*4.5. Extension of the Model*

As mentioned above, the minimum support number of the lower approximation set *N* and variable precision *β* are set in advance. The classification quality reflects the degree of dependence of decisive attribute D on the conditional attribute C and the uncertainty of decision system. The classification quality is inversely proportional to the uncertainty. In the model, the classification quality largely depends on the value of *β* [31]. Practically, the user does not always know how to set the value of *β* to obtain the maximum accuracy model. The minimum support number of the lower approximation set *N* also influences the accuracy of the model. Similarly, the user always sets the value of *N* in accordance with their experience. To determine the best value of *β* and *N*, we propose an algorithm. We set up a loop traverses all the values between 0.5–1, adding 0.01 each time. The results of the different *N* values are shown in Figure 2.

**Figure 2.** (**a**) Results when *N* = 2; (**b**) Results when *N* = 1; (**c**) Results when *N* = 1; (**d**) Results when *N* = 1.

As shown in Figure 3, it is obvious that with the increase in *N*, the average of λ decreases and the variance of λ increases.

**Figure 3.** Comparison of the average and variance of λ.

Although the model's accuracy increases when *β* decreases, for this model, the smaller *β* does not make the model better. *β* represents the tolerance degree of the rough set to noisy and incorrect information in the dataset, while a smaller *β* tolerates more noisy data, but this violates our purpose in building MILP-FRST. Therefore, how to balance *β* against λ requires further research. As for the minimum support number *N*, our results indicate that the smaller the *N*, the better the λ. This rule is useful in practical applications, but when using this rule, the characteristics of the object should be considered to determine the appropriate *N*.

#### **5. Conclusions**

The purpose of this paper is to establish an accurate decision-making method between the quality level of the diesel engine and the parameters of assembly clearance. Therefore, a novel mixed-integer linear programming model for the rough set-based classification with flexible attribute selection, called MILP-FRST, is presented. First, the correlation between the conditional attribute set and decisive attribute set according to the inference of the rough set and function dependence is calculated. Second, by integrating the data mining model with the mixed integer linear programming theory, the optimization method is studied with regard to the sensitivity of the rough set to noisy data. Integrating the *MILP* model with the rough set model, the related theories and concepts of the rough set are implemented in the model, and the extension of the rough set research is completed. Finally, a case study on test data from a diesel engine is carried out. Experiments show that the decision-making method proposed in this paper can realize the quantitative discussion of the relationship between assembly clearance and the quality level of the whole machine. Also, the effectiveness of and the advantages of the MILP-FRST are verified. Furthermore, the extension of the MILP-FRST indicates that the usage of the minimum support number *N* and related topics are worthy of more in-depth study.

**Author Contributions:** Conceptualization, X.Y. and W.C.; methodology, Y.X. and X.Y.; software, X.Y. and J.L.; validation, X.Y. and S.Z.; formal analysis, S.Z.; investigation, J.L. and Y.W.; resources, W.C. and Y.W.; data curation, Y.X.; writing—original draft preparation, X.Y.; writing—review and editing, S.Z.; visualization, J.L.; supervision, S.Z.; project administration, W.C.; funding acquisition, W.C.

**Funding:** This work is supported by the National Natural Science Foundation of China (Grant No.71501007 & 71672006 & 71871003). The study is also sponsored by the Aviation Science Foundation of China (2017ZG51081), the Technical Research Foundation (JSZL2016601A004).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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