# Patron–Prophet Artificial Bee Colony Approach for Solving Numerical Continuous Optimization Problems

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## Abstract

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## 1. Introduction

- The concept of donor–acceptor, termed Patron–Prophet, is introduced to the ABC using the scout bee strategy.
- A self-adaptive model is proposed to adapt the coefficient values based on the balance between intensification and diversification.
- The introduced model is evaluated with different mathematical benchmark problems and associated with other techniques to prove its significance.
- Along with standard performance metrics and statistical performance indicators, the Wilcoxon Signed Rank test is utilized to evaluate the significance.

## 2. Related Works

## 3. Patron–Prophet Artificial Bee Colony Algorithm

#### 3.1. Standard ABC

#### 3.1.1. Initialization

#### 3.1.2. Employee Bee Phase

#### 3.1.3. Probability Calculation

Algorithm 1: Computation of Probability Values for Every Solution. |

Begin for$\text{}i$ = 1: $\mathcal{F}$ perform $fi{t}_{i}=\{\begin{array}{c}\frac{1}{1+{f}_{i}},{f}_{i}\ge 0\\ 1+abs\left({f}_{i}\right),{f}_{i}0\end{array}\phantom{\rule{0ex}{0ex}}{P}_{i}=\frac{fi{t}_{i}}{{{\displaystyle \sum}}_{j=1}^{\mathcal{F}}fi{t}_{j}}$ end for End |

#### 3.1.4. Onlooker Bee Phase

#### 3.1.5. Scout Bee Phase

#### 3.2. Drawbacks of Standard ABC

#### 3.2.1. Non-Cooperative Behaviour in Scout Bee Phase

#### 3.2.2. Non-Balanced Diversification and Intensification

#### 3.3. Proposed Patron–Prophet ABC

#### 3.3.1. Patron–Prophet Strategy

#### 3.3.2. Self-Adaptability

Algorithm 2: PP-ABC. |

Input: Lower ${x}_{min}\text{}$ and upper ${x}_{max}\text{}$ bound of every dimension, # of the individual in a population ($\mathcal{F}$), the total number of dimensions (S), Population Initialization For i = 1: $\mathcal{F}$, do |

For j = 1: S, do |

Create ${x}_{i,j}$ individual |

${x}_{i,j}={x}_{min,j}\pm rand\text{}\left(0,1\right)\ast \left({x}_{max,j}-{x}_{min,j}\right)$ |

End for End for |

// Population fitness evaluation using Algorithm 1 |

$t=1$ |

Repeat |

{ // Employee individual strategy For each $\mathcal{F}$,$\text{}i\text{}$ do ${v}_{i,j}={x}_{i,j}+{\varnothing}_{i,j}\left({x}_{i,j}-{x}_{k,j}\right)$ |

Select between ${v}_{i}$ and ${x}_{i}$ End For |

// Onlooker individual strategy Set $r=0$ While ($r<=\mathcal{F}$) If$\text{}rand$(0,1) <${P}_{i}$ with Algorithm 1, then ${v}_{i,j}={x}_{i,j}+\alpha \text{}\left({x}_{i,j}-{x}_{k,j}\right)$ Select between ${v}_{i}$ and ${x}_{i}$ r = r + 1 End if End while |

// Scout individual strategy for $i=1\text{}tosize\left({\dot{\mathbb{D}}}_{t,K}\right)$ $\Delta X=\Vert \sqrt{\frac{{{\displaystyle \sum}}_{j=1}^{m}{\left({x}_{j}-{x}_{i}\right)}^{2}}{m}}\Vert $ where $m\text{}\in {\mathbb{Q}}_{t,K}$ ${x}_{i}={x}_{new}+\Delta {X}_{i}$ end |

Remember the best individual position obtained so far |

$t=t+1$ |

} |

Until ($t\le \text{}Ma{x}_{Iteration}$ ) |

#### 3.4. The Working Process of PP-ABC

## 4. Experimental Procedure and Result Analysis

#### 4.1. Experimental Setup

#### 4.1.1. Analysis of the Intensification Capability of PP-ABC

#### 4.1.2. Analysis of Diversification Capability of PP-ABC

#### 4.1.3. Analysis of Skipping Capability from Local Optima of PP-ABC

- A.
- Statistical analysis of the mathematical benchmark function results

^{-6}is rounded and represents 0.

#### 4.2. Time Complexity Analysis of Patron–Prophet ABC

- Initial phase: For population initialization, the time complexity is $O\left(\mathcal{F}\ast \mathrm{S}\right)$.
- Employee bee phase: In the employee bee phase, all the individuals take part in the computation of another individual and hence the time complexity of $O\left(\mathcal{F}\ast \mathrm{S}\right)$.
- Onlooker bee phase: Only in the onlooker bee phase, the selected individuals take part in the generation of solutions for the subsequent iterations, and hence the time complexity can be an average of $\frac{O\left(\mathcal{F}\ast \mathrm{S}\right)}{2}$ based on asymptotic notations, and it is expressed as $O\left(\mathcal{F}\ast \mathrm{S}\right)$.
- Scout bee phase: Only in the scout bee phase, the unimproved solutions are subject to improvisation. Since the balancing factor between intensification and diversification is handled efficiently, on every iteration, the scout bee phase obtains its computation half the way lower than the previous iteration. However, during each computation, all abandoned solutions act as a source of information for every newly generated key. Since every time the quantity of solution obtains half, we can define $T\left(\mathcal{F}\ast \mathrm{S}\right)$ as $\frac{T\left(\mathcal{F}\ast \mathrm{S}\right)}{2}$ and the computation in every turn of the scout bee phase is $\left(\mathcal{F}\ast \mathrm{S}\right)\mathrm{log}\left(\mathcal{F}\ast \mathrm{S}\right)$; the final statement is $T\left(\mathcal{F}\ast \mathrm{S}\right)$ and for the scout bee phase is $\frac{T\left(\mathcal{F}\ast \mathrm{S}\right)}{2}+\left(\mathcal{F}\ast \mathrm{S}\right)\mathrm{log}\left(\mathcal{F}\ast \mathrm{S}\right)$, which results in $O(\left(\mathcal{F}\ast \mathrm{S}\right){\mathrm{log}}^{2}\left(\mathcal{F}\ast \mathrm{S}\right))$.
- Fitness computation: The computational complexity for the fitness calculation is $O(\mathcal{F})$.

