## 2 Method Description
### The Second Phase of the Proposed Method  
In the second phase of the algorithm, the value interval for the network parameters is constructed using a technique based on Simulated Annealing. Simulated Annealing is an optimization procedure used in a variety of cases, such as resource allocation [33], portfolio problems [34], energy problems [35], biology [36], etc. The algorithm starts from a large value of a factor called temperature, which is gradually decreased. For large values of temperature, the algorithm performs a wide exploration of the search space, and at low values it focuses around some minimum of the objective function. The Simulated Annealing variant used here minimizes interval functions denoted as  
\[f(x)=\left[f_{\textrm{min}}(x),f_{\textrm{max}}(x)\right] \tag{5}\]  
Also, in order to compare two intervals \(a=[a_{1},a_{2}]\) and \(b=[b_{1},b_{2}]\), the operator \(D(a,b)\) is incorporated with the following definition:  
\[D(a,b)= \begin{cases}\textrm{TRUE},&a_{1}<b_{1}\;\textrm{OR}\;(a_{1}=b_{1}\; \textrm{AND}\;a_{2}<b_{2})\\ \textrm{FALSE},&\textrm{otherwise}\end{cases}  \tag{6}\]  
The method used below assumes that there are value intervals for the network parameters. The main steps of the method used in the second phase are shown as a procedure in Algorithm 3. The calculation of the functional values for such value intervals is performed using Algorithm 4. The algorithm starts from an initial value interval, which is based on the result of the first phase of the overall process. At each iteration it generates random value intervals that are close to the current value interval. If a value interval with a smaller functional value than the current one is presented, it is accepted; otherwise it will be accepted with a probability that is high for large temperature values and decreases significantly as the temperature value drops. This means that the algorithm makes a wide exploration of the search space for high temperature values and centers around some minimum as the temperature value decreases. ``` Function\(\textrm{simMethod}\left(\overrightarrow{z},F,T_{0},r_{T},N_{\textrm{eps}},a\right)\) Inputs: The vector \(\overrightarrow{z}\) with the initial set of parameters, the limit factor \(F>1\), the initial temperature \(T_{0}\), the reduction factor for temperature \(r_{T},\;r_{T}>0,\;r_{T}<1\), the number of samples taken at every iteration denoted as \(N_{\textrm{eps}}\) and \(a>0\) the perturbation factor. 1. Construct the bound vectors \(L^{*},\;R^{*}\) using the vector \(\overrightarrow{z}\) of the first phase as follows: \[ \begin{array}{rcl}L^{*}_{i}&=&-F\times z_{i}&,&i=1,\ldots,n\\ R^{*}_{i}&=&F\times z_{i}&,&i=1,\ldots,n\end{array} \]

The parameter \(F\) is a positive number greater than \(1\) and is used as a limiting factor for the network parameters. Figure 3: The scheme of the particles in the proposed method. ```
2. Set \(k=0\)
3. Set \(x_{b}=[L^{*},R^{*}],y_{b}=\text{fitness}(L^{*},R^{*},N_{s})\). This function utilizes the function fitness() outlined in Algorithm 4. 4. Set \(x_{c}=x_{b},y_{c}=y_{b}\)
5. For\(i=1,\dots,N_{\text{eps}}\)do 1. **Produce** a random sample \(x_{t}=(L_{t},R_{t})\) using the procedure of Algorithm 5 using as inputs to the function the interval \(x_{c}=[L_{c},R_{c}]\) and the double precision number \(a\). 2. **Set \(y_{t}=\text{fitness}(L_{t},R_{t},N_{s})\)** which calls the function provided in Algorithm 4. 3. **If \(D(y_{t},y_{c})=\text{TRUE}\) then Set \(x_{c}=x_{t},y_{c}=y_{t}\)** 4. **Else Set \(x_{c}=x_{t}\)** with probability \(\min\Bigl{\{}1,\exp\Bigl{(}-\frac{f_{t}-f_{c}}{T_{k}}\Bigr{)}\Bigr{\}}\) 5. **End if** 6. **If \(D(x_{c},x_{b})=\text{TRUE}\) then Set \(x_{b}=x_{c},y_{b}=y_{c}\)** 6. **End For** 7. **Set \(T_{k+1}=T_{k}r_{T}\)** 8. **Set \(k=k+1\). 9. **If \(T_{k}\leq\epsilon\) stop** and return\(x_{b}=[L_{b},R_{b}]\) as the best located interval of bounds. 1. Go to step 5. End function
```

**Algorithm 4** The algorithm used to calculate the function value for intervals of parameters. ```
function\(\text{fitness}(L,R,N_{s})\)
1.