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Article

# A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures

1
School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an 710021, China
2
Shaanxi Joint Laboratory of Artificial Intelligence, Shaanxi University of Science and Technology, Xi’an 710021, China
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(6), 275; https://doi.org/10.3390/axioms11060275
Received: 8 May 2022 / Revised: 30 May 2022 / Accepted: 1 June 2022 / Published: 6 June 2022
(This article belongs to the Special Issue Soft Computing with Applications to Decision Making and Data Mining)

## Abstract

:
Rough set theory provides a useful tool for data analysis, data mining and decision making. For multi-criteria decision making (MCDM), rough sets are used to obtain decision rules by reducing attributes and objects. However, different reduction methods correspond to different rules, which will influence the decision result. To solve this problem, we propose a novel method for MCDM based on rough sets and a fuzzy measure in this paper. Firstly, a type of non-additive measure of attributes is presented by the importance degree in rough sets, which is a fuzzy measure and called an attribute measure. Secondly, for a decision information system, the notion of the matching degree between two objects is presented under an attribute. Thirdly, based on the notions of the attribute measure and matching degree, a Choquet integral is constructed. Moreover, a novel MCDM method is presented by the Choquet integral. Finally, the presented method is compared with other methods through a numerical example, which is used to illustrate the feasibility and effectiveness of our method.

## 1. Introduction

In 1982, Pawlak [1,2] proposed rough set theory, as a mathematical tool, to deal with various kinds of data in data mining. It has been applied in various issues, such as attribute reduction [3,4,5], rule extraction [6,7,8], knowledge discovery [9,10,11] and feature selection [12,13,14]. To broaden the application ability of Pawlak’s rough set theory in practical problems [11,15], it has been extended by generalized relations [16,17], various coverings [18,19,20] and several types of neighborhoods [4,21]. Moreover, it has been combined with several theories, including lattice theory , matrix theory , fuzzy set theory  and others [25,26].
In multi-criteria decision-making (MCDM) problems , it is difficult to obtain the optimal attribute weight. Hence, different attribute weights will influence the decision results. Pawlak’s rough sets can obtain decision rules to make decisions, which can solve the issue above. Therefore, the decision-making methods based on Pawlak’s rough sets have received more and more attention [28,29]. The decision rules are obtained by reducing attributes and objects in Pawlak’s rough sets. Hence, there are many attribute and object reduction methods, such as the discernibility matrix method [30,31], positive region method [32,33], information entropy method [34,35] and other methods [36,37]. Different reduction methods correspond to different rules, which will influence the decision result. Hence, for the existing rule extraction algorithms of rough sets, the decision value will be not unique. For example, we use the hiring dataset taken from Komorowski et al. in , where all the attributes have nominal values. We use two famous rule extraction algorithms of rough sets, which are the CN2 algorithm  and the LEM2 algorithm , to illustrate this statement. We use the R programming language for these two algorithms (the CN2 algorithm  and the LEM2 algorithm  are at pages 97 and 105 in the package ‘RoughSets’, respectively). The package ‘RoughSets’ can be downloaded from https://CRAN.R-project.org/package=RoughSets accessed on 23 May 2022, and the steps of them are shown as follows: firstly, we use the first seven records to obtain rules by the CN2 algorithm  and the LEM2 algorithm , respectively. Then, we use the obtained rules to make decisions for the eight records $x 8$. The corresponding results are shown in Section 5.3, and we find that the predicted decision value of $x 8$ is not unique by using different values in the CN2 algorithm  and the LEM2 algorithm , respectively. Hence, for a problem of MCDM, different decision rules will influence the decision results. It is necessary to seek a new method of decision making by rough sets.
The research motivations of this paper are listed as follows:
• In rough set theory, the common decision-making method is using decision rules. It is difficult to find the best decision rules, because different methods can obtain different rules, which will influence the decision result. Hence, a new decision-making method based on rough sets should be presented, which will be independent of decision rules.
• In decision-making theory, attribute weights are needed in almost all decision-making methods, such as the WA, OWA and TOPSIS methods. However, it is difficult to obtain the optimal weight value, and many weight values are given artificially. To solve this problem, Choquet integrals can be used to aggregate decision information without attribute weights.
In this paper, a novel MCDM method based on rough sets and fuzzy measures is presented. Firstly, to show the correlation between attributes in a decision information system, a type of non-additive measure of attributes is presented by the importance degree in rough sets. It is called an attribute measure, and some properties of it are presented. Secondly, to describe how close any two objects are to each other in a decision information system, the notion of the matching degree between two objects is presented under an attribute. Thirdly, a Choquet integral is constructed based on the notions of attribute measure and matching degree above. Moreover, a novel MCDM method is presented by the Choquet integral, which can aggregate all information between two objects. Finally, to illustrate the feasibility and effectiveness of our method above, our method is compared with other methods through a numerical example. By the corresponding analysis, our method can address the deficiency of the existing methods well.
The rest of this article is organized as follows: Section 2 recalls several basic notions about Pawlak’s rough sets, fuzzy measures and Choquet integrals. In Section 3, a type of non-additive measure of attributes is presented by the importance degree in rough sets. Moreover, the notion of the matching degree between two objects is presented under an attribute, as well as corresponding Choquet integrals. In Section 4, a novel MCDM method is presented by the Choquet integral. In Section 5, we show the effectiveness and the efficiency of our method by a numerical example. Section 6 concludes this article and indicates further works.

## 2. Basic Definitions

In this section, we recall several concepts in Pawlak’s rough sets, fuzzy measures and Choquet integrals.

