# Dwell Time Estimation of Import Containers as an Ordinal Regression Problem
## 2 Related Work  
Some authors have proposed solutions for estimating the dwell time using statistical and supervised learning algorithms [7; 14; 15; 16; 17; 18]. For example, [PERSON] et al. [14], [PERSON] et al. [16], and [PERSON] [17] stated the estimation of the dwell time as supervised classification problems. Later, [PERSON] et al. [7] also modeled this problem as a re gression problem and evaluated the performance of regression algorithms in addition to supervised classification algorithms.  
[PERSON] et al. [14] compared the performance of the algorithms naive Bayes, decision tree, and a hybrid called NB-decision tree to estimate the dwell time. The authors proposed as the class attribute the number of days that a container stays in the yard. They used three metrics to assess the performance of the algorithms, the percentage of instances correctly classified, the Kappa statistic, and the root mean square error.  
[PERSON] et al. [16] evaluated the performance of the algorithms k-nearest neighbor, naive Bayes, One Rule, Repeated Incremental Pruning to Produce Error Reduction, K*, Decision Table, and Zero Rule estimating the dwell time. Their class attribute includes approximately 175 classes obtained from a discretization of the days those containers stay in the yard. To measure the performance of these algorithms, the authors computed the percentage of correctly classified instances, Kappa, and medium square error.  
[PERSON] [9] implemented an artificial neural network to estimate the dwell time for a container terminal. The class attribute proposed was the number of days that a container stays in the yard. The author considered three evaluation measures, correctly classified instances, Kappa statistic, and root mean squared error.  
[PERSON] et al. [7] explored the performance of three algorithms, multiple linear regression, decision trees, and random forest, estimating the dwell time of containers. They compared the mean absolute percentage error for regression algorithms and balanced accuracies for supervised classifiers. The authors assumed the dwell time as a continuous attribute for regression algorithms. However, they discretized the dwell time in three classes (less than a week, between one and two weeks, and more than two weeks) for classification algorithms.  
These authors have mainly focused on the selection of the algorithm and the selection of the performance metrics. Nevertheless, a third concern about the dwell time estimation consist of determining what type of machine-learning problem yields the best estimation. By addressing this issue, we can narrow the algorithms and the performance metrics to consider.  
Often, the operators measure the dwell time in the number of days that the container stays in the yard [7; 9; 20]. The number of days is an attribute with an order for which the difference between each value is relevant. [PERSON] and [PERSON] [25] provided a formal definition of ordinal variables (called attributes in our problem). They illustrated their definition with variables such as school grades, ages, or the number of children. These variables should be considered ordinal realizations of underlying continuous variables [25].  
Other problems similar to the dwell time estimation for import containers have been solved by adopting artificial neural networks as ordinal regressors. Among them are age estimation [26; 27], monocular depth estimation [28], and historical image dating [27]. For these problems, the values of the class attribute describe an order. The authors [26; 27; 28] reported lower errors when stating and solving them as ordinal regression problems. Similarly, the dwell time of import containers in a terminal describes an order. Thus, we can assume that estimating the dwell time is an ordinal regression problem [25]. However, we have not found a proposal modeling the estimation of the dwell time as an ordinal regression problem.