.945 | 34.92% | 1 | 28.10% | 1 | −5.62% | 0.638 | |

IDBO-BP-GWO-BP | 8.62% | 0.945 | 45.27% | 1 | 30.68% | 0.42 | −0.63% | 1 | 20.39% | 0.061 | 36.20% | 1 | 31.75% | 0.035 | −8.97% | 0.02 | |

IDBO-BP-TSSA-BP | 30.25% | 0.035 | 42.55% | 1 | 30.38% | 0.035 | −3.44% | 0.945 | 22.87% | 0.035 | 29.84% | 1 | 28.42% | 0.061 | −5.64% | 1 | |

IDBO-BP-BP | 32.18% | 0.001 | −5117.57% | 1 | 31.91% | 0.001 | −2.80% | 0.103 | 31.25% | 0.006 | −2013.85% | 1 | 38.16% | 0 | −4.48% | 0.061 |

^{a}Significance values have been adjusted with the Bonferroni correction for multiple tests.

Parms | Model | Train | Test | Avg. Rank | Overall Rank | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MSE | MAPE | R² | MAE | MSE | MAPE | R² | Include MSE | Exclude MSE | Include MSE | Exclude MSE | ||

LCS | BP | 5 | 1 | 5.42 | 2.17 | 4.83 | 1 | 5.08 | 2.25 | 2.24 | 2.65 | 5 | 6 |

TSSA-BP | 2.67 | 4 | 3.08 | 3.92 | 3.25 | 4.08 | 3.5 | 3.58 | 1.64 | 0.83 | 2 | 2 | |

GWO-BP | 3.83 | 4.5 | 4.08 | 3.33 | 4.08 | 4.75 | 4.08 | 2.58 | 2.43 | 1.69 | 6 | 5 | |

IGWO-BP | 3.5 | 4.42 | 3.58 | 3.33 | 3.33 | 4.17 | 3.67 | 3.75 | 1.95 | 1.17 | 3 | 3 | |

DBO-BP | 3.75 | 3.92 | 3.83 | 3.67 | 3.58 | 4.25 | 3.67 | 3.67 | 1.96 | 1.25 | 4 | 4 | |

IDBO-BP | 2.25 | 3.17 | 1 | 4.58 | 1.92 | 2.75 | 1 | 5.17 | 0.29 | −0.60 | 1 | 1 | |

TRS | BP | 3.25 | 1 | 5.17 | 3.25 | 4 | 1 | 5.08 | 2.92 | 1.67 | 1.89 | 4 | 6 |

TSSA-BP | 4.08 | 4.33 | 3.5 | 3 | 3.92 | 4.25 | 3.75 | 3.33 | 2.19 | 1.49 | 6 | 5 | |

GWO-BP | 4.17 | 4.25 | 3.83 | 3.17 | 3.58 | 4.08 | 3.42 | 3.58 | 2.07 | 1.38 | 5 | 4 | |

IGWO-BP | 2.83 | 3.75 | 2.67 | 4.33 | 3.5 | 4 | 3.25 | 3 | 1.58 | 0.82 | 2 | 2 | |

DBO-BP | 3.17 | 3.58 | 2.75 | 3.67 | 3.5 | 3.92 | 3.33 | 3.42 | 1.65 | 0.94 | 3 | 3 | |

IDBO-BP | 3.5 | 4.08 | 3.08 | 3.58 | 2.5 | 3.75 | 2.17 | 4.75 | 1.34 | 0.49 | 1 | 1 | |

TME | BP | 4.08 | 1 | 4.75 | 3.83 | 4.67 | 1 | 5.08 | 2.83 | 1.74 | 1.99 | 3 | 6 |

TSSA-BP | 3.58 | 3.75 | 3.17 | 3.67 | 3.83 | 4.5 | 3.5 | 3.08 | 1.95 | 1.22 | 4 | 3 | |

GWO-BP | 3.17 | 4.67 | 3.67 | 2.5 | 3.67 | 5 | 4.33 | 2.33 | 2.46 | 1.67 | 6 | 5 | |

IGWO-BP | 3.25 | 4 | 3 | 3.75 | 3.33 | 3.58 | 3 | 4 | 1.55 | 0.81 | 2 | 2 | |

DBO-BP | 3.75 | 4.17 | 3.5 | 3.25 | 3.75 | 4.25 | 3.5 | 3.75 | 1.99 | 1.25 | 5 | 4 | |

IDBO-BP | 3.17 | 3.42 | 2.92 | 4 | 1.75 | 2.67 | 1.58 | 5 | 0.81 | 0.07 | 1 | 1 | |

RH | BP | 4.58 | 1 | 5.08 | 3.42 | 5.17 | 1 | 5.25 | 3.75 | 1.86 | 2.15 | 4 | 6 |

TSSA-BP | 3.17 | 5.08 | 3.75 | 2.33 | 3 | 5.17 | 4.25 | 2.08 | 2.50 | 1.63 | 6 | 5 | |

GWO-BP | 3.5 | 3.83 | 3.08 | 3.42 | 2.92 | 3.5 | 3 | 3.58 | 1.60 | 0.92 | 3 | 3 | |

IGWO-BP | 3.5 | 4.08 | 3.25 | 3.58 | 3.92 | 4.25 | 3.42 | 3.67 | 1.90 | 1.14 | 5 | 4 | |

DBO-BP | 3.25 | 3.5 | 3 | 4 | 2.92 | 3.33 | 2.42 | 3.83 | 1.32 | 0.63 | 2 | 2 | |

IDBO-BP | 3 | 3.5 | 2.83 | 4.25 | 3.08 | 3.75 | 2.67 | 4.08 | 1.31 | 0.54 | 1 | 1 | |

TH | BP | 4.08 | 1 | 4.75 | 2.08 | 4.17 | 1 | 4.33 | 2.25 | 1.88 | 2.17 | 4 | 6 |

TSSA-BP | 4.17 | 4.5 | 3.92 | 3 | 3.83 | 4.08 | 3.08 | 3.33 | 2.16 | 1.45 | 5 | 4 | |

GWO-BP | 3.33 | 3.83 | 3 | 3.67 | 3.92 | 4.17 | 3.75 | 3.58 | 1.84 | 1.13 | 3 | 3 | |

IGWO-BP | 3.83 | 4.42 | 3.58 | 3.08 | 4.08 | 4.5 | 4 | 3.17 | 2.27 | 1.54 | 6 | 5 | |

DBO-BP | 3.42 | 3.67 | 3 | 4 | 2.58 | 3.42 | 2.42 | 3.83 | 1.34 | 0.60 | 2 | 2 | |