#### 2.1. Pawlak’s Rough Sets

We show some notions about Pawlak’s rough sets in [1,41] as follows:
Let $S = ( U , A )$ be an information system, where U is a nonempty finite set of objects and called the universe, and A is a nonempty finite set of attributes such that $a : U → V a$ for any $a ∈ A$, where $V a$ is called the value set of a. The indiscernibility relation induced by A is defined as follows
$I N D ( A ) = { ( x , y ) ∈ U × U : ∀ a ∈ A , a ( x ) = a ( y ) } .$
For every $X ⊆ U$, a pair of approximations $A ¯ ( X )$ and $A ̲ ( X )$ of X are denoted as
$A ¯ ( X ) = { x ∈ U : [ x ] A ⋂ X ≠ ∅ } , A ̲ ( X ) = { x ∈ U : [ x ] A ⊆ X } ,$
where $[ x ] A = { y ∈ U : ( x , y ) ∈ I N D ( A ) }$ and $U / A = { [ x ] A : x ∈ U }$. $A ¯$ and $A ̲$ are called the upper and lower approximation operators with respect to A, respectively.
Let be the empty set and $− X = U − X$. We have the following conclusions about $A ¯$ and $A ̲$.
Proposition 1
([1,41]). Let $S = ( U , A )$ be an information system. For any $X , Y ⊆ U$,
 (1L) $A ̲ ( U ) = U$ (1H) $A ¯ ( U ) = U$ (2L) $A ̲ ( ϕ ) = ϕ$ (2H) $A ¯ ( ϕ ) = ϕ$ (3L) $A ̲ ( X ) ⊆ X$ (3H) $X ⊆ A ¯ ( X )$ (4L) $A ̲ ( X ⋂ Y ) = A ̲ ( X ) ⋂ A ̲ ( Y )$ (4H) $A ¯ ( X ⋃ Y ) = A ¯ ( X ) ⋃ A ¯ ( Y )$ (5L) $A ̲ ( A ̲ ( X ) ) = A ̲ ( X )$ (5H) $A ¯ ( A ¯ ( X ) ) = A ¯ ( X )$ (6L) $X ⊆ Y ⇒ A ̲ ( X ) ⊆ A ̲ ( Y )$ (6H) $X ⊆ Y ⇒ A ¯ ( X ) ⊆ A ¯ ( Y )$ (7L) $A ̲ ( − A ̲ ( X ) ) = − A ̲ ( X )$ (7H) $A ¯ ( − A ¯ ( X ) ) = − A ¯ ( X )$ (8LH) $A ̲ ( − X ) = − A ¯ ( X )$ (9LH) $A ̲ ( X ) ⊆ A ¯ ( X )$
Moreover, Let $S = ( U , A )$ be an information system. For any $B , C ∈ A$ and $X ∈ U$,
$B ̲ ( X ) ⋃ C ̲ ( X ) ⊇ B ⋃ C ̲ ( X ) , B ̲ ( X ) ⋂ C ̲ ( X ) ⊆ B ⋂ C ̲ ( X ) .$
Then, $S = ( U , A ⋃ D )$ is called a decision information system, where A is a conditional attribute set and D is a decision attribute set. The notions of dependency degree and importance degree in the decision information system are shown in the following definition.
Definition 1
([1,41]). Let $S = ( U , A ⋃ D )$ be a decision information system. Then, the dependency degree of D with regard to A in S is
$γ D ( A ) = | P O S A ( D ) | | U | = ∑ X ∈ U / D | A ̲ ( X ) | | U | ,$
where $P O S A ( D ) = ⋃ X ∈ U / D A ̲ ( X )$. For any $B ∈ A$, the importance degree of D with regard to B in S is
$S i g D ( B ) = γ D ( A ) − γ D ( A − B ) .$

#### 2.2. Fuzzy Measures and Choquet Integrals

Firstly, the definition of the fuzzy measure is shown in Definition 2.
Definition 2
([42,43]). Given a universe U and a set function $m$: $P ( U ) → [ 0 , 1 ]$, where $P ( U )$ is the power set of U, $m$ is called a fuzzy measure on U if the following statements hold:
(1)
$m ( ∅ ) = 0$, $m ( U ) = 1$;
(2)
$A , B ⊆ U$, $A ⊆ B$, which implies $m ( A ) ≤ m ( B )$.
Inspired by the notion of the fuzzy measure, a type of fuzzy integral is proposed in Definition 3.
Definition 3
([44,45]). Given a real-valued function $f : U → [ 0 , 1 ]$ with $U = { x 1 , x 2 , ⋯ , x n }$, the Choquet integral of f with respect to the fuzzy measure $m$ is defined as:
$∫ f d m = ∑ i = 1 n [ m ( X ( i ) ) − m ( X ( i + 1 ) ) ] f ( x ( i ) ) ,$
where ${ x ( 1 ) , x ( 2 ) , ⋯ , x ( n ) }$ is a permutation of ${ x 1 , x 2 , ⋯ , x n }$ such that $f ( x ( 1 ) ) ≤ f ( x ( 2 ) ) ≤ ⋯ ≤ f ( x ( n ) )$, $X ( i ) = { x ( i ) , x ( i + 1 ) , ⋯ ,$$x ( n ) }$ and $X ( n + 1 ) = ∅$.
In Definition 3, the real-valued function $f : U → [ 0 , 1 ]$ is called a measurable function, which can be seen as a fuzzy set.

## 3. Fuzzy Rough Measures and Choquet Integrals

In this section, the notions of the attribute measure and matching degree between two objects are presented in a decision information system. The key work of this section is to induce the fuzzy measure and the measurable function from a discrete data table. Based on these new notions, a Choquet integral is constructed.

