Krill Herd Optimization Algorithm

The KHO algorithm simulated the krill behavior whereas every individual of the KH created their contribution with the process of moving [26 ]. The best solution is acquired when the krill individuals identify the food center. The KHO algorithm is according to the lagrangian as well as krill individual’s evolution behavior with the capability to do exploitation and exploration in the optimization issues. In this KHO algorithm, the random value performs an important role. The time-dependent position of the individual’s krill is calculated on the basis of three actions and they are; random diffusion, foraging activity and the movement induction of the krill individual. The KHO algorithm adopts the d-dimensional search space lagrangian design and it is expressed in the below equation; $$\frac{\partial Z_j}{\partial t}=O{\kappa }_j+G{\kappa }_j+E{\kappa }_j$$
Here, the terms $O{\kappa }_j$ , $G{\kappa }_j$ and $E{\kappa }_j$ indicates search agent’s movement induction, foraging behavior and random diffusion, respectively. The optimization commences by initializing the parameters such as maximum diffusion speed $E_{\text{MAX}}$ , krill position $Z_j$ , maximal foraging speed ${\nu }_g$ , maximal induced speed $o^{\text{MAX}}$ , the maximal number of iterations $J_{\text{MAX}}$ and numbers of krill $O$ . Compute the movement for every krill. The communal effects between krill individuals lead to movements and they attempt to conserve the higher density. ${\sigma }_j$ denote the motion induction direction and which is evaluated with the local target and the density of the repulsive swarm. Then it is expressed as; $$O{\kappa }_j^{\text{NEW}}=o^{\text{MAX}}{\sigma }_j+x_{o}O{\kappa }_j^{\text{old}}$$
The inertia weight is represented by $x_o$ , the last motion is denoted by $O{\kappa }_j^{\text{OLD}}$ and maximal induced speed is represented as $o^{\text{MAX}}$ . $${\sigma }_j={\sigma }_j^{\text{LO}}+{\sigma }_j^{\tau }$$ $${\sigma }_j^{\text{LO}}=\sum _{k=1}^{\mathit{Oo}}{\widehat{\kappa }}_{j,k}{\widehat{Y}}_{j,k}$$ $${\widehat{Y}}_{j,k}=\frac{Y_k-Y_j}{∥Y_k-Y_j∥+a}$$ $${\widehat{\kappa }}_{j,k}=\frac{{\kappa }_j-{\kappa }_k}{{\kappa }^{\text{WO}}-{\kappa }^{\text{BEST}}}$$
The worst and best krill individuals are represented by ${\kappa }^{\text{BEST}}$ and ${\kappa }^{\text{WO}}$ . The smaller positive number is represented by $a$ . The numbers of neighbors are denoted by $O_o$ . Also, the terms ${\sigma }_j^{\tau }$ and ${\sigma }_j^{\text{LO}}$ denote target direction effect and local effect offered by neighbors, respectively; ${\kappa }_j$ and ${\kappa }_{\text{k}}$ depicts fitness function of $j\text{th}$ krill and $k\text{th}$ neighbor, respectively; $Y_{\text{k}}$ and $Y_j$ indicates the corresponding positions of $j\text{th}$ krill and $k\text{th}$ neighbor, respectively. $${\sigma }_j^{\tau }=D^{\text{BEST}}{\widehat{\kappa }}_{j,\text{BEST}}{\widehat{Y}}_{j,\text{BEST}}$$
$D^{\text{BEST}}$ represents the krill individual efficient individual through best fitness is expressed in the below equation; $$D^{\text{BEST}}=2\left(\Re +\frac{J}{J_{\text{MAX}}}\right)$$ $$E_{t,j}=\frac{1}{5O}\sum _{k=1}^O{Y}_j-Y_k$$ where $\Re$ depicts random value which lies in the range [0,1], $J$ signifies current iteration, $E_{t,j}$ indicates sensing distance.
The foraging movement is defined as which is according to food location and previous experiences with the food location. Then it is expressed as; $$G{\kappa }_j={\nu }_g{\eta }_j+x_{g}G{\kappa }_j^{\text{OLD}}$$
Inertia weight with foraging movement is represented by $x_g$ , ${\eta }_j$ signifies fitness value of $j\text{th}$ krill and the krill best objective is represented by ${\eta }_j^{\text{BEST}}$ . $${\eta }_j={\eta }_j^{\text{FOOD}}+{\eta }_j^{\text{BEST}}$$ $${\eta }_j^{\text{FOOD}}=D^{\text{FOOD}}{\widehat{\kappa }}_{j,\text{FOOD}}{\widehat{Y}}_{j,\text{FOOD}}$$
The food coefficients are represented by $D^{\text{FOOD}}$ and it is expressed as; $$D^{\text{FOOD}}=2\left(1-\frac{J}{J_{\text{MAX}}}\right)$$
The $j\text{th}$ krill individual best objectives are determined by ${\eta }_j^{\text{BEST}}$ and it is expressed in the below equation $${\eta }_j^{\text{BEST}}={\widehat{\kappa }}_{j,j\text{BEST}}{\widehat{Y}}_{j,j\text{BEST}}$$
The food centers for iterations are computed and it is expressed as; $$Y^{\text{FOOD}}=\frac{{\sum }_{j=1}^O\frac{1}{{\kappa }_j}{Y}_j}{{\sum }_{j=1}^O\frac{1}{{\kappa }_j}}$$
The physical diffusion movement is described on the basis of the diffusion speed for maximum and the random direction vectors are expressed in the below equation; $$E{\kappa }_j=E^{\text{MAX}}\left(1-\frac{J}{J_{\text{MAX}}}\right)\mathrm{\Phi }$$
The random direction vector is represented by $\mathrm{\Phi }$ which lies between − 1 and 1.
To enhance the performance of KHO, the genetic reproduction mechanisms mutation and crossover are merged through KHO.
The below expression represents the crossover function of $Y_j\prime sn\text{th}$ component, $$Y_{j,n}=\left\{\begin{array}{cc}{Y}_{s,n},\hfill & {\Re }_{j,n}<c_o\\ Y_{j,n},\hfill & \text{else}\end{array}\right)$$
Then, crossover probability $c_o=0.2{\widehat{\kappa }}_{j,j\text{BEST}}$ $$Y_{j,n}=\left\{\begin{array}{cc}{Y}_{h\text{BEST},n}+\nu \left(Y_{q,n}-Y_{r,n}\right)\hfill & {\Re }_{j,n}<\text{Nu}\\ Y_{j,n}\hfill & \text{else}\end{array}\right)$$
The mutation probability is denoted by $N_u$ and it is set to $N_u=0.05/{\widehat{\kappa }}_{j,j\text{BEST}}$ .
The krill’s position vector in the interval $\left|t+\mathrm{\Delta }t\right|$ is found using below expression, $$Y_j\left(t+\mathrm{\Delta }t\right)=Y_j\left(t\right)+\mathrm{\Delta }t\frac{\partial Y_J}{\partial t}$$