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

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## Abstract

**:**

## 1. Introduction

- 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.

## 2. Basic Definitions

#### 2.1. Pawlak’s Rough Sets

**Proposition**

**1**

**.**Let $S=(U,A)$ be an information system. For any $X,Y\subseteq U$,

(1L) $\underline{A}(U)=U$ | (1H) $\overline{A}(U)=U$ |

(2L) $\underline{A}(\varphi )=\varphi $ | (2H) $\overline{A}(\varphi )=\varphi $ |

(3L) $\underline{A}(X)\subseteq X$ | (3H) $X\subseteq \overline{A}(X)$ |

(4L) $\underline{A}(X\bigcap Y)=\underline{A}(X)\bigcap \underline{A}(Y)$ | (4H) $\overline{A}(X\bigcup Y)=\overline{A}(X)\bigcup \overline{A}(Y)$ |

(5L) $\underline{A}(\underline{A}(X))=\underline{A}(X)$ | (5H) $\overline{A}(\overline{A}(X))=\overline{A}(X)$ |

(6L) $X\subseteq Y\Rightarrow \underline{A}(X)\subseteq \underline{A}(Y)$ | (6H) $X\subseteq Y\Rightarrow \overline{A}(X)\subseteq \overline{A}(Y)$ |

(7L) $\underline{A}(-\underline{A}(X))=-\underline{A}(X)$ | (7H) $\overline{A}(-\overline{A}(X))=-\overline{A}(X)$ |

(8LH) $\underline{A}(-X)=-\overline{A}(X)$ | (9LH) $\underline{A}(X)\subseteq \overline{A}(X)$ |

**Definition**

**1**

**.**Let $S=(U,A\bigcup D)$ be a decision information system. Then, the dependency degree of D with regard to A in S is

#### 2.2. Fuzzy Measures and Choquet Integrals

**Definition**

**2**

**.**Given a universe U and a set function $\mathfrak{m}$: $P(U)\to [0,1]$, where $P(U)$ is the power set of U, $\mathfrak{m}$ is called a fuzzy measure on U if the following statements hold:

- (1)
- $\mathfrak{m}(\varnothing )=0$, $\mathfrak{m}(U)=1$;
- (2)
- $A,B\subseteq U$, $A\subseteq B$, which implies $\mathfrak{m}(A)\le \mathfrak{m}(B)$.

**Definition**

**3**

**.**Given a real-valued function $f:U\to [0,1]$ with $U=\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}$, the Choquet integral of f with respect to the fuzzy measure $\mathfrak{m}$ is defined as:

## 3. Fuzzy Rough Measures and Choquet Integrals

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

**Definition**

**4.**

**Example**

**1.**

- The conditional attribute ‘${a}_{1}=Weatherprediction$’ has values: “Clear = 1”, “Cloudy = 2”, “Rain = 3”.
- The conditional attribute ‘${a}_{2}=Airtemperature$’ has values: “Hot = 1”, “Warm = 2”, “Cool = 3”.
- The conditional attribute ‘${a}_{3}=Windiness$’ has values: “Yes = 0”, “No = 1”.
- The conditional attribute ‘${a}_{4}=Humidity$’ has values: “Wet = 1”, “Normal = 2”, “Dry = 3”.
- The conditional attribute ‘D’ has values: “Yes = 1”, “No = 0”.

**Proposition**

**2.**

- (1)
- $\mu (\varnothing )=0$ and $\mu (A)=1$;
- (2)
- For any $B,C\subseteq A$, $B\subseteq C$ implies $\mu (B)\le \mu (C)$.

**Proof.**

**Example**

**2**

**.**Let $C=\{{a}_{1},{a}_{2},{a}_{3}\}$. $\underline{A-C}({X}_{1})=\{{x}_{1},{x}_{3}\}$ and $\underline{A-C}({X}_{2})=\varnothing $. By Definition 1, $PO{S}_{A-C}(D)=\underline{A-C}({X}_{1})\bigcup \underline{A-C}({X}_{2})=\{{x}_{1},{x}_{3}\}$, i.e., ${\gamma}_{D}(C)=\frac{|PO{S}_{A-C}(D)|}{\left|U\right|}=\frac{2}{7}=0.2857$. $Si{g}_{D}(C)={\gamma}_{D}(A)-{\gamma}_{D}(A-C)=1-0.2857=0.7143$. Hence,

**Proposition**

**3.**

**Proof.**

**Example**

**3**

**.**In Examples 1 and 2, $\mu (B)=0.4286$ and $\mu (C)=0.7143$. Hence, $0\le \mu (B),\mu (C)\le 1$.

**Proposition**

**4.**

**Proof.**

**Example**

**4**

**.**In Examples 1 and 2, $\mu (B)=0.4286$ and $\mu (C)=0.7143$. Since $\mu (B\bigcap C)=0.4286$, $\mu (B)+\mu (C)\ge 2\mu (B\bigcap C)$.