#### 3.1. Fuzzy Rough Measures Based on Attribute Importance Degrees

In this subsection, a type of non-additive measure of attributes is presented by the importance degree in rough sets, which is a fuzzy measure and called an attribute measure. Moreover, several properties of the attribute measure are proposed. Firstly, the notion of the attribute measure is proposed.
Definition 4.
Let $S = ( U , A ⋃ D )$ be a decision information system. For any $B ⊆ A$, we call $μ ( B )$ an attribute measure of B in S, where
$μ ( B ) = S i g D ( B ) S i g D ( A ) .$
By Definition 4, the notion of the attribute measure reflects the degree of correlation between attribute subset B and attribute set A. It will be a useful tool for describing relational data in rough set theory.
Example 1.
Let $S = ( U , A ⋃ D )$ be a decision information system that provides 7 days’ meteorological observation data, as shown in Table 1, where A is the set of four attributes of weather, and D denotes whether to hold a meeting. The detailed description of each attribute is as follows:
• The conditional attribute ‘$a 1 = W e a t h e r p r e d i c t i o n$’ has values: “Clear = 1”, “Cloudy = 2”, “Rain = 3”.
• The conditional attribute ‘$a 2 = A i r t e m p e r a t u r e$’ has values: “Hot = 1”, “Warm = 2”, “Cool = 3”.
• The conditional attribute ‘$a 3 = W i n d i n e s s$’ has values: “Yes = 0”, “No = 1”.
• The conditional attribute ‘$a 4 = H u m i d i t y$’ has values: “Wet = 1”, “Normal = 2”, “Dry = 3”.
• The conditional attribute ‘D’ has values: “Yes = 1”, “No = 0”.
Then, $U / D = { X 1 , X 2 }$, where $X 1 = { x 1 , x 2 , x 3 , x 4 }$, $X 2 = { x 5 , x 6 , x 7 }$. Hence,
By Definition 1,
Thus,
$S i g D ( A ) = γ D ( A ) − γ D ( ∅ ) = 1 − 0 = 1 .$
Suppose $B = { a 1 , a 2 }$. We have
By Definition 1,
Hence,
$S i g D ( B ) = γ D ( A ) − γ D ( A − B ) = 1 − 0.5714 = 0.4286 .$
Therefore, by Definition 4, we have
$μ ( B ) = S i g D ( B ) S i g D ( A ) = 0.4286 1 = 0.4286 .$
Several properties of the attribute measure in Definition 4 are proposed below.
Proposition 2.
Let $S = ( U , A ⋃ D )$ be a decision information system, and $μ ( B )$ be a attribute measure for any $B ⊆ A$. Then,
(1)
$μ ( ∅ ) = 0$ and $μ ( A ) = 1$;
(2)
For any $B , C ⊆ A$, $B ⊆ C$ implies $μ ( B ) ≤ μ ( C )$.
Proof.
(1) By Definition 1 and Proposition 1, we have that $γ D ( ∅ ) = 0$ and $γ D ( A ) ≠ 0$. Hence,
(2) For any $B , C ⊆ A$ and $X ∈ U$, if $B ⊆ C$, then $A − C ̲ ( X ) ⊆ A − B ̲ ( X )$ by Proposition 1. Hence, $γ D ( A − C ) ≤ γ D ( A − B )$, i.e., $S i g D ( B ) ≤ S i g D ( C )$. Therefore,
□
Example 2
(Continued from Example 1). Let $C = { a 1 , a 2 , a 3 }$. $A − C ̲ ( X 1 ) = { x 1 , x 3 }$ and $A − C ̲ ( X 2 ) = ∅$. By Definition 1, $P O S A − C ( D ) = A − C ̲ ( X 1 ) ⋃ A − C ̲ ( X 2 ) = { x 1 , x 3 }$, i.e., $γ D ( C ) = | P O S A − C ( D ) | | U | = 2 7 = 0.2857$. $S i g D ( C ) = γ D ( A ) − γ D ( A − C ) = 1 − 0.2857 = 0.7143$. Hence,
$μ ( C ) = S i g D ( C ) S i g D ( A ) = 0.7143 / 1 = 0.7143 .$
Therefore, $B ⊆ C$ implies $μ ( B ) ≤ μ ( C )$.
Proposition 3.
Let $S = ( U , A ⋃ D )$ be a decision information system, and $μ ( B )$ be a attribute measure for any $B ⊆ A$. Then, $0 ≤ μ ( B ) ≤ 1$.
Proof.
By Proposition 1 and the statement $( 2 )$ in Proposition 2, $μ ( ∅ ) ≤ μ ( B ) ≤ μ ( A )$. According to $( 1 )$ in Proposition 2, $0 ≤ μ ( B ) ≤ 1$. □
Example 3
(Continued from Example 1). In Examples 1 and  2, $μ ( B ) = 0.4286$ and $μ ( C ) = 0.7143$. Hence, $0 ≤ μ ( B ) , μ ( C ) ≤ 1$.
Proposition 4.
Let $S = ( U , A ⋃ D )$ be a decision information system, and $μ ( B )$ and $μ ( C )$ be two attribute measures for any $B , C ⊆ A$. Then, $μ ( B ) + μ ( C ) ≥ 2 μ ( B ⋂ C )$.
Proof.
By the statement $( 2 )$ in Proposition 2, $μ ( B ) ≥ μ ( B ⋂ C )$ and $μ ( C ) ≥ μ ( B ⋂ C )$. Hence, $μ ( B ) + μ ( C ) ≥ 2 μ ( B ⋂ C )$. □
Example 4
(Continued from Example 1). In Examples 1 and  2, $μ ( B ) = 0.4286$ and $μ ( C ) = 0.7143$. Since $μ ( B ⋂ C ) = 0.4286$, $μ ( B ) + μ ( C ) ≥ 2 μ ( B ⋂ C )$.
Proposition 5.
Let $S = ( U , A ⋃ D )$ be a decision information system, and $μ ( B )$ and $μ ( C )$ be two attribute measures for any $B , C ⊆ A$. Then, $μ ( B ) + μ ( C ) ≤ 2 μ ( A ⋃ B )$.
Proof.
By the statement $( 2 )$ in Proposition 2, $μ ( B ) ≤ μ ( B ⋃ C )$ and $μ ( C ) ≤ μ ( B ⋃ C )$. Hence, $μ ( B ) + μ ( C ) ≤ 2 μ ( B ⋃ C )$. □
Example 5
(Continued from Example 1). In Examples 1 and  2, $μ ( B ) = 0.4286$ and $μ ( C ) = 0.7143$. Since $μ ( B ⋃ C ) = 0.7143$, $μ ( B ) + μ ( C ) ≤ 2 μ ( B ⋃ C )$.
Theorem 1.
Let $S = ( U , A ⋃ D )$ be a decision information system, and $μ ( B )$ be a attribute measure for any $B ⊆ A$. Then, μ is a fuzzy measure on A.
Proof.
By Proposition 3, we find that $μ$ is a set function where $μ : P ( A ) → [ 0 , 1 ]$. According to Proposition 2, the statements (1) and (2) in Definition 4 hold for $μ$. Hence, $μ$ is a fuzzy measure on A. □
Inspired by Theorem 1, we also call $μ$ a fuzzy rough measure in a decision information system $S = ( U , A ⋃ D )$. In Example 5, we find that $μ ( B ) + μ ( C ) ≠ μ ( B ⋃ C )$. Hence, $μ$ is a non-additive measure, which shows that attributes are related in the decision information system $S = ( U , A ⋃ D )$.

