# Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Bees Algorithm

#### Previous Modifications to the Bees Algorithm

#### 2.2. The Proposed Grouped Bees Algorithm

^{m}; Min < X < Max}. So, assuming a fitness function is defined as f(X): U → R, each potential solution (bee) is formulated as an m-dimensional vector of input variables X = {x

_{1}, ..., x

_{j}, ..., x

_{m}} where x

_{j}is between min

_{j}and max

_{j}, corresponding to the decision variable number j. The search neighbourhood for the whole set of bees belonging to the same group of i is defined as an m-dimensional ball with the radius vector of Ngh(i) = {ngh(i)

_{1}, ... ngh(i)

_{j}, ..., ngh(i)

_{m}} which is centered on the scout bees in the group i. In fact, each (recruited) bee will search on its own neighbourhood (patch), but the neighbourhood size is the same for all bees belonging to the same group. Now, considering the first rule, it is suggested each radius of ngh(i)

_{j}should have the following relationship with the group number, i:

^{2}, while the total number of all the scout bees who search either randomly or strategically should be equal to n. To that end, the area under the graph of k·x

^{2}over the interval [1, groups + 1] should sum up to n. So, the problem is finding a coefficient, k, which makes the definite integral of the function of f(x) = k·x

^{2}with respect to x from 1 to groups + 1 equal to n, i.e., to find k like so:

^{+}= max(x, 0). To reiterate, Figure 3 illustrates the pseudo code of the Grouped Bees Algorithm.

## 3. Results

#### 3.1. Benchmark Set

#### 3.2. Speed Evaluation

#### 3.3. Accuracy Evaluation

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Basic Bees Algorithm flowchart as reported in [48].

**Figure 2.**Pseudo code of Enhanced Bees Algorithm (EBA) as reported in [46].

**Figure 4.**Search Strategy of GBA and EBA [46] on Shekel Function.

Parameter | Definition |
---|---|

n | Total number of scout bees * |

groups | Total number of groups, excluding the random group * |

i | Group number (index) starting from 1 |

j | Decision input variable index from 1 to m |

g(i) | Number of scout bees (i.e., patches) in the i-th group |

g_{rnd} | Number of scout bees for the random group, who are searching randomly |

rec(i) | Number of recruited bees for patches in i-th group |

Ngh | Radius vector of the neighbourhood for bees in the first group * |

Ngh(i) | Radius vector of the neighbourhood for the i-th group |

ngh(i)_{j} | Radius for the j-th decision variable for bees in the i-the group |

Function | Interval | Equation | Global |
---|---|---|---|

1. Martin & Gaddy 2D | [0, 10] | $\mathrm{min}f({x}_{1},{x}_{2})={({x}_{1}-{x}_{2})}^{2}+{(\frac{{x}_{1}+{x}_{2}-10}{3})}^{2}$ | $\overrightarrow{x}=(5,5)\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

2. Branin 3D | [−5, 10] | $\mathrm{min}f({x}_{1},{x}_{2})=a({x}_{2}-b{{x}_{1}}^{2}+c{x}_{1}-d)\phantom{\rule{0ex}{0ex}}\text{}+e(1-f)\mathrm{cos}({x}_{1})+e\phantom{\rule{0ex}{0ex}}a=1,b=\frac{5.1}{4}{(\frac{7}{22})}^{2},c=\frac{5}{22}*7,d=6,e=10,f=\frac{1}{8}*\frac{7}{22}$ | $\overrightarrow{x}=(-\frac{22}{7},12.275)\phantom{\rule{0ex}{0ex}}\overrightarrow{x}=(\frac{22}{7},2.275)\phantom{\rule{0ex}{0ex}}\overrightarrow{x}=(\frac{66}{7},2.475)\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0.3977272$ |

3. Rosenbrock 4D | [−1.2, 1.2] | $\mathrm{min}f(\overrightarrow{x})={{\displaystyle \sum}}_{i=1}^{3}(100{({{x}_{i}}^{2}-{x}_{i+1})}^{2}+{(1-{x}_{i})}^{2}$) | $\overrightarrow{x}=(\overrightarrow{1})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

4. Hypersphere 6D | [−5.12, 5.12] | $\mathrm{min}f(\overrightarrow{x})={\displaystyle {\displaystyle \sum}_{i=1}^{6}}{{x}_{i}}^{2}$ | $\overrightarrow{x}=(\overrightarrow{0})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

5a. Rosenbrock 2D | [−10, 10] | $\mathrm{min}f({x}_{1},{x}_{2})=100{({{x}_{1}}^{2}-{x}_{2})}^{2}+{(1-{x}_{1})}^{2}$ | $\overrightarrow{x}=(\overrightarrow{1})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

5b. Rosenbrock 2D | [−1.2, 1.2] | $\mathrm{min}f({x}_{1},{x}_{2})=100{({{x}_{1}}^{2}-{x}_{2})}^{2}+{(1-{x}_{1})}^{2}$ | $\overrightarrow{x}=(\overrightarrow{1})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

