# MAKHA—A New Hybrid Swarm Intelligence Global Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. The Monkey Algorithm (MA)

- 1)
- The climb process: In this exploitation process, monkeys search the local optimum solution extensively in a close range.
- 2)
- The watch-jump process: In this process, monkeys look for new solutions with objective value higher than the current ones. It is considered an exploitation and intensification method.
- 3)
- The somersault process: This process is for exploration and it prevents getting trapped in a local optimum. Monkeys search for new points in other search domains. In nature, each monkey attempts to reach the highest mountaintop, which corresponds to the maximum value of the objective function. The fitness of the objective function simulates the height of the mountaintop, while the decision variable vector is considered to contain the positions of the monkeys. Changing the sign of the objective function allows the algorithm to find the global minimum instead of the global maximum. The pseudo-code for this algorithm is shown in Figure 1.

- a)
- Random generation of α from the somersault interval [c, d] where c and d governs the maximum distance that the monkey can somersault.
- b)
- Create a pivot P by the following equation:$${P}_{i}=\frac{1}{NP-1}{\displaystyle \sum _{l=1}^{NP}\left({\displaystyle \sum _{i=1}^{NP}{X}_{lj}-{X}_{ij}}\right)}$$
- c)
- Get y (Monkey new position) from$${Y}_{i}={X}_{i}+\alpha \left|{P}_{i}-{X}_{ij}\right|$$
- d)
- Update X
_{i}with Y_{i}if feasible (within boundary limits) or repeat until feasible.

## 3. The Krill Herd Algorithm (KHA)

- a)
- The movement induced by the presence of other individuals.
- b)
- The foraging activity.
- c)
- Random diffusion.

## 4. MAKHA Hybrid Algorithm

- The watch-jump process.
- The foraging activity process.
- The physical random diffusion process.
- The genetic mutation and crossover process.
- The somersault process.

- Initialization procedure:
- -
- Random generation of population in which the positions of the hybrid agent (monkey/krill) are created randomly, X
_{i}= (X_{i1}, X_{i2}, …, X_{i(NV)}) where i = 1 to NP, which represents the number of hybrids, while NV represents the dimension of the decision variable vector.

- The fitness evaluation and sorting:
- -
- H
_{i}=f(X_{i}) where H stands for hybrid fitness and f is the objective function used.

- The watch-jump process:
- -
- Random generation of X
_{i}from (X_{ij}− b, X_{ij}+ b) where b is the eyesight of the hybrid (monkey in MA) which indicates the maximal distance the hybrid can watch and Y_{i}= (Y_{i}_{1}, Y_{i}_{2}, …, Y_{i(NV)}), which are the new hybrid positions. - -
- If −f (Y
_{i}) ≥ −f (X_{i}) then update X_{i}with Y_{i}if feasible (i.e., within limits).

- Foraging motion:
- -
- Depends on food location and the previous experience about the location.
- -
- Calculate the food attractive ${\text{\beta}}_{i}^{food}$ and the effect of best fitness so far ${\beta}_{i}^{Best}$$${\beta}_{i}^{food}={C}^{food}{\widehat{H}}_{i,food}{\widehat{X}}_{i,food}$$$${\beta}_{i}^{Best}={\widehat{H}}_{i,ibest}{\widehat{X}}_{i,ibest}$$
^{food}is the food coefficient, which decreases with time and is calculated from:$${\text{C}}^{food}=2(1-\text{I}/{\text{I}}_{max})$$_{max}is the maximum number of iterations. - -
- The center of food density is estimated from the following equation:$${X}^{food}=\frac{{\displaystyle \sum _{i=1}^{NP}\frac{1}{{H}_{i}}}{X}_{i}}{{\displaystyle \sum _{i=1}^{NP}\frac{1}{{H}_{i}}}}$$
_{ibest}is the best previously visited position. - -
- $\widehat{H}$ and $\widehat{X}$ are unit normalized values obtained from this general form:$${\widehat{X}}_{i,j}=\frac{{X}_{j}-{X}_{i}}{\Vert {X}_{j}-{X}_{i}\Vert +\epsilon}$$$${\widehat{H}}_{i,j}=\frac{{H}_{j}-{H}_{i}}{{H}_{worst}-{H}_{best}}$$
_{best}and H_{worst}are the best and the worst fitness values, respectively, of the hybrid agents so far. H stands for the hybrid fitness and was used as K symbol in krill herd method. - -
- The foraging motion is defined as$${F}_{i}={V}_{f}{\beta}_{i}+{w}_{f}{F}_{i}^{old}$$
_{f}is the foraging speed, w_{f}is the inertia weight of the foraging motion in the range [0, 1], and ${F}_{i}^{old}$ is the last foraging motion.