#### 3.2. Choquet Integrals under Fuzzy Rough Measures

In this subsection, for a decision information system, the notion of the matching degree between two objects is presented under an attribute. Based on the notions of attribute measure and matching degree, a Choquet integral is constructed.
Definition 5.
Let $S = ( U , A ⋃ D )$ be a decision information system. For any $x , y ∈ U$ and $a ∈ A$, we call $f ( x , y ) ( a )$ the matching degree between x and y with respect to a, where
$f ( x , y ) ( a ) = 1 1 + | a ( x ) − a ( y ) | .$
Example 6
(Continued from Example 1). By Definition 5, we have
$f ( x 1 , x 2 ) ( a 1 ) = 1 1 + | a 1 ( x 1 ) − a 1 ( x 2 ) | = 0.5 . f ( x 1 , x 2 ) ( a 2 ) = 1 1 + | a 2 ( x 1 ) − a 2 ( x 2 ) | = 0.5 . f ( x 1 , x 2 ) ( a 3 ) = 1 1 + | a 3 ( x 1 ) − a 3 ( x 2 ) | = 1.0 . f ( x 1 , x 2 ) ( a 4 ) = 1 1 + | a 4 ( x 1 ) − a 4 ( x 2 ) | = 0.5 .$
Theorem 2.
Let $S = ( U , A ⋃ D )$ be a decision information system with $A = { a 1 , a 2 , ⋯ , a n }$, and μ be a fuzzy rough measure in $S = ( U , A ⋃ D )$. Then, for any $x , y ∈ U$,
$∫ f ( x , y ) d μ = ∑ i = 1 n [ μ ( A ( i ) ) − μ ( A ( i + 1 ) ) ] f ( x , y ) ( a ( i ) ) ,$
is a Choquet integral of $f ( x , y )$ with respect to the fuzzy rough measure μ on A, where ${ a ( 1 ) , a ( 2 ) , ⋯ , a ( n ) }$ is a permutation of ${ a 1 , a 2 , ⋯ , a n }$ such that $f ( x , y ) ( a ( 1 ) ) ≤ f ( x , y ) ( a ( 2 ) ) ≤ ⋯ ≤ f ( x , y ) ( a ( n ) )$, $A ( i ) = { a ( i ) , a ( i + 1 ) , ⋯ , a ( n ) }$ and $A ( n + 1 ) = ∅$.
Proof.
By Theorem 1, we know that the fuzzy rough measure $μ$ is a fuzzy measure. Hence, it is immediate by Definition 3. □
Remark 1.
In Theorem 2, we find that $μ ( A ( i ) )$, $μ ( A ( i + 1 ) )$ and $a ( i )$ are related to $f ( x , y )$. Therefore, we denote $μ ( A ( i ) )$, $μ ( A ( i + 1 ) )$ and $a ( i )$ by $μ ( A ( i ) f ( x , y ) )$, $μ ( A ( i + 1 ) f ( x , y ) )$ and $a ( i ) f ( x , y )$ in the following discussion.
Example 7
(Continued from Example 1). By Example 6, we have
$f ( x 1 , x 2 ) ( a 1 ) ≤ f ( x 1 , x 2 ) ( a 2 ) ≤ f ( x 1 , x 2 ) ( a 4 ) ≤ f ( x 1 , x 2 ) ( a 3 ) .$
Hence, for $f ( x 1 , x 2 )$, we obtain
Hence,
Therefore, by Definition 4, we have
By Theorem 2,
$∫ f ( x 1 , x 2 ) d μ = ( 1 − 0.8571 ) × 0.5 + ( 0.8571 − 0 ) × 0.5 + 0 × 0.5 + 0 × 1 = 0.5 .$
In the same way, we have
$∫ f ( x 2 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 2 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 2 , x 2 ) ) ] f ( x 2 , x 2 ) ( a ( i ) f ( x 2 , x 2 ) ) = 1.0 , ∫ f ( x 3 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 3 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 3 , x 2 ) ) ] f ( x 3 , x 2 ) ( a ( i ) f ( x 3 , x 2 ) ) = 0.8571 , ∫ f ( x 4 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 4 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 4 , x 2 ) ) ] f ( x 4 , x 2 ) ( a ( i ) f ( x 4 , x 2 ) ) = 0.6429 , ∫ f ( x 5 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 5 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 5 , x 2 ) ) ] f ( x 5 , x 2 ) ( a ( i ) f ( x 5 , x 2 ) ) = 0.5 , ∫ f ( x 6 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 6 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 6 , x 2 ) ) ] f ( x 6 , x 2 ) ( a ( i ) f ( x 6 , x 2 ) ) = 0.5 , ∫ f ( x 7 , x 2 ) d μ = ∑ i = 1 4 [ μ ( A ( i ) f ( x 7 , x 2 ) ) − μ ( A ( i + 1 ) f ( x 7 , x 2 ) ) ] f ( x 7 , x 2 ) ( a ( i ) f ( x 7 , x 2 ) ) = 0.6429 .$
In Example 7, we have that $∫ f ( x 2 , x 2 ) d μ = 1.0$, which is greater than other values of $∫ f ( x j , x 2 ) d μ$ ($j = 1 , 3 , 4 , 5 , 6 , 7$). $∫ f ( x 2 , x 2 ) d μ = 1.0$ means that $x 2$ is the best match to itself, which is consistent with actual logic.