IDBO-BP | 3.17 | 3.58 | 2.75 | 3.47 | 2.42 | 3.43 | 2.45 | 3.99 | 1.29 | 0.56 | 1 | 1 |

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**Figure 3.**Population initialization in PWLCM: (

**a**) scatter map; (

**b**) frequency distribution histogram.

**Figure 8.**Schematic diagram of the IDBO-BP model for the mechanical properties of transverse rupture strength.

**Figure 10.**Comparison of results for various models based on predictions and actual values: (

**a**) longitudinal compressive strength; (

**b**) transverse rupture strength; (

**c**) transverse modulus of elasticity; (

**d**) radial hardness; (

**e**) tangential hardness.

Function | Dim | Range | ${\mathbf{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ |
---|---|---|---|

${f}_{1}\left(x\right)=\sum _{i=1}^{n}{x}_{i}^{2}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{2}\left(x\right)=\sum _{i=1}^{n}\left|{x}_{i}\right|+\prod _{i=1}^{n}\left|{x}_{i}\right|$ | 30/50/100 | [−10, 10] | 0 |

${f}_{3}\left(x\right)$=$\sum _{i=1}^{n}{\left(\sum _{j=1}^{i}{x}_{j}\right)}^{2}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{4}\left(x\right)=max\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{5}\left(x\right)$=$\sum _{i=1}^{n-1}\left[{100({x}_{i+1}-{x}_{{i}^{2}})}^{2}+{({x}_{i}-1)}^{2}\right]$ | 30/50/100 | [−30, 30] | 0 |

${f}_{6}\left(x\right)$=$\sum _{i=1}^{n}{(\left[{x}_{i}+0.5\right])}^{2}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{7}\left(x\right)$=${{x}_{1}}^{2}$+${10}^{6}\sum _{i=2}^{n}{{x}_{i}}^{2}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{8}\left(x\right)$=$\sum _{i=1}^{n}{{x}_{i}}^{2}+{\left(\sum _{i=1}^{n}0.5i{x}_{i}\right)}^{2}+{\left(\sum _{i=1}^{n}0.5i{x}_{i}\right)}^{4}$ | 30/50/100 | [−5, 10] | 0 |

Function | Dim | Range | ${\mathbf{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ |
---|---|---|---|

${f}_{9}\left(x\right)$=$\sum _{i=1}^{n}\left[{{x}_{i}}^{2}-10\mathit{cos}\left(2\pi {x}_{i}\right)+10\right]$ | 30/50/100 | [−5.12, 5.12] | 0 |

${f}_{10}\left(x\right)$=$\sum _{i=1}^{n}\left|{x}_{i}\mathit{sin}\left({x}_{i}\right)+0.1{x}_{i}\right|$ | 30/50/100 | [−10, 10] | 0 |

${f}_{11}\left(x\right)$=$\frac{\pi}{n}\left\{10\mathit{sin}\left(\pi {y}_{1}\right)+\sum _{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathit{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+\sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)$, ${\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}y}_{i}$= 1$+\frac{{x}_{i}+1}{4}$, for all i = 1, …, n $u\left({x}_{i},a,k,m\right)$=$\left\{\right)separators="|">\begin{array}{c}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0-a{x}_{i}a\\ k{\left({-x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}$ | 30/50/100 | [−50, 50] | 0 |

${f}_{12}\left(x\right)$ = 0.1$\left\{{\mathit{sin}}^{2}\left(3\pi {x}_{1}\right)+\sum _{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathit{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathit{sin}}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+\sum _{i=1}^{n}u\left({x}_{i},\mathrm{5,100,4}\right)$ | 30/50/100 | [−50, 50] | 0 |

${f}_{13}\left(x\right)$=${\left[\frac{1}{n-1}\sum _{i=1}^{n-1}\left(\sqrt{{s}_{i}}\times (\mathit{sin}\left(50{s}_{i}^{0.2}\right)+1\right)\right]}^{2}$ ${s}_{i}=\sqrt{{x}_{i}^{2}+{x}_{i+1}^{2}}$ | 30/50/100 | [−100, 100] | 0 |

