# Enhancing Backtracking Search Algorithm using Reflection Mutation Strategy Based on Sine Cosine

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- A new reflection mutation strategy based on sine cosine is proposed to balance exploration and exploitation ability of BSA. To improve exploration ability, the best global individual is used to guide search direction, while sine and cosine math functions are used to enhance exploitation ability of BSA. Based on the strategy, a novel backtracking search algorithm with reflection mutation strategy based on sine cosine (RSCBSA) is proposed to solve global optimization problems.
- In the above strategy, the center of a unit simplex constructed by three individuals selected randomly is employed to enhance diversity of population, since it considers more information of individuals. In addition, in RSCBSA, crossover operator of BSA is replaced with that of DE.
- A comprehensive experiment is designed to verity the effectiveness of the proposed RSCBSA. In addition, a new parameter in RSCBSA is analyzed to set suitable values so that the performance of RSCBSA is the best.

## 2. Backtracking Search Optimization Algorithm

**Initialization:**In the stage, BSA randomly produces the initial population P using Equation (1).

**Selection-I:**In the BSA’s selection-I stage, the initial historical population $oldP$ is generated randomly using Equation (2).

**Mutation and Crossover:**Mutation and crossover operators aim at producing a new individual. Equation (5) involving historical population and current population is used to generate a trail individual. Then, crossover operator is performed by Equation (6). From Equation (6), a binary integer-value map is employed to guide the crossover direction.

**Selection-II:**In the selection-II stage, BSA adopts a greedy selection mechanism to pick out a new solution ${P}_{i}^{new}$. For the minimum problem, if the fitness value of the trial individual V is better than that of the current individual ${P}_{i}$, V is selected, as shown in Equation (7);

## 3. The Proposed Algorithm

#### 3.1. Initialization

#### 3.2. Reflection Mutation Strategy Based on Sine Cosine

#### 3.3. Crossover Operator

#### 3.4. The Framework of The Proposed Algorithm

Algorithm 1: Framework of the Proposed Algorithm |

#### 3.5. Complex Analysis of The Proposed Algorithm

## 4. Experimental Simulations

#### 4.1. Benchmark Test Suit

#### 4.2. Parameter Setting

## 5. Experimental Results

#### 5.1. Compared with State-of-the-Art Algorithms

#### 5.2. Convergence Analysis

#### 5.3. Parameter Sensitivity Analysis

#### 5.4. Runtime Analysis

#### 5.5. Remarks

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Convergence figures on test functions F1–F23, where (

**a**–

**w**) indicate the convergence curves on the functions F1–F23 respectively.

**Figure 2.**Convergence curves on test functions F1–F23 for different a values, where (

**a**–

**w**) indicate the convergence curves on the functions F1–F23 respectively.

Benchmark | Function | PSO | ABC | DE | CA | BSA | RSCBSA |
---|---|---|---|---|---|---|---|

F1 | best | $2.6078\times {10}^{-21}$ | $5.3796\times {10}^{-16}$ | $2.5816\times {10}^{-46}$ | $2.5562\times {10}^{2}$ | $1.4976\times {10}^{-17}$ | $0.0000\times {10}^{0}$ |

mean | $3.9736\times {10}^{-7}$ | $7.5685\times {10}^{-16}$ | $5.4588\times {10}^{-45}$ | $1.6712\times {10}^{3}$ | $2.4454\times {10}^{-15}$ | $0.0000\times {10}^{0}$ | |

worst | 3.6505 × 10${}^{-6}$ | 5.4117 × 10${}^{-15}$ | 2.4629 × 10${}^{-44}$ | 4.1270 × 10${}^{3}$ | 2.8521× 10${}^{-14}$ | $0.0000\times {10}^{0}$ | |

std | 1.3868 × 10${}^{-12}$ | 6.8910 × 10${}^{-31}$ | 2.6423 × 10${}^{-89}$ | 9.5577 × 10${}^{5}$ | 3.0966 × 10${}^{-29}$ | $0.0000\times {10}^{0}$ | |

