# Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1.**

^{2}→ [0, 1] is called an overlap function if it satisfies the following requirements: for any x, y ∈ [0, 1]:

- (i)
- O is commutative, that is, O(x, y) = O(y, x);
- (ii)
- O(x, y) = 0 if and only if x y = 0;
- (iii)
- O(x, y) = 1 if and only if x y = 1;
- (iv)
- O is non-decreasing; and
- (v)
- O is an continuous function [23].

**Definition 2.**

**Example 1.**

- (1)
- The binary function is defined by

- (2)
- The binary function is defined by

^{p}, y

^{p})

**Definition 3.**

_{X}over X; and

_{X}) a Q(X, ℘

_{X}) ∈ M(X, ℘ X)

_{X}) is usually abbreviated as Q

_{X}when the Borel field does not need to be specifically indicated. Given X as a discourse domain, if E represents individuals in X that have a specific attribute A, then Q

_{X}(E) is seen as the truth value of the linguistic quantified statement “Q Xs are As”.

**Example 2.**

_{X}(E) ∈ {0, 1}.

**Example 3.**

**Definition 4:**

^{X}, m) is a fuzzy measure space. If h: X → [0, 1] is a measurable function, then the Sugeno integral of h over A ∈ ℘ is defined as follows [11]:

**Theorem 1.**

_{λ}= {x∈X: h(x) ≥ λ} for every λ ∈ [0, 1] [11].

_{A}h

_{°}m will be abbreviated as ∫ h

_{°}m whenever A = X.

## 3. Sugeno Integrals Based on Overlap Functions

**Definition 5.**

^{2}→ [0, 1] as an overlap function, if h: X → [0, 1] is a ℘ measurable function, then the Sugeno integral based on the overlap function (O-Sugeno integral) of h over A ∈ ℘ is defined by

_{λ}= {x∈X: h(x) ≥ λ} for every λ ∈ [0, 1].

**Theorem 2.**

^{2}→ [0, 1] is an overlap function. If ℘ = 2

^{X}, then for any ℘ measurable function h: X → [0, 1] and any subset A of X, we have

_{λ}= {x∈X: h(x) ≥ λ} for every λ∈ [0, 1].

**Proof of Theorem 2.**

- (1)
- ∀F ⊆ X, Let λ′ = $\underset{x\in F}{\mathrm{inf}}h(x)$.

_{λ}

_{′}= X, so F ⊆ h

_{λ}

_{′};

_{λ}

_{′}.

- (2)
- ∀λ ∈ [0, 1], Let F′ = h
_{λ}, then ∀x ∈ F′, h(x) ≥ λ, so $\underset{x\in {F}^{\prime}}{\mathrm{inf}}h(x)\ge \lambda $.

**Theorem 3.**

_{1}, …, x

_{n}} as a finite set, and ℘ = 2

^{X}, (X, ℘, m) as a fuzzy measure space, and O: [0, 1]

^{2}→ [0, 1] as an overlap function, if a ℘ measurable function h: X → [0, 1] such that h(x

_{i}) ≤ h(x

_{i+1}), for 1 ≤ i ≤ n − 1 (if not, rearrange h(x

_{i}), 1 ≤ i ≤ n). Then the O-Sugeno integral of h over A is further simplified as follows:

_{i}= {x

_{j}: i ≤ j ≤ n}, 1 ≤ i ≤ n.

**Proof of Theorem 3.**

**Theorem 4.**

^{X}and O: [0, 1]

^{2}→ [0, 1] as an overlap function, for arbitrary ℘ measurable functions h, h

_{1}, and h

_{2}and arbitrary subset A of X, the following conclusion is established:

- (1)
- If h
_{1}≤ h_{2}(i.e., for any x ∈ X, h_{1}(x) ≤ h_{2}(x)), then it holds that

- (2)
- If m(A) = 0, then it holds that

- (3)
- If a constant c ∈ [0, 1], then it holds that

- (4)
- If a constant c ∈ [0, 1], and for any x ∈ X, max(c, h)(x) = max{c, h(x)}, then it holds that

- (5)
- If A
_{1}⊆ A_{2}, then it holds that

- (6)

**Proof of Theorem 4.**

- (1)
- For any λ ∈ [0, 1], it holds that for any x ∈ X, λ ≤ h
_{1}(x) ≤ h_{2}(x).

