# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 4 Results and Analysis  
The real situation of daily demand for one of the apple juices is shown in Figure 3, which shows the weekly demand, and the forecasting results are represented by the trend line. Demand in the summer is generally higher than demand in the winter. The demand in the last two weeks was extremely low because of the end of the year holidays. Therefore, these two weeks were omitted from the calculations. The forecasting model can be as follows:  
\[\text{Weekly demand}=230.6+11.7\ \text{Week}-0.22\ \text{Week}^{2}\]  
The forecasting model performance is not very high. However, it is better than assuming the demand is stable. In reality, there were 17 deliveries in the year, where each lot is from 500 to 1500 boxes. Assume the following data \(H=5\$\), \(S=125\$\), \(P_{0}=752\), \(V=1500\), and the detailed demand data is in Table 1.  
DP Model was used to find the results in Table 2. The table shows that there should be 18 lots during the year in weeks 3, 6, 9, 12, 15, 18, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48. The initial inventory is enough to cover the demand for the first two weeks. After satisfying the demand, 257 units are expected to be available at the beginning of the third \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Week** & **Actual Demand** & **Expected Demand** & **Standard Deviation** & **Week** & **Actual Demand** & **Expected Demand** & **Standard Deviation** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: Demand information of the apple juice of the drinks’ company in a year. Figure 3: Weekly demand of apple juice with forecasting. week. This number of units is not the actual one that will be known later after two weeks. During the first two weeks, the inventory holding costs are 97. Table 2 shows that the safety stock values are considerably larger when the variability is larger. For example, \(SS_{35}\) is larger than \(SS_{68}\). By looking at Table 1, the first few weeks have larger standard deviation values than the later ones. The \(Q_{ij}\) values are different from the summation of the demand in the weeks from \(i\) to \(j\). Moreover, the results are just the initial plan. Later on, based on the actually inventory size, the plan is updated. For example, in Table 2, the number of orders is 18. Later on, the results will show that the number of orders will be 21. According to this initial plan, the size of the inventory depending on the actual consumption will be as in Figure 4. Four weeks have negative values indicating shortage. To fix such a problem, two possible ways are available. The first method is that the service level can be increased to 99% or more. However, the second method which is the dynamic replenishment policy is better. It depends on the last information of the inventory size. For example, after the first week, the initial inventory \(P_{0}\) will be 450. Then, after the second week, the \(P_{0}\) value will be 288. Therefore, the plan is evaluated to obtain a new lot size. Instead of 720, the updated plan contains a lot size of 689 units in week 3. Figure 5 shows the inventory size for the dynamic replenishment policy (the next week is week 35), where every week the plan will be updated. The shortage is zero now.