## 2 Mathematical Model for Scheduling of a Pipeline-Processing Workshop in a Shipyard
### Operations of Triangular Fuzzy Numbers  
When solving the fFJSP, all processing times are represented by triangular fuzzy numbers, so it is necessary to determine the basic operational rules of triangular fuzzy numbers, including addition, maximum operation, and comparison operations. For two arbitrary triangular fuzzy numbers \(\widetilde{A}=(a_{1},a_{2},a_{3})\) and \(\widetilde{B}=(b_{1},b_{2},b_{3})\), the following operational rules apply:  
1. Four operations: This is used to calculate the fuzzy completion time of each process. Addition operation: \(\widetilde{A}+\widetilde{B}=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3})\). Subtraction operation: \(\widetilde{A}-\widetilde{B}=(a_{1}-b_{1},a_{2}-b_{2},a_{3}-b_{3})\). Multiplication operation: \(\widetilde{A}\times\widetilde{B}=(a_{1}\times b_{1},a_{2}\times b_{2},a_{3} \times b_{3})\). Division operation: \(\widetilde{A}\div\widetilde{B}=(a_{1}\div b_{1},a_{2}\div b_{2},a_{3}\div b_{3})\). 2. Maximum operation: This is used to determine the fuzzy start time of each process. The currently used maximum operation strategies include [PERSON]'s criterion [14] and [PERSON]'s criterion [15]. To ensure that the fuzzy number always has a triangular characteristic and to facilitate subsequent calculations, this paper adopts a combination of [PERSON]'s criterion and comparison operation rules for fuzzy maximum operation. If \(\widetilde{A}>\widetilde{B}\), then \(\widetilde{A}\vee\widetilde{B}=\widetilde{A}\); otherwise, \(\widetilde{A}\vee\widetilde{B}=\widetilde{B}\), where the symbol \(\vee\) represents the maximum operation operator. 3. Comparison operation: This is used to compare the fuzzy end times of processes. The purpose is to obtain the maximum fuzzy completion time, which is compared by the following three criteria for two triangular fuzzy numbers: Criterion one: first, calculate the value of \(C\big{(}\widetilde{A}\big{)}\) and \(C\big{(}\widetilde{B}\big{)}\) according to Equation (9). Here, the weight of the most likely time is set to 2, and the weights of the minimum and maximum processing times are 1. If \(C\big{(}\widetilde{A}\big{)}>(<)C\big{(}\widetilde{B}\big{)}\), then \(\widetilde{A}>(<)\widetilde{B}\): \[\left\{ \begin{array}{l}C(\widetilde{A})=\frac{a_{1}+2a_{2}+a_{3}}{a_{3}},\\ C(\widetilde{B})=\frac{b_{1}+2b_{2}+b_{3}}{b_{1}+b_{2}+b_{3}},\\ \end{array} \right..\] (9) \begin{table}
\begin{tabular}{c c} \hline
**Variables** & **Meanings** \\ \hline \(n\) & The total number of workpieces \\ \(m\) & The total number of machines \\ \(l\) & The set of workpieces to be processed \\ \(M\) & The set of available machines \\ \(i\) & The number of the workpiece \\ \(j\) & The number of the process \\ \(k\) & The number of the machine \\ \(N\) & The total number of processes for all workpieces \\ \(n_{i}\) & The number of processes for workpiece \(i\) \\ \(O_{ij}\) & The \(j\)-th process of workpiece \(i\) \\ \(M_{ij}\) & The set of machinable machines for process \(O_{ij}\) \\ \(\widetilde{P}_{ijk}\) & Fuzzy processing time of process \(O_{ij}\) on machine \(M_{k}\) \\ \(\widetilde{C}_{ij}\) & Fuzzy completion time of workpiece \(i\) \\ \(\widetilde{S}_{ij}\) & Fuzzy start time of workpiece \(i\) \\ \(\widetilde{C}_{ij}\) & Fuzzy completion time of process \(O_{ij}\) \\ \(\widetilde{C}_{\max}\) & Fuzzy maximum completion time \\ \(\widetilde{S}_{ij}\) & Fuzzy start time of process \(O_{ij}\) \\ \(x_{ijk}\) & When it is 1, it indicates that process \(O_{ij}\) is being processed on machine \(M_{k}\) \\ \(y_{ijink}\) & When it is 1, it indicates that process \(O_{ij}\) is processed before process \(O_{ll}\) on machine \(M_{k}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Related symbols and meanings of fuzzy scheduling models for flexible job workshops. Criterion two: when \(C\left(\widetilde{A}\right)=C\left(\widetilde{B}\right)\), compare the cores of the two fuzzy numbers. If \(a_{2}>(<)b_{2}\), then \(\widetilde{A}>(<)\widetilde{B}\). Criterion three: if \(a_{2}=b_{2}\), compare the spans of the two triangular fuzzy numbers.