F2 | best | 5.4694 × 10${}^{-28}$ | 1.6002 × 10${}^{-15}$ | 4.1335 × 10${}^{-28}$ | 5.8462 × 10${}^{0}$ | 5.0408 × 10${}^{-10}$ | 4.3459× 10${}^{-218}$ |

mean | 1.3657 × 10${}^{-12}$ | 1.8228 × 10${}^{-15}$ | 2.0667 × 10${}^{-27}$ | 2.2100 × 10${}^{1}$ | 3.2572 × 10${}^{-9}$ | 3.9314 × 10${}^{-214}$ | |

worst | 1.5274 × 10${}^{-11}$ | 3.3803 × 10${}^{-15}$ | 5.6259 × 10${}^{-27}$ | 6.9183 × 10${}^{1}$ | 1.0338 × 10${}^{-8}$ | 3.1240 × 10${}^{-213}$ | |

std | 1.5008 × 10${}^{-23}$ | 1.6937 × 10${}^{-31}$ | 1.2307 × 10${}^{-54}$ | 2.3890 × 10${}^{2}$ | 5.5714 × 10${}^{-18}$ | 0.0000 × 10${}^{0}$ | |

F3 | best | 3.9653 × 10${}^{-103}$ | 3.5792 × 10${}^{3}$ | 7.3893 × 10${}^{3}$ | 8.3334 × 10${}^{3}$ | 3.0345 × 10${}^{1}$ | 8.1950 × 10${}^{-204}$ |

mean | 6.4404 × 10${}^{-93}$ | 5.9449 × 10${}^{3}$ | 1.1857 × 10${}^{4}$ | 2.0627 × 10${}^{4}$ | 1.7266 × 10${}^{2}$ | 7.7379 × 10${}^{-188}$ | |

worst | 1.5655 × 10${}^{-91}$ | 1.1140 × 10${}^{4}$ | 1.9801 × 10${}^{4}$ | 4.0887 × 10${}^{4}$ | 3.7642 × 10${}^{2}$ | 2.3203 × 10${}^{-186}$ | |

std | 8.1160 × 10${}^{-184}$ | 3.2828 × 10${}^{6}$ | 6.0971 × 10${}^{6}$ | 7.8083× 10${}^{7}$ | 6.6997 × 10${}^{3}$ | 0.0000 × 10${}^{0}$ | |

F4 | best | 7.8193 × 10${}^{-83}$ | 2.8317 × 10${}^{1}$ | 5.1221 × 10${}^{-4}$ | 3.0161 × 10${}^{1}$ | 9.0018 × 10${}^{-1}$ | 8.1338 × 10${}^{-152}$ |

mean | 1.5871 × 10${}^{-72}$ | 3.9440 × 10${}^{1}$ | 1.0355 × 10${}^{-3}$ | 4.6255 × 10${}^{1}$ | 2.1297 × 10${}^{0}$ | 4.7338 × 10${}^{-144}$ | |

worst | 2.4649 × 10${}^{-71}$ | 5.0921 × 10${}^{1}$ | 2.2349 × 10${}^{-3}$ | 7.9715 × 10${}^{1}$ | 3.9292 × 10${}^{0}$ | 8.8191 × 10${}^{-143}$ | |

std | 2.1488 × 10${}^{-143}$ | 3.7018 × 10${}^{1}$ | 1.5905 × 10${}^{-7}$ | 1.4438 × 10${}^{2}$ | 4.0870 × 10${}^{-1}$ | 2.6901 × 10${}^{-286}$ | |

