# Decision-theoretic rough sets-based three-way approximations of interval-
valued fuzzy sets

Guangming Langa
a School of Mathematics and Computer Science, Changsha University of Science
and Technology
Changsha, Hunan 410082, P.R. China
b College of Mathematics and Econometrics, Hunan University
Changsha, Hunan 410082, P.R. China
Corresponding author. Tel./fax: +86 731 8822855,
E-mail address: langguangming1984@126.com(G.M.Lang).

> Abstract. In practical situations, interval-valued fuzzy sets are frequently
> encountered. In this paper, firstly, we present shadowed sets for
> interpreting and understanding interval fuzzy sets. We also provide an
> analytic solution to computing the pair of thresholds by searching for a
> balance of uncertainty in the framework of shadowed sets. Secondly, we
> construct errors-based three-way approximations of interval-valued fuzzy
> sets. We also provide an alternative decision-theoretic formulation for
> calculating the pair of thresholds by transforming interval-valued loss
> functions into single-valued loss functions, in which the required
> thresholds are computed by minimizing decision costs. Thirdly, we compute
> errors-based three-way approximations of interval-valued fuzzy sets by using
> interval-valued loss functions. Finally, we employ several examples to
> illustrate that how to take an action for an object with interval-valued
> membership grade by using interval-valued loss functions.
>
> Keywords: Decision-theoretic rough sets; Interval-valued fuzzy sets;
> Interval-valued loss function; Shadowed set
>

##  1 Introduction

Interval-valued fuzzy sets[31], as an extension of fuzzy sets[39], is a
powerful mathematical tool to describe uncertainty information, in which the
concept of the membership function using the subintervals of the interval
$[0,1]$ as the set of membership grades is a fundamental notion. It has been
intensively investigated, not only its theoretical aspects, but also its
numerous applications, and the approximations of interval-valued fuzzy sets by
using several levels of membership grades have became an important research
direction.

Recently, researchers[1, 5, 7, 9, 10, 12, 22, 14, 15, 16, 20, 21, 40] have
investigated fuzzy sets from different aspects. For example, Pedrycz[22]
proposed shadowed sets for interpreting fuzzy sets by using several levels of
membership grades, in which, if the membership grade of an element is close to
$1$, it would be considered to be the same as $1$ and elevated to $1$; If the
membership grade is close to $0$, it would be considered to be the same as $0$
and is reduced to $0$; If the membership grade is neither close to $0$ nor
close to $1$, it would be put into a shadowed region, in which the elevation
and reduction operations use thresholds that provide semantically meaningful
and acceptance levels of degree of closeness of membership values to $1$ and
$0$, respectively. Sequently, a lot of investigations[23, 24, 25, 26, 27, 2,
3, 4, 8, 11, 19, 32, 41] have been done on shadowed sets. For instance, Deng
et al.[7] computed a pair of thresholds, whose interpretation and
determination is a fundamental issue for expressing fuzzy sets, for three-way
approximations of fuzzy sets by using loss functions, and classify a set of
objects into three regions by using the pair of thresholds. In practical
situations, interval-valued fuzzy sets whose membership functions are using
the subintervals of the interval $[0,1]$ are of interest because such type of
sets are frequently encountered. So far we have not seen the similar
investigation on interval-valued fuzzy sets. Therefore, it is of interest to
investigate that how to express interval-valued fuzzy sets as fuzzy sets.

To computing and interpreting a pair of thresholds, a lot of
investigations[13, 14, 15, 33, 34, 35, 36, 37, 16, 17, 18] have been done on
three-way decision-theory by using loss functions in literatures. For example,
Li et al.[13] evaluated the cost and benefit of assigning an instance to a
specific subcategory and defined a general loss function for supervised
leaning. Liang et al.[14, 15] presented triangular fuzzy decision-theoretic
rough sets and systematic studies on three-way decisions with interval-valued
decision-theoretic rough sets. Liu et al.[17] proposed stochastic decision-
theoretic rough sets, interval-valued decision-theoretic rough sets, fuzzy
decision-theoretic rough sets and dynamic decision-theoretic rough sets. In
practical situations, interval-valued loss functions as interval-valued
numbers[28, 29, 16] are of interest because such functions are frequently
encountered. Although interval-valued loss functions are complex in practice,
we have not seen enough investigations on interval-valued fuzzy sets by using
interval-valued loss functions so far. Therefore, it is urgent to further
study interval-valued loss functions for making a decision by using three-way
decision-theory.

The purpose of this paper is to further investigate interval-valued fuzzy
sets. Section 2 introduces the basic principles of decision-theoretic rough
sets, shadowed sets and decision-theoretic three-way approximations of fuzzy
sets. Section 3 presents shadowed sets of interval-valued fuzzy sets. Section
4 is devoting to errors-based interpretation of shadowed sets of interval-
valued fuzzy sets. Section 5 presents decision-theoretic rough sets-based
three-way approximations of interval-valued fuzzy sets by using transforming
interval-valued loss functions into single loss functions. Section 6
investigates decision-theoretic rough sets-based three-way approximations of
interval-valued fuzzy sets by using interval-valued loss functions from
another view. The conclusion comes in Section 7.

##  2 Preliminaries

In this section, we review some concepts of fuzzy sets, interval-valued fuzzy
sets, shadowed sets and decision-theoretic three-way approximations of fuzzy
sets.

###  2.1 Shadowed sets of fuzzy sets

In [39], Zadeh presented the concept of fuzzy sets for interpreting
uncertainty problems.

###### Definition 2.1

[39] Let $\mu_{A}$ be a mapping from $U$ to $[0,1]$ such as
$\mu_{A}:U\longrightarrow[0,1]:$ $x\longrightarrow\mu_{A}(x),$ where $x\in U$,
$\mu_{A}$ is the membership function. Then $A$ is referred to as a fuzzy set.

In [22], Pedrycz presented the concept of shadowed sets for expressing fuzzy
sets.

###### Definition 2.2

[22] Let $A$ be a fuzzy set, the shadowed set $S_{\mu_{A}}$ of $A$ is defined
as

$S_{\mu_{A}}(x)=\left\\{\begin{array}[]{ccc}1,&{\rm}\hfil&\mu(x)\geq\alpha;\\\
0,&{\rm}\hfil&\mu(x)\leq\beta;\\\
$[$0,1$]$,&{\rm}\hfil&\beta<\mu(x)<\alpha.\end{array}\right.$

In Pedrycz’s model, an optimal pair of thresholds is computed by minimizing
the absolute difference as

$\displaystyle V_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$
$\displaystyle|\text{Elevated Area}_{(\alpha,\beta)}(\mu_{A})+\text{Reduced
Area}_{(\alpha,\beta)}(\mu_{A})-\text{Shadowed
Area}_{(\alpha,\beta)}(\mu_{A})|$ $\displaystyle=$
$\displaystyle|\sum_{\mu_{A}(x)\geq\alpha}(1-\mu_{A}(x))+\sum_{\mu_{A}(x)\leq\beta}(\mu_{A}(x))-Card(\\{x\in
U|\beta<\mu_{A}(x)<\alpha\\})|,$

where card$(\cdot)$ denotes the cardinality of a set $\cdot$, and an optimal
pair of thresholds $\alpha$ and $\beta$ can be derived by minimizing the
objective function $V_{(\alpha,\beta)}(\mu_{A})$. Similarly, it is also
difficult to compute the pair of thresholds $\alpha$ and $\beta$ since
minimizing $V_{(\alpha,\beta)}(\mu_{A})$ involves two parameters $\alpha$ and
$\beta$. For convenience, by using $\alpha+\beta=1$, the objective function is
simplified into

$\displaystyle V_{(\alpha,1-\alpha)}(\mu_{A})$ $\displaystyle=$
$\displaystyle|\text{Elevated Area}_{(\alpha,1-\alpha)}(\mu_{A})+\text{Reduced
Area}_{(\alpha,1-\alpha)}(\mu_{A})-\text{Shadowed
Area}_{(\alpha,1-\alpha)}(\mu_{A})|$ $\displaystyle=$
$\displaystyle|\sum_{\mu_{A}(x)\geq\alpha}(1-\mu_{A}(x))+\sum_{\mu_{A}(x)\leq
1-\alpha}(\mu_{A}(x))-Card(\\{x\in U|1-\alpha(<\mu_{A}(x)<\alpha\\})|.$

###  2.2 Decision-theoretic three-way approximations of fuzzy sets

In terms of the errors, Deng et al.[7] expressed the objective function to
further investigate shadowed sets of fuzzy sets as

$\displaystyle V_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$
$\displaystyle|E_{e}(\mu_{A})+E_{r}(\mu_{A})-E_{s}(\mu_{A})|$ $\displaystyle=$
$\displaystyle|\sum_{\mu_{A}(x)\geq\alpha}(1-\mu_{A}(x))+\sum_{\mu_{A}(x)\leq\beta}(\mu_{A}(x))-\sum_{\beta<\mu_{A}(x)<\alpha}(1-\mu_{A}(x))+\sum_{\beta<\mu_{A}(x)<\alpha}(\mu_{A}(x))|.$

The objective function is constructed on elevated area, reduced area and
shadowed area, and it is necessary to investigate that which numeric value is
meaningful to the membership grade of elements in the shadowed area.

By replacing the unit interval $[0,1]$ with $0.5$, Deng et al. provided

$\displaystyle
T_{\mu_{A}}(x)=\left\\{\begin{array}[]{ccc}1,&{\rm}\hfil&\mu(x)\geq\alpha;\\\
0,&{\rm}\hfil&\mu(x)\leq\beta;\\\
0.5,&{\rm}\hfil&\beta<\mu(x)<\alpha.\end{array}\right.$

Subsequently, by analyzing $T_{\mu_{A}}(x)$, we have

$\displaystyle E_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$ $\displaystyle
E_{e}(\mu_{A})+E_{r}(\mu_{A})+E_{s_{0.5}}(\mu_{A})$ $\displaystyle=$
$\displaystyle\sum_{\mu_{A}(x)\geq\alpha}(1-\mu_{A}(x))+\sum_{\mu_{A}(x)\leq\beta}(\mu_{A}(x))-\sum_{0.5<\mu_{A}(x)<\alpha(t)}(\mu_{A}(x)-0.5)+\sum_{\beta<\mu_{A}(x)<0.5}(0.5-\mu_{A}(x)).$

Correspondingly, the total error as the summation of errors of all objects are
expressed as

$\displaystyle E_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$
$\displaystyle\sum_{x\in U}E_{(\alpha,\beta)}(\mu_{A}(x)),$

where

$\displaystyle
E_{(\alpha,\beta)}(\mu_{A}(x))=\left\\{\begin{array}[]{ccc}1-\mu(x),&{\rm}\hfil&\mu(x)\geq\alpha;\\\
0.5-\mu(x),&{\rm}\hfil&\beta<\mu(x)\leq 0.5;\\\
\mu(x)-0.5,&{\rm}\hfil&0.5<\mu(x)<\alpha;\\\
\mu(x)-0,&{\rm}\hfil&\mu(x)\leq\beta.\end{array}\right.$

The total error is minimized by minimizing the error of each individual
object, and we search for a pair of thresholds $\alpha$ and $\beta$ such that
$E_{(\alpha,\beta)}(\mu_{A}(x))$ is minimized for each object. We consider the
following actions and associated errors for minimizing the error of each
object:

$\displaystyle(1):\text{elevate to }1:1-\mu_{A}(x);(2):\text{reduce to
}0:\mu_{A}(x)-0;(3):\text{reduce or elevate to }0.5:|\mu_{A}(x)-0.5|.$

That is, the absolute differences between $\mu_{A}(x)$ and three values 1, 0.5
and 0, respectively, are the associated errors. A minimized difference is
obtained if $\mu_{A}(x)$ is changed into a value that is closest to
$\mu_{A}(x)$.