- Physical diffusion:This is an exploration step that is used at high dimensional problem, then$${\text{D}}_{i}={\text{D}}_{max}(1-\text{I}/{\text{I}}_{max})\delta $$
_{max}is the maximum diffusion speed and δ is the random direction vector. - Calculate the time interval Δt$$\text{\Delta}t={C}_{t}{\displaystyle \sum _{L=1}^{NV}\left(U{B}_{L}-L{B}_{L}\right)}$$
_{t}is constant. - The step for position is calculated through:$$\frac{d{X}_{i}}{dt}={F}_{i}+{D}_{i}$$$${X}_{i}(t+\text{\Delta}t)={X}_{i}(t)+\text{\Delta}t\frac{d{X}_{i}}{dt}$$
- Implementation of genetic operator:
- -
- Crossover$${X}_{i,m}=\left\{\begin{array}{l}{X}_{r,m},\text{}randomCr\\ {X}_{i,m,}\text{}otherwise\end{array}\right\}$$$${C}_{r}=0.8+0.2{\widehat{K}}_{i,best}$$
- -
- Mutation$${X}_{i,m}=\left\{\begin{array}{l}{X}_{gbest,m}+\mu ({X}_{p,m}-{X}_{q,m}),\text{}randomMu\\ {X}_{i,m},\text{}otherwise\end{array}\right\}$$$$Mu=0.8+0.05{\widehat{H}}_{i,best}$$$${\widehat{H}}_{i,best}=({H}_{i}-{H}_{gb})/({H}_{worst}-{H}_{gb})$$
_{gb}is the best global fitness of the hybrid so far and X_{gbest}is its position.

- The somersault process:
- -
- α is generated randomly from [c, d] where c and d are somersault interval. Two different implementations of the somersault process can be used:Somersault I
- -
- Create the pivot P [2]:$${P}_{i}=\frac{1}{NP}{\displaystyle \sum _{i=1}^{NP}{X}_{ij}}\text{where}{P}_{i}=({P}_{1},{P}_{2},\mathrm{...},{P}_{NV})$$$${Y}_{i}={X}_{i}+\alpha ({P}_{i}-{X}_{ij})$$
- -
- Update X
_{i}with Y if feasible or repeat until feasible.Somersault II - -
- Create a pivot P by this equation used in MA:$${P}_{i}=\frac{1}{NP-1}{\displaystyle \sum _{l}^{NP}\left({\displaystyle \sum _{i=1}^{NP}{X}_{lj}-{X}_{ij}}\right)}$$
- -
- Get Y (i.e., the hybrid new position)$${Y}_{i}={X}_{i}+\alpha \left|P-_{i}{X}_{ij}\right|$$
- -
- Update X
_{i}with Y if feasible and repeat until feasible.

## 5. Numerical Experiments

Name | Objective Function | ||
---|---|---|---|

Ackley [34] | ${f}_{1}=20\left(1-{e}^{-0.2\sqrt{0.5\left({x}_{1}^{2}+{x}_{2}^{2}\right)}}\right)-{e}^{0.5\left(\mathrm{cos}2\pi {x}_{1}+\mathrm{cos}2\pi {x}_{2}\right)}+{e}^{1}$ | ||

Beale [35] | ${f}_{2}={\left(1.5-{x}_{1}+{x}_{1}{x}_{2}\right)}^{2}+{\left(2.25-{x}_{1}+{x}_{1}{x}_{2}^{2}\right)}^{2}+{\left(2.625-{x}_{1}+{x}_{1}{x}_{2}^{3}\right)}^{2}$ | ||

Bird [36] | ${f}_{3}=\mathrm{sin}({x}_{1}){e}^{{(1-\mathrm{cos}{x}_{2})}^{2}}+\mathrm{cos}({x}_{2}){e}^{{(1-\mathrm{sin}{x}_{1})}^{2}}+{({x}_{1}-{x}_{2})}^{2}$ | ||

