## 4 Chance-Constrained Approach for Processing Time Predictions  
In this article, the chance-constrained programming method proposed by [PERSON] and [PERSON] [28] is adopted to mitigate the impact of uncertainties arising from fluctuations in processing times. First, by introducing confidence levels, uncertain constraints are transformed into deterministic ones, enabling effective modeling and prediction of processing time variations. Then, a robust pre-scheduling plan can be generated based on this predictive method.  
The statistical results of the FJSP reported in the literature [29; 30; 31] demonstrate that operation processing times typically follow either a gamma distribution or a normal distribution in practical production contexts. Therefore, in this study, it is assumed that the actual processing time \(\widetilde{p}_{ij}^{lk}\) of each operation \(O_{ij}\) follows a normal distribution:  
\[\widetilde{p}_{ij}^{lk}\sim N(\mu_{ij}^{lk},\sigma_{ij}^{lk}), \tag{12}\]  
where \(\mu_{ij}^{lk}\) is the theoretical processing time of the given instance \(p_{ij}^{lk}\), while \(\sigma_{ij}^{lk}\) is defined as \(\kappa\cdot p_{ij}^{lk}\), where \(\kappa\) represents the degree of random fluctuation.  
The actual processing time can be expressed as:  
\[\widetilde{p}_{ij}^{lk}=p_{ij}^{lk}+\sigma_{ij}^{lk}\times Z,\;\forall i,j,k, l,\;\;\;Z\sim N(0,1). \tag{13}\]  
The completion time \(C_{ij}^{lk}\) of operation \(O_{ij}\) is then given by:  
\[C_{ij}^{lk}=S_{ij}^{lk}+\widetilde{p}_{ij}^{lk},\;\forall i,j,l,k. \tag{14}\]  
Since \(\widetilde{p}_{ij}^{lk}\) is a random variable, \(C_{ij}^{lk}\) also possesses uncertainty. To limit the deviation caused by this uncertainty, Equation (14) is transformed into the following inequality constraint:  
\[(S_{ij}^{lk}-C_{ij}^{lk})+\widetilde{p}_{ij}^{lk}\leq\varsigma,\;\forall i,j, k,l, \tag{15}\]  
where \(\varsigma\) is a small positive value.  
Substituting Equation (13) into Equation (15) yields the following:  
\[(S_{ij}^{lk}-C_{ij}^{lk}+p_{ij}^{lk})+\sigma_{ij}^{lk}\times Z\leq\varsigma,\ \forall i,j,k,l, \tag{16}\]  
Here, based on the chance-constrained programming method, we account for the possibility that decisions may fail to satisfy the constraints under adverse conditions. To address this, a principle is adopted that allows a certain degree of violation of constraints. Specifically, the constraints are satisfied with at least a probability of \(\alpha\), which can be expressed as:  
\[\Pr((S_{ij}^{lk}-C_{ij}^{lk}+p_{ij}^{lk})+\sigma_{ij}^{lk}\times Z\leq \varsigma)\geq\alpha. \tag{17}\]  
By standardizing the above equation, we transform it into the standard normal distribution as follows:  
\[\Pr\left(Z\leq\frac{\varsigma+(C_{ij}^{lk}-S_{ij}^{lk}-p_{ij}^{lk})}{\sigma_{ ij}^{lk}}\right)\geq\alpha. \tag{18}\]  
Since \(Z\sim N(0,1)\), the corresponding \(Z\) value can be determined using the standard normal distribution table. This allows us to further simplify the constraint:  
\[Z_{\alpha}\leq\frac{\varsigma+(C_{ij}^{lk}-S_{ij}^{lk}-p_{ij}^{lk})}{\sigma_{ ij}^{lk}}. \tag{19}\]  
Thus, we have  
\[(S_{ij}^{lk}-C_{ij}^{lk}+p_{ij}^{lk})+\sigma_{ij}^{lk}\times Z_{\alpha}\leq \varsigma,\ \forall i,j,k,l, \tag{20}\]  
where \(Z_{\alpha}\) is the value corresponding to the cumulative probability \(\alpha\) in the standard normal distribution. By substituting the quantile \(Z_{\alpha}\), the random fluctuation component of the processing time can be replaced, and the random variable \(\widetilde{p}_{ij}^{lk}\) is transformed into a deterministic variable:  
\[\widetilde{p}_{ij}^{lk}=p_{ij}^{lk}+\sigma_{ij}^{lk}\times Z_{\alpha},\ \forall i,j,k,l, \tag{21}\]  
Through the above derivation, the uncertainty model containing random variables is transformed into a deterministic model. This ensures the feasibility of the constraints at the confidence level \(\alpha\), effectively addressing the uncertainties caused by fluctuations in the processing time.