## 1 Introduction  
Modern manufacturing enterprises are facing significant challenges, which are reflected by the rising production costs and equipment costs within the enterprise. At the same time, the market has more stringent requirements for product quality than ever [1]. How to reduce costs and improve profits is an essential task for production enterprises at this stage. The optimization of the production system includes two aspects of optimization: production process, and maintenance process. Currently, the optimization of the production process involves production planning and production scheduling. [PERSON] et al. [2] and [PERSON] et al. [3] studied the production plan or material demand plan at the system level. The conclusions provided guidance for the long-term production of enterprises. For short- and medium-term production, [PERSON] et al. [4] established a multi-objective scheduling model to minimize total idle time, total throughput time, and total tardiness and determined the optimal order scheduling strategy. [PERSON] et al. [5] considered the actual production constraints with a practical case and proposed a method to solve the actual production and scheduling problems. There are many enterprises for whom mass production is a reality. The traditional Economic Production Quantity (EPQ) model was widely used in the industrial field, but it ignored many practical factors in the actual production. Many scholars adjusted the EPQ model according to the actual production situation, such as variable demand rate [6, 7], adjustable productivity [8], dynamic price [9], etc. The above articles ignored the objective fact of equipment deterioration in the production system when considering short-term production. In the actual production scene, the equipment will wear and tear, and the state of the equipment will deteriorate [10]. If the equipment's state is not considered in the production process, the probability of sudden failure of the equipment will be significant. The equipment will fail when any critical component failure causes the equipment to stop working [11]. Therefore, a good production plan should fully consider the deterioration of the equipment and maintain the equipment in optimal operation. Equipment maintenance includes preventive maintenance (PM) before failure and corrective maintenance (CM) after failure [12]. Due to the development of modern sensor technology and the cost of CM being generally far higher than that of PM, PM is more common in contemporary research. [PERSON] et al. [13] studied the production scheduling and PM decision-making process of a single piece of equipment. The optimal production sequence and PM interval were obtained. [PERSON] et al. [14] and [PERSON] et al. [15] established a joint model of regular PM and mass production to obtain optimal parameters for production and maintenance plans. Some scholars proposed multi-objective models that allow policymakers to seek optimal solutions between production and equipment maintenance objectives [16; 17; 18]. Later, with the upgrading of production equipment, the component structure became complicated, and it is more meaningful to study more practical equipment. Many scholars extended their research objects into multi-unit systems. [PERSON] et al. [19], considered the economic dependency of multi-unit system maintenance and proposed a numerical algorithm to accurately calculate the cost. In the study of multi-component system maintenance operation, there is not only economic dependency but also structural dependency between each unit. [PERSON] et al. [20] proposed a grouping maintenance strategy for multi-component systems considering the structural dependency. On the basis of [20], [PERSON] et al. [21] measured the important role of the components in the system structure with Birnbaum's importance. [PERSON] et al. [22] studied the condition-based maintenance decision of complex systems considering structural and economic dependency. [PERSON] et al.