## 4 Discrete Shop Capacity Model Driven by Digital Twins  
The discrete workshop capacity analysis based on digital twins is built on a data-driven twin system simulation model. The mathematical relationship between products and production capacity can be established by simulating the production process of twin systems and iterating the data. The first step in this analysis is to construct a model for discrete workshop production capacity. Figure 1: Workshop capacity prediction framework driven by digital twin data. The occurrence of production capacity disturbance factors in discrete manufacturing workshops is random and variable. Twin data can be used in the model based on the fast simulation iteration ability of twin models to iterate production line operation information, collect and analyze production process data, and analyze data based on material pulling. Neural networks and other methods can be used to model the impact of disturbance factors on the production status and then achieve the model establishment of disturbance constraints. The disturbance constraint conditions include explicit and implicit disturbances. The explicit disturbance factors include product line changes, equipment failures, material shortages, and product quality non-conformities. The implicit disturbance factors include encompassing work-hour fluctuations and deviations from the workpiece completion time. Constraints caused by abnormal factors are established by varying the impact of multiple uncertain disturbances on the available production capacity time of the processes in the production system. Hence, this paper presents a mathematical modeling roadmap for workshop capacity driven by digital twins, which is shown in Figure 2. By clarifying the capacity reduction caused by disturbance factors, a mathematical model for discrete manufacturing workshop capacity is constructed, and prediction algorithms are employed to predict changes in workshop capacity. Capacity loss involves the identification of production bottlenecks, which can be attributed to various factors, such as delays in work-in-progress flow, process planning, process delays, material storage conditions, and equipment utilization. In a customized environment, the workshop requires considering the priority of orders and the urgency of  
Figure 2: Mathematical modeling of workshop capacity driven by digital twins. delivery times. The workshop digital twin system pre-simulates the disturbance factors that affect capacity loss through the previously collected workshop data and calculates the identification and transfer of capacity loss or production bottlenecks caused by uncertain disturbance factors. The production capacity model of a discrete workshop is a typical multi-constraint function constrained by production bottlenecks, as shown in Equation (1). \[C_{obj}=FC-F_{MR}-F_{ER}-F_{PR} \tag{1}\]  
where \(F_{ER}\) represents the production capacity reduced by insufficient equipment utilization, \(F_{PR}\) indicates the production capacity reduced by process fluctuations, and \(C_{obj}\) denotes the actual production capacity of the workshop. Material shortages are an important factor in reducing workshop production capacity. For each product, the material requirements of the product, \(F_{MR}\), are determined by the product type and product process, as shown in Equation (2):  
\[F_{MR}=\sum\limits_{i=1}^{l}\sum\limits_{j=1}^{n}SP_{(i,j)} \tag{2}\]  
where \(SP_{(i,j)}\) represents the finished product caused by missing materials, \(l\) is the type of product in the workshop, and \(n\) symbolizes the number of processes in the finished product.