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Soft Computing (2019) 23:6441–6453

https://doi.org/10.1007/s00500-018-3298-3

METHODOLOGIES AND APPLICATION

Markov chains and rough sets

Kavitha Koppula1·Babushri Srinivas Kedukodi1·Syam Prasad Kuncham1

Published online: 11 June 2018

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract

In this paper, we present a link between markov chains and rough sets. A rough approximation framework (RAF) gives a

set of approximations for a subset of universe. Rough approximations using a collection of reference points gives rise to a

RAF. We use the concept of markov chains and introduce the notion of a Markov rough approximation framework (MRAF),

wherein a probability distribution function is obtained corresponding to a set of rough approximations. MRAF supplements

well-known multi-attribute decision-making methods like TOPSIS and VIKOR in choosing initial weights for the decision

criteria. Further, MRAF creates a natural route for deeper analysis of data which is very useful when the values of the ranked

alternatives are close to each other. We give an extension to Pawlak’s decision algorithm and illustrate the idea of MRAF with

explicit example from telecommunication networks.

Keywords Rough set ·Markov chain ·Rough approximation framework ·Ring

1 Introduction

Rough set, as introduced by Pawlak (1982) during early

1980s, is an approximation of a subset of a universe by a

pair of two sets known as the lower approximation and the

upper approximation. The lower approximation of a set X

is a collection of all points in the universe whose equiva-

lence classes are contained in X. The upper approximation

of a set Xis a collection of all the points in the universe

whose equivalence classes have a non-empty intersection

with X.AsetXis said to be rough, if the difference between

the upper approximation and lower approximation is non-

empty. If the universe possesses some algebraic properties,

then these properties get induced into rough approximations.

Hence, several authors replaced the universe with different

algebraic structures such as groups (Kuroki and Mordeson

Communicated by V. Loia.

BBabushri Srinivas Kedukodi

babushrisrinivas.k@manipal.edu

Kavitha Koppula

kavitha.koppula@manipal.edu

Syam Prasad Kuncham

syamprasad.k@manipal.edu

1Department of Mathematics, Manipal Institute of Technology,

Manipal Academy of Higher Education (MAHE), Manipal,

Karnataka 576104, India

1997), rings (Davvaz 2004; Kedukodi et al. 2010) and mod-

ules (Davvaz and Mahdavipour 2006), etc.

Ciucci (2008) deﬁned rough approximation framework as

a construct wherein multiple approximations are possible on

the same set. Later, Kedukodi et al. (2010) have shown that

a rough approximation framework can be formed using the

concept of reference points.

In the year 1907, Andrei Andreevich Markov proposed

a stochastic mathematical process, widely known as the

markov chain. A markov chain consists of a set of states

together with a set of transition probabilities which describe

the probability of movement from one state to another in a

dynamic system. In a markov chain, the serial dependence is

only between adjacent periods, very much like in a typical

chain. Markov chains are used to develop various models for

decision making in applications involving uncertainty such as

queuing systems (Sharma 1995), inventory systems (Ching

et al. 2003), data mining (Ching and Ng 2003). It is interest-

ing to note that the PageRank algorithm (Page et al. 1998)

is based on markov chains. In this paper, we address the

problem of assigning probability (weights) to the choice of

approximation in a rough approximation framework. Such an

assignment of probability is needed even in case of a regular

rough approximation framework, wherein all the approxi-

mations of a set can be inscribed in one another. To realize

this, we introduce the concept of Markov rough approxima-

tion framework (MRAF), in which a markov chain model

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