Hybridize the Dwarf Mongoose Optimization (DMO) Algorithm to Obtain the Optimal Solution for Solve Optimization Problems

It is one of the mathematical methods used to solve problems of finding the minimum or maximum of functions. This method is usually used in problems searching for solutions to systems of linear equations or improving performance in numerical research problems. The conjugate gradient method is based on the use of conjugate directives to quickly optimize the function being optimized. Instead of using traditional regression directions, the conjugate gradient method uses directions that are interconnected with each other. You start moving towards the initial downhill direction. A conjugate direction of progress is then determined based on the previous step and the interconnected trends. This process is repeated until the solutions improve simultaneously in the directions of all regressions.

Definition 1: The optimization [14]

It means finding the best solution to the given problem, and finding the minimum or maximum value of a function consisting of n variables, where n can be any integer greater than zero.                             

Definition 2: Global and local minimum [14]

A- The global minimum value: represents the lowest value of the function in the entire research field.

B- Local minimum value: represents the lowest value of the function in specific locations within the search field. Algorithms that trend toward a global minimum are known as globally convergent algorithms, while algorithms that trend toward a local minimum are known as locally convergent algorithms.

2.1 Derivation of new conjugation coefficients

In three term conjugates gradient direction Babaie-Kafaki and Ghanbari [15] proposed a new value of $t^*$ as follow:

$t^*=\frac{\left\|y_k\right\|}{\left\|s_k\right\|}+\frac{s_k^T y_k}{\left\|s_k\right\|^2}$

Ibrahim and Shareef [16] proposed another value of $t$ as follow:

$t_k=\gamma \frac{\left\|y_k\right\|}{\left\|s_k\right\|}+(1-\gamma) \frac{s_k^T y_k}{\left\|s_k\right\|^2}$, where $\gamma \in(0,1)$

Here, we drive a new conjugacy coefficient with a new value of $t$ as [16, 17]:

$\alpha_k=t_k=\frac{\left(s_k^T g_k / 2\right)^2}{\left(f_{k+1}-f_k\right)^2}$

$d_{k+1}^{C G_3}=-g_{k+1}+\beta_k d_k-t_k\left(\frac{g_{k+1}^T d_k}{d_k^T y_k}\right) y_k$

Let $d_{k+1}^{C G_2}=d_{k+1}^{C G_3}$

$-g_{k+1}+\beta_k^{\text {New }} d_k=-g_{k+1}+\beta_k d_k-t_k\left(\frac{g_{k+1}^T d_k}{d_k^T y_k}\right) y_k$

Multiply both sides of the above equation by $y_k^T$ we get:

$\begin{aligned}-y_k^T g_{k+1}+\beta_k^{N e w} y_k^T d_k & =-y_k^T g_{k+1} \\ & +\beta_k y_k^T d_k-t_k\left(\frac{g_{k+1}^T d_k}{d_k^T y_k}\right)\left\|y_k\right\|^2\end{aligned}$

$\beta_k^{\text {New }}=\beta_k-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2$    (1)

where, $t_k=\frac{\left(s_k^T g_k / 2\right)^2}{\left(f_{k+1}-f_k\right)^2}>0$

From Ibrahim and Shareef [16],

The first case: If $\beta_k=\beta^{H S}=\frac{g_{k+1}^T y_k}{d_k^T y_k}$ then Eq. (1) become:

$\beta_k^{\text {New } 1}=\frac{g_{k+1}^T y_k}{d_k^T y_k}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2$

The second case: If $\beta_k=\beta^{F R}=\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}$ then Eq. (1) become:

$\beta_k^{\text {New } 2}=\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2$

The third case: If $\beta_k=\beta^{P R}=\frac{g_{k+1}^T y_k}{g_k^T g_k}$ then Eq. (1) become:

$\beta_k^{N e w 3}=\frac{g_{k+1}^T y_k}{g_k^T g_k}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2$

2.2 Convergence analysis of the new conjugate vector method

In this section, we will show that the proposed algorithm that has been identified achieves the property of sufficient regression that satisfies the property of convergence.

