## 4 MAKHA Hybrid Algorithm
* _The fitness evaluation and sorting:_
* _H_=_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 \(\left(X_{i\bar{y}}-b,\,X_{i\bar{y}}+b\right)\) where \(b\) is the eyesight of the hybrid (monkey in MA) which indicates the maximal distance the hybrid can watch and \(Y_{i}=\left(Y_{i1},\,Y_{i2},\,...,\,Y_{i(N)j}\right)\), which are the new hybrid positions. * If \(\
eg f\left(Y_{i}\right)\geq\
eg f\left(X_{i}\right)\) 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 \(\beta_{i}^{\mathit{food}}\) and the effect of best fitness so far \(\beta_{i}^{\mathit{Best}}\) \[\beta_{i}^{\mathit{food}}=C^{\mathit{food}}\hat{H}_{i,\mathit{food}}\hat{X}_{i,\mathit{food}}\] (3) \[\beta_{i}^{\mathit{Best}}=\hat{H}_{i,\mathit{best}}\hat{X}_{i,\mathit{best}}\] (4) where \(C^{\mathit{food}}\) is the food coefficient, which decreases with time and is calculated from: \[\mathrm{C}^{\mathit{food}}=2(1-1/\,\mathrm{I}_{\mathit{max}})\] (5) where \(I\) is the iteration number and \(I_{\mathit{max}}\) is the maximum number of iterations. * The center of food density is estimated from the following equation: \[X^{\mathit{food}}=\frac{\sum_{i=1}^{N\!p}\frac{1}{H_{i}}\,X_{i}}{ \sum_{i=1}^{N\!p}\frac{1}{H_{i}}}\] (6) and \(H_{\mathit{best}}\) is the best previously visited position. * \(\hat{H}\) and \(\hat{X}\) are unit normalized values obtained from this general form: \[\hat{X}_{i,j}=\frac{X_{j}-X_{i}}{\left|X_{j}-X_{i}\right|+\epsilon}\] (7) \[\hat{H}_{i,j}=\frac{H_{j}-H_{i}}{H_{\mathit{worst}}-H_{\mathit{best}}}\] (8)
where \(\varepsilon\) is a small positive number that is added to avoid singularities. \(H_{best}\) 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 \[F_{i}=V_{r}\beta_{i}+w_{f}F_{i}^{old}\] (9) where \(V_{f}\) 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 \[\mathrm{D_{i}}=\mathrm{D_{max}}(1-1/\mathrm{I_{max}})\delta\] (10) where \(D_{max}\) is the maximum diffusion speed and \(\delta\) is the random direction vector. * _Calculate the time interval \(\Delta t\)_ \[\Delta t=C_{i}\sum_{L=1}^{Nv}\left(UB_{L}-LB_{L}\right)\] (11) where \(C_{i}\) is constant. * _The step for position is calculated through:_ \[\beta_{i}^{\mathit{food}}=C^{\mathit{food}}\hat{H}_{i,\mathit{food}}\hat{X}_{i,\mathit{food}}\]0 (12) \[\beta_{i}^{\mathit{food}}=C^{\mathit{food}}\hat{H}_{i,\mathit{food}}\hat{X}_{i,\mathit{food}}\]1 (13) where \(\frac{dX_{i}}{dt}\) represents the velocity of the hybrid agent (Krill/Monkey). * _Crossover_ \[X_{i,m}=\left\{ \begin{array}{ll}X_{r,m},&random<Cr\\ X_{i,m,}&otherwise\end{array} \right\}\] (14) where \(r\)\(\in\) {l,2,...,\(i-1,i+1\),..., \(NP\)} and \(Cr\) is the crossover probability \[C_{r}=0.8+0.2\hat{K}_{i,best}\] (15)
- _Mutation_ \[X_{i,m}=\left\{ \begin{array}{ll}X_{gbest,m}+\mu(X_{p,m}-X_{q,m}),&random<Mu \\ X_{i,m},&otherwise\end{array} \right\}\] (16) where \(\mu\) is a random number, \(p,q\)\(\in\) {l,2,...,\(i-1,i+1\),..., \(NP\)} and \(Mu\) is the mutation probability: \[Mu=0.8+0.05\hat{H}_{i,best}\] (17)
\[\hat{H}_{i,best}=(H_{i}-H_{gb})/(H_{worst}-H_{gb}) \tag{18}\]  
where \(H_{gb}\) is the best global fitness of the hybrid so far and \(X_{gbest}\) is its position. * _The somersault process:_
* \(a\) is generated randomly from [\(c\), \(d\)] where \(c\) and \(d\) are somersault interval.