PSO-SVM Based Performance-Driving Scheduling Method for Semiconductor Manufacturing Systems

Featured Application

This method is developed to solve the dynamic scheduling problem of semiconductor manufacturing processes and can be extended to most mechanical processing scenarios and resource scheduling scenarios.

Abstract

There are currently many studies on data-driven optimization scheduling, but only a few studies have combined “closed-loop optimization” with “performance-driven”. Therefore, this research proposed a PSO-SVM-based (particle swarm optimization optimized support vector machine) scheduling method that reconciles the composite dispatching rules (CDR), performance-driving ideology, and feedback mechanism ideology. Firstly, the composite dispatching rules coalesce flexible equipment maintenance, multiple process constraints, and dynamic dispatching. Secondly, the performance-driving ideology is carried out through two learning models based on the PSO-SVM algorithm, based on targeted optimizing performances. Thirdly, the feedback mechanism ideology makes the scheduling method realize closed-loop optimizations adaptively. Finally, the superiority of the proposed scheduling method is validated in a semiconductor manufacturing system in China. Compared with CDR, the proposed scheduling method combines MOV, PR, and EU, respectively EU_ O, EU_ P, PCSR and ODR increased by 7.85%, 5.11%, 8.76%, 8.14%, 6.60%, and 7.33%, indicating the superiority of this method.

1. Introduction

A semiconductor manufacturing system is a typical complex manufacturing system containing hundreds or thousands of processing steps [1,2]. Unlike conventional manufacturing and assembly systems, semiconductor manufacturing involves a re-entrant phenomenon, in which a workpiece may queue for its turn to be processed by one piece of equipment and visit this equipment repeatedly [3]. In addition, complex products and processes, as well as expensive equipment, make managing semiconductor manufacturing systems a challenge [4].

Semiconductor manufacturing is a capital-intensive industry [5], which has high standards for overall equipment effectiveness, a high on-time delivery rate, quick cycle time, etc. Semiconductor manufacturing scheduling is a complicated and difficult task with multiple crucial characteristics, such as changing demands, varying task priorities, imbalanced capacity, moving bottlenecks, etc. [6]. More unwanted outcomes, task rework or discard, and other unfavorable uncertainties will arise from the increase in process complexity and the decrease in feature size [7]. To choose acceptable principles for semiconductor scheduling, Korytkowski et al. [8] suggested an evolutionary optimization technique based on simulation. To solve material handling problems, Qin et al. [9] put forward a dynamic simulation-based dispatching rule. Simultaneously considering the dispatching urgency degree and load level of downstream equipment, Li et al. [10] proposed an original dynamic dispatching rule and verified its effectiveness. Therefore, scheduling policies must be able to respond to rework phenomena and interruption conditions. In addition, due to the complex coupling relationship between upstream and downstream equipment, researchers prefer fab-widely dynamic scheduling methods.

Unlike the above, data-based methods pay more attention to the knowledge hidden in manufacturing systems. Applying data-driving methods to solve scheduling problems is a burgeoning study direction [11]. By analyzing production data, researchers can excavate hidden information about relevant problems and then turn it into available knowledge through modeling. If available knowledge is used to identify and solve problems, they rely on data mining rather than human experience. In this way, manufacturing knowledge can be utilized more effectively [12]. Although some significant successes in dynamic scheduling, goal-driving, and closed-loop mechanisms have been achieved, they are often considered separate decisions in scheduling problems. Thus, scheduling methods are unable to adapt well to real-time changes in production environments. Therefore, this paper proposes a PSO-SVM-based scheduling method with a performance-driving and feedback mechanism.

The arrangement of this paper is as follows. Section 2 describes the proposed scheduling system. The proposed scheduling system is introduced in Section 3, which includes composite dispatching rules, feature selection, states-performance prediction, and the (states and performance)-parameters optimization. Section 4 describes the simulation platform, experiment design and arrangement, and discussion of the simulation results. Section 5 concludes this research and future works.

2. Literature Review

2.1. Scheduling in Semiconductor Manufacturing

Manufacturing systems are affected by real-time events, which, in turn, impact manufacturing schedules. Mathematical programming models have been used to solve semiconductor scheduling problems. Eivazy et al. [13] introduced a dynamic scheduling approach to prioritize the release policies of the hybrid make-to-stock/make-to-order (MTS/MTO) issue. By using heuristic algorithms, Bahaji et al. [14] evaluated dispatching rules in two low-mix, high-volume wafer fabrication facilities. Based on rigorous modeling and statistical analysis, Wu [15] proposed a hybrid intelligent algorithm coalesced with a GA (genetic algorithm) and a novel encoding method named bi-vector to solve scheduling problems with multi-mode resource constraints. Che et al. [16] created a bi-objective mixed integer linear programming (MILP) model to shorten the cycle time and boost resilience. A Pareto-optimal solution can be obtained by solving the single-objective MILP model to reduce the cycle time with fixed robustness or to maximize robustness for a given cycle time. The suggested method’s efficacy was confirmed by simulation studies conducted on benchmarks and a small number of other randomly generated projects. Wang et al. [17] created a MIP (mixed integer programming) issue to reduce makespan based on machine eligibility requirements, release time, and parallel batch and then used a heuristic approach to find the optimal solution. Kim [18] proposed a mathematical model to determine the number of wafers to be processed in the photolithography area, in which the setup time is incurred when the type of wafers was changed on the equipment to maintain the target work-in-process level, minimize cycle time, and to maximize throughput. Chien et al. [19] developed a mini-max regret approach for the improvement of equipment utilization in semiconductor manufacturing. Based on the binary integer programming method, Ham et al. [20] proposed a heuristic scheduling method to reduce the cycle time of the parallel-batch processing in flow shops. They obtained optimal real-time scheduling performance by accounting for necessary batch processing. Although many studies have addressed performance optimization, most employed open-loop ideology and were not performance-driven. In other words, they optimized performances aimlessly and could not form a closed-loop system, which is self-adaptive to the capricious manufacturing environment.