_{1}

_{λ}= {x ∈ X: h

_{1}(x) ≥ λ} ⊆ {x ∈ X: h

_{2}(x) ≥ λ} = h

_{2}

_{λ}

_{1}

_{λ}) ≤ m(A ∩ h

_{2}

_{λ}).

_{1}

_{λ})) ≤ O(λ, m(A ∩ h

_{2}

_{λ}))

- (2)
- From m(A) = 0, we have m(A ∩ h
_{λ}) = 0.

- (3)
- For any x ∈ X, we define h(x) = c.

_{λ}= X. So,

_{λ})] = O[λ, m(A ∩ X)] = O[λ, m(A)].

_{λ}= Φ. So,

- (4)
- If λ ≤ c, then for every x ∈ X,

_{λ}= X.

_{λ})] = O[λ, m(A ∩ X)] = O[λ, m(A)].

_{λ}= h

_{λ}.

_{λ})] = O[λ, m(A ∩ h

_{λ})].

- (5)
- For any λ ∈ [0, 1] and A
_{1}⊆ A_{2}, we have

_{1}∩ h

_{λ}) ⊆ (A

_{2}∩ h

_{λ}), then m(A

_{1}∩ h

_{λ}) ≤ m(A

_{2}∩ h

_{λ}).

- (6)
- For any x ∈ X, it holds that

_{1}, h

_{2})(x) = max(h

_{1}(x), h

_{2}(x)).

_{1}, h

_{2})(x) ≥ h

_{1}(x) ≥ λ and max(h

_{1}, h

_{2})(x) ≥ h

_{2}(x) ≥ λ.

_{1}

_{λ}⊆ max(h

_{1}, h

_{2})

_{λ}and h

_{2}

_{λ}⊆ max(h

_{1}, h

_{2})

_{λ}.

_{1}

_{λ}) ≤ m(A ∩ max(h

_{1}, h

_{2})

_{λ}) and m(A ∩ h

_{2}

_{λ}) ≤ m(A ∩ max(h

_{1}, h

_{2})

_{λ}).

**Example 4.**

_{1}, s

_{2}, and s

_{3}. Let the attribute set X = {s

_{1}, s

_{2}, s

_{3}}. The importance of each attribute is determined by experts and house buyers as follows:

_{1}}) = 0.7, m({s

_{2}}) = 0.5, m({s

_{3}}) = 0.4, m({s

_{1}, s

_{2}}) = 0.9, m({s

_{1}, s

_{3}}) = 0.6, m({s

_{2}, s

_{3}}) = 0.8, m({s

_{1}, s

_{2}, s

_{3}}) = 1

_{1}({s

_{1}}) = 0.9, h

_{1}({s

_{2}}) = 0.8, h

_{1}({s

_{3}}) = 0.5; second property: h

_{2}({s

_{1}}) = 0.6, h

_{2}({s

_{2}}) = 0.9, h

_{2}({s

_{3}}) = 0.7.

_{1}and h

_{2}scores of the two properties are regarded as functions of the property set X. The overlap function is defined by

^{2}y

^{2}, for any x, y ∈ [0, 1].

_{1}over X, as follows:

_{2}over X, as follows:

## 4. O-Sugeno Integral Semantics of Fuzzy Quantifiers

_{q}with fuzzy quantifiers.

**Definition 6.**

_{q}contains the following:

- (i)
- An enumerable set of individual variables: x
_{0}, x_{1}, x_{2}; - (ii)
- A set of predicate symbols: $F={\cup}_{n=0}^{\infty}{F}_{n}$, where F
_{n}indicates the set of all n-place predicate symbols for every n ≥ 0, assuming that ${\cup}_{n=0}^{\infty}{F}_{n}\ne \mathsf{\Phi}$; - (iii)
- Propositional connectors: ~ and ∧; and
- (iv)
- Parentheses: ( ) [41].