Booth [35] | ${f}_{4}={\left({x}_{1}+2{x}_{2}-7\right)}^{2}+{\left(2{x}_{1}+{x}_{2}-5\right)}^{2}$ | ||

Bukin 6 [35] | ${f}_{5}=100\sqrt{\left|{x}_{2}-0.01{x}_{1}^{2}\right|}+0.01\left|{x}_{1}+10\right|$ | ||

Carrom table [37] | ${f}_{6}=-{\left[\mathrm{cos}{x}_{1}\mathrm{cos}{x}_{2}{e}^{\left|1-\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}/\pi \right|}\right]}^{2}/30$ | ||

Cross-leg table [37] | ${f}_{7}=-{\left[\left|\mathrm{sin}\left({x}_{1}\right)\mathrm{sin}\left({x}_{2}\right){e}^{\left|100-\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}/\pi \right|}\right|+1\right]}^{-0.1}$ | ||

Generalized egg holder [35] | ${f}_{8}={\displaystyle \sum _{i=1}^{m-1}}\left\{\begin{array}{l}-\left({x}_{i+1}+47\right)\mathrm{sin}\left(\sqrt{\left|{x}_{i+1}+{x}_{i}/2+47\right|}\right)+\\ \mathrm{sin}\left[\sqrt{\left|{x}_{i}-\left({x}_{i+1}+47\right)\right|}\right]\left(-{x}_{i}\right)\end{array}\right\}$ | ||

Goldstein–Price [38] | $\begin{array}{l}{f}_{9}={(1+({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}))\\ (30+{(2{x}_{1}-3{x}_{2})}^{2}(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}))\end{array}$ | ||

Himmelblau [39] | ${f}_{10}={\left({x}_{1}^{2}+{x}_{2}-11\right)}^{2}+{\left({x}_{2}^{2}+{x}_{1}-7\right)}^{2}$ | ||

Levy 13 [40] | ${f}_{11}={\text{sin}}^{2}\left(3\pi {x}_{1}\right)+{\left({x}_{1}-1\right)}^{2}\left[1+{\text{sin}}^{2}\left(3\pi {x}_{2}\right)\right]+{\left({x}_{2}-1\right)}^{2}\left[1+{\text{sin}}^{2}\left(2\pi {x}_{2}\right)\right]$ | ||

Schaffer [37] | ${f}_{12}=0.5+\frac{si{n}^{2}\left[\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right]-0.5}{{\left[0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)+1\right]}^{2}}$ | ||

Zettl [41] | ${f}_{13}){({x}_{1}^{2}+{x}_{2}^{2}-2{x}_{1})}^{2}+\frac{{x}_{1}}{4}$ | ||

Helical valley [42] | ${f}_{14}=100\left[{\left({x}_{3}-10\theta \right)}^{2}+{\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}-1\right)}^{2}\right]+{x}_{3}^{2}$, $2\pi \text{\theta}={\mathrm{tan}}^{-1}\left(\frac{{x}_{1}}{{x}_{2}}\right)$ | ||

Powell [43] | ${f}_{15}={\left({x}_{1}+10{x}_{2}\right)}^{2}+5{\left({x}_{3}-{x}_{4}\right)}^{2}+{\left({x}_{2}-2{x}_{3}\right)}^{4}+10{\left({x}_{1}-{x}_{4}\right)}^{4}$ | ||

Wood [44] | $\begin{array}{l}{f}_{16}=100{\left({x}_{1}^{2}-{x}_{2}\right)}^{2}+{\left({x}_{1}-1\right)}^{2}+{\left({x}_{3}-1\right)}^{2}+90{\left({x}_{3}^{2}-{x}_{4}\right)}^{2}+\\ 10.1\left[{\left({x}_{2}-1\right)}^{2}+{\left({x}_{4}-1\right)}^{2}\right]+19.8\left({x}_{2}-1\right)\left({x}_{4}-1\right)\end{array}$ | ||

Extended Cube [45] | ${f}_{17}={\displaystyle \sum _{i=1}^{m-1}}100{\left({x}_{i+1}-{x}_{i}^{3}\right)}^{2}+{\left(1-{x}_{i}\right)}^{2}$ | ||

