## 5 Case Study from Practical Business  
In this section, we describe the results of validity and efficiency tests of the proposed VRPSDP-TW-LO model based on the practical distribution business of Company A. Company A is currently supporting the convenience stores in a region of Pusan, Korea. A vehicle visits an average of 30 stores a day. This company has a similar problem due to the left-over trays, which could not picked up by the vehicle. They get a lot of complaints from the store, costs from the missed or damaged trays, and they usually dispatch additional vehicles to pick up the left-over trays. To make the experiments simple without loss of generality, it was assumed that the capacity of the vehicles is 30 trays, the total number of visiting nodes is 12, and the maximum operating time is 240 min (6 h). The daily demand of delivery to each node was randomly generated from the uniform distribution, from 1 to 8 trays. From the actual location of convenience stores, the distances between visiting nodes were calculated considering the actual travelling route using the software, ArcGIS. The demand for pick-up is determined as the number of trays which are previously delivered and left over, not having been picked up yet. To analyze the cumulative effect of daily operations, the performance over a consecutive 30 day period was analyzed. For the first day, it was set that there were no left-over trays. Figure 4 shows the daily total operating cost including travelling, penalties for violating time windows, and penalties for left-over trays for 30 days. Figure 3 shows one example of 30 daily operations. For both VRPSDP-TW and VRPSDP-TW-LO, all 12 stores in a same region are supported by three vehicles. The routes for each vehicle are different. The results of daily operations for 30 days were acquired based on these different vehicle routes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Mean Difference** & \(p\)**-Value** \\ \hline VRPSDP-TW-LO-VRPSDP-TW & \(-\)2445.7 & 0.014 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Result of paired \(t\)-test. Figure 2: Difference of performance between vehicle routing problem with simultaneous delivery and pick-up under a time window, VRPSDP-TW, and considering left over cost, VRPSDP-TW-LO. The difference between the two models in terms of total cost on each day were not significant during the initial 10 days; no greater than 14.9%. However, it gradually increased after 10 days. Figure 4: Daily total cost of VRPSDP-TW-LO and VRPSDP-TW. Figure 3: Example of different routes for VRPSDP-TW and VRPSDP-TW-LO. Eventually, the total cost became 61.8% on the 30 th and final day. This means that VRPSDP-TW-LO can save more than 60% of the cost of VRPSDP-TW. It can be inferred that the cost difference arises from the number of left-over trays. Figure 5 shows the number of time window violations (a), and number of left-over trays (b). Figure 6 shows the result of the integrated comparison of the differences in travelling distance, number of time window violations, and number of left-over trays. Where the number of time window violations was not significantly different, the difference of the number of left-over trays gets bigger as time goes on. In addition, the total travelling distance did not show significant differences between the two models; it was less than 10 km. As previously described, VRPSDP-TW does not consider the number of left-over trays for the pick-up service and load as many as the remained capacity of the vehicle after delivery service. Therefore, the number of left-over trays gets bigger as time goes on. In addition, there is a limitation to the trays which each distribution center can keep and manage; the additional vehicles should be arranged to only pick up the left-over trays. This causes additional costs and several issues for keeping and managing trays.