Abstract: Rough Set based concepts of Span and Spanning Sets were recently proposed to deal with uncertainties in data. Here, this paper, presents novel concepts for generic decision-making process using Rough Set based span for a decision table. Majority of problems in Artificial Intelligence deal with decision making. This paper provides real life applications of proposed Rough Set based span for decision tables. Here, novel concept of span for a decision table is proposed, illustrated with real life example of flood relief and rescue team assignment. Its uses, applications and properties are explored. The key contribution of paper is primarily to study decision making using Rough Set based Span for a decision tables, as against an information system in prior works. Here, the main contribution is that decision classes are automatically learned by the technique of Rough Set based span, for a particular problem, hence automating the decision-making process. These decision-making tools based on span can guide an expert in taking decisions in tough and time-bound situations.
Keywords. Rough Sets, Span, Spanning Sets, Decision Table, Reduct, Decision Making, AI
Uncertainty  in data and decision making have always posed challenges for mankind to take right decisions. There have been various studies in decision making in uncertainty [3, 5, 16], still much more progress and research is as much needed as before. Not only to assist humans but also to help machines take their own decisions, to some extent. Fuzzy Sets  and Rough Sets  are two popular techniques to deal with uncertainty occurring in data and hence in decision making.
The concept of uncertainty handling based on Fuzzy Sets  rely on membership value of each element of a universe to a set X. While the uncertainty handling via Rough Sets deal with subsets of universe and the granulation of universe in equivalence classes. The partial and complete belonging of these information granules in a subset determines the upper and lower approximations of the subset. The boundary region consists of those granules which are not in lower approximation and have some information common with the set under consideration.
Rough Sets have been used since its inception in 1982 by Pawlak [16, 17] in variety of applications. Some of the key areas where Rough Sets have been applied are feature selection [2, 11, 12, 19], classification , text summarization [20, 21], financial data analysis , medicine , data mining [8, 14], clustering , information retrieval  to mention a few.
The motivation of this paper is three-fold. Firstly, it presents real-life problems that require AI based automation using previously defined concept of Rough Set based span. Secondly, to propose the concept of span for a decision table, previous work, deals only with an information system with X, a subset of U, as a variable. While here the full decision class is a variable. The aim is to direct the search of right decision class via the concept of finding best span for the decision system. This can be considered as a stochastic search technique automating the decision category of each object of the universe. And hence assist in automatic unsupervised directed learning.
The paper is organized as follows. Section 2 discusses previous work pertaining to the proposed decision-making system and introduces the problem of flood rescue and relief team dispatch. The proposed novel concepts of span of a decision table are introduced and illustrated for example and applications in Section 3. Section 4 presents the effect of feature selection via reducts on the span of a decision table. Conclusion and future work are given in Section 5.
- Previous Work
Rough Set or any decision-making system depends on an information system, in case of unsupervised learning and on a decision system, for a supervised learning system. The information system consists of measurements or values of objects of universe for various features called attributes. The information system can be represented in a tabular form in which the rows depict the objects and the columns represents the properties, measurements, physical attribute or any other representational data about the objects. All objects are typically represented with the same attributes, otherwise a null symbolic value is inserted in the place, which is dealt in machine learning, AI and data mining in altogether separate way.
Yadav, N. et al. (2019) defined span and spanning set for a universe U and a set of attributes P for an information system. Here, the paper provides a modified version of the same definition explicitly for a complete information system, complete in sense that full attribute set R is considered in the definition. Hence in this definition the only variable is X, since attributes are fixed as the complete attribute set. Span of a set X given an information system is defined as follows.
Definition 1. Given a universe 𝑈 and an Information System (𝑈, 𝑅). The span of a subset 𝑋 of 𝑈 is defined as:
δ𝑋 = (𝑤1 ∗ |𝑅𝑋 | + 𝑤2 ∗ |𝐵𝑁𝑅(𝑋)|)/|U| where 𝑤1, 𝑤2 ε [0,1], 𝑤1+𝑤2=1.
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