Assumption (1): If f is bounded on the set $s=\left\{x \in R^n: f(x) \leq f\left(x_0\right)\right\}$ and is differentiable with the gradient $\nabla f$ and there is a Lipchitz Constant $L>0$ , then

$\|\nabla f(x)-\nabla f(y)\|<L\|x-y\|$ for all $x, y \in s$.

Theorem (1): The search direction $d_k$ generated by the proposed algorithm of the developed CG achieves the property of sufficient steepness for each k, when the step size $\lambda_k$ satisfies Wolfe conditions.

Proof: Using the principle of mathematical induction.

When k=0, $d_0=-g_0 \Rightarrow d_0^T g_0=-\left\|g_k\right\|^2<0$ then the theorem is true.

Now we assume that the theorem is true for all values of $k \geq 0$, i.e

$g_k^T d_k<0, g_k^T d_k \leq-c\left\|g_k\right\|^2, c>0$

We will now explain that the theorem is true when k+1.

$d_{k+1}=-g_{k+1}+\beta_k d_k$    (2)

By multiplying both sides of Eq. (2) above by $g_{k+1}^T$ we get:

$g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2+\beta_k g_{k+1}^T d_k$    (3)

The first case: If $\beta_k=\beta_k^{\text {New } 1}$

$g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2+\beta_k^{\text {New } 1} g_{k+1}^T d_k$

$\begin{aligned} & g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2 \\ & \quad+\left[\frac{g_{k+1}^T y_k}{d_k^T y_k}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right] g_{k+1}^T d_k\end{aligned}$    (4)

From Wolff's second strong condition, as shown in Eq. (5) below:

$\begin{aligned} & g\left(x_k+\alpha_k d_k\right)^T d_k \leq-\sigma g_k^T d_k \\ & \quad \Rightarrow g_{k+1}^T d_k \leq-\sigma g_k^T d_k\end{aligned}$    (5)

It is known that

$g_{k+1}^T y_k=g_{k+1}^T\left(g_{k+1}-g_k\right)\left\|g_{k+1}\right\|^2-g_{k+1}^T g_k$

Taking advantage of one direction of Powell's recovery condition, as shown in Eq. (6):

$g_{k+1}^T y_k \leq\left\|g_{k+1}\right\|^2+0.2\left\|g_{k+1}\right\|^2=1.2\left\|g_{k+1}\right\|^2$    (6)

From Wolff's strong condition we get the following Eq. (7):

$-(1-\sigma) g_k^T d_k \leq y_k^T d_k \leq-(1+\sigma) g_k^T d_k$    (7)

Substituting Eqs. (5), (6) and (7) into Eq. (4) results in:

$g_{k+1}^T d_{k+1} \leq-\left\|g_{k+1}\right\|^2$

$+\frac{c \sigma\left\|g_k\right\|^2}{d_k^T y_k}\left[1.2\left\|g_{k+1}\right\|^2-t_k \frac{c \sigma\left\|g_k\right\|^2}{d_k^T y_k} \cdot\left\|y_k\right\|^2\right]$

Since $d_k^T y_k>0, c>0,0.5<\sigma<1$, therefore the part is $W=\frac{c \sigma\left\|g_k\right\|^2}{d_k^T y_k}>0$. This leads to

$g_{k+1}^T d_{k+1} \leq-\left\|g_{k+1}\right\|^2+1.2 W\left\|g_{k+1}\right\|^2-t_k W^2 .\left\|y_k\right\|^2$

Since $t_k>0, t_k W^2 .\left\|y_k\right\|^2>0$, this leads to

$g_{k+1}^T d_{k+1} \leq-(1-1.2 W)\left\|g_{k+1}\right\|^2$

$g_{k+1}^T d_{k+1} \leq-\Omega_1\left\|g_{k+1}\right\|^2$

where, $\Omega_1=1-1.2 W$, when $1-1.2 W>0$.