F5 | best | 7.9093 × 10${}^{-2}$ | 8.6115 × 10${}^{-3}$ | 2.3793 × 10${}^{1}$ | 1.0168 × 10${}^{5}$ | 8.6456 × 10${}^{-1}$ | 2.3440× 10${}^{1}$ |

mean | 2.2878 × 10${}^{0}$ | 5.3359 × 10${}^{-1}$ | 3.3665 × 10${}^{1}$ | 1.2073 × 10${}^{6}$ | 5.5375 × 10${}^{1}$ | 2.4658 × 10${}^{1}$ | |

worst | 7.0303 × 10${}^{0}$ | 1.8079 × 10${}^{1}$ | 8.8254 × 10${}^{1}$ | 4.4121 × 10${}^{6}$ | 8.6258 × 10${}^{1}$ | 2.6975 × 10${}^{1}$ | |

std | 3.1431 × 10${}^{0}$ | 1.3289 × 10${}^{1}$ | 4.0340 × 10${}^{2}$ | 1.0354 × 10${}^{12}$ | 9.4312 × 10${}^{2}$ | 6.2485 × 10${}^{1}$ | |

F6 | best | 0.0000 × 10${}^{0}$ | 5.5033 × 10${}^{-16}$ | 0.0000 × 10${}^{0}$ | 4.3707 × 10${}^{2}$ | 2.2988 × 10${}^{-17}$ | 8.6469 × 10${}^{-7}$ |

mean | 1.9105 × 10${}^{-32}$ | 7.4696 × 10${}^{-16}$ | 0.0000 × 10${}^{0}$ | 1.8850 × 10${}^{3}$ | 4.8673 × 10${}^{16}$ | 1.4989 × 10${}^{-1}$ | |

worst | 1.1093 × 10${}^{-31}$ | 2.5076 × 10${}^{-15}$ | 0.0000 × 10${}^{0}$ | 3.1993 × 10${}^{3}$ | 1.9639× 10${}^{-15}$ | 5.1162× 10${}^{-1}$ | |

std | 8.1244× 10${}^{64}$ | 1.3274 × 10${}^{-31}$ | 0.0000 × 10${}^{0}$ | 5.4730 × 10${}^{5}$ | 2.3180 × 10${}^{-31}$ | 2.7930 × 10${}^{-2}$ | |

F7 | best | 1.1152 × 10${}^{-4}$ | 9.4208 × 10${}^{-2}$ | 3.1487 × 10${}^{-3}$ | 3.5096 × 10${}^{-1}$ | 4.3966 × 10${}^{-3}$ | 3.3201 × 10${}^{-5}$ |

mean | 7.4359 × 10${}^{-4}$ | 2.0345 × 10${}^{-1}$ | 8.2541 × 10${}^{-3}$ | 1.4979 × 10${}^{0}$ | 1.4448 × 10${}^{-2}$ | 1.7506 × 10${}^{-4}$ | |

worst | 2.1471 × 10${}^{-3}$ | 3.5513 × 10${}^{-1}$ | 1.2403 × 10${}^{-2}$ | 3.4948 × 10${}^{0}$ | 2.1994 × 10${}^{-2}$ | 4.4544 × 10${}^{-4}$ | |

std | 3.1481 × 10${}^{-7}$ | 3.7637 × 10${}^{-3}$ | 4.3611 × 10${}^{-6}$ | 8.3832 × 10${}^{-1}$ | 1.6176× 10${}^{5}$ | 1.1159 × 10${}^{-8}$ |

Algorithm | Ranking |
---|---|

PSO | 2.3571 |

ABC | 3 |

DE | 3.4286 |

CA | 6 |

BSA | 4.0714 |

RSCBSA | 2.1429 |

Benchmark | Function | PSO | ABC | DE | CA | BSA | RSCBSA |
---|---|---|---|---|---|---|---|