Table 1: Loss function. Action | Fuzzy set membership grade | Three-way membership grade | Error | Loss
---|---|---|---|---
$a_{e}$ | $\mu_{A}(x)\geq\alpha$ | $1$ | $1-\mu_{A}(x)$ | $\lambda_{e}$
$a_{r}$ | $\mu_{A}(x)\leq\beta$ | $0$ | $\mu_{A}(x)$ | $\lambda_{r}$
$a_{s_{\downarrow}}$ | $0.5\leq\mu_{A}(x)<\alpha$ | $0.5$ | $\mu_{A}(x)-0.5$ | $\lambda_{s_{\downarrow}}$
$a_{s_{\uparrow}}$ | $\beta<\mu_{A}(x)<0.5$ | $0.5$ | $0.5-\mu_{A}(x)$ | $\lambda_{s_{\uparrow}}$

By considering various costs of the actions of elevation and reduction, Deng
et al. presented an analytic solution of computing the pair of thresholds
$\alpha$ and $\beta$ by using loss functions. In Table $2$, the set of actions
$\\{a_{e},a_{r},a_{s_{\downarrow}},a_{s_{\uparrow}}\\}$ describes four
possible actions on changing the membership grade. The fuzzy membership grade
$\mu_{A}(x)$ represents the state of object in the second column, and the
errors of different actions are given in the fourth column, and the losses of
different actions are given in the fifth column.

Suppose
$\lambda_{e}>0,\lambda_{r}>0,\lambda_{s_{\downarrow}}>0,\lambda_{s_{\uparrow}}>0,\lambda_{s_{\downarrow}}\leq\lambda_{r}$
and $\lambda_{s_{\uparrow}}\leq\lambda_{e}$, we immediately have three rules
as (E) If $\mu_{A}(x)\geq\alpha$, then $T_{\mu_{A}}(x)=1$; (R) If
$\mu_{A}(x)\leq\beta$, then $T_{\mu_{A}}(x)=0$; (S) If
$\beta<\mu_{A}(x)<\alpha$, then $T_{\mu_{A}}(x)=0.5$, where

$\displaystyle\alpha=\frac{2\lambda_{e}+\lambda_{s_{\downarrow}}}{2(\lambda_{e}+\lambda_{s_{\downarrow}})}\text{
and
}\beta=\frac{\lambda_{s_{\uparrow}}}{2(\lambda_{r}+\lambda_{s_{\uparrow}})}.$

##  3 Shadowed sets of interval-valued fuzzy sets and its errors-based
interpretations

In this section, we present the concept of shadowed sets of interval-valued
fuzzy sets and its errors-based interpretations for illustrating interval-
valued fuzzy sets.

###  3.1 Shadowed sets of interval-valued fuzzy sets

In this subsection, we present the concept of shadowed sets of interval-valued
fuzzy sets.

###### Definition 3.1

Let $D_{[0,1]}$ be the set of closed subintervals of the interval $[0,1]$. An
interval-valued fuzzy set $A$ in $U$ is given by
$A=\\{(x,\widetilde{\mu}_{A}(x))|x\in U\\}$, where
$\widetilde{\mu}_{A}:X\longrightarrow
D_{[0,1]}:x\longrightarrow\widetilde{\mu}_{A}(x)=[\mu^{-}_{A}(x),\mu^{+}_{A}(x)].$

###### Definition 3.2

Let $A$ be an interval-valued fuzzy set,
$\widetilde{\mu}_{A}(x)=[\mu^{-}_{A}(x),\mu^{+}_{A}(x)]$ be a membership grade
of $x\in U$, and $\theta\in[0,1]$. Then the transformed formula of
$\widetilde{\mu}_{A}(x)$ is
$m_{\theta}(\widetilde{\mu}_{A}(x))=(1-\theta)\cdot\mu^{-}_{A}(x)+\theta\cdot\mu^{+}_{A}(x);$
Furthermore, $A_{\theta}=\\{(x,m_{\theta}(\widetilde{\mu}_{A}(x)))|x\in U\\}.$

###### Example 3.3

$(1)$ Let $\widetilde{\mu}_{A}(x)=[0.1,0.2]$ and
$\widetilde{\mu}_{B}(x)=[0.15,0.25]$ for $x\in U$, and $\theta=0.5$. Then

$\displaystyle m_{\theta}(\widetilde{\mu}_{A}(x))$ $\displaystyle=$
$\displaystyle(1-0.5)\times 0.1+0.5\times 0.2=0.15;$ $\displaystyle
m_{\theta}(\widetilde{\mu}_{B}(x))$ $\displaystyle=$
$\displaystyle(1-0.5)\times 0.15+0.5\times 0.25=0.2.$

$(2)$ Let
$A=\frac{x_{1}}{\widetilde{\mu}_{A}(x_{1})}+\frac{x_{2}}{\widetilde{\mu}_{A}(x_{2})}+\frac{x_{3}}{\widetilde{\mu}_{A}(x_{3})}+\frac{x_{4}}{\widetilde{\mu}_{A}(x_{4})}$
be an interval-valued fuzzy set, where
$\widetilde{\mu}_{A}(x_{1})=[0.1,0.2],\widetilde{\mu}_{A}(x_{2})=[0.6,0.8],\widetilde{\mu}_{A}(x_{3})=[0.3,0.5]$
and $\widetilde{\mu}_{A}(x_{4})=[0.8,0.1].$ If we take $\theta=0.5$, then
$A_{\theta}=\frac{x_{1}}{0.15}+\frac{x_{2}}{0.7}+\frac{x_{3}}{0.4}+\frac{x_{4}}{0.45}.$

For simplicity, we denote $\widetilde{\mu}_{A}$ as $\mu_{A}$ in the following
discussion.

###### Definition 3.4

Let $A$ be an interval-valued fuzzy set, then the shadowed set $S_{\mu_{A}}$
of $A$ is defined as

$S_{\mu_{A}}(x)=\left\\{\begin{array}[]{ccc}1,&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\geq\alpha;\\\
0,&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\leq\beta;\\\
$[$0,1$]$,&{\rm}\hfil&\beta<m_{\theta}(\mu_{A}(x))<\alpha.\end{array}\right.$

For an object $x$, we elevate the membership grade from $\mu_{A}(x)$ to $1$ if
$m_{\theta}(\mu_{A}(x))\geq\alpha$; We reduce the membership grade from
$\mu_{A}(x)$ to 0 if $m_{\theta}(\mu_{A}(x))\leq\beta$; We change the
membership grade from $\mu_{A}(x)$ to $[0,1]$ if
$\beta<m_{\theta}(\mu_{A}(x))<\alpha$.

The pair of thresholds $\alpha$ and $\beta$ are important for computing three-
way approximations of interval-valued fuzzy sets. In what follows, we
introduce a systematic way to compute the pair of thresholds $\alpha$ and
$\beta$ by minimizing an objective function as

$\displaystyle V_{(\alpha,\beta)}(A)$ $\displaystyle=$
$\displaystyle|\text{Elevated Area}_{(\alpha,\beta)}(A)+\text{Reduced
Area}_{(\alpha,\beta)}(A)-\text{Shadowed Area}_{(\alpha,\beta)}(\mu_{A})|$
$\displaystyle=$
$\displaystyle|\sum_{m_{\theta}(\mu_{A}(x))\geq\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{m_{\theta}(\mu_{A}(x))\leq\beta}(m_{\theta}(\mu_{A}(x)))-Card(\\{x\in
U|\beta<m_{\theta}(\mu_{A}(x))<\alpha\\})|,$

where card$(\cdot)$ denotes the cardinality of a set $\cdot$, and an optimal
pair of thresholds $\alpha$ and $\beta$ can be derived by minimizing the
objective function $V_{(\alpha,\beta)}(A)$. Similarly, minimizing
$V_{(\alpha,\beta)}(A)$ involves two parameters $\alpha$ and $\beta$. For
convenience, by assuming that $\alpha+\beta=1$, the objective function is
simplified into

$\displaystyle V_{(\alpha,1-\alpha)}(A)$ $\displaystyle=|\text{Elevated
Area}_{(\alpha,1-\alpha)}(m_{\theta}(\mu_{A}(x)))+\text{Reduced
Area}_{(\alpha,1-\alpha)}(m_{\theta}(\mu_{A}(x)))-\text{Shadowed
Area}_{(\alpha,1-\alpha)}(m_{\theta}(\mu_{A}(x)))|$
$\displaystyle=|\sum_{m_{\theta}(\mu_{A}(x))(x)\geq\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{m_{\theta}(\mu_{A}(x))\leq
1-\alpha}(m_{\theta}(\mu_{A}(x)))-Card(\\{x\in
U|1-\alpha<m_{\theta}(\mu_{A}(x))<\alpha\\})|.$

There exist two interpretations of shadowed sets of interval-valued fuzzy
sets. In a wide sense, a shadowed set is a three-valued fuzzy set, which is
used to approximate an interval-valued fuzzy set. In a narrow sense, we
interpret the notion of a shadowed set according to its exact formulation,
namely, the choice of the set of membership grades $\\{0,[0,1],1\\}$ and the
objective function. Therefore, shadowed sets of interval-valued fuzzy sets are
examples of three-way approximations of interval-valued fuzzy sets.

###  3.2 Errors-based interpretation of shadowed sets for interval-valued
fuzzy sets

In this section, we present a detailed analysis of a objective function for
shadowed sets of interval-valued fuzzy sets in terms of errors of
approximations. We also provide a new objective function by the total error of
approximations for determining the thresholds $\alpha$ and $\beta$.

To further study shadowed sets of interval-valued fuzzy sets, we express the
objective function in terms of the errors introduced by a shadowed set
approximation. For an object $x$ with membership grade
$m_{\theta}(\mu_{A}(x))$, the elevation operation changes the membership grade
from $m_{\theta}(\mu_{A}(x))$ to $1$, the reduction operation changes the
membership grade from $m_{\theta}(\mu_{A}(x))$ to $0$, and the errors induced
by elevation and reduction are shown as

$\displaystyle
E_{e}(\mu_{A}(x))=1-m_{\theta}(\mu_{A}(x)),E_{r}(\mu_{A}(x))=m_{\theta}(\mu_{A}(x)).$

The errors $E_{e}(\mu_{A})$ and $E_{r}(\mu_{A})$ induced by the elevation and
reduction operations for an interval-valued fuzzy set $A$ of the universe $U$,
respectively, are shown as

$\displaystyle
E_{e}(\mu_{A})=\sum_{m_{\theta}(\mu_{A}(x))\geq\alpha}(1-m_{\theta}(\mu_{A}(x))),E_{r}(\mu_{A}(x))=\sum_{m_{\theta}(\mu_{A}(x))\leq\beta}m_{\theta}(\mu_{A}(x)).$

The error for the shadowed area is not clear because of the unit interval
$[0,1]$ as the membership grade when $\beta<m_{\theta}(\mu_{A}(x))<\alpha$. By
computing the difference between $m_{\theta}(\mu_{A}(x))$ and the maximum $1$
and the minimum value $0$ and summarizing them up, we have

$\displaystyle
E_{s}(\mu_{A})=\sum_{\beta<m_{\theta}(\mu_{A}(x))<\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{\beta<m_{\theta}(\mu_{A}(x))<\alpha}m_{\theta}(\mu_{A}(x)).$

Subsequently, we express the objective function in terms of errors by using
the error-based interpretation of the three areas as

$\displaystyle V_{(\alpha,\beta)}(\mu_{A})$
$\displaystyle=|E_{e}(\mu_{A})+E_{r}(\mu_{A})-E_{s}(\mu_{A})|$
$\displaystyle=|\sum_{m_{\theta}(\mu_{A}(x))\geq\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{m_{\theta}(\mu_{A}(x))\leq\beta}m_{\theta}(\mu_{A}(x)-\sum_{\beta<m_{\theta}(\mu_{A}(x))<\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{\beta<m_{\theta}(\mu_{A}(x))<\alpha}(m_{\theta}\mu_{A}(x))|.$

The objective function is a kind of trade-off of errors produced by three
regions. But the rationale for such a trade-off is not entirely clear. On one
hand, $E_{s}(m_{\theta}(\mu_{A}(x)))$ consists the errors of elevation and
reduction operations, and it is impossible to elevate $m_{\theta}(\mu_{A}(x))$
to $1$ and reduce $m_{\theta}(\mu_{A}(x))$ to $0$ if
$\beta<m_{\theta}(\mu_{A}(x))<\alpha$ simultaneously. On the other hand, we
are not able to allocate any numeric membership grade for the elements in the
shadowed area. In other words, any numeric value of the unit interval $[0,1]$
could be permitted to reflect the uncertainty. Therefore, it is necessary to
investigate that which numeric value is meaningful to the membership grade of
elements in the shadowed area.

