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# Markov frameworks and stock market decision making

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In this paper, we present applications of Markov rough approximation framework (MRAF). The concept of MRAF is defined based on rough sets and Markov chains. MRAF is used to obtain the probability distribution function of various reference points in a rough approximation framework. We consider a set to be approximated together with its dynamacity and the effect of dynamacity on rough approximations is stated with the help of Markov chains. An extension to Pawlak’s decision algorithm is presented, and it is used for predictions in a stock market environment. In addition, suitability of the algorithm is illustrated in a multi-criteria medical diagnosis problem. Finally, the definition of fuzzy tolerance relation is extended to higher dimensions using reference points and basic results are established.
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Soft Computing (2020) 24:16413–16424
https://doi.org/10.1007/s00500-020-04950-4
METHODOLOGIES AND APPLICATION
Markov frameworks and stock market decision making
Kavitha Koppula1·Babushri Srinivas Kedukodi1·Syam Prasad Kuncham1
Published online: 18 May 2020
Abstract
In this paper, we present applications of Markov rough approximation framework (MRAF). The concept of MRAF is deﬁned
based on rough sets and Markov chains. MRAF is used to obtain the probability distribution function of various reference
points in a rough approximation framework. We consider a set to be approximated together with its dynamacity and the
effect of dynamacity on rough approximations is stated with the help of Markov chains. An extension to Pawlak’s decision
algorithm is presented, and it is used for predictions in a stock market environment. In addition, suitability of the algorithm
is illustrated in a multi-criteria medical diagnosis problem. Finally, the deﬁnition of fuzzy tolerance relation is extended to
higher dimensions using reference points and basic results are established.
Keywords Rough set ·Markov chain ·Rough approximation framework
1 Introduction
Pawlak (1982) introduced the notion of rough sets by deﬁn-
ing the lower approximation of a set Xas the collection of all
the elements of the universe whose equivalence classes are
contained in X,and the upper approximation of Xas the set
of all the elements of the universe whose equivalence classes
have a non-empty intersection with X. We often consider
the universe to be an algebraic structure and study the corre-
sponding algebraic properties of rough approximations. The
concept of rough approximation framework was deﬁned by
Ciucci (2008), basically as a collection of rough approxima-
tions of the set. A rough approximation framework is said to
be regular if all the approximations of the set are inscribed in
one another. An illustration of rough approximation frame-
work was given by Kedukodi et al. (2010) using reference
points.
Communicated by V. Loia.
BBabushri Srinivas Kedukodi
babushrisrinivas.k@manipal.edu
Kavitha Koppula
kavitha.koppula@manipal.edu
1Department of Mathematics, Manipal Institute of Technology,
Manipal Academy of Higher Education (MAHE), Manipal,
Karnataka 576104, India
In Markov chains, the probabilities which ﬁnally decide
the stability of a system are represented in terms of a matrix
known as transition probability matrix. Markov chains have
been used in several applications to predict the future pos-
sibilities in dynamic and uncertain systems. One such area
that has been the focus of intense research is the prediction
of the performance of stock markets. In a typical stock mar-
ket environment, customer’s either SELL, BUY or HOLD
a particular stock by assessing and predicting its perfor-
mance utilizing previous and current performance of the
stock, inputs from rating agencies, etc (Tavana et al. 2017).
Such an assessment of past performance of the stock with the
available empirical data of a stock and predicting the future
performance or value of the stock is a challenging task due
to its very dynamic nature and multiple variables that affect
its performance.
Despite this, a variety of mathematical models have been
proposed in the literature (Aleksandar et al. 2018; Xiongwen
et al. 2020; Emrah and Taylan 2017; Sudan et al. 2019; Gour
et al. 2018; Prasenjit and Ranadive 2019, etc.) Such methods
often apply a number of ideas from Markov chains, fuzzy sets
(Chan 2015;Chojietal.2013; Rezaie et al. 2013), rough sets,
artiﬁcial neural network (Suk et al. 2010) or other interesting
methods (Chen et al. 2007; Gong et al. 2019; Markovic et al.
2017).
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
123
16414 K. Koppula et al.
probabilities for various reference points in the rough approx-
imation framework. In the present work, we present explicit
examples of MRAF. The ﬁrst example focuses on explaining
the mathematical ideas involved, and the second example
demonstrates how to apply the concept in a practical sit-
uation. In the second example, we use MRAF along with
Pawlak decision algorithm (Pawlak 2002) to analyze data
from the different rating agencies (reference points) in the
stock market environment. We arrive at a recommendation
on the future prospects of a set of stocks along with the prob-
abilities for the suggested recommendation to be correct.
