# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 4 Results and Analysis
\begin{table}
\begin{tabular}{c c} \hline \hline
**Result** & **Value** \\ \hline TC & 5150 \\ \hline \(C\) & 97.0 \\ \hline \(O\) & 18 \\ \hline \(x_{ij}=1\) & \(x_{35}\), \(x_{68}\), \(x_{911}\), \(x_{1214}\), \(x_{1517}\), \(x_{1820}\), \(x_{2122}\), \(x_{2324}\), \(x_{2526}\), \(x_{2728}\), \(x_{2930}\), \(x_{3132}\), \(x_{3335}\), \\  & \(x_{3638}\), \(x_{3941}\), \(x_{4244}\), \(x_{4547}\), \(x_{4850}\) \\ \hline \(Q_{\bar{i}}\) & \(Q_{35}=720\), \(Q_{68}=893\), \(Q_{911}=975\), \(Q_{1214}=1040\), \(Q_{1517}=1110\), \(Q_{1820}=1113\), \\  & \(Q_{2122}=733\), \(Q_{2324}=798\), \(Q_{2526}=789\), \(Q_{2728}=753\), \(Q_{2930}=784\), \(Q_{3132}=777\), \\  & \(Q_{3335}=1127\), \(Q_{3638}=1088\), \(Q_{3941}=1052\), \(Q_{4244}=1000\), \(Q_{4547}=954\), \\  & \(Q_{4850}=911\) \\ \hline \(SS_{35}=155\), \(SS_{68}=142\), \(SS_{911}=138\), \(SS_{1214}=139\), \(SS_{1517}=160\), \(SS_{1820}=147\), \\  & \(SS_{2122}=114\), \(SS_{2324}=138\), \(SS_{2526}=149\), \(SS_{2728}=123\), \(SS_{2930}=130\), \\  & \(SS_{3132}=137\), \(SS_{3335}=127\), \(SS_{3638}=111\), \(SS_{3941}=103\), \(SS_{4244}=100\), \\  & \(SS_{4547}=118\), \(SS_{4850}=173\) \\ \hline Residual initial & 257 \\ inventory & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial results. Figure 4: Inventory level according to initial plan with a service level of 95%. Figure 6 shows the lot size according to the dynamic plan and assumes that no further information is known for the demand in the next year. The lots assignment is changed with time, with the most apparent changes occurring after week 30 where lot sizes are lower than usual, and in smaller cycle times, to respond to the sudden increase in the demand. In this study, it was assumed that there is no restriction on the lower size of the lot. Further research can investigate this point. The average lot size was found to be 808, and the number of orders is 21 instead of the 18 in the original plan. Therefore, the ordering cost is 2625. Further changes are expected for the rest of the year. The total cost of the dynamic plan is 5394.2, which is more than the original cost of 5150. That is an increase of about 5%. However, there is no shortage in the dynamic plan. Such a shortage should be included in the calculations of the costs of the initial plan. The results of the paper show that the responsive dynamic plan is better than the static plan. The service rate was found to be 100% in the case study instead of 92% for the static plan. DP model is faster than the MIP model. However, MIP is useful to describe and model the system. Then, a comparison is made between the results of the two models on a small case. Results are identical. Repeating the model 35 times was not a problem regarding the CPU time. Results are obtained in few seconds. The fixed calculations are for the next week, where the primary question was focused on whether we need a new delivery or not in the next week, and if so, what size lot we would need. Later decisions are obtained after obtaining the information about the actual demand. However, the calculated lot sizes of the future weeks can be useful, especially when the initial forecast is sent to suppliers about the needed deliveries in the next month for example. Assuming that the demand is constant as in the traditional models does not work at all in the current case study. Large safety stocks are needed in this case, and a lot of shortages are expected. Since the model in this study is the first one that considers the discrete nature of delivery, there are no previous similar studies to compare with. Therefore, only a partial comparison can be made. For example, the effectiveness of DP to solve MIP problems was  
Figure 5: Inventory level according to dynamic plan until week 35. Figure 6: Dynamic replenishment plan until week 35.  
found in some studies such as [PERSON] and [PERSON] [54,56]. Moreover, many studies found that variable quantity size and period between deliveries give the optimal solution [57], which was exactly found in this study.