###### Abstract  
In this paper, a hybrid bat optimization algorithm based on variable neighbourhood structure and two learning strategies is proposed to solve a three-stage distributed assembly permutation flowshop scheduling problem to minimize the makespan. The algorithm is firstly designed to increase the population diversity by classifying the populations, which solves the difficult trade-off between convergence and diversity of the bat algorithm. Secondly, a selection mechanism is used to update the bat's velocity and location, solving the difficulty of the algorithm to trade-off exploration and mining capacity. Finally, the Gaussian learning strategy and elite learning strategy assist the whole population to jump out of the local optimal frontier. The simulation results demonstrate that the algorithm proposed in this paper can well solve the DAPFSP. In addition, compared with other metaheuristic algorithms, IHBA has better performance and gives full play to its advantage of finding optimal solutions.  
hybrid bat algorithm; optimization problem; the distributed assembly permutation flowshop scheduling problem; variable neighborhood descent +
Footnote †: journal: Applied Sciences  
17 September 2021  
22 October 2021  
as quickly as possible. This allows the objective function to minimize the flow of work and efficiently improve resource utilization. DPFSP is an NP-hard with \(m\) greater than or equal to 2, and is a derivation and extension of the traditional permutation flow shop scheduling problem with this criterion. Similarly, DAPFSP can be seen as an extension of DPFSP, which is more complex than DPFSP. Therefore, the DAPFSP with completion time as the objective function is also a strong NP-hard problem. In recent years, as a simple yet efficient method, the bat algorithm has solved many scheduling problems in academia, including the job shop scheduling problem [1], the open shop scheduling problem [2], the task scheduling problem [3], and the production scheduling problem [4], by continuously performing local search in the neighborhood of the current solution to obtain the optimal solution.  
In this paper, an improved hybrid bat algorithm, IHBA, is proposed for solving the DAPFSP with maximum makespan based on the original bat algorithm. The main contributions of this paper are:  
* The algorithm firstly designs the population classification to increase the diversity of the population and solve the difficult trade-off between convergence and diversity of the bat algorithm.
* A selection mechanism is used to update the speed and location of bats, solving the difficulty of the algorithm to trade-off exploration and mining capacity.
* Gaussian learning strategy and elite learning strategy assist the whole population to jump out of the local optimal frontier.  
This paper is organized as described below. Section 2 reviews the closely related literature. Section 3 presents the DAPFSP problem and its mathematical model. In Section 4, the hybrid bat algorithm proposed in this paper is described in detail. In addition, the parameter calibration and simulation calculation results are analyzed in Section 5. In the last section, the paper is summarized, and future work is provided.