Usually, the rating agencies (experts) do not get evaluated
for the quality of their recommendations. However, the algo-
rithm proposed in this paper takes care of this aspect through
the idea of Markov chain transition probability matrix which
yields a telescopic way of assignment of weights. This
method naturally evaluates the quality of recommendations
made by the rating agencies (experts). Further, the suitabil-
ity of the proposed algorithm is also validated by evaluating
its efﬁcacy in arriving at a decision in a multi-criteria medi-
cal problem. We extend the fuzzy similarity-based rough set
approach used in multi-criteria decision problems to identify
key or crucial attribute among multiple attributes.
2 Basic deﬁnitions and preliminaries
In this section, we provide basic deﬁnitions and results. We
refer Pawlak (1982), Ciucci (2008), Kedukodi et al. (2010),
Davaaz (2004,2006) for basic deﬁnitions of rough set and
rough approximation framework.
Deﬁnition 2.1 (Medhi 2009) The stochastic process
{Xn,n=0,1,2,...}is called a Markov chain; if, for
j,k,j1,... jn1N,Pr{Xn=k|Xn1=j,Xn2=
j1,...,X0=jn1}=Pr{Xn=k|Xn1=j}=Pjk
(say) whenever the ﬁrst member is deﬁned. The transition
probabilities Pjk satisfy Pjk 0,
kPjk =1 for all j.
Deﬁnition 2.2 (Ching and Ng 2006) A vector π=0
1,
...,π
k1)tis said to be a stationary distribution of a ﬁnite
Markov chain if it satisﬁes:
(i) πi0 and k1
i=0πi=1.
(ii) k1
j=0Pijπj=πi.
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. (2007,
2009), Kedukodi et al. (2017), Koppula et al. (2019), Kun-
cham et al. (2016,2017), Nayak et al. (2018) and Akram and
Dudek (2008).
Deﬁnition 2.3 (Koppula et al. 2018)LetS={B1,B2,...,
Bm}be a collection of reference points in the universe U.
Consider a sequence of trails Yn,n1 of selection of ref-
erence points from the set S. Let the trails dependence be
connected by a Markov chain with a transition probability
matrix
C=Yn1
Yn
B1B2... Bm
B1q11 q12 ... q1m
B2q21 q22 ... q2m
... ... ... ... ...
... ... ... ... ...
Bmqm1qm2... qmm
Let Ebe the rough approximation framework formed by S.
Then (E,C)is called a Markov rough approximation frame-
work (MRAF).
The size of MRAF is the size of the rough approximation
framework E.
Deﬁnition 2.4 (Huang 1992) quadruple IS =(U,AT,
V,h)is called an information system, where
U={u1,u2,...,un}is a non-empty ﬁnite set of objects,
called the universe of discourse, AT ={a1,,a2,...,an}is
a non-empty ﬁnite set of attributes. V=∪
aAT Vawhere Va
is the set of attribute values associated with each attribute
aAT and h:U×AT Vis an information function
that assigns particular values to the objects against attribute
set such that aAT ,uU,h(u,a)Va.
Deﬁnition 2.5 (Huang 1992) In an information system, if
each attribute has a single entity as attribute value, then
it is called single-valued information system, otherwise
it is known as set-valued information system. Set-valued
information system is a generalization of the single-valued
information system, in which an object can have more than
one attribute values.
Deﬁnition 2.6 (Orlowska 1985) For a set-valued information
system IS =(U,AT ,V,h), bAT and ui,ujU,
tolerance relation is deﬁned as
Tb={(ui,uj)|b(ui)b(uj)= φ}
For BAT ,a tolerance relation is deﬁned as
TB={(ui,uj)|b(ui)b(uj)= φ,bB}=
bB
Tb,
where (ui,uj)TBimplies that uiand ujare indis-
cernible(tolerant) with respect to a set of attributes B.
Deﬁnition 2.7 (Shivani et al. 2020)LetSV I S =(U,AT ,V,
h), bAT be a set-valued information system, then a fuzzy
123
Markov frameworks and stock market decision making 16415
relation Rbis deﬁned as
μRb(ui,uj)=2|b(ui)b(uj)|
|b(ui)|+|b(uj)|.
For a set of attributes BA,a fuzzy relation RBcan be
deﬁned as μRB(ui,uj)=inf
bBμRb(ui,uj)
Deﬁnition 2.8 (Shivani et al. 2020) Binary relation using a
threshold value αis deﬁned as T
b={(ui,uj)|μRb(ui,uj)
α},where α(0,1)is a similarity threshold, which gives
a level of similarity for insertion of objects within tolerance
classes.
For a set of attributes BAT ,the binary relation is
deﬁned as T
B={(ui,uj)|μRB(ui,uj)α}.
