![Transform outcomes of P(A∣[xi]R^)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathcal {A} \mid [x_{i}]_{\widehat{\mathcal {R}}})$$\end{document} with different values of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}](https://www.researchgate.net/profile/Prasenjit-Mandal-3/publication/324113909/figure/fig2/AS:963461311905793@1606718485061/Transform-outcomes-of-PAxiRdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Decision rules with different values of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}](https://www.researchgate.net/profile/Prasenjit-Mandal-3/publication/324113909/figure/fig3/AS:963461311893532@1606718485828/Decision-rules-with-different-values-of-thdocumentclass12ptminimal_Q320.jpg)
Content uploaded by Prasenjit Mandal
Author content
All content in this area was uploaded by Prasenjit Mandal on Mar 31, 2018
Content may be subject to copyright.
Vol.:(0123456789)
1 3
Granular Computing
https://doi.org/10.1007/s41066-018-0090-9
ORIGINAL PAPER
Multi‑granulation interval‑valued fuzzy probabilistic rough sets
andtheir corresponding three‑way decisions based oninterval‑valued
fuzzy preference relations
PrasenjitMandal1 · A.S.Ranadive2
Received: 15 December 2017 / Accepted: 21 March 2018
© Springer International Publishing AG, part of Springer Nature 2018
Abstract
In this paper, we study interval-valued fuzzy probabilistic rough sets (IVF-PRSs) based on multiple interval-valued fuzzy
preference relations (IVFPRs) and consistency matrices, i.e., the multi-granulation interval-valued fuzzy preference relation
probabilistic rough sets (MG-IVFPR-PRSs). First, in the proposed study, we convert IVFPRs into fuzzy preference relations
(FPRs), and then construct the consistency matrix, the collective consistency matrix, the weighted collective preference rela-
tions, and the group collective preference relation (GCPR). Using this GCPR, four types of MG-IVFPR-PRSs are founded in
terms of different constraints on parameter. Finally, to find a suitable way of explaining and determining these parameters in
each model, three-way decisions are studied based on Bayesian minimum-risk procedure, i.e., the multi-granulation interval-
valued fuzzy preference relation decision-theoretic rough set approach. An example is included to show the feasibility and
potential of the theoretic results obtained.
Keywords Interval-valued fuzzy probabilistic rough set· Interval-valued fuzzy preference relation· Multi-granulation·
Three-way decisions
1 Introduction
The idea of rough set theory was basically proposed by Paw-
lak in (1982). A key notion of that theory is the approxima-
tion of a subset of objects by a pair of definable sets called
lower and upper approximations. It is characterized by a zero
tolerance of errors; that is, an object in the lower approxi-
mation which certainly belongs to set and an object in the
complement of the upper approximation which certainly
does not belongs to the set. For that reason, many different
generalizations of rough sets are proposed. By introducing
certain levels of errors, probabilistic rough sets (PRSs) (Yao
2008) are quantitative generalization of rough sets. Although
several specific models of PRSs had been considered by
some studies (Yao 1998, 2003; Ma and Sun 2012; Ziarko
2005, 2008; Wang and Xu 2002; Yao etal. 2008; Pawlak
etal. 1988; Polkowski 1996), a more general model, called
decision-theoretic rough set (DTRS) model, was proposed
by Yao and Wong (1992) and Yao (2009). With the aid of
Bayesian minimum-risk decision procedure, DTRS model
offers mathematical way to interpret and determine the
required thresholds in PRS. This model is fulfilled by split-
ting the approximated set into three regions, conducted by
the idea of 3WDs (Yao 2010). The 3WDs consists of three
different kinds of rules—positive rules (corresponding to
positive region), negative rules (corresponding to the nega-
tive region), and boundary rules (corresponding to boundary
region). In DTRS model, based on Bayesian minimum risk,
conditional probability and loss function play an important
role in determining thresholds from the given cost function.
Different thresholds for different PRSs can be deduced from
appropriate cost functions.
However, the PRSs and DTRS model cannot deal with
numerical data directly. For that reason, many studies usu-
ally adopted in real applications, which define all types of
relations rather than equivalence relations, e.g., tolerance
* Prasenjit Mandal
prasenjitmandal08@yahoo.com
A. S. Ranadive
asranadive04@yahoo.com
1 Bhalukdungri Jr. High School, Raigara, Purulia,
WestBengal723153, India
2 Department ofPure andApplied Mathematics, Guru
Ghasidas University, Bilaspur, Chhattisgarh, India
Granular Computing
1 3
relations (Liang etal. 2012), similarity measures (Liu etal.
2016), dominance relations (Du and Hu 2016), covering
(Wang etal. 2015), inclusion measures (Zhang etal. 2016),
fuzzy relations (Wei and Zhang 2004; Sun etal. 2014; Yang
etal. 2013), intuitionistic fuzzy relations (Liang and Liu
2015; Zhang etal. 2017), and bipolar-valued fuzzy relations
(Mandal and Ranadive 2017). They can be used to measure
and represent various real values.
To handling all types of real data, Zadeh (1965) intro-
duced fuzzy sets. The remarkable applications of fuzzy sets
are done in Horng et al. (2005), Chen and Kao (2013), Chen
and Hong (2014), Chen (1996), Chen and Chen (2011), and
Chen etal. (2009). However, it is well known that interval-
valued fuzzy set (IVFS) is more effective than fuzzy set for
imprecise evaluation (Chen and Horng 1996; Chen 1997;
Chen etal. 1997; Chen and Hsiao 2000; Chen and Chen
2008, 2009; Wei and Chen 2009; Chen and Wang 2009;
Chen and Sanguansat 2011; Mendel etal. 2006; Chen and
Lee 2010; Lee and Chen 2008). Thus, studies on combina-
tion of IVFS and rough set theory have been considered
to be significant approach to rough set theory. Liang and
Liu 2014 combined IVFS and DTRS, and then study on
3WDs with interval-valued decision-theoretic rough sets
(IVDTRS). Zhao and Hu (2015, 2016) study interval-valued
fuzzy decision-theoretic rough set (IVF-DTRS) approaches
based on fuzzy probability measure and IVF-PRSs and their
corresponding 3WDs.
Granular computing (Pedrycz and Chen 2011, 2015a,
b) is an emerging computing paradigm of information pro-
cessing. It concerns the processing of complex information
entities called granules, which arise in the process of data
abstraction and derivation of knowledge from information
or data. Therefore, in the viewpoint of granular computing,
Qian etal. (2010b) generalized Pawlak rough set model to a
multi-granulation rough set (MGRS) model using multiple
equivalence relations instead of single equivalence relation.
In the MGRS model, two basic models are mainly pro-
posed, such as optimistic and pessimistic MGRS (Qian etal.
2010b). Since then, MGRSs have been developed quickly
(Qian etal. 2010a; Liu etal. 2014; Lin etal. 2012; Xu etal.
2011, 2012, 2013, 2014). The combination of MGRSs and
DTRSs is an important topic among these developments (Lv
etal. 2013; Feng and Mi 2016; Qian etal. 2014).
However, preference analysis is a class of important
issues in group decision-making for pairwisely comparing
alternatives. The decision makers are expressed his/her opin-
ions as preference relations. Then, they are used to deriving
the weights of alternatives, and thus, the best alternative is
chosen. Based on fuzzy set theory (Zadeh 1965), the prefer-
ence degree of the alternative
xi
over the alternative
xj
can
be expressed as
rij ∈[0, 1]
. Then, a fuzzy binary relation
on
U={x1,x2,…,xn}
is defined and a preference matrix
R(xi,xj)=(rij)n×n
with the entries
rij
is given. The preference
matrix is called an FPR (Tanio 1984; Chen and Niou 2011),
where
rij ∈[0, 1]
and
rii =0.5
, since the preference value is
exact real numbers. However, owing to the complexity and
uncertainty of the real-world decision-making problems, it
is difficult to provide precise preference value to evaluate the
judgments. For that reason, it is popular to study decision-
making models and their applications, where the judgments
of decision makers are expressed as interval-valued com-
parison matrix (Xu 2011; Chen etal. 2015; Liu etal. 2018).
On the basis of above-mentioned analysis, IVF-PRSs
considered the incomplete available information and the
possible existence of random available information of the
objects, while it cannot consider multi-granulation and pref-
erence analysis. In group decision-making problem using
IVFPRs, the decision results are obtained pairwise compar-
ing the objects according to derive its weights, while it can-
not consider the incomplete available information and the
possible existence of random available information of the
objects. Therefore, the motivation of this paper is to study
IVF-PRSs based on multiple IVFPRs and consistency matri-
ces, i.e., MG-IVFPR-PRSs. In fact, the consistency matrices
are constructed to avoid the self-contradiction of the objects.
Based on MG-IVFPR-PRSs, we develop a new approach
for group decision-making which is considered the incom-
plete available information, the possible existence of ran-
dom available information, and preference analysis. First,
a multi-granulation interval-valued fuzzy preference rela-
tion probabilistic approximation space (MG-IVFPR-PAS) is
defined using m IVFPRs. Then, we aggregate the m IVFPRs
given by m experts and construct the group collective prefer-
ence relation (GCPR) according to the method as given in
Chen etal. (2015). Second, we investigate four types of MG-
IVFPR-PRSs within the frameworks of MG-IVFPR-PAS and
GCPR in terms of different constraints on parameters. Third,
Bayesian decision procedure within the MG-IVFPR-PRS,
i.e., the MG-IVFPR-DTRS approach is studied. Finally, we
develop a new approach for group decision-making using
MG-IVFPR-PRSs and their corresponding 3WDs. What we
want to do is shown in Fig.1.
The paper is organized as follows: in Sect.2, some basic
concepts about fuzzy probability and FPRs are briefly repre-
sented. Section3 is devoted to MG-FPR-PRS models as well
as associated 3WDs in the framework of MG-IVFPR-PAS.
In Sect.4, we propose a new approach for group decision-
making based on MG-IVFPR-PRSs including practical
applications. Section5 concludes the paper.
2 Preliminaries
In this section, we will review some basic concepts such as
interval-valued fuzzy sets, interval-valued fuzzy probability,
and interval-valued fuzzy preference relations, which have
Granular Computing
1 3
been addressed in Zadeh (1965, 1968), Zhao and Hu (2015,
2016), Tanio (1984), Lee (2012), and Chen etal. (2014).
2.1 Fuzzy sets andprobability theory
Let U be a universe of discourse. A fuzzy
set A is a mapping from U into [0, 1], i.e.,
A∶U
→
[0, 1],x
↦
A(x)∈[0, 1],∀x∈U
. The family of
all fuzzy sets on U is denoted by F(U). Given a fuzzy set
A∈F(U)
and
𝛼∈[0, 1]
, the
𝛼
-cut set of A is defined as
A𝛼={x∈U∶A(x)≥𝛼}
, which is a classical subset of U.
Denition 1 (Zadeh 1968) Let
(U,𝙰,P)
be a probability
space in which U is a sample space of events,
𝙰
is a
𝜎
-alge-
bra of events (i.e., the measurable subsets of U), and P is a
probability measure defined on
𝙰
. For a fuzzy set
A∈F(U)
,
if
A∈𝜁(𝙰)={A∈F(U)∶A𝛼∈𝙰,𝛼∈[0, 1]}
, then A is a
fuzzy event on U. The probability of fuzzy event A (called
the fuzzy probability of A) is
P
(A)=
∫U
A(x)
dP
.
