# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 1 Introduction  
Manufacturing systems rarely perform exactly as expected and predicted, since they may experience disorders from many unexpected events, such as equipment break-downs, delays, defectives, deterioration, etc., as reported in [PERSON] et al. [1]. In this context, quality, production planning and maintenance define the fundamental functions to achieve success in the manufacturing industry, implying resource efficiency along the product, process, and production system life-cycle. Therefore, integrated operations management approaches are needed to have a global vision of the company by taking into account the interactions between the different key functions. This paper aims to develop an effective method for decision making on industrial strategies under integrated approach of the production control, quality, and maintenance planning. There is a broad variety of practical problems dealing with the association between production and quality. It is clear that equipment availability, product quality, and productivity are strongly interrelated. However, these fields have been traditionally treated by manufacturers and researchers almost in isolation. Some authors have proposed frameworks for the joint production-qualityrelationship as in [PERSON] and [PERSON] [2], who presented an analytical method for evaluating the performance of production systems, jointly considering quality and production performance indices. [PERSON] et al. [3] studied a production unit that randomly shifts from an in-control to an out-of control state, where at the end of the production cycle, maintenance activities are performed depending on the state of the unit. The simulation work proposed by [PERSON] et al. [4] addressed the problem of an unreliable manufacturing system that produces conforming and non-conforming parts, where due to the wear of the system, the authors considered the use of external production to supplement the limited production capacity. An algorithm integrating production and quality issues were presented by [PERSON] et al. [5], where they determined the buffer sizing and inspection positioning problem of large production lines, identifying promising locations for the inspection stations. Recently, [PERSON] et al. [6] investigated the joint design and optimization of a continuous sampling plan, make-to-stock production and maintenance. They defined the number of successive items clear of defects required to discontinue rigorous inspection, the fraction of product sampling, the maintenance period and the amount of inventory needed as protections against disruptions. According to these studies, the relation between production and quality exists in several ways. Nevertheless, the model developed in our paper is different, because we take into account the fact that production at high rates accelerates the machine degradation and thus increases the total cost of repairs, defectives, production, etc. Therefore, the decisions involved in our formulation seek how to balance production, quality, and maintenance activities for efficient operations management. The issues related to the maintenance of manufacturing systems are relevant to our research because in modern production systems their components are usually unreliable and so maintenance decisions should be integrated in the decision-making to properly estimate their global effect, as in the work of [PERSON] et al. [7]. Who developed a method to find the optimal production rate for a manufacturing system, which produces several products in order to satisfy random demands; also, they established an economical scheduling for preventive maintenance. The study of [PERSON] et al. [8] addressed the problem of a production system that is continuously monitored and subject to stochastic degradation. To assess such degradation, the system undergoes preventive maintenance whenever its reliability reaches an appropriate value. [PERSON] et al.