# Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems
## 6 Discussion and Conclusions  
From the perspective of numerous or few relationships between any pair of patterns, this study used GRA to identify direct influences among patterns. A total influence matrix was generated to verify the direct/indirect influences among them. The total influence formed the foundation of the proposed GTI-TRS associated with the construction of the TRSC. The idea is that the higher the value of \(g_{ij}\), the more similar \(\mathbf{x}_{i}\) is to \(\mathbf{x}_{j}\). We noted that [19] had proposed grey tolerance rough sets (GTRS), which defined an overall similarity measure \(S_{A}^{G}(\mathbf{x}_{i},\mathbf{x}_{j})\), so that \(S_{A}^{G}(\mathbf{x}_{i},\mathbf{x}_{j})=\gamma(\mathbf{x}_{i},\mathbf{x}_{j})\). Therefore, the main difference between the GTRS and the GTI-TRS is that the former considers only direct relationships, but the latter considers direct as well as indirect relationships using the proposed grey total influence matrix. In particular, a GA was implemented to determine the optimal parameter specification for the proposed GTI-TRSC that cannot be easily determined by users.  
Even though parameter specifications for GA are subjective, experimental results showed that the chosen parameters were acceptable. Indeed, the proposed GTI-TRSC is sufficiently simple to implement as a computer program without conforming to any statistical assumptions. Experimental results obtained by the proposed GTI-TRSC are promising. We see that GTI-TRSC-CO and GTI-TRSC-SU produced satisfactory results compared with the other rough-set-based classification methods considered. In particular, these two classifiers were superior in terms of classification performance to the TRSC. However, it should be noted that there is no best classifier [42].  
This study has motivated us to investigate the subject further. As mentioned above, the grey DEMATEL has been an important issue for multiple attribute decision making. It is interesting to examine a distinctive version of the grey DEMATEL built on the proposed grey direct influence matrix. First, a novel DEMATEL-based Analytic Network Process (DANP) proposed in [20] was used to avoid agonizing pairwise comparisons for the ANP by replacing the total influence matrix produced by the DEMATEL by directly using the unweighted supermatrix of the ANP. The DANP has gained considerable research attention in recent years due to its convenience [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52]. It is interesting to explore its applicability to practical problems by incorporating the new version of the grey DEMATEL into DANP.  
Second, considering decision-making in a fuzzy environment, the linguistic interpretation of elements of the grey direct influence matrix is challenging. In practical applications, it may be desirable that a linguistic term be generated from numerical data [41]. Linguistic values such as \"low influence\", \"medium influence\", and \"high influence\" associated with fuzzy sets can be considered. Such a linguistic grey direct influence matrix can then be incorporated into the DEMATEL for further processing.  
Finally, the traditional GRG is implemented using the weighted-average method, where dependency among the attributes is not considered. However, an assumption of additivity may not be realistic in practical applications [53]. Thus, it is more useful to employ a nonadditive grey total influence matrix with a nonadditive GRG [54], and check the resultant impact on the performance of GTI-TRSC. The aforementioned grey DANP, fuzzy-grey-DEMATEL, and nonadditive grey DEMATEL will be investigated in future studies.  
Conceptualization, Supervision, and Methodology: [PERSON]; Validation and Investigation: [PERSON].  
Research was funded by the Ministry of Science and Technology, Taiwan, under grant MOST 106-2410-H-033-006-MY2.  
The authors declare no conflict of interest.