The second case: If $\beta_k=\beta_k^{\text {New } 2}$ in Eq. (3), we get:

$g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2+\beta_k^{\text {New } 2} g_{k+1}^T d_k$

$\begin{gathered}g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2 \\ +\left[\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right] g_{k+1}^T d_k\end{gathered}$    (8)

Substituting Eq. (5) into Eq. (8) we get:

$\begin{gathered}g_{k+1}^T d_{k+1} \leq-\left\|g_{k+1}\right\|^2 \\ +\left[\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{c \sigma\left\|g_k\right\|^2}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right]\left(c \sigma\left\|g_k\right\|^2\right)\end{gathered}$

$\begin{gathered}g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2 \\ +c \sigma\left\|g_{k+1}\right\|^2-t_k\left(\frac{c^2 \sigma^2\left\|g_k\right\|^4}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\end{gathered}$

Since $t_k>0, t_k\left(\frac{c^2 \sigma^2\left\|g_k\right\|^4}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2>0$. This leads to

$g_{k+1}^T d_{k+1} \leq-(1-c \sigma)\left\|g_{k+1}\right\|^2$

$g_{k+1}^T d_{k+1} \leq-\Omega_2\left\|g_{k+1}\right\|^2$

where, $\Omega_2=1-c \sigma$, when $1-c \sigma>0, c>0,0.5<\sigma<1$.

The third case: If $\beta_k=\beta_k^{\text {New } 3}$ in Eq. (3) we get:

$g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2+\beta_k^{N e w 3} g_{k+1}^T d_k$

$\begin{aligned} & g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2 \\ & \quad+\left[\frac{g_{k+1}^T y_k}{\left\|g_k\right\|^2}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right] g_{k+1}^T d_k\end{aligned}$    (9)

Substituting Eqs. (5) and (6) into Eq. (9) we get:

$\begin{gathered}g_{k+1}^T d_{k+1} \leq-\left\|g_{k+1}\right\|^2 \\ +\left[\frac{1.2\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{c \sigma\left\|g_k\right\|^2}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right]\left(c \sigma\left\|g_k\right\|^2\right) \\ g_{k+1}^T d_{k+1}=-\left\|g_{k+1}\right\|^2+1.2 c \sigma\left\|g_{k+1}\right\|^2 \\ -t_k\left(\frac{c^2 \sigma^2\left\|g_k\right\|^4}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\end{gathered}$

Since $t_k>0, t_k\left(\frac{c^2 \sigma^2\left\|g_k\right\|^4}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2>0$. This leads to

$\begin{gathered}g_{k+1}^T d_{k+1} \leq-(1-1.2 c \sigma)\left\|g_{k+1}\right\|^2 \\ g_{k+1}^T d_{k+1} \leq-\Omega_3\left\|g_{k+1}\right\|^2\end{gathered}$

where, $\Omega_3=1-1.2 c \sigma$, when $1-1.2 c \sigma>0, c>0$, $0.5<\sigma<1$

2.3 Comprehensive convergence analysis of the proposed algorithm

Lemma (1): Suppose that Assumption (1) is fulfilled and that the conjugate gradient method is fulfilled, since $d_k$ is a slope search direction, and $\propto_k$ is generated from the strong Wolff conditions (SWP) if  $\sum_{K=1}^{\infty} \frac{1}{\left\|d_{k+1}\right\|^2}=\infty$ then $\lim _{k \rightarrow \infty} \inf \left\|g_k\right\|=0$.

Theorem (2): Suppose that Assumption (1) is fulfilled and that the proposed conjugate gradient method is fulfilled in the direction of the search slope $d_k$, and that the step length $\propto_k$ is generated from the conditions (SWP) then $\lim _{k \rightarrow \infty} \inf \left\|g_k\right\|=0$.

Proof: Using Lemma (1), and since the algorithm fulfills theorem (1), and if $g_{k+1} \neq 0$, then we must prove that $\left\|d_{k+1}\right\|$ is constrained from above, we take $\|\cdot\|$ for both sides of the Eq. (2), we get

$\left\|d_{k+1}\right\|=\left\|-g_{k+1}+\beta_k d_k\right\|$

$\Rightarrow\left\|d_{k+1}\right\| \leq\left\|g_{k+1}\right\|+\left|\beta_k\right|\left\|d_k\right\|$    (10)

The first case: If $\beta_k=\beta_k^{\text {New } 1}$

$\Rightarrow\left\|d_{k+1}^{N e w 1}\right\| \leq\left\|g_{k+1}\right\|+\mid \beta_k^{\text {New } 1}\|\| d_k \|$