## 4. A Novel Decision-Making Method Based on Fuzzy Rough Measures and Choquet Integrals

In this section, a novel MCDM method is presented by the Choquet integral, which can aggregate all information between two objects.

#### 4.1. The Problem of Decision Making

Let $S = ( U , A ⋃ D )$ be a decision information system, which is shown in Table 2, where $U = { x 1 , ⋯ , x m }$ is the set of objects, $A = { a 1 , ⋯ , a n }$ is a conditional attribute set, D is a decision attribute, $x j i = a i ( x j )$ is the attribute value of $x j$ under conditional attribute $a j$, and $d j$ is the decision value of $x j$ under decision attribute D. For a new object $x m + 1$, we take the value of each conditional attribute to be $( a 1 ( x m + 1 ) )$, $( a 2 ( x m + 1 ) )$, ⋯, $( a n ( x m + 1 ) )$. Then, the decision maker should give the decision value of $x m + 1$ according to $S = ( U , A ⋃ D )$.

#### 4.2. The Novel Decision-Making Method

Based on Theorems 1 and 2, we present a novel method to solve the issue of MCDM by using fuzzy rough measures and Choquet integrals. We show this novel method as follows, for the problem of decision making in Section 4.1:
Step 1: For any $x j ∈ U$ ($j = 1 , 2 , ⋯ , m$) and $a i ∈ A$ ($i = 1 , 2 , ⋯ , n$), we calculate all matching degrees $f ( x j , x m + 1 ) ( a i ) = 1 1 + | a j ( x j ) − a i ( x m + 1 ) |$, which are shown in Table 3.
Step 2: For any $x j ∈ U$ ($j = 1 , 2 , ⋯ , m$), we calculate all Choquet integrals under fuzzy rough measures $μ$, which are shown as follows:
$s ( x 1 , x m + 1 ) = ∫ f ( x 1 , x m + 1 ) d μ = ∑ i = 1 n [ μ ( A ( i ) f ( x 1 , x m + 1 ) ) − μ ( A ( i + 1 ) f ( x 1 , x m + 1 ) ) ] f ( x 1 , x m + 1 ) ( a ( i ) f ( x 1 , x m + 1 ) ) , ⋮ s ( x j , x m + 1 ) = ∫ f ( x j , x m + 1 ) d μ = ∑ i = 1 n [ μ ( A ( i ) f ( x j , x m + 1 ) ) − μ ( A ( i + 1 ) f ( x j , x m + 1 ) ) ] f ( x j , x m + 1 ) ( a ( i ) f ( x j , x m + 1 ) ) , ⋮ s ( x m , x m + 1 ) = ∫ f ( x m , x m + 1 ) d μ = ∑ i = 1 n [ μ ( A ( i ) f ( x m , x m + 1 ) ) − μ ( A ( i + 1 ) f ( x m , x m + 1 ) ) ] f ( x m , x m + 1 ) ( a ( i ) f ( x m , x m + 1 ) ) .$
Step 3: We obtain the ranking of all alternatives by the value of $s ( x j , x m + 1 )$. Moreover, the decision maker chooses the best one whose decision value is the same as that of $x m + 1$.
For steps 1–3 above, the MCDM algorithm by fuzzy rough measures and Choquet integrals is shown in Algorithm 1.
 Algorithm 1 The MCDM algorithm by fuzzy rough measures and Choquet integrals Input: A decision information system $S = ( U , A ⋃ D )$ and a new decision object $x m + 1$, where $U = { x 1 , ⋯ , x m }$, $A = { a 1 , ⋯ , a n }$.Output: The decision value of $x m + 1$.(1)  for $j = 1 → m$(2)       for $i = 1 → m$(3)          Compute $f ( x j , x m + 1 ) ( a i )$;(4)       end(5)  Compute $s ( x j , x m + 1 ) = ∫ f ( x j , x m + 1 ) d μ$;(6)  end(7)  for $j = 1 → m$(8)  Obtain the ranking of all $s ( x j , x m + 1 )$;(9)  end(10)  Give the decision value of $x m + 1$ by the ranking of all $s ( x j , x m + 1 )$.

## 5. Comparison and Analysis

To illustrate the feasibility and effectiveness of our method above, it is compared with other methods through a numerical example in this section.

#### 5.1. Hiring Dataset

In this section, we list the hiring dataset taken from Komorowski et al. in , where all the attributes have nominal values, which is shown in Table 4. It contains 8 objects with 4 conditional attributes and 1 decision attribute. The detailed description of each attribute is as follows:
• The conditional attribute ‘Diploma’ has values: “MBA”, “MSc”, “MCE”.
• The conditional attribute ‘Experience’ has values: “High”, “Low”, “Medium”.
• The conditional attribute ‘French’ has values: “Yes”, “No”.
• The conditional attribute ‘Reference’ has values: “Excellent”, “Good”, “Neutral”.
• The conditional attribute ‘Decision’ has values: “Accept”, “Reject”.