Shekel 5[46]^{*} | ${f}_{18}=-{\displaystyle \sum _{i=1}^{M}({\displaystyle \sum _{j=1}^{4}{({x}_{j}-{C}_{ji})}^{2}+{\beta}_{i}{)}^{-1}}}$ | ||

Sphere [47] | ${f}_{19}={\displaystyle \sum _{i=1}^{m}}{x}_{i}^{2}$ | ||

Hartman 6 * [48] | ${f}_{20}=-{\displaystyle \sum _{i=1}^{4}{\alpha}_{i}\mathrm{exp}(-{\displaystyle \sum _{j=1}^{6}{A}_{ij}{({x}_{j}-{O}_{ij})}^{2})}}$ | ||

Griewank [49] | ${f}_{21}=\frac{1}{4000}\left[{\displaystyle {\displaystyle \sum}_{i=1}^{m}}{\left({x}_{i}-100\right)}^{2}\right]-\left[{\displaystyle {\displaystyle \prod}_{i=1}^{m}}\mathrm{cos}\left(\frac{{x}_{i}-100}{\sqrt{i}}\right)\right]+1$ | ||

Rastrigin [50] | ${f}_{22}={\displaystyle {\displaystyle \sum}_{i=1}^{m}}\left({x}_{i}^{2}-10\text{}\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right)$ | ||

Rosenbrock [51] | ${f}_{23}={\displaystyle {\displaystyle \sum}_{i=1}^{m-1}}100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}$ | ||

Sine envelope sine wave [37] | ${f}_{24}={\displaystyle {\displaystyle \sum}_{i=1}^{m-1}}\left\{0.5+\frac{si{n}^{2}\left[\sqrt{{x}_{i+1}^{2}+{x}_{i}^{2}}\right]-0.5}{{\left[0.001\left({x}_{i+1}^{2}+{x}_{i}^{2}\right)+1\right]}^{2}}\right\}$ | ||

Styblinski–Tang [52] | ${f}_{25}=0.5{\displaystyle \sum _{i=1}^{m}}({x}_{i}^{4}-16{x}_{i}^{2}+5{x}_{i})$ | ||

Trigonometric [53] | ${f}_{26}={\displaystyle {\displaystyle \sum}_{i=1}^{m}}{\left[m+i\left(1-\mathrm{cos}{x}_{i}\right)-\mathrm{sin}{x}_{i}-{\displaystyle {\displaystyle \sum}_{j=1}^{m}}\mathrm{cos}{x}_{j}\right]}^{2}$ | ||

Zacharov [54] | ${f}_{27}={\displaystyle {\displaystyle \sum}_{i=1}^{m}}{x}_{i}^{2}+{\left({\displaystyle {\displaystyle \sum}_{i=1}^{m}}0.5i{x}_{i}\right)}^{2}+{\left({\displaystyle {\displaystyle \sum}_{i=1}^{m}}0.5i{x}_{i}\right)}^{4}$ |

**Table 2.**Decision variables, global optimum of benchmark functions and number of iterations used for testing the performance of MA, KHA, and MAKHA.

Objective Function | NV | Search Domain | Global Minimum | Iterations | ||||||
---|---|---|---|---|---|---|---|---|---|---|