Below, we present a three-way approximation of an interval-valued fuzzy set by
replacing the unit interval $[0,1]$ with $0.5$ as,

$\displaystyle
T_{\mu_{A}}(x)=\left\\{\begin{array}[]{ccc}1,&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\geq\alpha;\\\
0,&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\leq\beta;\\\
0.5,&{\rm}\hfil&\beta<m_{\theta}(\mu_{A}(x))<\alpha.\end{array}\right.$

By analyzing $T_{\mu_{A}}(x)$, we see that the correspondences between areas
of elevation and reduction and errors of elevation and reduction remain to be
the same. But we need to revise the errors of the shadowed region as

$\displaystyle
E_{s_{0.5}}(\mu_{A})=\sum_{0.5<m_{\theta}(\mu_{A}(x))<\alpha}(1-m_{\theta}(\mu_{A}(x)))+\sum_{\beta<m_{\theta}(\mu_{A}(x))<0.5}(m_{\theta}(\mu_{A}(x))).$

By using $E_{e}(\mu_{A}),E_{r}(\mu_{A})$ and $E_{s_{0.5}}(\mu_{A})$, we have

$\displaystyle E_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$ $\displaystyle
E_{e}(\mu_{A})+E_{r}(\mu_{A})+E_{s_{0.5}}(\mu_{A})$ $\displaystyle=$
$\displaystyle\sum_{m_{\theta}(\mu_{A}(x))\geq\alpha}(1-m_{\theta}(\mu_{A}(x))+\sum_{m_{\theta}(\mu_{A}(x))\leq\beta}(m_{\theta}(\mu_{A}(x)))-\sum_{0.5<m_{\theta}(\mu_{A}(x))<\alpha(t)}(m_{\theta}(\mu_{A}(x))-0.5)$
$\displaystyle+\sum_{\beta<m_{\theta}(\mu_{A}(x))<0.5}(0.5-m_{\theta}(\mu_{A}(x))).$

The total errors of the three areas are minimized instead of searching for a
trade-off between different areas. Correspondingly, we express the total error
as the summation of errors of all objects as

$\displaystyle E_{(\alpha,\beta)}(\mu_{A})$ $\displaystyle=$
$\displaystyle\sum_{x\in U}E_{(\alpha,\beta)}(m_{\theta}(\mu_{A}(x))),$

where

$\displaystyle
E_{(\alpha,\beta)}(\mu_{A}(x))=\left\\{\begin{array}[]{ccc}1-m_{\theta}(\mu_{A}(x)),&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\geq\alpha;\\\
0.5-m_{\theta}(\mu_{A}(x)),&{\rm}\hfil&\beta<m_{\theta}(\mu_{A}(x))\leq
0.5;\\\
m_{\theta}(\mu_{A}(x))-0.5,&{\rm}\hfil&0.5<m_{\theta}(\mu_{A}(x))<\alpha;\\\
m_{\theta}(\mu_{A}(x)),&{\rm}\hfil&m_{\theta}(\mu_{A}(x))\leq\beta.\end{array}\right.$

The total error will be minimized by minimizing the error of each individual
object, and we can search for a pair of thresholds $\alpha$ and $\beta$ such
that $E_{(\alpha,\beta)}(\mu_{A}(x))$ is minimized for each object. We
consider the following actions and associated errors for minimizing the error
of each object:

$\displaystyle(1):\text{elevate to
}1:1-m_{\theta}(\mu_{A}(x));(2):\text{reduce to
}0:m_{\theta}(\mu_{A}(x))-0;(3):\text{reduce or elevate to
}0.5:|m_{\theta}(\mu_{A}(x))-0.5|.$

That is, the absolute differences between $m_{\theta}(\mu_{A}(x))$ and three
values $1$, $0.5$ and $0$, respectively, are the associated errors. A
minimized difference is obtained if $m_{\theta}(\mu_{A}(x))$ is changed into a
value that is closest to $m_{\theta}(\mu_{A}(x))$.

##  4 Decision-theoretic rough sets-based three-way approximations of
interval-valued fuzzy sets

In this section, we introduce a framework for decision-theoretic rough sets-
based three-way approximations of interval-valued fuzzy sets.

###  4.1 Cost-sensitive three-way approximations of interval-valued fuzzy sets

In Section 3, we investigate three-way approximation of interval-valued fuzzy
sets by using three membership grades of $0$, $0.5$ and $1$. We take one of
the following three actions for an object with a membership grade: elevate the
membership grade to 1, reduce the membership grade to 0, and change the
membership grade to 0.5. More specially, there are two situations for the
third case: reduce the membership grade to 0.5 if $m_{\theta}(\mu_{A}(x))\geq
0.5$ and elevate the membership grade to 0.5 if $m_{\theta}(\mu_{A}(x))<0.5$.
Each action will incur error and the costs of different actions are not
necessarily the same.

Table 2: Loss function. Action | Membership grade | Three-way membership grade | Error | Loss
---|---|---|---|---
$a_{e}$ | $m_{\theta}(\mu_{A}(x))\geq\alpha$ | $1$ | $1-m_{\theta}(\mu_{A}(x))$ | $\lambda_{e}$
$a_{r}$ | $m_{\theta}(\mu_{A}(x))\leq\beta$ | $0$ | $m_{\theta}(\mu_{A}(x))$ | $\lambda_{r}$
$a_{s_{\downarrow}}$ | $0.5\leq m_{\theta}(\mu_{A}(x))<\alpha$ | $0.5$ | $m_{\theta}(\mu_{A}(x))-0.5$ | $\lambda_{s_{\downarrow}}$
$a_{s_{\uparrow}}$ | $\beta<m_{\theta}(\mu_{A}(x))<0.5$ | $0.5$ | $0.5-m_{\theta}(\mu_{A}(x))$ | $\lambda_{s_{\uparrow}}$

Table $2$ summarizes information about three-way approximations of an
interval-valued fuzzy set. Concretely, the set of actions
$\\{a_{e},a_{r},a_{s_{\downarrow}},a_{s_{\uparrow}}\\}$ describes four
possible actions on changing the membership grade. For simplicity, we also use
$\\{e,r,s_{\downarrow},s_{\uparrow}\\}$ to denote the four actions. The
elevation action $a_{e}$ elevate the membership grade of $x$ from
$m_{\theta}(\mu_{A}(x))$ to $1$, the reduction action $a_{r}$ reduce the
membership grade of $x$ from $m_{\theta}(\mu_{A}(x))$ to $0$, the elevation
$a_{s_{\uparrow}}$ elevate the membership grade of $x$ from
$m_{\theta}(\mu_{A}(x))$ to $0.5$ if $m_{\theta}(\mu_{A}(x))<0.5$, the
reduction $a_{s_{\downarrow}}$ reduce the membership grade of $x$ from
$m_{\theta}(\mu_{A}(x))$ to $0.5$ if $m_{\theta}(\mu_{A}(x))>0.5$. The fuzzy
membership grade $m_{\theta}(\mu_{A}(x))$ represents the state of object in
the second column, and the errors of different actions are given in the fourth
column, and the losses of different actions are given in the fifth column.

Each of the four losses $\lambda_{e},\lambda_{r},\lambda_{s_{\downarrow}}$ and
$\lambda_{s_{\uparrow}}$ provides the unit cost, and the actual cost of each
action is weighted by the magnitude of its error. Suppose
$R_{a}(x)=\lambda_{a}E_{a}(\mu_{A}(x))$ denote the loss for taking actions
$\\{e,r,s_{\downarrow},s_{\uparrow}\\}$, the losses of four actions for an
object can be computed as

$\displaystyle R_{e}(x)$ $\displaystyle=$
$\displaystyle\lambda_{e}E_{e}(\mu_{A}(x))=(1-m_{\theta}(\mu_{A}(x)))\lambda_{e};$
$\displaystyle R_{r}(x)$ $\displaystyle=$
$\displaystyle\lambda_{r}E_{r}(\mu_{A}(x))=m_{\theta}(\mu_{A}(x))\lambda_{r};$
$\displaystyle R_{s_{\downarrow}}(x)$ $\displaystyle=$
$\displaystyle\lambda_{s_{\downarrow}}E_{s_{\downarrow}}(m_{\theta}(\mu_{A}(x)))=(m_{\theta}(\mu_{A}(x))-0.5)\lambda_{s_{\downarrow}};$
$\displaystyle R_{s_{\uparrow}}(x)$ $\displaystyle=$
$\displaystyle\lambda_{s_{\uparrow}}E_{s_{\uparrow}}(m_{\theta}(\mu_{A}(x)))=(0.5-m_{\theta}(\mu_{A}(x)))\lambda_{s_{\uparrow}}.$

Since only an action is taken for each object, the total loss of the
approximation is computed by

$\displaystyle R=\sum_{x\in U}R_{a}(x)=\sum_{x\in
U}\lambda_{a}E_{a}(\mu_{A}(x)).$

To minimize the total loss $R$, we take an action $\tau(x)$ that minimizes the
loss $R_{a}(x)$ for each object, and $\tau(x)$ is a solution to the following
minimization problem as

$\text{arg }min_{a\in action}R_{a}(x),$

where $a\in\\{e,r,s_{\downarrow},s_{\uparrow}\\}$.

According to the value $\mu_{A}(x)$ of an object $x$, we have two groups of
decision rules for obtaining three-way approximations of an interval-valued
fuzzy set as follows:

$(1)$ When $m_{\theta}(\mu_{A}(x))\geq 0.5$, $(E1)$ If $R(a_{e}|x)\leq
R(a_{r}|x)$ and $R(a_{e}|x)\leq R(a_{s_{\downarrow}}|x)$, then take action
$a_{e}$; $(R1)$ If $R(a_{r}|x)\leq R(a_{e}|x)$ and $R(a_{r}|x)\leq
R(a_{s_{\downarrow}}|x)$, then take action $a_{r}$; $(S1)$ If
$R(a_{s_{\downarrow}}|x)\leq R(a_{e}|x)$ and $R(a_{s_{\downarrow}}|x)\leq
R(a_{r}|x)$, then take action $a_{s_{\downarrow}}$.

$(2)$ When $m_{\theta}(\mu_{A}(x))<0.5$, $(E2)$ If $R(a_{e}|x)\leq R(a_{r}|x)$
and $R(a_{e}|x)\leq R(a_{s_{\uparrow}}|x)$, then take action $a_{e}$; $(R2)$
If $R(a_{r}|x)\leq R(a_{e}|x)$ and $R(a_{r}|x)\leq R(a_{s_{\uparrow}}|x)$,
then take action $a_{r}$; $(S2):$ If $R(a_{s_{\uparrow}}|x)\leq R(a_{e}|x)$
and $R(a_{s_{\uparrow}}|x)\leq R(a_{r}|x)$, then take action
$a_{s_{\uparrow}}$.

