## 2 Problem Description
### Formulation of the PMPP  
#### 2.2.1 Production Policy  
A production system describes an integrated set of processes that allow us to produce a good service that meets the primary objective of the company, which is customer satisfaction. The coordination of different activities such as production and maintenance is necessary in order to achieve the profit maximization. The aim of this subsection is to present the production plan that minimizes the production and holding costs. Assuming that the horizon is partitioned equally into \(H\) periods of length \(\Delta t\). The demand \(d(k)\) is a random variable that follows a normal distribution. Let \(\{f_{k}(\ ),\ k=1,\ \ldots,\ H\}\) represent the cost function of the production and storage. The following sequential stochastic linear programming problem provides an optimal production plan over the planning horizon. \[Min\,E\left\{\sum_{k=0}^{H-1}f_{k}(S(k),U(k))+f_{H}(S(H))\right\} \tag{1}\]  
Subject to:  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]  
\[P\left[S(k+1)\geq 0\right]\geq\alpha\qquad\qquad\qquad k=0,1,\ldots H-1 \tag{3}\]  
\[0\leq\ U(k)\ \leq\ U_{\max}\qquad\qquad\qquad\qquad k=0,1,\ldots,H-1 \tag{4}\]  
The first constraint, (2), defines the inventory balance equation for each production period. The second constraint, (3), imposes a service level requirement in each period, which in turn induces a safety stock. Finally, constraints (4) define lower and upper bounds on the production rate during each period. The following equation represents the expected production and holding costs for each period \(k\). We use the quadratic form for our expression, which allows both an excess and shortage of inventory to be penalized [18]. \[f_{k}(S(k),U(k))=\ C_{s}\,\times\,E[S(k)^{2}]+\ C_{pr}\,\times\ U(k)^{2} \tag{5}\]  
Hence, over the finite horizon \(H\). \(\Delta t\), the total expected cost of production and inventory can be expressed as:\[F(U)\,=\,\sum_{k=0}^{H}\{f_{k}(S(k),U(k))\}=\,C_{s}\,\times E[S(H)^{2}]\,+\,\sum_{k= 0}^{H-1}\,[C_{s}\,\times\,E[S(k)^{2}]+\,C_{pr}\,\times U(k)^{2}] \tag{6}\]  
Note that \(U(H)\) is not included in the cost formulation, because we do not consider the production order at the end of the horizon \(H.\Delta t\). In practice, this constrained stochastic nonlinear programming problem is generally difficult to solve. Therefore, we transformed it into an equivalent deterministic problem, which becomes easier to solve [18]). Note that \(V(S(0))=0\) and let \(V_{d}(k)\) be constant and equal to \(V_{d}\) for all of \(k\). After simplifying, which is described in more detail in [PERSON] et al. [18], we obtain the following expression:  
\[Min\,F(U)=C_{s}\,\cdot\,\hat{S}(H)^{2}+\sum_{k=0}^{H-1}\Bigl{[}C_{s}\cdot\hat {S}(k)^{2}+C_{pr}\cdot U(k)^{2}\Bigr{]}\,+C_{s}\cdot V_{d}\cdot\frac{H(H+1)}{2} \tag{7}\]  
Subject to:  
\[\hat{S}(k+1)=\hat{S}(k)+U(k)-\hat{\hat{d}}(k)\ \ \ k=\{0,1,\ldots,H-1\} \tag{8}\]  
\[\mathrm{P}[S(k+1)\geq 0]\geq\alpha\Rightarrow U(k)\geq\varphi_{d,k}^{-1}( \alpha)V_{d}(k)-S(k)+\hat{d}(k)\ \ k=\{0,1,\ldots,H-1\} \tag{9}\]  
\[0\leq U(k)\leq U_{\max}\ \ k=\{0,1,\ldots,H-1\} \tag{10}\]  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]0  
The transformation from stochastic to the deterministic problem is described as follows. We have:  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]1  
and:  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]2  
The storage variable \(S(k)\) is described statistically by the mean \(E[S(k)]=\hat{\hat{S}}(k)\). And the variance \(E[(S(k)-\hat{\hat{S}}(k))^{2}]=Var(S(k))\). Thus, the relation between \(\hat{\hat{S}}(k)\) and \(\hat{\hat{S}}(k+1)\) is defined by the following equation:  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]3  
\[S(k+1)=\ S(k)\ +\ U(k)\ -\ d(k)\qquad k=0,1,\ldots,H-1 \tag{2}\]4  
Indeed, this equation represents the average evolution of storage variable for each period \(k\), \(k\in\{1\ldots H-1\}\).