MA | KHA | MAKHA | ||||||||

Ackley | 2 | [−35, 35] | 0 | 25 | 3000 | 1000 | ||||

Beale | 2 | [−4.5, 4.5] | 0 | 25 | 3000 | 1000 | ||||

Bird | 2 | [−2π, 2π] | −106.765 | 25 | 3000 | 1000 | ||||

Booth | 2 | [−10, 10] | 0 | 25 | 3000 | 1000 | ||||

Bukin 6 | 2 | [−15, 3] | 0 | 25 | 3000 | 1000 | ||||

Carrom table | 2 | [−10, 10] | −24.15681 | 25 | 3000 | 1000 | ||||

Cross-leg table | 2 | [−10, 10] | −1 | 25 | 3000 | 1000 | ||||

Generalized egg holder | 2 | [−512, 512] | −959.64 | 124 | 15,000 | 5000 | ||||

Goldstein-Price | 2 | [−2, 2] | 3 | 25 | 3000 | 1000 | ||||

Himmelblau | 2 | [−5, 5] | 0 | 25 | 3000 | 1000 | ||||

Levy 13 | 2 | [−10, 10] | 0 | 25 | 3000 | 1000 | ||||

Schaffer | 2 | [−100, 100] | 0 | 199 | 15,000 | 8000 | ||||

Zettl | 2 | [−5, 5] | −0.003791 | 25 | 3000 | 1000 | ||||

Helical valley | 3 | [−1000, 1000] | 0 | 25 | 3000 | 1000 | ||||

Powell | 4 | [−1000, 1000] | 0 | 50 | 6000 | 2000 | ||||

Wood | 4 | [−1000, 1000] | 0 | 25 | 3000 | 1000 | ||||

Extended Cube | 5 | [−100, 100] | 0 | 25 | 3000 | 1000 | ||||

Shekel 5 | 4 | [0, 10] | −10.1532 | 25 | 3000 | 1000 | ||||

Sphere | 5 | [−100, 100] | 0 | 75 | 9000 | 1000 | ||||

Hartman 6 | 6 | [0,1] | −3.32237 | 25 | 3000 | 1000 | ||||

Griewank | 50 | [−600, 600] | 0 | 124 | 15000 | 5000 | ||||

Rastrigin | 50 | [−5.12, 5.12] | 0 | 124 | 15000 | 5000 | ||||

Rosenbrock | 50 | [−50, 50] | 0 | 124 | 15000 | 5000 | ||||

Sine envelope sine wave | 50 | [−100, 100] | 0 | 124 | 15000 | 5000 | ||||

Styblinski-Tang | 50 | [−5,5] | −1958.2995 | 124 | 15000 | 5000 | ||||

Trigonometric | 50 | [−1000, 1000] | 0 | 124 | 15000 | 5000 | ||||

Zacharov | 50 | [−5,10] | 0 | 25 | 3000 | 1000 |

Method | Condition | Parameter | Selected value |
---|---|---|---|

MA | b | 1 | |

R ≥ 100 | b | 10 | |

c | −1 | ||

d | 1 | ||

R ≥ 500 | c | −10 | |

R ≥ 500 | d | 30 | |

N_{C} | 30 | ||

KHA | D_{max} | [0.002, 0.01] | |

C_{t} | 0.5 | ||

V_{f} | 0.02 | ||

N_{max} | 0.01 | ||

w_{f} and w_{N} | [0.1, 0.8] | ||

MAKHA I | b | 1 | |

R < 2 | b | 0.5*R | |

R ≥ 100 | b | 10 | |

c | −0.1 | ||

d | 0.1 | ||

D_{max} | 0 | ||

C_{t} | 0.5 | ||

V_{f} | 0.2 | ||

w_{f} | 0.1 | ||

Somersault I is used | |||

MAKHA II (NV = 50) | b | 0.3*R | |

c | −R | ||

d | R | ||

D_{max} | [0.002, 0.01] | ||

C_{t} | 0.5 | ||

V_{f} | 0.02 | ||

w_{f} | [0.1, 0.8] | ||

Somersault II is used |

**Figure 4.**Surface plots of the two-variable benchmark functions used in this study: (

**a**) Ackley, (

**b**) Beale, (

**c**) Booth, (

**d**) Carrom table, (

**e**) Cross-leg table, (

**f**) Himmelblau, (

**g**) Levy 13, and (

**h**) Schaffer.

_{s}solvers (i.e., optimization methods) to be tested over a set of n

_{p}problems. For each problem p and solver s, the performance metric t

_{ps}must be defined. In our study, reliability of the stochastic method in accurately finding the global minimum of the objective function is considered as the principal goal, and hence the performance metric is defined as

_{calc}is the mean value of that objective function calculated by the stochastic method over several runs. In our study, f

_{calc}is calculated from 30 runs to solve each test problem by each solver; note that each run is different because of the random number seed used and the stochastic nature of the method. So, the focus is on the average performance of stochastic methods, which is desirable for comparison purposes.

_{ps}, is used to compare the performance on problem, p, by solver, s, with the best performance by any solver on this problem. This performance ratio is given by

_{ps}is 1 for the solver that performs the best on a specific problem p. To obtain an overall assessment of the performance of solvers on n

_{p}problems, the following cumulative function for r

_{ps}is used:

_{ps}within a factor of ς of the best possible ratio. The PP of a solver is a plot of ρ

_{s}(ς) versus ς; it is a non-decreasing, piece-wise constant function, continuous from the right at each of the breakpoints.