3 Examples of Markov frameworks
We begin with a mathematical example.
Proposition 3.1 Let U =R×Rand a,bU.Deﬁne a
relation R on U by aRb if there exists a square S centered at
to be the distance from the center (0,0) of the square to one
of the vertices) such that a,b lie on the square. Then R is an
equivalence relation on U .
Proof (i) Let a=(x,y)Uand δx
yare perpendicular
distances from ato Xaxis and Yaxis, respectively. Take
δa=max{δx
y}.Form a square Swith centre (0,0)and
radius 2δa.We claim that alies on S.
Let δyδx.Then δa=δy. This implies |y|=δa,δa
xδa.Hence alies on one of its sides (x
y)or (x,δy).
Similarly, when δxδywe can prove that alies on one of
its sides x,y)or (δx,y)of the square S.Therefore aRa.
Hence Ris reﬂexive.
(ii) Let aRb. Then there exists a square centered at the origin
with radius rsuch that a,blie on the square. This implies
bRa.Hence Ris symmetric.
(iii) Let aRb,bRc.Then there exist square S1centered at the
origin with radius k1such that a,blie on S1and square S2
centered at the origin with radius k2such that b,clie on S2.
This implies k1=k2(because blies on S1and S2). Hence
S1=S2.Therefore aRc.Hence Ris transitive. Therefore R
is an equivalence relation on U.
Notation 3.2 Let aU.Denote the rotations (in anticlock-
wise sense) of a square centered at the origin with radius
2δaby an angle θ(reference points) as S(a)
,0θ2π.
In particular, the square deﬁned by Proposition 3.1 is
denoted by S(a,0)(as shown in Fig. 1).
Proposition 3.3 Let U =R×Rand x,yU.Deﬁne a
relation Rθon U by xR
θy if there exists θsuch that S(x) =
Fig. 1 Square with rotation zero degrees
S(y)
.Then Rθdeﬁnes a MRAF of size greater than or equal
to 1on U corresponding to the set X k={(x,y)|x2+y2
k}.
Proof First we prove that Rθis an equivalence relation on
U.
(i) Let xU.Take θ=0.By Proposition 3.1,xlies on the
square S(x,0).This implies xR
θx.Hence Rθis reﬂexive.
(ii) Let xR
θy.Then there exists θsuch that S(x) =S(y)
.
This implies yR
θx.Hence Rθis symmetric.
(iii) Let xR
θyand yR
θz.Then there exists θsuch that
S(x) =S(y)and S(y) =S(z)
.This implies S(x) =
S(z)
.Hence Rθis transitive. Therefore Rθis an equivalence
relation on U.
The equivalence classes corresponding to the above equiv-
alence relation are given by [x]θ={yU|xR
θy}=
{S(x)|0θ2π}.
The sets L(Xk)=∪{xU|[x]θXk}=∪{xU|
S(x,θ) Xk}={(x,y)||x|=k
2,|y|≤k
2or |y|=
k
2,|x|≤k
2}and U(Xk)=∪{xU|[x]θXk= φ}=
∪{xU|S(x ) Xk= φ}={(x,y)||x|=k,|y|≤kor
|y|=k,|x|≤k}are, respectively, the lower approximation
and upper approximations of the set Xk.Note that whenever
the number of rotations of a square is zero, we get MRAF
of size 1. If the number of rotations of a square is more than
one, then we get a MRAF of size greater than one.
Suppose we take four rotations of the square given by
θ={0,22.5,45,67.5}in Proposition 3.3 with the transition
probability matrix given by
T=
022.54567.5
0a1b1c1d1
22.5a2b2c2d2
45 a3b3c3d3
67.5a4b4c4d4
123
16416 K. Koppula et al.
Fig. 2 MRAf of size 4
with ai+bi+ci+di=1,i=1,2,3,4.Then (Sθ,T)is a
MRAF of size 4. A visual representation is shown in Fig. 2,
wherein a circle approximated by MRAF of size 4.
Proposition 3.4 Let U =R×Rand a,bU.Deﬁne a
relation R0on U by aR
0b if there exists a circle centered at
the origin with radius r(> 0)such that a,b lie on the circle.
Then R0deﬁnes MRAF of size 1 on U corresponding to the
set Xk={(x,y)||x|=k,|y|≤kor|y|=k,|x|≤k}.
Proof We ﬁrst prove that R0is an equivalence relation on U.
(i) Let a=(x,y)U.Then alies on the circle centered
at the origin with radius (x0)2+(y0)2. Therefore
aR
0a.Hence R0is reﬂexive.
(ii) Let aR
0b.Then there exists a circle centered at the
origin with radius rsuch that a,blie on the circle. This
implies bR
0a.Hence R0is symmetric.