F8 | best | −3.2818 × 10${}^{3}$ | −1.2570 × 10${}^{4}$ | −1.2570 × 10${}^{4}$ | −1.2049 × 10${}^{4}$ | −1.2569 × 10${}^{4}$ | −1.0278 × 10${}^{4}$ |

mean | −2.4289 × 10${}^{3}$ | −1.2541 × 10${}^{4}$ | −1.2980 × 10${}^{4}$ | −9.8465 × 10${}^{3}$ | −1.2569 × 10${}^{4}$ | −8.8889 × 10${}^{3}$ | |

worst | −1.6827 × 10${}^{3}$ | −1.2209 × 10${}^{4}$ | −1.2214 × 10${}^{4}$ | −6.9449 × 10${}^{3}$ | −1.2569 × 10${}^{4}$ | −8.0066 × 10${}^{3}$ | |

std | 9.9518 × 10${}^{4}$ | 8.0666 × 10${}^{3}$ | −1.2442 × 10${}^{4}$ | 3.3642 × 10${}^{6}$ | 2.2234 × 10${}^{-1}$ | 2.3757 × 10${}^{5}$ | |

F9 | best | 5.9698 × 10${}^{0}$ | 1.1369 × 10${}^{-13}$ | 6.8781 × 10${}^{-12}$ | 7.7160 × 10${}^{1}$ | 1.0854 × 10${}^{-1}$ | 0.0000 × 10${}^{0}$ |

mean | 1.1840 × 10${}^{1}$ | 1.4061 × 10${}^{-8}$ | 5.0229 × 10${}^{0}$ | 1.4651 × 10${}^{2}$ | 3.3603 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | |

worst | 2.0894 × 10${}^{1}$ | 9.9496 × 10${}^{-1}$ | 2.6971 × 10${}^{1}$ | 2.5216 × 10${}^{2}$ | 6.6977 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | |

std | 1.5334 × 10${}^{1}$ | 6.2573 × 10${}^{-2}$ | 6.5267 × 10${}^{1}$ | 1.8300 × 10${}^{3}$ | 3.1008 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | |

F10 | best | 4.4409 × 10${}^{-15}$ | 4.7074 × 10${}^{-14}$ | 7.9936 × 10${}^{-15}$ | 5.5878 × 10${}^{0}$ | 2.1721 × 10${}^{-9}$ | 8.8818 × 10${}^{-16}$ |

mean | 3.8505 × 10${}^{-2}$ | 6.0574 × 10${}^{-14}$ | 7.9936 × 10${}^{-15}$ | 1.0163 × 10${}^{1}$ | 2.5030 × 10${}^{-8}$ | 8.8818 × 10${}^{-16}$ | |

worst | 1.1552 × 10${}^{0}$ | 1.5721 × 10${}^{-13}$ | 7.9936 × 10${}^{-15}$ | 1.3951× 10${}^{1}$ | 7.7180 × 10${}^{-8}$ | 8.8818 × 10${}^{-16}$ | |

std | 4.2996× 10${}^{-2}$ | 5.4819 × 10${}^{-28}$ | 2.2403 × 10${}^{-59}$ | 3.9939 × 10${}^{0}$ | 5.5106 × 10${}^{-16}$ | 3.8894 × 10${}^{-62}$ | |

F11 | best | 3.9319 × 10${}^{-2}$ | 1.1102 × 10${}^{-15}$ | 0.0000 × 10${}^{0}$ | 5.5845 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ |

mean | 9.3162 × 10${}^{-2}$ | 3.5826 × 10${}^{-4}$ | 0.0000 × 10${}^{0}$ | 1.5086 × 10${}^{1}$ | 1.6533 × 10${}^{-11}$ | 0.0000 × 10${}^{0}$ | |

worst | 2.2875 × 10${}^{-1}$ | 3.5406 × 10${}^{-2}$ | 0.0000 × 10${}^{0}$ | 4.6938 × 10${}^{1}$ | 4.9135 × 10${}^{-10}$ | 0.0000× 10${}^{0}$ | |

std | 1.9610 × 10${}^{-3}$ | 7.4801 × 10${}^{-5}$ | 0.0000 × 10${}^{0}$ | 8.2619 × 10${}^{1}$ | 7.7747 × 10${}^{-21}$ | 0.0000× 10${}^{0}$ | |

