# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 1 Introduction
[9] study a manufacturing system composed by a failure prone-machine, a manufacturing store, and a purchase warehouse with service level, where a preventive maintenance plan is provided in order to decrease the failure rate. In the study of [PERSON] et al. [10], the authors dealt with a preventive maintenance strategy and the determination of an economical production plan. Their model defines the optimal maintenance interval at which machines are maintained simultaneously. A common feature of the above papers is that they have mainly studied the joint production scheduling and maintenance planning, which has received much attention in the literature, but this does not necessarily lead to an optimal solution. Since they have disregarded the importance of quality aspects in their results. Hence, taking into account the interrelations between production, quality, and maintenance, traditionally approaches may be modified. In the context of deteriorating systems, machine failure is probably one of the most frequently observed disruption that does deteriorate the system performance. A considerable amount of research has been spurred to address time-dependent failures. However, in most manufacturing systems is often more realistic to assume that machine reliability does depend on the degree of utilization of the machine. Thus, operation-dependent failures are common in such systems, and this assumption renders the problem much more involved, as indicated by [PERSON] et al. [11], who provided the structure of a policy minimizing the long-term average backlog and inventory cost for an unreliable machine, where the failure rate is a piecewise constant function of the production rate. In the same vein, [PERSON] et al. [12] dealt with the problem of dependence between production and failure rates in the context of single randomly failing and repairable manufacturing system producing two products. [PERSON] et al. [13] were interested in the study of a production unit that aims to satisfy the deterministic market demands for multiple products. They considered that the production cost depends on the usingrate of the machine, and that such machine deteriorates with increased use. Other researchers have treated the problem of production-dependent failure rates, for example [PERSON] et al. [14] analyzed a manufacturing system comprising parallel machines with failure rate depending on their productivity. They determined the productivity of the main and the supporting machine. From the discussed papers, it is evident that models considering operation-dependent failures, are rarely studied in the literature, and their focus have been mainly on the dependence between production and failure rate. Thus, one drawback of these papers is that the connection between productivity and quality deterioration have not been considered. In contrast, since deterioration is a common industrial phenomenon, our model aims to extend the concept of deterioration to state that indeed production at higher rates accelerates the machine degradation, and their effect not only may increase the failure rate but also may decline product quality. Our research aims to generalize previous assumptions and extent several conjectures reported in the literature. In particular, we extend the work of [PERSON] et al. [11], [PERSON] et al. [9], and [PERSON] et al. [14] in several directions: (i) at presenting an integrated production-maintenance-quality approach which serves to analyze the interactions between these three key functions; (ii) at studying the impact of a double deterioration process with continuous deterioration of part quality and reliability; (iii) at considering the dependence between productivity and product quality, leading to define operation-dependent defectives. We note that these set of characteristics have not been treated simultaneously in the literature yet. We developed a stochastic optimal control model to determine the structure of the control policy. Moreover, the obtained results are examined thorough an extensive sensitivity analysis.