## 3 Data-Driven Robust Scheduling Method
### Kernel-Based Estimation of Uncertain Parameters  
In the actual production process, uncertain parameters are usually regarded as bounded uncertainties, and managers often define the bounds of uncertain parameters based on experience. In the absence of domain-specific knowledge, the analysis of historical data provides a practical way to characterize uncertain intervals. For example, the upper and lower bounds of the interval can be specified as the maximum and minimum of data samples. However, this method may be too conservative because the worst-case scenarios at the boundary are very unlikely to occur in practice.  
In this paper, we decompose the uncertain parameters \(\widehat{SU}_{i}\) and \(\widehat{PT}_{i}\) in the following forms:  
\[\widehat{SU}_{i}=SU_{i}+\xi_{1,i}\widehat{SU}_{i} \tag{8}\]  
\[\widehat{SU}_{i}=SU_{i}+\xi_{1,i}\hat{PT}_{i} \tag{9}\]  
where \(SU_{i}\) and \(PT\) represent the nominal value of the uncertain parameter; \(\hat{SU}_{i}\) and \(\hat{PT}_{i}\) denote the magnitude of the random perturbation, which can be approximated by the bounds of the samples; \(\xi_{i}\) is a normalized random variable, which controls the range of uncertainties. By specifying the interval of \(\xi_{i}\), inherent uncertain parameters can be transformed to bounded uncertainties that are easy to model mathematically.  
In recent years, some researchers have proposed to adopt statistical analysis and machine learning methods to learn the distributional information from the data and then establish a data-driven uncertainty set to reduce conservatism. As a well-known nonparametric estimation approach, kernel density estimation (KDE) has proven to be an effective tool for modeling probability density distribution [29]. Assume that we have the data of uncertain parameters drawn i.i.d. according to distribution \(\mathbb{P}\). In this paper, we propose to use the confidence region of the probability density function to quantify the uncertain interval by KDE.  
Given historical data of uncertain parameters, KDE learns the probability density function of the samples according to  
\[p(x)=\frac{1}{nh}\sum_{i=1}^{n}\varphi(\frac{x-x_{i}}{h}) \tag{10}\]  
where \(p(x)\) denotes the estimated density function of uncertain parameter \(x\); \(h\) is a positive parameter representing the bandwidth; n is the number of samples; \(\varphi(\cdot)\) is a kernel function, which satisfies  
\[\varphi(u)\geq 0 \tag{11}\]  
\[\int\varphi(u)du=1 \tag{12}\]  
In our research, the Gaussian function commonly used in engineering problems is introduced as the kernel function, and the value of bandwidth \(h\) is determined through pilot experiments.  
Based on the probability density functions of uncertainties, the confidence interval of the uncertain values can be calculated through interval estimation. To reduce the conservatism of the uncertain interval, we introduce an adjustable parameter \(\sigma\) to control the number of uncertain values that lie within the uncertain interval. Taking the uncertain parameter \(\widehat{SU}\) as an example, the corresponding \(\sigma\) can be described as  
\[\sigma_{i}=\frac{SU_{i}^{1-\eta}}{SU_{i}^{max}} \tag{13}\]  
where \(\hat{SU}_{i}^{1-\eta}\) represents the value of \(\hat{SU}_{i}\) under the confidence level \(1-\eta\), which can be calculated by the quartile function of \(\widehat{SU}_{i}\)[30], and \(\hat{SU}_{i}^{max}\) denotes the maximum range of uncertainty. When the confidence level \(1-\eta\) equals 1, all of the sampled points are contained in the uncertain interval and \(\hat{SU}_{i}^{1-\eta}=\hat{SU}_{i}^{max}\). Conversely, as the value of \(1-\eta\) falls to 0, fewer samples are considered (see Figure 2). Thus, the region of \(\widehat{SU}_{i}\) determined by \(\sigma_{i}\) can be described as:  
\[\widehat{SU}_{i}=SU_{i}+\xi_{1,i}\hat{SU}_{i}^{max}\quad\xi_{1i}\in[-\sigma_ {i},\sigma_{i}] \tag{14}\]  
In the following optimization algorithm, the above data-driven uncertain interval is used to assist the encoding and decoding procedure.