_{s}(ς) for all solvers and to select the highest one, which is the probability that a specific solver will “win” over the rest of solvers used. In our case, the PP plot compares how accurately the stochastic methods can find the global optimum value relative to one another, and so the term “win” refers to the stochastic method that provides the most accurate value of the global minimum in the benchmark problems used.

## 6. Results and Discussion

^{−10}. The progress of the mean values is presented in Figure 5, Figure 6, Figure 7 and Figure 8 for each benchmark function and a brief discussion of those results follows.

**Figure 5.**Evolution of mean best values for MA, KHA and MAKHA for: (

**a**) Ackley, (

**b**) Beale, (

**c**) Bird (

**d**) Booth, (

**e**) Bukin6 (

**f**) Carrom table, (

**g**) Cross-leg table, and (

**h**) Generalized Eggholder functions.

**Figure 6.**Evolution of mean best values for MA, KHA and MAKHA for: (

**a**) Goldstein–Price (

**b**) Himmelblau, (

**c**) Levy 13, (

**d**) Schaffer functions, (

**e**) Zettl, (

**f**) Helical Valley, (

**g**) Powell and (

**h**) Wood.

**Figure 7.**Evolution of mean best values for MA, KHA and MAKHA for: (

**a**) Extended cube, (

**b**) Shekel, (

**c**) Sphere, (

**d**) Hartman, (

**e**) Griewank (

**f**) Rastrigin, (

**g**) Rosenbrock, and (

**h**) Sine Envelope Sine functions.

**Figure 8.**Evolution of mean best values for MA, KHA and MAKHA for: (

**a**) Styblinski-Tang, (

**b**) Trigonometric and (

**c**) Zacharov functions.

**Table 4.**Values of the mean minima (f

_{calc}) and standard deviations (σ) obtained by the MAKHA, MA and KHA algorithms for the benchmark problems used in this study.