###  4.2 Single-valued loss functions-based three-way approximations of
interval-valued fuzzy sets

In this subsection, we consider loss functions satisfying certain properties
for obtaining an analytic solution defining a three-way approximation.

Suppose
$(c1):\lambda_{e}>0,\lambda_{r}>0,\lambda_{s_{\downarrow}}>0,\lambda_{s_{\uparrow}}>0;$
$(c2):\lambda_{s_{\downarrow}}\leq\lambda_{r};$
$(c3):\lambda_{s_{\uparrow}}\leq\lambda_{e}$, Condition $(c1)$ requires that
all costs are nonnegative; Condition $(c2)$ illustrates that reducing a
membership grade $\mu_{A}(x)\geq 0.5$ to $0.5$ represents a smaller adjustment
than reducing it to $0$, and a smaller cost is associated with action
$a_{s_{\downarrow}}$; Condition $(c3)$ illustrates that elevating a membership
grade $\mu_{A}(x)<0.5$ to $0.5$ represents a smaller adjustment than elevating
it to $1$, and a smaller cost is associated with action $a_{s_{\uparrow}}$.
With the assumptions $(c1)-(c3)$, we simplify the decision rules as follows:

$(1)$ When $m_{\theta}(\mu_{A}(x))\geq 0.5$, the rule $(E1)$ is expressed as

$\displaystyle R(a_{e}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda_{e}\leq
m_{\theta}(\mu_{A}(x))\lambda_{r}$ $\displaystyle\Leftrightarrow$
$\displaystyle\mu_{A}(x)\geq\frac{\lambda_{e}}{\lambda_{e}+\lambda_{r}}=\gamma;$
$\displaystyle R(a_{e}|x)\leq R(a_{s_{\downarrow}}|x)$
$\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda_{e}\leq(m_{\theta}(\mu_{A}(x))-0.5)\lambda_{s_{\downarrow}}$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{2\lambda_{e}+\lambda_{s_{\downarrow}}}{2(\lambda_{e}+\lambda_{s_{\downarrow}})}=\alpha.$

The rule $R(1)$ is expressed by

$\displaystyle R(a_{r}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle m_{\theta}(\mu_{A}(x))\leq\gamma;$ $\displaystyle
R(a_{r}|x)\leq R(a_{s_{\downarrow}}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda_{r}\leq(m_{\theta}(\mu_{A}(x))-0.5)\lambda_{s_{\downarrow}}$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\leq\frac{-\lambda_{s_{\downarrow}}(t)}{2(\lambda_{r}-\lambda_{s_{\downarrow}}(t))}=\gamma^{-}.$

The rule $S(1)$ is expressed by

$\displaystyle R(a_{s_{\downarrow}}|x)\leq R(a_{e}|x)\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\alpha;R(a_{s_{\downarrow}}|x)\leq
R(a_{s_{\downarrow}}|x)\Leftrightarrow m_{\theta}(\mu_{A}(x))\geq\gamma^{-}.$

Since $\gamma^{-}\leq 0$ contradicts with the assumption
$m_{\theta}(\mu_{A}(x))\geq 0.5$, it is impossible to apply rule $(R1)$ for
reducing membership values. Therefore, when $m_{\theta}(\mu_{A}(x))\geq 0.5$,
the rules are simply expressed as (E1) If $m_{\theta}(\mu_{A}(x))\geq\alpha$,
then $T_{\mu_{A}}(x)=1$; (S1) If $0.5\leq m_{\theta}(\mu_{A}(x))<\alpha$, then
$T_{\mu_{A}}(x)=0.5$.

$(2)$ When $m_{\theta}(\mu_{A}(x))<0.5$, the rule $(E2)$ is expressed as

$\displaystyle R(a_{e}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda_{e}\leq
m_{\theta}(\mu_{A}(x))\lambda_{r}$ $\displaystyle\Leftrightarrow$
$\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{\lambda_{e}}{\lambda_{e}+\lambda_{r}}=\gamma;$
$\displaystyle R(a_{e}|x)\leq R(a_{s_{\uparrow}}|x)$
$\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda_{e}\leq(0.5-m_{\theta}(\mu_{A}(x)))\lambda_{s_{\uparrow}}$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{\lambda_{e}-0.5\lambda_{s_{\uparrow}}}{\lambda_{e}-\lambda_{s_{\uparrow}}}=\gamma^{+}.$

The rule $R(2)$ is expressed as

$\displaystyle R(a_{r}|x)\leq R(a_{e}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle m_{\theta}(\mu_{A}(x))\leq\gamma;$ $\displaystyle
R(a_{r}|x)\leq R(a_{s_{\uparrow}}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda_{r}\leq(0.5-m_{\theta}(\mu_{A}(x)))\lambda_{s_{\uparrow}}$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\leq\frac{\lambda_{s_{\uparrow}}}{2(\lambda_{r}+\lambda_{s_{\uparrow}})}=\beta.$

The rule $S(2)$ is expressed as

$\displaystyle R(a_{s_{\uparrow}}|x)\leq R(a_{e}|x)\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\gamma^{+};R(a_{s_{\uparrow}}|x)\leq
R(a_{s_{\uparrow}}|x)\Leftrightarrow m_{\theta}(\mu_{A}(x))\geq\beta.$

Since $\gamma^{+}\geq 1$ contradicts with the assumption
$m_{\theta}(\mu_{A}(x))<0.5$, it is impossible to apply rule $(E2)$ for
elevating membership values. Therefore, when $m_{\theta}(\mu_{A}(x))<0.5$, the
remaining rules are simply expressed as (R2) If
$m_{\theta}(\mu_{A}(x))\leq\beta$, then $T_{\mu_{A}}(x)=0$; (S2) If $\beta\leq
m_{\theta}(\mu_{A}(x))<0.5$, then $T_{\mu_{A}}(x)=0.5$.

By combining the two sets of rules, we immediately have three rules as (E) If
$m_{\theta}(\mu_{A}(x))\geq\alpha(t)$, then $T_{\mu_{A}}(x)=1$; (R) If
$m_{\theta}(\mu_{A}(x))\leq\beta$, then $T_{\mu_{A}}(x)=0$; (S) If
$\beta<m_{\theta}(\mu_{A}(x))<\alpha$, then $T_{\mu_{A}}(x)=0.5$, where

$\displaystyle\alpha=\frac{2\lambda_{e}+\lambda_{s_{\downarrow}}}{2(\lambda_{e}(t)+\lambda_{s_{\downarrow}})}\text{
and
}\beta=\frac{\lambda_{s_{\uparrow}}}{2(\lambda_{r}+\lambda_{s_{\uparrow}})}.$

##  5 Interval-valued loss functions-based three-way approximations of
interval-valued fuzzy sets: I

In this section, we introduce a framework for interval-valued loss functions-
based three-way approximations of interval-valued fuzzy sets.

###  5.1 Cost-sensitive three-way approximations of interval-valued fuzzy sets

In Section 4, we investigate three-way approximation of interval-valued fuzzy
sets by using three membership grades of $0$, $0.5$ and $1$. We take one of
the following three actions for an object with a membership grade: elevate the
membership grade to 1, reduce the membership grade to 0, and change the
membership grade to 0.5. More specially, there are two situations for the
third case: reduce the membership grade to 0.5 if $m_{\theta}(\mu_{A}(x))\geq
0.5$ and elevate the membership grade to 0.5 if $m_{\theta}(\mu_{A}(x))<0.5$.
Each action will incur error and the costs of different actions are not
necessarily the same.

Table 3: Interval-valued loss function. Action | Membership grade | Three-way membership grade | Error | Loss
---|---|---|---|---
$a_{e}$ | $m_{\theta}(\mu_{A}(x))\geq\alpha$ | $1$ | $1-m_{\theta}(\mu_{A}(x))$ | $\widetilde{\lambda}_{e}=[\lambda^{-}_{e},\lambda^{+}_{e}]$
$a_{r}$ | $m_{\theta}(\mu_{A}(x))\leq\beta$ | $0$ | $m_{\theta}(\mu_{A}(x))$ | $\widetilde{\lambda}_{r}=[\lambda^{-}_{r},\lambda^{+}_{r}]$
$a_{s_{\downarrow}}$ | $0.5\leq m_{\theta}(\mu_{A}(x))<\alpha$ | $0.5$ | $m_{\theta}(\mu_{A}(x))-0.5$ | $\widetilde{\lambda}_{s_{\downarrow}}=[\lambda^{-}_{s_{\downarrow}},\lambda^{+}_{s_{\downarrow}}]$
$a_{s_{\uparrow}}$ | $\beta<m_{\theta}(\mu_{A}(x))<0.5$ | $0.5$ | $0.5-m_{\theta}(\mu_{A}(x))$ | $\widetilde{\lambda}_{s\uparrow}=[\lambda^{-}_{s\uparrow},\lambda^{+}_{s\uparrow}]$

Table $3$ summarizes information about three-way approximations of an
interval-valued fuzzy set. Concretely, the set of actions
$\\{a_{e},a_{r},a_{s_{\downarrow}},a_{s_{\uparrow}}\\}$ describes four
possible actions on changing the membership grade. For simplicity, we also use
$\\{e,r,s_{\downarrow},s_{\uparrow}\\}$ to denote the four actions.
Concretely, the elevation action $a_{e}$ elevate the membership grade of $x$
from $m_{\theta}(\mu_{A}(x))$ to $1$, the reduction action $a_{r}$ reduce the
membership grade of $x$ from $m_{\theta}(\mu_{A}(x))$ to $0$, the elevation
$a_{s_{\uparrow}}$ elevate the membership grade of $x$ from
$m_{\theta}(\mu_{A}(x))$ to $0.5$ if $m_{\theta}(\mu_{A}(x))<0.5$, the
reduction $a_{s_{\downarrow}}$ reduce the membership grade of $x$ from
$m_{\theta}(\mu_{A}(x))$ to $0.5$ if $m_{\theta}(\mu_{A}(x))>0.5$. The fuzzy
membership grade $m_{\theta}(\mu_{A}(x))$ represents the state of object in
the second column, and the errors of different actions are given in the fourth
column, and the losses of different actions are given in the fifth column.

Each of the four losses
$m_{\theta}(\widetilde{\lambda}_{e}),m_{\theta}(\widetilde{\lambda}_{r}),m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})$
and $m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})$ provides the unit cost,
and the actual cost of each action is weighted by the magnitude of its error.
Suppose $R_{a}(x)=m_{\theta}(\widetilde{\lambda}_{a})E_{a}(\mu_{A}(x))$ denote
the loss for taking actions $\\{e,r,s_{\downarrow},s_{\uparrow}\\}$, the
losses of four actions for an object can be computed as

$\displaystyle R_{e}(x)$ $\displaystyle=$ $\displaystyle
m_{\theta}(\widetilde{\lambda}_{e})E_{e}(\mu_{A}(x))=(1-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{e});$
$\displaystyle R_{r}(x)$ $\displaystyle=$ $\displaystyle
m_{\theta}(\widetilde{\lambda}_{r})E_{r}(\mu_{A}(x))=m_{\theta}(\mu_{A}(x))m_{\theta}(\widetilde{\lambda}_{r});$
$\displaystyle R_{s_{\downarrow}}(x)$ $\displaystyle=$ $\displaystyle
m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})E_{s_{\downarrow}}(m_{\theta}(\mu_{A}(x)))=(m_{\theta}(\mu_{A}(x))-0.5)m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}});$
$\displaystyle R_{s_{\uparrow}}(x)$ $\displaystyle=$ $\displaystyle
m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})E_{s_{\uparrow}}(m_{\theta}(\mu_{A}(x)))=(0.5-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}}).$

Since only an action is taken for each object, the total loss of the
approximation is computed by

$\displaystyle R=\sum_{x\in U}R_{a}(x)=\sum_{x\in
U}m_{\theta}(\widetilde{\lambda}_{a})E_{a}(\mu_{A}(x)).$

To minimize the total loss $R$, we take an action $\tau(x)$ that minimizes the
loss $R_{a}(x)$ for each object, and $\tau(x)$ is a solution to the following
minimization problem as

$\text{arg }min_{a\in action}R_{a}(x),$

where $a\in\\{e,r,s_{\downarrow},s_{\uparrow}\\}$.

