## 1 Introduction  
During the construction of ships and offshore equipment, pipeline processing in the workshop is an important part of the whole construction project. The pipeline manufacturing and installation workload accounts for about 23-30% of the workload of the whole ship, and its production progress and quality directly affect the smooth advancement of the major nodes of the ship construction process [1]. Under the modern shipbuilding mode, with a reasonable arrangement of the production plan, reducing the waiting time is the top priority of the whole production scheduling. For the shipyard pipeline-processing workshop scheduling problem, it can be regarded as a flexible job-shop scheduling problem (FISP) [2]. The FISP was first proposed by [PERSON] and [PERSON] in 1990. Currently, we have several production-scheduling methods, including mathematical planning, rule scheduling, simulation scheduling, and group intelligent optimization. Among these, the group intelligence algorithm stands out for its accuracy and efficiency. By imitating the behavioral patterns of nature or living organisms, it can obtain an ideal scheduling plan. Common examples of this approach include the genetic algorithm, artificial fish swarm algorithm, ant colony algorithm, and particle swarm algorithm. For the FISP problem, a variety of different optimization algorithms have been used to solve this type of problem. For instance, [PERSON] [3] proposed a new objective function using the improved firefly algorithm to solve FJSP with the objective of minimizing the maximum completion time and minimizing the total machine load. [PERSON] [4] used a multi-objective sparrow search algorithm based on non-dominated sorting to solve the single objective FJSP. [PERSON] [5] used the bat algorithm to solve the FJSP by analyzing the relationship between the initially selected machine and the scheduled completion time of each process. [PERSON] [6] introduced an adaptive mechanism in the swarm algorithm to enhance the local intensification of the algorithm capability, providing a new idea for solving FJSP. All of the above studies were based on deterministic process processing times; however, the actual production environment is dynamic, and accurate process processing times are often difficult to obtain in advance, which impacts the actual utility of scheduling plans. Therefore, the study of the fuzzy flexible job-shop scheduling problem (fFJSP) with an uncertain processing time is of greater practical significance. At present, the use of triangular fuzzy numbers to describe the processing time of the workpiece has become the main research direction to solve the fFJSP, and scholars worldwide have carried out extensive research and exploration in this regard. For example, [PERSON] [7] expressed the processing time, processing cost, and raw-material cost in terms of a triangular fuzzy number and solved the fFJSP by using an adaptive discrete flower-pollination algorithm. Furthermore, [PERSON] [8] used a decomposition-based multi-objective evolutionary algorithm that is a variable neighborhood search based on four neighborhood actions to solve fFJSP. Moreover, [PERSON] [9] constructed a mechanism for interconverting scheduling solutions and whale individuals for the fFJSP scheduling problem to innovate the solution of the fFJSP scheduling problem. [PERSON] et al. [10] applied the teach-and-learn algorithm to the fFJSP by adding a search framework based on the teach-and-learn mechanism and a special local search operator to enhance the algorithm. A two-stage crossover scheme is added to the search framework based on the teach-and-learn mechanism and special local search operators to enhance the local search capability of the algorithm. Many researchers have also carried out related work in this area and have made some important advances [11; 12; 13]. The processing time of different pipelines varies widely, ranging from a few minutes to several hours.