2.2. Prediction Problems in Semiconductor Scheduling

For the past few years, in both industry and academia, related industries have paid more and more attention to the application of various data mining methods in performance index prediction and expected to seek statistical rules from the manufacturing data of daily production activities to accurately predict the key performance indicators of semiconductor production line under the current scheduling scheme. As performance indicators control accurately and precisely, they help companies to develop scientific and effective daily production plans, optimize scheduling strategies, and grasp the real-time status of wafers throughout the life cycle of the production line.

Chien et al. [21] designed a multi-principal component analysis model to detect the cause of defects and establish corresponding defect classification rules. The framework simplifies the traditional defect monitoring and classification system with fewer key indicators. Based on the particle swarm optimization (PSO) algorithm and k-means clustering method, Hsu et al. [22] proposed a default detection and prediction model to mine the underlying knowledge behind the data while analyzing the data collected on the production line. Fan et al. [23] put forward a model of key production factor identification for wafer defects; the PCA methodology was initially used to pick features, and the AdaBoost classifier and SVM classifier were then used to categorize the data. The difficulty in selecting a kernel function acceptable for a given situation and the lengthy process of solving model parameter equations are two drawbacks of this approach. Based on the logistic regression method, Wang et al. [24] classified the input performances. Finally, they selected 108 indicators from 774 that could affect the cycle time, and the prediction accuracy outperformed other back propagation neural networks (BPNN) and traditional linear regression methods. Tirkel et al. [25] used clustering methods to screen historical production data, making the accuracy of traditional decision-making methods based on decision trees and neural networks up to 76.7% and 88.2%. Their research first selected features in the dataset and then established the processing cycle prediction model. This may lead to incompatibility between the prediction method and the previous clustering method. Based on real-time data from complex production lines, Liu et al. [26] proposed a real-time evaluation method for complex production line health to quantitatively analyze the behavior patterns and correlations among attributes within a system through big data analysis methods.

Soft computing achieves low-cost solutions and robustness by tolerating uncertainties, inaccuracies, and incomplete truth values [27]. To efficiently complete daily duties, it imitates the molecular mechanisms of intelligent systems seen in nature (human perception, brain structure, evolution, immunity, etc.) [28]. Fuzzy logic, artificial neural networks, genetic algorithms, support vector machines, and chaos theory are only a few examples of the complimentary and interdependent computational patterns that make up soft computing [29]. Due to its capacity for nonlinear regression and prediction based on limited-sample statistical learning and structural risk reduction, support vector machine (SVM) has been used in machine learning for pattern identification, portrait recognition, and text categorization [30]. SVMs have made great progress in nonlinear regression problems and prediction problems, especially in landslide displacement prediction [31]. However, because parameter selection is so sensitive, SVMs must use efficient and shrewd approaches to find the best parameters [32]. Zhang et al. [33] built a novel SVM prediction model with parameter optimization based on obtained characteristic spectra and provided the basis for the establishment of a photosynthesis prediction model.

The identification of kernel functions and other pertinent parameters is the most significant issue for SVMs during model construction [34]. A grid search is a popular technique for choosing parameters, but it has certain drawbacks, including the need for a large amount of training data and a dramatic rise in time costs as searching grids become denser. The optimization mechanism related to the hyperparameter setting of kernel optimization in SVM training may be realized using the Particle Swarm Optimization (PSO) method [35,36,37]. As with other swarm intelligence (SI) algorithms like Ant Colony Optimization (ACO) and Artificial Bee Colony (ABC), PSO is one of the most popular SI-based bios-inspired algorithms that makes use of the social sharing of information model [38,39,40]. The PSO algorithm, which belongs to a common class of intelligent algorithms and is typically used to calculate the parameters of an SVM model for simplicity and speed, was introduced by Kennedy et al. [41]. Liu et al. [42] suggested a novel IPSO-SVM model for online defect detection issues in eddy current testing, which might increase recognition efficiency and accuracy. Based on the above-mentioned, a novel hybrid algorithm coalescing a PSO and SVM will be used in this research to build the performance prediction model and the dispatching-parameters optimization model.