#### 4.3. Three-Bar Truss Design Optimization Problem

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Convergence rate of the (

**a**) F1 and (

**b**) F2 benchmark functions. Convergence rate of the (

**c**) F3, (

**d**) F4, (

**e**) F5, (

**f**) F6, (

**g**) F7, and (

**h**) F8 benchmark functions. Convergence rate of the (

**i**) F9, (

**j**) F10, (

**k**) F11, (

**l**) F12, (

**m**) F13, and (

**n**) F14 benchmark functions. Convergence rate of the (

**o**) F15 benchmark function.

Type | Method |
---|---|

Individuals in a population | 30 |

$Dimension\text{}\left(D\right)$ | 10 & 30 |

Termination Criteria ($Ma{x}_{Iteration}$) | 1000 × D |

Runs | 25 |

$C$ | 1 |

$\phi $ | 2 |

$\alpha $ | 0.1 (initially) |

$\omega $ | 2 |

Function | Mathematical Formulation | Global Optimum | Range |
---|---|---|---|

F1 | ${\displaystyle \sum}_{i=1}^{D}}{z}_{i}^{2}\text{},z=X-O,\text{}O=\left[{O}_{1},{O}_{2},\text{}\dots ,\text{}{O}_{D}\right]$ | 0 | [−100, 100]D |

F2 | ${\displaystyle \sum}_{i=1}^{D}}{\left({\displaystyle {{\displaystyle \sum}}_{j=1}^{i}}{z}_{i}\right)}^{2}\text{},\text{}z=X-O,\text{}O=\left[{O}_{1},{O}_{2},\text{}\dots ,\text{}{O}_{D}\right]$ | 0 | [−100, 100]D |

F3 | ${\displaystyle \sum}_{i=1}^{D-1}}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right],$ | (1, 1 …, 1) | [−100, 100]D |

F4 | ${\displaystyle \sum}_{i=1}^{D}}{\left({\displaystyle {{\displaystyle \sum}}_{j=1}^{i}}{z}_{i}\right)}^{2}\left(1+0.4\left|N\left(0,1\right)\right|\right)\text{},\text{}z=X-O$ | 0 | [−100, 100]D |

F5 | $-20\text{}exp\left(-0.2\text{}\sqrt{\frac{1}{D}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{z}_{i}^{D}}\right)-exp\left(\frac{1}{D}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}\mathrm{cos}2\pi {z}_{i}\right)+20+e,\text{}z=X-O$ | 0 | [−32, 32]D |

F6 | $-20\text{}exp\left(-0.2\text{}\sqrt{\frac{1}{D}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{z}_{i}^{D}}\right)-exp\left(\frac{1}{D}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}\mathrm{cos}2\pi {z}_{i}\right)+20+e,\text{}z=M\left(X-O\right),cond\left(M\right)=1$ | 0 | [−32, 32]D |

F7 | $\frac{1}{400}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{z}_{i}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{D}}cos\left(\frac{{z}_{i}}{\sqrt{i}}\right)+1z=X-O$ | 0 | [0, 600]D |

F8 | $\frac{1}{400}{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{z}_{i}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{D}}cos\left(\frac{{z}_{i}}{\sqrt{i}}\right)+1,z=M\left(X-O\right),cond\left(M\right)=3$ | 0 | [0, 600]D |

F9 | ${\displaystyle \sum}_{i=1}^{D}}\left[{z}_{i}^{2}-10\mathrm{cos}\left(2\pi {z}_{i}\right)+10\right],\text{}z=X-O$ | 0 | [−5, 5]D |

F10 | ${\displaystyle \sum}_{i=1}^{D}}\left[{z}_{i}^{2}-10\mathrm{cos}\left(2\pi {z}_{i}\right)+10\right],\text{}z=M\left(X-O\right),cond\left(M\right)=2$ | 0 | [−5, 5]D |

F11 | $418.9828\ast D-{\displaystyle {\displaystyle \sum}_{i=1}^{D}}{x}_{i}\mathrm{sin}\left({\left|{x}_{i}\right|}^{\frac{1}{2}}\right)$ | (420.96, …, 420.96) | [−500, 500]D |

F12 | $\frac{\pi}{D}\left\{10\text{}si{n}^{2}\left(\pi {y}_{i}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{D-1}}{\left({y}_{i}-1\right)}^{2}\left[1+si{n}^{2}\left(\pi {y}_{i}+1\right)\right]+{\left(yD-1\right)}^{2}+{\displaystyle {\displaystyle \sum}_{i=1}^{D}}u\left({x}_{i},10,100,4\right)\right\}$ $y=1+\frac{{x}_{i+1}}{4}\text{}u\left({x}_{i}a,k,m\right)=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0-a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}$ | 0 | [−50, 50]D |