According to the value $\mu_{A}(x)$ of an object $x$, we have two groups of
decision rules for obtaining three-way approximations of an interval-valued
fuzzy set as follows:

$(1)$ When $m_{\theta}(\mu_{A}(x))\geq 0.5$, $(E1)$ If $R(a_{e}|x)\leq
R(a_{r}|x)$ and $R(a_{e}|x)\leq R(a_{s_{\downarrow}}|x)$, then take action
$a_{e}$; $(R1)$ If $R(a_{r}|x)\leq R(a_{e}|x)$ and $R(a_{r}|x)\leq
R(a_{s_{\downarrow}}|x)$, then take action $a_{r}$; $(S1)$ If
$R(a_{s_{\downarrow}}|x)\leq R(a_{e}|x)$ and $R(a_{s_{\downarrow}}|x)\leq
R(a_{r}|x)$, then take action $a_{s_{\downarrow}}$.

$(2)$ When $m_{\theta}(\mu_{A}(x))<0.5$, $(E2)$ If $R(a_{e}|x)\leq R(a_{r}|x)$
and $R(a_{e}|x)\leq R(a_{s_{\uparrow}}|x)$, then take action $a_{e}$; $(R2)$
If $R(a_{r}|x)\leq R(a_{e}|x)$ and $R(a_{r}|x)\leq R(a_{s_{\uparrow}}|x)$,
then take action $a_{r}$; $(S2):$ If $R(a_{s_{\uparrow}}|x)\leq R(a_{e}|x)$
and $R(a_{s_{\uparrow}}|x)\leq R(a_{r}|x)$, then take action
$a_{s_{\uparrow}}$.

###  5.2 Loss functions-based three-way approximations of interval-valued
fuzzy sets

In this subsection, we consider interval-valued loss functions satisfying
certain properties for obtaining an analytic solution defining a three-way
approximation.

Suppose
$(c1):m_{\theta}(\widetilde{\lambda}_{e})>0,m_{\theta}(\widetilde{\lambda}_{r})>0,m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})>0,m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})>0;$
$(c2):m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})\leq
m_{\theta}(\widetilde{\lambda}_{r});$
$(c3):m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})\leq
m_{\theta}(\widetilde{\lambda}_{e})$, Condition $(c1)$ requires that all costs
are nonnegative; Condition $(c2)$ illustrates that reducing a membership grade
$\mu_{A}(x)\geq 0.5$ to $0.5$ represents a smaller adjustment than reducing it
to $0$, and a smaller cost is associated with action $a_{s_{\downarrow}}$;
Condition $(c3)$ illustrates that elevating a membership grade
$\mu_{A}(x)<0.5$ to $0.5$ represents a smaller adjustment than elevating it to
$1$, and a smaller cost is associated with action $a_{s_{\uparrow}}$. With the
assumptions $(c1)-(c3)$, we simplify the decision rules as follows:

$(1)$ When $m_{\theta}(\mu_{A}(x))\geq 0.5$, the rule $(E1)$ is expressed as

$\displaystyle R(a_{e}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{e})\leq(m_{\theta}(\mu_{A}(x))-0)m_{\theta}(\widetilde{\lambda}_{r})$
$\displaystyle\Leftrightarrow$
$\displaystyle\mu_{A}(x)\geq\frac{m_{\theta}(\widetilde{\lambda}_{e})}{\lambda_{e}m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{r})}=\gamma;$
$\displaystyle R(a_{e}|x)\leq R(a_{s_{\downarrow}}|x)$
$\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{e})\leq(m_{\theta}(\mu_{A}(x))-0.5)m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{2m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})}{2(m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})}=\alpha.$

The rule $R(1)$ is expressed by

$\displaystyle R(a_{r}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle m_{\theta}(\mu_{A}(x))\leq\gamma;$ $\displaystyle
R(a_{r}|x)\leq R(a_{s_{\downarrow}}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda_{r}\leq(m_{\theta}(\mu_{A}(x))-0.5)m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\leq\frac{-m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})}{2(m_{\theta}(\widetilde{\lambda}_{r})-m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})(t))}=\gamma^{-}.$

The rule $S(1)$ is expressed by

$\displaystyle R(a_{s_{\downarrow}}|x)\leq R(a_{e}|x)\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\alpha;R(a_{s_{\downarrow}}|x)\leq
R(a_{s_{\downarrow}}|x)\Leftrightarrow m_{\theta}(\mu_{A}(x))\geq\gamma^{-}.$

Since $\gamma^{-}\leq 0$ contradicts with the assumption
$m_{\theta}(\mu_{A}(x))\geq 0.5$, it is impossible to apply rule $(R1)$ for
reducing membership values. Therefore, when $m_{\theta}(\mu_{A}(x))\geq 0.5$,
the rules are simply expressed as (E1) If $m_{\theta}(\mu_{A}(x))\geq\alpha$,
then $T_{\mu_{A}}(x)=1$; (S1) If $0.5\leq m_{\theta}(\mu_{A}(x))<\alpha$, then
$T_{\mu_{A}}(x)=0.5$.

$(2)$ When $m_{\theta}(\mu_{A}(x))<0.5$, the rule $(E2)$ is expressed as

$\displaystyle R(a_{e}|x)\leq R(a_{r}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{e})\leq(m_{\theta}(\mu_{A}(x))-0)m_{\theta}(\widetilde{\lambda}_{r})$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{m_{\theta}(\widetilde{\lambda}_{e})}{m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{r})}=\gamma;$
$\displaystyle R(a_{e}|x)\leq R(a_{s_{\uparrow}}|x)$
$\displaystyle\Leftrightarrow$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{e})\leq(0.5-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\geq\frac{m_{\theta}(\widetilde{\lambda}_{e})-0.5m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})}{m_{\theta}(\widetilde{\lambda}_{e})-m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})}=\gamma^{+}.$

The rule $R(2)$ is expressed as

$\displaystyle R(a_{r}|x)\leq R(a_{e}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle m_{\theta}(\mu_{A}(x))\leq\gamma;$ $\displaystyle
R(a_{r}|x)\leq R(a_{s_{\uparrow}}|x)$ $\displaystyle\Leftrightarrow$
$\displaystyle
m_{\theta}(\mu_{A}(x))m_{\theta}(\widetilde{\lambda}_{r})\leq(0.5-m_{\theta}(\mu_{A}(x)))m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})$
$\displaystyle\Leftrightarrow$ $\displaystyle
m_{\theta}(\mu_{A}(x))\leq\frac{m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})}{2(m_{\theta}(\widetilde{\lambda}_{r})+m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}}))}=\beta.$

The rule $S(2)$ is expressed as

$\displaystyle R(a_{s_{\uparrow}}|x)\leq R(a_{e}|x)\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\gamma^{+};R(a_{s_{\uparrow}}|x)\leq
R(a_{s_{\uparrow}}|x)\Leftrightarrow m_{\theta}(\mu_{A}(x))\geq\beta.$

Since $\gamma^{+}\geq 1$ contradicts with the assumption
$m_{\theta}(\mu_{A}(x))<0.5$, it is impossible to apply rule $(E2)$ for
elevating membership values. Therefore, when $m_{\theta}(\mu_{A}(x))<0.5$, the
remaining rules are simply expressed as (R2) If
$m_{\theta}(\mu_{A}(x))\leq\beta$, then $T_{\mu_{A}}(x)=0$; (S2) If $\beta\leq
m_{\theta}(\mu_{A}(x))<0.5$, then $T_{\mu_{A}}(x)=0.5$.

By combining the two sets of rules, we immediately have three rules as (E) If
$m_{\theta}(\mu_{A}(x))\geq\alpha(t)$, then $T_{\mu_{A}}(x)=1$; (R) If
$m_{\theta}(\mu_{A}(x))\leq\beta$, then $T_{\mu_{A}}(x)=0$; (S) If
$\beta<m_{\theta}(\mu_{A}(x))<\alpha$, then $T_{\mu_{A}}(x)=0.5$, where

$\displaystyle\alpha=\frac{2m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}})}{2(m_{\theta}(\widetilde{\lambda}_{e})+m_{\theta}(\widetilde{\lambda}_{s_{\downarrow}}))}\text{
and
}\beta=\frac{m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}})}{2(m_{\theta}(\widetilde{\lambda}_{r})+m_{\theta}(\widetilde{\lambda}_{s_{\uparrow}}))}.$

##  6 Interval-valued loss functions-based three-way approximations of
interval-valued fuzzy sets: II

In this section, we introduce another framework for decision-theoretic rough
sets-based three-way approximations of interval-valued fuzzy sets.

###### Definition 6.1

Let $\widetilde{\mu}_{1}=[\mu^{-}_{1},\mu^{+}_{1}]$ and
$\widetilde{\mu}_{2}=[\lambda^{-}_{2},\mu^{+}_{2}]$ be interval-valued sets,
then the degree of possibility of $\widetilde{\mu}_{1}\geq\widetilde{\mu}_{2}$
and $\widetilde{\mu}_{2}\geq\widetilde{\mu}_{1}$ are defined as

$\displaystyle p(\widetilde{\mu}_{1}\geq\widetilde{\mu}_{2})$ $\displaystyle=$
$\displaystyle
max\\{1-max\\{\frac{\mu_{2}^{+}-\mu_{1}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}},0\\},0\\};$
$\displaystyle p(\widetilde{\mu}_{2}\geq\widetilde{\mu}_{1})$ $\displaystyle=$
$\displaystyle
max\\{1-max\\{\frac{\mu_{1}^{+}-\mu_{2}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}},0\\},0\\}.$

In the sense of Definition 6.1, we have

$p(\widetilde{\mu}_{1}\geq\widetilde{\mu}_{2})=\left\\{\begin{array}[]{ccc}0,&{\rm}\hfil&\frac{\mu_{2}^{+}-\mu_{1}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}}\geq
1;\\\
1-\frac{\mu_{2}^{+}-\mu_{1}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}},&{\rm}\hfil&0<\frac{\mu_{2}^{+}-\mu_{1}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}}<1;\\\
1,&{\rm}\hfil&\frac{\mu_{2}^{+}-\mu_{1}^{-}}{\mu_{1}^{+}-\mu_{1}^{-}+\mu_{2}^{+}-\mu_{2}^{-}}\leq
0.\end{array}\right.$

Furthermore, we have the complementary matrix of the preference as

$\displaystyle
P_{\widetilde{\mu}_{1}\widetilde{\mu}_{2}\widetilde{\mu}_{3}}=\left[\begin{array}[]{ccc}p(\widetilde{\mu}_{1}\geq\widetilde{\mu}_{1})&p(\widetilde{\mu}_{1}\geq\widetilde{\mu}_{2})&p(\widetilde{\mu}_{1}\geq\widetilde{\mu}_{3})\\\
p(\widetilde{\mu}_{2}\geq\widetilde{\mu}_{1})&p(\widetilde{\mu}_{2}\geq\widetilde{\mu}_{2})&p(\widetilde{\mu}_{2}\geq\widetilde{\mu}_{3})\\\
p(\widetilde{\mu}_{3}\geq\widetilde{\mu}_{1})&p(\widetilde{\mu}_{3}\geq\widetilde{\mu}_{2})&p(\widetilde{\mu}_{3}\geq\widetilde{\mu}_{3})\\\
\end{array}\right].$

