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Markov chains and rough sets

DOI:10.1007/s00500-018-3298-3
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Kavitha Koppula at Manipal Academy of Higher Education
Kavitha Koppula
Babushri Srinivas Kedukodi
Babushri Srinivas Kedukodi
Babushri Srinivas Kedukodi
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Kuncham Syam Prasad at Manipal Academy of Higher Education
Kuncham Syam Prasad
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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.
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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
123
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... Recently, Koppula et al. (2018) introduced the concept of Markov rough approximation framework (MRAF) by using Markov chains and rough sets. MRAF helps to assign the probabilities for various reference points in the rough approximation framework. ...
... The results on fuzzy ideals of semirings, near-rings and the ideals of seminear-rings can be found in Bhavanari et al. (2010), Jagadeesha et al. (2016a, b), Kedukodi et al. (2007Kedukodi et al. ( , 2009, Kedukodi et al. (2017), Koppula et al. (2019), Kuncham et al. (2016), Nayak et al. (2018 and Akram and Dudek (2008). Definition 2.3 (Koppula et al. 2018) Let S = {B 1 , B 2 , . . . , B m } be a collection of reference points in the universe U . ...
... , C 15 ] from four different rating agencies (R 1 , R 2 , R 3 and R 4 ) with In Fig. 5, as the target price projections by various rating agencies are different, an expected target price is computed using MRAF. For the sake of simplicity, we consider MRAF given in the example of Koppula et al. (2018) and the corresponding stationary probability distribution [1/6, 2/6, 1/6, 2/6] for the rating agencies R 1 , R 2 , R 3 and R 4 , respectively. Both current market price and the calculated expected target price are compared as per conditions in Fig. 6. ...
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