(iii) Let aR
0b,bR
0c.Then there exist circle C1centered at
the origin with radius k1such that a,blie on C1and circle C2
centered at the origin with radius k2such that b,clie on C2.
This implies k1=k2(because blies on C1and C2). Hence
C1=C2.Therefore aR
0c.Hence R0is transitive. Therefore
R0is an equivalence relation on U.
Let aU.Denote Caas the circle centered at the origin
and radius is the distance between (0,0)and a.Then the
equivalence classes corresponding to the above equivalence
relation are given by
[a]={bU|aR
0b}=Ca.
The sets
L(Xk)=∪{aU|[a]⊆Xk}=∪{aU|CaXk}
={(x,y)|x2+y2k}
and
U(Xk)=∪{aU|[a]∩Xk= φ}=∪{aU|CaXk= φ}
={(x,y)|x2+y22k}
Fig. 3 MRAf of size 1
Fig. 4 Telescoping MRAF (Recursive implementation of Figs. 2band
3b)
are, respectively, called lower approximation and upper
approximation of the set Xk.We get MRAF of size 1 because
all the rotations of circle yield the same circle. Thus we get
a trivial transition probability matrix having just an entry of
1init.
A visual representation of MRAF of size 1 is shown in Fig. 3,
wherein a collection of squares are approximated by MRAF
of size 1.
Figure 4represents telescoping MRAF, wherein circles
and squares are alternatively rough approximations of each
other.
Example 3.5 Now we illustrate an application of MRAF in
the stock market environment. In the process, we give an
extension to Pawlak’s decision algorithm and address the
problem of decision making in a situation involving data
from several sources (rating agencies/experts) with varying
123
Markov frameworks and stock market decision making 16417
criteria (attributes) for a particular stock (company/service
provider). Usually, listed companies in a share market release
their quarterly performance reports in terms of EBITDA
(Earnings Before Interest, Tax, Depreciation and Amortiza-
tion), EBITDA Margin, EBIT (Earnings Before Interest and
Tax), EBIT Margin, PAT (Proﬁt After Tax), Adjusted PAT,
Revenue, Sales, etc (attributes). Among these, various rat-
ing agencies/experts select few parameters as performance
indicators of a given stock, predict its future performance
for the next quarter and give suggestions to the existing or
prospective customers either to BUY, SELL or HOLD the
shares of a given company. From the customer point of view,
making a decision either to SELL, HOLD or BUY the shares
of a particular company is a challenging task due to the dis-
parity in performance indicators selected by various rating
agencies/experts (reference points) and also because of their
contrary views sometimes on the same stock. MRAF together
with Pawlak’s decision algorithm is very useful to tackle such
problems.
Algorithm 3.6 Extension of Pawlak’s Decision Algorithm:
Step 1: Gather information from multiple sources (refer-
ence points) R1,R2,..., Rpfor U={u1,u2,...,um}with
respect to the attributes A={ak
1,ak
2,...,ak
n}and tabulate
them as a decision table for the reference point Rk.
Step 2: Form decision rules associated with the decision table
for each reference point Rksatisfying the conditions given
by Pawlak (2002).
Step 3: Supports(, ) =Supps(, ) ×P(Rk)=
0,Supps(, ) is given by Pawlak (2002), P(Rk)is the
probability of the reference point Rkobtained from MRAF.
Step 4: Combine the decision rules from the various decision
algorithms with respect to various reference points and form
decision table.
Step 5: Verify whether the decision rules satisfy the Pawlak’s
(2002) decision algorithm.
Step 6: Calculate strength, certainty and coverage factors for
the decision algorithm given in Step 5 by using the following
formulae.
Cers(, ) =|(S)|
|(S)|=supports(,)
|(S)|
Covs(, ) =|(S)|
|(S)|=supports(,)
|(S)|,
where s= φ, s= φ.
Step 7: Arrive at decision using the values of certainty
and coverage factors.
Now to demonstrate the utility of the above algorithm with a
stock market case scenario as described above, we have col-
lected relevant data of 15 companies [C1,C2,C3,...,C15]
from four different rating agencies (R1,R2,R3and R4)with
Fig. 5 Quarterly performance of a company according to rating agen-
cies (CMP current market price, TP target price, Rec. recommendation)
Fig. 6 Criteria for company performance based on E(TP) and CMP
respect to various attributes. Representative data obtained for
a single company (C4)are presented in Fig. 5.
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 con-
sider 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 R1,R2,R3and
R4, respectively. Both current market price and the calculated
expected target price are compared as per conditions in Fig.
6.
Subsequently, all the companies are grouped based on the
similarity among the various attributes along with the likely
scenario (Fig. 6) of the stock and is presented in Fig. 7.