2.3. Goal-Driving Methods in Scheduling Problems

Several studies on scheduling methods with goal-driving approaches in semiconductor manufacturing systems exist. To improve the ODR (on-time delivery rate) of wafers with high priorities in order-driving fabrication, Seo et al. [43] put forward a novel scheduling method, which contains wafers with high priorities for reservation and equipment for reservation. Experimental results showed that the dispatching rule is superior to conventional rules in terms of the ODR of wafers with high priorities. Considering external interference in scheduling problems, Ni et al. [44] introduced a novel scheduling method with real-time order driving to aid manufacturers in responding to real-time orders and then coalesce them into the scheduling problems. Through data-driving methods with two different training patterns, Luo et al. [45] developed a learning framework with two stages to predict the parameters of the degradation. The effectiveness of the proposed method was verified through many experiments on simulation plants of a semiconductor manufacturing system. Kück et al. [46] presented an approach that dealt with scheduling problems, in which schemes were implemented by coalescing a heuristic optimization algorithm with the simulation model.

3. Framework of the Proposed Scheduling Method

3.1. Variable Definition

Variable definition of this research are listed in

Table 1. Variable definition.

Variable	Definition
$I$	set of equipment, $i\in I$
$J$	set of workpieces, $j\in J$
$C$	set of process constraints
$H_C$	the time constraint of the constraint $C$
$t_i^a$	daily available time of equipment $i$
$t_j^D$	due date of workpiece $j$
$t_j^{r\_p}$	remaining processing time of workpiece $j$
$t_j^i$	waiting and processing time of workpiece $j$ on equipment $i$
$t_M^E$	earliest start time of equipment maintenance
$t_M^L$	latest finish time of equipment maintenance
$t_M^{MAX}$	longest maintenance time of equipment $i$
$t_M^{MIN}$	shortest maintenance time of equipment $i$
${\alpha }_n$	parameters of the dispatching rule ($n\in \left\{\mathrm{1,2},\dots ,5\right\},i\in \left\{\mathrm{1,2},\dots ,5\right\}$)
${\tau }_i^j\left(t\right)$	priority of workpiece $j$ for processing on equipment $i$

.

3.2. Assumptions

(1)

Dispatching-related information is known or can be obtained from the manufacturing execution system, including workpiece processing time, maximum queue size (max. work in process, WIP) in front of a workstation, and the time each workstation is available;

(2)

Each station can process one workpiece lot at a time;

(3)

The transportation time of workpieces is not considered;

(4)

The processing of a workpiece cannot be interrupted.

In consideration of the complexity of semiconductor manufacturing systems, a PSO-SVM-based scheduling method with a performance-driving approach and feedback mechanism is proposed. This system can adjust dispatching parameters according to varying manufacturing environments (operating status), such as changes in the number of WIPs, available time of equipment, the workload of downstream equipment, and intervals between two successive workpieces. Figure 1 illustrates the framework of the proposed scheduling method.

The proposed scheduling method mainly contains four research contents: (1) composite dispatching rules (CDR), (2) feature selection, (3) performance prediction model, and (4) parameter optimization model. The “simulation model” is a virtual semiconductor manufacturing system, which is built according to an actual semiconductor manufacturing system in our previous work.

CDR is a dispatching policy that is linked with considerations for process restrictions, equipment maintenance, and other factors. We may access a large number of samples that are kept in the “database” based on the simulation model. Feature selection (offline work, shown by broken lines in Figure 1) is performed before creating performance prediction models and parameter optimization models to decrease redundant features and computation time. The performance prediction model is a relational model that generates anticipant performances according to the operational condition of the manufacturing system. The parameter optimization model is also a relational model that outputs optimized parameters based on the operating status and estimated performances. Subsequently, the weight parameters of CDR are correlated with the selected features and six performances of the manufacturing system. Thus, the scheduling system can adaptively update dispatching parameters, form the most suitable dispatching rules, and implement the adaptive scheduling course.

3.3. Composite Dispatching Rules (CDR)

CDR considers the characteristics related to process constraints, such as the intervals between two successive workpieces, the processing time of workpieces, and the processing sequence. In addition, to better prioritize workpiece dispatching and equipment maintenance, CDR considers the characteristics related to equipment maintenance and workpiece dispatching, such as workload of downstream equipment, index of equipment maintenance, and due date of workpiece processing. A flowchart of CDR is illustrated in Figure 2.

Step 1: Check whether workpiece j satisfies the process constraint $G_{j,C}^c<\left(H_C-T_j^C\right),$ where $G_{j,C}^c$ represents the total remaining processing time of constraint $C$, $H_C$ represents the time constraint of $C$, and $T_j^C$ represents the processing completion time of workpiece $j$ in constraint $C$.

Step 2: Compute the index ($P_1$) of process constraints:

$$P_1=\left\{\begin{array}{c}{P}_{MAX}, G_{j,C}^c\times \Omega \ge H_C-T_n^C\\ \frac{G_{j,C}^c\times \Omega }{H_C-T_n^C}, G_{j,C}^c\times \Omega <H_C-T_n^C\end{array}\right.,$$

(1)

where $\Omega$ is a constant that can be obtained from the manufacturing execution system.