_{q}:

**Definition 7.**

- (i)
- For every n ≥ 0, if F is an n-place predicate symbol and y
_{1}, …, y_{n}are individual variables, then F(y_{1}, …, y_{n}) is a well-formed formula; - (ii)
- If Q is a quantifier, x is an individual variable, and φ is a well-formed formula, then (Qx) φ is also a well-formed formula; and
- (iii)
- If φ, φ
_{1}, and φ_{2}are all well-formed formulas, then ~φ, φ_{1}, and ∧φ_{2}are also well-formed formulas [41].

_{q}:

**Definition 8.**

- (i)
- A measurable space (X, ℘), which is called the domain of the interpretation;
- (ii)
- For every n ≥ 0, there exists an element x
_{i}^{I}in X corresponding to the individual variable x_{i}; and - (iii)
- For every n ≥ 0 and any F ∈ F
_{n}, there exists a ℘^{n}-measurable function F^{I}: X^{n}→ [0, 1] [41].

**Definition 9.**

_{I}(φ) of formula φ under I based on O-Sugeno integrals is defined recursively as follows:

- (i)
- If φ = F(y
_{1}, …, y_{n}), then

^{I}(φ) = F(y

_{1}

^{I}, …, y

_{n}

^{I}).

- (ii)
- If φ = (Qx) y, then

_{I {. / x}}: X → [0, 1] is a mapping such that

_{I {. / x}}(φ)(u) = T

_{I {u/x}}(φ), for every u ∈ X,

^{I {u/x}}= y

^{I}and x

^{I {u/x}}= u, for every x, y ∈ X and x ≠ y.

- (iii)
- If φ = ~ ψ, then

_{I}(φ) = 1 − T

_{I}(ψ),

_{1}∧ φ

_{2}, then

**Proposition 1.**

_{I}((Qx)φ) = T

_{I}(φ).

**Proof of Proposition 1.**

**Proposition 2.**

- (1)
- ${T}_{I}((\exists x)\phi )=\underset{u\in X}{\mathrm{sup}}{T}_{I\{u/x\}}(\phi )$ and
- (2)
- ${T}_{I}((\forall x)\phi )=\underset{u\in X}{\mathrm{inf}}{T}_{I\{u/x\}}(\phi )$.

**Proof of Proposition 2.**

**Example 5.**

_{I}((Qx)H(x)) represents the truth value of QxH(x) under I calculated by the O-Sugeno integral. According to Theorem 2 in Section 3 and Definition 4 in Section 4, we have

_{i}) is the result of rearranging the possible values of H(x) in non-decreasing order, and X

_{i}= {x

_{j}: i ≤ j ≤ 10} for 1 ≤ i ≤ 10. Table 2 presents the rearranged truth values h(x

_{i}) for 1 ≤ i ≤ 10.

_{i}as follows:

_{X}(X

_{i}) = (|X

_{i}|/|X|)

^{3/2}= [(11 − i)/10]

^{3/2}, for 1 ≤ i ≤ 10

^{2}y

^{2}for every x, y ∈ [0, 1], the truth value of QxH(x) is calculated as follows:

**Example 6.**

_{1}and P

_{2}represent respectively the linguistic predicates “to be cloudy” and ”to be cold”. The respective weather conditions of the week are indicated in Table 3. Suppose Q is a fuzzy quantifier, “most”, then the formula φ = (Qx)ψ = (Qx) (P

_{1}(x)∧~ P

_{2}(x)) represents “many days (in this week) are cloudy but not cold”.

_{1}and P

_{2}given in Table 3, then T

_{I}(φ) = T

_{I}((Qx) (P

_{1}(x) ∧~P

_{2}(x))) represents the truth value of φ = (Qx) (P

_{1}(x)∧~ P

_{2}(x)) under interpretation I about the O-Sugeno integral. According to Theorem 2 in Section 3 and Definition 4 in Section 4, we have

_{i}) is the result of the rearranged possible values of T

_{I}(P

_{1}(x)∧~ P

_{2}(x)) in non-decreasing order, and X

_{i}= {x

_{j}: i ≤ j ≤ 7} for 1 ≤ i ≤ 7. Table 4 presents the rearranged truth values h(x

_{i}) for 1 ≤ i ≤ 7.

_{i}are calculated as follows:

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) = 6/7, Q

_{X}(X

_{3}) = 5/7, Q

_{X}(X

_{4}) = 4/7, Q

_{X}(X

_{5}) = 3/7, Q

_{X}(X

_{6}) = 2/7, Q

_{X}(X

_{7}) = 1/7.