^{*}Numerical Performance of | ||||||
---|---|---|---|---|---|---|

MA | KHA | MAKHA | ||||

Objective function | f_{calc} | σ | f_{calc} | σ | f_{calc} | σ |

Ackley | 4.8E−8 | 0 | 0.00129 | 0 | 0 | 0 |

Beale | 0.084 | 0.125 | 0.0508 | 0.193 | 0 | 0 |

Bird | −105.326 | 1.45 | −103.52 | 7.4 | −106.7645 | 0 |

Booth | 0.179 | 0.172 | 1.28E−12 | 0 | 0 | 0 |

Bukin 6 | 3.487 | 1.77 | 0.074 | 0.029 | 0.0267 | 0.0157 |

Carrom table | −23.9138 | 0.436 | −24.1568 | 0 | −24.1568 | 0 |

Cross-leg table | −0.002 | 4E−3 | −0.00035 | 0 | −0.9985 | 0 |

Generalized egg holder | −949.58 | 0 | −862.1 | 0 | −959.641 | 0 |

Goldstein-Price | 3.051 | 0.055 | 3 | 0 | 3 | 0 |

Himmelblau | 0.179 | 0.187 | 7.4E−13 | 0 | 3.7E−31 | 0 |

Levy 13 | 0.0616 | 0.08 | 2.1E−7 | 0 | 1.35E−31 | 0 |

Schaffer | 0 | 0 | 1.7E−6 | 0 | 0 | 0 |

Zettl | −0.0037 | 1E−3 | −0.00379 | 0 | −0.00379 | 0 |

Helical valley | 136 | 250 | 9.8E−5 | 3E−3 | 0 | 0 |

Powell | 18.46 | 49 | 1.9E−5 | 0 | 0 | 0 |

Wood | 113.6 | 256 | 0.698 | 1.6 | 0 | 0 |

Extended Cube | 3.568 | 0.8 | 1.658 | 5.28 | 0 | 0 |

Shekel 5 | −10.139 | 0.06 | −5.384 | 3.1 | −10.1532 | 0 |

Sphere | 1.4E−14 | 0 | 1E−10 | 0 | 0 | 0 |

Hartman 6 | −2.7499 | 0.2 | −3.2587 | 0.06 | −3.2627 | 0.06 |

Griewank | 0.0165 | 0.032 | 0.0769 | 0.095 | 8.4E−8 | 0 |

Rastrigin | 1.3E-9 | 0 | 89.38 | 48.38 | 6.47E−6 | 0 |

Rosenbrock | 47.45 | 7.5 | 52.593 | 18.97 | 1E−5 | 0 |

Sine envelope sine wave | 3.3E−11 | 0 | 16.107 | 5.94 | 3.46E−6 | 0 |

Styblinski-Tang | −1916.84 | 28.1 | −1645.42 | 36.9 | −1958.31 | 0 |

Trigonometric | 1.35E−5 | 0 | 285.43 | 545.5 | 0 | 0 |

Zacharov | 1.46E−10 | 0 | 1E−3 | 5.5E−3 | 8.4E−6 | 0 |

**Figure 9.**Performance profiles of the MAKHA, MA, and KHA methods for the global optimization of the twenty-seven benchmark problems used in this study.

- (a)
- Exploration or diversification feature: The watch-jump process (MA), physical random diffusion (KHA), the somersault process (MA), and genetic operators (KHA).
- (b)
- Exploitation or intensification feature: The climb process (MA), the watch-jump process (MA), the induced motion (KHA), and the foraging activity (KHA).

## 7. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

A | Hartman’s recommended constants |

a | Pseudo-gradient monkey step |

b | Eyesight of the monkey (hybrid), which indicates the maximum distance the monkey (hybrid) can watch. |

C | Shekel’s recommended constants |

C^{food} | Food coefficient |

Cr | Crossover probability |

C_{t} | Empirical and experimental Constants (Time constant) |

c | Somersault interval |

D_{i} | Physical diffusion of krill (hybrid) number i |

D_{max} | Maximum diffusion speed |

d | Somersault interval |

ds_{i} | Sensing distance of the krill |

ds_{ij} | Distance between each 2 krill positions |

f | Objective function |

F_{i} | Foraging motion |

G | Global minimum |

H | Fitness value of the hybrid in MAKHA |

I, i, j and l | Counters for any value |

K | Fitness value of the krill in KHA |

M | Number of local minima in Shekel function |

LB | Lower boundaries and low limit of decision variable |

Mu | Mutation probability |

m | Dimension of the problem, i.e., number of variables. |

N | Induced speed for KHA |

N_{c} | Number of climb cycles |

N_{max} | Maximum induced speed |

NP | Population size (number of points) |

NV | Dimension of the problem, i.e., number of variables. |

n | A counter |

n_{p} | Number of problems |

n_{s} | Number of solvers |

O | Hartman’s recommended constants |

P | Pivot value |

R | The half range of boundaries between the lower boundary and the upper boundary of the decision variables (X) |

r_{ps} | The performance ratio |

T | Time taken by krill or hybrid |

t_{ps} | Performance metric |

UB | Upper boundaries and high limit of decision variable |

V_{f} | Foraging speed |

w_{f} or w_{N} | Inertia weight |

X | Decision variable matrix |

X^{food} | Centre of food density |

x | Decision variable |

Y | Decision variable matrix |

## Greek Letters

α | Somersault interval random output. |

β | Shekel’s recommended constant |

β^{food} | Food attractive factor |

δ | Random direction vector |

∆t | Incremental period of time |

ε | Small positive number to avoid singularity |

ζζ | The simulating value of r _{ps} |

ζmax | The maximum assumed value of rps |

ρ | The cumulative probabilistic function of r _{ps} and the fraction of the total number of problems |

σ | Standard deviation |

ςς | The counter of ρ points |

## References

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**MDPI and ACS Style**

Khalil, A.M.E.; Fateen, S.-E.K.; Bonilla-Petriciolet, A. MAKHA—A New Hybrid Swarm Intelligence Global Optimization Algorithm. *Algorithms* **2015**, *8*, 336-365.
https://doi.org/10.3390/a8020336

**AMA Style**

Khalil AME, Fateen S-EK, Bonilla-Petriciolet A. MAKHA—A New Hybrid Swarm Intelligence Global Optimization Algorithm. *Algorithms*. 2015; 8(2):336-365.
https://doi.org/10.3390/a8020336

**Chicago/Turabian Style**

Khalil, Ahmed M.E., Seif-Eddeen K. Fateen, and Adrián Bonilla-Petriciolet. 2015. "MAKHA—A New Hybrid Swarm Intelligence Global Optimization Algorithm" *Algorithms* 8, no. 2: 336-365.
https://doi.org/10.3390/a8020336