## 5 Resolution Methods
### Uncertain Simulation (99-Method)  
* The first row represents the values of the uncertain distribution.
* The second row contains the associated inverse values.  
This approach is illustrated in Table 1, which provides a detailed view of how uncertain variables are managed in the context of total cost calculations.  
The value \(M\{\text{TC}(x,\Theta)<\text{TC}_{q}\}\) corresponds to \(\Phi(\text{TC}(x,\text{TC}_{q}))\), which can be approximately estimated as follows:  
\[D(x,\text{TC}_{q})=\frac{k}{999}\quad\text{if}\quad s_{k}<\text{TC}_{q}<s_{k+1 }\text{ for some }k\]  
where \(s_{1},\dots,s_{999}\) are given by Table 1.  
Pseudocode for Uncertain Simulation (99-Method)  
The main steps of uncertain simulation (99-method) are explained as follows (see the Algorithm 2):  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\alpha\) & **0.01** & **0.02** & **0.999** \\ \hline \(II^{-1}(x,\alpha)\) & \(s_{1}\) & \(s_{2}\) & \(s_{999}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Inverse uncertainty distribution of \(II^{-1}(x,\alpha)\).  
Figure 1: Supply chain network.  
```
Initialize parameters:  Set \(N=\) number_of_iterations  Set \(M=\) number_of_samples_per_iteration  Initialize results_list Define the uncertain variables and their distributions:  Define the distribution for uncertain variable \(\xi\) Main simulation loop: for\(i=1\) to \(N\)do  Initialize \(e=0\) Sample uncertain variables: for\(k=1\) to \(M\)do  Generate a sample \(\xi_{k}\) from the distribution of \(\xi\)  Compute \(TC(Q;\xi_{k})\) using the given model Check condition for uncertain value: if\(TC(Q;\xi_{k})<TC_{0}\)then  Increment \(e\) by \(1\) endif endfor Store the result for this iteration:  Append \(e/M\) to results_list endfor
```  
**Algorithm 3** Uncertain Simulation for Chance Value  
Algorithm 3 Uncertain Simulation for Chance Value  
The uncertain simulation process involves the following steps:  
1. **Sampling**: Generate a sample of uncertain values.
2. **Cost calculation**: Compute the total cost (TC) for each sample.
3. **Threshold check**: Verify if the total cost is below a predefined threshold.
4. **Repetition**: Repeat the sampling and checking process \(N\) times.
5. **Proportion calculation**: Determine the proportion of samples for which the total cost is below the threshold. This method evaluates how often the total cost meets the threshold under uncertainty (see the Algorithm 3).  
```
Initialize \(e\) to \(0\) for\(i\gets 1\) to \(N\)do  Generate sample \(\Theta\) based on probability \(P_{r}\)  Define \(\omega\) as a set of uncertain variables for each element \(\omega_{j}\) in \(\omega\)do  Calculate \(TC\_value=TC(x,\Theta\{\omega_{j}\})\) if\(TC\_value<TC_{0}\)then  Increment \(e\) by \(1\) endif endfor return\(e/N\)
```  
**Algorithm 4** Uncertain Simulation of \(M\{\text{TC}(x,\Theta)<\text{TC}_{q}\}\)  
To estimate the probability distribution based on the 99-method, Algorithm 2 follows these steps:  
1. **Value iteration**: Iterate over the values in the uncertainty table.
2. **Distribution lookup**: For each value, find the corresponding distribution value in the inverse uncertainty table.  
3. **Threshold comparison**: Compare the total cost (TC) with the predefined threshold \(\text{TC}_{q}\).
4. **Distribution value determination**: Determine the appropriate distribution value based on the threshold comparison.  
This algorithm helps in estimating the probability distribution by checking how often the total cost is below the threshold using the inverse uncertainty table ( see the Algorithm 4).  
```
Initialize \(DC\) to 0  Set \(u\) to 1 while\(u<999\)do if\(s_{u}<TC_{q}<s_{u+1}\)then  Set \(DC\) to \(u/999\)  Break loop else  Increment \(u\) by 1 endif endwhile return\(DC\)
```  
**Algorithm 4** Uncertain Simulation  
The flowchart illustrates the new framework developed in this work to solve the proposed uncertain dependent chance goal programming problem (see Figure 2).  
Figure 2: Flowchart of the proposed framework.