_{1}(x)∧~P

_{2}(x)) is calculated as follows:

## 5. Applying Integral Semantics of Fuzzy Quantifiers to MADM

_{1}, s

_{2}, …, s

_{m}} is a set of m decision objects (also known as feasible alternatives), G = {g

_{1}, g

_{2}, …, g

_{n}} is a set of n evaluation indicators (also called attributes), and Q represents the fuzzy quantifiers such as most, many, more than half, etc.

_{i}), where for 1 ≤ i ≤ n − 1, h(x

_{i}) ≤ h(x

_{i+1}).

_{X}(X

_{i}), where for 1 ≤ i ≤ n, X

_{i}= {x

_{j}: i ≤ j ≤ n}.

_{i}) of proposition (Qx)φ(x) under its interpretation for each decision-making object s

_{i}∈ S:

**Example 7.**

_{i}) for 1 ≤ i ≤ 5. Table 6 presents the rearranged truth values.

_{i}= {x

_{j}: i ≤ j ≤ 5} for 1 ≤ i ≤ 5 about the fuzzy quantifier as follows:

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) = 0.64, Q

_{X}(X

_{3}) = 0.36, Q

_{X}(X

_{4}) = 0.16, Q

_{X}(X

_{5}) = 0.04.

_{3}) can be obtained by ranking these true values, so student s

_{3}is the best student.

**Example 8.**

_{1}, s

_{2}, s

_{3}, and s

_{4}. The decision-maker evaluates these suppliers in four aspects, which are called decision attributes: product price, product quality, service level, and reputation. The specific data are given in Table 7.

_{i}) for 1 ≤ i ≤ 4. Table 8 presents the rearranged truth values.

_{i}= {x

_{j}: i ≤ j ≤ 4} for 1 ≤ i ≤ 4 about the fuzzy quantifier as follows:

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) = (3/4)

^{3/2}≈ 0.650, Q

_{X}(X

_{3}) = (1/2)

^{3/2}≈ 0.354, Q

_{X}(X

_{4}) = (1/4)

^{3/2}≈ 0.125.

^{2}→ [0, 1] be defined as

_{2}has the highest score, thus supplier s

_{2}should be selected.

**Example 9.**

_{i}, for 1 ≤ i ≤ 8. The decision-maker evaluates these alternatives in 12 aspects, which are called decision attributes: urban support, traffic conditions, geological environment, land price, urban traffic improvement, convenient delivery, surrounding facilities, neighboring enterprises, talent attraction, logistics development space, prospect of environmental development, and predicted economic development. The specific data are given in Table 9.

_{i}) for 1 ≤ i ≤ 12. Table 10 presents the rearranged truth values.

_{i}= {x

_{j}: i ≤ j ≤ 12} for 1 ≤ i ≤ 12 about the fuzzy quantifier as follows:

_{X}(X

_{i}) = (|X

_{i}|/|X|)

^{3/2}= [(13 − i)/12]

^{3/2}, for 1 ≤ i ≤ 12.

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) ≈ 0.878, Q

_{X}(X

_{3}) ≈ 0.761, Q

_{X}(X

_{4}) ≈ 0.650, Q

_{X}(X

_{5}) ≈ 0.544, Q

_{X}(X

_{6}) ≈ 0.446,

_{X}(X

_{7}) ≈ 0.354, Q

_{X}(X

_{8}) ≈ 0.269, Q

_{X}(X

_{9}) ≈ 0.192, Q

_{X}(X

_{10}) = 0.125, Q

_{X}(X

_{11}) ≈ 0.068, Q

_{X}(X

_{12}) ≈ 0.024.

_{3}has the highest score, thus alternative s

_{3}should be selected.

**Example 10.**

_{i}, for 1 ≤ i ≤ 4. In order to purchase a satisfactory car, the customer browsed the comments of each alternative on various network platforms and evaluated them from seven aspects (attributes): appearance, interior, space, comfort, power, operation difficulty, and cost performance. Through text sentiment analysis, all evaluation information is converted into specific data, as shown in Table 12.

_{i}) for 1 ≤ i ≤ 7. Table 13 presents the rearranged truth values.

