# Heuristic Approaches to Attribute Reduction for Generalized Decision Preservation
## 5 Conclusions and Future Researches  
Developing efficient algorithms for attribute reduction for decision systems is an important issue in many research fields, such as knowledge representation, multiple attribute decision making, and artificial intelligence. Discernibility matrix-based attribute reduction for generalized decision preservation has low computational efficiency and is intolerable for processing a large amount of data. To deal with this issue, a monotonic similarity measure for generalized decision preservation is introduced for attribute reduction. By using the proposed similarity measure, two types of heuristic attribute reduction algorithms (FGRAG and BGRAG) have been designed to obtain the reducts for generalized decision preservation in this paper. One is a forward attribute reduction algorithm based on the add-deleting strategy, the other is a backward attribute reduction algorithm based on the deleting strategy. Results of comparative experiments indicate that both FGRAG and BGRAG can significantly reduce the running time of attribute reduction, while retaining the generalized decision of each object in the universe.  
The effectiveness of the proposed heuristic algorithms in machine learning and multi-attribute decision making is presented as follows: (1) In practical applications, the original data sets often contain redundant or iterative attributes. These attributes may deteriorate the efficiency of learning algorithms. The proposed algorithms FGRAG and BGRAG can remove the redundant attributes from the original data sets quickly, and obtain the simplified data sets which only have indispensable attributes. These simplified data sets can reduce the cost of storage and improve the efficiency of learning algorithms, as well as providing an understandable approach for analyses of data set structures. (2) For a decision system, objects with the same condition attribute values may have different decision values. The rules with respect to these objects are referred to as uncertain rules. By the attribute reduction methods proposed in this paper, the possible decision values of each uncertain rule are left unchanged. Thus, the uncertain information in a decision system can be retained. Based on the reducts calculated by the  
Figure 3: Time consumption comparison with the universe increasing.  
proposed algorithms, simplified uncertain rules can be extracted from a decision system. These rules can be useful for multiple criteria decision making and decision conflict analyses.  
Some future research directions are: (1) Attribute reduction for generalized decision preservation based on metaheuristic methods has not been proposed, so far. Based on studies of metaheuristic methods [25; 27; 29], attribute reduction for generalized decision preservation based on metaheuristic methods (i.e., bee colony or ant colony optimization) will be studied in the future. Some comparative experiments will be also conducted to verify that the proposed algorithms are effective. (2) The heuristic attribute reduction algorithms proposed in this paper are based on the equivalence relation and can be only used in classical Pawlak decision systems. However, in practical applications, there are many generalized decision systems, such as set-valued decision systems, interval-valued decision systems, and fuzzy decision systems. Heuristic attribute reduction algorithms for generalized decision preservation in generalized decision systems will be investigated in future studies.  
[PERSON] designed the research, wrote the whole paper and designed the experiments. [PERSON] gave some theoretical suggestions on the similarity degree and experiments. [PERSON] conducted the experiments.  
We gratefully thank the anonymous referees for their comments and suggestions. Our works in this paper are supported by the National Natural Science Foundation of China (Nos. 61403329, 61572418, 61572419, and 11801491), the Natural Science Foundation of Shandong Province (No. ZR2018 BA004).  
The authors declare no conflict of interest.