###### Abstract  
Hybrid flow-shop scheduling based on the parallel sequential movement mode (HFSP-PSMM) is an extended application of hybrid flow-shop scheduling that ensures that the equipment works continuously during the processing cycle. However, current research has only investigated the flow-shop scheduling of single-equipment processing, and ignores the effect of auxiliary time. Therefore, this paper investigates a multi-equipment hybrid flow-shop scheduling problem based on the parallel sequential movement mode and considers the setup time and handling time. The mathematical model of the HFSP-PSMM was developed with the handling and makespan numbers as the optimization objectives. The NSGA-II-V based on the NSGA-II was designed by combining the problem characteristics. New crossover, mutation, and selection strategies were proposed and variable neighborhood search operations were implemented for the optimal set of Pareto solutions. Finally, through an algorithm comparison, performance testing, and an example simulation, the effectiveness of the NSGA-II-V for solving the HFSP-PSMM was verified. hybrid flow-shop scheduling; handling time; setup time; parallel sequential mode +
Footnote †: journal: Applied  
the mean continuous running time as the optimization objectives to improve the local search capability of the genetic algorithm. [PERSON] et al. [7] proposed a traveling salesman algorithm and a genetic algorithm with the optimization objectives of the makespan, mean flow time, and machine idle time. [PERSON] et al. [8] proposed an improved particle swarm optimization algorithm with the optimization objectives of the makespan and total flow time. [PERSON] et al. [9] used an improved genetic algorithm to solve the no-wait HFSP with the objective of minimizing the maximum completion time and the earliest start time of each piece of equipment. In the HFSP, there are three modes of job organization: the sequential movement mode, the parallel movement mode, and the parallel sequential movement mode. the sequential movement mode can be described as a process in which all the workpieces of the current process are machined before the next step can be performed. The parallel movement mode can be described as the workpiece being ready for the next step once the current workpiece has been machined. The parallel sequential movement mode requires each machine to be continuously processed without allowing idle time during processing. This reduces the machine failure rate due to frequently switching on and off, reducing energy consumption. Therefore, the HFSP based on the parallel sequential movement mode can significantly improve the utilization of processing equipment. Moreover, there are certain scenarios in which production processing can only be performed in the parallel movement mode. For example, in the production of glass fibers, integrated circuits, and ceramic frits, the processing equipment cannot be idle once it is started until all the workpieces are processed due to limitations in the production processes, technical specifications, or manufacturing costs. The parallel sequential movement mode idea is similar to the no-idle flow-shop scheduling problem (NFSP), in that the goal is to ensure that there is no idle processing equipment. [PERSON] et al. [10] studied the multi-objective flow-shop batch-scheduling problem by considering a separable processing time and adjustment time in the parallel sequential movement pattern, constructing a multi-objective model, and proposing a separable forbidden search algorithm to solve the problem. [PERSON] et al. [11] constructed a mathematical model of flow-shop scheduling in the parallel sequential movement mode and designed a hybrid genetic algorithm to solve the model. In the study of the NFSP, [PERSON] et al. [12] proposed a hybrid discrete differential evolutionary algorithm for solving the no-idle permutation flow-shop scheduling problem. [PERSON] et al.