F12 | best | 4.7116 × 10${}^{-32}$ | 6.1436 × 10${}^{-16}$ | 1.5705 × 10${}^{-32}$ | 6.2196 × 10${}^{1}$ | 2.8984 × 10${}^{-19}$ | 6.0657 × 10${}^{-7}$ |

mean | 5.1561 × 10${}^{-32}$ | 7.8009 × 10${}^{-16}$ | 1.5705 × 10${}^{-32}$ | 2.7468 × 10${}^{5}$ | 2.8011 × 10${}^{-17}$ | 8.8653 × 10${}^{-3}$ | |

worst | 1.2553 × 10${}^{-31}$ | 5.2760 × 10${}^{-13}$ | 1.5705 × 10${}^{-32}$ | 1.7374 × 10${}^{6}$ | 4.4505 × 10${}^{-16}$ | 2.3351 × 10${}^{-2}$ | |

std | 2.1613 × 10${}^{-64}$ | 8.9331 × 10${}^{-27}$ | 2.9963 × 10${}^{-95}$ | 1.7490 × 10${}^{11}$ | 6.7812 × 10${}^{-33}$ | 4.6445 × 10${}^{-5}$ | |

F13 | best | 1.0987 × 10${}^{-2}$ | 5.3680 × 10${}^{-16}$ | 1.3498 × 10${}^{-32}$ | 6.4179 × 10${}^{3}$ | 2.2895 × 10${}^{-18}$ | 1.1010 × 10${}^{-2}$ |

mean | 1.0987× 10${}^{-3}$ | 8.3837 × 10${}^{-16}$ | 1.3498 × 10${}^{-32}$ | 2.6594 × 10${}^{6}$ | 3.7103 × 10${}^{-16}$ | 5.0903 × 10${}^{-1}$ | |

worst | 1.0987 × 10${}^{-2}$ | 2.2538 × 10${}^{-15}$ | 1.3498 × 10${}^{-32}$ | 1.2544 × 10${}^{7}$ | 2.9765 × 10${}^{-15}$ | 1.4244 × 10${}^{0}$ | |

std | 1.0865 × 10${}^{-5}$ | 1.0659 × 10${}^{-31}$ | 0.0000 × 10${}^{0}$ | 1.0542 × 10${}^{13}$ | 4.7169 × 10${}^{-31}$ | 1.0055 × 10${}^{-1}$ |

Algorithm | Ranking |
---|---|

PSO | 4.5833 |

ABC | 2.5833 |

DE | 2.25 |

CA | 5.6667 |

BSA | 2.5833 |

RSCBSA | 3.3333 |

Benchmark | Function | PSO | ABC | DE | CA | BSA | RSCBSA |
---|---|---|---|---|---|---|---|

F14 | best | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ |

mean | 3.4567× 10${}^{0}$ | 9.9800 × 10${}^{-1}$ | 1.1624 × 10${}^{0}$ | 6.6697 × 10${}^{0}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | |

worst | 1.2671 × 10${}^{1}$ | 9.9800 × 10${}^{-1}$ | 5.9289 × 10${}^{0}$ | 1.6441 × 10${}^{1}$ | 9.9800 × 10${}^{-1}$ | 9.9800 × 10${}^{-1}$ | |

std | 9.3179 × 10${}^{0}$ | 4.4373 × 10${}^{-31}$ | 7.8343 × 10${}^{-1}$ | 1.9194 × 10${}^{1}$ | 1.9722 × 10${}^{-31}$ | 1.9722 × 10${}^{-31}$ | |