Suppose $\widetilde{R}_{a}(x)=\widetilde{\lambda}_{a}E_{a}(\mu_{A}(x))$ denote
the loss for taking actions $\\{e,r,s_{\downarrow},s_{\uparrow}\\}$, the
losses of four actions for an object can be computed as

$\displaystyle\widetilde{R}_{e}(x)$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{e}E_{e}(\mu_{A}(x))=[(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e},(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}];$
$\displaystyle\widetilde{R}_{r}(x)$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{r}E_{r}(\mu_{A}(x))=[m_{\theta}(\mu_{A}(x))\lambda^{-}_{r},m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}];$
$\displaystyle\widetilde{R}_{s_{\downarrow}}(x)$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{s_{\downarrow}}E_{s_{\downarrow}}(m_{\theta}(\mu_{A}(x)))=[(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}},(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}];$
$\displaystyle\widetilde{R}_{s_{\uparrow}}(x)$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{s_{\uparrow}}E_{s_{\uparrow}}(m_{\theta}(\mu_{A}(x)))=[(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}},(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}].$

Since only an action is taken for each object, the total loss of the
approximation is computed by

$\displaystyle\widetilde{R}=\sum_{x\in U}\widetilde{R}_{a}(x)=\sum_{x\in
U}\widetilde{\lambda}_{a}E_{a}(m_{\theta}(\mu_{A}(x))).$

To minimize the total loss $\widetilde{R}$, we take an action $\tau(x)$ that
minimizes the loss $\widetilde{R}_{a}(x)$ for each object, and $\tau(x)$ is a
solution to the following minimization problem as

$\text{arg }min_{a\in action}\widetilde{R}_{a}(x),$

where $a\in\\{e,r,s_{\downarrow},s_{\uparrow}\\}$.

According to the value $\mu_{A}(x)$ of an object $x$, we have two groups of
decision rules for obtaining three-way approximations of an interval-valued
fuzzy set.

In what follows, in light of complementary matrix of the preference, we
discuss the ranking of the expected loss $\widetilde{R}_{a}(x)$ and generate
decision rules in the context of interval-valued fuzzy sets. Concretely, there
are two situations to discuss: $m_{\theta}(\mu_{A}(x))\geq 0.5$ and
$m_{\theta}(\mu_{A}(x))<0.5$.

###  6.1 Situation 1: $m_{\theta}(\mu_{A}(x))\geq 0.5$

If $m_{\theta}(\mu_{A}(x))\geq 0.5$, then we have the complementary matrix of
the preference as

$\displaystyle
P_{ers_{\downarrow}}=\left[\begin{array}[]{ccc}p_{ee}&p_{er}&p_{es_{\downarrow}}\\\
p_{re}&p_{rr}&p_{rs_{\downarrow}}\\\
p_{s_{\downarrow}e}&p_{s_{\downarrow}r}&p_{s_{\downarrow}s_{\downarrow}}\\\
\end{array}\right].$

According to the properties of the degree of possibilities, we have
$p_{ee}=p_{rr}=p_{s_{\downarrow}s_{\downarrow}}=0.5,p_{er}+p_{re}=1,p_{es_{\downarrow}}+p_{s_{\downarrow}e}=1$
and $p_{rs_{\downarrow}}+p_{s_{\downarrow}r}=1.$ Then we simplify the
complementary matrix as

$\displaystyle
P_{ers_{\downarrow}}=\left[\begin{array}[]{ccc}0.5&p_{er}&p_{es_{\downarrow}}\\\
1-p_{er}&0.5&p_{rs_{\downarrow}}\\\
1-p_{es_{\downarrow}}&1-p_{rs_{\downarrow}}&0.5\\\ \end{array}\right].$

In light of the complementary matrix $P_{ers_{\downarrow}}$, all elements in
each line of the matrix are summarized as

$\displaystyle
p_{e}=0.5+p_{er}+p_{es_{\downarrow}};p_{r}=0.5-p_{er}+p_{rs_{\downarrow}};p_{s_{\downarrow}}=2.5-p_{es_{\downarrow}}-p_{rs_{\downarrow}},$

where $p_{e}$ is the total degree of preference of $\widetilde{R}_{e}(x);$
$p_{r}$ is the total degree of preference of $\widetilde{R}_{r}(x);$
$p_{s_{\downarrow}}$ is the total degree of preference of
$\widetilde{R}_{s_{\downarrow}}(x)$. The values of $p_{e},p_{r}$ and
$p_{s_{\downarrow}}$ depend on $p_{er},p_{es_{\downarrow}}$ and
$p_{rs_{\downarrow}}$. We immediately have three rules as (E) If $p_{e}\leq
p_{r}$ and $p_{e}\leq p_{s_{\downarrow}}$, then $T_{\mu_{A}}(x)=1$; (S) If
$p_{s_{\downarrow}}\leq p_{e}$ and $p_{s_{\downarrow}}\leq p_{r}$, then
$T_{\mu_{A}}(x)=0.5$;(R) If $p_{r}\leq p_{e}$ and $p_{r}\leq
p_{s_{\downarrow}}$, then $T_{\mu_{A}}(x)=0$.

Table 4: The complementary matrix for situation 1. p | $\widetilde{R}_{e}(x)$ | $\widetilde{R}_{r}(x)$ | $\widetilde{R}_{s_{\downarrow}}(x)$
---|---|---|---
$\widetilde{R}_{e}(x)$ | $p_{ee}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{e}(x))$ | $p_{er}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{r}(x))$ | $p_{es_{\downarrow}}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{s_{\downarrow}}(x))$
$\widetilde{R}_{r}(x)$ | $p_{re}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{e}(x))$ | $p_{rr}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{r}(x))$ | $p_{rs_{\downarrow}}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{s_{\downarrow}}(x))$
$\widetilde{R}_{s_{\downarrow}}(x)$ | $p_{s_{\downarrow}e}=p(\widetilde{R}_{s_{\downarrow}}(x)\geq\widetilde{R}_{e}(x))$ | $p_{s_{\downarrow}r}=p(\widetilde{R}_{s_{\downarrow}}(x)\geq\widetilde{R}_{r}(x))$ | $p_{s_{\downarrow}s_{\downarrow}}=p(\widetilde{R}_{s_{\downarrow}}(x)\geq\widetilde{R}_{s_{\downarrow}}(x))$

In consideration of Definition 6.1,
$p_{er}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{r}(x))$ has three kinds of
possible results: (I): $p_{er}=0$, (II): $0<p_{er}<1,$ and (III): $p_{er}=1$.
Furthermore, we have the similar results for $p_{es_{\downarrow}}$ and
$p_{rs_{\downarrow}}$.

(1) For $p_{er},$ if $p_{er}=0$, we have

$\displaystyle\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}\geq
1.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\geq(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{\lambda^{+}_{e}}{\lambda^{-}_{r}+\lambda^{+}_{e}}.$

If $1>p_{er}>0$, we have

$\displaystyle
1>\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}>0.$

In other words, we have

$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}$ $\displaystyle>$
$\displaystyle m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\Leftrightarrow
m_{\theta}(\mu_{A}(x))<\frac{\lambda^{+}_{e}}{\lambda^{-}_{r}+\lambda^{+}_{e}};$
$\displaystyle m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}$ $\displaystyle>$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))<\frac{\lambda^{-}_{e}}{\lambda^{+}_{r}+\lambda^{-}_{e}}.$

If $p_{er}=1$, we have

$\displaystyle\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}\leq
0.$

In other words, we have

$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\geq
m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{\lambda^{-}_{e}}{\lambda^{+}_{r}+\lambda^{-}_{e}}.$

(2) For $p_{es_{\downarrow}},$ if $p_{es_{\downarrow}}=0$, we have

$\displaystyle\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}\geq
1.$

In other words, we have

$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}\geq(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{0.5\lambda^{-}_{s_{\downarrow}}+\lambda^{+}_{e}}{\lambda^{-}_{s_{\downarrow}}+\lambda^{+}_{e}}.$

If $1>p_{es_{\downarrow}}>0$, we have

$\displaystyle
1>\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}>0.$

In other words, we have

$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}$
$\displaystyle>$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{0.5\lambda^{+}_{s_{\downarrow}}+\lambda^{-}_{e}}{\lambda^{-}_{e}+\lambda^{+}_{s_{\downarrow}}};$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}$ $\displaystyle>$
$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))<\frac{0.5\lambda^{-}_{s_{\downarrow}}+\lambda^{+}_{e}}{\lambda^{-}_{s_{\downarrow}}+\lambda^{+}_{e}}.$

If $p_{es_{\downarrow}}=1$, we have

$\displaystyle\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}\leq
0.$

In other words, we have

$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}\leq(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\frac{0.5\lambda^{+}_{s_{\downarrow}}+\lambda^{-}_{e}}{\lambda^{+}_{s_{\downarrow}}-\lambda^{+}_{r}}.$

(3) For $p_{rs_{\downarrow}},$ if $p_{rs_{\downarrow}}=0$, we have

$\displaystyle\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}\geq
1.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}\leq(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\frac{0.5\lambda^{-}_{s_{\downarrow}}}{\lambda^{-}_{s_{\downarrow}}-\lambda^{+}_{r}}.$

If $0<p_{rs_{\downarrow}}<1$, we have

$\displaystyle
1>\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}>0.$

In other words, we have

$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}$
$\displaystyle>$ $\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\Leftrightarrow
m_{\theta}(\mu_{A}(x))<\frac{0.5\lambda^{+}_{s_{\downarrow}}}{\lambda^{+}_{s_{\downarrow}}-\lambda^{-}_{r}};$
$\displaystyle m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}$ $\displaystyle>$
$\displaystyle(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{0.5\lambda^{-}_{s_{\downarrow}}}{\lambda^{+}_{s_{\downarrow}}-\lambda^{+}_{r}}.$

If $p_{rs_{\downarrow}}=1$, we have

$\displaystyle\frac{(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{m_{\theta}(\mu_{A}(x))\lambda^{+}_{e}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{e}+(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}-(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{-}_{s_{\downarrow}}}\leq
0.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\geq(m_{\theta}(\mu_{A}(x))-0.5)\lambda^{+}_{s_{\downarrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{0.5\lambda^{+}_{s_{\downarrow}}}{\lambda^{+}_{s_{\downarrow}}-\lambda^{-}_{r}};$