F13 | $0.1\left\{10\text{}si{n}^{2}\left(\pi {y}_{i}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{D-1}}{\left({y}_{i}-1\right)}^{2}\left[1+10si{n}^{2}\left(\pi {y}_{i}+1\right)\right]+{\left(yD-1\right)}^{2}+{\displaystyle {\displaystyle \sum}_{i=1}^{D}}u\left({x}_{i},10,100,4\right)\right\}$ | 0 | [−50, 50]D |

F14 | Ten sphere functions | 0 | [−5, 5]D |

F15 | Ten different benchmark functions (i.e., 2 rotated Rastrigin’s procedures, 2 rotated Weier stress functions, 2 rotated Griewank’s procedures, 2 rotated Ackley’s procedures, and two turned Sphere functions) | 0 | [−5, 5]D |

Patron–Prophet | Self-Adaptability | |||||
---|---|---|---|---|---|---|

Min | Mean | Std.dev. | Min | Mean | Std.dev. | |

F1 | 0 | 0 | 0 | 0 | 0 | 0 |

F2 | 0 | 0 | 0 | 1.26 × 10^{0} | 3.45 × 10^{0} | 3.56 × 10^{0} |

F3 | 1.13 × 10^{−6} | 3.11 × 10^{−6} | 1.34 × 10^{−8} | 5.22 × 10^{−5} | 3.26 × 10^{−1} | 7.27 × 10^{−1} |

F4 | 4.76 × 10^{−10} | 7.90 × 10^{−4} | 3.24 × 10^{−6} | 5.87 × 10^{−8} | 8.01 × 10^{−6} | 4.36 × 10^{−2} |

F5 | 0 | 8.65 × 10^{−14} | 4.62 × 10^{−16} | 0 | 1.14 × 10^{−14} | 9.62 × 10^{−12} |

F6 | 6.83 × 10^{−6} | 3.76 × 10^{−5} | 4.60 × 10^{−6} | 7.94 × 10^{−6} | 4.87 × 10^{−6} | 5.60 × 10^{−5} |

F7 | 0 | 6.47 × 10^{−8} | 3.70 × 10^{−10} | 0 | 7.58 × 10^{−9} | 4.71 × 10^{−9} |

F8 | 3.72 × 10^{−5} | 9.26 × 10^{−1} | 2.53 × 10^{−2} | 4.83 × 10^{−4} | 6.37 × 10^{−3} | 3.64 × 10^{−1} |

F9 | 0 | 0 | 0 | 0 | 0 | 0 |

F10 | 4.86 × 10^{0} | 6.45 × 10^{0} | 3.76 × 10^{0} | 5.12 × 10^{0} | 6.21 × 10^{0} | 4.12 × 10^{0} |

F11 | 0 | 0 | 0 | 0 | 0 | 0 |

F12 | 0 | 2.09 × 10^{−32} | 3.75 × 10^{−32} | 0 | 3.10 × 10^{−31} | 4.77 × 10^{−28} |

F13 | 2.23 × 10^{−42} | 2.76 × 10^{−30} | 2.61 × 10^{−25} | 3.34 × 10^{−40} | 3.87 × 10^{−32} | 3.72 × 10^{−32} |

F14 | 6.97 × 10^{−6} | 6.75 × 10^{−6} | 2.70 × 10^{−6} | 9.66 × 10^{−7} | 3.85 × 10^{−8} | 5.42 × 10^{−6} |

F15 | 1.62 × 10^{−1} | 3.65 × 10^{−1} | 1.15 × 10^{0} | 7.23 × 10^{−1} | 5.44 × 10^{−1} | 6.21 × 10^{0} |

PP-ABC | DGABC | APABC | ABC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | |

F1 | 0 | 0 | 0 | 0 | 2.22 × 10^{−21} | 2.0 × 10^{−21} | 0 | 0 | 0 | 0 | 0 | 0 |

F2 | 0 | 0 | 0 | 0 | 1.47 × 10^{−10} | 6.94 × 10^{−11} | 2.66 × 10^{−2} | 9.14 × 10^{−2} | 6.99 × 10^{−2} | 2.46 × 10^{0} | 5.75 × 10^{0} | 2.54 × 10^{0} |

F3 | 4.58 × 10^{−8} | 2.56 × 10^{−2} | 5.24 × 10^{−2} | 3.97 × 10^{0} | 4.55 × 10^{0} | 1.96 × 10^{0} | 1.03 × 10^{−2} | 7.21 × 10^{−1} | 6.21 × 10^{−1} | 8.57 × 10^{−3} | 3.97 × 10^{−1} | 2.85 × 10^{−1} |

F4 | 3.65 × 10^{−12} | 6.89 × 10^{−5} | 2.14 × 10^{−5} | 9.86 × 10^{−7} | 4.54 × 10^{−5} | 2.51 × 10^{−5} | 8.55 × 10^{−1} | 3.21× 10^{−1} | 2.39× 10^{−1} | 2.54 × 10^{2} | 4.25 × 10^{2} | 2.11 × 10^{2} |

F5 | 0 | 9.54 × 10^{−16} | 3.51 × 10^{−18} | 6.90 × 10^{−7} | 5.35 × 10^{−5} | 2.00 × 10^{−3} | 0 | 5.65 × 10^{−15} | 1.90 × 10^{−15} | 5.45 × 10^{−17} | 7.65× 10^{−15} | 2.50 × 10^{−14} |

F6 | 5.72 × 10^{−8} | 2.65 × 10^{−7} | 3.59 × 10^{−7} | 1.14 × 10^{−9} | 2.61 × 10^{−8} | 3.75 × 10^{−8} | 5.27 × 10^{−8} | 3.52 × 10^{−7} | 5.95 × 10^{−7} | 2.65 × 10^{−1} | 3.52 × 10^{−1} | 4.52 × 10^{−1} |