If
U={x1,x2,…,xn}
, and
pi=P({xi})
(i=1, 2, …,n)
,
then
Denition 2 (Zadeh 1968) Let
(U,𝙰,P)
be a probability
space and A,B be two fuzzy events on U. If
P(B)≠0
, then
(1)
P
(A)=
n
∑
i=1
A(xi)pi
.
the conditional probability of A given B (called fuzzy con-
ditional probability) is defined as
P
(A∣B)=
P(AB)
P(B)
.
If
U={x1,x2,…,xn}
, and
pi=P({xi})
(i=1, 2, …,n)
,
then
for all fuzzy events
A,B∈𝜁(𝙰)
.
Proposition 1 (Zhao and Hu 2016) Let (U,
𝙰
, P) be a
probability space and A,B,C be three fuzzy events on U. If
P(A)≠0
, then the following conclusions hold.
(1)
P(� ∣ A)=0, P(U∣A)=1
;
(2) If
B⊆C
, then
P(B)≤P(C)
and
P(B∣A)≤P(C∣A)
;
(3)
P(B∣A)+P(Bc∣A)=1
.
2.2 Interval‑valued fuzzy sets andprobability
theory
Let
𝐈ℝ
+
= {[a−,a+]∶0≤a−≤a+,a−,a+∈
ℝ
+}
be the
family of all positive interval numbers. We denote by
a=[a,a]
the degenerate interval numbers. The arithmetic
(2)
P
(A∣B)=
P(AB)
P(B)
=
∑
n
i=1A(xi)B(xi)pi
∑
n
i=1
B(x
i
)p
i
Fig. 1 Proposed work in this paper
Granular Computing
1 3
operation of interval numbers is defined as follows: for all
[a−
,a+],[b−
,b+]∈
𝐈
ℝ+
,
and if
b−≠0
, the division is defined as follows:
For simplicity, symbol
⊗
is omitted in the sequel.
If
0≤a−≤a+≤1
, it is defined that
1−[a−,a+]=[1−a+,1 −a−]
.
The order relation on
𝐈ℝ+
is defined as: for all
[a−
,a+],[b−
,b+]∈𝐈ℝ+
,
Obviously, the order on
𝐈ℝ+
is partial. The strict order of
interval numbers is defined as:
[a−
,a+]<[b−
,b+]
if and only
if
a−<b−
,
a+≤b+
or
a−≤b−
,
a+<b+
.
Based on the order relation defined in (6) for posi-
tive interval numbers, the corresponding join,
∨
, and
meet,
∧
, are defined, respectively, as follows: for all
[a−
,a+],[b−
,b+]∈𝐈ℝ
+
.
Let
𝐈[0,1]
= {[a
−
,a
+
]∶0
≤
a
−≤
a
+≤
1
}
be the set of all
interval numbers on [0,1]. Let U be a universe of discourse.
An interval-valued fuzzy set (IVFS) on U is a mapping that
assigns each object in U a unique interval value from
𝐈[0,1]
,
that is
Let
(x)=[a−
,a+]≜[A−(x),A+(x)]
for each
x∈U
.
Then, two fuzzy sets
A−
and
A+
on U, called lower and upper
fuzzy sets of
, are derived with
A−⊆A+
. Subsequently, an
IVFS
is always denoted by
[A−
,A+]
. Let
F
𝐈
[
𝟎
,
𝟏
](U)
be the
family of all IVFSs on U. The operators for IVFSs are
defined through respective operations on lower and upper
fuzzy sets. For all IVFSs
,∈F
𝐈
[
𝟎
,
𝟏
](U)
and each
x∈U
,
(3)
[a−
,a+]⊕[b−
,b+]=[a−+b−
,a++b+],
(4)
[a−
,a+]⊗[b−
,b+]=[a−b−
,a+b+],
(5)
[
a
−
,a
+
]
[b
−
,b
+]=
[
a
−
b
−∧a
+
b
+,
a
−
b
−∨a
+
b
+
].
(6)
[a−
,a+]≤[b−
,b+]
⇔
a−≤b−
,a+≤b+
.
[
a
−
,a
+
]∧[b
−
,b
+
]=[a
−
∧b
−
,a
+
∧b
+
]
,
[a
−
,a
+
]∨[b
−
,b
+
]=[a
−
∨b
−
,a
+
∨b
+
].
∶
U→𝐈
[0,1]
,x↦
(
x
)=[
a
−
,a
+].
(∩)(x)
= [(A−∩B−)(x),(A+∩B+)(x)]
=[A
−
(x)∧B
−
(x),A
+
(x)∧B
+
(x)],
(∪)(x)
= [(A−∪B−)(x),(A+∪B+)(x)]
=[A
−
(x)∨B
−
(x),A
+
(x)∨B
+
(x)],
The order of IVFSs is defined as: for all
,∈F
𝐈
[
𝟎
,
𝟏
](U)
,
⊆
⇔
A−(x)≤B−(x),A+(x)≤B+(x)
,
∀x∈U
.
Denition 3 (Zadeh 1968) Let
(U,𝙰,P)
be a probability
space in which U is a sample space of events,
𝙰
is a
𝜎
-alge-
bra of events (i.e., the measurable subsets of U), and P is a
probability measure defined on
𝙰
. For an IVFS
on U, if
where
(A−)𝛼
and
(A+)𝛼
are
𝛼
-cut sets of lower and upper
fuzzy sets of
, respectively, and then,
is an interval-val-
ued fuzzy event on U. The probability of an interval-valued
fuzzy event
(called the interval-valued fuzzy probability
of
) is
where
=[A−
,A+]
.
If
U={x1,x2,…,xn}
, and
pi=P({xi})
(i=1, 2, …,n)
,
then Definition1 that
for each interval-valued fuzzy event
=[A−
,
A+]∈F
𝐈
[𝟎,𝟏](U).
Denition 4 (Zhao and Hu 2015) Let
(U,𝙰,P)
be a proba-
bility space and
=[A−
,A+]
,
=[B−
,B+]
be two interval-
valued fuzzy events on U with
P(B−)≠0
. The conditional
probability of
given
(called interval-valued fuzzy con-
ditional probability) is defined by
c
(x)=[1−A
+
(x),1−A
−
(x)],
()(x)=[A
−
(x)B
−
(x),A
+
(x)B
+
(x)].
(7)
∈𝜁
[0,1]
(
𝙰
)={∈
F𝐈[𝟎,𝟏]
(
U
)∶=[
A
−
,A
+],
(A−)
𝛼
,(A+)
𝛼
∈𝙰,𝛼∈[0, 1]},
(8)
P
(
)=
�
U
(x)dP
=[�U
A−(x)dP,�U
A+(x)dP
]
=[P(A
−
),P(A
+
)],
(9)
P
(
)=
n
∑
i=1
(xi)pi
=
[
n
∑
i=1
A−(xi)pi,
n
∑
i=1
A+(xi)pi
]
(10)
P
(
∣
)=
P()
P().
Granular Computing
1 3
Proposition 2 (Zhao and Hu 2015) Let (U,
𝙰
, P) be a
probability space and
=[A−
,A+]
,
=[B−
,B+]
be two
interval-valued fuzzy events on U with
P(B−)≠0
, the fol-
lowing holds:
If
U={x1,x2,…,xn}
, and
pi=P({xi})
(i=1, 2, …,n)
,
then
Proposition 3 (Zhao and Hu 2016) Let (U,
𝙰
, P) be a
probability space and
=[A−
,A+]
,
=[B−
,B+]
and
=[C−
,C+]
be three interval-valued fuzzy events on U. If
P(A−)≠0
, then the following conclusions hold.
(1)
P(� ∣ )=
0, P(U∣)=
1
; and
(2) If
⊆
, then
P(∣)≤P(∣)
.
According to Propositions1 and 2 that
2.3 Fuzzy andinterval‑valued fuzzy preference
relations
Denition 5 (Tanio 1984) Let R be a fuzzy preference rela-
tion (FPR) for the set
U={x1
,
x2
,
⋯
,
xn}
, shown as follows:
where
rij
denotes the degree of preference of alternative
xi
over alternative
xj
,
rij ∈[0, 1]
,
rii =0.5
,
1≤i≤n
and
1≤j≤n
. Especially,
rij =0.5
indicates that there is no difference between
alternative
xi
and alternative
xj
;
rij >0.5
indicates that alternative
xi
is better than alterna-
tive
xj
;
rij <0.5
indicates that alternative
xj
is better than alterna-
tive
xi
;
P
(
∣
)=[P(A
−
∣B
−
)∧P(A
+
∣B
+
),
P(A
−
∣B
−
)∨P(A
+
∣B
+
)].
P
(A−∣B−)=
∑n
i=1A
−
(xi)B
−
(xi)pi
∑n
i=1B−(xi)pi
,
P
(A+∣B+)=
∑
n
i=1A+(xi)B+(xi)pi
∑
n
i=1
B+(x
i
)p
i
.
P(c∣)=[
1
−
P
(
A
+∣
B
−)∨
P
(
A
−∣
B
+)
,
1−P(A
+
∣B
−
)∧P(A
−
∣B
+
)].
R(
xi,x
j
)=(
rij
)
n×n
=
R(xi,y
j)x1x2···xn
x1r11 r12 ···r1n
x2r21 r22 ···r2n
········· ··· ···
xnrn1rn2
···
rnn
,
(11)
rij =1
indicates that alternative
xi
is absolutely better than
alternative
xj
;
rij =0
indicates that alternative
xj
is absolutely better than
alternative
xi
;
where
1≤i≤n
and
1≤j≤n
.
Denition 6 (Lee 2012) Given a fuzzy preference relation
R=(rij)n×n
, where
rij
denotes the fuzzy preference value
(FPV) for alternative
xi
over alternative
xj
,
rij +rji =1
,
rii =0.5
,
1≤i≤n
and
1≤j≤n
. The consistency matrix
R
=(r
ik
)
n×n
is constructed based on the FPR R, shown as
follows:
The consistency matrix
R
=(
rik
)
n×n
has the following prop-
erties (Lee 2012):
(1)
rik +rki =1
,
(2)
rii =0.5
,
(3)
rik =rij +rjk −0.5
,
(4)
rik ≤ris
for all
i∈{1, 2, …,n}
, where
k∈{1, 2, …,n}
and
s∈{1, 2, …,n}
.
Denition 7 (Chen etal. 2014) Let
R=
(rik)n×n
be a consist-
ency matrix constructed by a FPR
R=(rij)n×n
given by an
expert. The consistency degree d between R and
R
is defined
as follows:
where
d∈[0, 1]
,
rij
denotes the FPV in the FPR R for alter-
native
xi
over alternative
xj
, and
rij
denote an FPV in the
consistency matrix
R
for alternative
xi
over alternative
xj
,
1≤i≤n
, and
1≤j≤n
. The larger the value of d, the more
consistent the FPR given by the expert. If the value of d is
close to one, then the information of the FPR given by the
expert is more consistent.
