# PSO-SVM Based Performance-Driving Scheduling Method for Semiconductor Manufacturing Systems
## 3 Framework of the Proposed Scheduling Method
### Pso-svm
Assume one swarm has \(n\) particles, and each particle has a position vector and a velocity vector. Take particle \(i\) as an example: the position vector \(X_{i}=(x_{i1},x_{i2},\ldots,x_{iD})\), velocity vector \(V_{i}=(v_{i1},v_{i2},\ldots,v_{iD})\), where \(i=1,2,\ldots,n\). In the D-dimensional search space, each particle will be regarded as a potential solution to the problem to be solved. In each generation, particles are accelerated toward the previous best position and the global best position, where the previous best position of particle \(i\) will be denoted as \(P_{i}=(p_{i1},p_{i2},\ldots,p_{iD})\), the global best position will be denoted as \(P_{g}=\left(p_{g1},p_{g2},\ldots,p_{gD}\right)\). The new velocity value \(id\) will be used to compute the next position of the particle. The process will be iterated until either the maximum iterations or the minimum error is reached. Update the velocity and position of particles according to (18) and (19):  
\[v_{id}^{k+1}=\omega\times v_{id}^{k}+c_{1}\times rd_{1}^{k}\times\left(p_{id}^ {k}-x_{id}^{k}\right)+c_{2}\times rd_{2}^{k}\times\left(p_{gd}^{k}-x_{gd}^{k}\right) \tag{18}\]  
\[x_{id}^{k+1}=v_{id}^{k+1}+x_{id}^{k} \tag{19}\]  
where \(\omega\) is a coefficient; both \(c_{1}\) and \(c_{2}\) are learning factors; both \(rd_{1}^{k}\) and \(rd_{2}^{k}\) are positive random numbers within the normal distribution [0, 1]; \(k\) represents the \(k\)_th_ iteration; \(x_{id}^{k}\) denotes the position of the particle \(i\), i.e., the current value of the SVM parameters; \(v_{id}\in[v_{min},v_{max}]\) denotes the velocity of particle \(i\); \(i=1,2,\ldots,n\); \(d=1,2,\ldots,D\). For SVMs, there exist multiple methods to determine the optimal parameters, such as genetic algorithm (GA), particle swarm optimization (PSO), artificial bee colony (ABC), and so on. In comparison, PSO is powerful and easy to implement. Thus, this paper will adopt PSO to optimize the parameters for SVMs. In SVMs, different kernel functions will arise from different models. There exist a few popular kernel functions, such as the linear kernel function, sigmoid kernel function, RBF kernel function, and polynomial kernel function. The RBF kernel function not only has better performance but also can analyze higher-dimensional data. Therefore, this research will adopt RBF as the kernel function. The flowchart of the PSO-SVM algorithm is shown in Figure 3.  
The workflow of the PSO-SVM model mainly contains the following four steps:  
Step 1: Data preprocessing. Prepare the training set data and the testing set data, as well as normalize and visualize them. Step 2: Parameter optimization. Initialize model parameters, such as parameter \(C\), parameter \(\gamma\), population size, and the maximum iteration. Adopt the PSO algorithm to search the global optimal solution through cross-validation. Step 3: Construction of the PSO-SVM model. Put the training data set into the SVM model with the optimized parameters to obtain the trained PSO-SVM model. Step 4: Testing the PSO-SVM model. The testing data set will be used to verify the performance of the trained PSO-SVM model.