F7 | 0 | 5.36 × 10^{−10} | 2.69 × 10^{−10} | 5.12 × 10^{−2} | 7.64 × 10^{−2} | 2.99 × 10^{−2} | 7.66 × 10^{−8} | 2.02 × 10^{−7} | 2.65 × 10^{−7} | 5.74 × 10^{−4} | 2.65 × 10^{−3} | 4.96 × 10^{−3} |

F8 | 2.61 × 10^{−5} | 8.15 × 10^{−2} | 1.42 × 10^{−2} | 7.59 × 10^{−2} | 9.55 × 10^{−2} | 2.65 × 10^{−2} | 1.16 × 10^{−1} | 1.21 × 10^{−1} | 4.65 × 10^{−2} | 3.86 × 10^{−2} | 8.65 × 10^{−2} | 3.75 × 10^{−2} |

F9 | 0 | 0 | 0 | 5.21 × 10^{0} | 6.75 × 10^{0} | 2.01 × 10^{0} | 0 | 0 | 0 | 0 | 0 | 0 |

F10 | 4.35 × 10^{0} | 7.36 × 10^{0} | 2.65 × 10^{0} | 1.21 × 10^{1}E+01 | 1.45 × 10^{1} | 3.55 × 10^{0} | 4.27 × 10^{0} | 9.95 × 10^{0} | 1.97 × 10^{0} | 1.19 × 10^{1} | 3.26 × 10^{1} | 1.30 × 10^{1} |

F11 | 0 | 0 | 0 | 2.21 × 10^{2} | 3.93 × 10^{2} | 1.92 × 10^{2} | 0 | 0 | 0 | 0 | 0 | 0 |

F12 | 0 | 1.98× 10^{0} | 2.65 × 10^{−40} | 9.55 × 10^{−17} | 6.93 × 10^{−16} | 2.33 × 10^{−15} | 0 | 4.82 × 10^{−32} | 1.65 × 10^{−46} | 1.26 × 10^{−32} | 4.99 × 10^{−32} | 4.42 × 10^{−32} |

F13 | 1.12 × 10^{−46} | 1.65 × 10^{−27} | 1.50 × 10^{−26} | 1.15 × 10^{−19} | 1.75 × 10^{−19} | 3.54 × 10^{−19} | 0 | 1.89 × 10^{−3} | 1.05 × 10^{−32} | 0.00 × 10^{0} | 1.66 × 10^{−32} | 2.97 × 10^{−48} |

F14 | 7.86 × 10^{−8} | 5.64 × 10^{−7} | 1.69 × 10^{−7} | 2.66 × 10^{−7} | 4.75 × 10^{−7} | 1.85 × 10^{−6} | 4.79 × 10^{−3} | 2.55 × 10^{−2} | 6.97 × 10^{−2} | 1.55 × 10^{−4} | 3.85 × 10^{−4} | 1.20 × 10^{−3} |

F15 | 9.57 × 10^{−2} | 2.54 × 10^{−1} | 9.85 × 10^{−1} | 5.48 × 10^{−1} | 2.01 × 10^{0} | 9.55 × 10^{−1} | 1.88 × 10^{0} | 5.94 × 10^{0} | 3.98 × 10^{0} | 1.36 × 10^{1} | 1.59 × 10^{1} | 6.46 × 10^{0} |

ACoM-ABC | SABC-SG | KFABC | MPABC | |||||||||

Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | |

F1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

F2 | 1.12 × 10^{−23} | 1.55 × 10^{−23} | 3.56 × 10^{−23} | 0 | 0 | 0 | 2.61 × 10^{−12} | 6.52 × 10^{−12} | 2.64 × 10^{−13} | 0 | 0 | 0 |

F3 | 1.64 × 10^{−7} | 2.45 × 10^{−7} | 7.34 × 10^{−8} | 2.69 × 10^{−7} | 2.45 × 10^{−6} | 7.34 × 10^{−7} | 2.57 × 10^{−6} | 7.32 × 10^{−6} | 9.82 × 10^{−7} | 8.37 × 10^{−6} | 5.47 × 10^{−5} | 4.57 × 10^{−6} |

F4 | 4.62 × 10^{−21} | 9.68 × 10^{−21} | 3.52 × 10^{−22} | 5.92 × 10^{−18} | 8.72 × 10^{−18} | 2.32 × 10^{−19} | 6.47 × 10^{−10} | 8.12 × 10^{−10} | 6.25 × 10^{−11} | 3.97 × 10^{−7} | 5.47 × 10^{−6} | 2.64 × 10^{−7} |

F5 | 0 | 0 | 0 | 6.38 × 10^{−13} | 9.42 × 10^{−12} | 4.25 × 10^{−13} | 2.46 × 10^{−11} | 3.64 × 10^{−11} | 9.24 × 10^{−12} | 4.57 × 10^{−15} | 6.54 × 10^{−15} | 9.87 × 10^{−16} |

F6 | 3.62 × 10^{−15} | 3.62 × 10^{−15} | 0 | 4.62 × 10^{−3} | 4.62 × 10^{−3} | 0 | 2.64 × 10^{−2} | 7.58 × 10^{−2} | 5.62 × 10^{−2} | 6.42 × 10^{−3} | 7.24 × 10^{−3} | 4.68× 10^{−4} |

F7 | 0 | 0 | 0 | 6.25 × 10^{−6} | 8.27 × 10^{−5} | 6.24 × 10^{−5} | 5.12 × 10^{−2} | 8.36 × 10^{−2} | 6.25 × 10^{−3} | 2.64 × 10^{−2} | 5.92 × 10^{−2} | 5.14 × 10^{−2} |