F15 | best | 3.0749 × 10${}^{-4}$ | 4.1171 × 10${}^{-4}$ | 3.0749 × 10${}^{-4}$ | 4.8171 × 10${}^{-4}$ | 3.0749 × 10${}^{-4}$ | 3.0749 × 10${}^{-4}$ |

mean | 3.3567 × 10${}^{-3}$ | 5.5933 × 10${}^{-4}$ | 4.7100 × 10${}^{-4}$ | 2.2035 × 10${}^{-3}$ | 3.0749 × 10${}^{-4}$ | 3.5056 × 10${}^{-4}$ | |

worst | 2.0363 × 10${}^{-2}$ | 1.0451 × 10${}^{-3}$ | 7.8431 × 10${}^{-4}$ | 1.4641 × 10${}^{-2}$ | 3.0749 × 10${}^{-4}$ | 1.2232 × 10${}^{-3}$ | |

std | 4.4754 × 10${}^{-5}$ | 1.3891 × 10${}^{-8}$ | 2.6811 × 10${}^{-8}$ | 6.8205 × 10${}^{-6}$ | 1.1755 × 10${}^{-38}$ | 2.8670 × 10${}^{-8}$ | |

F16 | best | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ |

mean | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | |

worst | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | −1.0316 × 10${}^{0}$ | |

std | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 1.9722 × 10${}^{-31}$ | 1.9722 × 10${}^{-31}$ | |

F17 | best | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ |

mean | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | |

worst | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | 3.9789 × 10${}^{-1}$ | |

std | 2.7733 × 10${}^{-32}$ | 2.7733 × 10${}^{-32}$ | 2.7733 × 10${}^{-32}$ | 2.7733 × 10${}^{-32}$ | 3.0815 × 10${}^{-33}$ | 1.9600 × 10${}^{-13}$ | |

F18 | best | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ |

mean | 3.0000 × 10${}^{0}$ | 3.0002 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | |

worst | 3.0000 × 10${}^{0}$ | 3.0039 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | 3.0000 × 10${}^{0}$ | |

std | 0.0000 × 10${}^{0}$ | 5.0468 × 10${}^{-7}$ | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 0.0000 × 10${}^{0}$ | 2.0556 × 10${}^{-13}$ | |

F19 | best | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | -3.8628 × 10${}^{0}$ |

mean | −3.8370 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | |

worst | −3.0898 × 10${}^{0}$ | -3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | −3.8628 × 10${}^{0}$ | |

std | 1.9255 × 10${}^{-2}$ | 1.9722 × 10${}^{-31}$ | 1.9722 × 10${}^{-31}$ | 1.9722 × 10${}^{-31}$ | 3.1554 × 10${}^{-30}$ | 3.1554 × 10${}^{-30}$ | |

F20 | best | −3.3220 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ |

mean | −3.2863 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.3212 × 10${}^{0}$ | −3.2809 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.2863 × 10${}^{0}$ | |

worst | −3.2031 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.2974 × 10${}^{0}$ | −3.1993 × 10${}^{0}$ | −3.3220 × 10${}^{0}$ | −3.2031 × 10${}^{0}$ | |

std | 2.9688 × 10${}^{-3}$ | 3.1554 × 10${}^{-30}$ | 1.9578 × 10${}^{-5}$ | 3.1319 × 10${}^{-3}$ | 7.8886 × 10${}^{-31}$ | 2.9685 × 10${}^{-3}$ | |

F21 | best | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ |

mean | −4.7418 × 10${}^{0}$ | −1.0153 × 10${}^{1}$ | −9.7358 × 10${}^{0}$ | −6.4744 × 10${}^{0}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | |

worst | −2.6305 × 10${}^{0}$ | −1.0153 × 10${}^{1}$ | −2.6829 × 10${}^{0}$ | −2.6305 × 10${}^{0}$ | −1.0153 × 10${}^{1}$ | −1.0153 × 10${}^{1}$ | |

std | 1.0831 × 10${}^{1}$ | 1.2622 × 10${}^{-29}$ | 2.5369 × 10${}^{0}$ | 1.1149 × 10${}^{1}$ | 1.2622 × 10${}^{-29}$ | 4.2873 × 10${}^{-8}$ | |