${f}_{14}\left(x\right)={\mathit{sin}}^{2}\left(\pi {y}_{1}\right)+{\displaystyle \sum _{i=1}^{n-1}}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathit{sin}}^{2}\left(\pi {y}_{i}+1\right)\right]$ $+{\left({y}_{n}-1\right)}^{2}\left[1+{\mathit{sin}}^{2}\left(2\pi {y}_{n}\right)\right],$ ${\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}y}_{i}=1+\frac{{x}_{i}-1}{4},$ for all i = 1, …, n | 30/50/100 | [−10, 10] | 0 |

Algorithm | Parameter | Setting |
---|---|---|

WOA | a | Gradually reduced from 2 to 0 |

GWO | a | Uniformly lowered from 2 to 0 |

PSO | ${C}_{1}$ and ${C}_{2}$ | 2 |

Inertia weight | Linearly decreased from 0.9 to 0.1 | |

DBO | $\mathsf{\alpha}\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\beta}$ | 0.1 |

a and b | 0.3 and 0.5 | |

IDBO | $\mathsf{\alpha}\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\beta}$ | 0.1 |

a and b | 0.3 and 0.5 | |

The proportions of the ball-rolling dung beetle, the brood ball, the small dung beetle and the thief were $\left[0.40.2\right]$, 0.2, 0.2 and $\left[0.20.4\right]$ |

F | D | Index | WOA | GWO | PSO | DBO | IDBO |
---|---|---|---|---|---|---|---|

F1 | 30 | Best | $3.94\times 1{0}^{-83}$ | $7.29\times 1{0}^{-30}$ | $6.50\times 1{0}^{2}$ | $1.03\times 1{0}^{-176}$ | $5.01\times 1{0}^{-201}$ |

Mean | $7.03\times 1{0}^{-74}$ | $9.20\times 1{0}^{-28}$ | $1.77\times 1{0}^{3}$ | $1.77\times 1{0}^{-89}$ | $8.31\times 1{0}^{-151}$ | ||

STD | $3.41\times 1{0}^{-73}$ | $9.69\times 1{0}^{-28}$ | $6.99\times 1{0}^{2}$ | $9.68\times 1{0}^{-89}$ | $4.46\times 1{0}^{-150}$ | ||

50 | Best | $4.13\times 1{0}^{-89}$ | $1.29\times 1{0}^{-29}$ | $6.20\times 1{0}^{2}$ | $1.71\times 1{0}^{-163}$ | $1.64\times 1{0}^{-199}$ | |

Mean | $6.15\times 1{0}^{-72}$ | $9.52\times 1{0}^{-28}$ | $1.93\times 1{0}^{3}$ | $1.71\times 1{0}^{-110}$ | $8.00\times 1{0}^{-151}$ | ||

STD | $4.33\times 1{0}^{-71}$ | $1.13\times 1{0}^{-27}$ | $8.76\times 1{0}^{2}$ | $1.21\times 1{0}^{-109}$ | $5.66\times 1{0}^{-150}$ | ||

100 | Best | $3.31\times 1{0}^{-88}$ | $2.43\times 1{0}^{-29}$ | $7.52\times 1{0}^{2}$ | $2.11\times 1{0}^{-182}$ | $1.89\times 1{0}^{-208}$ | |

Mean | $2.61\times 1{0}^{-72}$ | $1.20\times 1{0}^{-27}$ | $2.09\times 1{0}^{3}$ | $4.94\times 1{0}^{-105}$ | $6.48\times 1{0}^{-152}$ | ||

STD | $2.36\times 1{0}^{-71}$ | $2.64\times 1{0}^{-27}$ | $7.95\times 1{0}^{2}$ | $4.90\times 1{0}^{-104}$ | $6.48\times 1{0}^{-151}$ | ||

F2 | 30 | Best | $1.71\times 1{0}^{-57}$ | $1.67\times 1{0}^{-17}$ | $1.41\times 1{0}^{1}$ | $7.24\times 1{0}^{-81}$ | $3.30\times 1{0}^{-111}$ |

Mean | $8.35\times 1{0}^{-50}$ | $9.15\times 1{0}^{-17}$ | $2.03\times 1{0}^{1}$ | $6.05\times 1{0}^{-55}$ | $1.98\times 1{0}^{-83}$ | ||

STD | $4.56\times 1{0}^{-49}$ | $5.34\times 1{0}^{-17}$ | $3.98\times 1{0}^{0}$ | $3.31\times 1{0}^{-54}$ | $9.03\times 1{0}^{-83}$ | ||

50 | Best | $2.53\times 1{0}^{-58}$ | $2.48\times 1{0}^{-17}$ | $9.95\times 1{0}^{0}$ | $1.67\times 1{0}^{-83}$ | $1.64\times 1{0}^{-103}$ | |

Mean | $9.04\times 1{0}^{-51}$ | $9.55\times 1{0}^{-17}$ | $1.95\times 1{0}^{1}$ | $2.02\times 1{0}^{-48}$ | $1.46\times 1{0}^{-80}$ | ||

STD | $4.88\times 1{0}^{-50}$ | $6.54\times 1{0}^{-17}$ | $4.77\times 1{0}^{0}$ | $1.43\times 1{0}^{-47}$ | $1.04\times 1{0}^{-79}$ | ||

100 | Best | $5.90\times 1{0}^{-59}$ | $1.06\times 1{0}^{-17}$ | $1.05\times 1{0}^{1}$ | $5.00\times 1{0}^{-84}$ | $8.80\times 1{0}^{-107}$ | |

