## 3 System Model
### System Actors and Their Relationships
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline \((\ldots)^{+}\) & The saturation function restricting the argument to non-negative values \\ \(t=1,2,\ldots,T\) & Period, where \(T\) is the planning horizon \\ \(\Theta_{N}=\{1,2,\ldots,N\}\) & The set containing the controlled node indices, where \(N\) is their total number \\ \(\Theta_{M}=\{1,2,\ldots,M\}\) & The set containing the external source indices, where \(M\) is their total number \\ \(\alpha_{ij}\) & Nominal supplier fraction between nodes \(i\) and \(j\), i.e., the part of the replenishment signals \\  & generated by node \(j\) that is requested from supplier \(i\) \\ \(\alpha_{ij}(t)\) & Supplier fraction between nodes \(i\) and \(j\) in period \(t\) \\ \(\beta_{ij}\) & Lead-time delay between nodes \(i\) and \(j\), i.e., the time from placing the order to its delivery \\ \(\mathrm{B}\) & The highest lead-time delay between any two nodes in the system \\ \(l_{i}(t)\) & The on-hand stock level at node \(i\) in period \(t\) \\ \(d_{i}(t)\) & The external demand imposed on node \(i\) in period \(t\) \\ \(s_{i}(t)\) & The external demand satisfied by node \(i\) in period \(t\) \\ \(o_{i}(t)\) & The amount of goods to be ordered by node \(i\) at the end of period \(t\) from its suppliers \\ \(\Omega_{i}^{S}(t)\) & The quantity of goods in replenishment orders sent by node \(i\) to its neighbors in period \(t\) \\ \(\Omega_{i}^{S}(t)\) & The quantity of goods in replenishment orders received by node \(i\) from its suppliers in period \(t\) \\ \(l(t)\) & The vector containing on-hand stock levels at the controlled nodes in period \(t\) \\ \(d(t)\) & The vector containing the external demands imposed on the controlled nodes in period \(t\) \\ \(s(t)\) & The vector containing the external demands satisfied by the controlled nodes in period \(t\) \\ \(o(t)\) & The vector containing the replenishment signals generated by the controlled nodes in period \(t\) \\ \hline \end{tabular}
\end{table} 
Table 1: Mathematical preliminaries. Introducing the combined set of indices \(\Theta\ =\ \Theta_{N}\ \cup\ \Theta_{\mathbf{M}}\), the quantity of goods received by node \(i\) in period \(t\) may be written as:  
\[\Omega_{i}^{R}(t)=\sum_{j\in\Theta}\alpha_{\bar{j}}\Big{(}t\ -\ \beta_{\bar{j}} \Big{)}o_{i}\Big{(}t\ -\ \beta_{\bar{j}}\Big{)}. \tag{2}\]  
Consequently, using (1) and (2), the stock level evolution can be described by  
\[l_{i}(t\ +\ 1)=\Big{(}l_{i}(t)+\Omega_{i}^{R}(t)-d_{i}(t)\Big{)}^{+}-\Omega_{i }^{S}(t), \tag{3}\]  
where \(0^{+}\) is the saturation function restricting the argument to non-negative values. This function presents a major non-linearity in the considered class of lost-sales systems. With the demand that is actually satisfied by node \(i\) at time \(t\) denoted by:  
\[s_{i}(t)=\ \min\{l_{i}(t)+\Omega_{i}^{R}(t),\ d_{i}(t)\}, \tag{4}\]  
the stock level evolution can be expressed through the following relationship:  
\[l_{i}(t\ +\ 1)=\ l_{i}(t)+\Omega_{i}^{R}(t)-s_{i}(t)-\Omega_{i}^{S}(t). \tag{5}\]  
In the nominal state, each node is able to fulfill both the external demand and internal orders originating from other controlled nodes. However, when the external demand is satisfied, yet the stock is insufficient to complete the internal orders, their quantity is accordingly reduced, leading to yet another capacity restriction:  
\[\bigvee_{i}0\leq\sum_{j\in\Theta}\alpha_{\bar{j}}(t)\leq\ 1. \tag{6}\]  
The external sources, which are not subject to capacity limitations, are always able to fulfill all the orders placed by the controlled nodes.