$\left|\beta_k^{\text {New } 1}\right|=\left|\frac{1}{d_k^T y_k}\left[g_{k+1}^T y_k-t_k\left(\frac{g_{k+1}^T d_k}{d_k^T y_k}\right)\left\|y_k\right\|^2\right]\right|$

Using Eqs. (5) and (6) we get the following

$\left|\beta_k^{\text {New } 1}\right| \leq\left|\frac{1}{d_k^T y_k}\left[1.2\left\|g_{k+1}\right\|^2-t_k \frac{-\sigma g_k^T d_k}{d_k^T y_k} \cdot\left\|y_k\right\|^2\right]\right|$

$\Rightarrow\left|\beta_k^{\text {New } 1}\right| \leq \mathrm{A}_1$

$\Rightarrow\left\|d_{k+1}^{N e w 1}\right\| \leq\left\|g_{k+1}\right\|+\mathrm{A}_1\left\|d_k\right\|$

$\Rightarrow\left\|d_{k+1}^{N e w 1}\right\| \leq T_1+\mathrm{A}_1 \mathrm{U}_1 \leq \mathrm{M}_1$

$\Rightarrow\left\|d_{k+1}^{\text {New } 1}\right\| \leq \mathrm{M}_1 \Rightarrow \frac{1}{\left\|d_{k+1}^{\text {New } 1}\right\|} \geq \frac{1}{\mathrm{M}_1}$

$\Rightarrow \sum_{k=1}^{\infty} \frac{1}{\left\|d_{k+1}^{\text {New } 1}\right\|^2} \geq \sum_{k=1}^{\infty} \frac{1}{\mathrm{M}_1{ }^2}=\frac{1}{\mathrm{M}_1{ }^2} \sum_{k=1}^{\infty} 1=+\infty$

According to Lemma (1), this leads to

$\lim _{k \rightarrow \infty} \inf \left\|g_k\right\|=0$

The second case: If $\beta_k=\beta_k^{\text {New } 2}$ in Eq. (10) we get:

$\Rightarrow\left\|d_{k+1}^{N e w 2}\right\| \leq\left\|g_{k+1}\right\|+\mid \beta_k^{\text {New2 }}\|\| d_k \|$

$\left|\beta_k^{\text {New } 2}\right|=\left|\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right|$

Using Eq. (5) we get the following

$\left|\beta_k^{\text {New } 2}\right| \leq\left|\frac{\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{c \sigma\left\|g_k\right\|^2}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right|$

$\Rightarrow\left|\beta_k^{\text {New } 2}\right| \leq \mathrm{A}_2$

$\Rightarrow\left\|d_{k+1}^{\text {New }}\right\| \leq\left\|g_{k+1}\right\|+\mathrm{A}_2\left\|d_k\right\|$

$\Rightarrow\left\|d_{k+1}^{N e w}\right\| \leq T_2+\mathrm{A}_2 \mathrm{U}_2 \leq \mathrm{M}_2$

$\Rightarrow\left\|d_{k+1}^{\text {New } 2}\right\| \leq \mathrm{M}_2 \Rightarrow \frac{1}{\left\|d_{k+1}^{\text {New }}\right\|} \geq \frac{1}{\mathrm{M}_2}$

$\Rightarrow \sum_{k=1}^{\infty} \frac{1}{\left\|d_{k+1}^{\text {New2 }}\right\|^2} \geq \sum_{k=1}^{\infty} \frac{1}{\mathrm{M}_2{ }^2}=\frac{1}{\mathrm{M}_2{ }^2} \sum_{k=1}^{\infty} 1=+\infty$

According to Lemma (1), this leads to

$\lim _{k \rightarrow \infty} \inf \left\|g_k\right\|=0$

The third case: If $\beta_k=\beta_k^{\text {New } 3}$ in Eq. (10) we get:

$\Rightarrow\left\|d_{k+1}^{\text {New } 3}\right\| \leq\left\|g_{k+1}\right\|+\left|\beta_k^{\text {New } 3}\right|\left\|d_k\right\|$

