# Dwell Time Estimation of Import Containers as an Ordinal Regression Problem
## 5 Conclusions  
In this work, we aimed to show that estimating the dwell time of import containers in a yard is an ordinal regression problem. Thus, we modeled and solved the problem as both an ordinal regression problem and a supervised classification problem; this last is the trending approach in the literature. To corroborate our hypothesis, we compute the \(MAE\mu\), and the reshuffles provoked during the container stacking using the dwell time estimated by each approach. As a result of our research, we noticed that the ordinal regression algorithm achieved the lowest \(MAE\mu\) in 15 (71%) of our 21 experiments. Similarly, the ordinal regression algorithm yielded the lowest reshuffles in 14 (67%) experiments. The statistical test corroborated significant differences between both methods regarding the \(MAE\mu\), while differences were not significant regarding the reshuffles. Since different optimal allocations of the containers can lead to different numbers of reshuffles, we examined the relationship between \(MAE\mu\) and reshuffles. We noticed that a lower \(MAE\mu\) conducts to a lower number of reshuffles in 16 (76%) experiments. Therefore, we can state that by modeling the dwell time estimation  
Figure 7: Bar graphs with rank sums computed with the Wilcoxon signed ranks test comparing the ordinal regression method vs. the classification method regarding \(MAE\mu\) (**a**) and reshuffles (**b**). Positive ranks specify a better performance for the ordinal regression method. as an ordinal regression problem and decreasing the \(MAE\mu\), we can reduce the number of reshuffles generated during the stacking of import containers in a yard. We found a subset of six attributes relevant for estimating the dwell time of import containers in a terminal from a set of 35 evaluated. Such attributes are \"Days_in_country\", \"Arrival_week_day\", \"Arrival_month_day\", \"Destination\", \"Product\", and \"Predicted_dwell_time\". The attribute \"Predicted_dwell_time\" approximates the dwell time using the formula to compute the CPU burst time in groups of similar containers. Attributes \"Days_in_country\", \"Arrival_week_day\", \"Arrival_month_day\" are derived from the arrival date of the containers to the country. \"Destination\" and \"Product\" were obtained from the containers' information. Moreover, we evaluated 25 attributes related to the weather forecast and the distance between the yard and the destination of the container. Such an evaluation showed that they were irrelevant for the dwell time estimation of the containers in our datasets. We observed that all the algorithms produced a higher \(MAE\mu\) and reshuffles for the examples in the dataset that excluded the attribute \"Product\". Hence, we recommend including the attribute \"Product\" for estimating the dwell time of import containers in a yard. Nevertheless, further experiments are needed to support this recommendation. Our results can be applied on dwell time estimation problems where the dwell time is an ordinal variable. Two additional examples of these problems are the dwell time estimation of ships in docks and the dwell time of buses in a bus workshop for repairing. For these examples, an accurate dwell time estimation may conduct to saving resources, as our research work does. We propose evaluating other ordinal regression algorithms, such as deep neural networks for ordinal regression, as future work. Moreover, we shall be working on a new dataset about import containers available for the research community to reduce the lack of available datasets. All the authors have contributed to the results reported in this research as the following distribution. Conceptualization, L.D.A.J., M.A.M.-P., R.M. and C.M.-P.; methodology, L.D.A.J., M.A.M.-P., R.M. and D.V.-R.; software, D.V.-R. and L.D.A.J., validation, L.D.A.J., M.A.M.-P. and D.V.-R.; formal analysis, L.D.A.J., M.A.M.-P., R.M. and D.V.-R.; investigation, L.D.A.J. ; data curation, L.D.A.J. and D.V.-R.; writing--original draft, D.V.-R. and L.D.A.J. ; writing--review and editing, M.A.M.-P., R.M.