Algorithm 3 Simulated Annealing Algorithm

Let$x$=$x_0$.

n = length($x_0$)

$T=T_0$

M = 100, % m is the iteration times at different temperatures

$T_{min}$ = 1e-3

Partition = [[];

While $T$ > $T_{min}$

Temp = zeros(m,n + 1);

for k = 1:m

If rand(1) > 0.6

RC_PNB search algorithm,$x_{new}$←neighbor($x$)

else

PN_VC search algorithm,$x_{new}$←neighbor($x$)

end

If F($x_{new}$) < F($x_0$)

$x$ ← $x_{new}$

Elseif /(1+ $e^{\frac{F\left(x_{new}\right)- F\left(x_0\right)}{T}}$ ) ≥ random(0, 1):

$x$ ← $x_{new}$

End

temp(k,:) = [$x$, F($x$)];

end

[$F_{min}$,index] = min(temp(:,end)); %Find the optimal partition at current temperature.

Partition = [partition; temp(index,end)];

$T$ = $T\times \sigma$; here $\sigma$ is cooling rate

End

[ $F_{opt}$ ,index1] = min(Partition (:,end));

$x_{opt}$ = Partition (index1,1:end-1); %the final state is $x_{opt}$ , optimal value of objective function is $F_{opt}$