## 1 Introduction
In [18], the authors treat the production system as a single-component system and propose an integrated model for lot size and maintenance planning for random failures to jointly optimize PM cycle length and maximum production capacity. In [19], the authors consider different environmental factors (pollutants, emissions, etc.) in manufacturing systems and propose a production and maintenance optimization strategy that combines emission control based on the effects of system degradation to optimize emissions and PM quantities. The authors of [20] propose a joint optimization model of PM quantity and production lot sizing considering two failure modes, hard and soft failures, and state that the manufacturer will gain the maximum profit by using the optimal lot size and maintenance strategy [21] proposed a joint optimization model that considers both production quantity and PM interval, and showed that production lot size and preventive maintenance are mutually influential in terms of cost and profit, so they should be jointly optimized. The authors of [22] considered the production, maintenance, and quality control problems in a production system and proposed a joint optimization model to determine the optimal production lot size, inventory threshold, and maintenance threshold by minimizing the cost. The authors of [23] developed a model to optimize the production lot size by considering CBM activities and obtained the long-term average cost rate of a degraded manufacturing system using renewal theory. The authors of [24] considered the joint optimization of the economic production lot sizing and CBM for the production equipment. The degradation process is determined by age and covariate values, which are modeled as a Markov process. The problem is formulated and solved in the framework of a semi-Markovian decision process. The study in [25] investigated the optimal lot sizing and maintenance strategy for a partially observable production system by using multivariate Bayesian control methods. The literature mentioned above is based on certain failure modes or maintenance methods to optimize production planning and maintenance decisions. In terms of maintenance decisions, few of them consider PM thresholds. However, a comprehensive strategy to study production planning and PM thresholds through continuous system degradation is worth considering. Moreover, it has been shown that the joint optimization of production planning and maintenance strategies is superior to separate or sequential optimization strategies in terms of cost [26]. Therefore, in this paper, a joint optimization model of PM threshold and lot sizing based on system degradation is proposed. With the objective of maximizing profit per unit time, the optimal combination of production lot sizing and PM threshold is found. The solution algorithm for the optimal batch size and PM threshold is given in conjunction with the actual situation. Finally, the validity of the model is illustrated by numerical arithmetic examples. The main contribution of this paper to integrated production-maintenance scheduling is outlined below:  
1. Determining thresholds is a major challenge for maintenance management and this paper considers production lot sizing jointly with PM thresholds for more practicality. 2. Based on the nonlinear degradation of the system, a joint optimization model of production lot sizing and PM threshold is developed and a solution algorithm is given. The rest of the paper is structured as follows. Section 2 describes the production system under study. Section 3 develops a joint lot-size-maintenance optimization model and provides the corresponding solution algorithm. Section 4 presents a case study on a centrifugal system and tests the performance of the model. Section 5 concludes the study and provides recommendations for future research.