5c. Rosenbrock 2D | [−2.048, 2.048] | $\mathrm{max}f({x}_{1},{x}_{2})=(3905.93)-100{({{x}_{1}}^{2}-{x}_{2})}^{2}-{(1-{x}_{1})}^{2}$ | $\overrightarrow{x}=(\overrightarrow{1})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=3905.93$ |

6. Goldstein & Price 2D | [−2, 2] | $\mathrm{min}f({x}_{1},{x}_{2})=A({x}_{1},{x}_{2})B({x}_{1},{x}_{2})\phantom{\rule{0ex}{0ex}}\begin{array}{c}A({x}_{1},{x}_{2})=1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{{x}_{1}}^{2}\\ \text{}-14{x}_{2}+6{x}_{1}{x}_{2}+3{{x}_{2}}^{2})\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{c}B({x}_{1},{x}_{2})=30+{(2{x}_{1}-3{x}_{2})}^{2}(18-32{x}_{1}+12{{x}_{1}}^{2}\\ \text{}+48{x}_{2}-36{x}_{1}{x}_{2}+27{{x}_{2}}^{2})\end{array}$ | $\overrightarrow{x}=(0,-1)\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=3$ |

7. Schwefel 6D | [−500, 500] | $\mathrm{max}f(\overrightarrow{x})={\displaystyle {\displaystyle \sum}_{i=1}^{6}}{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})$ | $\overrightarrow{x}=(\overrightarrow{420.9687})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})\approx -2513.9$ |

8. Shekel Foxholes 2D * | [−65.536, 65.536] | $\mathrm{min}f({x}_{1},{x}_{2})=-{\displaystyle {\displaystyle \sum}_{j=1}^{25}}\frac{1}{j+{({x}_{1}-{A}_{1j})}^{6}+{({x}_{2}-{A}_{2j})}^{6}}$ | $\overrightarrow{x}=(-32,-32)\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})\approx -1$ |

9. Steps 5D | [−5.12, 5.12] | $\mathrm{min}f(\overrightarrow{x})={\displaystyle {\displaystyle \sum}_{i=1}^{5}}\mathrm{int}({x}_{i})$ | $\overrightarrow{x}\in \overrightarrow{\left[-5.12,-5\right]}\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=-25$ |

10. Rosenbrock 5D | [−2.48, 2.48] | $\mathrm{min}f(\overrightarrow{x})={{\displaystyle \sum}}_{i=1}^{4}(100{({{x}_{i}}^{2}-{x}_{i+1})}^{2}+{(1-{x}_{i})}^{2}$) | $\overrightarrow{x}=(\overrightarrow{1})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

11. Griewangk 10D | [−600, 600] | $\mathrm{min}f(\overrightarrow{x})=\frac{1}{4000}.{\displaystyle {\displaystyle \sum}_{i=1}^{10}}{{x}_{i}}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{10}}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}})+1$ | $\overrightarrow{x}=(\overrightarrow{0})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

12. Rastrigin 20D | [−5.12, 5.12] | $\mathrm{min}f(\overrightarrow{x})={\displaystyle {\displaystyle \sum}_{i=1}^{20}}({{x}_{i}}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10)$ | $\overrightarrow{x}=(\overrightarrow{0})\phantom{\rule{0ex}{0ex}}f(\overrightarrow{x})=0$ |

**Table 3.**Mean number of evaluations of functions 1–7 for BA, EBA [46] and GBA.

Function | Mean Number of Evaluations | ||
---|---|---|---|

Basic BA | Enhanced BA | Grouped BA | |

1. Martin & Gaddy 2D | 526 | 124 | 114 |

2. Branin 3D | 1657 | 184 | 216 |

3. Rosenbrock 4D | 28,529 | 33,367 | 29,601 |

4. Hypersphere 6D | 7113 | 526 | 565 |

5a. Rosenbrock 2D | 2306 | 1448 | 1026 |

5b. Rosenbrock 2D | 631 | 689 | 580 |

5c. Rosenbrock 2D | 868 | 830 | 679 |

6. Goldstein & Price 2D | 999 | 212 | 273 |

7. Schwefel 6D | Approx. 3 × 10^{6} | N/A | Approx. 1 × 10^{6} |

Function | Percentage Change in GBA w.r.t. | |
---|---|---|

Basic BA | Enhanced BA | |

1. Martin & Gaddy 2D | −78.33% | −8.06% |

2. Branin 3D | −86.96% | 17.39% |

3. Rosenbrock 4D | 3.76% | −11.29% |

4. Hypersphere 6D | −92.06% | 7.41% |

5a. Rosenbrock 2D | −55.51% | −29.14% |

5b. Rosenbrock 2D | −8.08% | −15.82% |

5c. Rosenbrock 2D | −21.77% | −18.19% |

6. Goldstein & Price 2D | −72.67% | 28.77% |

Average | −51.45% | −3.62% |

**Table 5.**The parameters used by GBA in the speed experiment. * The Ngh vector has the same scalar value in all dimensions.

