## 2 Literature Review  
Regarding scheduling problems with machine availability constraints, hundreds of contributions have been developed in the literature. Most of the machine availability is caused by preventive maintenance. Preventive maintenance (PM) is designed as a prior measure to reduce the probability of failure or degradation, and activating PM tasks in scheduling problems are usually classified into two categories: (i) PM tasks are performed at a fixed interval or within a time window or (ii) PM tasks are carried out depending on certain monitored conditions. Table 1 exhibits a brief review of work on single-machine scheduling problems with PM tasks based on the two categories. As seen from Table 1, the previous major studies focused on the first category with different objective functions. Additionally, [PERSON] et al. [8] provided a detailed review and classification of papers that dealt with deterministic scheduling problems related to fixed PM tasks in different manufacturing shop floors. This information indicated increasing interest in studying this field over the past several decades. The references most pertinent to our considered paper are [PERSON] et al. [9], [PERSON] and [PERSON] [2], and [PERSON] and [PERSON] [5]. In these studies, the PM task is driven by monitoring the current machine's working time to ensure that it does not exceed a preset critical time threshold and to initialize the status of the machine. [PERSON] et al. [9] proposed three heuristic algorithms and a branch and bound (BAB) method to minimize the total completion time. [PERSON] and [PERSON] [2] considered the two categories mentioned in Table 1 and proposed the BAB method to minimize maximum tardiness. The two above studies assumed that all jobs were ready at time 0. [PERSON] and [PERSON] [5] first considered the dynamic case of jobs' release time in the single-machine problem. They proposed a mixed integer programming (MIP), a BAB method and a heuristic algorithm to minimize the makespan. Another PM task motivated by the wafer cleaning operation of a semiconductor manufacturing factory was proposed by [PERSON] and [PERSON] [10], where a machine has to be maintained periodically so that the amount of dirt left on the machine does not exceed a preset critical dirt threshold. [PERSON] and [PERSON] [10] developed a MIP, a dynamic programming-based heuristic algorithm to minimize the total absolute deviation of job completion times. Later, [PERSON] et al. [11] extended the single-machine problem to a parallel machine problem and developed a MIP model and three heuristic algorithms to minimize the number of tardy jobs. [PERSON] et al. [6] extended the study of [PERSON] and [PERSON] [10] to consider job release time and bicriteria (total weighted tardiness and job completion time). They proposed a scatter simulated annealing (SSA) algorithm to obtain nondominated solutions. From Table 1, it is evident that our considered problem, i.e., the dynamic single-machine scheduling problem with PM tasks, where PM tasks are driven by the threshold value of the machine's continuous working time and the total weighted tardiness as the objective, has not been studied so far. The considered problem is NP-hard since the static single-machine problem with the objective of the total weighted tardiness, where the machine is always available, has proven to be NP-hard [7]. For this kind of NP-hard problem, applying traditional methodologies, such as heuristic algorithms or exact algorithms, suffers either from solution effectiveness or computational efficiency. In recent years, various GAs based on global exploration and local exploitation search mechanisms, due to their flexibility, have been utilized more successfully than traditional approaches in solving NP-hard problems [12]. Compared with classical scheduling problems, employing meta-heuristic algorithms to solve single-machine scheduling problems with machine unavailability constraints has been very limited, with only a few studies to date. [PERSON] et al.