This implies that, for workpiece $j$, which is subjected to process constraint $C$, the greater the ratio of $G_{j,C}^c$ to $H_C-T_n^C$ the higher the probability of workpiece $j$ violating constraint $C$, and the higher priority of workpiece $j$ to be selected and processed on equipment $i$. If $(G_{j,C}^c\times \Omega )\ge (H_C-T_n^C)$, workpiece $j$ will have a greater probability of violating constraint $C$ and become a hot lot, i.e., it will have the highest priority ($P_{MAX}=1$) on any equipment.

Step 3: Compute the index of the due date ($P_3$) for workpiece $j$

$$P_3=t_j^{r_p}/(t_j^D-t)$$

(2)

where $t_j^{r_p}$ represents the remaining processing time of workpiece $j$, $t_j^{r_p}$ represents the due date of workpiece $j$, and $t$ r represents the current decision time.

Step 4: Compute the workpiece priority ($P_{LOT}$):

$$P_{LOT}={\sum }_{n=1}^3\left({\alpha }_n·P_n\right)$$

(3)

where ${\alpha }_n,n\in \{1,2,3\}$ r represents the nth parameter of $P_{LOT}$$P_1$ is the index of process constraints, $P_2$ is an inherent index (1, 4) of customer priority, and $P_3$ is the index of due date.

Step 5: Check whether the current decision time is within the equipment maintenance period:

$$\left\{\begin{array}{c}t-t_M^E\ge 0.\\ t_M^L-t\ge t_M^{MAX}\end{array}\right.$$

(4)

where $t_M^E$ and $t_M^L$ respectively represent the earliest start time and latest end time of equipment maintenance; $t_M^{MAX}$ represents the maximum maintenance time. If the current decision time is within the equipment maintenance period, go to Step 6; otherwise, dispatch workpieces according to $P_{LOT}$.

Step 6: Compute the index of equipment maintenance ($P_4$):

$$P_4=t_M^{MAX}/(t_M^L-t)$$

(5)

Step 7: Compute the index of workload ($P_5$) for downstream equipment:

$$P_5=\sum t_j^{i+1}/t_{i+1}^a$$

(6)

where $t_j^{i+1}$ represents the waiting and processing time of workpiece j non-downstream equipment; $t_{i+1}^a$ represents the daily available time of downstream equipment.

Clearly, if ${\tau }_i^n\left(t\right)\ge 1$, the workload of the downstream equipment will exceed its daily available time; that is, the equipment will become a bottleneck. If there exist several pieces of equipment that complete the same process, $t_{i+1}^a$ represents the available processing time of all of them.

Step 8: Compute the maintenance priority ($P_{PM}$):

$$P_{PM}={\sum }_{n=4}^5\left({\alpha }_n·P_n\right)$$

(7)

Step 9: Normalize the workpiece priority ($P_{LOT}$) and maintenance priority ($P_{PM}$). The results are denoted as $P_{LOT}^{\prime }$ and $P_{PM}^{\prime }$ respectively.

Step 10: Compare $P_{LOT}^{\prime }$ with $P_{PM}^{\prime }$ If $P_{LOT}^{\prime }>P_{PM}^{\prime }$, dispatch workpieces; otherwise, maintain equipment.

3.4. 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 $\left(x_1,y_1\right),\left(x_2,y_2\right),\dots ,\left(x_k,y_k\right)$ is input vectors with n-dimension, where $\left(x_i,y_i\right)$ 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):

$$f\left(x\right)=\omega ·\phi \left(x\right)+b$$

(8)

where $\phi \left(x\right)$ 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):

$$\frac{1}{2}{‖\omega ‖}^2+C\frac{1}{k}{\sum }_{i=1}^k{L}_{\epsilon }\left(y_i,f\left(x_i\right)\right)$$

(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. $\epsilon$ is a user-prescribed parameter. $C\frac{1}{k}{\sum }_{i=1}^k{L}_{\epsilon }\left(y_i,f\left(x_i\right)\right)$ denotes the empirical error computed through the $\epsilon —\mathrm{insensitive}$ loss function which is shown in (10):

$$L_{\epsilon }\left(y_i,f\left(x_i\right)\right)=\left\{\begin{array}{cc}\left|y_i-f\left(x_i\right)\right|-\epsilon ,\hfill & \left|y_i-f\left(x_i\right)\right|\ge \epsilon \hfill \\ 0,\hfill & \left|y_i-f\left(x_i\right)\right|<\epsilon \hfill \end{array}\right.$$

(10)

In the case of infeasibility, one can introduce slack variables $({\xi }_i,{\xi }_i^{\mathrm{*}})$ to cope with the infeasible constraints of the optimization problem. Then, the above problem can be formalized as:

$$\mathrm{m}\mathrm{i}\mathrm{n}\frac{1}{2}{‖\omega ‖}^2+C{\sum }_{i=1}^k\left({\xi }_i+{\xi }_i^{*}\right)$$

(11)

subject to:

$$\left\{\begin{array}{c}{y}_i-\omega ·\phi \left(x_i\right)-b\le \epsilon +{\xi }_i\\ \omega ·\phi \left(x_i\right)+b-y_i\le \epsilon +{\xi }_i^{*}\\ {\xi }_i,{\xi }_i^{*}\ge 0\end{array}\right.$$

(12)