_{i}= {x

_{j}: 1 ≤ i ≤ 7} for 1 ≤ i ≤ 7 about fuzzy quantifier as follows:

_{X}(X

_{i}) = (|X

_{i}|/|X|)

^{2}= [(8 − i)/7]

^{2}, for 1 ≤ i ≤ 7.

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) = 0.735, Q

_{X}(X

_{3}) = 0.510, Q

_{X}(X

_{4}) = 0.327, Q

_{X}(X

_{5}) = 0.184, Q

_{X}(X

_{6}) = 0.082, Q

_{X}(X

_{7}) = 0.020

_{1}has the highest score, thus alternatives s

_{1}should be selected.

**Example 11.**

_{i}) for 1 ≤ i ≤ 4. Table 16 presents the rearranged truth values.

_{i}= {x

_{j}: 1 ≤ i ≤ 9} for 1 ≤ i ≤ 9 about the fuzzy quantifier as follows:

_{X}(X

_{1}) = 1, Q

_{X}(X

_{2}) = 64/81, Q

_{X}(X

_{3}) = 49/81, Q

_{X}(X

_{4}) = 4/9, Q

_{X}(X

_{5}) = 25/81, Q

_{X}(X

_{6}) = 16/81, Q

_{X}(X

_{7}) = 1/9, Q

_{X}(X

_{8}) = 4/81, Q

_{X}(X

_{9}) = 1/81.

_{4}has the highest score, and the alternatives s

_{4}should be selected.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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s_{1} | s_{2} | s_{3} | s_{4} | s_{5} | s_{6} | s_{7} | s_{8} | s_{9} | s_{10} | |
---|---|---|---|---|---|---|---|---|---|---|

H (x) | 0.95 | 0.1 | 0.73 | 1 | 0.84 | 0.7 | 0.67 | 0.9 | 1 | 0.81 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

h (x_{i}) | 0.1 | 0.67 | 0.7 | 0.73 | 0.81 | 0.84 | 0.9 | 0.95 | 1 | 1 |

Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | |
---|---|---|---|---|---|---|---|

P_{1}^{I} | 0.1 | 0 | 0.5 | 0.8 | 0.6 | 1 | 0.2 |

P_{2}^{I} | 1 | 0.9 | 0.4 | 0.7 | 0.3 | 0.4 | 0 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

h(x_{i}) | 0 | 0 | 0.2 | 0.3 | 0.5 | 0.6 | 0.6 |

Mathematics | Physics | Biology | Chemistry | Literature | |
---|---|---|---|---|---|

s_{1} | 0.75 | 0.85 | 0.95 | 0.90 | 0.86 |

s_{2} | 0.85 | 0.92 | 0.91 | 0.95 | 0.86 |

s_{3} | 0.92 | 0.87 | 0.90 | 0.89 | 0.91 |

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

s_{1} (h_{1}(x_{i})) | 0.75 | 0.85 | 0.86 | 0.9 | 0.95 |

s_{2} (h_{2}(x_{i})) | 0.85 | 0.86 | 0.91 | 0.92 | 0.95 |

s_{3} (h_{3}(x_{i})) | 0.87 | 0.89 | 0.90 | 0.91 | 0.92 |

Product Price | Product Quality | Service Level | Reputation | |
---|---|---|---|---|

s_{1} | 0.95 | 0.71 | 0.85 | 0.8 |

s_{2} | 0.8 | 0.76 | 0.92 | 0.83 |

s_{3} | 0.85 | 0.81 | 0.7 | 0.86 |

s_{4} | 0.76 | 0.9 | 0.75 | 0.84 |

1 | 2 | 3 | 4 | |
---|---|---|---|---|

s_{1} (h_{1}(x_{i})) | 0.71 | 0.8 | 0.85 | 0.95 |

s_{2} (h_{2}(x_{i})) | 0.76 | 0.8 | 0.83 | 0.92 |

s_{3} (h_{3}(x_{i})) | 0.7 | 0.81 | 0.85 | 0.86 |

s_{4} (h_{4}(x_{i})) | 0.75 | 0.76 | 0.84 | 0.9 |

s_{1} | s_{2} | s_{3} | s_{4} | s_{5} | s_{6} | s_{7} | s_{8} | |
---|---|---|---|---|---|---|---|---|