F8 | 2.47 × 10^{−2} | 5.24 × 10^{−2} | 2.40 × 10^{−2} | 6.30 × 10^{−2} | 7.21 × 10^{−1} | 2.61 × 10^{−2} | 2.47 × 10^{−1} | 5.93 × 10^{−1} | 6.42 × 10^{−1} | 2.61 × 10^{−2} | 8.15 × 10^{−2} | 1.42 × 10^{−3} |

F9 | 0 | 0 | 0 | 2.62 × 10^{0} | 5.84 × 10^{0} | 2.14 × 10^{0} | 4.95 × 10^{0} | 1.26 × 10^{1} | 7.35 × 10^{0} | 0 | 0 | 0 |

F10 | 8.24 × 10^{0} | 1.27 × 10^{1} | 2.70 × 10^{0} | 1.26 × 10^{1} | 2.74 × 10^{1} | 1.64 × 10^{1} | 1.62 × 10^{1} | 2.94 × 10^{1} | 1.57 × 10^{1} | 1.50 × 10^{1} | 2.65 × 10^{1} | 1.43 × 10^{1} |

F11 | 1.40 × 10^{2} | 2.28 × 10^{2} | 4.26 × 10^{1} | 2.67 × 10^{2} | 3.64 × 10^{2} | 1.24 × 10^{2} | 0 | 0 | 0 | 2.64 × 10^{−16} | 7.65 × 10^{−16} | 5.61 × 10^{−16} |

F12 | 3.67 × 10^{−32} | 5.65 × 10^{−32} | 1.96 × 10^{−47} | 6.47 × 10^{−16} | 7.57 × 10^{−15} | 5.47 × 10^{−16} | 2.28 × 10^{−24} | 2.28 × 10^{−24} | 0 | 3.47 × 10^{−32} | 6.49 × 10^{−32} | 4.62 × 10^{−38} |

F13 | 1.74 × 10^{−32} | 2.64 × 10^{−32} | 2.52 × 10^{−48} | 1.82 × 10^{−16} | 5.62 × 10^{−16} | 3.43 × 10^{−32} | 1.54 × 10^{−19} | 4.62 × 10^{−19} | 7.52 × 10^{−20} | 2.64 × 10^{−24} | 7.53 × 10^{−24} | 2.67 × 10^{−25} |

F14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5.62 × 10^{−2} | 8.62 × 10^{−2} | 1.69 × 10^{−3} |

F15 | 1.70 × 10^{−1} | 5.20 × 10^{−1} | 9.10 × 10^{−1} | 6.40 × 10^{−1} | 7.60 × 10^{−1} | 2.60 × 10^{−2} | 5.61 × 10^{0} | 1.25 × 10^{1} | 4.64 × 10^{0} | 1.24 × 10^{0} | 5.62 × 10^{0} | 2.50 × 10^{0} |

PP-ABC | DGABC | APABC | ABC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | |

F1 | 0 | 0 | 0 | 2.21 × 10^{−24} | 2.87 × 10^{−23} | 2.65 × 10^{−23} | 0 | 0 | 0 | 0 | 0 | 0 |

F2 | 1.26 × 10^{−1} | 4.75 × 10^{−1} | 1.47 × 10^{1} | 1.09 × 10^{−1} | 3.55 × 10^{−1} | 2.46 × 10^{−1} | 6.39 × 10^{2} | 7.85 × 10^{2} | 1.45 × 10^{2} | 2.07 × 10^{3} | 3.21 × 10^{3} | 1.15 × 10^{3} |

F3 | 1.97 × 10^{−2} | 6.75 × 10^{−1} | 6.55 × 10^{−1} | 1.36 × 10^{1} | 2.46 × 10^{1} | 7.45 × 10^{0} | 6.93× 10^{−1} | 4.55 × 10^{0} | 3.85 × 10^{0} | 6.87 × 10^{−6} | 5.47 × 10^{−4} | 3.46× 10^{−5} |

F4 | 5.48 × 10^{2} | 1.56 × 10^{3} | 5.68 × 10^{2} | 1.27 × 10^{3} | 2.13 × 10^{3} | 8.55 × 10^{2} | 6.86× 10^{3} | 7.96 × 10^{3} | 1.13 × 10^{3} | 2.32 × 10^{4} | 2.87 × 10^{4} | 5.48 × 10^{3} |

F5 | 1.64 × 10^{−14} | 1.95 × 10^{−14} | 3.15 × 10^{−15} | 5.82 × 10^{−3} | 7.46 × 10^{−2} | 2.11 × 10^{−1} | 2.96 × 10^{−25} | 5.70 × 10^{−24} | 6.98 × 10^{−23} | 5.48 × 10^{−16} | 3.48 × 10^{−15} | 3.66 × 10^{−15} |

F6 | 5.87 × 10^{−15} | 5.87 × 10^{−15} | 0 | 6.48 × 10^{−11} | 2.66 × 10^{−10} | 9.02 × 10^{−10} | 5.78 × 10^{−4} | 3.25 × 10^{−3} | 3.25 × 10^{−3} | 1.71× 10^{1} | 1.80 × 10^{1} | 8.65 × 10^{−1} |

F7 | 0 | 0 | 0 | 2.87 × 10^{−18} | 1.66 × 10^{−17} | 5.70 × 10^{−17} | 4.69 × 10^{−14} | 2.55 × 10^{−17} | 7.54 × 10^{−17} | 0 | 0 | 0 |

F8 | 0 | 0 | 0 | 1.02 × 10^{−3} | 1.56 × 10^{−3} | 2.58 × 10^{−3} | 9.87 × 10^{−4} | 3.59 × 10^{−2} | 1.99 × 10^{−2} | 3.29 × 10^{−5} | 1.99 × 10^{−4} | 1.66 × 10^{−4} |