#### 5.2. An Applied Example

For the hiring dataset , which is shown in Table 4 in the paper, we denote the first seven records as the original decision information, and the eighth record $x 8$ as a new object (we suppose that we do not know the decision value of $x 8$). In order to facilitate the calculation, we perform the following for Table 4:
• The conditional attribute ‘Diploma = $a 1$’ has values: “MBA = 1”, “MSc = 2”, “MCE = 3”.
• The conditional attribute ‘Experience = $a 2$’ has values: “Medium = 1”, “High = 2”, “Low = 3”.
• The conditional attribute ‘French = $a 3$’ has values: “Yes = 1”, “No = 0”.
• The conditional attribute ‘Reference = $a 4$’ has values: “Excellent = 1”, “Neutral = 2”, “Good = 3”.
• The conditional attribute ‘Decision = D’ has values: “Accept = 1”, “Reject = 0”.
It can be denoted as in Table 5.
Then, we use our method to predict the decision value of $x 8$, i.e., we should predict the “?” in Table 5.
Example 8.
Let $I S = ( U , A )$ be an information system, which is the first seven records shown in the hiring dataset . For a decision object $x 8$, we take the value of each conditional attribute to be $a 1 ( x 8 ) = 3$, $a 2 ( x 8 ) = 3$, $a 3 ( x 8 ) = 0$, $a 4 ( x 8 ) = 1$. It is shown in Table 5. Then, we use the following steps to give the decision value of $x 8$ according to $S = ( U , A ⋃ D )$.
Step 1: For any $x j ∈ U$ ($j = 1 , 2 , ⋯ , 7$) and $a i ∈ A$ ($i = 1 , 2 , 3 , 4$), we calculate all matching degrees $f ( x j , x 8 ) ( a i )$, which are shown in Table 6.
Step 2: For $f ( x 1 , x 8 )$, we have
$f ( x 1 , x 8 ) ( a 1 ) ≤ f ( x 1 , x 8 ) ( a 2 ) ≤ f ( x 1 , x 8 ) ( a 3 ) ≤ f ( x 1 , x 8 ) ( a 4 ) .$
Hence, we obtain
Hence,
Therefore, by Definition 4, we have
In the same way, for any $f ( x j , x 8 )$ ($j = 1 , 2 , ⋯ , 7$), we can obtain all permutations of ${ a 1 , a 2 , a 3 , a 4 }$ in Table 7.
By Table 7, we can calculate all $μ ( A ( i ) f ( x j , x 8 ) )$ in Table 8, where $i ∈ { 1 , 2 , 3 , 4 }$, $j = 1 , 2 , ⋯ , 7$ and $μ ( A ( 5 ) f ( x j , x 8 ) ) = 0$.
By Table 8 and Theorem 2, we calculate
$s ( x 1 , x 8 ) = ∫ f ( x 1 , x 8 ) d μ = ( 1 − 0.8571 ) × 0.3333 + ( 0.8571 − 0 ) × 0.3333 + 0 × 0.5 + 0 × 1 = 0.3333 ; s ( x 2 , x 8 ) = ∫ f ( x 2 , x 8 ) d μ = ( 1 − 0.8571 ) × 0.5 + ( 0.8571 − 0 ) × 0.5 + 0 × 0.5 + 0 × 0.5 = 0.5000 ; s ( x 3 , x 8 ) = ∫ f ( x 3 , x 8 ) d μ = ( 1 − 0.8571 ) × 0.5 + ( 0.8571 − 0 ) × 0.5 + 0 × 0.5 + 0 × 1 = 0.5000 ; s ( x 4 , x 8 ) = ∫ f ( x 4 , x 8 ) d μ = ( 1 − 0.8571 ) × 0.3333 + ( 0.8571 − 0.2857 ) × 0.3333 + ( 0.2857 − 0 ) × 0.5 + 0 × 1 = 0.3810 ; s ( x 5 , x 8 ) = ∫ f ( x 5 , x 8 ) d μ = ( 1 − 0.8571 ) × 0.3333 + ( 0.8571 − 0.7143 ) × 0.5 + ( 0.7143 − 0.2857 ) × 0.5 + ( 0.2857 − 0 ) × 1 = 0.6190 ; s ( x 6 , x 8 ) = ∫ f ( x 6 , x 8 ) d μ = ( 1 − 0.7143 ) × 0.3333 + ( 0.7143 − 0.4286 ) × 0.5 + ( 0.4286 − 0 ) × 1 + 0 × 1 = 0.6667 ; s ( x 7 , x 8 ) = ∫ f ( x 7 , x 8 ) d μ = ( 1 − 0.2857 ) × 0.3333 + ( 0 . 0.2857 − 0 ) × 0.5 + 0 × 0.5 + 0 × 0.5 = 0.3810 .$
Step 3: We obtain the ranking of all alternatives by the value of $s ( x j , x m + 1 )$, where $s ( x 6 , x 8 )$ is the best one. Hence, the decision value of $x 8$ is the same as that of $x 6$, which is 0.