Table 5: Types of operations for situation 1 |  $p_{er}$ |  $p_{es_{\downarrow}}$ |  $p_{rs_{\downarrow}}$ |  $T_{\mu_{A}}(x)$
---|---|---|---|---
$1$ |  I |  I |  I |  1
$2$ |  I |  I |  II |  1
$3$ |  I |  I |  III |  1
$4$ |  I |  II |  I |  1
$5$ |  I |  II |  II |  1 or  0.5
$6$ |  I |  II |  III |  1 or  0.5
$7$ |  I |  III |  I |  1
$8$ |  I |  III |  II |  0.5
$9$ |  I |  III |  III |  0.5
$10$ |  II |  I |  I |  1 or  0
$11$ |  II |  I |  II |  1 or  0  or  0.5
$12$ |  II |  I |  III |  1
$13$ |  II |  II |  I |  1 or  0
$14$ |  II |  II |  II |  1 or  0.5 or  0
$15$ |  II |  II |  III |  1 or  0.5
$16$ |  II |  III |  I |  0
$17$ |  II |  III |  II |  0.5 or  0
$18$ |  II |  III |  III |  0.5
$19$ |  III |  I |  I |  0
$20$ |  III |  I |  II |  0
$21$ |  III |  I |  III |  1
$22$ |  III |  II |  I |  0
$23$ |  III |  II |  II |  0.5 or  0
$24$ |  III |  II |  III |  0.5
$25$ |  III |  III |  I |  0
$26$ |  III |  III |  II |  0.5 or  0
$27$ |  III |  III |  III |  0.5
Table 6: Special types of operations for situation 1. Type  |  Condition  |  $T_{\mu_{A}}(x)$
---|---|---
$5$ |  $2p_{es_{\downarrow}}+p_{rs_{\downarrow}}\leq 2$ |  1
|  $2p_{es_{\downarrow}}+p_{rs_{\downarrow}}>2$ |  0.5
$6$ |  $p_{es_{\downarrow}}\leq\frac{1}{2}$ |  0
|  $p_{es_{\downarrow}}>\frac{1}{2}$ |  0.5
$10$ |  $p_{er}\leq\frac{1}{2}$ |  1
|  $p_{er}>\frac{1}{2}$ |  0
$11$ |  $2p_{es_{\downarrow}}-p_{rs_{\downarrow}}\leq 1$ |  1
|  $2p_{es_{\downarrow}}-p_{rs_{\downarrow}}>1$ |  0
$13$ |  $2p_{er}+p_{rs_{\downarrow}}\leq 1\wedge p_{er}+2p_{es_{\downarrow}}\leq 2$ |  1
|  $2p_{er}+p_{rs_{\downarrow}}\leq 1\wedge-p_{er}+2p_{es_{\downarrow}}\leq 1$ |  0
$14$ |  $2p_{er}+p_{es_{\downarrow}}-p_{rs_{\downarrow}}\leq 1\wedge p_{er}+2p_{es_{\downarrow}}+p_{rs_{\downarrow}}\leq 2$ |  1
|  $1\leq 2p_{er}+p_{es_{\downarrow}}-p_{rs_{\downarrow}}\leq 1\wedge p_{es_{\downarrow}}-p_{er}+2p_{rs_{\downarrow}}\leq 1$ |  0
|  $2\leq p_{er}+2p_{es_{\downarrow}}+p_{rs_{\downarrow}}\wedge 1\leq p_{es_{\downarrow}}-p_{er}+2p_{rs_{\downarrow}}\leq 1$ |  0
$15$ |  $2p_{er}+p_{es_{\downarrow}}\leq 2\wedge p_{er}+2p_{rs_{\downarrow}}\leq 1$ |  1
|  $1\leq 2p_{er}+p_{es_{\downarrow}}\wedge p_{er}-p_{es_{\downarrow}}\leq 1$ |  0.5
$17$ |  $2p_{rs_{\downarrow}}-p_{er}\leq 0$ |  0
|  $2p_{rs_{\downarrow}}-p_{er}>0$ |  0.5
$23$ |  $2p_{rs_{\downarrow}}+p_{es_{\downarrow}}\leq 2$ |  0
|  $2p_{rs_{\downarrow}}+p_{er}>2$ |  0.5
$26$ |  $p_{rs_{\downarrow}}\leq 0.5$ |  0
|  $p_{rs_{\downarrow}}>0.5$ |  0.5

###### Example 6.2

(Continuation of Example 3.3) Let
$\widetilde{\lambda}_{e}=[1,2],\widetilde{\lambda}_{r}=[5,6]$ and
$\widetilde{\lambda}_{s_{\downarrow}}=[3,4]$ when $m_{\theta}(\mu_{A}(x))\geq
0.5,$ we have that

$\displaystyle\widetilde{R}_{e}(x_{2})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{e}E_{e}(m_{\theta}(\mu_{A}(x_{2})))=[0.3\lambda^{-}_{e},0.3\lambda^{+}_{e}]=[0.3,0.6];$
$\displaystyle\widetilde{R}_{r}(x_{2})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{r}E_{r}(m_{\theta}(\mu_{A}(x_{2})))=[0.7\lambda^{-}_{r},0.7\lambda^{+}_{r}]=[3.5,4.2];$
$\displaystyle\widetilde{R}_{s_{\downarrow}}(x_{2})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{s_{\downarrow}}E_{s_{\downarrow}}(m_{\theta}(\mu_{A}(x_{2})))=[0.2\lambda^{-}_{s_{\downarrow}},0.2\lambda^{+}_{s_{\downarrow}}]=[0.6,0.8].$

In light of the complementary matrix $P_{ers_{\downarrow}}$, we have

$\displaystyle
P_{ers_{\downarrow}}=\left[\begin{array}[]{ccc}p_{ee}&p_{er}&p_{es_{\downarrow}}\\\
p_{re}&p_{rr}&p_{rs_{\downarrow}}\\\
p_{s_{\downarrow}e}&p_{s_{\downarrow}r}&p_{s_{\downarrow}s_{\downarrow}}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0.5&0&0\\\ 1&0.5&1\\\ 1&0&0.5\\\
\end{array}\right].$

Therefore, we have that elevating the membership grade of $x_{2}$ to 1 is the
best choice.

###  6.2 Situation 2: $m_{\theta}(\mu_{A}(x))<0.5$

For $m_{\theta}(\mu_{A}(x))\leq 0.5$, we have

$\displaystyle
P_{ers_{\uparrow}}=\left[\begin{array}[]{ccc}p_{ee}&p_{er}&p_{es_{\uparrow}}\\\
p_{re}&p_{rr}&p_{rs_{\uparrow}}\\\
p_{s_{\uparrow}e}&p_{s_{\uparrow}r}&p_{s_{\uparrow}s_{\uparrow}}\\\
\end{array}\right].$

According to the properties of the degree of possibilities, we have
$p_{ee}=p_{rr}=p_{s_{\uparrow}s_{\uparrow}}=0.5,p_{er}+p_{re}=1,p_{es_{\uparrow}}+p_{s_{\uparrow}e}=1$
and $p_{rs_{\uparrow}}+p_{s_{\uparrow}r}=1.$ Then we simplify the matrix as

$\displaystyle
P_{ers_{\uparrow}}=\left[\begin{array}[]{ccc}0.5&p_{er}&p_{es_{\uparrow}}\\\
1-p_{er}&0.5&p_{rs_{\uparrow}}\\\
1-p_{es_{\uparrow}}&1-p_{rs_{\uparrow}}&0.5\\\ \end{array}\right].$

In light of the complementary matrix $P_{ers_{\uparrow}}$, all elements in
each line of the matrix are summarized as

$\displaystyle
p_{e}=0.5+p_{er}+p_{es_{\uparrow}};p_{r}=0.5-p_{er}+p_{rs_{\uparrow}};p_{s_{\uparrow}}=2.5-p_{es_{\uparrow}}-p_{rs_{\uparrow}},$

where $p_{e}$ is the total degree of preference of $\widetilde{R}_{e}(x);$
$p_{r}$ is the total degree of preference of $\widetilde{R}_{r}(x);$
$p_{s_{\uparrow}}$ is the total degree of preference of
$\widetilde{R}_{s_{\uparrow}}(x)$. The values of $p_{e},p_{r}$ and
$p_{s_{\uparrow}}$ depend on $p_{er},p_{es_{\uparrow}}$ and
$p_{rs_{\uparrow}}$. We immediately have three rules as (E) If $p_{e}\leq
p_{r}$ and $p_{e}\leq p_{s_{\uparrow}}$, then $T_{\mu_{A}}(x)=1$; (S) If
$p_{s_{\uparrow}}\leq p_{e}$ and $p_{s_{\uparrow}}\leq p_{r}$, then
$T_{\mu_{A}}(x)=0.5$;(R) If $p_{r}\leq p_{e}$ and $p_{r}\leq p_{s_{v}}$, then
$T_{\mu_{A}}(x)=0$.

Table 7: The complementary matrix for situation 2. p | $\widetilde{R}_{e}(x)$ | $\widetilde{R}_{r}(x)$ | $\widetilde{R}_{s_{\uparrow}}(x)$
---|---|---|---
$\widetilde{R}_{e}(x)$ | $p_{ee}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{e}(x))$ | $p_{er}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{r}(x))$ | $p_{es_{\uparrow}}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{s_{\uparrow}}(x))$
$\widetilde{R}_{r}(x)$ | $p_{re}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{e}(x))$ | $p_{rr}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{r}(x))$ | $p_{rs_{\uparrow}}=p(\widetilde{R}_{r}(x)\geq\widetilde{R}_{s_{\uparrow}}(x))$
$\widetilde{R}_{s_{\uparrow}}(x)$ | $p_{s_{\uparrow}e}=p(\widetilde{R}_{s_{\uparrow}}(x)\geq\widetilde{R}_{e}(x))$ | $p_{s_{\uparrow}r}=p(\widetilde{R}_{s_{\uparrow}}(x)\geq\widetilde{R}_{r}(x))$ | $p_{s_{\uparrow}s_{\uparrow}}=p(\widetilde{R}_{s_{\uparrow}}(x)\geq\widetilde{R}_{s_{\uparrow}}(x))$

In consideration of Definition 6.1,
$p_{er}=p(\widetilde{R}_{e}(x)\geq\widetilde{R}_{r}(x))$ have three kinds of
possible results: (I): $p_{er}=0$, (II): $0<p_{er}<1;$ (III): $p_{er}=1$.
Furthermore, we have the similar results for $p_{es_{\downarrow}}$ and
$p_{rs_{\downarrow}}$.

(1) For $p_{er},$ if $p_{er}=0$, we have

$\displaystyle\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}\geq
1.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\geq(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{\lambda^{+}_{e}}{\lambda^{-}_{r}+\lambda^{+}_{e}}.$

If $0<p_{er}<1$, we have

$\displaystyle
1>\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}>0.$

In other words, we have

$\displaystyle m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}$ $\displaystyle>$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{\lambda^{+}_{e}}{\lambda^{+}_{e}+\lambda^{-}_{r}};$
$\displaystyle m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}$ $\displaystyle>$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{\lambda^{-}_{e}}{\lambda^{+}_{r}+\lambda^{-}_{e}}.$

If $p_{er}=1$, we have

$\displaystyle\frac{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}\leq
0.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}\leq(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\frac{\lambda^{-}_{e}}{\lambda^{+}_{r}+\lambda^{-}_{e}}.$

(2) For $p_{es_{\uparrow}},$ if $p_{es_{\uparrow}}=0$, we have

$\displaystyle\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}\geq
1.$

In other words, we have

$\displaystyle(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}\geq(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{0.5\lambda^{-}_{s_{\uparrow}}-\lambda^{+}_{e}}{\lambda^{-}_{s_{\uparrow}}-\lambda^{+}_{e}}.$

If $1>p_{es_{\uparrow}}>0$, we have

$\displaystyle
1>\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}>0.$

In other words, we have

$\displaystyle(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}$
$\displaystyle>$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{\lambda^{-}_{e}-0.5\lambda^{+}_{s_{\uparrow}}}{\lambda^{-}_{e}-\lambda^{+}_{s_{\uparrow}}};$
$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}$ $\displaystyle>$
$\displaystyle(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{0.5\lambda^{-}_{s_{\uparrow}}-\lambda^{-}_{e}}{\lambda^{-}_{s_{\uparrow}}-\lambda^{+}_{e}}.$

If $p_{es_{\uparrow}}=1$, we have

$\displaystyle\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{e}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}\leq
0.$