F9 | 0 | 0 | 0 | 4.28 × 10^{1} | 4.94 × 10^{1} | 6.55 × 10^{0} | 0 | 0 | 0 | 0 | 0 | 0 |

F10 | 1.97 × 10^{1} | 5.75 × 10^{1} | 2.69 × 10^{1} | 1.13 × 10^{2} | 1.28 × 10^{2} | 1.54 × 10^{1} | 7.80 × 10^{1}1 | 9.47 × 10^{1} | 1.67 × 10^{1} | 2.67 × 10^{2} | 2.96 × 10^{2} | 2.97 × 10^{1} |

F11 | 0 | 0 | 0 | 3.09 × 10^{3} | 3.66 × 10^{3} | 4.12 × 10^{2} | 5.72 × 10^{−14} | 1.99 × 10^{−13} | 6.11 × 10^{−13} | 9.72 × 10^{−13} | 1.54 × 10^{−12} | 5.70 × 10^{−13} |

F12 | 5.43 × 10^{−56} | 5.48 × 10^{−56} | 1.75 × 10^{−64} | 1.15 × 10^{−2} | 2.55 × 10^{−2} | 3.70 × 10^{−2} | 4.63 × 10^{−32} | 2.66 × 10^{−31} | 2.70 × 10^{−31} | 1.7 × 10^{−32} | 1.70 × 10^{−32} | 5.69 × 10^{−49} |

F13 | 1.76 × 10^{−17} | 1.78 × 10^{−17} | 2.46 × 10^{−27} | 3.56 × 10^{−17} | 1.57 × 10^{−17} | 5.13 × 10^{−17} | 2.87 × 10^{−31} | 1.60 × 10^{−30} | 1.31 × 10^{−30} | 1.52 × 10^{−32} | 1.52 × 10^{−32} | 2.66 × 10^{−48} |

F14 | 0 | 0 | 0 | 2.09 × 10^{−13} | 5.68 × 10^{−14} | 2.66 × 10^{−13} | 4.95 × 10^{−7} | 1.25 × 10^{−6} | 3.01 × 10^{−6} | 0 | 0 | 0 |

F15 | 4.87 × 10^{0} | 1.60 × 10^{1} | 5.69 × 10^{0} | 1.55 × 10^{1} | 2.07 × 10^{1} | 5.15 × 10^{0} | 5.58 × 10^{0} | 7.57 × 10^{0} | 1.99 × 10 ^{0} | 1.07 × 10^{0} | 1.36 × 10^{1} | 2.87 × 10^{0} |

ACoM-ABC | SABC-SG | KFABC | MPABC | |||||||||

Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | Min | Mean | Std.dev | |

F1 | 0 | 0 | 0 | 0 | 0 | 0 | 4.22 × 10^{−6} | 5.24 × 10^{−6} | 3.66 × 10^{−6} | 0 | 0 | 0 |

F2 | 2.57 × 10^{−5} | 9.46 × 10^{−5} | 7.88 × 10^{−5} | 2.54 × 10^{0} | 1.82 × 10^{1} | 1.45 × 10^{1} | 3.64 × 10^{0} | 2.16 × 10^{1} | 1.65 × 10^{1} | 2.65 × 10^{0} | 3.54 × 10 ^{0} | 1.25 × 10^{0} |

F3 | 5.54 × 10^{−2} | 5.66 × 10^{−2} | 3.52 × 10^{−3} | 5.47 × 10^{0} | 6.54 × 10^{0} | 1.25 × 10^{0} | 8.37 × 10^{0} | 9.54 × 10^{0} | 2.54 × 10^{0} | 0 | 2.65 × 10^{−30} | 5.82 × 10^{−30} |

F4 | 1.25 × 10^{3} | 1.6 × 10^{3} | 2.0× 10^{2} | 8.5 × 10^{2} | 1.46× 10^{3} | 5.36 × 10^{3} | 9.47 × 10^{3} | 1.76 × 10^{3} | 6.87 × 10^{3} | 9.56 × 10^{2} | 1.53 × 10^{3} | 2.74 × 10^{2} |

F5 | 2.54 × 10^{−13} | 4.89 × 10^{−13} | 1.96 × 10^{−13} | 5.65 × 10^{−9} | 8.24 × 10^{−9} | 3.54 × 10^{−9} | 3.65 × 10^{−10} | 6.74 × 10^{−10} | 4.74 × 10^{−11} | 7.25 × 10^{−10} | 9.15 × 10^{−9} | 4.25 × 10^{−9} |

F6 | 4.13 × 10^{−15} | 4.13 × 10^{−15} | 0 | 2.47 × 10^{−11} | 7.41 × 10^{−11} | 4.21 × 10^{−12} | 4.57 × 10^{−9} | 4.57 × 10^{−9} | 0 | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 0 |

F7 | 0 | 0 | 0 | 3.65 × 10^{−12} | 5.96 × 10^{−12} | 2.34 × 10^{−13} | 0 | 0 | 0 | 0 | 0 | 0 |

F8 | 0 | 0 | 0 | 7.00 × 10^{−5} | 9.87 × 10^{−5} | 6.47 × 10^{−6} | 7.00 × 10^{−5} | 9.87 × 10^{−5} | 6.47 × 10^{−6} | 0 | 0 | 0 |

F9 | 0 | 0 | 0 | 1.75 × 10^{1} | 2.98 × 10^{1} | 1.24 × 10^{1} | 2.14 × 10^{1} | 3.65 × 10^{−1} | 1.15 × 10^{1} | 0 | 0 | 0 |