$\left|\beta_k^{N e w 3}\right|=\left|\frac{g_{k+1}^T y_k}{g_k^T g_k}-t_k\left(\frac{g_{k+1}^T d_k}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right|$

Using Eqs. (5) and (6) we get the following

$\left|\beta_k^{\text {New } 3}\right| \leq\left|\frac{1.2\left\|g_{k+1}\right\|^2}{\left\|g_k\right\|^2}-t_k\left(\frac{c \sigma\left\|g_k\right\|^2}{\left(d_k^T y_k\right)^2}\right)\left\|y_k\right\|^2\right|$

$\Rightarrow\left|\beta_k^{\text {New } 3}\right| \leq \mathrm{A}_3$

$\Rightarrow\left\|d_{k+1}^{\text {New } 3}\right\| \leq\left\|g_{k+1}\right\|+\mathrm{A}_3\left\|d_k\right\|$

$\Rightarrow\left\|d_{k+1}^{\text {New } 3}\right\| \leq T_3+\mathrm{A}_3 \mathrm{U}_3 \leq \mathrm{M}_3$

$\Rightarrow\left\|d_{k+1}^{\text {New } 3}\right\| \leq \mathrm{M}_3 \Rightarrow \frac{1}{\left\|d_{k+1}^{\text {New } 3}\right\|} \geq \frac{1}{\mathrm{M}_3}$

$\Rightarrow \sum_{k=1}^{\infty} \frac{1}{\left\|d_{k+1}^{N e w 3}\right\|^2} \geq \sum_{k=1}^{\infty} \frac{1}{\mathrm{M}_3{ }^2}=\frac{1}{\mathrm{M}_3{ }^2} \sum_{k=1}^{\infty} 1=+\infty$

According to Lemma (1), this leads to

$\lim _{k \rightarrow \infty} \inf \left\|g_k\right\|=0$

The habitat of the dwarf mongoose typically includes regions abundant in termite mounds, rocks, and hollow trees, providing ample shelter, particularly in semi-desert areas and savannah shrublands across Africa. Remarkably, the dwarf mongoose ranks among the smallest carnivorous animals known. Its body length is approximately 47 cm. The weight of an adult ferret is about 400 grams. Females outperform males and children outperform their older brothers. Division of labor and altruism are the highest recorded in mammals. The dwarf mongoose relies on a semi-nomadic lifestyle, covering a relatively long distance. Therefore, it detects hunting areas better without returning to its previous sleeping mound, so it does not exploit it well and does not exhaust all its prey. The group of dwarf mongooses operates cohesively during feeding, guided by the chirps emitted by the alpha female. The distance covered by a dwarf mongoose group is contingent upon factors such as the presence of offspring and the overall size of the group. Figure 2 shows The optimization procedures of the DMO.

2.jpg

Figure 2. The optimization procedures of the proposed DMO

4.1 Mathematical model for DMO

The initial population is randomly initialized within the search space defined by the upper bound (UB) and lower bound (LB) for the given optimization problem. This process is represented by Eq. (17).

$\mathrm{X}=\left[\begin{array}{ccccc}\mathrm{x}_{1,1} & \mathrm{x}_{1,2} & \cdots & \mathrm{x}_{1, \mathrm{~d}-1} & \mathrm{x}_{1, \mathrm{~d}} \\ \mathrm{x}_{2,1} & \mathrm{x}_{2,2} & \cdots & \mathrm{x}_{2, \mathrm{~d}-1} & \mathrm{x}_{2, \mathrm{~d}} \\ \vdots & \vdots & \cdots & \vdots & \vdots \\ \mathrm{x}_{\mathrm{n}, 1} & \mathrm{x}_{\mathrm{n}, 2} & \cdots & \mathrm{x}_{\mathrm{n}, \mathrm{d}-1} & \mathrm{x}_{\mathrm{n}, \mathrm{d}}\end{array}\right]$    (17)

The population of current candidates, denoted as X, is randomly generated using Eq. (18), where, $x_{i, j}$ represents the position of dimension (j) of the population (i), n refers to the size of the population, and d refers to the dimension of the problem.

