## 1 Introduction  
The flow shop scheduling problem (FSSP) plays an important role in manufacturing systems. Optimizing multiple criteria of the FSSP can help reduce manufacturing costs and improve the manufacturing efficiency of enterprises. In recent years, many variants of the FSSP have emerged and many methods have been proposed to optimize some of its criteria. The permutation flow shop scheduling problem (PFSP) is a classical form of the FSSP, which was first introduced and formulated by [PERSON] [1]. The problem with more than three machines is shown to be NP-hard [2]. The goal of this problem is to schedule operations on the machines to optimize one or more performance criteria, such as minimizing the makespan, mean tardiness, total late work, and total flow time of all jobs. A rational scheduling algorithm can not only improve the efficiency and performance of the system but also reduce the cost of machines. In the past decades, various exact or heuristic algorithms for solving scheduling problems have been suggested [3]. Researchers have generalized the PFSP problem into multiple variants to simulate real production scenarios, such as no-wait flow shop, blocking flow shop, no-idle flow shop, and energy-efficient flow shop. To solve problems of different scales, many effective exact methods, heuristics, and meta-heuristic algorithms have been proposed to optimize various criteria for the PFSP. Exact methods based on enumeration with an integer programming formulation are usually employed to find the optimal solution for PFSP. When dealing with large-scale problems, the solution space gets bigger and the use of exact algorithms will always lead to the combinatorial explosion so that the computation time is usually unacceptable. The most commonly applied approaches for large-scale problems are heuristic approaches such as NEH [4] and CDS [5], which are capable of real-time decision making. However, hand-crafted heuristic algorithms only consider limited information which leads to the unstable performance of the algorithm. Meta-heuristic algorithms such as the genetic algorithm (GA) [6], tabu search algorithm (TS) [7] and the particle swarm optimization algorithm (PSO) [8; 9] are a class of algorithmic frameworks that are problem-independent. The performance of these algorithms depends on adequate parameter tuning and initial solutions. Moreover, these algorithms have a slow convergence speed on problems that with high computational complexity. With the development of deep learning, reinforcement learning, and some high-performance search algorithms, methods applying these techniques have achieved better performance compared to traditional algorithms on some complex combinatorial optimization problems. The late work appears to be important in production planning from the perspective of the customer as well as the manager. Customers are often concerned about the completion of the order, because those tasks that are not completed by the due dates need to be processed additionally. Managers are also interested in minimizing the late work criteria, as delays may cause financial losses. The late work phrase was first introduced in literature [10] and is defined as the number of tardy job units. The symbol \(Y\) is first used to represent it. [PERSON] et al. describe the difference between late work and other performance criteria such as makespan, tardiness, lateness, and proves that problems with a late work objective function are at least as difficult as problems with a maximum delay criterion [11]. Since then, many exact and heuristic algorithms have been proposed for single machine [12], parallel machine [13], and dedicated machine problems [14] with late work objective functions. The problem of minimizing late work has been demonstrated in many fields such as chip manufacturing [15], computer integrated manufacturing (CIM) [14], supply chain management [16], and software development processes [17].