F10 | 8.15 × 10^{1} | 9.16 × 10^{−1} | 2.04 × 10^{1} | 4.60 × 10^{1} | 7.85 × 10^{1} | 1.16 × 10^{1} | 0 | 0 | 0 | 5.42 × 10^{1} | 6.51 × 10 ^{1} | 2.15 × 10^{1} |

F11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

F12 | 2.65 × 10^{−32} | 2.88 × 10^{−32} | 2.34 × 10^{−47} | 2.64 × 10^{−26} | 6.57 × 10^{−26} | 7.68 × 10^{−42} | 3.65 × 10^{−18} | 8.24 × 10^{−18} | 5.43 × 10^{−32} | 1.57 × 10^{−32} | 1.57 × 10^{−32} | 5.24 × 10^{−48} |

F13 | 1.66 × 10^{−16} | 2.68 × 10^{−16} | 2.70 × 10^{−25} | 6.92 × 10^{−8} | 9.38 × 10^{−8} | 1.26 × 10^{−10} | 4.28 × 10^{−10} | 6.47 × 10^{−10} | 2.67 × 10^{−11} | 5.47 × 10^{−12} | 8.75 × 10^{−12} | 1.11 × 10^{−12} |

F14 | 0 | 0 | 0 | 5.97 × 10^{−12} | 7.54 × 10^{−12} | 1.62 × 10^{−12} | 2.21 × 10^{−14} | 4.92 × 10^{−14} | 1.21 × 10^{−15} | 0 | 0 | 0 |

F15 | 1.05 × 10^{1} | 1.27 × 10^{1} | 3.25 | 1.86 × 10^{1} | 5.72 × 10^{1} | 2.65 × 10^{1} | 2.13 × 10^{1} | 6.41 × 10^{1} | 2.67 × 10 ^{1} | 7.54 × 10 ^{0} | 1.25 × 10 ^{1} | 3.21 × 10^{0} |

Function | PP-ABC vs DGABC | PP-ABC vs APABC | PP-ABC vs ABC | |||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | |||||||||||||||||

F1 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | ||||||||||||||||

F2 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F3 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F4 | 3.38 × 10^{−3} | 375 | 90 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F5 | 0 | 0 | 465 | + | 2.88 × 10^{−6} | 5 | 460 | + | 0 | 0 | 465 | + | ||||||||||||||||

F6 | 0 | 465 | 0 | − | 6.42 × 10^{−3} | 100 | 365 | + | 0 | 0 | 465 | + | ||||||||||||||||

F7 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F8 | 1.83 × 10^{−3} | 81 | 384 | + | 1.80 × 10^{−5} | 24 | 441 | + | 4.68 × 10^{−3} | 95 | 370 | + | ||||||||||||||||

F9 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | ||||||||||||||||

F10 | 0 | 0 | 465 | + | 4.11 × 10^{−3} | 93 | 372 | + | 0 | 0 | 465 | + | ||||||||||||||||

F11 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | ||||||||||||||||

F12 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F13 | 0 | 0 | 465 | + | 0 | 465 | 0 | − | 0 | 465 | 0 | − | ||||||||||||||||

F14 | 3.7 × 10 ^{−2} | 276 | 189 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

F15 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||||||

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

Function | PP-ABC vs ACoM-ABC | PP-ABC vs SABC-SG | PP-ABC vs KFABC | PP-ABC vs MPABC | ||||||||||||||||||||||||

p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | |||||||||||||

F1 | 1 | 0 | 0 | = | 1 | 0 | 0 | = | 1 | 0 | 0 | = | 1 | 0 | 0 | = | ||||||||||||

F2 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 1 | 0 | 0 | = | ||||||||||||

F3 | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 0 | 465 | 0 | − | ||||||||||||

F4 | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 3.52 × 10^{−4} | 458 | 7 | − | ||||||||||||

F5 | 0 | 465 | 0 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F6 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F7 | 0 | 465 | 0 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F8 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F9 | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | ||||||||||||

F10 | 2.35 × 10^{−6} | 3 | 462 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F11 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 0 | 0 | 465 | + | ||||||||||||

F12 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F13 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

F14 | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 0 | 465 | 0 | − | 0 | 0 | 465 | + | ||||||||||||

F15 | 5.75 × 10^{−6} | 12 | 453 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | ||||||||||||

+/=/− | 8/2/5 | 10/2/3 | 10/2/3 | 10/3/2 |

Function | PP-ABC vs. DGABC | PP-ABC vs. APABC | PP-ABC vs. ABC | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | ||||||||||||||||

F1 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | |||||||||||||||

F2 | 1.65 × 10^{-1} | 300 | 165 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F3 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F4 | 1.48 × 10^{-4} | 48 | 417 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F5 | 0 | 0 | 465 | + | 0 | 465 | 0 | − | 0 | 0 | 465 | + | |||||||||||||||

F6 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||||||

F7 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 465 | 0 | − | |||||||||||||||

F8 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F9 | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | |||||||||||||||

F10 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F11 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F12 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||||||

F13 | 6.44 × 10^{−1} | 255 | 210 | − | 0 | 465 | 0 | − | 0 | 465 | 0 | − | |||||||||||||||

F14 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||||||

F15 | 9.84 × 10^{−2} | 107 | 358 | + | 6.98 × 10^{−6} | 451 | 14 | − | 1.04 × 10^{−2} | 357 | 108 | − | |||||||||||||||

+/=/− | 13/0/2 | 10/2/3 | 9/3/3 | ||||||||||||||||||||||||

Function | PP-ABC vs. ACoM-ABC | PP-ABC vs. SABC-SG | PP-ABC vs. KFABC | PP-ABC vs. MPABC | |||||||||||||||||||||||

p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | p-Value | T+ | T− | Winner | ||||||||||||