In other words, we have

$\displaystyle(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{e}\geq(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\frac{0.5\lambda^{+}_{s_{\uparrow}}-\lambda^{-}_{e}}{\lambda^{+}_{s_{\uparrow}}-\lambda^{-}_{e}}.$

(3) For $p_{rs_{\uparrow}},$ if $p_{rs_{\uparrow}}=0$, we have

$\displaystyle\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}\geq
1.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}\leq(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\leq\frac{0.5\lambda^{-}_{s_{\uparrow}}}{\lambda^{-}_{s_{\uparrow}}+\lambda^{+}_{r}}.$

If $1>p_{rs_{\downarrow}}>0$, we have

$\displaystyle
1>\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}>0.$

In other words, we have

$\displaystyle(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}$
$\displaystyle>$ $\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\Leftrightarrow
m_{\theta}(\mu_{A}(x))<\frac{0.5\lambda^{+}_{s_{\uparrow}}}{\lambda^{+}_{s_{\uparrow}}+\lambda^{-}_{r}};$
$\displaystyle(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}$
$\displaystyle>$ $\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{+}_{r}\Leftrightarrow
m_{\theta}(\mu_{A}(x))>\frac{0.5\lambda^{-}_{s_{\uparrow}}}{\lambda^{-}_{s_{\uparrow}}+\lambda^{+}_{r}}.$

If $p_{rs_{\downarrow}}=1$, we have

$\displaystyle\frac{(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}}{(1-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{r}-(1-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{r}+(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}-(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{-}_{s_{\uparrow}}}\leq
0.$

In other words, we have

$\displaystyle
m_{\theta}(\mu_{A}(x))\lambda^{-}_{r}\geq(0.5-m_{\theta}(\mu_{A}(x)))\lambda^{+}_{s_{\uparrow}}\Leftrightarrow
m_{\theta}(\mu_{A}(x))\geq\frac{0.5\lambda^{+}_{s_{\uparrow}}}{\lambda^{-}_{r}+\lambda^{+}_{s_{\uparrow}}}.$

Table 8: Types of operations for situation 2. |  $p_{er}$ |  $p_{es_{\uparrow}}$ |  $p_{rs_{\uparrow}}$ |  $T_{\mu_{A}}(x)$
---|---|---|---|---
$1$ |  I |  I |  I |  1
$2$ |  I |  I |  II |  1
$3$ |  I |  I |  III |  1
$4$ |  I |  II |  I |  1
$5$ |  I |  II |  II |  1 or  0.5
$6$ |  I |  II |  III |  1 or  0.5
$7$ |  I |  III |  I |  1
$8$ |  I |  III |  II |  0.5
$9$ |  I |  III |  III |  0.5
$10$ |  II |  I |  I |  1 or  0
$11$ |  II |  I |  II |  1 or  0  or  0.5
$12$ |  II |  I |  III |  1
$13$ |  II |  II |  I |  1 or  0
$14$ |  II |  II |  II |  1 or  0.5 or  0
$15$ |  II |  II |  III |  1 or  0.5
$16$ |  II |  III |  I |  0
$17$ |  II |  III |  II |  0.5 or  0
$18$ |  II |  III |  III |  0.5
$19$ |  III |  I |  I |  0
$20$ |  III |  I |  II |  0
$21$ |  III |  I |  III |  1
$22$ |  III |  II |  I |  0
$23$ |  III |  II |  II |  0.5 or  0
$24$ |  III |  II |  III |  0.5
$25$ |  III |  III |  I |  0
$26$ |  III |  III |  II |  0.5 or  0
$27$ |  III |  III |  III |  0.5
Table 9: Special types of operations for situation 2. Type  |  Condition  |  $T_{\mu_{A}}(x)$
---|---|---
$5$ |  $2p_{es_{\uparrow}}+p_{rs_{\uparrow}}\leq 2$ |  1
|  $2p_{es_{\uparrow}}+p_{rs_{\uparrow}}>2$ |  0.5
$6$ |  $p_{es_{\uparrow}}\leq\frac{1}{2}$ |  0
|  $p_{es_{\uparrow}}>\frac{1}{2}$ |  0.5
$10$ |  $p_{er}\leq\frac{1}{2}$ |  1
|  $p_{er}>\frac{1}{2}$ |  0
$11$ |  $2p_{es_{\uparrow}}-p_{rs_{\uparrow}}\leq 1$ |  1
|  $2p_{es_{\uparrow}}-p_{rs_{\uparrow}}>1$ |  0
$13$ |  $2p_{er}+p_{rs_{\uparrow}}\leq 1\wedge p_{er}+2p_{es_{\uparrow}}\leq 2$ |  1
|  $2p_{er}+p_{rs_{\uparrow}}\leq 1\wedge-p_{er}+2p_{es_{\uparrow}}\leq 1$ |  0
$14$ |  $2p_{er}+p_{es_{\uparrow}}-p_{rs_{\uparrow}}\leq 1\wedge p_{er}+2p_{es_{\uparrow}}+p_{rs_{\uparrow}}\leq 2$ |  1
|  $1\leq 2p_{er}+p_{es_{\uparrow}}-p_{rs_{\uparrow}}\leq 1\wedge p_{es_{\uparrow}}-p_{er}+2p_{rs_{\downarrow}}\leq 1$ |  0
|  $2\leq p_{er}+2p_{es_{\downarrow}}+p_{rs_{\downarrow}}\wedge 1\leq p_{es_{\uparrow}}-p_{er}+2p_{rs_{\uparrow}}\leq 1$ |  0
$15$ |  $2p_{er}+p_{es_{\uparrow}}\leq 2\wedge p_{er}+2p_{rs_{\uparrow}}\leq 1$ |  1
|  $1\leq 2p_{er}+p_{es_{\uparrow}}\wedge p_{er}-p_{es_{\uparrow}}\leq 1$ |  0.5
$17$ |  $2p_{rs_{\uparrow}}-p_{er}\leq 0$ |  0
|  $2p_{rs_{\uparrow}}-p_{er}>0$ |  0.5
$23$ |  $2p_{rs_{\uparrow}}+p_{es_{\uparrow}}\leq 2$ |  0
|  $2p_{rs_{\uparrow}}+p_{er}>2$ |  0.5
$26$ |  $p_{rs_{\uparrow}}\leq 0.5$ |  0
|  $p_{rs_{\uparrow}}>0.5$ |  0.5

###### Example 6.3

(Continuation of Example 3.3) Let
$\widetilde{\lambda}_{e}=[5,6],\widetilde{\lambda}_{r}=[1,2]$ and
$\widetilde{\lambda}_{s_{\uparrow}}=[3,4]$ when $m_{\theta}(\mu_{A}(x))<0.5,$
we have

$\displaystyle\widetilde{R}_{e}(x_{3})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{e}E_{e}(m_{\theta}(\mu_{A}(x_{3})))=[0.6\lambda^{-}_{e},0.6\lambda^{+}_{e}]=[3,3.6];$
$\displaystyle\widetilde{R}_{r}(x_{3})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{r}E_{r}(m_{\theta}(\mu_{A}(x_{3})))=[0.4\lambda^{-}_{r},0.4\lambda^{+}_{r}]=[0.4,0.8];$
$\displaystyle\widetilde{R}_{s_{\uparrow}}(x_{3})$ $\displaystyle=$
$\displaystyle\widetilde{\lambda}_{s_{\uparrow}}E_{s_{\uparrow}}(m_{\theta}(\mu_{A}(x_{3})))=[0.1\lambda^{-}_{s_{\uparrow}},0.1\lambda^{+}_{s_{\uparrow}}]=[0.3,0.4].$

In light of the complementary matrix $P_{ers_{\uparrow}}$, we have

$\displaystyle
P_{ers_{\uparrow}}=\left[\begin{array}[]{ccc}p_{ee}&p_{er}&p_{es_{\uparrow}}\\\
p_{re}&p_{rr}&p_{rs_{\uparrow}}\\\
p_{s_{\uparrow}e}&p_{s_{\uparrow}r}&p_{s_{\uparrow}s_{\uparrow}}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0.5&1&1\\\ 0&0.5&1\\\ 0&0&0.5\\\
\end{array}\right].$

Therefore, we have that elevating the membership of $x_{3}$ to 0.5 is the best
choice.

##  7 Four semantics issues of this model

In this section, we investigate four semantics issues of decision-theoretic
three-way approximations of interval-valued fuzzy sets.

(1) Interpretations of interval-valued loss functions

In decision-theoretic three-way approximations of interval-valued fuzzy sets,
the pair of thresholds depends on the choice of interval-valued loss functions
which are fundamental notions of the decision-theoretic model. In other words,
given an interval-valued loss function, the pair of thresholds can be computed
accordingly. On the other hand, if the pair of thresholds is interpreted in
terms of an interval-valued loss function, then the user can provide a better
estimation of the thresholds in time. Therefore, the decision-theoretic model
gives an interpretation of the pair of thresholds, and it is important to
discuss approximations of interval-valued fuzzy sets by using interval-valued
loss functions.

(2) Relationships to shadowed sets of fuzzy sets

In Sections 3 and 4, we see that three regions of decision-theoretic rough
set-based three-way approximation $T_{\mu_{(A)}}$ and shadowed set
$S_{\mu_{(A)}}$ are both defined through a pair of thresholds $\alpha$ and
$\beta$. For shadowed sets, the objective function is given with respect to
the membership functions, and different membership functions will produce
different shadowed sets. In contrast, the objective function of the decision-
theoretic framework is given with respect to interval-valued loss functions,
which is independent of any particular fuzzy membership functions.

(3) Relationships to decision-theoretic rough sets

In Section 4, we adopt the main ideas from decision-theoretic rough set in
developing decision-theoretic rough set-based three-way approximations of
interval-valued fuzzy sets. A rough membership function can be viewed as a
fuzzy membership function. There are some differences between three-way
approximations of interval-valued fuzzy sets and decision-theoretic rough
sets. For decision-theoretic rough sets, we deal with two-state three-way
decision problems. A rough membership function denotes the probability that an
object is in the set. On the other hand, three-way approximations of interval-
valued fuzzy sets are a many-state decision problem.

(4) Relationships to decision-theoretic three-way approximations of fuzzy sets

In [7], Deng et al. discussed three-way approximations of fuzzy sets by using
loss functions. In practice, there are a lot of interval-valued loss
functions. Compare with Deng’s model, we discuss three-way approximations of
interval-valued fuzzy sets by using interval-valued loss functions.

##  8 Conclusions

Many researchers have investigated approximations of interval-valued fuzzy
sets. In this paper, firstly, we have presented shadowed sets for interpreting
and understanding interval-valued fuzzy sets. Secondly, we have constructed
decision-theoretic rough set-based three-way approximations of interval-valued
fuzzy sets. Thirdly, we have computed the pair of thresholds for decision-
theoretic rough set-based three-way approximations of interval-valued fuzzy
sets by using interval-valued loss functions. Fourthly, we have constructed
approximations of interval-valued fuzzy sets by using interval-valued loss
functions from another view. Finally, we have employed several examples to
illustrate that how to make a decision for interval-valued fuzzy sets by using
interval-valued loss functions.

There are still many interesting topics deserving further investigations on
fuzzy sets. For example, there are many types of fuzzy sets and loss
functions, and it is of interest to investigate loss functions-based three-way
approximations of interval-valued fuzzy sets. In the future, we will further
investigate interval-valued fuzzy sets and discuss its application in
knowledge discovery.

##  Acknowledgments

We would like to thank the anonymous reviewers very much for their
professional comments and valuable suggestions. This work is supported by the
National Natural Science Foundation of China (NO. 11371130, 11401052,
11401195), the Scientific Research Fund of Hunan Provincial Education
Department (No.14C0049).

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