From Fig. 7, a decision algorithm is formed as shown in
Fig. 8.
The certainty and coverage factors for the above decision
algorithm are calculated using Algorithm 3.6 and are shown
in Fig. 9.
Information in Fig. 9can be used to arrive at best possible
decision as per the decision algorithm presented in Fig. 8.
For example, if (EBITDA, inline), (PAT, beat) and (CMP-
TP, closer to E(TP)) then the Recommendation is BUY with
the probability 0.909.
If (EBITDA, miss), (PAT, inline) and (CMP-TP, lower to
E(TP)) then the Recommendation is HOLD with probability
0.667.
If Recommendation is SELL, then (EBITDA margin,
miss), (EBIT margin, miss) and (CMP-TP, higher to E(TP))
with probability 0.5.
123
16418 K. Koppula et al.
Fig. 7 Grouping of companies based on similarity criteria (numerical in
the bracket indicates number of companies satisfying the corresponding
decision criteria)
Fig. 8 Decision algorithm associated with decision Fig. 7
To validate the results of Example 3.5, we have consid-
ered three companies (viz. ACC Limited, Wipro Limited
and Maruti Suzuki India Limited designated as C4,C9,C15,
respectively) and the data from their quarterly reports. Also,
recommendation reports from four different rating agencies
(reference points) are noted. From these data, expected values
for various parameters are computed and a recommendation
is suggested based on MRAF. Subsequently, actual share
prices of these companies are plotted and compared with
the suggestion arrived from MRAF. The validation is shown
in Fig. 10.
The graph indicates that the recommendation derived from
MRAF and extended Pawlak’s decision algorithm seems
Fig. 9 Certainty and coverage factors for the decision algorithm (Fig.
8). Support is obtained by multiplying the number (Fig. 7)with
P(Ri), i=1,2,3,4 as deﬁned in Algorithm 3.6
Fig. 10 Graph depicting the actual share prices of selected companies
from July to November of 2016 along with benchmark index Nifty
(Source: www.nseindia.com)
to be appropriate for the selected companies as indicated
by share price variations of these companies. (An increase
in share price of Maruti Suzuki Ltd was observed and an
appropriate recommendation ‘BUY’ is obtained from the
algorithm. Similarly, changes in share price of ACC cements
are not signiﬁcant and the recommendation obtained is
‘HOLD’.) Example 3.5 can be extended to a telescoping
MRAF over a period of 4 quarters in a typical ﬁnancial year.
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Markov frameworks and stock market decision making 16419
Fig. 11 The expected values of criteria
Fig. 12 The linguistic terms to the expected values
It is to be noted that the rating agencies (experts) do not
get evaluated for their recommendations. However, in our
method Markov chain transition probability matrix yields
a telescopic way of assignment of weights which naturally
evaluates the quality of recommendations made by the rating
agencies (experts). The above example indicates the potential
applications of MRAF in real world situations.
Example 3.7 In order to demonstrate that the proposed
method can be useful in other similar uncertain and dynamic
situations, we consider Example 4.3 from Bingzhen et al.
(2019) that describes a multi-criteria decision-making prob-
lem and utilizes multi-granulation vague rough set method
to arrive at a decision in medical diagnosis. With the help
of MRAF, probability distribution function was assigned to
experts R1,R2,R3as [1/3 1/6 1/2]. Later, the expected values
of criteria are calculated using the probabilities of R1,R2,R3
and are shown in Fig. 11.
The expected values of various criteria are denoted as Very
Low, Low, Average, High and Very High if 0.1 expected
value <0.25, 0.25 expected value <0.34, 0.34 expected
value <0.49, 0.49 expected value <0.64 and expected
value 0.64, respectively. The values of Recommendation
are taken as the sum of all the expected values corresponding
to each alternative and denote them as ‘Yes’ if Recommen-
dation value 2 and ‘No’ if Recommendation value <2as
shown in Fig. 12.
The decision rules are formed for the criteria given in
decision Fig. 12 and are presented in Fig. 13.
Coverage and certainty factors for the above decision algo-
rithm are calculated using the proposed algorithm and are
shown in Fig. 14.
From the above values, we can observe that if (E(y1),
Average), (E(y2),Very High) and (E(y4), Very Low), then
the Recommendation is Yes to say that the patient has gas-
Fig. 13 The decision algorithm associated with Fig. 12
Fig. 14 Certainty and coverage factors for the decision algorithm in
Fig. 13
troenteritis with probability 1. This is in agreement with
the decision obtained using multi-granulation vague rough
set theory, thus conﬁrming the usefulness of MRAF and
Pawlak’s decision algorithm.
