2.2.1. Clan-updating Operator

According to the natural habits of elephants, a matriarch leads the elephants in each clan. Therefore, the new position of each elephant *ci* is influenced by matriarch *ci*. The elephant *j* in clan *ci* can be calculated by Equation (1).

$$\mathbf{x}\_{nm\mu;i,j} = \mathbf{x}\_{ci,j} + a \times (\mathbf{x}\_{best,ci} - \mathbf{x}\_{ci,j}) \times r \tag{1}$$

where *xnew,ci,j* and *xci,j* present the new position and old position for elephant *j* in clan *ci*, respectively. *xbest,ci* is matriarch *ci* which represents the best elephant in the clan. *a* ∈ [0,1] indicates a scale factor, *r* ∈ [0,1]. The best elephant can be calculated by Equation (2) for each clan.

$$\mathbf{x}\_{n\mathbf{w},ci,j} = \boldsymbol{\beta} \times \mathbf{x}\_{c\text{center},ci} \tag{2}$$

where β ∈ [0,1] represents a factor which determines the influence of the *xcenter,ci* on *xnew,ci,j*. *xnew,ci,j* is the new individual. *xcenter,ci* is the center individual of clan *ci*. It can be calculated by Equation (3) for the *d*-th dimension.

$$\mathbf{x}\_{\text{center},ci,d} = \frac{1}{n\_{ci}} \times \sum\_{j=1}^{n\_{ci}} \mathbf{x}\_{ci,j,d} \tag{3}$$

where 1 ≤ *d* ≤ *D* and *nci* indicate the number of elephants in clan *ci*. *xci,j,d* represents the *d*-th dimension of elephant individual *xci,j*. *xcenter,ci* is the center of clan *ci* and it can be updated by Equation (3).