Denition 8 (Xu 2011) Let
be an interval-valued fuzzy
preference relation (IVFPR) for the set
U={x1,x2,…,xn}
,
shown as follows:
(12)
r
ik =1
n
n
∑
j=1
(rij +rjk)−
0.5.
(13)
d
=1−2
n(n−1)
n
∑
i=1
n
∑
j=1,j≠i|
|
|
rij −rij
|
|
|,
R(
xi,x
j
)=(∇
ij
)
n×n
=
R(xi,y
j)x1x2···xn
x1∇11 ∇12 ···∇
1n
x2∇21 ∇22 ···∇
2n
········· ······
xn
∇
n1
∇
n2
···∇
nn
,
(14)
Granular Computing
1 3
where
∇
ij =[r
−
ij
,r
+
ij ]
denotes an interval-valued fuzzy prefer-
ence value (IVFPV) for alternative
xi
over alternative
xj
.
Then,
0≤r−
ij ≤r+
ij ≤1
,
∇
ji =
1−∇
ij =[1−r
+
ij
,1−r
−
ij ]
,
r+
ii
=r
−
ii
=
0.5
,
1≤i≤n
and
1≤j≤n
.
3 Multi‑granulation interval‑valued fuzzy
preference relation probabilistic rough
sets
This section proposes the model of multi-granulation inter-
val-valued fuzzy preference relation probabilistic rough sets
(MG-IVFPR-PRSs), within the framework of multi-granula-
tion interval-valued fuzzy preference relation probabilistic
approximation space.
Denition 9 Let
U={x1,x2,…,xn}
be a non-empty finite
universe.
q(1≤q≤m)
be m IVFPRs for the set U. P
be a probability measure defined on the
𝜎
-algebra formed
by the image of element
xi∈U(1≤i≤n)
. Then, we call
(U,q(1≤k≤m),P)
a multi-granulation interval-valued
fuzzy preference relation probabilistic approximation space
(MG-IVFPR-PAS).
For aggregating the m IVFPRs
Rq(1≤q≤m)
given by
m experts
eq(1≤q≤m)
, we adopt the following algorithm
as given in Chen etal. (2015).
Algorithm1 Assume that there are m IVFPRs
1
,
2
,
…
,
m
given by m experts
e1
,
e2
,
…
,
em
, respectively. Assume
that the IVFPR
q
given by expert
eq
for alternative
xi
over
alternative
xj
is shown as follows:
where
∇q
ij
is an IVFPV,
∇q
ij
=[r
−
ij
q
,r
+
ij
q]
,
0≤
r
−
ij
q≤
r
+
ij
q≤1
,
∇q
ji
=
1−∇
q
ij
=[1−r
+
ij
q
,1−r
−
ij
q
]
,
∇+
ii
q
=∇
−
ii
q
=
0.5
,
1≤i≤n
,
1≤j≤n
, and
1≤q≤m
.
Step 1: For the IVFPRs given by expert
eq
, construct the
FPR
Rq
=(r
q
ij
)n×
n
for expert
eq
, construct the consistency
matrix
Rq
=(r
q
ij
)n×
n
for each expert
eq
, construct the collec-
tive consistency matrix
R
∗
=(
r
∗
ij
)n×
n
, and calculate the con-
sistency degree
dq
of expert
eq
, where
R
q(xi,x
j)=(∇q
ij )n×n
=
Rq(xi,x
j)x1x2···xn
x1∇q
11 ∇q
12 ···∇
q
1n
x2∇q
21 ∇q
22 ···∇
q
2n
········· ······
xn
∇
q
n1
∇
q
n2
···∇
q
nn
,
(15)
where
1≤i≤n
,
1≤j≤n
,
1≤q≤m
and
𝜃∈[0, 1]
. In
Eq. (16), we let
rq
ij
=
1
2
(r−
ij
q+r+
ij
q)
, where
rq
ij ∈[
0, 1
]
,
1≤i≤n
,
1≤j≤n
and
1≤q≤m
, such that we can trans-
form the IVFPR
q
(xi,xj)=(∇
q
ij
)n×
n
given by expert
eq
into
the FPR R
q
=(r
q
ij
)n×
n
, where
rq
ij
∈[0, 1
]
,
1≤i≤n
,
1≤j≤n
,
1≤q≤m
and m is the number of decision makers. In
Eq.(17), we adopt Eq.(12) to construct the consistency
matrix
Rq
=(r
q
ij
)n×
n
for expert
eq
, where
rq
ij
∈[0, 1
]
,
1≤i≤n
,
1≤j≤n
1≤q≤m
, and m is the number of deci-
sion makers. In Eq.(18), we adopt Eq.(13) to calculate the
consistency degree
dq
for expert
eq
, where
dq∈[0, 1]
,
1≤q≤m
, and m is the number of decision makers. In
Eq.(19), we let
r∗
ij
=
1
m∑m
q=1
r
q
ij
to construct the collective
consistency matrix
R∗
=(r∗
ij
)n×
n
for all experts, where
rq
ij
∈[0, 1
]
,
1≤i≤n
,
1≤j≤n
,
1≤q≤m
, and m is the
number of decision makers.
Step 2: Calculate the weight
𝜆q
of expert
eq
based on
the obtained consistency degrees of the experts, shown as
follows:
where
dq
is the consensus degree of expert
eq
and
1≤q≤m
.
Step 3: Construct the weighted collective preference rela-
tion
∗= (∇∗
ij)
n×
n
for all experts, shown as follows:
where
1≤i≤n
,
1≤j≤n
and
1≤q≤m
.
Step 4: Construct the group collective preference relation
(GCPR)
=(
∇
ij
)
n×n
for all experts, shown as follows:
(16)
r
q
ij =
1
2
(r−
ij
q+r+
ij
q)
,
(17)
r
q
ij =1
n
n
∑
k=1
(rq
ik +rq
kj)−
0.5,
(18)
d
q=1−2
n(n−1)
n
∑
i=1
n
∑
j=1,j≠i|
|
|
rq
ij −rq
ij
|
|
|,
(19)
r
∗
ij =1
m
m
∑
q
=
1
rq
ij
,
(20)
𝜆
q=
d
q
∑
m
t=1
d
t
,
(21)
∇
∗
ij =
m
∑
q=1
𝜆q∇q
ij =
[
m
∑
q=1
𝜆qr−
ij
q,
m
∑
q=1
𝜆qr+
ij
q
]
= [∇−
ij
∗,∇+
ij
∗],
Granular Computing
1 3
where
1≤i≤n
,
1≤j≤n
, and
1≤q≤m
. Equa-
tion(22) shows that the GCPR is also IVFS, so we denote
=[
−
,
+
]
.
Denition 10 Let
q(1≤q≤m)
be m IVFPRs for the set
U given by the experts
eq(1≤q≤m)
, respectively. Let
be
the GCPR constructed by m IVFPRs
q(1≤q≤m)
. Then,
for each
xi∈U(1≤i≤n)
, the IVFS
[xi]
is defined as
for all
xj∈U(1≤j≤n)
. In Eq.(23), we adopt Algorithm3
to calculate the fuzzy set
[xi]
(1≤i≤n)
.
3.1 MG‑IVFPR‑PRSs based oninterval‑valued fuzzy
probability
The aim of proposing MG-IVFPR-PRSs is to characterize
fuzzy events in terms of the available knowledge represented
by m IVFPRs given by the m experts.
Denition 11 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
, and
=[A
−
,A
+
](∈ F𝐈
[0,1]
(U
))
be a interval-
valued fuzzy event. For a pair of parameters
[𝛼−
,𝛼+]
and
[𝛽−
,𝛽+]
with
0≤[𝛽−,𝛽+]<P()<[𝛼−,𝛼+]≤
1
, the
[𝛼−
,𝛼+]
-multi-granulation interval-valued fuzzy preference
relation probabilistic lower approximation and
[𝛽−
,𝛽+]
-multi-granulation interval-valued fuzzy preference relation
probabilistic upper approximation of
are defined, respec-
tively, as follows:
The pair
(
Apr[𝛼−,𝛼+]
(
), Apr[
𝛽−
,
𝛽+
]
(
))
is called
([𝛼−
,𝛼+],[𝛽−
,𝛽+])
-multi-granulation interval-valued fuzzy
preference relation probabilistic rough set
(22)
∇
ij =
[
∇
−
ij
∗
+r
∗
ij
2,
∇
+
ij
∗
+r
∗
ij
2
]
=[
∇−
ij
,
∇+
ij
],
(23)
[
xi]
(xj)=
(xi,xj)
=[
−(xi,xj),
+(xi,xj
)]
=(
∇
ij
)
n
×
n
Apr[𝛼−
,
𝛼+]
(
)={xi∈U(1
≤
i
≤
n)∶
P(
∣[x
i
]
)
≥
[𝛼−,𝛼+]}
,
Apr[𝛽−,𝛽+]
�
(
)={xi∈U(1
≤
i
≤
n)∶
P(
∣[x
i
]
�
)>[𝛽−,𝛽+]}
.
(
([𝛼−
,𝛼+],[𝛽−
,𝛽+])
-MG-IVFPR-PRS) of
. The positive,
negative, and boundary regions of
are defined, respec-
tively, as follows:
Remark 1 If
U={x1,x2,…,xn},
and
P(xi)
=
pi(i=1, 2, …,n)
, then we have
where
Proposition 4 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
,∈F
I
[0,1](U)
with
P()≠
0
. The fol-
lowing properties hold.
(1) If
0≤[𝛽−,𝛽+]<P()<[𝛼−,𝛼+]≤
1
, then
Apr
[𝛼−,𝛼+]
�
(
)⊆Apr
[𝛽−,𝛽+]
�
(
)
.
(2)
Apr[𝛼−,𝛼+]
(�) = �
,
Apr[𝛽−
,
𝛽+]
(U)=
U
for all
0<[𝛼−,𝛼+]≤
1
and
0≤[𝛽−,𝛽+]<
1.
POS[𝛼−
,
𝛼+]
(
)=Apr
[𝛼−,𝛼+]
(
)
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
)
≥
[𝛼−,𝛼+]}
,
NEG
[𝛽−,𝛽+](
)=(Apr
[𝛽−
,
𝛽+]
(
))c
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
)
≤
[𝛽−,𝛽+]}
,
BND([𝛼−,𝛼+],[𝛽−,𝛽+])
(
)
=Apr[𝛽−,𝛽+]
�
()−Apr[𝛼−,𝛼+]
�
()
={xi∈U(1≤i≤n)∶
[𝛽−,𝛽+]<P(
∣[x
i
]
�
)<[𝛼−,𝛼+]}
.
(24)
P(∣[x
i
]
)
=[P(A−∣[xi]
−)∧P(A+∣[xi]
+)
,
P(A−∣[x
i
]
−)∨P(A+∣[x
i
]
+)]
,
(25)
P
(A−∣[xi]
−)=
∑
n
j=1
−(xi,xj)A−(xj)pj
∑
n
j=1
−(xi,xj)pj
,
(26)
P
(A+∣[xi]
+)=
∑n
j=1
+(xi,xj)A+(xj)pj
∑
n
j=1
+(xi,xj)pj
.