Urban support | 0.912 | 0.97 | 0.824 | 0.706 | 0.964 | 0.556 | 0.656 | 0.734 |

Traffic conditions | 0.9 | 0.846 | 0.786 | 0.93 | 0.824 | 0.972 | 0.738 | 0.892 |

Geological environment | 0.89 | 0.876 | 0.93 | 0.824 | 0.772 | 0.932 | 0.936 | 0.814 |

Land price | 0.69 | 0.574 | 0.856 | 0.712 | 0.592 | 0.93 | 0.726 | 0.794 |

Urban traffic improvement | 0.7 | 0.624 | 0.858 | 0.89 | 0.652 | 0.978 | 0.972 | 0.904 |

Convenient delivery | 0.85 | 0.864 | 0.904 | 0.774 | 0.902 | 0.606 | 0.596 | 0.912 |

Surrounding facilities | 0.648 | 0.774 | 0.912 | 0.842 | 0.804 | 0.67 | 0.806 | 0.796 |

Neighboring enterprises | 0.806 | 0.828 | 0.912 | 0.774 | 0.812 | 0.604 | 0.772 | 0.804 |

Talent attraction | 0.846 | 0.972 | 0.826 | 0.774 | 0.962 | 0.604 | 0.796 | 0.806 |

Logistics development space | 0.796 | 0.712 | 0.912 | 0.804 | 0.806 | 0.608 | 0.778 | 0.952 |

Prospect of environmental development | 0.792 | 0.774 | 0.956 | 0.796 | 0.846 | 0.734 | 0.752 | 0.846 |

Predicted economic development | 0.808 | 0.808 | 0.816 | 0.842 | 0.792 | 0.774 | 0.804 | 0.912 |

s_{1} (h_{1}(x_{i})) | s_{2} (h_{2}(x_{i})) | s_{3} (h_{3}(x_{i})) | s_{4} (h_{4}(x_{i})) | s_{5} (h_{5}(x_{i})) | s_{6} (h_{6}(x_{i})) | s_{7} (h_{7}(x_{i})) | s_{8} (h_{8}(x_{i})) | |
---|---|---|---|---|---|---|---|---|

1 | 0.648 | 0.574 | 0.786 | 0.706 | 0.592 | 0.556 | 0.596 | 0.734 |

2 | 0.69 | 0.624 | 0.816 | 0.712 | 0.652 | 0.604 | 0.656 | 0.794 |

3 | 0.7 | 0.712 | 0.824 | 0.774 | 0.772 | 0.604 | 0.726 | 0.796 |

4 | 0.792 | 0.774 | 0.826 | 0.774 | 0.792 | 0.606 | 0.738 | 0.804 |

5 | 0.796 | 0.774 | 0.856 | 0.774 | 0.804 | 0.608 | 0.752 | 0.806 |

6 | 0.806 | 0.808 | 0.858 | 0.796 | 0.806 | 0.67 | 0.772 | 0.814 |

7 | 0.808 | 0.828 | 0.904 | 0.804 | 0.812 | 0.734 | 0.778 | 0.846 |

8 | 0.846 | 0.846 | 0.912 | 0.824 | 0.824 | 0.774 | 0.796 | 0.892 |

9 | 0.85 | 0.864 | 0.912 | 0.842 | 0.846 | 0.93 | 0.804 | 0.904 |

10 | 0.89 | 0.876 | 0.912 | 0.842 | 0.902 | 0.932 | 0.806 | 0.912 |

11 | 0.9 | 0.97 | 0.93 | 0.89 | 0.962 | 0.972 | 0.936 | 0.912 |

12 | 0.912 | 0.972 | 0.956 | 0.93 | 0.964 | 0.978 | 0.972 | 0.952 |

s_{1} | s_{2} | s_{3} | s_{4} | s_{5} | s_{6} | s_{7} | s_{8} | |
---|---|---|---|---|---|---|---|---|

