# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 3 Notation and Problem Statement
### Problem Formulation  
We start by conjecturing that the manufacturing system analyzed in this paper consists of an unreliable machine subject to a double deterioration process producing a single part type. The machine mode is described by the stochastic process \(\xi(t)\in\Omega\ =\{1,2,3,4\}\). More precisely, the machine is available when it is operational (\(\xi(t)=1\)), an unavailable when it is at failure (\(\xi(t)=2\)). Once at failure, the decision-maker must decide between two types of maintenance actions available. When \((\xi(t)=3)\), a minimal repair is conducted where the machine has the same failure rate as before failure, in other words, it restores the system to ABAO conditions. Furthermore, when \((\xi(t)=4)\) a major maintenance is performed mitigating all the effects of the deterioration process, thus restoring the system to AGAN conditions. The transitions rates of the system \(\lambda_{\alpha\alpha^{\prime}}\), from state \(\alpha\) to \(\alpha^{\prime}\), are statistically described by the state probabilities:  
\[ \begin{split} P\big{[}\xi(t+\delta\ t)&=\ \alpha\ |\ \xi(t)=\alpha^{\prime},x(t)=x,\ a(t)=a\big{]}\\ &=\left\{\begin{array}{ll}\lambda_{\alpha\alpha^{\prime}}( \cdot)\delta\ t+o(x,a,\delta\ t)&if\ \alpha\ \
eq\alpha^{\prime}\\ 1+\lambda_{\alpha\alpha^{\prime}}(\cdot)\delta\ t+o(x,a,\delta\ t)&if\ \alpha\ =\alpha^{\prime}\end{array}\right.\end{split}  \tag{1}\]  
with  
\[\lim_{\delta\ t\to 0}\frac{o(\delta\ t)}{\delta\ t}=0;\ \lambda_{\alpha \alpha^{\prime}}(\cdot)=-\sum_{\alpha\ \
eq\alpha^{\prime}}\lambda_{\alpha\alpha^{\prime}}(\cdot) \tag{2}\]  
\[\lambda_{\alpha\alpha^{\prime}}(\cdot)\geq 0,\ (\alpha\ \
eq\alpha^{\prime}),\quad \forall\ \alpha,\alpha^{\prime}\in\ \Omega. \tag{3}\]  
Figure 1: Manufacturing system under study. The stochastic process defines a generator matrix \(\mathbf{Q}(\cdot)=(\lambda_{aa^{\prime}}(\cdot))\), which is defined as follows:  
\[\mathbf{Q}(\cdot)= \begin{bmatrix}\lambda_{11}&\lambda_{12}(a)&0&0\\ 0&\lambda_{22}&0&\lambda_{34}\\ \lambda_{31}&0&\lambda_{33}&0\\ \lambda_{41}&0&0&\lambda_{44}\end{bmatrix} . \tag{4}\]  
The transition diagram of the system is presented in Figure 2.  
The transition rate \(\lambda_{12}(a)\) implies that the failure rate of the machine depends on its age. The rate \(\lambda_{23}\) defines the transition from the failure mode to the minimal repair mode. The inverse \([\lambda_{23}\cdot(1-w(t))]\) represents the expected delay between a call for the technician and his arrival. A similar delay is represented by the reciprocal \([\lambda_{24}\cdot w(t)]\) when the machine is send to major maintenance. Transitions \(\lambda_{31}\) and \(\lambda_{41}\) implies that the maintenance durations are defined by an exponential random variable with constant mean. Additionally, we define a binary variable \(w(t)\in\{0,1\}\) that allows us to properly synchronize the transitions to the maintenance options available, as denoted in the following expression:  
\[\omega(t)=\left\{ \begin{array}{rl}0&if\ minimal\ repair\ is\ performed\\ 1&if\ major\ maintenance\ is\ conducted\end{array} \right.. \tag{5}\]  
One noteworthy feature of the model is the assumption of production-dependent defectives, which implies that when the machine operates at a higher production rate, it is more likely to deteriorate faster, generating more defectives. Hence, to make this more precise, we state that the defectives rate \(\beta(\cdot)\) depends on the production rate \(u(t)\) according to the following expression:  
\[\beta(u(t))=\left\{ \begin{array}{ll}b_{1}&if\ u(t)<u_{1}\\ b_{2}&if\ u(t)\in\ (u_{1},u_{2}]\\ \cdots&\cdots\\ b_{k}&if\ u(t)\in(u_{k-1},u_{k}]\\ \cdots&\cdots\\ b_{n}&if\ u(t)\in(u_{n-1},u_{max}]\end{array} \right. \tag{6}\]  
Figure 2: Transition diagram. with \(b_{n}\geq\ldots\geq b_{2}\geq b_{1}\), and \(0\leq u_{1}\leq u_{2}\leq\ldots\leq u_{max}\). Where \(b_{k}\) and \(u_{k}\) are given constants. More precisely, the value of constants \(b_{k}\) of the defectives rate has the general form, ([PERSON] et al.