# JIT Production Strategy and Maintenance for Quality Deteriorating Systems
## 3 Problem Statement  
Let us introduce the manufacturing system under study that consists of an unreliable production unit producing one part type. Due to the random availability of the unit, its uptimes and downtimes are assumed to follow Markov dynamics, with states in \(\Omega=\{1,2,3\}\) and generator matrix \(Q(\cdot)\) such that \(Q(\cdot)=\{q_{\alpha\alpha^{\prime}}\}\), where \(q_{\alpha\alpha^{\prime}}\) denotes the transition rates from state \(\alpha\) to \(\alpha^{\prime}\), \(q_{\alpha\alpha^{\prime}}\geqslant 0\), \(q_{\alpha\alpha}=-\sum\limits_{\alpha^{\prime}\
eq\alpha}q_{\alpha\alpha^{ \prime}}\), and \(\alpha\), \(\alpha^{\prime}\in\Omega\). Thus, \(\alpha(t)\) denotes a continuous-time discrete space stochastic process that describes the modes of the production unit, such that: \(\alpha(t)=1\), the production unit is operational; \(\alpha(t)=2\), the unit is at failure, where a minimal repair is conducted, leaving the unit in as-bad-as-old (ABAO) conditions; and \(\alpha(t)=3\), the unit is under major maintenance that restores the system to as-good-as-new (AGAN) conditions. The major maintenance implies a perfect repair that serves to mitigate all the effects of the deterioration process. In this case, after the conduction of a major maintenance, the rate of defectives is restored to initial conditions. The dynamic behavior of the inventory level \(x(t)\) can be described by the following differential Equations:  
\[\frac{\partial x(t)}{\partial t}=u(t)-\frac{d}{(1-\beta(a))} \tag{1}\]  
where \(u(t)\) represents the production rate at time \(t\), \(d\) is the demand rate, and \(\beta(a)\) denotes the defectives rate at age \(a\). Note that in Equation (1) the demand rate is adjusted by \(1/(1-\beta(a))\), since the production unit must produce more to compensate for the increasing presence of defective units originated by the deterioration process. The production rate at any instant in time must satisfy the capacity constraint \(0\leqslant u(t)\leqslant u_{max}\), where \(u_{max}\) denotes the maximum production rate. When processing parts, it is assumed that the production system progressively ages. Thus, we define \(a(t)\) as the age of the production unit at time \(t\). This age \(a(t)\) is measured by the cumulative number of parts produced at time \(t\) since the last conduction of a major maintenance, and it is calculated using the following Equation:  
\[\frac{\partial a(t)}{\partial t}=\eta_{0}\cdot u(t) \tag{2}\]  
\[a(T)=0 \tag{3}\]  
where \(\eta_{0}\) is a given positive constant and \(T\) refers to the last restart time of the production unit. One noteworthy feature of the model is the incorporation of the effects of a deterioration process in its formulation. In this regard, we assume that the rate of defectives increases as the machine ages [3], as denoted in the following Equation:  
\[\beta(a)=b_{0}+b_{1}\Big{[}1-e^{-\eta_{1}\theta a(t)^{\eta_{2}}}\Big{]} \tag{4}\]  
More formally, \(b_{0}\) is the rate of defectives at AGAN conditions, \(b_{1}\) is the upper limit considered in the deterioration process, and \(\eta_{1}\) and \(\eta_{2}\) are non-negative constants. \(\theta\) is an adjustment parameter with value in \(0\leqslant\theta\leqslant 1\), which is useful to adjust the pace of increment of the defectives rate for a particular production system. Equation (4) allows us to model the quality deterioration phenomenon, which in this case implies that the rate of defectives \(\beta\) increases as the machine ages. Due to the random availability of the production unit, its steady state availability \(\pi_{1}\) can be calculated as follows:\[\pi_{i}\cdot Q(\cdot)=0\text{ and }\sum_{i=1}^{3}\pi_{i}=1 \tag{5}\]  
In view of Equations (5), for the operational mode we have:  
\[\pi_{1}=\frac{1}{1+q_{12}/q_{21}+q_{13}/q_{31}} \tag{6}\]  
Additionally, when considering the effects of the deterioration process on the rate of defectives, the decision-maker must ensure that the productions system is able to satisfy product demand at high levels of deterioration.