## 5 Computational Experiments
### Application to Chemical Batch Plants  
We apply the proposed methods to a production scheduling problem in a chemical batch plant. In this paper, we use the lubricating oil filling schedule in chemical batch plants. The job contents and constraints will be briefly explained. A total of 34 filling operations are given. The filling amount, the drum to be filled, and the delivery date are shown for each. There are 9 types of drums to fill. The filling speed differs depending on the type. Filling works consist of the following three steps. 1. Cleaning and draining oil from the line, 2. Product edge, and 3. Filling. There are two main time constraints. Filling work start time is 09:05. Break times are 10:30-11:00, 12:00-13:00, and 15:00-15:30. Filling work cannot be done during the above break time. If the type of oil is different from the previous oil during cleaning, cleaning costs (setup costs) are incurred. Two objective functions are used in chemical batch plant scheduling: the sum of delivery delays and the sum of setup costs. The objective function is expressed in Equation (5). The required amount of filling of each oil differs from day to day. An approximate solution method is used for the input/output data. We considered the following features: 5. The values of the sum of delivery delay, 2. The values of the sum of setup costs, and 7. The sum of completion time of each machine. We prepared 270 data sets. Machine learning evaluates the data by performing 5-cross-validation. The case where only the objective function values were used as features was compared with the case where features were selected by SFS from the features extracted in Section 4.2.1. The hyperparameters for machine learning and inverse optimization are the same as in Section 5.  
Figures 6 and 7 show the Gantt chart when the weighting factor is \((w_{1},\ w_{2})=(0.5,\ 0.5)\) and \((w_{1},\ w_{2})=(0.9,\ 0.1)\). The gray bar sections on the Gantt chart represent break time. Table 12 shows the features selected by SFS in the order in which they were selected. Table 13 shows the MSE of the estimation results.  
Since the number of tanks that can be assigned to a job is predetermined, only seven machines were used in this case. Figure 7 shows that many machines have shorter completion times compared to Figure 6. As a result, the value of the sum of delivery delays is smaller. Table 12 shows that all features are selected. The order in which the features are selected is the same as for the large-scale problem. Since the objective function is the same, the order of effective features is also the same. Furthermore, the MSE is smaller than when only the objective function value is used as features. It was confirmed that feature extraction is effective even when the data is close to real data. In inverse optimization, estimation accuracy was better than machine learning. However, both were less accurate than when the scheduling problem was generated randomly. In the randomly generated problems, setup costs varied slowly. The real data, however, do not vary slowly because the setup cost is predetermined to a certain value due to the nature of the problem. This is the reason why the accuracy of the inverse optimization was lower than that of the randomly generated problems.