Granular Computing
1 3
(3) If
⊆
and
[𝛽−
,𝛽+]<P()∧P(),
P()∨P()<[𝛼−
,𝛼+]
, then
Apr[𝛼−,𝛼+]
()
⊆
Apr[𝛼−,𝛼+]
()
and
Apr[𝛽−
,
𝛽+]
()
⊆
Apr[𝛽−
,
𝛽+]
()
.
(4) If
P
(
)<[𝛼
−
1
,𝛼
+
1
]
≤
[𝛼
−
2
,𝛼
+
2
]
≤
1
and
0
≤[𝛽−
1
,𝛽
+
1]
≤
[𝛽
−
2
,𝛽
+
2
]<P(
)
, then
Apr[𝛼−
2,
𝛼+
2
]
�
(
)⊆Apr
[𝛼−
1,
𝛼+
1
]
�
(
)
and
Apr[𝛽−
2,
𝛽+
2
]
()
⊆
Apr[𝛽−
1,
𝛽+
1
]
()
.
Proof
(1) If
x∈Apr
[𝛼−,𝛼+]
(
)
,
∀x∈U
, then
P(∣[x]
)≥[𝛼−
,
𝛼+
]
>
[
𝛽−
,
𝛽+]
. It shows that
x
∈Apr
[𝛽−
,
𝛽+]
(
)
. There-
fore, we prove:
Apr[𝛼−,𝛼+]
(
)⊆Apr
[𝛽−
,
𝛽+]
�
(
)
.
(2) From Proposition 3(1), we have
(3) If
x
∈Apr
[𝛼−,𝛼+]
(
)
,
∀x∈U
, then
P(∣[x]
)≥[𝛼−
,
𝛼+]
. Since
⊆
, then from Proposition 3(2), we have
P(∣[x]
)≤P(∣[x]
)
. Therefore, we have
P(∣[
x
]
)≥[𝛼−
,
𝛼+]
. It shows that
x
∈Apr
[𝛼−,𝛼+]
(
)
.
Therefore, we prove
Apr[𝛼−,𝛼+]
()
⊆
Apr[𝛼−,𝛼+]
()
.
Similarly, we can prove
Apr[𝛽−
,
𝛽+]
()
⊆
Apr[𝛽−
,
𝛽+]
()
.
(4) If
x
∈Apr
[𝛼−
2
,𝛼+
2
]
(
)
,
∀x∈U
, then
P(∣[x]
)≥[𝛼−
2,
𝛼+
2
]≥[𝛼
−
1
,𝛼
+
1]
. It shows that
x
∈Apr
[𝛼−
1
,𝛼+
1
]
(
)
. There-
fore, we can prove
Apr
[
𝛼−
2,
𝛼+
2
]
(
)⊆Apr[
𝛼−
1,
𝛼+
1]
�
(
)
. Sim-
ilarly, we can prove and
Apr
[
𝛽−
2,
𝛽+
2
]
()
⊆
Apr
[
𝛽−
1,
𝛽+
1
]
()
.
A special case of
([𝛼−
,𝛼+],[𝛽−
,𝛽+])
-MG-IVFPR-PRS
with
[𝛽−,𝛽+]=
1−[𝛼−,𝛼+]<P()<[𝛼−,𝛼+]
is referred
to as symmetric MG-IVFPR-PRS. One of the advantages
is that only one parameter needs to be decided. This would
deduce the cost in evaluation of values of parameters.
Apr[𝛼−
,
𝛼+]
�
(�)
={x∈U∶P(�∣[x]�
)≥[𝛼−,𝛼+
]}
={x∈U∶
0≥[𝛼−,𝛼+]}
=�,
Apr[𝛽−,𝛽+]
�
(U)
={x∈U∶P(U∣[x]�
)>[𝛽−,𝛽+
]}
={x∈U∶
1≥[𝛼−,𝛼+]}
=U.
Denition 12 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
=[A−,A+](∈ F
I
[0,1](U))
be a interval-
valued fuzzy event. For
0.5 <[𝛼−,𝛼+]≤
1
, the
[𝛼−
,𝛼+]
-multi-granulation interval-valued fuzzy preference relation
probabilistic lower and upper approximations of
are
defined, respectively, as follows:
The pair
(
Apr[𝛼−,𝛼+]
(
),Apr
[𝛼−
,
𝛼+]
(
))
, is called
[𝛼−
,𝛼+]
-multi-granulation interval-valued fuzzy preference relation
probabilistic rough set (
[𝛼−
,
𝛼+]
-MG-IVFPR-PRS) of
. The
positive, negative and boundary regions of
are defined,
respectively, as follows:
From Definition12, the following assertions are clear.
Proposition 5 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
,∈F
I
[0,1](U)
with
0.5 <[𝛼−,𝛼+]≤
1
.
The following properties hold.
(1)
Apr
[𝛼−,𝛼+]
�
(
)⊆Apr
[𝛼−
,
𝛼+]
�
(
)
.
Apr[𝛼−
,
𝛼+]
(
)={xi∈U(1
≤
i
≤
n)∶
P(
∣[x
i
]
)
≥
[𝛼−,𝛼+]}
,
Apr[𝛼−,𝛼+]
�
(
)={xi∈U(1
≤
i
≤
n)∶
P(
∣[x
i
]
�
)>
1−[𝛼−,𝛼+]}
.
POS
[
𝛼−
,
𝛼+
](
)=Apr
[𝛼−,𝛼+]
(
)
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
)
≥
[𝛼−,𝛼+
]}
NEG
[𝛼−,𝛼+](
)=(Apr
[𝛼−,𝛼+]
(
))c
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
)
≤
[𝛼−,𝛼+]}
,
BND[𝛼−,𝛼+]
(
)
=Apr[𝛼−,𝛼+]
�
()−Apr[𝛼−,𝛼+]
�
()
={xi∈U(1≤i≤n)∶
1−[𝛼−,𝛼+]<P(
∣[x
i
]
�
)<[𝛼−,𝛼+]}
.
Granular Computing
1 3
(2)
Apr
[𝛼−,𝛼+]
(�) = Apr
[𝛼−
,
𝛼+]
(�) = � ,
Apr
[
𝛼−
,
𝛼+
]
(U
)=
Apr
[
𝛼−
,
𝛼+
]
(U)=U
.
(3) If
⊆
, then
Apr[𝛼−,𝛼+]
�
(
)⊆Apr
[𝛼−,𝛼+]
�
(
)
and
Apr[𝛼−
,
𝛼+]
�
(
)⊆Apr
[𝛼−
,
𝛼+]
�
(
)
.
(4) If
0.5
<[𝛼
−
1
,𝛼
+
1
]
≤
[𝛼
−
2
,𝛼
+
2
]
≤
1
, then
Apr
[
𝛼−
2,
𝛼+
2]
(
)
⊆
Apr[
𝛼−
1,
𝛼+
1]
�
(
)
and
Apr[𝛼−
1,
𝛼+
1
]
�
(
)⊆Apr
[𝛼−
2,
𝛼+
2
]
�
(
)
.
The above two kinds of MG-IVFPR-PRSs,
([𝛼−
,𝛼+],[𝛽−
,𝛽+])
-MG-IVFPR-PRS and
[𝛼−
,
𝛼+]
-MG-
IVFPR-PRS are parameter-related, i.e., all needs to evaluate
values of parameters when applying them. In the follow-
ing, we introduce another two kinds of MG-IVFPR-PRSs,
their parameter free, i.e., which do not have undetermined
parameter.
Denition 13 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
, and
=[A
−
,A
+
](∈ F𝐈
[0,1]
(U
))
be a interval-
valued fuzzy event. The
0.5
-multi-granulation interval-val-
ued fuzzy preference relation probabilistic lower and upper
approximations of
are defined, respectively, as follows:
The pair
(
Apr
0.5
�
(
),Apr
0.5
�
(
))
is called
0.5
-multi-granula-
tion interval-valued fuzzy preference relation probabilistic
rough set (
0.5
-MG-IVFPR-PRS) of
. The positive, nega-
tive, and boundary regions of
are defined, respectively, as
follows:
Apr
0.5
�
(
)={xi∈U(1
≤
i
≤
n
)∶
P(
∣[x
i
]
�
)>
0.5}
,
Apr
0.5
�
(
)={xi∈U(1
≤
i
≤
n
)∶
P(∣[x
i
]�
)≥
0.5}
.
POS
0.5(
)=Apr
0.5
�
(
)
={xi∈U(1≤i≤n
)∶
P(∣[x
i
]�
)>
0.5}
;
NEG
0.5(
)=(Apr
0.5
�
(
))c
={xi∈U(1≤i≤n
)∶
P(
∣[x
i
]
�
)<
0.5}
;
From Definition13, the following assertions are clear.
Proposition 6 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
,∈F
I
[0,1](U)
. The following proper-
ties hold.
(1)
Apr
0.5
�
(
)⊆Apr
0.5
�
(
)
.
(2)
Apr
0.5
�
(�) = Apr
0.5
R(�) = � ;
Apr
0.5
�
(U)=Apr
0.5
�
(U)=U
.
(3) If
⊆
, then
Apr
0.5
�
(
)⊆Apr
0.5
�
(
)
and
Apr
0.5
�
(
)⊆Apr
0.5
�
(
)
.
Denition 14 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
=[
A
−
,A
+](∈
FI
[0,1](
U
))
be a interval-
valued fuzzy event. The Bayesian multi-granulation interval-
valued fuzzy preference relation probabilistic lower and
upper approximations of
are defined, respectively, as
follows:
The pair
(
AprP(
)
(
),Apr
P()
(
))
, is called Bayesian multi-
granulation interval-valued fuzzy preference relation proba-
bilistic rough set (B-MG-IVFPR-PRS) of
. The positive,
negative, and boundary regions of
are defined, respec-
tively, as follows:
BND
0.5(
)=Apr
0.5
�
(
)−Apr
0.5
�
(
)
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
�
)=
0.5}.
AprP()
�
(
)={xi∈U(1
≤
i
≤
n)∶
P(∣[x
i
]�
)>P(}
,
Apr
P()
(
)={xi∈U(1
≤
i
≤
n)∶
P(
∣[x
i
]
)
≥
P(
)}
.
POS
P(
)(
)=Apr
P()
�
(
)
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]�
)>P(
)}
;
Granular Computing
1 3
From Definition14, the following assertions are clear.
Proposition 7 Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-
PAS,
be the GCPR constructed by m IVFPRs
q(1≤q≤m)
and
∈F
I
[0,1](U)
. The following properties
hold.
(1)
Apr
P(
)
�
(
)⊆Apr
P()
�
(
)
.
(2)
AprP()
(�) = Apr
P()
(U
)=�,
Apr
P()
(�) = Apr
P()
(U)=
U
.
3.2 Bayesian decision procedure
withinMG‑IVFPR‑PAS
We mainly discuss, in this section, DTRS approach for
IVFSs in the framework of MG-IVFPR-PAS, i.e., the MG-
IVFPR-DTRS approach.