To obtain the solution to Equation (11), the Lagrangian function $\mathrm{L}$ is introduced:

$$\begin{array}{cc}\mathrm{L}=& \frac{1}{2}{‖\omega ‖}^2+C{\sum }_{i=1}^k\left({\xi }_i+{\xi }_i^{*}\right)-{\sum }_{i=1}^k{a}_i\left(\epsilon +{\xi }_i-y_i+\omega ·\phi \left(x\right)+b\right)\hfill \\ & -{\sum }_{i=1}^k\left({\eta }_i{\xi }_i+{\eta }_i^{*}{\xi }_i^{*}\right)-{\sum }_{i=1}^k{a}_i^{*}\left(\epsilon +{\xi }_i^{*}-y_i-\omega ·\phi \left(x\right)+b\right)\hfill \end{array}$$

(13)

After taking the Lagrangian and conditions for optimality, this optimization formulation can be transferred into the following dual problem shown in (14):

$$\begin{array}{cc}maxW\left(a_i,a_i^{*}\right)=& {\sum }_{i=1}^k{y}_i\left(a_i-a_i^{*}\right)-\epsilon {\sum }_{i=1}^k\left(a_i+a_i^{*}\right)\hfill \\ & -\frac{1}{2}{\sum }_{i=1}^k{\sum }_{i=1}^k\left(a_i-a_i^{*}\right)\left(a_j-a_j^{*}\right)\left(\phi \left(x_i\right)·\phi \left(y_i\right)\right)\hfill \end{array}$$

(14)

subject to:

$${\sum }_{i=1}^k\left(a_i-a_i^{*}\right)=0,a_i,a_i^{*}\in \left[0,C\right]$$

(15)

where $a_i,a_i^{*}$ are Lagrangian multipliers. Then, the dual representation of the model becomes:

$$f\left(x\right)={\sum }_{i=1}^k\left(a_i-a_i^{*}\right)\phi \left(x_i\right)·\phi \left(x\right)+b$$

(16)

Equation (16) can be rewritten as Equation (17) by invoking the kernel function $K\left(x_{i,}{x}_j\right)$

$$f\left(x\right)={\sum }_{i=1}^k\left(a_i-a_i^{*}\right)K\left(x_{i,}{x}_j\right)+b$$

(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. 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 $X_i=(x_{i1},x_{i2},\dots ,x_{iD})$, velocity vector $V_i=(v_{i1},v_{i2},\dots ,v_{iD})$, where $i=\mathrm{1,2},\dots ,n.$ 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 $P_i=(p_{i1},p_{i2},\dots ,p_{iD})$, the global best position will be denoted as $P_g=(p_{g1},p_{g2},\dots ,p_{gD})$. 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):

$$v_{id}^{k+1}=\omega \times v_{id}^k+c_1\times {rd}_1^k\times \left(p_{id}^k-x_{id}^k\right)+c_2\times {rd}_2^k\times \left(p_{gd}^k-x_{gd}^k\right)$$

(18)

$$x_{id}^{k+1}=v_{id}^{k+1}+x_{id}^k$$

(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 $i$; $i=\mathrm{1,2},\dots ,n$; $d=\mathrm{1,2},\dots ,D$.

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 γ, 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.

3.5. Performance Prediction and Parameter Optimization

To choose investigated performances scientifically, we performed a performance correlation analysis in our previous work. Based on this analysis, in this study, we selected six types of performance that belonged to the following three categories: equipment-related performances, fab-wide performances, and wafer-related performances; they are listed in

Table 2. Selected performance.

Category	Performance
Equipment related	EU_P (equipment utilization of photolithography area)
	EU_O (equipment utilization of oxidation area)
Fab-wide related	MOV (movement of the manufacturing system)
	PR (production rate)
Wafer related	PCSR (process constraint satisfaction rate)
	ODR (on-time delivery rate)

.

This research used a PSO-SVM to build the performance prediction model and the parameter optimization model. The workflow is illustrated in Figure 4 and Figure 5, respectively.

In performance prediction, the input data is the real-time production status, which can be obtained in the production line; the output data is the predicted performance that we expected. Through the performance prediction illustrated in Figure 4, we can obtain the expected performances according to the real-time production status.

In parameter optimization, the input data includes two parts: the real-time production status and the predicted performances. The output data is the five dispatching parameters $({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_5)$ Through parameter optimization illustrated in Figure 5, we can obtain the expected dispatching parameters according to the real-time production status and the predicted performances.

In this section, the above-mentioned real-time production status is filtered by the immune-algorithm-based attribute selection method. The above-mentioned six predicted performances were chosen through performance correlation analysis in our previous work.

4. Empirical Study

To evaluate the efficiency of the CDR and the PSO-SVM-based scheduling method, we embedded the CDR into the simulation model, which was built according to an industrial semiconductor manufacturing system (a mixed 5″ and 6″ semiconductor manufacturing system of a semiconductor manufacturer in Shanghai) and conducted a series of simulation experiments. The simulation platform was built in the Tecnomatix Plant Simulation software 2022, which belongs to Siemens. The control layer and the function layer of the simulation platform are shown in Figure 6. In terms of experimental hardware, three Nvidia GeForce RTX 4000 GPUs (Nvidia, Santa Clara, CA, USA) were used to train and validate corresponding models.

