# PSO-SVM Based Performance-Driving Scheduling Method for Semiconductor Manufacturing Systems
## 3 Framework of the Proposed Scheduling Method
### Pso-svm  
Based on the minimization induction principle of structural risk, the SVM aims to minimize the upper bound of generalization error, which includes the whole training error and a confidence level. In SVMs, the solution process relies on the subset of the training data points which are called support vectors. The SVM prediction function can be formulated as follows:  
Assume that \((x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{k},y_{k})\) is input vectors with n-dimension, where \((x_{i},y_{i})\) denotes a set of data points, \(x_{i}\) denotes the travel time at interval \(i\), \(y_{i}\) denotes the desired value, such as the observed travel time at the next interval (\(i+1\)), \(X\) denotes the input vector space, and \(Y\) denotes the output variable space. The approximate function is shown in (8):  
\[f(x)=\omega\cdot\varphi(x)+b \tag{8}\]  
where \(\varphi(x)\) denotes the high-dimensional feature space nonlinearly mapped from \(X\). Both coefficient \(\omega\) and coefficient \(b\) can be estimated by minimizing the regularized risk function (9):  
\[\frac{1}{2}\|\omega\|^{2}+C\frac{1}{k}{\sum_{i=1}^{k}{L_{\varepsilon}(y_{i},f( x_{i}))}} \tag{9}\]  
where \(\|\omega\|^{2}\) is the regularized term. By minimizing \(\|\omega\|^{2}\), it will make the function as flat as possible to control the functional capacity. \(C\) is a user-prescribed parameter used to trade the model smoothness and the empirical risk. \(\varepsilon\) is a user-prescribed parameter. \(C\frac{1}{k}{\sum_{i=1}^{k}{L_{\varepsilon}(y_{i},f(x_{i}))}}\) denotes the empirical error computed through the \(\varepsilon\)'insensitive loss function which is shown in (10):  
\[L_{\varepsilon}(y_{i},f(x_{i}))=\left\{ \begin{array}{cl}|y_{i}-f(x_{i})|- \varepsilon,&|y_{i}-f(x_{i})|\geq\varepsilon\\ 0,&|y_{i}-f(x_{i})|<\varepsilon\end{array} \right. \tag{10}\]  
In the case of infeasibility, one can introduce slack variables \(\left(\xi_{i},\xi_{i}^{*}\right)\) to cope with the infeasible constraints of the optimization problem. Then, the above problem can be formalized as:  
\[\min\frac{1}{2}\|\omega\|^{2}+C{\sum_{i=1}^{k}{(\xi_{i}+\xi_{i}^{*})}} \tag{11}\]  
subject to:  
\[\left\{ \begin{array}{l}y_{i}-\omega\cdot\varphi(x_{i})-b\leq\varepsilon+ \xi_{i}\\ \omega\cdot\varphi(x_{i})+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0\end{array} \right. \tag{12}\]  
To obtain the solution to Equation (11), the Lagrangian function L is introduced:  
\[ \begin{array}{ll}\text{L}=&\frac{1}{2}\|\omega\|^{2}+C{\sum_{i=1}^{k}{(\xi_ {i}+\xi_{i}^{*})}}-{\sum_{i=1}^{k}{a_{i}(\varepsilon+\xi_{i}-y_{i}+\omega\cdot \varphi(x)+b)}}\\ &-{\sum_{i=1}^{k}{(\eta_{i}\xi_{i}+\eta_{i}^{*}\xi_{i}^{*})}}-{\sum_{i=1}^{k} {a_{i}^{*}(\varepsilon+\xi_{i}^{*}-y_{i}-\omega\cdot\varphi(x)+b)}}\end{array}  \tag{13}\]  
After taking the Lagrangian and conditions for optimality, this optimization formulation can be transferred into the following dual problem shown in (14):  
\[ \begin{array}{ll}maxW\left({a_{i},a_{i}^{*}}\right)=&{\sum_{i=1}^{k}{y_{i} \big{(}{a_{i}-a_{i}^{*}}\big{)}}-\varepsilon{\sum_{i=1}^{k}{\big{(}{a_{i}+a_{ i}^{*}}\big{)}}}}\\ &-\frac{1}{2}{\sum_{i=1}^{k}{\sum_{i=1}^{k}{\big{(}{a_{i}-a_{i}^{*}}\big{)} \big{(}{a_{j}-a_{j}^{*}}\big{)}}\big{(}{\varphi(x_{i})\cdot\varphi(y_{i})} \big{)}}}\end{array}  \tag{14}\]  
subject to:  
\[{\sum_{i=1}^{k}{({a_{i}-a_{i}^{*}})}}=0,{a_{i},a_{i}^{*}}\in[0,C] \tag{15}\]  
where \(a_{i},a_{i}^{*}\) are Lagrangian multipliers. Then, the dual representation of the model becomes:  
\[f(x)={\sum_{i=1}^{k}{({a_{i}-a_{i}^{*}})\varphi(x_{i})\cdot\varphi(x)+b}} \tag{16}\]  
Equation (16) can be rewritten as Equation (17) by invoking the kernel function \(K\big{(}{x_{i},x_{j}}\big{)}\)  
\[f(x)={\sum_{i=1}^{k}{({a_{i}-a_{i}^{*}})K\big{(}{x_{i},x_{j}}\big{)}}}+b \tag{17}\]  
PSO, coming from the behavioral study on birds' predation, is an evolutionary computing technique that imitates natural swarm behavior. The basic idea is to find the optimal solution through cooperation and information sharing among individuals in the swarm.