# Estimation of Distribution Algorithms with Fuzzy Sampling for Stochastic Programming Problems
## 2 Estimation of Distribution Algorithms
This means that the observed linear dependencies are measured by their correlation coefficient with each other, as follows: \[corr(x_{i},x_{j})=\frac{cov(x_{i},x_{j})}{\sigma_{i}\sigma_{j}},\] (3) where \(corr(x_{i},x_{j})\) is the linear correlation coefficient between \(x_{i}\) and \(x_{j}\), \(cov(x_{i},x_{j})\) is their covariance, \(\sigma_{i}\) and \(\sigma_{j}\) are their standard deviations, respectively, and \(i,j=1,\ldots,n\). Briefly, all variables are divided into two sets: \(W\) and \(S\), where \(W\) is the set of weakly dependence variables, and \(S\) is the set of strong dependent variables. These variable sets are defined as follows: \[W = \{x_{i}:|corr(x_{i},x_{j})|\leq\theta,\ \ \forall j=1,\ldots,n,j \
eq i,i=1,\ldots,n\},\] (4) \[S = \{x_{i}:x_{i}\
otin W,\ \ i=1,\ldots,n\},\] (5) where \(\theta\) is a threshold \((0\leq\theta\leq 1)\), and this reflects how much the user trusts the univariate model in the problem. Algorithm 2 shows the main flow of the WI strategy. ```
```
1:Use \(m\) individuals to calculate the correlation matrix \(C=(c_{ij})\), where \(c_{ij}=corr(x_{i},x_{j}),i,j=1,\ldots,n\). 2:Use \(C\) to construct \(W\) and \(S\) as defined in Equations (4) and (5), respectively. 3:Estimate a univariate model for \(W\) based on the \(m\) selected individuals. ```  
**Algorithm 2** Pseudo-code for WI* **Subspace Modeling (SM). ** This strategy is applied on the \(S\) set. The performance of the multivariate model needs a large population size and the complexity of computations increases frequently. The SM strategy is applied on the variables set in \(S\), which preferably has a limited number of variables. If the size \(|S|\) of set \(S\) is not limited, then the population points are projected to several subspaces of the \(n\) dimensional search space. Then, a multivariate model can be built for each subspace, which means that the dependence is considered only between variables in the same subspace. The main steps of the SM strategy are explained in Algorithm 3. ```
1: Construct \(S\) as in Equation (5). 2: Randomly partition \(S\) into \(\lceil|S|/c\rceil\) nonintersected subsets: \(S_{1},S_{2},\ldots S_{\lceil|S|/c\rceil}\). 3: Estimate a multivariate model for each subset based on \(m\) selected individuals. ```  
**Algorithm 3** Pseudo-code for SM  
Parameter \(c\) is a predefined one that controls the number of the subspaces, where \((1\leq c\leq n)\). After carrying out the WI and SM strategies, the final joint probability distribution function (pdf) has the following form:  
\[f(\mathbf{x}_{i})=\prod_{x_{i}\in W}\phi_{i}(\mathbf{x}_{i})\cdot\prod_{k=1}^{ \lceil|S|/c\rceil}\psi_{k}(\mathbf{s}_{k}), \tag{6}\]  
where \(\phi_{i}(\cdot)\) is the univariate pdf of variable \(x_{i}\in W\), and \(\psi_{k}(\cdot)\) is the multivariate pdf of the variables in \(S_{k}\). The main steps of the EDA-MCC method are illustrated in Algorithm 4. ```
1: Generate an initial population \(P\) of \(M\) individuals. 2:repeat
3: Select \(m\leq M\) individuals from \(P\). 4: Call Algorithms 2 and 3 sequentially to build a model, as in Equation (6). 5: Generate new individuals \(P^{\prime}\): Variable values of an individual are generated independently from \(\phi_{i}(\cdot)\) and \(\psi_{k}(\cdot)\). Then, combine all generated variable values together to produce an individual. 6:\(P\gets P+P^{\prime}\). 7:until a stopping criterion is met. ```  
**Algorithm 4** Pseudo-code for EDA-MCC