With an average WIP inventory of more than 80,000 pieces, the manufacturing system consists of 11 processing areas, which are oxidation, photolithography, injection, epitaxial growth, dry etching, deposition, sputtering, wet cleaning, and three non-processing areas, namely, the virtual machine, testing, and outsourcing. Among these processing areas, the oxidation area and the photolithography area are bottleneck processing areas. In addition, the production comprises more than 800 machines, belonging to five categories: single-workpiece processing equipment, batch-processing equipment, multi-workpiece processing equipment, cluster tools, and tanks.

4.1. Feature Selection

The investigated semiconductor manufacturing system has 67 features. Using feature selection based on the immune algorithm, we selected 11 as the characteristic features, which are shown in

Table 3. Selected features.

NO.	Selected Features
1	the number of WIPs in a 5″ production line.
2	the ratio of the WIP quantity in the oxidation area to that in the whole manufacturing system.
3	the ratio of the WIP quantity in the photolithography area to that in the whole manufacturing system.
4	the ratio of the WIP quantity in the dry-etching area to that in the whole manufacturing system.
5	the ratio of the number of bottleneck equipment to that of the whole manufacturing system.
6	the ratio of the number of available bottleneck equipment in the oxidation and diffusion area to that in the whole manufacturing system.
7	the ratio of the number of available bottleneck equipment in the photolithography area to that in the whole manufacturing system.
8	the ratio of the number of workpieces processed by the manufacturing system to the throughput.
9	average remaining processing time of workpieces.
10	number of urgent workpieces (hot lots) in the whole manufacturing system.
11	number of urgent workpieces in the oxidation and photolithography area.

.

When creating a PSO-SVM model, randomize parameter initialization, obtain the statistical parameters, and choose the best values. Set the top and lower bounds for the kernel parameters and SVM parameters at random, then use the PSO method to find the ideal parameters between the upper and lower bounds. Compute the SVM’s associated statistical parameters for the acquired parameter. In this study, we choose as the ideal parameters those that have lower RMSE, MAE, and SI values and higher CC values. The final optimal values for the SVM and kernel parameters are displayed in

Table 4. Optimal parameters of the PSO-SVM.

Kernel Type	RBF	Polynomial	Linear
$C$	18.7	183.78	1500
$\epsilon$	0.0827	0.0000538	0.0885
$\gamma$	2.6	-	-
$d$	-	3	-

.

The parameters of PSO-SVM are listed in

Table 5. Statistical parameters for different kernel types of PSO-SVM.

Kernel Type	RBF		Polynomial		Linear
	Train	Test	Train	Test	Train	Test
CC	0.911	0.902	0.932	0.925	0.870	0.849
RMSE	0.048	0.064	0.024	0.035	0.075	0.076
MAE	0.040	0.052	0.016	0.028	0.064	0.065
SI	0.129	0.173	0.082	0.096	0.204	0.204

. Compared with other kernel functions, PSO-SVM with linear kernel function results in lower CC.

4.2. Experimental Design

To test the availability of our proposed method, we designed multiple simulation-based case studies. Because the proposed method was a prototype system, which contained multiple methodologies, we designed different case studies for different methodologies and the whole proposed method.

In Section 4.3.1, we conduct case studies and comparisons between the proposed composite dispatching rules (CDR) and the other three widely used heuristic dispatching rules, FIFO (first in, first out), EDD (earliest due date), and CR, to determine if CDR is preferable (critical ratio).

In Section 4.3.2, we conduct case studies and comparisons between CDR and the proposed scheduling method to evaluate the efficacy of the proposed “performance-driving” and “feedback” concepts. Different dispatching parameters will lead to different performances, so we randomly set the parameters’ values and gathered nearly twenty groups of experimental data to avoid experimental coincidence.

In this research, to carry out “performance-driving”, we selected six kinds of performances from three aspects: equipment-related (EU_P and EU_O), fab-wide (MOV and PR), and wafer-related (PCSR and ODR). To investigate scheduling effects generated by different “performance-driving”, in Section 4.3.2, we make multiple case studies with “equipment-related performance-driving method”, “fab-wide performance-driving method”, “wafer-related performance-driving method”, and “all six kinds of performance-driving method”. Like Section 4.3.1, we randomly set the parameters’ values and gathered nearly twenty groups of experimental data to avoid experimental coincidence.

4.3. Experimental Results

To test the availability of our proposed method, we designed multiple simulation-based case studies. Because the proposed method was a prototype system, which contained multiple methodologies, we designed different case studies for different methodologies and the whole proposed method.

4.3.1. Composite Dispatching Rules (CDR)

In addition to the simulation results of CDR, we also recorded the outcomes of the other three widely used dispatching rules: FIFO (first in first out), EDD (earliest due date), and CR (critical ratio); these outcomes are shown in

Table 6. PSO-SVM Simulation results of CDR, FIFO, EDD, and CR.

Dispatching Rules	Running Time (s)	MOV (Steps)	PCSR (%)	CT (Days)	ODR (%)	EU (%)
CDR	4.372	90,238	91.08	42.98	84.23	69.56
FIFO	2.047	88,012	89.73	43.30	82.99	67.82
EDD	4.284	88,627	91.48	43.81	84.68	66.25
CR	3.293	89,401	89.62	43.04	83.28	66.01

to demonstrate the effectiveness of CDR. We perform normalization based on CDR, and the normalization results are shown in Figure 7, to clearly illustrate the differences between these dispatching rules.