Let
(U,q(1≤k≤m),P)
be a MG-IVFPR-PAS,
be
the GCPR constructed by m IVFPRs
q(1≤q≤m)
. The
Bayesian decision procedure adopts two states and three
actions to describe the decision process. The set of states
is given by
Ω={,c}
, where
is an IVFS. The set of
actions is
{aP,aN,aB}
, where
aP
,
aN
, and
aB
represent the
three actions in classifying an object, namely, deciding
POS
()
, deciding NEG
()
, and deciding BND
()
, respec-
tively. The interval-valued loss function
𝜆
is given by a
3×2
interval-valued matrix shown in Table1.
The expected losses of taking the individual actions for
element x are computed (expressed) as follows:
NEG
P(
)(
)=(Apr
P()
�
(
))c
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
�
)<P(
)}
;
BND
P(
)(
)=Apr
P()
(
)−AprP(
)
(
)
={xi∈U(1≤i≤n)∶
P(
∣[x
i
]
)=P(
)}.
(27)
P
=(a
P
∣[x
i
]
)
=𝜆
PP
P(
∣[x
i
]
)+𝜆
PN
P(
c∣[x
i
]
)
;
(28)
N
=(
aN
∣[
xi
]
)
=𝜆
NP
P(
∣[x
i
]
)+𝜆
NN
P(
c∣[x
i
]
)
;
(29)
B
=(
aB
∣[
xi
]
)
=𝜆
BP
P(
∣[x
i
]
)+𝜆
BN
P(
c∣[x
i
]
)
.
Note that, for example, due to the interval values of
𝜆PP
and
P(∣[xi]
)
, the product of Eq.(27) is defined by for-
mula(4) and the addition operator,
⊕
, is defined by for-
mula(3). For each element
xi∈U(1≤i≤n)
, the IVFS
[xi]
is adopted as the description of
xi(1≤i≤n)
and defined in
Definition 3.3. The Bayesian decision procedure leads to the
following three minimum-risk decision rules:
(P1) If
P≤
B
and
P≤
N
, then decide
xi∈POS()
.
(N1) If
N≤
P
and
N≤
B
, then decide
xi∈NEG()
.
(B1) For remaining elements
xi∈U(1≤i≤n)
satisfying
neither (P1) nor (N1), decide
xi∈BND()
.
Since the equation
P(∣[xi]
)+P(c∣[xi]
)
=
1
does not
hold in general. Therefore, to simplify the following discus-
sion, we denote
Then, by Proposition2 and Remark1, we have
As a result, the losses for taking actions
aP
,
aB
and
aN
can
be expressed as
From Eqs.(32)–(34), we have the following equivalences:
(1)
(
a
P∣[
x
i]
)
≤
(
a
B∣[
x
i]
)
⇔
(𝜆−
BP −𝜆−
PP)(
P1
∧
p2
)
≥(𝜆−
PN −𝜆−
BN )(1−p
3
∨p
4
)
,
(
𝜆
+
BP
−𝜆
+
PP
)(P1∨p2)
≥
(𝜆−
PN −𝜆−
BN )(1−p
3
∧p
4
)
;
(2)
(aP∣[xi]
)≤(aN∣[xi]
)
⇔
(𝜆−
NP −𝜆−
PP)(P
1
∧p
2
)
≥(𝜆−
PN −𝜆−
NN )(1−p
3
∨p
4
)
,
(
𝜆
+
NP
−𝜆
+
PP
)(P1∨p2)
≥
(𝜆−
PN −𝜆−
NN )(1−p
3
∧p
4
)
;
P
(A
−
∣[xi]
−)=p1,P(A
+
∣[xi]
+)=p2
,
P
(A+∣[x
i
]
−)=p
3
,P(A−∣[x
i
]
+)=p
4.
(30)
P(∣[xi]
)=[p1∧p2,p1∨p2],
(31)
P(c∣[xi]
)=[1−p3∨p4,1−p3∧p4].
(32)
(a
P
∣[x
i
]
)
=[𝜆−
PP(P1∧p2)+𝜆−
PN (1−p3∨p4)
,
𝜆+
PP
(P
1
∨p
2
)+𝜆+
PN
(1−p
3
∧p
4
)]
;
(33)
(a
N
∣[x
i
]
)
=[𝜆−
NP(P1∧p2)+𝜆−
NN (1−p3∨p4)
,
𝜆+
NP
(P
1
∨p
2
)+𝜆+
NN
(1−p
3
∧p
4
)]
;
(34)
(a
B
∣[x
i
]
)
=[𝜆−
BP(P1∧p2)+𝜆−
BN (1−p3∨p4)
,
𝜆+
BP
(P
1
∨p
2
)+𝜆+
BN
(1−p
3
∧p
4
)]
.
Granular Computing
1 3
(3)
(aN∣[xi]
)≤(aP∣[xi]
)
⇔
(𝜆−
NP −𝜆−
PP)(P
1
∧p
2
)
≥
(𝜆
−
PN
−𝜆
−
NN
)(1−p
3
∨p
4
)
,
(
𝜆
+
NP
−𝜆
+
PP
)(P1∨p2)
≥
(𝜆−
PN −𝜆−
NN )(1−p
3
∧p
4
);
(4)
(
a
N∣[
x
i]
)
≤
(
a
B∣[
x
i]
)
⇔
(𝜆−
NP −𝜆−
BP)(
P1
∧
p2
)
≥(𝜆−
BN −𝜆−
NN )(1−p
3
∨p
4
),
(
𝜆
+
NP
−𝜆
+
BP
)(P1∨p2)
≥
(
𝜆
−
BN
−𝜆
−
NN
)(1−p3∧p4)
.
Consider a special kind of loss function:
That is, the loss of classifying an object
xi
in state
into the
positive region POS
()
is less than or equal to the loss of
classifying
xi(1≤i≤n)
into the boundary region BND
()
,
and both of these losses are less than the loss of classifying
xi(1≤i≤n)
into the negative region NEG
()
. The reverse
order of losses is used for classifying an object that does not
in state
.
From Eq.(35) that
(35)
𝜆∗
PP
<𝜆
∗
BP
<𝜆
∗
NP
,
𝜆
∗
NN
<𝜆
∗
BN
<𝜆
∗
PN
,(∗= −,+)
.
(a
P
∣[x
i
]
)≤(a
B
∣[x
i
]
)
⇔p1∧p2≥
𝜆−
PN −𝜆−
BN
𝜆−
BP −𝜆−
PP
(1−p3∨p4)
,
p1∨p2≥
𝜆+
PN −𝜆+
BN
𝜆+
BP
−𝜆+
PP
(1−p3∧p4);
(a
P
∣[x
i
]
)≤(a
N
∣[x
i
]
)
⇔p1∧p2≥
𝜆−
PN −𝜆−
NN
𝜆−
NP −𝜆−
PP
(1−p3∨p4)
,
p1∨p2≥
𝜆+
PN −𝜆+
NN
𝜆+
NP
−𝜆+
PP
(1−p3∧p4);
(a
N
∣[x
i
]
)≤(a
P
∣[x
i
]
)
⇔p1∧p2≤
𝜆−
PN −𝜆−
NN
𝜆−
NP −𝜆−
PP
(1−p3∨p4)
,
p1∨p2≤
𝜆+
PN −𝜆+
NN
𝜆+
NP
−𝜆+
PP
(1−p3∧p4);
Denote parameters
Then, we have the following:
According to condition(35), decision rules (P1)–(N1) can be
equivalently rewritten as follows based on aforementioned
analyses:
(P2) I f
P(∣[
x
i]
)≥[𝛼−
,
𝛼+]
P
(c∣[
x
i]
)
and
P(∣[xi]
)≥[𝛾−
,𝛾+]P(c∣[xi]
)
, then decide
xi∈POS()
;
(a
N
∣[x
i
]
)≤(a
B
∣[x
i
]
)
⇔p1∧p2≤
𝜆−
BN −𝜆−
NN
𝜆−
NP −𝜆−
BP
(1−p3∨p4)
,
p1∨p2≤
𝜆+
BN −𝜆+
NN
𝜆+
NP
−𝜆+
BP
(1−p3∧p4).
(36)
𝛼
−=
𝜆−
PN −
𝜆−
BN
𝜆−
BP
−𝜆−
PP
,𝛼+=
𝜆+
PN −
𝜆+
BN
𝜆+
BP
−𝜆+
PP
,
(37)
𝛾
−=
𝜆−
PN −
𝜆−
NN
𝜆−
NP
−𝜆−
PP
,𝛾+=
𝜆+
PN −
𝜆+
NN
𝜆+
NP
−𝜆+
PP
,
(38)
𝛽
−=
𝜆−
BN −
𝜆−
NN
𝜆−
NP
−𝜆−
BP
,𝛽+=
𝜆+
BN −
𝜆+
NN
𝜆+
NP
−𝜆+
BP
.
(a
P
∣[x
i
]
)≤(a
B
∣[x
i
]
)
⇔P(∣[x
i
]
)≥[𝛼−
,𝛼+]P(c∣[x
i
]
)
,
(a
P
∣[x
i
]
)≤(a
N
∣[x
i
]
)
⇔P(∣[x
i
]
)≥[𝛾−
,𝛾+]P(c∣[x
i
]
)
,
(a
N
∣[x
i
]
)≤(a
P
∣[x
i
]
)
⇔P(
∣[x
i
]
)
≤
[𝛾−
,𝛾+]P(
c∣[x
i
]
)
,
(a
N
∣[x
i
]
)≤(a
B
∣[x
i
]
)
⇔P(
∣[x
i
]
)
≤
[𝛽−
,𝛽+]P(
c∣[x
i
]
)
.
Table 1 Interval-valued loss
function,
[𝜆]
, for IVFS
: positive
c
: negative
aP
: accept
𝜆
PP
=
𝜆
(
aP
∣
)=[
𝜆
−
PP
,𝜆
+
PP]
𝜆
PN
=
𝜆
(
aP
∣
c)=[
𝜆
−
PN
,𝜆
+
PN ]
aN
: reject
𝜆
NP
=
𝜆
(
aN
∣
)=[
𝜆
−
NP,
𝜆
+
NP]
𝜆
NN
=
𝜆
(
aN
∣
c)=[
𝜆
−
NN ,
𝜆
+
NN ]
aB
: defer
𝜆
BP =𝜆(aB∣)=[𝜆
−
BP
,𝜆
+
BP]
𝜆
BN =𝜆(aB∣
c
)=[𝜆
−
BN
,𝜆
+
BN ]
Granular Computing
1 3
(N2) I f
P(∣[
x
i]
)≤[𝛾−
,
𝛾+]
P
(c∣[
x
i]
)
and
P(∣[xi]
)≤[𝛽−
,𝛽+]P(c∣[xi]
)
, then decide
xi∈NEG()
;
(B2) For remaining elements
xi∈U(1≤i≤n)
satisfying
neither (P2) nor (N2), decide
xi∈BND()
.
If the loss function still satisfies the following conditions:
then it follows that
𝛼∗>𝛾
∗>𝛽
∗
. For the same time, we have
Thus, the decision rules (P2)–(N2) can be obtained under
conditions(35) and (39):
(P3) If
P(∣[
x
i]
)≥[𝛼−
,
𝛼+]
P
(c∣[
x
i]
)
, then decide
xi∈POS()
;
(N3) If
P(∣[
x
i]
)≤[𝛽−
,
𝛽+]
P
(c∣[
x
i]
)
, then decide
xi∈NEG()
;
(B3) For remaining elements
xi∈U(1≤i≤n)
satisfying
neither (P3) nor (N3), decide
xi∈BND()
.