Mean | $3.65\times 1{0}^{-50}$ | $9.57\times 1{0}^{-17}$ | $1.94\times 1{0}^{1}$ | $2.17\times 1{0}^{-56}$ | $1.80\times 1{0}^{-80}$ | ||

STD | $3.22\times 1{0}^{-49}$ | $7.80\times 1{0}^{-17}$ | $4.20\times 1{0}^{0}$ | $1.71\times 1{0}^{-55}$ | $1.45\times 1{0}^{-79}$ | ||

F3 | 30 | Best | $8.29\times 1{0}^{3}$ | $5.82\times 1{0}^{-9}$ | $2.23\times 1{0}^{3}$ | $1.61\times 1{0}^{-148}$ | $2.25\times 1{0}^{-187}$ |

Mean | $4.90\times 1{0}^{4}$ | $1.64\times 1{0}^{-5}$ | $4.87\times 1{0}^{3}$ | $5.61\times 1{0}^{-79}$ | $3.79\times 1{0}^{-95}$ | ||

STD | $1.40\times 1{0}^{4}$ | $3.50\times 1{0}^{-5}$ | $1.71\times 1{0}^{3}$ | $3.07\times 1{0}^{-78}$ | $2.08\times 1{0}^{-94}$ | ||

50 | Best | $1.48\times 1{0}^{4}$ | $2.04\times 1{0}^{-9}$ | $1.34\times 1{0}^{3}$ | $9.33\times 1{0}^{-144}$ | $6.40\times 1{0}^{-178}$ | |

Mean | $4.36\times 1{0}^{4}$ | $3.33\times 1{0}^{-5}$ | $5.71\times 1{0}^{3}$ | $5.63\times 1{0}^{-81}$ | $4.74\times 1{0}^{-99}$ | ||

STD | $1.31\times 1{0}^{4}$ | $1.23\times 1{0}^{-4}$ | $2.02\times 1{0}^{3}$ | $2.82\times 1{0}^{-80}$ | $3.35\times 1{0}^{-98}$ | ||

100 | Best | $1.47\times 1{0}^{4}$ | $3.82\times 1{0}^{-9}$ | $2.16\times 1{0}^{3}$ | $2.31\times 1{0}^{-157}$ | $9.16\times 1{0}^{-183}$ | |

Mean | $4.27\times 1{0}^{4}$ | $3.43\times 1{0}^{-5}$ | $5.39\times 1{0}^{3}$ | $9.29\times 1{0}^{-56}$ | $1.63\times 1{0}^{-85}$ | ||

STD | $1.26\times 1{0}^{4}$ | $1.57\times 1{0}^{-4}$ | $1.60\times 1{0}^{3}$ | $9.29\times 1{0}^{-55}$ | $1.63\times 1{0}^{-84}$ | ||

F4 | 30 | Best | $2.60\times 1{0}^{-1}$ | $4.49\times 1{0}^{-8}$ | $1.79\times 1{0}^{1}$ | $2.07\times 1{0}^{-77}$ | $1.13\times 1{0}^{-93}$ |

Mean | $5.48\times 1{0}^{1}$ | $6.54\times 1{0}^{-7}$ | $2.85\times 1{0}^{1}$ | $3.47\times 1{0}^{-50}$ | $3.07\times 1{0}^{-63}$ | ||

STD | $2.85\times 1{0}^{1}$ | $5.12\times 1{0}^{-7}$ | $6.42\times 1{0}^{0}$ | $1.90\times 1{0}^{-49}$ | $1.68\times 1{0}^{-62}$ | ||

50 | Best | $8.90\times 1{0}^{-1}$ | $2.11\times 1{0}^{-8}$ | $1.60\times 1{0}^{1}$ | $2.22\times 1{0}^{-85}$ | $5.72\times 1{0}^{-100}$ | |

Mean | $4.97\times 1{0}^{1}$ | $7.76\times 1{0}^{-7}$ | $2.74\times 1{0}^{1}$ | $1.01\times 1{0}^{-51}$ | $1.35\times 1{0}^{-68}$ | ||

STD | $2.53\times 1{0}^{1}$ | $1.22\times 1{0}^{-6}$ | $5.21\times 1{0}^{0}$ | $7.18\times 1{0}^{-51}$ | $9.52\times 1{0}^{-68}$ | ||

100 | Best | $2.73\times 1{0}^{-1}$ | $5.70\times 1{0}^{-8}$ | $1.57\times 1{0}^{1}$ | $3.24\times 1{0}^{-81}$ | $1.78\times 1{0}^{-100}$ | |

Mean | $5.02\times 1{0}^{1}$ | $6.89\times 1{0}^{-7}$ | $2.84\times 1{0}^{1}$ | $4.82\times 1{0}^{-48}$ | $4.69\times 1{0}^{-66}$ | ||

STD | $2.65\times 1{0}^{1}$ | $8.72\times 1{0}^{-7}$ | $4.94\times 1{0}^{0}$ | $4.38\times 1{0}^{-47}$ | $4.42\times 1{0}^{-65}$ | ||

F5 | 30 | Best | $2.71\times 1{0}^{1}$ | $2.59\times 1{0}^{1}$ | $3.58\times 1{0}^{4}$ | $2.54\times 1{0}^{1}$ | $2.48\times 1{0}^{1}$ |