Function | n | Groups | Ngh * |
---|---|---|---|

1. Martin & Gaddy 2D | 6 | 3 | 0.13 |

2. Branin 3D | 8 | 3 | 0.05 |

3. Rosenbrock 4D | 4 | 3 | 0.001 |

4. Hypersphere 6D | 4 | 3 | 0.035 |

5a. Rosenbrock 2D | 5 | 3 | 0.11 |

5b. Rosenbrock 2D | 6 | 3 | 0.08 |

5c. Rosenbrock 2D | 4 | 3 | 0.09 |

6. Goldstein & Price 2D | 9 | 3 | 0.006 |

7. Schwefel 6D | 500 | 3 | 0.2 |

Function | Grouped BA | Standard BA | Modified BA | |||||
---|---|---|---|---|---|---|---|---|

Eval. | Iter. | Eval. | Iter. | Diff.% | Eval. | Iter. | Diff.% | |

5c. Rosenbrock 2D | 480 | 32 | 494 | 19 | 2.92% | 503 | 50 | 4.79% |

6. Goldstein & Price 2D | 1020 | 51 | 1040 | 40 | 1.96% | 1026 | 100 | 0.59% |

7. Schwefel 6D | 1972 | 16 | 2002 | 77 | 1.52% | 2011 | 200 | 1.98% |

8. Shekel Foxholes 2D | 988 | 19 | 1040 | 40 | 5.26% | 1026 | 100 | 3.85% |

9. Steps 5D | 120 | 8 | 130 | 5 | 8.33% | 126 | 10 | 5% |

10. Rosenbrock 5D | 1012 | 11 | 1040 | 40 | 2.77% | 1026 | 100 | 1.38% |

11. Griewangk 10D | 1020 | 68 | 1040 | 40 | 1.96% | 1026 | 100 | 0.59% |

12. Rastrigin 20D | 1015 | 35 | 1040 | 40 | 2.46% | 1026 | 100 | 1.08% |

Function | Grouped BA | Standard BA | Modified BA | |||
---|---|---|---|---|---|---|

Mean | Median | Mean | Median | Mean | Median | |

5c. Rosenbrock 2D | 0.0019 | 0.0002 | 0.0224 | 0.0078 | 0.0014 | 0.0003 |

6. Goldstein & Price 2D | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0011 | 0.0000 |

7. Schwefel 6D | 453.2246 | 454.0311 | 620.3443 | 572.2723 | 93.0656 | 73.4942 |

8. Shekel Foxholes 2D | 0.2397 | 0.0000 | 0.0213 | 0.0095 | 0.1525 | 0.0000 |

9. Steps 5D | 2.1500 | 2.0000 | 4.2000 | 4.5000 | 3.8500 | 4.0000 |

10. Rosenbrock 5D | 1.3319 | 1.1019 | 1.2090 | 1.3728 | 1.3038 | 0.854 |

11. Griewangk 10D | 0.9297 | 0.9483 | 1.0774 | 1.0887 | 1.0774 | 1.0718 |

12. Rastrigin 20D | 83.6515 | 79.322 | 116.2691 | 120.6735 | 83.7651 | 77.871 |

Wins | 5 | 3 | 3 |

**Table 8.**The parameters used by GBA in the accuracy experiment. * The Ngh vector has the same scalar value in all dimensions.

Function | n | Groups | Ngh * |
---|---|---|---|

5c. Rosenbrock 2D | 4 | 3 | 0.2 |

6. Goldstein & Price 2D | 9 | 3 | 0.006 |

7. Schwefel 6D | 20 | 6 | 0.5 |

8. Shekel Foxholes 2D | 40 | 2 | 0.3 |

9. Steps 5D | 4 | 3 | 3 |

10. Rosenbrock 5D | 7 | 6 | 0.02 |

11. Griewangk 10D | 4 | 3 | 10 |

12. Rastrigin 20D | 15 | 3 | 0.025 |

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**MDPI and ACS Style**

Nasrinpour, H.R.; Bavani, A.M.; Teshnehlab, M. Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm. *Computers* **2017**, *6*, 5.
https://doi.org/10.3390/computers6010005

**AMA Style**

Nasrinpour HR, Bavani AM, Teshnehlab M. Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm. *Computers*. 2017; 6(1):5.
https://doi.org/10.3390/computers6010005

**Chicago/Turabian Style**

Nasrinpour, Hamid Reza, Amir Massah Bavani, and Mohammad Teshnehlab. 2017. "Grouped Bees Algorithm: A Grouped Version of the Bees Algorithm" *Computers* 6, no. 1: 5.
https://doi.org/10.3390/computers6010005