## 1 Introduction  
Most systems will degrade over time due to a variety of internal factors, such as component aging, mechanical wear, and external factors, such as shock and vibration [1, 2, 3]. Failure occurs when degradation accumulates beyond an acceptable or safe level (threshold) [4]. Obviously, unexpected downtime poses certain safety risks as well as serious economic losses. Lack of timely maintenance of the system may also lead to significant losses in production and reduced profits [5]. Maintenance can be defined as all activities required to keep a system working properly and may include inspection, lubrication, adjustment, repair, and replacement [6]. Maintenance is an important measure to prevent potential failures. When and how to implement maintenance based on system characteristics and fault evolution is a critical and widely studied topic in system operation research. In some industrial systems, such as manufacturing systems, defense systems, and power generation systems, system maintenance is required between successive tasks. There may be different maintenance schemes for different system states, such as do nothing, minimal repair (repair as old), preventive maintenance (PM), corrective maintenance (CM), time-based maintenance (TBM), condition-based maintenance (CBM), and condition-based maintenance, but it is not possible to perform all maintenance actions during intervals, so the best decision needs to be made in conjunction with the actual situation. In [7], a weather-centered opportunistic maintenance strategy was developed for flexible implementation of wind turbine operations and maintenance. In [8], three types of maintenance windows (regular, opportunistic, and postponed) are planned to ensure flexible arrangements for inspections and spare parts. Preventive maintenance is gaining momentum and has been applied in many industrial systems [9; 10; 11], but how to determine the optimal preventive maintenance threshold is an interesting issue in practice. CM is maintenance that restores a system to a specified functional state after repairing a faulty component. TBM is time- or cycle-based preventive maintenance. Since TBM follows a set schedule, it is likely to lead to under- or over-maintenance. CBM is equipment condition-based preventive maintenance that makes decisions on real-time diagnostic information about impending failures and, therefore, if properly applied, it can effectively reduce system downtime compared to other maintenance strategies [12; 13; 14]. Unlike CM and TBM, CBM relies on the state of the monitored system over time. Maintenance management has a significant impact on the reliability and availability of production systems [15]. In a production system, production activities and maintenance plans are inseparable and they are two factors that affect each other and are interrelated. The approach of studying either factor alone without considering the other one cannot optimize the goals of the production system. Therefore, it is worthwhile to consider the production and maintenance plans together, to solve the conflicts between them rationally and improve the efficiency of the production system effectively. A modern manufacturing system needs to face the market demand of multi-variety and small lot size production, how to reduce cost, reduce inventory, and improve efficiency against fierce market competition, which determines the viability of an enterprise in the whole environment. Bi-objective optimization can more accurately optimize the reliability of the system [16] and there are some studies that determine production planning and maintenance strategies through bi-objective optimization and multi-objective optimization. As such, [17] first proposes a joint model of production and maintenance and classifies them into three categories, explaining that in some cases, integrating maintenance and production planning is an effective method to reduce the total cost of production activities, PM activities, and costs associated with equipment failure.