Mean | $2.79\times 1{0}^{1}$ | $2.69\times 1{0}^{1}$ | $4.52\times 1{0}^{5}$ | $2.58\times 1{0}^{1}$ | $2.52\times 1{0}^{1}$ | ||

STD | $4.45\times 1{0}^{-1}$ | $7.15\times 1{0}^{-1}$ | $4.03\times 1{0}^{5}$ | $1.83\times 1{0}^{-1}$ | $3.06\times 1{0}^{-1}$ | ||

50 | Best | $2.72\times 1{0}^{1}$ | $2.59\times 1{0}^{1}$ | $4.35\times 1{0}^{4}$ | $2.52\times 1{0}^{1}$ | $2.47\times 1{0}^{1}$ | |

Mean | $2.82\times 1{0}^{1}$ | $2.72\times 1{0}^{1}$ | $4.51\times 1{0}^{5}$ | $2.58\times 1{0}^{1}$ | $2.52\times 1{0}^{1}$ | ||

STD | $4.52\times 1{0}^{-1}$ | $7.16\times 1{0}^{-1}$ | $3.86\times 1{0}^{5}$ | $2.68\times 1{0}^{-1}$ | $3.10\times 1{0}^{-1}$ | ||

100 | Best | $2.69\times 1{0}^{1}$ | $2.53\times 1{0}^{1}$ | $3.65\times 1{0}^{4}$ | $2.53\times 1{0}^{1}$ | $2.47\times 1{0}^{1}$ | |

Mean | $2.80\times 1{0}^{1}$ | $2.70\times 1{0}^{1}$ | $4.21\times 1{0}^{5}$ | $2.58\times 1{0}^{1}$ | $2.52\times 1{0}^{1}$ | ||

STD | $4.45\times 1{0}^{-1}$ | $7.56\times 1{0}^{-1}$ | $3.26\times 1{0}^{5}$ | $2.17\times 1{0}^{-1}$ | $2.55\times 1{0}^{-1}$ | ||

30 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $8.05\times 1{0}^{2}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $2.56\times 1{0}^{3}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $8.87\times 1{0}^{2}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

50 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $9.63\times 1{0}^{2}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $2.60\times 1{0}^{3}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $8.53\times 1{0}^{2}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

100 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $8.35\times 1{0}^{2}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $2.37\times 1{0}^{3}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $1.09\times 1{0}^{3}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

F7 | 30 | Best | $3.68\times 1{0}^{-77}$ | $4.83\times 1{0}^{-23}$ | $8.52\times 1{0}^{8}$ | $1.93\times 1{0}^{-159}$ | $9.32\times 1{0}^{-196}$ |

Mean | $1.85\times 1{0}^{-66}$ | $1.04\times 1{0}^{-21}$ | $1.75\times 1{0}^{9}$ | $6.52\times 1{0}^{-108}$ | $9.14\times 1{0}^{-140}$ | ||

STD | $9.98\times 1{0}^{-66}$ | $1.42\times 1{0}^{-21}$ | $5.93\times 1{0}^{8}$ | $2.48\times 1{0}^{-107}$ | $5.01\times 1{0}^{-139}$ | ||

50 | Best | $1.87\times 1{0}^{-81}$ | $8.86\times 1{0}^{-24}$ | $6.67\times 1{0}^{8}$ | $1.04\times 1{0}^{-158}$ | $8.72\times 1{0}^{-212}$ | |

Mean | $3.32\times 1{0}^{-66}$ | $7.31\times 1{0}^{-22}$ | $1.78\times 1{0}^{9}$ | $1.48\times 1{0}^{-105}$ | $3.17\times 1{0}^{-148}$ | ||

STD | $2.35\times 1{0}^{-65}$ | $1.05\times 1{0}^{-21}$ | $8.39\times 1{0}^{8}$ | $9.64\times 1{0}^{-105}$ | $2.23\times 1{0}^{-147}$ | ||

100 | Best | $1.50\times 1{0}^{-82}$ | $2.06\times 1{0}^{-23}$ | $3.42\times 1{0}^{8}$ | $2.07\times 1{0}^{-173}$ | $1.84\times 1{0}^{-200}$ | |

Mean | $4.67\times 1{0}^{-68}$ | $9.61\times 1{0}^{-22}$ | $1.83\times 1{0}^{9}$ | $4.86\times 1{0}^{-93}$ | $2.34\times 1{0}^{-141}$ | ||

STD | $3.28\times 1{0}^{-67}$ | $1.86\times 1{0}^{-21}$ | $8.66\times 1{0}^{8}$ | $4.86\times 1{0}^{-92}$ | $1.68\times 1{0}^{-140}$ | ||

F8 | 30 | Best | $7.79\times 1{0}^{-86}$ | $3.58\times 1{0}^{-31}$ | $6.83\times 1{0}^{0}$ | $2.68\times 1{0}^{-173}$ | $2.53\times 1{0}^{-198}$ |

Mean | $8.40\times 1{0}^{-77}$ | $2.89\times 1{0}^{-29}$ | $3.41\times 1{0}^{1}$ | $3.19\times 1{0}^{-109}$ | $6.83\times 1{0}^{-161}$ | ||

STD | $4.17\times 1{0}^{-76}$ | $5.64\times 1{0}^{-29}$ | $1.84\times 1{0}^{1}$ | $1.68\times 1{0}^{-108}$ | $2.95\times 1{0}^{-160}$ | ||