#### 5.3. Comparison with Other Methods

We use the R programming language for dealing with Example 8 by the AQ algorithm , the CN2 algorithm  and the LEM2 algorithm , respectively. The AQ algorithm , the CN2 algorithm ) and the LEM2 algorithm  are at pages 96, 97 and 105 in the the package ’RoughSets’, respectively. The package ’RoughSets’ can be downloaded from https://CRAN.R-project.org/package=RoughSets, accessed on 23 May 2022. In fact, Table 1 and $x 8$ are taken from the hiring dataset in , where the actual decision value of $x 8$ is 0. Then, we use some existing algorithms to predict the decision value of $x 8$ according to Table 1. All results are shown in Table 9.
As shown in Table 9, we find that our method is effective, since the predicted value is equal to the actual value. In the AQ algorithm , we use “nOFItervales = 3”, “confidence = 0.8” and “timescovered = 3”, and then we obtain 6 rules to make a decision. In the CN2 algorithm , we use “nOFItervales = 3”, and then we obtain two rules to make a decision. In the LEM2 algorithm , we use “maxNOfCuts = 1”, and then we obtain two rules to make a decision. The AQ algorithm , the CN2 algorithm  and the LEM2 algorithm  all depend on the corresponding rules, which are obtained through rough sets. Although the CN2 algorithm  and the LEM2 algorithm  can also obtain the predicted value 0 for $x 8$, the predicted value will be changed by different threshold values. We present some discussions on this statement.
For the AQ algorithm , the predicted value is also 1, which does not equal the actual value, although we changed “nOFItervales”, “confidence” and “timescovered”. For example, we use “nOFItervales = 3”, “confidence = 0.9” and “timescovered = 8”, and we obtain 16 rules and the predicted value 1; we use “nOFItervales = 1”, “confidence = 0.9” and “timescovered = 3”, and we obtain 15 rules and the predicted value 1; we use “nOFItervales = 3”, “confidence = 0.98” and “timescovered = 28”, and we obtain 56 rules and the predicted value 1. Hence, we only present the CN2 algorithm  and the LEM2 algorithm  in Table 10.
As shown in Table 10, we find that the predicted decision value of $x 8$ is changed by using different values in the CN2 algorithm  and the LEM2 algorithm , respectively. However, our method uses the matching degree between any original object $x j ∈ U$ ($j = 1 , 2 , ⋯ , 7$) and the decision object $x 8$, and then corresponding Choquet integrals are used to aggregate them. Hence, the result of our method is unique. In particular, our method is more stable than others. For the above comparative analysis, our method is more feasible and effective than others under the hiring dataset .

## 6. Conclusions

In this article, we combine rough sets and fuzzy measures to solve the problem of MCDM, which can well avoid the limitations of the existing decision-making method under rough sets. The contributions of this paper are listed as follows:
• The notion of the attribute measure is presented based on the importance degree in rough sets, which can illustrate the non-additive relationship of two attributes in rough sets. By the new notion, we can find that attributes are related to each other in information systems. It can also be used to construct the corresponding Choquet integral.
• Then, a type of nonlinear aggregation operator (i.e., Choquet integral) is constructed, which can aggregate all information between two objects in a decision information system. Moreover, a method based on the Choquet integral is proposed to deal with the problem of MCDM, which is inspired by case-based reasoning theory. This novel method can address the deficiency of the existing methods well. It can solve the issue of attribute association in MCDM.
In further research, the following topics can be considered: other integrals and generalized rough set models [47,48,49] will be connected with the research content of this article. The novel method can be combined with other decision-making and aggregation methods [50,51,52].

## Author Contributions

This paper was written with the contribution of all authors. The individual contributions and responsibilities of all authors can be described as follows: J.W. analyzed the existing work on rough sets and fuzzy measures and wrote the paper. X.Z. put forward the idea of this paper and also completed the preparatory work of the paper. All authors have read and agreed to the published version of the manuscript.

## Funding

This research was funded by the National Natural Science Foundation of China under Grant No. 61976130, and the Natural Science Foundation of Education Department of Shaanxi Province under Grant No. 20JK0506.

Not applicable.

Not applicable.

## Data Availability Statement

In this paper, we use the hiring dataset taken from Komorowski et al. in . It is also at page 106 in the the package ’RoughSets’, respectively. The package ’RoughSets’ can be downloaded from https://CRAN.R-project.org/package=RoughSets accessed on 23 May 2022.

## Conflicts of Interest

The authors declare no conflict of interest.

