## 1 Introduction  
In this paper, we consider the single-machine problem with job release times and machine unavailable periods, where machine unavailable periods are caused by flexible preventive maintenance (PM) activities. For classical single-machine scheduling problems, most research assumes that all jobs are ready for processing simultaneously or that machines are always available to simplify the complexity of scheduling problems. These two assumptions may impede many possible practical applications, and some studies have demonstrated that there is a need to consider the dynamic job release time [1] or machine unavailable periods [2, 3]. Both are common phenomena in the real world and are significant factors in production scheduling decisions. That is, taking into consideration jobs' release time and machine unavailable periods, a given production scheduling problem can be solved more realistically. For the considered problem, more precisely, there are \(n\) jobs with different release times to the production system and waiting to be processed on a single machine without preemption. The machine is not always available; it needs to be maintained periodically to prevent its continuous working time from exceeding a specific threshold value and to initialize the machine's status. To the best of our knowledge, there are only three studies considering dynamic job release time and machine availability constraints simultaneously for the single-machine scheduling problem. [PERSON] [4] was the first to consider this type of problem and proposed a MIP model to minimize the weighted number of late jobs. In this study, the machine unavailable periods were fixed and known in advance. [PERSON] and [PERSON] [5] considered the dynamic job release time for the machine availability schedulingproblem. The main difference from the study of [PERSON] [4] is that, when implementing PM as a decision variable in scheduling planning, it is not fixed and known. For this problem, the researchers proposed a MIP model, heuristic algorithms and the branch and bound (BAB) method to minimize the makespan. [PERSON] et al. [6] considered the single-machine maintenance scheduling problem with dynamic job release time, and this study was motivated by a clean operation of semiconductor manufacturing in which the machine had to stop to remove the dirt in the machine as a clean agent. For this problem, the researchers proposed a scatter simulated annealing algorithm to simultaneously minimize the total weighted tardiness and total completion time. Our considered problem is the same as that of [PERSON] and [PERSON] [5] mentioned above; the objective of [PERSON] and [PERSON] [5] is to minimize the makespan, implying maximizing the throughput of the system. With the increasing importance of time-related competition and customer satisfaction, production performance based on due-dates becomes more significant. Thus, the objective we adopted is to minimize the total weighted tardiness (_TWT_) for responding to the needs of on-time delivery in just-in-time (JIT) production, which is one of the important, relevant objectives for today's manufacturing environments. Moreover, the _TWT_ objective has not been considered as often by researchers in this problem. This problem is also NP-hard because the special case without machine maintenance constraints, that is, the single-machine scheduling problem to minimize the total weighted tardiness has proven to be NP-hard [7]. For the NP-hard problem considered in this paper, we propose a trajectory-based immigration strategy genetic algorithm. The main reason is that different genetic algorithms (GAs) have been implemented successfully in many complicated scheduling problems but are seldom applied to the considered problem. Additionally, an immigration strategy is one of the common ways to keep the diversity of the population and avoid local convergence.