Evaluation value | 0.837 | 0.844 | 0.903 | 0.872 | 0.872 | 0.778 | 0.852 | 0.891 |

s_{1} | s_{2} | s_{3} | s_{4} | |
---|---|---|---|---|

Appearance | 0.8149 | 0.7320 | 0.8352 | 0.6786 |

Interior | 0.6890 | 0.7302 | 0.7056 | 0.6810 |

Space | 0.5969 | 0.3858 | 0.2555 | 0.3183 |

Comfort | 0.7058 | 0.6030 | 0.7398 | 0.6429 |

Power | 0.5708 | 0.5227 | 0.6259 | 0.4488 |

Operation difficulty | 0.6632 | 0.6041 | 0.4893 | 0.4579 |

Cost performance | 0.6765 | 0.4597 | 0.6123 | 0.4090 |

s_{1} (h_{1}(x_{i})) | s_{2} (h_{2}(x_{i})) | s_{3} (h_{3}(x_{i})) | s_{4} (h_{4}(x_{i})) | |
---|---|---|---|---|

1 | 0.5708 | 0.3858 | 0.2555 | 0.3183 |

2 | 0.5969 | 0.4597 | 0.4893 | 0.4090 |

3 | 0.6632 | 0.5227 | 0.6123 | 0.4488 |

4 | 0.6765 | 0.6030 | 0.6259 | 0.4579 |

5 | 0.6890 | 0.6041 | 0.7056 | 0.6429 |

6 | 0.7058 | 0.7302 | 0.7398 | 0.6786 |

7 | 0.8149 | 0.7320 | 0.8352 | 0.6810 |

s_{1} | s_{2} | s_{3} | s_{4} | |
---|---|---|---|---|

Evaluation value | 0.773 | 0.714 | 0.714 | 0.670 |

s_{1} | s_{2} | s_{3} | s_{4} | |
---|---|---|---|---|

Fixed acidity | 0.9689 | 0.9222 | 0.8210 | 0.8327 |

Volatile acidity | 0.4055 | 0.9843 | 0.7323 | 0.7323 |

Citric acid | 0.6648 | 0.5369 | 0.8750 | 0.7983 |

Residual sugar | 0.8147 | 0.6207 | 0.8922 | 0.9914 |

Chlorides | 0.7727 | 0.4513 | 0.8312 | 0.7581 |

Free sulfur dioxide | 0.8285 | 0.7531 | 0.3430 | 0.9456 |

Total sulfur dioxide | 0.7143 | 0.8937 | 0.6246 | 0.9867 |

Sulfate | 0.6903 | 0.8148 | 0.8726 | 0.9580 |

Alcohol | 0.7607 | 0.7855 | 0.8020 | 0.8682 |

s_{1} (h_{1}(x_{i})) | s_{2} (h_{2}(x_{i})) | s_{3} (h_{3}(x_{i})) | s_{4} (h_{4}(x_{i})) | |
---|---|---|---|---|

Fixed acidity | 0.4055 | 0.4513 | 0.3430 | 0.7323 |

Volatile acidity | 0.6648 | 0.5369 | 0.6246 | 0.7581 |

Citric acid | 0.6903 | 0.6207 | 0.7323 | 0.7983 |

Residual sugar | 0.7143 | 0.7531 | 0.8020 | 0.8327 |

Chlorides | 0.7607 | 0.7855 | 0.8210 | 0.8682 |

Free sulfur dioxide | 0.7727 | 0.8148 | 0.8312 | 0.9456 |

Total sulfur dioxide | 0.8147 | 0.8937 | 0.8726 | 0.9580 |

Sulfate | 0.8285 | 0.9222 | 0.8750 | 0.9867 |

Alcohol | 0.9689 | 0.9843 | 0.8922 | 0.9914 |

s_{1} | s_{2} | s_{3} | s_{4} | |
---|---|---|---|---|

Evaluation value | 0.815 | 0.778 | 0.790 | 0.871 |

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## Share and Cite

**MDPI and ACS Style**

Mao, X.; Temuer, C.; Zhou, H.
Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making. *Axioms* **2023**, *12*, 734.
https://doi.org/10.3390/axioms12080734

**AMA Style**

Mao X, Temuer C, Zhou H.
Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making. *Axioms*. 2023; 12(8):734.
https://doi.org/10.3390/axioms12080734

**Chicago/Turabian Style**

Mao, Xiaoyan, Chaolu Temuer, and Huijie Zhou.
2023. "Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making" *Axioms* 12, no. 8: 734.
https://doi.org/10.3390/axioms12080734