Let us consider another condition of loss function:
If the loss function satisfies conditions(35) and (40), it then
follows that
𝛼∗=𝛾∗=𝛽∗
. Thus, we have
When the losses of classifying
xi
into
POS()
and
BND()
are the same, we decide
xi∈BND()
. Similarly, when
the losses of classifying
xi
into
NEG()
and BND
()
are
the same, we decide
xi∈BND
()
. According to this tie-
breaking criteria and the equivalences(41) and (42), the
(39)
𝜆∗
PN
−𝜆∗
BN
𝜆
∗
BP
−𝜆∗
PP
>
𝜆∗
BN
−𝜆∗
NN
𝜆∗
NP
−𝜆∗
BP
,(∗= −,+)
,
P
(
∣[xi]
)
≥
[
𝛼−
,
𝛼+
]P(
c
∣[xi]
)
⇒P(
∣[x
i
]
)
≥
[𝛾−
,𝛾+]P(
c∣[x
i
]
)
,
P(∣[
xi
]
)≤[𝛽−
,
𝛽+]
P
(c∣[
xi
]
)
⇒P(
∣[x
i
]
)
≤
[𝛾−
,𝛾+]P(
c∣[x
i
]
)
.
(40)
𝜆∗
PN
−𝜆∗
BN
𝜆
∗
BP
−𝜆∗
PP
=
𝜆∗
BN
−𝜆∗
NN
𝜆∗
NP
−𝜆∗
BP
,(∗= −,+)
.
(41)
P
(
∣[xi]
)
≥
[
𝛼−
,
𝛼+
]P(
c
∣[xi]
)
⇔P(
∣[x
i
]
)
≥
[𝛾−
,𝛾+]P(
c∣[x
i
]
)
,
(42)
P
(
∣[xi]
)
≤
[
𝛾−
,
𝛾+
]P(
c
∣[xi]
)
⇔P(∣[xi]
)≤[𝛽−
,𝛽+]P(c∣[xi]
)
⇔P(
∣[x
i
]
)
≤
[𝛼−
,𝛼+]P(
c∣[x
i
]
)
.
following simplified and equivalent forms of decision rules
(P2)–(N2) are obtained:
(P4) If
P(∣[
x
i]�
)>[𝛼−
,
𝛼+]
, then decide
xi∈POS()
;
(N4) If
P(∣[xi]�
)<[𝛼−
,𝛼+]
, then decide
xi∈NEG()
;
(B4) For remaining elements
xi∈U(1≤i≤n)
satisfying
neither (P4) nor (N4), decide
xi∈BND()
.
Remark 2 Since the order relation defined in Eq.(6) is par-
tial. Therefore, the situation in some cases must be arise two
intervals cannot compare directly. In this cases, we adopt the
comparing methods as given in Liang and Liu (2014). The
method is described as follows:
Let
[a−
,a+],[b−
,b+]∈𝐈ℝ+
and
𝜃∈[0, 1]
is constant.
If
M𝜃([a−
,a+]) ≤M𝜃([b−
,b+])
, then
[a−
,a+]≤[b−
,b+]
,
and vice versa, where
M𝜃([a−,a+])=(1−𝜃)a−+𝜃a+
and
M𝜃([b−
,b+]
)=(1−𝜃)b−+𝜃b+
. Where
M𝜃
is a transformed
outcome and
𝜃
reflects the risk attitude of decision maker.
The above compare method is applied only for those situ-
ations when the two intervals cannot be compare directly.
4 An approach togroup decision‑making
based onMG‑IVFPR‑PRS
Based on the MG-IVFPR-PRS model in Sect.3, it is requi-
site to consider their applications in group decision-making
problems. In what follows, we give the algorithm to solve
the group decision-making problem with MG-IVFPR-PRS.
4.1 An algorithm
With the help of the results in Sect.3, we design the algo-
rithm of group decision-making based on the MG-IVFPR-
PRS model and their corresponding 3WDs. The key steps
are elaborated as follows:
Step 1: Suppose that a group decision-making problem
has a set of alternatives
U={x1,x2,…,xn}.
Assume that
there are m decision makers
e1,e2,…,em
provides m IVF-
PRs
1,2,…,m
to evaluate his/her judgments on U.
Step 2: Using Algorithm1, to construct GCPR
for all
decision makers.
Step 3: Presenting the values of
is the incomplete
available information of all the characteristic factors
xi(i=1, 2, …,n)
.
Step 4: Presenting the values of
P(xi)=pi(i=1, 2, …,n)
is the possible existence of random available information of
all of
xi(i=1, 2, …,n)
.
Step 5: Based on Eq.(24), computing the conditional
probability
P(∣[xi]
)
(i=1, 2, …,n)
.
Granular Computing
1 3
Step 6: Based on Eq.(36), computing the conditional
probability
P(c∣[xi]
)
(i=1, 2, …,n)
.
Step 7: Presenting the values of loss function
𝜆Δ∇
(Δ = P,B,N;∇=P,N)
according to Table1.
Step 8: If the values of loss functions satisfy the condi-
tions (35) and (39), then go to Step 9. Otherwise, we need
to assign a new values of loss functions and go to Step 5.
Step 9: Based on Eqs.(36)–(38), calculate the thresholds
[𝛼−
,𝛼+]
,
[𝛾−
,𝛾+]
and
[𝛽−
,𝛽+]
.
Step 10: Using Eq.(4), computing
[𝛼−
,
𝛼+]
P
(c∣[
x
i]
)
(i=1, 2, …,n)
and
[𝛽−
,𝛽+]P(c∣[xi]
)
(i=1, 2, …,n)
.
Step 11: If
P(∣[xi]
)
(i=1, 2, …,n)
com-
pare directly
[𝛼−
,𝛼+]P(c∣[xi]
)
(i=1, 2, …,n)
and
[𝛽−
,
𝛽+]
P
(c∣[
x
i]
)
(i=1, 2, …,n)
, then go to Step 13.
Otherwise, go to Step 12.
Step 12: Based on Remark2, transform
P(∣[xi]
)
(i=1, 2, …,n)
,
[𝛼−
,𝛼+]P(c∣[xi]
)
(i=1, 2, …,n)
and
[𝛽−
,
𝛽+]
P
(c∣[
x
i]
)
(i=1, 2, …,n)
to
M𝜃(P(∣[xi]
))
(i=1, 2, …,n)
,
M𝜃([𝛼−
,𝛼+]P(c∣[xi]
))
(i=1, 2, …,n)
and
M𝜃([𝛽−
,
𝛽+]
P
(c∣[
x
i]
))
(i=1, 2, …,n)
with a certain
value of
𝜃∈[0, 1]
. Then, go to Step 12.
Step 13: Making the decision according to the decision
rules (P2)–(N2).
4.2 An illustrative example
In this subsection, we apply the propose algorithm to a
real group decision-making. This example is about quick
decision-making based on a real investment context, under
MG-IVFPR-PRS environment. Furthermore, analysis is
done to provide to show the feasibility and reasonableness
of the proposed models.
4.2.1 Problem description
The various types of mutual funds (MFs) of different com-
panies listed in the Growth Enterprise Market board of the
India Stock Exchange are a popular investment source to
an investor as a long-term investment. However, the suffi-
cient knowledge about the various types of MFs of different
companies is always not possible for every investor. Our
proposed models are effective for those investors. Suppose
an investor plans to invest his/her money in MFs of differ-
ent companies, with the aim of high returns, while he/she
has no sufficient knowledge about all MFs. He/she chooses
initially five MFs according to the past performances, while
he/she invests his/her money to the best options out of these
five MFs. For this, he/she decides to take advice’s from
three stock market brokers in India. For making reasonable
options, we have the following decision analysis.
4.2.2 Decision analysis
We use the algorithm in Sect.4.1 of decision analysis based
on MG-IVFPR-PRSs for group decision-making.
Step 1: Suppose
U={x1,x2,x3,x4,x5}
be the five MFs
and
E={e1,e2,e3}
be the three brokers. Assume that these
three brokers provide his/her judgments using IVFPRs,
which represent in Eqs.(43)–(45).
R
1(xi,x
j)=
R
1(
xi,x
j
)
x1x2x3x4x5
x1[0.5,0.5] [0.6,0.8] [0.7,1] [0.2,0.3] [0.4,0.5]
x2[0.2,0.4] [0.5,0.5] [0.4,0.6] [0.7,0.8] [0.3,0.5]
x3[0,0.3] [0.4,0.6] [0.5,0.5] [0.6,0.9] [0.4,0.7]
x4[0.7,0.8] [0.2,0.3] [0.1,0.4] [0.5,0.5] [0.3,0.4]
x5[0.5,0.6] [0.5,0.7] [0.3,0.6] [0.6,0.7] [0.5,0.5]
,
(43)
R
2(xi,x
j)=
R
2(
xi,x
j
)
x1x2x3x4x5
x1[0.5,0.5] [0.5,0.7] [0.8,0.9] [0.3,0.5] [0.3,0.6]
x2[0.3,0.5] [0.5,0.5] [0.6,0.7] [0.5,0.6] [0.4,0.5]
x30.1,0.2] [0.3,0.4] [0.5,0.5] 0.7,0.9] [0.6,0.7]
x4[0.5,0.7] [0.4,0.5] [0.1,0.3] [0.5,0.5] [0.5,0.6]
x5[0.4,0.7] [0.5,0.6] [0.3,0.4] [0.4,0.5] [0.5,0.5]
,
(44)
R
3(xi,x
j)=
R
3(
xi,x
j
)
x1x2x3x4x5
x1[0.5,0.5] [0.7,0.9] [0.8,1] [0.4,0.5] [0.3,0.4]
x2[0.1,0.3] [0.5,0.5] [0.6,0.7] [0.4,0.7] [0.4,0.6]
x3[0,0.2] [0.3,0.4] [0.5,0.5] [0.7,0.8] [0.5,0.8]
x4[0.5,0.6] [0.3,0.6] [0.2,0.3] [0.5,0.5] [0.4,0.7]
x5[0.6,0.7] [0.4,0.6] [0.2,0.5] [0.3,0.6] [0.5,0.5]
.
(45)
Granular Computing
1 3
Step 2: Using Algorithm1, to construct GCPR
for all
brokers, which represent in Eq.(46).
Step 3: Let IVFS
be the quantitative description of all the characteristic factors
xi(i=1, 2, 3, 4, 5)
according to the available inaccurate and
insufficient information.
Step 4: Let
{0.15, 0.13, 0.24, 0.18, 0.3}
be the pos-
sible existence of random available information of
xi(i=1, 2, 3, 4, 5).