50 | Best | $1.90\times 1{0}^{-95}$ | $3.62\times 1{0}^{-31}$ | $7.56\times 1{0}^{0}$ | $1.65\times 1{0}^{-180}$ | $1.59\times 1{0}^{-215}$ | |

Mean | $1.34\times 1{0}^{-75}$ | $1.93\times 1{0}^{-29}$ | $2.92\times 1{0}^{1}$ | $1.24\times 1{0}^{-116}$ | $3.43\times 1{0}^{-153}$ | ||

STD | $5.27\times 1{0}^{-75}$ | $2.53\times 1{0}^{-29}$ | $2.23\times 1{0}^{1}$ | $6.24\times 1{0}^{-116}$ | $2.42\times 1{0}^{-152}$ | ||

100 | Best | $7.99\times 1{0}^{-93}$ | $2.83\times 1{0}^{-31}$ | $8.12\times 1{0}^{0}$ | $2.98\times 1{0}^{-177}$ | $3.62\times 1{0}^{-212}$ | |

Mean | $1.79\times 1{0}^{-74}$ | $2.53\times 1{0}^{-29}$ | $2.90\times 1{0}^{1}$ | $1.13\times 1{0}^{-103}$ | $8.64\times 1{0}^{-153}$ | ||

STD | $1.29\times 1{0}^{-73}$ | $4.61\times 1{0}^{-29}$ | $1.60\times 1{0}^{1}$ | $1.06\times 1{0}^{-102}$ | $8.64\times 1{0}^{-152}$ | ||

Rank | 30 | w/t/l | 0/1/7 | 0/1/7 | 0/0/8 | 0/1/7 | 7/1/0 |

50 | w/t/l | 0/1/7 | 0/1/7 | 0/0/8 | 0/1/7 | 7/1/0 | |

100 | w/t/l | 0/1/7 | 0/1/7 | 0/0/8 | 0/1/7 | 7/1/0 |

F | D | Index | WOA | GWO | PSO | DBO | IDBO |
---|---|---|---|---|---|---|---|

F9 | 30 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $7.52\times 1{0}^{1}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ |

Mean | $0.00\times 1{0}^{0}$ | $2.79\times 1{0}^{0}$ | $1.09\times 1{0}^{2}$ | $9.62\times 1{0}^{-1}$ | $0.00\times 1{0}^{0}$ | ||

STD | $0.00\times 1{0}^{0}$ | $3.25\times 1{0}^{0}$ | $1.72\times 1{0}^{1}$ | $3.66\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

50 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $7.23\times 1{0}^{1}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $3.41\times 1{0}^{-15}$ | $7.50\times 1{0}^{0}$ | $1.13\times 1{0}^{2}$ | $2.79\times 1{0}^{-1}$ | $0.00\times 1{0}^{0}$ | ||

STD | $1.78\times 1{0}^{-14}$ | $2.92\times 1{0}^{1}$ | $1.82\times 1{0}^{1}$ | $1.27\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

100 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $6.65\times 1{0}^{1}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $1.14\times 1{0}^{-15}$ | $2.47\times 1{0}^{0}$ | $1.07\times 1{0}^{2}$ | $3.38\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $8.00\times 1{0}^{-15}$ | $4.00\times 1{0}^{0}$ | $1.83\times 1{0}^{1}$ | $1.68\times 1{0}^{1}$ | $0.00\times 1{0}^{0}$ | ||

F10 | 30 | Best | $4.32\times 1{0}^{-58}$ | $1.97\times 1{0}^{-17}$ | $6.43\times 1{0}^{0}$ | $4.37\times 1{0}^{-89}$ | $9.47\times 1{0}^{-108}$ |

Mean | $3.25\times 1{0}^{-32}$ | $5.71\times 1{0}^{-4}$ | $1.08\times 1{0}^{1}$ | $1.26\times 1{0}^{-4}$ | $2.44\times 1{0}^{-81}$ | ||

STD | $1.78\times 1{0}^{-31}$ | $7.99\times 1{0}^{-4}$ | $2.60\times 1{0}^{0}$ | $3.44\times 1{0}^{-4}$ | $1.34\times 1{0}^{-80}$ | ||

50 | Best | $4.00\times 1{0}^{-58}$ | $1.75\times 1{0}^{-16}$ | $3.44\times 1{0}^{0}$ | $1.99\times 1{0}^{-88}$ | $2.32\times 1{0}^{-101}$ | |

Mean | $3.83\times 1{0}^{-1}$ | $4.95\times 1{0}^{-4}$ | $1.01\times 1{0}^{1}$ | $1.32\times 1{0}^{-1}$ | $8.49\times 1{0}^{-81}$ | ||

STD | $2.71\times 1{0}^{0}$ | $5.42\times 1{0}^{-4}$ | $2.85\times 1{0}^{0}$ | $9.17\times 1{0}^{-1}$ | $6.01\times 1{0}^{-80}$ | ||

100 | Best | $2.83\times 1{0}^{-60}$ | $3.32\times 1{0}^{-17}$ | $4.88\times 1{0}^{0}$ | $3.72\times 1{0}^{-87}$ | $1.69\times 1{0}^{-109}$ | |