From Table 6 and Figure 7, we can draw the following conclusions:

(1)

Compared to FIFO, CDR improved CT, MOV, ODR, EU, and PCSR by 0.74%, 2.53%, 1.49%, 2.57%, and 1.50%, respectively;

(2)

Compared to EDD, CDR improved CT, MOV, and EU by 1.89%, 1.82%, and 5.00%, respectively, while reducing ODR and PCSR by 0.53% and 0.44%, respectively;

(3)

Compared to CR, CDR improved CT, MOV, ODR, EU, and PCSR by 0.14%, 0.94%, 1.14%, 5.38%, and 1.63%, respectively;

(4)

In terms of CT, MOV, and EU, CDR performed better than FIFO, EDD, and CR. In terms of ODR and PCSR, EDD performed better than CDR. EDD selects wafers to be processed according to the urgency degree of the due date; therefore, the ODR and PCSR of EDD ought to be higher;

(5)

Simulation results showed that, compared to the other three common dispatching rules, CDR could satisfactorily improve performance. No dispatching rule is perfect, but, with overall consideration, CDR is outstanding.

4.3.2. The Proposed Scheduling Method

Simulation tests and comparisons with CDR serve to validate the proposed PSO-SVM-based scheduling solution. To avoid contingency, we conducted tens of experiments with various dispatching parameters. Different dispatching parameters could lead to different performances.

Table 7. Simulation results of the proposed scheduling method.

NO.	Fab-Wide		Equipment-Related		Wafer-Related
	MOV (Steps)	PR (%)	EU_P (%)	EU_O (%)	PCSR (%)	ODR (%)
1	96,682	39.61	77.11	73.28	96.04	89.71
2	97,571	39.73	73.94	72.79	96.91	90.26
3	97,849	39.60	74.94	73.97	97.04	90.41
4	97,269	39.65	74.26	72.12	97.12	90.78
5	97,379	38.62	75.88	72.24	97.02	90.75
6	97,339	39.86	75.11	73.09	97.18	90.46
7	97,544	39.42	76.42	74.88	96.83	90.36
8	97,547	39.81	74.30	72.66	96.89	90.51
9	97,434	39.47	76.00	75.23	97.34	88.95
10	97,179	38.99	76.43	75.73	97.58	91.48
11	97,223	40.07	74.13	74.14	97.27	89.77
12	96,961	39.57	76.94	75.99	97.53	90.59
13	97,658	39.61	75.56	74.91	96.91	91.05
14	96,979	39.07	77.23	72.68	97.13	90.45
15	96,871	39.19	76.11	74.31	96.96	90.31
16	97,734	39.51	76.04	72.83	97.62	90.57
Ave.	97,326	39.49	75.65	73.80	97.09	90.40
CDR	90,238	37.57	69.56	68.24	91.08	84.23
Imp. (%)	7.85	5.11	8.76	8.15	6.60	7.33

displays the simulation outcomes. The proposed scheduling method refers to the scheduling methodology that includes CDR, performance driving, and feedback mechanism. We perform normalization processing on the simulation results to examine the simulation findings. The results of normalization are displayed in Figure 8.

From Table 7 and Figure 8, we can draw the following conclusions:

• Compared to CDR, the proposed scheduling method improved MOV, PR, EU_P, EU_O, PCSR, and ODR by 7.85%, 5.11%, 8.76%, 8.15%, 6.60%, and 7.33%, respectively. This illustrates the usefulness and effectiveness of the “performance-driving” and “feedback mechanism”;
• The proposed scheduling method can satisfactorily improve performance and respond to the changing manufacturing environment;
• Different dispatching parameters could result in different performances. Therefore, we could improve performances by changing dispatching parameters appropriately. However, the production status is ever-changing, so we cannot confirm dispatching parameters that are suitable for all production statuses. We can make the scheduling system capable of adaptively adjusting dispatching parameters.

4.3.3. Comparisons

We ran three more simulation tests that were only based on equipment-related performances (EU_P and EU_O), fab-wide performances (MOV and PR), and wafer-related performances (PCSR and ODR) to further highlight the impact of the “performance-driving” design. Figure 9 presents the differences in the structure among the above three systems and the system “driven by all six selected performances”, “driven by equipment-related performances”, “driven by fab-wide performances”, and “driven by wafer-related performances”; they differed only in the input of the parameter optimization model.

(1)

Equipment-related performance driving

We recorded tens of experiments’ results with different dispatching parameters to avoid contingency. The results of equipment-related performance driving are shown in Figure 10.

From Figure 10, we can draw the following conclusions:

• Compared to CDR, the proposed scheduling method with equipment-related performance driving improved MOV, PR, EU_P, EU_O, PCSR, and ODR by 8.56%, 4.18%, 10.65%, 10.08%, 6.64%, and 7.28%, respectively;
• Performances changed with the change of dispatching parameters;
• The proposed scheduling method with equipment-related performance driving could largely improve performances, especially in equipment utilization (equipment-related performances).

