## 1 Introduction
[23] studied the joint optimization of production and maintenance of multi-component series-parallel systems. The above articles reflect that the research on multi-unit systems is a trend, and the structure of units in the system is an important factor affecting the maintenance strategy. One point in common is that scholars formulate strategies by assuming the deterioration of units before and after maintenance, and the deterioration of equipment is described by setting the deterioration function. The deterioration function is obtained according to the long-term data of the equipment, which is reasonable, but the function will change when the equipment maintenance is considered. [PERSON] et al. [24] proposed a deterioration state space partitioning (DSSP) method and established the maintenance model of the multi-unit series system. The steady-state probability density function of the system considering maintenance was derived and verified. The DSSP method was applied to the joint optimization of maintenance and spare parts supply strategies to determine the optimal maintenance and spare parts supply ordering activities [25; 26; 27]. [PERSON] et al. [28] studied the joint optimization of production scheduling and equipment maintenance using the DSSP method. The DSSP method was extended, and various maintenance models were established considering the maintenance characteristics of different systems [29]. A question to be considered is that the system studied before by the DSSP method is a series system. The structural relationship of critical components in many equipment is not a simple series relationship but a complex series-parallel structure. At the same time, the deterioration process of the units is affected by the relationship between the units in the equipment [22]. Therefore, a method is proposed considering the structural correlation between equipment units based on the DSSP method [24]. Based on the actual production situation, a three-unit series-parallel batch production system is taken as the research object. The system produces products of a single variety in large quantities. Units deteriorate in the production process, and the demand rate will change with the change of the system state [30; 31]. To avoid interfering with the normal production process, maintenance activities are arranged after the end of the batch. Since the structure of the three units in the system is not a simple series relationship, when the necessary unit exceeds the fault threshold, the system needs to be maintained, but when the non-essential unit exceeds the fault threshold, the system does not need to be maintained. Under the set strategy, there are 23 situations in production and maintenance actions. The probability of each situation can be solved by using the extended DSSP method to analyze the state probability of each unit. Considering the probability and cost of each situation, a model with the long-term expected cost rate as the objective function is established. The model is a complex nonlinear optimization problem with a large solution space. The genetic algorithm is selected because of its simplicity and optimization ability in solving production problems [32; 33]. Finally, the proposed model is tested through a case study and sensitivity analysis. The purpose is to reasonably arrange maintenance and production plans to ensure that the system operates at low cost and high efficiency. In our view, there are three main contributions of this paper. 1. Considering the structural relationship between units, the maintenance group of the system is divided, and the steady-state probability density function considering structural importance is calculated based on extended DSSP. 2. The deterioration of each unit leads to a decline in product quality, thus affecting the demand rate. The numerical relationship between each unit state and system demand rate is extended to the three-unit system considering that part of the unqualified products can be repaired. 3.