# Research on an Assembly Building Buffer Zone Based on Gray Critical Chain
## 2 Buffer Size Setting for Assembly Building Projects
### Determination of Safe Time  
The estimates for the duration of the process follow a probability distribution that is skewed towards the right. These duration estimates are determined using a three-point estimation method, which involves considering the most optimistic duration (a), the most probable duration (m), and the most pessimistic duration (b) for each process based on expert experience. The process time estimates have a roughly 95% guarantee, and the safe time is defined as the difference between the completion time that has a 95% probability and the completion time that has a 50% probability. The process safety time shown in Figure 2 is \(\sigma=\Delta\)t = t\({}^{95\%}\) - t\({}^{50\%}\)[37]. Determination of Comprehensive Impact Weight Coefficients for the Construction Schedule of Prefabricated Projects  
Compared to traditional cast-in-place buildings, assembly buildings are more susceptible to schedule variations due to their unique characteristics. This study searched through various data platforms such as CNKI, Wanfang Library, Web of Science, etc., and identified 306 articles published in the past three years related to \"assembly building schedule influencing factors\" and \"critical chain project management\". Among these articles, 157 were deemed relevant to the optimization of assembly building project schedules. The analysis identified five key influencing factors for the buffer zone size of assembly building projects, which are personnel experience (n\({}_{1}\)), prefabricated component production degree (n\({}_{2}\)), technological complexity (n\({}_{3}\)), construction plan reasonableness (n\({}_{4}\)), and construction space satisfaction degree (n\({}_{5}\)) as shown in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Type of Factor** & **Factors** \\ \hline \multirow{3}{*}{Personnel factor} & Lack of experienced workers U1 \\  & Inadequate safety precautions U2 \\  & Ineffective staffing U3 \\  & Unitinery transportation of prefabricated components U4 \\ Material factor & Prefabricated production inefficiencies U5 \\  & Unreasonable stacking on site U6 \\  & Difficult construction U7 \\ Technical factor & Failure of machinery and equipment U8 \\  & Technical complexity U9 \\  & Unitinery joining of work processes at the construction site U10 \\ Management factor & Poorly designed construction program and construction schedule U11 \\  & Inadequate coordination of on-site movement control U12 \\  & Policies and regulations U13 \\ Environmental factor & Insufficient construction space U14 \\  & Force majeure U15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of factors identified as influencing the schedule of assembly buildings. Figure 2: Probability distribution of process duration. In order to accurately express the quantitative results of the influencing factors, this paper proposes the joint use of the OPA-entropy weight method to improve the gray correlation analysis method. This allows for the calculation of the weights of the influencing factors and the scientific derivation of the comprehensive weight coefficient \(\eta_{i}\). Gray correlation analysis is an analytical method used to assess the degree of similarity and dissimilarity between factors. It measures the correlation between factors by comparing different situations that have been predetermined [38]. By adjusting the discrimination coefficient according to the weights, gray correlation analysis derives the gray correlation degree \(\delta_{i}\), which represents the integrated weight coefficient \(\eta_{i}\) of each process. The calculation steps are as follows:  
1. Construct a decision matrix. With m processes and n influencing factors, the evaluation of the n influencing factors on the m processes is used to build a decision matrix of size m \(\times\) n denoted as \(X\) Equation (1). The element \(x_{mn}\) represents the evaluation value of the nth influencing factor on the nth process. \[X=