# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 3 Notation and Problem Statement
### Problem Formulation
[14]):  
\[b_{k}=\eta_{0}(\frac{u_{k}}{u_{max}})^{\eta_{1}} \tag{7}\]  
where \(\eta_{0}\) and \(\eta_{1}\) are known positive constants and \(u_{max}\) is the maximum production rate. Equation (7) serves to define the value of constant \(b_{k}\). Figure 3 presents the trend of the rate of defectives \(\beta(\cdot)\) for different values of \(\eta_{0}\) and \(\eta_{1}\). We can observe in Figure 3, the considerable influence of the productivity of the machine on the rate of defectives. At considering the presence of defectives, the evolution of the stock level \(x(t)\) is defined by the following differential equation:  
\[\frac{dx(t)}{dt}=u(t)-d[1+\beta(u(t))],\quad x(0)=x_{0}. \tag{8}\]  
The constant \(x_{0}\) defines the initial stock level and \(d\) denotes the demand rate. Concerning the evolution of the age of the machine \(a(t)\), it implies an increasing function of the number of parts produced, and it is defined by the next differential equation:  
\[\frac{da(t)}{dt}=ku(t) \tag{9}\]  
\[a(T)=0 \tag{10}\]  
with \(k\) as a given positive constant and \(T\) is the last restart time of the machine. Furthermore, bearing in mind that the deterioration process also has an effect on the reliability of the machine, in particular in its failure rate \(\lambda_{12}(\cdot)\). Then the lifetime distribution of a new machine follows an increasing function, as in [PERSON] et al. [16]:  
\[\lambda_{12}(a(t))=\lambda_{1}+\lambda_{2}[1-e^{-r\theta\;a(t)^{\eta_{2}}}] \tag{11}\]  
where the parameter \(\theta\) is useful to adjust the trend of the failure rate and \(0\leq\theta\;\leq 1\), \(\lambda_{1}\) is the failure rate in AGAN conditions, \(\lambda_{2}\) is the limit considered in the deterioration process for the rate \(\lambda_{12}(\cdot)\), \(r\), and \(\eta_{2}\) are non-negative constants. At selecting appropriate values for \(r\), Equation (11) can model increasing functions similar to the Weibull distribution. We present in Figure 4, the trend of the failure rate for different values of the adjustment parameter \(\theta\). Figure 3: Defectives rate. The decision variables of the model are the production rate \(u(t)\), and the maintenance switching strategy \(\omega(t)\). Thus, the set of feasible control policies \(\Gamma(\alpha)\), including \((u(t),\omega(t))\) is given by:  
\[\Gamma(\alpha)=\Big{\{}(u(t),\omega(t))\in R^{2},\quad 0\leq u(t)\leq u_{max}, \quad\omega(t)\in\{0,1\}\,\Big{\}}. \tag{12}\]  
We are now able to define the cost rate of the model as:  
\[\mathrm{g}(\alpha,x,a)=c^{+}x^{+}+c^{-}x^{-}+\ c_{d}\cdot\beta(u(t))\cdot d+c_ {pro}\cdot u(t)+c_{r}\cdot Ind\{\xi(t)=3\}+c_{m}\cdot Ind\{\xi(t)=4\} \tag{13}\]  
with:  
\[x^{+}=max(0,x)\]  
\[b_{k}=\eta_{0}(\frac{u_{k}}{u_{max}})^{\eta_{1}} \tag{7}\]0  
\[Ind(\xi(t)=\alpha)=\left\{ \begin{array}{ll}1&\text{if }\xi(t)=\alpha\\ 0&\text{otherwise}\end{array} \right.\]  
where the cost parameters \(c^{+}\) and \(c^{-}\) are used to penalize inventory and backlog, respectively. The parameter \(c_{d}\) denotes the defective cost originated by the additional handling and inspection, \(c_{pro}\) denotes the production cost, \(c_{r}\) is the minimal repair cost and \(c_{m}\) is the major maintenance cost. The objective in our model implies the determination of the optimal control policies that minimizes the integral of the following expected discounted cost:  
\[v(\alpha,x,a)= \begin{array}{c}inf\\ (u(t),\omega(t))\in\Gamma(\alpha)\end{array} E\left[\int\limits_{0}^{\infty}e^ {-pt}g(\cdot)dt\mid\alpha(0),\ x(0),\ a(0)\right] \tag{14}\]  
where \(\rho\) denotes a positive discounted rate, and \(v(\cdot)\) defines the value function of the model. Based on the _optimality principle_, and at defining the cost-to-go function as \(v(\cdot,t)\), we can break-up the integral of Equation (14) as follows:  
\[v(\alpha,x,a,t)= \begin{array}{c}inf\\ u(t),\omega(t)\\ 0\leq t\leq\infty\end{array} E\left[\int\limits_{0}^{t}e^{-pt}g(\cdot)dt+\int \limits_{t}^{\infty}e^{-pt}g(\cdot)dt\mid\alpha(0),\ x(0),\ a(0)\right].