Fuzzy tolerance relation Shivani et al. (2020) introduced a
novel approach for attribute selection in set-valued informa-
tion system based on tolerance rough set theory and deﬁned
fuzzy relation between two objects using a similarity thresh-
old. In this paper, we generalized this relation for alternative
selection in set-valued information system by using the con-
cepts Markov rough approximation frame work (MRAF) and
reference points.
Deﬁnition 3.8 Let (U,AL,V,h), aAL be a set-valued
information system, where U={u1,u2,...,un}is a non-
empty ﬁnite set of objects (Criteria).
AL ={a1,a2,...,an}is a non-empty set of alternatives,
V=aAL Vawhere Vais the set of criteria values asso-
ciated with each alternative aAL,h:U×AL V
such that aAL,uU,h(u,a)=Vaand X=
{R1,R2,...Rn}is the collection of reference points from
the universe U,then the fuzzy relation is deﬁned as
n
k=1
μa(ui,uj)RkP(Rk)=
n
k=1
2|a(ui)Rka(uj)Rk|
|a(ui)Rk|+|a(uj)Rk|P(Rk),
where P(Rk), k=1,2,...,nis the probability distribution
for the reference points Rk,k=1,2,...nobtained from
MRAF.
The binary relation using a threshold value α(0,1)is
deﬁned as
T(a,α) =(ui,uj)|
n
k=1
μa(ui,uj)RkP(Rk)α.
123
16420 K. Koppula et al.
Lemma 3.9 T(a,α) is a tolerance relation.
Proof (1) Reﬂexive: Consider
n
k=1
μa(ui,ui)RkP(Rk)
=
n
k=1
2|a(ui)Rka(ui)Rk|
|a(ui)Rk|+|a(ui)Rk|P(Rk)
=2|a(ui)R1a(ui)R1|
|a(ui)R1|+|a(ui)R1|P(R1)
+2|a(ui)R2a(ui)R2|
|a(ui)R2|+|a(ui)R2|P(R2)+···
+2|a(ui)Rna(ui)Rn|
|a(ui)Rn|+|a(ui)Rn|P(Rn).
=2|a(ui)R1|
2|a(ui)R1|P(R1)+2|a(ui)R2|
2|a(ui)R2|P(R2)+···
+2|a(ui)Rn|
2|a(ui)Rn|P(Rn)
=P(R1)+P(R2)+···+P(Rn)
=1α.
This implies (ui,ui)T(a,α). Hence T(a,α) is reﬂexive.
(2) Symmetry: Let (ui,uj)T(a,α).
Then
n
k=1
μa(ui,uj)RkP(Rk)α
⇒[2|a(ui)R1a(uj)R1|
|a(ui)R1|+|a(uj)R1|P(R1)
+2|a(ui)R2a(uj)R2|
|a(ui)R2|+|a(uj)R2|P(R2)+···
+2|a(ui)Rna(uj)Rn|
|a(ui)Rn|+|a(uj)Rn|P(Rn)]≥α
⇒[2|a(uj)R1a(ui)R1|
|a(uj)R1|+|a(ui)R1|P(R1)
+2|a(uj)R2a(ui)R2|
|a(uj)R2|+|a(ui)R2|P(R2)+···
+2|a(uj)Rna(ui)Rn|
|a(uj)Rn|+|a(ui)Rn|P(Rn)]≥α
n
k=1
μa(uj,ui)RkP(Rk)α.
This implies (uj,ui)T(a,α).Hence T(a,α) is symmetric.
Thus T(a,α) is a tolerance relation.
The tolerance class for an attribute uiwith respect to T(a,α)
is [T(a,α)](ui)={ujU|uiT(a,α)uj}.
The lower and upper approximations of a non-empty sub-
set BUas:
T(a,α)(B)={uiU|[T(a,α)](ui)B}.
T(a,α)(B)={uiU|[T(a,α)](ui)B= φ}.
Properties of lower and upper approximations with respect
to fuzzy tolerance relation In the following, we assume that
(U,AL,V,h)be a set-valued information system, aAL
and BU(0,1).
Theorem 3.10 T(a,α) (B)BT(a,α) (B).
Notation 3.11 Let {R1,R2,...,Rn}be the reference points
in the universe U.Then
μa(ui,uj)min =Minimum of
{μa(ui,uj)R1
a(ui,uj)R2,...,μ
a(ui,uj)Rn}
μa(ui,uj)max =Maximum of
{μa(ui,uj)R1
a(ui,uj)R2,...,μ
a(ui,uj)Rn}
The binary relations using a threshold value α(0,1)are
deﬁned as
T(a,α) max ={(ui,uj)|μa(ui,uj)max α}.
T(a,α) min ={(ui,uj)|μa(ui,uj)min α}.
The tolerance classes for an attribute uiwith respect to T(a,α)
are
[T(a,α) max](ui)={ujU|uiT(a,α) maxuj}.
[T(a,α) min](ui)={ujU|uiT(a,α) minuj}.
The lower and upper approximations of a non-empty subset
BUare
T(a,α) max(B)={uiU|[T(a,α) max ](ui)B}.
T(a,α) max(B)={uiU|[T(a,α) max ](ui)B= φ}.
T(a,α) min (B)={uiU|[T(a,α) min ](ui)B}.
T(a,α) min (B)={uiU|[T(a,α) min ](ui)B= φ}
respectively.
Theorem 3.12 T(a,α) max (B)T(a,α) (B)T(a,α) min (B)
BT(a,α) min (B)T(a,α) (B)T(a,α) max (B).
Proof By the fundamental theorem of linear programming,
the expression n
k=1μa(ui,uj)RkP(Rk)attains its extreme
values at the vertices (corners) of the convex polytope
(polygon) formed by it. Clearly, the minimum value is
attained at μa(ui,uj)min and maximum value is attained at
μa(ui,uj)max.
123
Markov frameworks and stock market decision making 16421
Now, we prove that T(a,α) max(B)T(a,α)(B). Let z
T(a,α) max(B)⇒[T(a,α) max](z)B.Now, we prove that
[T(a,α)](z)B.Let y∈[T(a,α)](z). Then
n
k=1
μa(y,z)RkP(Rk)α,
As
μa(y,z)max
n
k=1
μa(y,z)RkP(Rk)α,
we get μa(y,z)max α. This implies
y∈[T(a,α) max](z).
This gives
[T(a,α)](z)⊆[T(a,α) max ](z).
As [T(a,α) max](z)B,we get [T(a,α) ](z)B.This
implies zT(a,α)(B). Now, we prove that T(a,α)(B)
T(a,α) min (B). Let zT(a,α) (B)⇒[T(a,α)](z)B.Now,
we prove that [T(a,α) min](z)B.Let y∈[T(a,α) min ](z).
Then μa(y,z)min α. As
n
k=1
μa(y,z)RkP(Rk)μa(y,z)min α,
we get
n
k=1
μa(y,z)RkP(Rk)α.
This implies
y∈[T(a,α)](z).
This gives
[T(a,α) min](z)⊆[T(a,α) ](z).
As [T(a,α)](z)B,we get [T(a,α) min ](z)B.This implies
zT(a,α) min (B).
Clearly, from Theorem 3.10,wehaveT(a,α) min (B)
BT(a,α) min (B). Now, we prove that T(a,α) min (B)
T(a,α)(B).
Let zT(a,α) min (B). Then [T(a,α) min](z)B= φ.
Now, take y∈[T(a,α) min ](z). Then μa(y,z)min α.
As n
k=1μa(y,z)RkP(Rk)μa(y,z)min α, we get
Fig. 15 Set-valued information system with respect to the reference
points R1and R2
y∈[T(a,α)](z). This implies [T(a,α) min ](z)⊆[T(a,α)](z).
As [T(a,α) min](z)B= φ, we get [T(a,α)](z)B= φ.
Hence zT(a,α)(B). Now, we prove that T(a,α)(B)
T(a,α) max(B). Let zT(a,α)(B). Then [T(a,α)](z)B= φ.
Now, take y∈[T(a,α)](z). Then n
k=1μa(y,z)RkP(Rk)
α. As μa(y,z)max n
k=1μa(y,z)RkP(Rk)α, we get
y∈[T(a,α) max](z). This implies [T(a,α) ](z)⊆[T(a,α) ](z).
As [T(a,α)](z)B= φ, we get [T(a,α) max ](z)B= φ.
Hence zT(a,α) max(B).
Here we provide an example to select the best alterna-
tive in multi-criteria decision-making problem by using the
tolerance relation and the proposed algorithm.
Example 3.13 The set-valued information system with alter-
natives a1,a2,a3,a4and their corresponding attributes
u1,u2,u3,u4,u5with respect to reference points R1and R2
is given in Fig. 15.
By using MRAF, the probability assigned to the reference
points [R1,R2]as [1
4,3
4].Then the values obtained by tol-
erance relation (Deﬁnition 3.8) are presented in Fig. 16.
In order to apply the decision algorithm, the sum of each
column in Fig. 16 is compiled and the compiled values
(ui,aj)are denoted as Low, Average, Good and Very Good
if (ui,aj)1.75 1.75 (ui,aj)<2.5, 2.5 (ui,aj)<
3.25, 3.25 (ui,aj)<4 , respectively. The values of Recom-
mendation are taken as the sum of all values corresponding
to each alternative and denoted them as ‘Yes’ if Recommen-
dation value 14 and ‘No’ if Recommendation value <14
and are presented in Fig. 17.