Step 5: Using Eq.(24), we compute the conditional prob-
ability
P(∣[xi]
)
(i=1, 2, 3, 4, 5)
as follows:
Step 6: Using Eq.(36), we compute the conditional prob-
ability
P(c∣[xi]
)
(i=1, 2, 3, 4, 5)
as follows:
Step 7: Suppose that the loss function is given as fol-
lows:
𝜆PP =[1, 2]
,
𝜆BP =[3, 5]
,
𝜆NP =[5, 8]
,
𝜆NN =[0, 1]
,
𝜆BN =[5, 7.6]
, and
𝜆PN =[11, 15.1]
.
Step 8: Since the given loss functions satisfy the condi-
tions (35) and (39).
Step 9: Based on Eqs. (36)–(38), we compute the thresh-
olds values:
[𝛼−,𝛼+]=[3
, 2.5],
[𝛾−,𝛾+]=[2.75, 2.35]
, and
[𝛽−,𝛽+]=[2.5, 2.2]
.
R
(xi,x
j)=
R(xi,x
j)x1x2x3x4x5
x1[0.5,0.5] [0.5873,0.6873][0.6768,0.7761][0.4613,0.5286][0.4499,0.5344]
x2[0.3127,0.4127] [0.5,0.5] [0.5053,0.5718][0.5402,0.6226][0.4268,0.5095]
x3[0.2239,0.3232] [0.4282,0.4947][0.5,0.5] [0.6018,0.7021][0.489,0.6045]
x4[0.4714,0.5387] [0.3774,0.4598][0.2979,0.3982][0.5,0.5] [0.4207,0.5031]
x
5
[0.4656,0.5501] [0.4905,0.5732][0.3955,0.511] [0.4969,0.5793][0.5,0.5]
.
(46)
=
[0.75, 0.85]
x1
+
[0.6, 0.65]
x2
+
[0.85, 0.9]
x3
+[0.25, 0.35]
x4
+[0.8, 0.9]
x5
P(∣[x
1
]
)=[0.6936, 0.7706],
P
(∣[x2]
)=[0.6631, 0.7469]
,
P
(∣[x3]
)=[0.657, 0.7372],
P
(∣[x4]
)=[0.6545, 0.7565]
,
P
(
∣[x
5
]
)=[0.6703, 0.75].
P(c∣[x
1
]
)=[0.2288, 0.3066],
P
(c∣[x2]
)=[0.2572, 0.334],
P
(c∣[x3]
)=[0.2619, 0.3458]
,
P
(c∣[x4]
)=[0.2603, 0.3273]
,
P
(
c∣[x
5
]
)=[0.2466, 0.3316]
.
Step 10: Based on Eq. (4), we com-
pute
[𝛼−
,
𝛼+]
P
(c∣[
x
i]
)
(i=1, 2, 3, 4, 5)
and
[𝛽−
,𝛽+]P(c∣[xi]
)
(i=1, 2, 3, 4, 5)
as follows:
and
Step 11: Since
P(∣[xi]
)
cannot be compared directly
[𝛼−
,
𝛼+]
P
(c∣[
x
i]
)
and
[𝛽−
,
𝛽+]
P
(c∣[
x
i]
)
for all
(i=1, 2, 3, 4, 5)
. Then, go to Step 12.
Step 12: Based on Remark2, we transform
P(∣[xi]
)
,
[𝛼−
,𝛼+]P(∣[xi]
)
, and
[𝛽−
,𝛽+]P(∣[xi]
)
with different
values of
𝜃
. The results are shown in Figs.2,3, and4.
Step 13: According to the decision rules (P2)–(N2), the
decision results are shown in Table2 with different values
of
𝜃
.
From Table2, we can see that the smaller value of
𝜃
, the
decision results may be different for the higher value of
𝜃
.
However, the decision results are unchanged from certain
stage with the increasing of
𝜃
. For clarity, we display the
decision results in Fig.8. In this paper, we suggest that an
appropriate value of
𝜃
is between 0.5 and 1 due to the fact
that it can get an appropriate decision results.
4.2.3 The influences oftheparameter
𝜃
During the decision analysis of group decision-making
based on MG-IVFPR-PRSs, it involves the parameter
𝜃
. In
what follows, we successively analyze this parameter to the
selection of MFs in details.
[
𝛼
−
,𝛼
+
]P(
c
∣[x1]
)=[0.6865, 7666],
[
𝛼−,𝛼+]P(c∣[x2]
)=[0.7716, 0.8349]
,
[
𝛼−,𝛼+]P(c∣[x3]
)=[0.7858, 0.8644]
,
[
𝛼−,𝛼+]P(c∣[x4]
)=[0.7808, 0.8182]
,
[
𝛼−,𝛼+]P(c∣[x
5
]
)=[0.7399, 0.8289]
,
[𝛽−
,
𝛽+
]P(
c
∣[x1]
)=[0.5721, 0.6746]
,
[
𝛽−,𝛽+]P(c∣[x2]
)=[0.6430, 0.7347]
,
[
𝛽−,𝛽+]P(c∣[x3]
)=[0.6548, 0.7607]
,
[
𝛽−,𝛽+]P(c∣[x4]
)=[0.6506, 0.7200]
,
[
𝛽−,𝛽+]P(
c∣[x
5
]
)=[0.6166, 0.7294]
.
Granular Computing
1 3
1. The transform outcomes values of
P(∣[xi]
)
,
[𝛼−
,𝛼+]P(∣[xi]
)
, and
[𝛽−
,𝛽+]P(∣[xi]
)
with dif-
ferent values of
𝜃
. Based on Remark2, the transform
outcomes of
P(∣[xi]
)
,
[𝛼−
,𝛼+]P(∣[xi]
)
, and
[𝛽−
,
𝛽+]
P
(∣[
x
i]
)
with different values of
𝜃
are fur-
ther discussed in Figs.5, 6, and 7. With regard to the
results of Figs.5, 6, and 7, we find that the transform
outcomes of
P(∣[xi]
)
,
[𝛼−
,𝛼+]P(∣[xi]
)
, and
[𝛽−
,
𝛽+]
P
(∣[
x
i]
)
are increasing with the increasing
of
𝜃
.
Fig. 2 Transform outcomes of
P(∣[x
i
]
)
with different values of
𝜃
Fig. 3 Transform outcomes of
[𝛼−
,
𝛼+]
P
(∣[
xi
]
)
with different val-
ues of
𝜃
Fig. 4 Transform outcomes of
[𝛽−
,
𝛽+]
P
(∣[
xi
]
)
with different val-
ues of
𝜃
Table 2 Decision results for different risk attitude of decision maker
𝜃
Positive region Negative region Boundary region
0
x1
∅
x2,x3,x4,x5
0.1
x1
∅
x2,x3,x4,x5
0.2
x1
x3
x2,x4,x5
0.3
x1
x3
x2,x4,x5
0.4
x1
x3
x2,x4,x5
0.5
x1
x3
x2,x4,x5
0.6
x1
x3
x2,x4,x5
0.7
x1
x3
x2,x4,x5
0.8
x1
x3
x2,x4,x5
0.9
x1
x3
x2,x4,x5
1
x1
x3
x2,x4,x5
Granular Computing
1 3
2. The decision rules with different values of
𝜃
Continu-
ing the discussion of Figs.5, 6, and 7, we deduce the
decision rules with different values of
𝜃
. On the basis of
the condition
[𝛼−
,𝛼+]>[𝛽−
,𝛽+]>[𝛾−
,𝛾+]
, we gener-
ate decision rules (P2)–(N2), i.e., POS
()
, NEG
()
and
BND
()
. With respect to (P2)–(N2), the three decision
regions rely on the values of
[𝛼−
,
𝛼+]
P
(∣[
x
i]
)
and
Fig. 5 Transform outcomes of
P(∣[x
i
]
)
with different values of
𝜃
Fig. 6 Transform outcomes of
[𝛼−
,
𝛼+]P(∣[x
i
]
)
with different values of
𝜃
.
Granular Computing
1 3
[𝛽−
,𝛽+]P(∣[xi]
)
. Hence, the decision rules with dif-
ferent values of
𝜃
are described in Fig.8.
4.3 Comparisons oftheproposed model andother
existing models ingroup decision‑making
problems
1. Comparison with IVF-PRS model The IVF-PRS model
in Zhao and Hu (2016) is established based on single
interval-valued fuzzy relation (IVFR), so the model in
Zhao and Hu (2016) cannot deal with group decision-
making problems with preference analysis. The MG-
IVFPR-PRS model proposed in the present paper is
based on multiple IVFPRs, which can deal group deci-
sion-making problems with preference analysis. Hence,
the application domain of the MG-IVFPR-PRS model is
wider than that of the IVF-PRS model in Zhao and Hu
(2016).
Fig. 7 Transform outcomes of
[𝛽−
,
𝛽+]
P
(∣[
xi
]
)
with different values of
𝜃
Fig. 8 Decision rules with dif-
ferent values of
𝜃
Granular Computing
1 3
2. Comparison with IVF-DTRS approach The interval-
valued decision-theoretic rough set approach in Liang
and Liu (2014) is established based on single classical
equivalence relation and consider that only the loss func-
tion is interval-valued. The IVF-DTRS approach in Zhao
and Hu (2016) is established based on single IVFR and
also consider that the loss function is interval-valued.
However, these approaches cannot be dealt with group
decision-making with preference analysis. Our proposed
approaches can do it.
3. Comparison with group decision-making method based
on IVFPRs The method for group decision-making in
Chen etal. (2015) is established on IVFPRs and consist-
ency matrices. Using this model, the decision results
are obtained only on the basis of experts’ suggestions
which cannot consider the incomplete available informa-
tion and the existence of random available information.
However, we cannot avoid it for obtaining more accurate
decision results. If we consider these two types available
information, the obtained decision results may be differ-
ent. For instance, if we obtain the decision results to pro-
pose example, using the method in Chen etal. (2015),
the best option is
x1
and the worst option is
x4
. However,
if we use the method proposed in the present paper, the
best option is
x1
(positive region) and the worst option
is
x3
(negative region).
5 Conclusion
This paper investigates MG-IVFPR-PRS models within
the frameworks of MG-IVFPR-PAS and GCPR. Using this
model, we have presented an approach for group decision-
making, which is basis of the experts’ suggestions, the
incomplete available information, and the existence of
random available information of the objects. The proposed
method provides us with a useful way for group decision-
making using MG-IVFPR-PRS model based on IVFPRs and
consistency matrices.
In this paper, we cannot consider the mathematical way to
find the optimal value of the parameter
𝜃
; however, it plays
important role for the decision analysis of group decision-
making based on MG-IVFPR-PRSs. To address this issue,
we use optimization techniques as given by Tsai etal. (2008,
2012), Chen and Huang (2003), Chen and Chung (2006) and
Chen and Chien (2011) in our future concerns.
Acknowledgements The authors would like to thank the Editor-
in-Chief and reviewers for their thoughtful comments and valuable
suggestions.
Compliance with ethical standards
Conflict of interest Prasenjit Mandal and A. S. Ranadive declare that
there is no conflict of interest.
Ethical approval This article does not contain any study performed on
humans or animals by the authors.
Informed consent Informed consent was obtained from all individual
participants included in the study.
References
Chen SM (1996) A fuzzy reasoning approach for rule-based systems
based on fuzzy logics. IEEE Trans Systems Man Cybern Part B
Cybern 26(5):769–778
Chen SM (1997) Interval-valued fuzzy hypergraph and fuzzy partition.