**Proposition**

**5.**

**Proof.**

**Example**

**5**

**.**In Examples 1 and 2, $\mu (B)=0.4286$ and $\mu (C)=0.7143$. Since $\mu (B\bigcup C)=0.7143$, $\mu (B)+\mu (C)\le 2\mu (B\bigcup C)$.

**Theorem**

**1.**

**Proof.**

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

**Definition**

**5.**

**Example**

**6**

**.**By Definition 5, we have

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Example**

**7**

**.**By Example 6, we have

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

#### 4.1. The Problem of Decision Making

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

**Step 1:**For any ${x}_{j}\in U$ ($j=1,2,\cdots ,m$) and ${a}_{i}\in A$ ($i=1,2,\cdots ,n$), we calculate all matching degrees $f}_{({x}_{j},{x}_{m+1})}({a}_{i})=\frac{1}{1+|{a}_{j}({x}_{j})-{a}_{i}({x}_{m+1})|$, which are shown in Table 3.

**Step 2:**For any ${x}_{j}\in U$ ($j=1,2,\cdots ,m$), we calculate all Choquet integrals under fuzzy rough measures $\mu $, which are shown as follows:

**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 ${x}_{m+1}$.

Algorithm 1 The MCDM algorithm by fuzzy rough measures and Choquet integrals |

Input: A decision information system $S=(U,A\bigcup D)$ and a new decision object ${x}_{m+1}$, where $U=\{{x}_{1},\cdots ,{x}_{m}\}$, $A=\{{a}_{1},\cdots ,{a}_{n}\}$.Output: The decision value of ${x}_{m+1}$.(1) for $j=1\to m$ (2) for $i=1\to m$ (3) Compute ${f}_{({x}_{j},{x}_{m+1})}({a}_{i})$; (4) end (5) Compute $s({x}_{j},{x}_{m+1})=\int {f}_{({x}_{j},{x}_{m+1})}d\mu $; (6) end (7) for $j=1\to m$ (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

#### 5.1. Hiring Dataset

- 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

- 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”.

**Example**

**8.**

**Step 1:**For any ${x}_{j}\in U$ ($j=1,2,\cdots ,7$) and ${a}_{i}\in A$ ($i=1,2,3,4$), 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

**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

## 6. Conclusions

- 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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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U | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | D |
---|---|---|---|---|---|

${x}_{1}$ | 1 | 1 | 1 | 1 | 1 |

${x}_{2}$ | 2 | 2 | 1 | 2 | 1 |

${x}_{3}$ | 2 | 2 | 1 | 1 | 1 |

${x}_{4}$ | 1 | 2 | 0 | 3 | 1 |

${x}_{5}$ | 1 | 3 | 1 | 2 | 0 |

${x}_{6}$ | 3 | 3 | 1 | 3 | 0 |

${x}_{7}$ | 2 | 1 | 1 | 2 | 0 |

U | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ⋯ | ${\mathit{a}}_{\mathit{n}}$ | D |
---|---|---|---|---|---|

${x}_{1}$ | ${x}_{11}$ | ${x}_{12}$ | ⋯ | ${x}_{1n}$ | ${d}_{1}$ |

${x}_{2}$ | ${x}_{21}$ | ${x}_{22}$ | ⋯ | ${x}_{2n}$ | ${d}_{2}$ |

⋮ | ⋮ | ⋮ | ⋯ | ⋮ | ⋮ |

${x}_{m}$ | ${x}_{m1}$ | ${x}_{m2}$ | ⋯ | ${x}_{mn}$ | ${d}_{m}$ |

U | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ⋯ | ${\mathit{a}}_{\mathit{n}}$ |
---|---|---|---|---|

${x}_{1}$ | ${f}_{({x}_{1},{x}_{m+1})}({a}_{1})$ | ${f}_{({x}_{1},{x}_{m+1})}({a}_{2})$ | ⋯ | ${f}_{({x}_{1},{x}_{m+1})}({a}_{n})$ |

${x}_{2}$ | ${f}_{({x}_{2},{x}_{m+1})}({a}_{1})$ | ${f}_{({x}_{2},{x}_{m+1})}({a}_{2})$ | ⋯ | ${f}_{({x}_{2},{x}_{m+1})}({a}_{n})$ |

⋮ | ⋮ | ⋮ | ⋯ | ⋮ |

${x}_{m}$ | ${f}_{({x}_{m},{x}_{m+1})}({a}_{1})$ | ${f}_{({x}_{m},{x}_{m+1})}({a}_{2})$ | ⋯ | ${f}_{({x}_{m},{x}_{m+1})}({a}_{n})$ |

**Table 4.**The hiring dataset [38].

U | Diploma | Experience | French | Reference | Decision |
---|---|---|---|---|---|