Mean | $2.28\times 1{0}^{-1}$ | $4.47\times 1{0}^{-4}$ | $1.08\times 1{0}^{1}$ | $1.99\times 1{0}^{-3}$ | $3.31\times 1{0}^{-80}$ | ||

STD | $2.28\times 1{0}^{0}$ | $5.21\times 1{0}^{-4}$ | $2.61\times 1{0}^{0}$ | $1.57\times 1{0}^{-2}$ | $2.91\times 1{0}^{-79}$ | ||

F11 | 30 | Best | $6.35\times 1{0}^{-3}$ | $1.31\times 1{0}^{-2}$ | $1.15\times 1{0}^{1}$ | $1.11\times 1{0}^{-7}$ | $4.56\times 1{0}^{-6}$ |

Mean | $2.76\times 1{0}^{-2}$ | $5.00\times 1{0}^{-2}$ | $1.07\times 1{0}^{3}$ | $3.57\times 1{0}^{-3}$ | $5.73\times 1{0}^{-5}$ | ||

STD | $2.02\times 1{0}^{-2}$ | $2.95\times 1{0}^{-2}$ | $4.83\times 1{0}^{3}$ | $1.89\times 1{0}^{-2}$ | $1.03\times 1{0}^{-4}$ | ||

50 | Best | $2.47\times 1{0}^{-3}$ | $1.22\times 1{0}^{-2}$ | $1.26\times 1{0}^{1}$ | $7.74\times 1{0}^{-8}$ | $3.29\times 1{0}^{-6}$ | |

Mean | $2.34\times 1{0}^{-2}$ | $4.54\times 1{0}^{-2}$ | $1.10\times 1{0}^{3}$ | $2.26\times 1{0}^{-3}$ | $3.79\times 1{0}^{-5}$ | ||

STD | $1.81\times 1{0}^{-2}$ | $2.56\times 1{0}^{-2}$ | $3.36\times 1{0}^{3}$ | $1.47\times 1{0}^{-2}$ | $4.79\times 1{0}^{-5}$ | ||

100 | Best | $3.42\times 1{0}^{-3}$ | $1.32\times 1{0}^{-2}$ | $7.66\times 1{0}^{0}$ | $5.39\times 1{0}^{-8}$ | $2.17\times 1{0}^{-6}$ | |

Mean | $2.26\times 1{0}^{-2}$ | $4.46\times 1{0}^{-2}$ | $1.61\times 1{0}^{3}$ | $9.76\times 1{0}^{-5}$ | $7.55\times 1{0}^{-5}$ | ||

STD | $1.92\times 1{0}^{-2}$ | $2.36\times 1{0}^{-2}$ | $6.56\times 1{0}^{3}$ | $7.08\times 1{0}^{-4}$ | $1.96\times 1{0}^{-4}$ | ||

F12 | 30 | Best | $9.44\times 1{0}^{-2}$ | $3.15\times 1{0}^{-1}$ | $7.93\times 1{0}^{2}$ | $1.79\times 1{0}^{-4}$ | $1.50\times 1{0}^{-4}$ |

Mean | $5.49\times 1{0}^{-1}$ | $6.39\times 1{0}^{-1}$ | $2.14\times 1{0}^{5}$ | $5.44\times 1{0}^{-1}$ | $3.32\times 1{0}^{-2}$ | ||

STD | $3.20\times 1{0}^{-1}$ | $1.90\times 1{0}^{-1}$ | $2.87\times 1{0}^{5}$ | $4.09\times 1{0}^{-1}$ | $4.58\times 1{0}^{-2}$ | ||

50 | Best | $1.81\times 1{0}^{-1}$ | $1.00\times 1{0}^{-1}$ | $6.48\times 1{0}^{1}$ | $7.70\times 1{0}^{-4}$ | $5.05\times 1{0}^{-5}$ | |

Mean | $6.09\times 1{0}^{-1}$ | $6.13\times 1{0}^{-1}$ | $1.89\times 1{0}^{5}$ | $6.14\times 1{0}^{-1}$ | $2.66\times 1{0}^{-2}$ | ||

STD | $2.75\times 1{0}^{-1}$ | $2.44\times 1{0}^{-1}$ | $3.94\times 1{0}^{5}$ | $4.19\times 1{0}^{-1}$ | $4.10\times 1{0}^{-2}$ | ||

100 | Best | $1.17\times 1{0}^{-1}$ | $1.02\times 1{0}^{-1}$ | $7.83\times 1{0}^{1}$ | $1.35\times 1{0}^{-3}$ | $6.25\times 1{0}^{-5}$ | |

Mean | $4.87\times 1{0}^{-1}$ | $6.46\times 1{0}^{-1}$ | $3.87\times 1{0}^{5}$ | $7.15\times 1{0}^{-1}$ | $3.72\times 1{0}^{-2}$ | ||

STD | $2.78\times 1{0}^{-1}$ | $2.30\times 1{0}^{-1}$ | $4.63\times 1{0}^{5}$ | $4.89\times 1{0}^{-1}$ | $6.20\times 1{0}^{-2}$ | ||

F13 | 30 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ |

Mean | $8.23\times 1{0}^{-5}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $3.25\times 1{0}^{-4}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

50 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $5.65\times 1{0}^{-5}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $3.11\times 1{0}^{-4}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

100 | Best | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | |

Mean | $7.81\times 1{0}^{-5}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

STD | $3.89\times 1{0}^{-4}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | $0.00\times 1{0}^{0}$ | ||