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Table 1. Weather observation data.
Table 1. Weather observation data.
U$a 1$$a 2$$a 3$$a 4$D
$x 1$11111
$x 2$22121
$x 3$22111
$x 4$12031
$x 5$13120
$x 6$33130
$x 7$21120
Table 2. A decision-making table.
Table 2. A decision-making table.
U$a 1$$a 2$$a n$D
$x 1$$x 11$$x 12$$x 1 n$$d 1$
$x 2$$x 21$$x 22$$x 2 n$$d 2$
$x m$$x m 1$$x m 2$$x m n$$d m$
Table 3. A matching degree table.
Table 3. A matching degree table.
U$a 1$$a 2$$a n$
$x 1$$f ( x 1 , x m + 1 ) ( a 1 )$$f ( x 1 , x m + 1 ) ( a 2 )$$f ( x 1 , x m + 1 ) ( a n )$
$x 2$$f ( x 2 , x m + 1 ) ( a 1 )$$f ( x 2 , x m + 1 ) ( a 2 )$$f ( x 2 , x m + 1 ) ( a n )$
$x m$$f ( x m , x m + 1 ) ( a 1 )$$f ( x m , x m + 1 ) ( a 2 )$$f ( x m , x m + 1 ) ( a n )$
Table 4. The hiring dataset .
Table 4. The hiring dataset .
UDiplomaExperienceFrenchReferenceDecision
$x 1$MBAMediumYesExcellentAccept
$x 2$MSCHighYesNeutralAccept
$x 3$MSCHighYesExcellentAccept
$x 4$MBAHighNoGoodAccept
$x 5$MBALowYesNeutralReject
$x 6$MCELowYesGoodReject
$x 7$MSCMediumYesNeutralReject
$x 8$MCELowNoExcellentReject
Table 5. A decision problem in the hiring dataset.
Table 5. A decision problem in the hiring dataset.
UDiploma ($a 1$)Experience ($a 2$)French ($a 3$)Reference ($a 4$)Decision (D)
An information system $I S = ( U ′ , A )$$x 1$MBA (1)Medium (1)Yes (1)Excellent (1)Accept (1)
$x 2$MSC (2)High (2)Yes (1)Neutral (2)Accept (1)
$x 3$MSC (2)High (2)Yes (1)Excellent (1)Accept (1)
$x 4$MBA (1)High (2)No (0)Good (3)Accept (1)
$x 5$MBA (1)Low (3)Yes (1)Neutral (2)Reject (0)
$x 6$MCE (3)Low (3)Yes (1)Good (3)Reject (0)
$x 7$MSC (2)Medium (1)Yes (1)Neutral (2)Reject (0)
A decision object$x 8$MCE (3)Low (3)No (0)Excellent (1)“?”
Table 6. $f ( x j , x 8 ) ( a i )$, ($i = 1 , 2 , 3 , 4$) and ($j = 1 , 2 , ⋯ , 7$).
Table 6. $f ( x j , x 8 ) ( a i )$, ($i = 1 , 2 , 3 , 4$) and ($j = 1 , 2 , ⋯ , 7$).
U$a 1$$a 2$$a 3$$a 4$
$f ( x 1 , x 8 ) ( a i )$$0.3333$$0.3333$$0.5000$$1.0000$
$f ( x 2 , x 8 ) ( a i )$$0.5000$$0.5000$$0.5000$$0.5000$
$f ( x 3 , x 8 ) ( a i )$$0.5000$$0.5000$$0.5000$$1.0000$
$f ( x 4 , x 8 ) ( a i )$$0.3333$$0.5000$$1.0000$$0.3333$
$f ( x 5 , x 8 ) ( a i )$$0.3333$$1.0000$$0.5000$$0.5000$
$f ( x 6 , x 8 ) ( a i )$$1.0000$$1.0000$$0.5000$$0.3333$
$f ( x 7 , x 8 ) ( a i )$$0.5000$$0.3333$$0.5000$$0.5000$
Table 7. ${ a ( 1 ) f ( x j , x 8 ) , a ( 2 ) f ( x j , x 8 ) , a ( 3 ) f ( x j , x 8 ) , a ( 4 ) f ( x j , x 8 ) }$ relates to any $f ( x j , x 8 )$ ($j = 1 , 2 , ⋯ , 7$).
Table 7. ${ a ( 1 ) f ( x j , x 8 ) , a ( 2 ) f ( x j , x 8 ) , a ( 3 ) f ( x j , x 8 ) , a ( 4 ) f ( x j , x 8 ) }$ relates to any $f ( x j , x 8 )$ ($j = 1 , 2 , ⋯ , 7$).
$a ( 1 ) f ( x j , x 8 )$$a ( 2 ) f ( x j , x 8 )$$a ( 3 ) f ( x j , x 8 )$$a ( 4 ) f ( x j , x 8 )$
$f ( x 1 , x 8 )$$a 1$$a 2$$a 3$$a 4$
$f ( x 2 , x 8 )$$a 1$$a 2$$a 3$$a 4$
$f ( x 3 , x 8 )$$a 1$$a 2$$a 3$$a 4$
$f ( x 4 , x 8 )$$a 1$$a 4$$a 2$$a 3$
$f ( x 5 , x 8 )$$a 1$$a 3$$a 4$$a 2$
$f ( x 6 , x 8 )$$a 4$$a 3$$a 1$$a 2$
$f ( x 7 , x 8 )$$a 2$$a 1$$a 3$$a 4$
Table 8. $μ ( A ( i ) f ( x j , x 8 ) )$ with $i ∈ { 1 , 2 , 3 , 4 }$ and $j = 1 , 2 , ⋯ , 7$.
Table 8. $μ ( A ( i ) f ( x j , x 8 ) )$ with $i ∈ { 1 , 2 , 3 , 4 }$ and $j = 1 , 2 , ⋯ , 7$.
$A ( 1 ) f ( x j , x 8 )$$A ( 2 ) f ( x j , x 8 )$$A ( 3 ) f ( x j , x 8 )$$A ( 4 ) f ( x j , x 8 )$
$μ ( A ( i ) f ( x 1 , x 8 ) )$1$0.8571$00
$μ ( A ( i ) f ( x 2 , x 8 ) )$1$0.8571$00
$μ ( A ( i ) f ( x 3 , x 8 ) )$1$0.8571$00
$μ ( A ( i ) f ( x 4 , x 8 ) )$1$0.8571$$0.2857$0
$μ ( A ( i ) f ( x 5 , x 8 ) )$1$0.8571$$0.7143$$0.2857$
$μ ( A ( i ) f ( x 6 , x 8 ) )$1$0.7143$$0.4286$0
$μ ( A ( i ) f ( x 7 , x 8 ) )$1$0.2857$00
Table 9. The decision results of $x 8$ utilizing different methods for Example 8.
Table 9. The decision results of $x 8$ utilizing different methods for Example 8.
MethodsThe Actual Decision Value of $x 8$The Predicted Decision Value of $x 8$
The AQ algorithm 01
The CN2 algorithm 00
The LEM2 algorithm 00
Algorithm 1 in this paper 00
Table 10. The decision results of $x 8$ utilizing different threshold values for Example 8.
Table 10. The decision results of $x 8$ utilizing different threshold values for Example 8.
Different Threshold Values in AlgorithmsRulesThe Predicted Decision Value Of $x 8$
“nOFItervales = 3” in the CN2 algorithm 20
“nOFItervales = 1” in the CN2 algorithm 61
“maxNOfCuts = 1” in the LEM2 algorithm 20
“maxNOfCuts = 3” in the LEM2 algorithm 31
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Wang, Jingqian, and Xiaohong Zhang. 2022. "A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures" Axioms 11, no. 6: 275. https://doi.org/10.3390/axioms11060275

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