## 2 Mathematical Model for Scheduling of a Pipeline-Processing Workshop in a Shipyard
### Fuzzy Scheduling Model for a Pipeline-Processing Workshop in a Shipyard
This article uses \(\widetilde{\mathcal{C}}_{\text{max}}\) to represent the maximum fuzzy completion time of each scheduling task, and the corresponding single-objective fFJSP optimization model is as follows:  
\[\left\{ \begin{array}{l}f=\min\left\{\widetilde{\mathcal{C}}_{\text{max}} \right\}\\ \widetilde{\mathcal{C}}_{\text{max}}=\underset{i=1}{n}\left\{\widetilde{ \mathcal{C}}_{i}\right\}\\ i=1,2\ldots n\end{array} \right., \tag{2}\]  
The solving process of single objective fFJSP can be expressed as follows: Under the condition of meeting actual production needs, solving the two subproblems of process sorting and machine allocation, \(\widetilde{\mathcal{C}}_{\text{max}}\) is minimized, and the scheduling solution that minimizes \(\widetilde{\mathcal{C}}_{\text{max}}\) is the optimal scheduling solution. In the fuzzy scheduling model of flexible job workshops, pipelines and processing machines must simultaneously follow the following constraints:  
1. The different processes of the same workpiece must be processed in the following sequence: \[\left\{ \begin{array}{l}\widetilde{\mathcal{S}}_{ij}+\widetilde{P}_{jik} \cdot x_{ijk}\leq\widetilde{\mathcal{C}}_{ij}\\ \widetilde{\mathcal{C}}_{ij}\leq\widetilde{S}_{i(j+1)}\end{array} \right..\] (3)
2. The fuzzy completion time of the last process of each workpiece should not exceed the fuzzy maximum completion time of the entire scheduling process as follows: \[\widetilde{\mathcal{C}}_{in_{i}}\leq\widetilde{\mathcal{C}}_{\text{max}}.\] (4)
3. The same machine cannot process multiple processes at the same time, where \(L\) is a sufficiently large positive integer, as seen below. \[\left\{ \begin{array}{l}\widetilde{S}_{ij}+\widetilde{P}_{jik}\leq\widetilde{ S}_{ll}+L\left(1-y_{ijllk}\right)\\ \widetilde{\mathcal{C}}_{ij}\leq\widetilde{S}_{i(j+1)}+L(1-y_{thi(j+1)k}) \end{array} \right..\] (5)
4. Each process cannot be processed by multiple machines at the same time, as follows: \[\sum\limits_{k=1}^{M_{ij}}x_{ijk}=1.\] (6)
5. Each machine is allowed to operate in a loop on the same machine while meeting the conditions of an optional set of machines as shown below. \[\left\{ \begin{array}{l}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n_{i}}x_{ ijllk}=x_{lhk}\\ \sum\limits_{l=1}^{n}\sum\limits_{k=1}^{n_{l}}x_{ijllk}=x_{ijk}\end{array}  \right..\] (7)
6. A process that is currently in processing status cannot be terminated midway. \[\widetilde{\mathcal{C}}_{ij}-\widetilde{S}_{ij}=\sum\limits_{k=1}^{m} \widetilde{P}_{ijk}\cdot x_{ijk}.\] (8)
7. All machines exist independently of each other, and all machines are in a machinable state at the initial moment of processing. 8. There is no strict processing sequence between different workpieces. Different processes of the same workpiece must be processed in sequence in fFJSP. Owing to the consideration of uncertain processing time, triangular fuzzy numbers are used to represent uncertain processing time and completion time parameters, and a flexible job-shop fuzzy scheduling model is established. The relevant parameters and the meanings involved in the model are shown in Table 1.