${x}_{1}$ | MBA | Medium | Yes | Excellent | Accept |

${x}_{2}$ | MSC | High | Yes | Neutral | Accept |

${x}_{3}$ | MSC | High | Yes | Excellent | Accept |

${x}_{4}$ | MBA | High | No | Good | Accept |

${x}_{5}$ | MBA | Low | Yes | Neutral | Reject |

${x}_{6}$ | MCE | Low | Yes | Good | Reject |

${x}_{7}$ | MSC | Medium | Yes | Neutral | Reject |

${x}_{8}$ | MCE | Low | No | Excellent | Reject |

U | Diploma (${\mathit{a}}_{1}$) | Experience (${\mathit{a}}_{2}$) | French (${\mathit{a}}_{3}$) | Reference (${\mathit{a}}_{4}$) | Decision (D) | |
---|---|---|---|---|---|---|

An information system $IS=({U}^{\prime},A)$ | ${x}_{1}$ | 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) | “?” |

U | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ |
---|---|---|---|---|

${f}_{({x}_{1},{x}_{8})}({a}_{i})$ | $0.3333$ | $0.3333$ | $0.5000$ | $1.0000$ |

${f}_{({x}_{2},{x}_{8})}({a}_{i})$ | $0.5000$ | $0.5000$ | $0.5000$ | $0.5000$ |

${f}_{({x}_{3},{x}_{8})}({a}_{i})$ | $0.5000$ | $0.5000$ | $0.5000$ | $1.0000$ |

${f}_{({x}_{4},{x}_{8})}({a}_{i})$ | $0.3333$ | $0.5000$ | $1.0000$ | $0.3333$ |

${f}_{({x}_{5},{x}_{8})}({a}_{i})$ | $0.3333$ | $1.0000$ | $0.5000$ | $0.5000$ |

${f}_{({x}_{6},{x}_{8})}({a}_{i})$ | $1.0000$ | $1.0000$ | $0.5000$ | $0.3333$ |

${f}_{({x}_{7},{x}_{8})}({a}_{i})$ | $0.5000$ | $0.3333$ | $0.5000$ | $0.5000$ |

**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 ${f}_{({x}_{j},{x}_{8})}$ ($j=1,2,\cdots ,7$).

${\mathit{a}}_{(1)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{a}}_{(2)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{a}}_{(3)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{a}}_{(4)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | |
---|---|---|---|---|

${f}_{({x}_{1},{x}_{8})}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ |

${f}_{({x}_{2},{x}_{8})}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ |

${f}_{({x}_{3},{x}_{8})}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ |

${f}_{({x}_{4},{x}_{8})}$ | ${a}_{1}$ | ${a}_{4}$ | ${a}_{2}$ | ${a}_{3}$ |

${f}_{({x}_{5},{x}_{8})}$ | ${a}_{1}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{2}$ |

${f}_{({x}_{6},{x}_{8})}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{1}$ | ${a}_{2}$ |

${f}_{({x}_{7},{x}_{8})}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{3}$ | ${a}_{4}$ |

${\mathit{A}}_{(1)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{A}}_{(2)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{A}}_{(3)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | ${\mathit{A}}_{(4)}^{{\mathit{f}}_{({\mathit{x}}_{\mathit{j}},{\mathit{x}}_{8})}}$ | |
---|---|---|---|---|

$\mu ({A}_{(i)}^{{f}_{({x}_{1},{x}_{8})}})$ | 1 | $0.8571$ | 0 | 0 |

$\mu ({A}_{(i)}^{{f}_{({x}_{2},{x}_{8})}})$ | 1 | $0.8571$ | 0 | 0 |

$\mu ({A}_{(i)}^{{f}_{({x}_{3},{x}_{8})}})$ | 1 | $0.8571$ | 0 | 0 |

$\mu ({A}_{(i)}^{{f}_{({x}_{4},{x}_{8})}})$ | 1 | $0.8571$ | $0.2857$ | 0 |

$\mu ({A}_{(i)}^{{f}_{({x}_{5},{x}_{8})}})$ | 1 | $0.8571$ | $0.7143$ | $0.2857$ |

$\mu ({A}_{(i)}^{{f}_{({x}_{6},{x}_{8})}})$ | 1 | $0.7143$ | $0.4286$ | 0 |

$\mu ({A}_{(i)}^{{f}_{({x}_{7},{x}_{8})}})$ | 1 | $0.2857$ | 0 | 0 |

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**MDPI and ACS Style**

Wang, J.; Zhang, X. A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures. *Axioms* **2022**, *11*, 275.
https://doi.org/10.3390/axioms11060275

**AMA Style**

Wang J, Zhang X. A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures. *Axioms*. 2022; 11(6):275.
https://doi.org/10.3390/axioms11060275

**Chicago/Turabian Style**

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