F1 | 1 | 0 | 0 | = | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||

F2 | 2.83 × 10^{−4} | 56 | 409 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||

F3 | 0 | 465 | 0 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 465 | 0 | − | |||||||||||

F4 | 1.71 × 10^{−1} | 166 | 299 | + | 2.18 × 10^{−2} | 344 | 121 | − | 4.07 × 10^{−2} | 133 | 322 | + | 5.44 × 10^{−1} | 203 | 262 | + | |||||||||||

F5 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||

F6 | 0 | 465 | 0 | - | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 465 | 0 | − | |||||||||||

F7 | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 1 | 0 | 0 | = | 1 | 0 | 0 | = | |||||||||||

F8 | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||

F9 | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||

F10 | 1.92 × 10^{−6} | 1 | 464 | + | 2.22 × 10^{−4} | 53 | 412 | + | 0 | 465 | 0 | − | 3.61 × 10^{−1} | 151 | 314 | + | |||||||||||

F11 | 1 | 0 | 0 | = | 1 | 0 | 0 | = | 1 | 0 | 0 | = | 1 | 0 | 0 | = | |||||||||||

F12 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||

F13 | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 0 | 0 | 465 | + | |||||||||||

F14 | 1 | 0 | 0 | = | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 1 | 0 | 0 | = | |||||||||||

F15 | 1.24 × 10^{−5} | 345 | 120 | − | 0 | 0 | 465 | + | 0 | 0 | 465 | + | 2.85 × 10^{−2} | 339 | 126 | − | |||||||||||

+/=/− | 6/6/3 | 12/2/1 | 12/2/1 | 6/6/3 |

**Table 8.**Category-based comparison for the proposed PP-ABC algorithm for benchmark functions with 10 dimensions.

Function Category | PP-ABC vs. DGABC | PP-ABC vs. APABC | PP-ABC vs. ABC | PP-ABC vs. ACoM-ABC | PP-ABC vs. SABC-SG | PP-ABC vs. KFABC | PP-ABC vs. MPABC |
---|---|---|---|---|---|---|---|

UM (F1–F4) | 3/0/1 | 3/1/0 | 3/1/0 | 1/1/2 | 0/2/2 | 1/1/2 | 0/2/2 |

MM (F5–F11) | 7/0/0 | 5/1/1 | 5/1/1 | 4/1/2 | 7/0/0 | 6/1/0 | 6/1/0 |

PF (F12, F13) | 1/0/1 | 1/0/1 | 1/0/1 | 2/0/0 | 2/0/0 | 2/0/0 | 2/0/0 |

CF (F14, F15) | 2/0/0 | 1/0/1 | 0/1/1 | 1/0/1 | 1/0/1 | 1/0/1 | 2/0/0 |

**Table 9.**Category-based comparison for the proposed PP-ABC algorithm for benchmark functions with 30 dimensions.

Function Category | PP-ABC vs. DGABC | PP-ABC vs. APABC | PP-ABC vs. ABC | PP-ABC vs. ACoM-ABC | PP-ABC vs. SABC-SG | PP-ABC vs. KFABC | PP-ABC vs. MPABC |
---|---|---|---|---|---|---|---|

UM (F1–F4) | 3/0/1 | 3/1/0 | 3/1/0 | 2/1/1 | 2/1/1 | 4/0/0 | 2/1/1 |

MM (F5–F11) | 6/0/1 | 5/2/0 | 5/2/0 | 2/4/1 | 6/1/0 | 4/2/1 | 2/4/1 |

PF (F12, F13) | 2/0/0 | 1/0/1 | 1/0/1 | 2/0/0 | 2/0/0 | 2/0/0 | 2/0/0 |

CF (F14, F15) | 1/0/1 | 2/0/0 | 2/0/0 | 0/1/1 | 2/0/0 | 2/0/0 | 0/1/1 |

Algorithm | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | Objective Function Value |
---|---|---|---|

PP-ABC | 0.7886 | 0.4082 | 263.895 |

WOAmM | 0.7894 | 0.4061 | 263.895 |

AAA | 0.7887 | 0.4081 | 263.895 |

TSA | 0.788 | 0.408 | 263.68 (infeasible) |

CS | 0.7887 | 0.4090 | 263.895 |

BAT | 0.7886 | 0.4084 | 263.895 |

MBA | 0.7886 | 0.4086 | 263.895 |

MVO | 0.7886 | 0.4084 | 263.895 |

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**MDPI and ACS Style**

Thirugnanasambandam, K.; Ramalingam, R.; Mohan, D.; Rashid, M.; Juneja, K.; Alshamrani, S.S.
Patron–Prophet Artificial Bee Colony Approach for Solving Numerical Continuous Optimization Problems. *Axioms* **2022**, *11*, 523.
https://doi.org/10.3390/axioms11100523

**AMA Style**

Thirugnanasambandam K, Ramalingam R, Mohan D, Rashid M, Juneja K, Alshamrani SS.
Patron–Prophet Artificial Bee Colony Approach for Solving Numerical Continuous Optimization Problems. *Axioms*. 2022; 11(10):523.
https://doi.org/10.3390/axioms11100523

**Chicago/Turabian Style**

Thirugnanasambandam, Kalaipriyan, Rajakumar Ramalingam, Divya Mohan, Mamoon Rashid, Kapil Juneja, and Sultan S. Alshamrani.
2022. "Patron–Prophet Artificial Bee Colony Approach for Solving Numerical Continuous Optimization Problems" *Axioms* 11, no. 10: 523.
https://doi.org/10.3390/axioms11100523