F14 | 30 | Best | $3.69\times 1{0}^{-1}$ | $8.22\times 1{0}^{-1}$ | $4.86\times 1{0}^{2}$ | $8.97\times 1{0}^{-2}$ | $2.88\times 1{0}^{-4}$ |

Mean | $9.42\times 1{0}^{-1}$ | $1.33\times 1{0}^{0}$ | $1.03\times 1{0}^{3}$ | $5.50\times 1{0}^{-1}$ | $8.37\times 1{0}^{-2}$ | ||

STD | $4.24\times 1{0}^{-1}$ | $2.92\times 1{0}^{-1}$ | $3.21\times 1{0}^{2}$ | $4.22\times 1{0}^{-1}$ | $9.28\times 1{0}^{-2}$ | ||

50 | Best | $2.55\times 1{0}^{-1}$ | $6.37\times 1{0}^{-1}$ | $3.73\times 1{0}^{2}$ | $2.69\times 1{0}^{-4}$ | $4.52\times 1{0}^{-4}$ | |

Mean | $9.42\times 1{0}^{-1}$ | $1.23\times 1{0}^{0}$ | $9.25\times 1{0}^{2}$ | $4.83\times 1{0}^{-1}$ | $1.05\times 1{0}^{-1}$ | ||

STD | $3.68\times 1{0}^{-1}$ | $2.13\times 1{0}^{-1}$ | $3.46\times 1{0}^{2}$ | $2.10\times 1{0}^{-1}$ | $1.11\times 1{0}^{-1}$ | ||

100 | Best | $1.93\times 1{0}^{-1}$ | $8.13\times 1{0}^{-1}$ | $4.09\times 1{0}^{2}$ | $1.40\times 1{0}^{-3}$ | $2.75\times 1{0}^{-4}$ | |

Mean | $8.70\times 1{0}^{-1}$ | $1.25\times 1{0}^{0}$ | $9.66\times 1{0}^{2}$ | $5.16\times 1{0}^{-1}$ | $7.45\times 1{0}^{-2}$ | ||

STD | $3.82\times 1{0}^{-1}$ | $2.29\times 1{0}^{-1}$ | $3.04\times 1{0}^{2}$ | $2.94\times 1{0}^{-1}$ | $7.85\times 1{0}^{-2}$ | ||

Rank | 30 | w/t/l | 0/1/5 | 0/1/5 | 0/1/5 | 0/1/5 | 4/2/0 |

50 | w/t/l | 0/0/6 | 0/1/5 | 0/1/5 | 0/1/5 | 4/2/0 | |

100 | w/t/l | 0/0/6 | 0/1/5 | 0/1/5 | 0/1/5 | 4/2/0 |

WOA | GWO | PSO | DBO | IDBO | |
---|---|---|---|---|---|

w/t/l | w/t/l | w/t/l | w/t/l | w/t/l | |

D = 30 | 0/2/12 | 0/2/12 | 0/1/13 | 0/2/12 | 11/3/0 |

D = 50 | 0/1/13 | 0/2/12 | 0/1/13 | 0/2/12 | 11/3/0 |

D = 100 | 0/1/13 | 0/2/12 | 0/1/13 | 0/2/12 | 11/3/0 |

Total | 0/4/38 | 0/6/36 | 0/3/39 | 0/6/36 | 33/9/0 |

TP | 9.52% | 14.29% | 7.14% | 14.29% | 100.00% |

Function | WOA | GWO | PSO | DBO |
---|---|---|---|---|

F1 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $6.07\times 1{0}^{-11}$ |

F2 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $4.20\times 1{0}^{-10}$ |

F3 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $2.57\times 1{0}^{-7}$ |

F4 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $1.61\times 1{0}^{-10}$ |

F5 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $4.18\times 1{0}^{-9}$ |

F6 | N/A | N/A | $1.21\times 1{0}^{-12}$ | N/A |

F7 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $1.61\times 1{0}^{-10}$ |

F8 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $2.61\times 1{0}^{-10}$ |

F9 | N/A | $1.16\times 1{0}^{-12}$ | $1.21\times 1{0}^{-12}$ | $3.34\times 1{0}^{-1}$ |

F10 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.69\times 1{0}^{-11}$ |

F11 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $1.99\times 1{0}^{-2}$ |

F12 | $3.69\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $1.33\times 1{0}^{-10}$ |

F13 | $8.15\times 1{0}^{-2}$ | N/A | N/A | N/A |

F14 | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $3.02\times 1{0}^{-11}$ | $1.09\times 1{0}^{-10}$ |

+/=/− | 11/3/0 | 12/2/0 | 13/1/0 | 12/2/0 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Zhu, Y.
Predicting the Mechanical Properties of Heat-Treated Woods Using Optimization-Algorithm-Based BPNN. *Forests* **2023**, *14*, 935.
https://doi.org/10.3390/f14050935

**AMA Style**

Zhang R, Zhu Y.
Predicting the Mechanical Properties of Heat-Treated Woods Using Optimization-Algorithm-Based BPNN. *Forests*. 2023; 14(5):935.
https://doi.org/10.3390/f14050935

**Chicago/Turabian Style**

Zhang, Runze, and Yujie Zhu.
2023. "Predicting the Mechanical Properties of Heat-Treated Woods Using Optimization-Algorithm-Based BPNN" *Forests* 14, no. 5: 935.
https://doi.org/10.3390/f14050935