## 2 Mathematical Model for Scheduling of a Pipeline-Processing Workshop in a Shipyard  
### Shipyard Pipeline-Processing Workshop Analysis  
Shipbuilding is a process with a long cycle and tedious tasks, and this process requires the cooperative production of various workshops and departments. Among them, the shipyard pipeline-processing workshop is an important workshop in the shipbuilding process, and its task is to process various kinds of marine pipes with different specifications. The process of shipyard pipeline processing usually includes six main processes, i.e., undercutting, bending, pipe calibration, welding, cleaning and grinding, and pump-pressure testing. According to the specific application requirements of the pipeline, some pipelines need to bend, while others do not. Figure 1 shows the typical process of shipyard pipeline processing.  
The operation status of the ship pipeline-processing workshop directly affects the overall production efficiency and quality of the ship. In the actual production process, some problems arise in the pipeline-processing workshop, which can have adverse consequences such as decreased production efficiency and increased costs. Through research on the shipyard, the following issues were found:  
* Low production efficiency: The speed of technological updates is fast, but shipyards have failed to keep up with the latest pipeline-processing technologies and methods in a timely manner. The backward level of technology has led to the inability to adopt more efficient production processes, and processing methods still remain in traditional and time-consuming operations.
* Improper production organization and management: The production-plan arrangement is unreasonable and lacks flexible production-scheduling capabilities. The current shipyard pipeline-processing workshop lacks scientific and effective methods for formulating plans. Planners usually arrange work plans based on past experience and only on the number of pipelines or segments. Insufficient coordination occurs between processes, often resulting in material waiting and idle equipment, greatly extending the production cycle.
* Low level of informatization: Owing to insufficient investment in information technology construction, workshops overly rely on traditional manual operations and experiential judgment in the formulation, execution, and adjustment of production plans. This approach is not only inefficient but also difficult to quickly and accurately respond to emergency changes or abnormal situations, resulting in production progress being affected and even production bottlenecks. The low level of informatization is also reflected in the lack of effective decision-support systems. The management lacks sufficient data support when making key decisions such as production scheduling, resource allocation, and cost control, resulting in strong subjectivity and high risk in the decision-making process.  
Under the constraints of the process flow and production resources in the production workshop, it is particularly important to allocate the processing order and machine selection of pipelines reasonably, adopt scientific production-scheduling schemes to optimize the production process, shorten the maximum completion time, reduce the flow time of pipelines in the manufacturing system, and improve machine utilization, among other scheduling objectives.