F22 | best | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ |

mean | −6.1650 × 10${}^{0}$ | −1.0403 × 10${}^{1}$ | −-1.0227 × 10${}^{1}$ | −6.7240 × 10${}^{0}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | |

worst | −1.8376 × 10${}^{0}$ | −1.0403 × 10${}^{1}$ | −5.1288 × 10${}^{0}$ | −2.7519 × 10${}^{0}$ | −1.0403 × 10${}^{1}$ | −1.0403 × 10${}^{1}$ | |

std | 1.2604 × 10${}^{1}$ | 5.0487 × 10${}^{-29}$ | 8.9629 × 10${}^{-1}$ | 1.2263 × 10${}^{1}$ | 0.0000 × 10${}^{0}$ | 6.5010 × 10${}^{-9}$ | |

F23 | best | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ |

mean | −6.7622 × 10${}^{0}$ | −1.0536 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −6.2481 × 10${}^{0}$ | −1.0536 × 10${}^{1}$ | −1.0133 × 10${}^{1}$ | |

worst | −2.4217E × 10${}^{0}$ | −1.0512 × 10${}^{1}$ | −1.0536 × 10${}^{1}$ | −2.4217 × 10${}^{0}$ | −1.0536 × 10 | -3.8354 × 10${}^{0}$ | |

std | 1.4389 × 10${}^{1}$ | 1.9658 × 10${}^{-5}$ | 7.8886 × 10${}^{-29}$ | 1.4417 × 10${}^{1}$ | 2.8399 × 10${}^{-29}$ | 2.3087 × 10${}^{0}$ |

Algorithm | Ranking |
---|---|

PSO | 4.85 |

ABC | 2.95 |

DE | 3.3 |

CA | 4.6 |

BSA | 2.35 |

RSCBSA | 2.95 |

Benchmark | RSCBSA vs. BSA | RSCBSA vs. PSO | RSCBSA vs. ABC | RSCBSA vs. DE | RSCBSA vs. CA | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H | p-Value | Winner | H | p-Value | Winner | H | p-Value | Winner | H | p-Value | Winner | H | p-Value | Winner | |

F1 | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + |

F2 | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + |

F3 | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + |

F4 | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + |

F5 | 1 | 1.0763 × 10${}^{-2}$ | + | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 1.2493 × 10${}^{-5}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + |

F6 | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 2.3692 × 10${}^{-11}$ | − | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 1.2118 × 10${}^{-12}$ | − | 1 | 3.0199 × 10${}^{-11}$ | + |

F7 | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 2.0283 × 10${}^{-7}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + | 1 | 3.0199 × 10${}^{-11}$ | + |

F8 | 1 | 2.2076 × 10${}^{-11}$ | − | 1 | 3.0104 × 10${}^{-11}$ | + | 1 | 1.4248 × 10${}^{-11}$ | − | 1 | 2.5416 × 10${}^{-11}$ | − | 1 | 3.8481 × 10${}^{-3}$ | − |

F9 | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.1661 × 10${}^{-12}$ | + | 1 | 1.0566 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + |

F10 | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.5702 × 10${}^{-13}$ | + | 1 | 9.7992 × 10${}^{-13}$ | + | 1 | 1.6853 × 10${}^{-14}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + |

F11 | 1 | 4.5700 × 10${}^{-12}$ | + | 1 | 1.2118 × 10${}^{-12}$ | + | 1 | 1.2078 × 10${}^{-12}$ | + | 0 | NaN | = | 1 | 1.2118 × 10${}^{-12}$ | + |

F12 | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 2.4291 × 10${}^{-11}$ | − | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 1.2118 × 10${}^{-12}$ | − | 1 | 3.0199 × 10${}^{-11}$ | + |

F13 | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 2.6537 × 10${}^{-11}$ | − | 1 | 3.0199 × 10${}^{-11}$ | − | 1 | 1.2118 × 10${}^{-12}$ | − | 1 | 3.0199 × 10${}^{-11}$ | + |

