Logarithmic Entropy Measures for Fuzzy Rough Set and their Application in Decision Making Problem

Introduction

In the past, the concept of uncertainty related to information had not achieved any special kind of attention. But in recent decades, the uncertainty and vagueness concept related to information achieved more focus due to the importance of information or data on research proposals or other problems. The main motive of this type of studies is to remove the uncertainties or ambiguities in the collected information, provided to find any result or take proper decision for related problems. In the conventional prospect of science, uncertainty characterizes an unfavorable situation, which must be prevented at all cost. To evade uncertainty, it is important to quantify the degree of uncertainty. (Shannon, 1948) used “entropy” to quantify the uncertain degree of the randomness in a probability distribution. The information contained in this experiment is given by which is known as Shannon Entropy. Later (Zadeh, 1965), introduced the concept of fuzzy set and defined the entropy of a fuzzy set which was distinct from the classical Shannon’s entropy, as no probabilistic notion was required in order to describe it. Fuzzy entropy measures the fuzziness of a fuzzy set which developed from the inherent vagueness or imprecision conducted by the fuzzy set.

In contrast to fuzzy set theory, to deal with the imprecise and unclear or incomplete information, (Pawlak, 1982), exploited another theory recognized as rough set theory but the beginning and highlighting of this theory are distinct from fuzzy set theory. According to distinct aspect of the prospects, these two theories may complement each other. As fuzzy set deal with vague data, while rough sets handle the incomplete information thus both theories are the complement to each other. Since datasets obtained from real-world applications are inherently prone to contain both vague and incomplete information like crops harmed by different types of insects or pests. This implies that no definite decision can be taken regarding the use of particular type of pesticide and insecticide on the basis of crops knowledge. Thus, to handle these types of problems which involves vagueness or incompleteness, the idea of hybridization of these two models into fuzzy-rough set was given by (Nakamura, 1988) and (Dubois & Prade, 1990) and has been widely used by (Nanda & Majumdar, 1992). A Fuzzy rough set sum up the connected but distinct notions of the vagueness of fuzzy set and indiscernibility of the rough set, both of which are opposite and can be met in real-life problems. A Fuzzy-rough set has a benefit over a rough set, while diminishing the information loss in real-valued datasets produced by the discretization of the rough set.

The entropy of a system introduced by (Shannon, 1948), defines the measure of uncertainty about its actual structure. It has been a helpful procedure for illustrating the information contents in several forms and applications in various separated fields. Various authors like (Gupta & Sheoran, 2014) and (Sharma et al., 2017) have used Shannon’s concept and its variants to calculate the uncertainty in the rough set and fuzzy set theory. Some authors like (Chengyi et al., 2004), (Qi et al., 2008), (Gupta et al., 2016) and (Sharma et al., 2017) used similarity measure to deal with the uncertainty and vagueness. (Sharma et al., 2018) proposed some trigonometric entropy measures for fuzzy rough set and discussed their application in medical area and data reduction. Some researchers like (Ren et al., 2017), (Zhu & Li, 2016), (Li & Ren, 2015), (Wan & Li, 2015) and (Li & Wan, 2017) solve the decision-making problem by using non-probabilistic information measures.

In this paper, some logarithmic non-probabilistic entropy measures have been proposed for fuzzy rough sets. Also, their axiomatic definition is given. The rest of the research paper is organized as: