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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

© The Author(s) 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

Syam Prasad Kuncham

syamprasad.k@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

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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,... jn−1∈N,Pr{Xn=k|Xn−1=j,Xn−2=

j1,...,X0=jn−1}=Pr{Xn=k|Xn−1=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=(π0,π

1,

...,π

k−1)tis said to be a stationary distribution of a ﬁnite

Markov chain if it satisﬁes:

(i) πi≥0 and k−1

i=0πi=1.

(ii) k−1

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,n≥1 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=Yn−1

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=∪

a∈AT Vawhere Va

is the set of attribute values associated with each attribute

a∈AT and h:U×AT →Vis an information function

that assigns particular values to the objects against attribute

set such that ∀a∈AT ,∀u∈U,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), ∀b∈AT and ui,uj∈U,

tolerance relation is deﬁned as

Tb={(ui,uj)|b(ui)∩b(uj)= φ}

For B⊆AT ,a tolerance relation is deﬁned as

TB={(ui,uj)|b(ui)∩b(uj)= φ,∀b∈B}=

b∈B

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), ∀b∈AT be a set-valued information system, then a fuzzy

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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 B⊆A,a fuzzy relation RBcan be

deﬁned as μRB(ui,uj)=inf

b∈BμRb(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 B⊆AT ,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,b∈U.Deﬁne a

relation R on U by aRb if there exists a square S centered at

the origin with radius r >0(the radius of a square deﬁned

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,δ

yare perpendicular

distances from ato X−axis and Y−axis, respectively. Take

δa=max{δx,δ

y}.Form a square Swith centre (0,0)and

radius √2δa.We claim that alies on S.

Let δy≥δx.Then δa=δy. This implies |y|=δa,−δa≤

x≤δa.Hence alies on one of its sides (x,δ

y)or (x,−δy).

Similarly, when δx≥δywe can prove that alies on one of

its sides (δx,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 a∈U.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,θ)

,0≤θ≤2π.

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,y∈U.Deﬁne a

relation Rθon U by xR

θy if there exists θsuch that S(x,θ) =

Fig. 1 Square with rotation zero degrees

S(y,θ)

.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 x∈U.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,θ) =S(y,θ)

.

This implies yR

θx.Hence Rθis symmetric.

(iii) Let xR

θyand yR

θz.Then there exists θsuch that

S(x,θ) =S(y,θ)and S(y,θ) =S(z,θ)

.This implies S(x,θ) =

S(z,θ)

.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={y∈U|xR

θy}=

{S(x,θ)|0≤θ≤2π}.

The sets L(Xk)=∪{x∈U|[x]θ⊆Xk}=∪{x∈U|

S(x,θ) ⊆Xk}={(x,y)||x|=k

2,|y|≤k

2or |y|=

k

2,|x|≤k

2}and U(Xk)=∪{x∈U|[x]θ∩Xk= φ}=

∪{x∈U|S(x,θ ) ∩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

⎞

⎟

⎟

⎟

⎟

⎠

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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,b∈U.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 (x−0)2+(y−0)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 a∈U.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]={b∈U|aR

0b}=Ca.

The sets

L(Xk)=∪{a∈U|[a]⊆Xk}=∪{a∈U|Ca⊆Xk}

={(x,y)|x2+y2≤k}

and

U(Xk)=∪{a∈U|[a]∩Xk= φ}=∪{a∈U|Ca∩Xk= φ}

={(x,y)|x2+y2≤√2k}

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

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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.

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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), ∀a∈AL 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=a∈AL Vawhere Vais the set of criteria values asso-

ciated with each alternative a∈AL,h:U×AL →V

such that ∀a∈AL,∀u∈U,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)Rk∩a(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.

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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)Rk∩a(ui)Rk|

|a(ui)Rk|+|a(ui)Rk|P(Rk)

=2|a(ui)R1∩a(ui)R1|

|a(ui)R1|+|a(ui)R1|P(R1)

+2|a(ui)R2∩a(ui)R2|

|a(ui)R2|+|a(ui)R2|P(R2)+···

+2|a(ui)Rn∩a(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)

=1≥α.

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)R1∩a(uj)R1|

|a(ui)R1|+|a(uj)R1|P(R1)

+2|a(ui)R2∩a(uj)R2|

|a(ui)R2|+|a(uj)R2|P(R2)+···

+2|a(ui)Rn∩a(uj)Rn|

|a(ui)Rn|+|a(uj)Rn|P(Rn)]≥α

⇒[2|a(uj)R1∩a(ui)R1|

|a(uj)R1|+|a(ui)R1|P(R1)

+2|a(uj)R2∩a(ui)R2|

|a(uj)R2|+|a(ui)R2|P(R2)+···

+2|a(uj)Rn∩a(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)={uj∈U|uiT(a,α)uj}.

The lower and upper approximations of a non-empty sub-

set B⊆Uas:

T(a,α)(B)={ui∈U|[T(a,α)](ui)⊆B}.

T(a,α)(B)={ui∈U|[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, a∈AL

and B⊆U,α∈(0,1).

Theorem 3.10 T(a,α) (B)⊆B⊆T(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,μ

a(ui,uj)R2,...,μ

a(ui,uj)Rn}

μa(ui,uj)max =Maximum of

{μa(ui,uj)R1,μ

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)={uj∈U|uiT(a,α) maxuj}.

[T(a,α) min](ui)={uj∈U|uiT(a,α) minuj}.

The lower and upper approximations of a non-empty subset

B⊆Uare

T(a,α) max(B)={ui∈U|[T(a,α) max ](ui)⊆B}.

T(a,α) max(B)={ui∈U|[T(a,α) max ](ui)∩B= φ}.

T(a,α) min (B)={ui∈U|[T(a,α) min ](ui)⊆B}.

T(a,α) min (B)={ui∈U|[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)

⊆B⊆T(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

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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 z∈T(a,α)(B). Now, we prove that T(a,α)(B)⊆

T(a,α) min (B). Let z∈T(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

z∈T(a,α) min (B).

Clearly, from Theorem 3.10,wehaveT(a,α) min (B)⊆

B⊆T(a,α) min (B). Now, we prove that T(a,α) min (B)⊆

T(a,α)(B).

Let z∈T(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 z∈T(a,α)(B). Now, we prove that T(a,α)(B)⊆

T(a,α) max(B). Let z∈T(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 z∈T(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

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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), ∀b∈AT 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=b∈AT Vbwhere Vbis

the set of alternative values associated with each attribute

b∈AT ,h:U×AT →Vsuch that ∀b∈AT ,∀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)Rk∩b(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 B⊆AT ,a fuzzy relation is deﬁned

as μB(ui,uj)=inf

b∈B(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

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Markov frameworks and stock market decision making 16423

Fig. 20 Compilation values of (ui,bj)

For a set of attributes B⊆AT ,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.

Ethical standard This article does not contain any studies with human

participants or animals performed by any of the authors.

Open Access This article is licensed under a Creative Commons

Attribution 4.0 International License, which permits use, sharing, adap-

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-

cate if changes were made. The images or other third party material

in this article are included in the article’s Creative Commons licence,

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-

right holder. To view a copy of this licence, visit http://creativecomm

ons.org/licenses/by/4.0/.

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