F14 | 0 | NaN | = | 1 | 9.7829 × 10${}^{-13}$ | + | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 2.7085 × 10${}^{-14}$ | + | 1 | 1.1642 × 10${}^{-12}$ | + |

F15 | 1 | 2.1633 × 10${}^{-11}$ | − | 0 | 9.7028 × 10${}^{-1}$ | = | 1 | 4.1804 × 10${}^{-9}$ | + | 1 | 8.8803 × 10${}^{-6}$ | + | 1 | 1.2050 × 10${}^{-10}$ | + |

F16 | 0 | NaN | = | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 1.6853 × 10${}^{-14}$ | = |

F17 | 1 | 6.6113 × 10${}^{-4}$ | = | 1 | 3.8943 × 10${}^{-13}$ | = | 1 | 3.8943 × 10${}^{-13}$ | = | 1 | 3.8943 × 10${}^{-13}$ | = | 1 | 3.8943 × 10${}^{-13}$ | = |

F18 | 1 | 4.1865 × 10${}^{-2}$ | = | 1 | 4.1865 × 10${}^{-2}$ | = | 0 | 4.6889 × 10${}^{-1}$ | = | 1 | 4.1865 × 10${}^{-2}$ | = | 1 | 4.1865 × 10${}^{-2}$ | = |

F19 | 0 | NaN | = | 1 | 2.7085 × 10${}^{-14}$ | + | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 1.6853 × 10${}^{-14}$ | = | 1 | 1.6853 × 10${}^{-14}$ | = |

F20 | 1 | 6.2958 × 10${}^{-4}$ | − | 1 | 5.4952 × 10${}^{-3}$ | = | 1 | 3.5049 × 10${}^{-13}$ | − | 1 | 7.8511 × 10${}^{-11}$ | − | 0 | 9.6974 × 10${}^{-1}$ | = |

F21 | 1 | 2.1150 × 10${}^{-6}$ | = | 1 | 6.7082 × 10${}^{-5}$ | + | 1 | 2.1150 × 10${}^{-6}$ | = | 1 | 1.9600 × 10${}^{-4}$ | + | 1 | 2.5975 × 10${}^{-2}$ | + |

F22 | 1 | 1.4331 × 10${}^{-4}$ | = | 1 | 1.2791 × 10${}^{-8}$ | + | 1 | 1.4992 × 10${}^{-5}$ | = | 1 | 1.1131 × 10${}^{-5}$ | + | 1 | 3.3861 × 10${}^{-8}$ | + |

F23 | 1 | 3.1216 × 10${}^{-4}$ | − | 1 | 2.0775 × 10${}^{-6}$ | + | 1 | 2.7674 × 10${}^{-3}$ | − | 1 | 2.7674 × 10${}^{-3}$ | − | 1 | 6.7273 × 10${}^{-7}$ | + |

+/−/= | 9/7/7 | 14/4/5 | 9/7/7 | 12/6/5 | 17/1/5 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, C.; Li, S.; Zhang, Y.; Chen, Z.; Zhang, C.
Enhancing Backtracking Search Algorithm using Reflection Mutation Strategy Based on Sine Cosine. *Algorithms* **2019**, *12*, 225.
https://doi.org/10.3390/a12110225

**AMA Style**

Zhou C, Li S, Zhang Y, Chen Z, Zhang C.
Enhancing Backtracking Search Algorithm using Reflection Mutation Strategy Based on Sine Cosine. *Algorithms*. 2019; 12(11):225.
https://doi.org/10.3390/a12110225

**Chicago/Turabian Style**

Zhou, Chong, Shengjie Li, Yuhe Zhang, Zhikun Chen, and Cuijun Zhang.
2019. "Enhancing Backtracking Search Algorithm using Reflection Mutation Strategy Based on Sine Cosine" *Algorithms* 12, no. 11: 225.
https://doi.org/10.3390/a12110225