Algorithm 1. The pseudo-code of HFBOA
	Generate the initialize population of the butterflies $X_i(i=1,2,\cdots ,n)$ randomly;
	Initialize the parameters ${\beta }_0,\gamma$, and $\mu$
	Define sensory modality c, power exponent a and switch probability p
	for $i=1:n$
	Calculate the fitness value of each butterflies
	end for
	while $t<T_{max}$
	for $i=1:n$
	Update the fragrance of current search agent by Equation (6)
	for $j=1:n$
	if $fitnes{s}_i<fitnes{s}_j$
	Update the attractiveness $\beta$ and $R_{ij}$ by Equation (9) and Equation (10) respectively
	else
	Continue
	end if
	% $\mathbf{Case}\mathbf{1}$
	if $rand<sp$
	Update the position using Equation (7)
	else
	Update the position using Equation (8)
	end if
	% $\mathbf{Case}\mathbf{2}$
	if $rand<sp$
	Update the position using Equation (3)
	else
	Update the position using Equation (8)
	end if
	Calculate the fitness value of each butterflies
	Find the best fitness $F_i$
	end for
	end for
	Update the value of power exponent c and parameter $\alpha$ using Equations (11) and (12)
	$t=t+1$
	end while
	Output the best fitness