# Proactive Production Scheduling Approach for Off-Site Construction with Due Date Uncertainty
## 3 Proactive PC Production Scheduling Model with Contractor Schedule Uncertainty
### MC-Method-Based PC Production Simulation Module  
PC production generally has six processes (P2,18): mold assembling (p1), placing of all embedded parts including rebars (p2), concrete casting (p3), concrete curing (p4), mold removal (p5), and product finishing (p6). All PC products follow the same production sequence regardless of their product types. Processing times are different depending on product types. The process can be classified in two ways. First, they can be categorized into uninterruptible and interruptible (11). In the case of interruptible processes like p1, p2, p5, and p6, they must stop when the working hours end. Conversely, uninterruptible processes like p3 and p4 cannot be halted until they are completed. Second, the processes can be classified into parallel and non-parallel types (11). A parallel process such as p4 immediately starts after p3 is completed without external resources, allowing multiple PC products to be concurrently processed. On the other hand, a non-parallel process cannot proceed with another product simultaneously; it must wait until the current PC product is fully completed. Based on the PC production process, the MC-method-based PC production simulation module simulates and evaluates the total schedule tardiness at a specified confidence level considering the contractor's schedule uncertainty (see Algorithm 1). This evaluation is conducted using the MC method, which is widely employed to numerically assess the expectations of functions of random variables (19). Firstly, the simulation module accepts a job sequence (PC production schedule), the contractor's due date reliability, the required confidence level (\(n\)), the number of simulations (S), and processing time (\(pt_{tp}\)) as input variables. In each simulation, the dataset for the adjusted due date is generated based on the contractor's due date uncertainty, modeled using a triangular distribution with three points (most pessimistic, most likely, and most optimistic values), which is particularly useful when a distribution is unknown due to data unavailability (20,21), in line 2. All processes except for concrete curing (p4) start the next job when the current job is finished. However, concrete curing (p4) will start immediately after the previous process (p3) has finished in lines 5-9. If the processing end time falls outside of working hours, the end time for each process is adjusted accordingly. For non-interruptible processes like concrete casting (p3), the end time is rescheduled to the start of the next working day, with the remaining processing time added in lines 10-12 since the process cannot be halted midway. The end time for concrete curing (p4), a non-interruptible process, is adjusted to the start time of the next day since curing can continue during non-working hours in lines 13-14. The end times for the other processes (p1, p2, p5, and p6) are extended to include non-working hours in line 16. Once the final process of the last job is completed in a simulation, tardiness is calculated using Equation (1) in line 19. \[Tardiness=\sum_{j=1}^{J}\max\bigl{(}ft_{bj}-cd_{j},0\bigr{)} \tag{1}\]  
Figure 1: Framework of the proposed model. where \(J\) represents the total number of jobs; \(j\) denotes the index of jobs; \(ft_{6j}\) refers to the finish time of job \(j\) in the 6 th process; \(cd_{j}\) denotes the changed due date of the \(j^{th}\) job.