## 3 System Model
### Order-Up-To Policy  
The OUT policy is one of the classical inventory control strategies considered in logistic applications. It aims at maintaining the stock, which is depleted to satisfy the demand imposed at the controlled node, at a chosen RSL. In each period, any controlled node governed according to the OUT policy checks its inventory position, which comprises the sum of the on-hand stock level and the amount of goods in transit, against its RSL. If the inventory position is below the RSL, the node will issue a replenishment order for its suppliers. The fundamental aspect of the OUT policy operation is thus to determine an appropriate RSL for each controlled node. The choice should enable one to satisfy the external demand, while simultaneously avoiding excessive stock accumulation. The standard OUT policy is assumed to be deployed in a distributed way, i.e., independently at each node, and each node may be subjected to the uncertain external demand.  
The stock replenishment process at a node (and thus the goods redistribution in the network) proceeds as a sequence of activities to be executed in each period (Figure 1):  
1. Register the shipments obtained from the node suppliers (external sources and neighbors in the network).
2. Fulfill the external demand, if possible.
3. Fulfill the orders originating from other nodes in the network, if possible.
4. Based on the discrepancy between the current stock and the RSL, generate a stock replenishment order.  
Let \(l^{r}_{i}\) be the RSL of node \(i\) and \(\Delta_{i}(t)\) denote the amount of goods in transit that are expected at this node. Then, according to the OUT policy, the amount of goods that should be ordered at the end of period \(t\) by node \(i\) from its suppliers is given by:  
\[o_{i}(t)=\ l^{r}_{i}-l_{i}(t)-\Delta_{i}(t). \tag{11}\]  
Introducing \(l^{r}\) as the vector of RSLs of all the controlled nodes, (11) may be written for the entire network as:  
\[\mathbf{o}(t)=\ l^{r}-l(t)-\sum_{\beta=1}^{B}\sum_{k=t-\beta}^{t-1}\mathbf{M}_{ \beta}(k)\mathbf{o}(k). \tag{12}\]  
During the optimization process discussed in the numerical section, RSLs will be sought such that a high service level (high satisfaction of the external demand) is guaranteed, while at the same time, excessive stock accumulation is avoided. As is customary in the subject literature [22], the service level is defined as the percentage of fulfilled demand with respect to the requests imposed externally on the network, i.e., the fill rate. It can be shown (see [55] for derivation details) that a 100% fill rate can be obtained with a sufficiently large, precisely determined RSL, even for the highest demand in the entire planning horizon. For such a \"worst-case\" scenario, the RSL vector is calculated using:  
\[l^{r}=\left(\mathbf{I}_{N}+\sum_{\beta=1}^{B}\beta\mathbf{M}_{\beta}\right) \mathbf{M}^{-1}d_{\text{max}}, \tag{13}\]  
where \(\mathbf{d}_{\text{max}}\) is the vector of the demand upper bounds. However, the value obtained using (13) overestimates the RSLs for time-varying demands. Therefore, in practice, one should adjust \(l^{r}\) for a particular type of demand profile, e.g., as analyzed in this work, through the simulations incorporating a CGA.