# Research on an Assembly Building Buffer Zone Based on Gray Critical Chain
## 4 Case Study
### Buffer Size Calculation
 \tag{17}\]  
(3) The OPA method was used to collect data on the ranking of the influencing factors by 25 experts (detailed information can be found in the Supplementary Material), and the results of the calculated weights were as follows: \(W_{j}=\) (0.2452, 0.1676, 0.2272, 0.2229, 0.1370)  
(4) According to the steps in Section 2.2 (4), the entropy weight method is used to calculate the entropy weight coefficients of each indicator, resulting in the indicator weight \(\alpha_{j}=\) (0.1488, 0.2587, 0.2158, 0.1771, 0.1995)  
(5) Gray correlation analysis is used to analyze the correlation between the influencing factors and the process, and the buffer size is calculated according to the steps described  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Work Sequence** & \(\mathbf{n_{1}}\) & \(\mathbf{n_{2}}\) & \(\mathbf{n_{3}}\) & \(\mathbf{n_{4}}\) & \(\mathbf{n_{5}}\) \\ \hline A & 4 & 2.4 & 2.4 & 3.4 & 3 \\ B & 3.4 & 2  
in Section 2.2 of this paper, and the project risk preference level coefficient \(\alpha=0.951\) is derived from Equation (12), and the resource tension \(\beta\) and process complexity \(\gamma\) are calculated according to Equations (13) and (14). The composite weight coefficient \(\varphi_{j}\), the discrimination coefficient \(\rho\), and the correlation coefficient \(\varepsilon_{ij}\) are calculated according to Equations (8)-(10), correlation matrix \(Y\), and gray correlation degree \(\delta_{i}\), and the results are shown in Equation (18) and Table 4.  
\[Y= \begin{bmatrix}0.632&0.439&0.510&0.625&0.385\\ 0.496&0.385&0.582&0.455&1.000\\ 0.696&0.472&0.807&0.625&0.610\\ 0.696&0.410&0.641&0.417&0.610\\ 0.696&0.510&0.758&0.556&0.472\\ 0.632&0.676&0.926&0.625&0.510\\ 0.696&0.676&0.714&0.833&0.758\\ 0.696&0.610&0.862&0.625&0.556\\ 0.534&0.610&0.862&1.000&0.676\\ 0.696&0.862&0.862&0.714&0.676\\ 0.534&1.000&1.000&0.714&0.472\\ 0.579&0.610&0.862&0.714&0.610\\ 0.775&0.758&0.807&1.000&0.610\\ 0.873&0.676&0.807&0.714&0.410\\ 0.696&0.610&0.758&0.500&0.610\\ 0.534&0.862&0.758&0.385&0.862\\ 0.632&0.510&0.862&0.556&0.410\\ 1.000&0.472&0.610&0.833&0.862\\ 0.385&0.676&0.758&0.500&0.610\\ 0.873&0.410&0.385&0.556&0.510\\ \end{bmatrix}  \tag{18}\]  
Analyzing and calculating the results of the coefficients derived from Table 4, the project buffer PB and sink buffer FB are, respectively, PB = 4.572 d, FB\({}_{1}\) = 0.735 d, FB\({}_{2}\) = 1.644 d and FB\({}_{3}\) = 3.571 d.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Work Sequence** & \(\delta i\) & \(\mathbf{i}_{1}^{50\%}\) & \(\mathbf{i}_{1}^{69\%}\) & \(\sigma\) & \(\beta\) & \(\gamma\) \\ \hline A & 0.517 & 2.382 & 3.146 & 0.764 & 0.250 & 0.000 \\ B & 0.582 & 2.775 & 3.613 & 0.837 & 0.333 & 0.134 \\ C & 0.641 & 4.275 & 5.113 & 0.837 & 0.800 & 0.205 \\ D & 0.554 & 2.419 & 3.500 & 1.081 & 1.000 & 0.334 \\ E & 0.597 & 3.6 Comparing the modified gray critical chain method of this paper with the traditional cut-and-paste method (C&PM) and root-square error method (RSEM) to derive the buffer sizes, the results are shown in Table 5.  
As can be seen from Table 5, the improved grey critical chain method in this paper is able to reduce the buffer size and optimize the assembly project duration compared to the traditional buffer size calculation methods (cut-and-paste method (C&PM) and root-square method (RSEM)).