## 2 Mathematical Model for Scheduling of a Pipeline-Processing Workshop in a Shipyard
### Fuzzy Scheduling Model for a Pipeline-Processing Workshop in a Shipyard  
Based on the description and reasonable assumptions of the scheduling problem in shipyard pipeline-processing workshops, it can be described as a flexible job-shop  
Figure 1: Shipyard pipeline-processing process. scheduling problem. Many uncertain factors often exist in actual production, such as equipment failures, changes in the material supply, and physical conditions of workers, all of which can affect the actual processing time of pipelines. Thus, the triangular fuzzy number in fuzzy set theory is introduced to allow the processing time to be a fuzzy variable, making it more in line with the actual production situation. This scheduling problem can ultimately be reduced to a fuzzy flexible job-shop scheduling problem. For fFJSP, the key is to determine the processing sequence of each workpiece and allocate machines for each process. After the completion of all processing tasks, the optimal scheduling solution is found by minimizing the maximum fuzzy completion time as the indicator, and the final production-scheduling plan is generated. The specific description can be as follows: There are \(n\) independent pipelines \(J=\{J_{1},J_{2},\ldots,J_{n}\}\) to be processed on m machines \(M=\{M_{1},M_{2},\ldots,M_{m}\}\). The total number of pipeline processes \(J_{i}\) is represented as \(n_{i}\), \(J_{i}\), and the \(j\) process is represented as \(O_{ij}\). \(O_{ij}\) means that only a portion of the machines can be selected for processing in the machine set, that the processing time on each machine varies, and that the processing time is not a fixed value. For the fFJSP, the problem needs to determine the processing order of all processes and allocate the processing machine for each process. After the completion of all the process processing tasks, the optimal scheduling solution is found by minimizing the maximum fuzzy completion time, thus generating the final production-scheduling plan. Specifically, it can be described as follows: there are \(n\) independent pipelines \(J=\{J_{1},J_{2},\ldots,J_{n}\}\) that need to be processed on \(M\) machines \(M=\{M_{1},M_{2},\ldots,M_{m}\}\). The total number of processes for pipeline \(J_{i}\) is represented as \(n_{i}\), and the \(j\)-th process of \(J_{i}\) is represented as \(O_{ij}\). \(O_{ij}\) can only select some machines from machine set \(M_{ij}\) for processing--that is \(\exists i,j,M_{ij}\subsetneq M\); the processing time on each machine is different, while the processing time is not a fixed value. The membership function image of triangular fuzzy numbers \(\widetilde{T}=(t_{1},t_{2},t_{3})\) in a two-dimensional coordinate system is a triangle, as shown in Figure 2. Compared with traditional fixed processing times, using triangular fuzzy numbers to represent uncertain processing times has greater practical significance. Its membership function is shown in Figure 2, corresponding to the following:  
\[\mu_{\widetilde{T}}(x)=\left\{ \begin{array}{ll}\frac{x-t_{1}}{t_{2}-t_{1}}, t_{1}\leq x<t_{2},\\ \frac{x_{3}-x}{t_{3}-t_{2}},t_{2}\leq x<t_{3},\\ 0,x<t_{1}\;or\;x\geq t_{3}.\end{array} \right. \tag{1}\]  
The processing time of the \(J\)-th operation of the pipeline \(J_{i}\) on machine \(M_{k}\) is represented by the triangular fuzzy number \(\widetilde{P}_{ijk}=(P^{1}_{ijk},P^{2}_{ijk},P^{3}_{ijk})\), where \(P^{1}_{ijk}\) represents the minimum processing time of the pipeline, \(P^{2}_{ijk}\) represents the most likely processing time of the pipeline, \(P^{2}_{ijk}\) corresponds to a membership function of 1, and \(P^{3}_{ijk}\) represents the maximum processing time of the pipeline. Figure 2: Fuzzy number membership function. The process of solving fFJSP can be briefly described as follows: under the premise of meeting production constraints, by reasonably arranging the processing sequence of workpieces and assigning corresponding machines to each process, we can ensure that the final scheduling plan meets specific indicators.