(2)

Fab-wide performances driving

Similarly, we recorded tens of experiments’ results with different dispatching parameters to avoid contingency. The results of fab-wide performance driving are shown in Figure 11.

From Figure 11, we can draw the following conclusions:

• Compared to CDR, the proposed scheduling method with fab-wide performance driving improved MOV, PR, EU_O, EU_P, PCSR, and ODR by 8.84%, 2.98%, 8.26%, 8.64%, 6.68%, and 6.42%, respectively;
• Performances changed with the change of dispatching parameters;
• The proposed scheduling method with fab-wide performance driving could largely improve performances, especially in MOV and PR (fab-wide performances).

(3)

Wafer-related performances driving

Similarly, we recorded tens of experiments’ results with different dispatching parameters to avoid contingency. The results of wafer-related performance driving are shown in Figure 12.

From Figure 12, we can draw the following conclusions:

• Compared to CDR, the proposed scheduling method with wafer-related performance driving improved MOV, PR, EU_O, EU_P, PCSR, and ODR by 7.55%, 4.13%, 7.22%, 7.23%, 7.37%, and 8.26%, respectively;
• Performances changed with the change of dispatching parameters;
• The proposed scheduling method with wafer-related performance driving could largely improve performances, especially in PCSR and ODR (wafer-related performances).

(4)

Further Discussion

To clarify the distinction among different performance-driving, we recorded the simulation results of CDR, scheduling methods with fab-wide performance-driving, equipment-related performance-driving, wafer-related performance-driving, and with all six selected performance-driving. Simulation results and normalization results are shown in

Table 8. Simulation results of the scheduling system with different performance-driving.

Methods	Running Time (s)	Fab-Wide		Equipment-Related		Wafer-Related
		MOV (Steps)	PR (%)	EU_P (%)	EU_O (%)	PCSR (%)	ODR (%)
CDR	4.37	90,238	37.57	69.56	68.24	91.08	84.23
Driven by Equipment-related performances	827.27	97,964	39.14	76.97	75.12	96.96	90.36
Driven by Fab-wide performances	748.44	98,216	40.69	72.57	70.88	95.66	88.64
Driven by Wafer-related performances	875.37	97,051	39.12	74.59	73.17	97.79	91.19
Driven by 6 performances (this research)	1038.27	97,326	39.49	75.65	73.80	97.09	90.40

and Figure 13, respectively.

From Table 8 and Figure 13, we can draw the following conclusions:

• Compared to CDR, the proposed scheduling method improved MOV, PR, EU_O, EU_P, PCSR, and ODR by 7.85%, 5.11%, 8.76%, 8.14%, 6.60%, and 7.33%, respectively. This can illustrate the usefulness and effectiveness of “performance driving” and “feedback mechanism”;
• The proposed scheduling method with fab-wide performance driving could maximally improve fab-wide performances;
• The proposed scheduling method with equipment-related performance driving could maximally improve equipment-related performances;
• The proposed scheduling method with wafer-related performance driving could maximally improve wafer-related performance;
• The simulation findings demonstrate that the performance-driving scheduling method could significantly enhance performance.

From all the experimental results mentioned above, it can be seen that this method can improve the CT, MOV, ODR, and other indicators of the production line, but it also has the disadvantage of long running time. Compared with other algorithms, although the calculation time of this algorithm is relatively long, the response time is kept within 17 min, which can fully meet the requirements of intelligent manufacturing systems for the response time of scheduling schemes. The simplified version of this algorithm has been successfully applied in a semiconductor enterprise in Shanghai. The challenges in practical application mainly focus on the quality of company data. If the quality of enterprise data is high, this method system is theoretically feasible and excellent.

5. Conclusions

To improve the performance of semiconductor manufacturing systems, this research proposed a PSO-SVM-based scheduling method with performance driving and a feedback mechanism. We designed a dynamic dispatching rule that integrated equipment maintenance, process constraints, and the correlated real-time status of the manufacturing system. Using the parameter optimization model, we could obtain the most suitable dispatching parameters to update the dispatching rule to adaptively acclimatize to the new production environment. The simulation results of this study indicated that the proposed PSO-SVM-based scheduling method outperformed several other conventional scheduling policies in average cycle time (CT), daily movement (MOV), average equipment utility (EU), on-time delivery rate (ODR) and process constraints satisfaction rate (PCSR). Compared to CDR, the proposed scheduling method improved MOV, PR, EU_O, EU_P, PCSR, and ODR by 7.85%, 5.11%, 8.76%, 8.14%, 6.60%, and 7.33%, respectively. Additionally, it may self-adaptively fulfill the manufacturing system’s dynamic environment and enhance overall system performances to produce semiconductors.

Although the proposed method performs better than other common heuristic scheduling methods, this research did not consider the adjustable parameters of the PSO-SVM, which could influence the performances of the whole manufacturing system. Owing to the research size and time limitation, this research did not consider other deep learning methods that have similar functions. In conclusion, the dynamically adjustable parameters of the PSO-SVM need to be considered to further improve the overall performances, and we will adopt other deep learning methods to complete the proposed methodology to make comparisons.