Then by using the proposed algorithm, the alternatives a2
and a3can be selected with the probability 1.
The above example clearly shows that fuzzy tolerance
relation can be used to arrive at a best alternative in multi-
criteria decision problems.
123
16422 K. Koppula et al.
Fig. 16 Tolerance relation values
Fig. 17 Compilation values of (ui,aj)
Fig. 18 Set-valued information system with respect to the reference
points R1and R2
Now, we generalize the fuzzy tolerance relation given by
Shivani et al. (2020) for selecting the attributes in set-valued
information system by using MRAF.
Deﬁnition 3.14 Let (U,AT,V,h), bAT be a set-valued
information system, where U={u1,u2,...,un}is a non-
empty ﬁnite set of objects, AT ={b1,b2,...,bn}is a
non-empty set of attributes, V=bAT Vbwhere Vbis
the set of alternative values associated with each attribute
bAT ,h:U×AT Vsuch that bAT ,u
U,h(u,b)=Vband X={R1,R2,...Rn}is the col-
lection of reference points from the universe U,then the
fuzzy relation is deﬁned as n
k=1μb(ui,uj)RkP(Rk)=
n
k=12|b(ui)Rkb(uj)Rk|
|b(ui)Rk|+|b(uj)Rk|P(Rk), where P(Rk), k=
1,2,...,nis the probability distribution for the reference
points Rk,k=1,2,...nobtained from MRAF.
For a set of alternatives BAT ,a fuzzy relation is deﬁned
as μB(ui,uj)=inf
bB(n
k=1μb(ui,uj)RkP(Rk)).
The binary relation using a threshold value α(0,1)is
deﬁned as
T(b,α) =(ui,uj)|
n
k=1
μb(ui,uj)RkP(Rk)α.
Fig. 19 Tolerance relation values
123
Markov frameworks and stock market decision making 16423
Fig. 20 Compilation values of (ui,bj)
For a set of attributes BAT ,we deﬁne a relation as
T(B,α) ={(ui,uj)|μB(ui,uj)α}.
Theorem 3.15 T(b,α) is a tolerance relation.
Example 3.16 The set-valued information system with crite-
ria b1,b2,b3,b4and their corresponding alternatives
u1,u2,u3,u4,u5with respect to reference points R1and R2
are given in Fig. 18.
By using MRAF, the probability assigned to the reference
points [R1,R2]as [1
3,2
3].Then the values obtained by tol-
erance relation are presented in Fig. 19.
In order to apply the decision algorithm, the sum of each
column in Fig. 19 is compiled and the compiled values
(ui,bj)are denoted as Low, Average and Good if 1 (ui,bj)
<2, 2 (ui,bj)<3 and 3, respectively. The values of
Recommendation are taken as the sum of all values corre-
sponding to each alternative and denoted them as ‘Yes’ if
Recommendation value 15 and ‘No’ if Recommendation
(Rec.) value <15 and are presented in Fig. 20.
Then by using the proposed algorithm, the attributes b1
and b2can be selected with the probability 1.
4 Conclusions
We have provided applications of Markov frameworks in
dealing with decision problems in dynamic and uncertain
environments.The MRAF enables the possibility of assigning
probabilities in a telescopic manner and Pawlak’s decision
algorithm enables in arriving at decisions in multi-criteria
decision problems. This method can also be used to sup-
plement other multi-criteria decision-making methods for
deeper analysis of the data especially when the values of
the ranked alternatives are very close to each other. The pro-
posed algorithm is explained in the stock market environment
to give the recommendation SELL, BUY or HOLD for a
particular stock. The proposed algorithm is validated in the
medical diagnosis problem to give the recommendation. In
addition, fuzzy tolerance relation is used along with MRAF
as a tool to ascertain critical attributes as well as alterna-
tives in multi-criteria decision problems. Methods presented
in this paper can be used in the development of other methods
which deal with multi-criteria decision problems.
Acknowledgements The authors acknowledge anonymous reviewers
and the editor for their valuable comments and suggestions. All authors
acknowledge the support and encouragement of Manipal Institute of
Technology, Manipal Academy of Higher Education (MAHE), India.
Funding Open access funding provided by Manipal Academy of Higher
Education, Manipal.
Compliance with ethical standards
Conﬂict of interest The authors declare that they have no conﬂict of
interest.
participants or animals performed by any of the authors.
tation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indi-
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unless indicated otherwise in a credit line to the material. If material
is not included in the article’s Creative Commons licence and your
intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copy-
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