$x_{i, j}=$ unifrnd(Varmin, Varmax,Varsize$)$    (18)

where, unifrnd is a random number uniformly distributed between the lower and upper limits of the problem (Varmin, Varmax). Varsize is the dimension size of the problem. The social structure of the dwarf mongoose is segmented into three groups: the alpha group, the scouts, and the babysitters. Each group plays a role in contributing to compensatory behavioral conditioning, as outlined below.

4.1.1 Alpha group

After preparing the population, the fitness of each solution is evaluated by calculating the population fitness using Eq. (19). Subsequently, the alpha female ($\alpha$) is selected based on this probability.

$\alpha=\frac{f_i t_i}{\sum_{i=1}^n f_i t_i}$    (19)

To determine the position of the food candidate $\left(x_{i+1}\right)$, DMO uses Eq. (20).

$x_{i+1}=x_i+p h_i *$ peep    (20)

So that $p h_i$ is a random number uniformly distributed between [-1,1]. The alpha female’s vocalization that keeps the family within a path is denoted by peep. The initial sleep stack is set to ∅ in which all the dwarf mongoose sleep. After each iteration, the sleep hill is calculated as shown in Eq. (21).

$s m_i=\frac{f_i t_{i+1}-f_i t_i}{\max \left\{\left|f_i t_{i+1}, f_i t_i\right|\right\}}$    (21)

The average value of the sleep hill is obtained using Eq. (22).

$\emptyset_i=\frac{\sum_{i=1}^n s m_i}{n}$    (22)

4.1.2 Scouting group

The dwarf mongoose optimization algorithm starts with the exploration phase, so that the scout group searches for the new sleeping pile, so that the mongoose does not visit the previous sleeping pile, which ensures the exploration process. This stage is to evaluate the success or failure in knowing where the next sleeping pile [20]. Eq. (23) simulates the scouting mongoose.

$\begin{gathered}x_{i+1}= \\ \left\{\begin{array}{c}x_i-c f * p h_i * \operatorname{rand}(0,1) *\left[x_i-m_i\right] \text { if } \emptyset_{i+1}>\emptyset_i \\ x_i+c f * p h_i * \operatorname{rand}(0,1) *\left[x_i-m_i\right] \text { else }\end{array}\right.\end{gathered}$    (23)

The variable rand represents a random number between 0 and 1. 1. $c f=\left(1-\frac{\text { iter }}{\text { max_iter }}\right)^{\left(\frac{\text { 2.iter }}{\text { max_iter }}\right)}$ It denotes the parameter responsible for the collective voluntary movement of the ferret group, and it decreases linearly with iterations. The vector m determines the direction of movement of the mongoose towards the new sleeping pile.

4.1.3 Babysitters group

The care of the young within the group is typically assigned to lower-ranking members who take turns in this role. This arrangement permits the alpha female, often the mother, to guide the others in daily food-gathering activities. She typically revisits the young midday or during the evening to nurse them. The quantity of caregivers is influenced by the overall number of the group. This dynamic is factored into the population algorithm, adjusting the total count in accordance with a predetermined percentage.

4.2 Algorithm for DMO

1-Start

2-Define algorithm parameters: [peep].

3-Create initial mongoose population (search agents): n.

4-Set the initial number of caregivers: bs.

5-Adjust the population: n = n - bs.

6-Define the caregiver rotation threshold L.

7-For $i=1$: maximum iteration.

8-Assess mongoose fitness and start the time counter.

9-Identify the alpha mongoose using Eq. (19).

10-Generate a potential food location according to Eq. (20).

11-Evaluate new fitness of $x_{i+1}$.

12-Determine the sleeping mound location with Eq. (21).

13-Calculate the average of the sleeping mound from Eq. (22).

14-Formulate the movement vector, $m=\sum_{i=1}^n \frac{x_i \mathrm{sm}_{\mathrm{i}}}{x_i}$.

15-Exchange babysitters if C > L and set.

16-Initialize bs position and calculate fitness, $f_i t_i \leq \alpha$.

17-Simulate the scout mongoose next position using Eq. (23).

18-Update best solution so far

19-End of Iteration Loop

20-Output Best Solution

21-End of Algorithm