IEEE Trans Systems Man Cybern Part B Cybern 27(4):725–733
Chen SJ, Chen SM (2008) Fuzzy risk analysis based on measures of
similarity between interval-valued fuzzy numbers. Comput Math
Appl 55:1670–1685
Chen SM, Chen JH (2009) Fuzzy risk analysis based on similarity
measures between interval-valued fuzzy numbers and interval-
valued fuzzy number arithmetic operators. Expert Syst Appl
36(3):6309–6317
Chen SM, Chen CD (2011) Handling forecasting problems based
on high-order fuzzy logical relationships. Expert Syst Appl
38(4):3857–3864
Chen SM, Chien CY (2011) Parallelized genetic ant colony systems
for solving the traveling salesman problem. Expert Syst Appl
38(4):3873–3883
Chen SM, Chung NY (2006) Forecasting enrollments of students using
fuzzy time series and genetic algorithms. Int J Inf Manag Sci
17(3):1–17
Chen SM, Hong JA (2014) Multicriteria linguistic decision making
based on hesitant fuzzy linguistic term sets and the aggregation
of fuzzy sets. Inf Sci 286:63–74
Chen SM, Horng YJ (1996) Finding inheritance hierarchies in interval-
valued fuzzy concept-networks. Fuzzy Sets Syst 84:75–83
Chen SM, Hsiao WH (2000) Bidirectional approximate reasoning for
rule-based systems using interval-valued fuzzy sets. Fuzzy Sets
Syst 113:185–203
Chen SM, Huang CM (2003) Generating weighted fuzzy rules from
relational database systems for estimating values using genetic
algorithms. IEEE Trans Fuzzy Syst 11(4):495–506
Chen SM, Kao PY (2013) Taiex forecasting based on fuzzy time
series, particle swarm optimization techniques and support vec-
tor machines. Inf Sci 247:62–71
Chen SM, Lee LW (2010) Fuzzy multiple criteria hierarchical group
decision-making based on interval type-2 fuzzy sets. IEEE Trans
Syst Man Cybern Part A Syst Hum 40(5):1120–1128
Chen SM, Niou SJ (2011) Fuzzy multiple attributes group decision-
making based on fuzzy preference relations. Expert Syst Appl
38(4):3865–3872
Chen SM, Sanguansat K (2011) Analyzing fuzzy risk based on similar-
ity measures between interval-valued fuzzy numbers. Expert Syst
Appl 38(7):8612–8621
Chen SM, Wang HY (2009) Evaluating students’ answerscripts
based on interval-valued fuzzy grade sheets. Expert Syst Appl
36(6):9839–9846
Granular Computing
1 3
Chen SM, Hsiao WH, Jong WT (1997) Bidirectional approximate
reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst
91:339–353
Chen SM, Wang NY, Pan JS (2009) Forecasting enrollments using
automatic clustering techniques and fuzzy logical relationships.
Expert Syst Appl 36(8):11,070–11,076
Chen SM, Lin TE, Lee LW (2014) Group decision making using
incomplete fuzzy preference relations based on the additive con-
sistency and the order consistency. Inf Sci 259:1–15
Chen SM, Cheng SH, Lin TE (2015) Group decision making systems
using group recommendations based on interval fuzzy preference
relations and consistency matrices. Inf Sci 298:555–567
Du WS, Hu BQ (2016) Dominance-based rough set approach to incom-
plete ordered information systems. Inf Sci 346:106–129
Feng T, Mi JS (2016) Variable precision multigranulation decision-
theoretic fuzzy rough sets. Knowl Based Syst 91:93–101
Horng YJ, Chen SM, Chang YC, Lee CH (2005) A new method for
fuzzy information retrieval based on fuzzy hierarchical clus-
tering and fuzzy inference techniques. IEEE Trans Fuzzy Syst
13(2):216–228
Lee LW, Chen SM (2008) Fuzzy multiple attributes group decision-
making based on the extension of TOPSIS method and interval
type-2 fuzzy sets. In: Proceedings of the 2008 international con-
ference on machine learning and cybernetics, Kunming, China,
vol 6, pp 3260–3265
Lee LW (2012) Group decision making with incomplete fuzzy prefer-
ence relations based on the additive consistency and the order
consistency. Expert Syst Appl 39(14):11,666–11,676
Liang D, Liu D (2014) Systematic studies on three-way decisions with
interval-valued decision-theoretic rough sets. Inf Sci 276:186–203
Liang DC, Liu D (2015) Deriving three-way decisions from intuition-
istic fuzzy decision-theoretic rough sets. Inf Sci 300:28–48
Liang JY, Li R, Qian YH (2012) Distance: a more comprehensible
perspective for measures in rough set theory. Knowl Based Syst
27:126–136
Lin G, Qian Y, Liang J (2012) Nmgrs: neighborhood-based multigranu-
lation rough sets. Int J Approx Reason 53:1404–1418
Liu C, Miao D, Qian J (2014) On multi-granulation covering rough
sets. Int J Approx Reason 55:1404–1418
Liu D, Liang DC, Wang CC (2016) A novel three-way decision model
based on incomplete information system. Knowl Based Syst
91:32–45
Liu F, Peng YN, Yu Q, Zhao H (2018) A decision-making model based
on interval additive reciprocal matrices with additive approxima-
tion-consistency. Inf Sci 422:161–176
Lv Y, Chen Q, Wu L (2013) Multi-granulation probabilistic rough set
model. Int Conf Fuzzy Syst Knowl Discov 10:146–151
Ma W, Sun B (2012) Probabilistic rough set over two universes and
rough entropy. Int J Approx Reason 53:608–619
Mandal P, Ranadive AS (2017) Multi-granulation bipolar-valued fuzzy
probabilistic rough sets and their corresponding three-way deci-
sions over two universes. Soft Comput. https ://doi.org/10.1007/
s0050 0-017-2765-6
Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems
made simple. IEEE Trans Fuzzy Syst 14(6):808–821
Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356
Pawlak Z, Wong SKW, Ziarko W (1988) Rough sets: probabilistic ver-
sus deterministic approach. Int J Man Mach Stud 29:81–95
Pedrycz W, Chen SM (2011) Granular computing and intelligent sys-
tems: design with information granules of high order and high
type. Springer, Heidelberg
Pedrycz W, Chen SM (2015a) Granular computing and decision-mak-
ing: interactive and iterative approaches. Springer, Heidelberg
Pedrycz W, Chen SM (2015b) Information granularity, big data, and
computational intelligence. Springer, Heidelberg
Polkowski L (1996) Rough mereology: a new paradigm for approxi-
mate reasoning. Int J Approx Reason 15:333–365
Qian Y, Liang J, Dang C (2010a) Incomplete multigranulation rough
set. IEEE Trans Syst Man Cybern 20:420–431
Qian YH, Liang JY, Yao YY, Dang CY (2010b) MGRS: a multigranu-
lation rough set. Inf Sci 180:949–970
Qian YH, Zhang H, Sang YL, Liang JY (2014) Multigranulation deci-
sion-theoretic rough sets. Int J Approx Reason 55:225–237
Sun B, Ma W, Zhao H (2014) Decision-theoretic rough fuzzy set model
and application. Inf Sci 283:180–196
Tanio T (1984) Fuzzy preference orderings in group decision making.
Fuzzy Sets Syst 12:117–131
Tsai PW, Pan JS, Chen SM, Liao BY, Hao SP (2008) Parallel cat swarm
optimization. In: Proceedings of the 2008 international conference
on machine learning and cybernetics, Kunming, China, vol 6, pp
3328–3333
Tsai PW, Pan JS, Chen SM, Liao BY (2012) Enhanced parallel cat
swarm optimization based on the taguchi method. Expert Syst
Appl 39(7):6309–6319
Wang JY, Xu LM (2002) Probabilistic rough set model. Comput Sci
29(8):76–78
Wang ZH, Wang H, Feng QR, Shu L (2015) The approximation num-
ber function and the characterization of covering approximation
space. Inf Sci 305:196–207
Wei SH, Chen SM (2009) Fuzzy risk analysis based on interval-valued
fuzzy numbers. Expert Syst Appl 16(2):2285–2299
Wei L, Zhang WX (2004) Probabilistic rough sets characterized by
fuzzy sets. Int J Uncertain Fuzziness Knowl Based Syst 12:47–60
Xu Z (2011) Consistency of interval fuzzy preference relations in group
decision making. Appl Soft Comput 11(5):3898–3909
Xu W, Wang Q, Zhang X (2011) Multi-granulation fuzzy rough sets in
a fuzzy tolerance approximation space. Int J Gen Syst 13:246–259
Xu W, Wang Q, Zhang X (2012) A generalized multi-granulation rough
set approach. Lect Notes Bioinf 1(6840):681–689
Xu W, Wang Q, Zhang X (2013) Multi-granulation rough sets based
on tolerance relations. Soft Comput 17:1241–1252
Xu W, Wang Q, Luo S (2014) Multi-granulation fuzzy rough sets. J
Intell Fuzzy Syst 26:1323–1340
Yang HL, Liao X, Wang S, Wang J (2013) Fuzzy probabilistic rough
set model on two universes and its applications. Int J Approx
Reason 54:1410–1420
Yao YY (1998) Relational interpretations of neighborhood operators
and rough set approximation. Inf Sci 111:239–259
Yao YY (2003) Probabilistic approaches to rough sets. Expert Syst
20:287–297
Yao YY (2008) Probabilistic rough set approximations. Int J Approx
Reason 49:255–271
Yao YY (2009) Three-way decision: an interpretation of rules in rough
set theory. Lect Notes Comput Sci 5589:642–649
Yao YY (2010) Three-way decisions with probabilistic rough sets. Inf
Sci 180:341–353
Yao YY, Wong SKM (1992) A decision theoretic framework for
approximating concepts. Int J Man Mach Stud 37:793–809
Yao JT, Yao YY, Ziarko W (2008) Probabilistic rough sets: approxima-
tions, decision-makings, and applications. Int J Approx Reason
49:253–254
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–358
Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal
Appl 23:421–427
Zhang HY, Yang SY, Ma JM (2016) Ranking interval sets based on
inclusion measures and applications to three-way decisions.
Knowl Based Syst 91:62–70
Zhang XX, Chen DG, Tsang ECC (2017) Generalized dominance
rough set models for the dominance intuitionistic fuzzy informa-
tion systems. Inf Sci 378:1–25
Granular Computing
1 3
Zhao XR, Hu BQ (2015) Fuzzy and interval-valued decision-theoretic
rough set approaches based on the fuzzy probability measure. Inf
Sci 298:534–554
Zhao XR, Hu BQ (2016) Fuzzy probabilistic rough sets and their cor-
responding three-way decisions. Knowl Based Syst 91:126–142
Ziarko W (2005) Probabilistic rough sets. Lect Notes Comput Sci
3641:283–293
Ziarko W (2008) Probabilistic approach to rough sets. Int J Approx
Reason 49:272–284
A preview of this full-text is provided by Springer Nature.
Content available from Granular Computing
This content is subject to copyright. Terms and conditions apply.
