Research on the problem of aggregation of multiple rules in fuzzy inference systems

Abstract

In this paper, we establish the matching relation between implication operator and aggregation operator, which provides a new solution for the design and construction of multi-rule fuzzy inference system. Firstly, according to the definition and monotonicity of implication operator, a new classification method of implication operator is proposed, and then the fuzzy inference process using different implication operators is classified. Then, dynamic maximum aggregation operator and dynamic minimum aggregation operator are proposed. Based on the compositional rule of inference (CRI) method, a matching method and basis of implication operator and aggregation operator for fuzzy inference systems is given and illustrated with examples. Finally, the applicability of the proposed method in this paper is further illustrated by comparing the method with existing methods in the literature and using the nearness degree as an evaluation index.

Keywords

Multi-rule fuzzy inference systems classification of fuzzy implication aggregation operators nearness degree

1 Introduction

Fuzzy inference system is a system with the ability to handle fuzzy information based on fuzzy set theory and fuzzy inference methods. Fuzzy inference system mainly consists of four parts: input, fuzzy rule base, fuzzy inference method and output [1–3]. Fuzzy inference system is also called multi-rule fuzzy inference system when there are multiple rules in the fuzzy rule base. Let the domains X = {x₁, x₂, ⋯ x_n}, Y = {y₁, y₂, ⋯ y_m}, $F (X)$ and $F (Y)$ denote the entire fuzzy sets defined on the domains X and Y. If given Ω (Ω ≥ 2) rules: if x is A_ω, then y is B_ω, where ω = 1, 2, …, Ω, and given system observation: x is A^*, then the multi-rule fuzzy inference model can be expressed as:

Rule 1:	If x is A₁, then y is B₁.
Rule 2:	If x is A₂, then y is B₂.
⋮	⋮
Rule Ω:	If x is A_Ω, then y is B_Ω.
Observation:	x is A^*
Result:	y is B^*

In the model, $A_{ω}, A^{*} \in ℱ (X)$ and $B^{*} \in F (Y)$ . References [5, 8] give the solution of the above model, the basic idea is to aggregate multiple rules into a single rule and then reasoning. References [6, 7] propose that the basic idea of the solution method is to use the reasoning results obtained from each rule, and aggregate them as the final reasoning results. Aggregation operators are used in these algorithms.

Aggregation operator is a mathematical function used to fuse different information, which has many applications in different fields. In decision-making, they can be used to evaluate options and select the best course of action based on fuzzy criteria [9, 10]. In pattern recognition, they can be used to classify objects or events according to their fuzzy features [11, 15]. In control systems, they can be used to adjust the behavior of the system according to fuzzy feedback [16]. In the multi-rule fuzzy reasoning system, the use of aggregation operators is a very common operation, so researchers put forward a large number of aggregation operators for fuzzy inference system. Depending on the fuzzy sets, there are type-2 fuzzy set aggregation operators [17, 18], interval fuzzy set aggregation operators [19, 20], intuitionistic fuzzy set aggregation operators [21, 22], Fermatean fuzzy set aggregation operators [23, 24] and complex probabilistic fuzzy set aggregation operators [29]. According to the number of aggregate elements, there are binary aggregate functions [30–32] and n-ary overlapping aggregate functions [33–35]. The min aggregation operators are used in systems based on Mamdani and Larsen reasoning methods because of their simple and fast operation [36–38]. The weighted average operator can highlight the information needed by people and is widely used to solve the aggregation problem in fuzzy systems based on Takagi-Sugeno reasoning method [39–41]. In recent years, fuzzy integral aggregation operator [42], fuzzy product aggregation operator [43] and fuzzy geometric aggregation operator [44] have been applied to fuzzy inference systems for specific problems or situations.

Fuzzy implication is the key operation in fuzzy inference systems to represent IF-THEN fuzzy rules and is also responsible for the propagation of uncertainty in fuzzy inference [4]. Moreover, it plays an important role in many application fields. For example, it provides semantic explanation for implicative connectives in fuzzy logic [45], constructs upper and lower approximation operators in fuzzy rough sets [46], constructs equation solutions in fuzzy relational equations [47], it is used to define fuzzy erosion operators in fuzzy mathematical morphology [48], to construct image similarity measures in image processing [49], and to deduce Galois connection in formal concept analysis [50]. The wide range of applications has driven the research on fuzzy implication theory. As new methods for constructing fuzzy implication operators continue to emerge [51], the number of fuzzy implication operators has increased.

Generally speaking, the inference results obtained by using different implication operators are different. For example, a symmetric implication fuzzy reasoning method is proposed in [52], and the reversibility of the reasoning method is investigated when 8 different implication operators are used. In the literature [53], a fuzzy reasoning method by optimizing the similarity of truth-tables was proposed and the inference algorithm using nine different implication operators was investigated. The implication operators suitable for different fuzzy inference methods are different, mainly in terms of whether the algorithms have good robustness. The robustness of the inverse triple-I algorithm based on three common interval residual implication operators was studied in the literature [54]. The literature [55] investigated interval-valued full implication algorithms based on residual implication of four significant intervals, and the results showed that the robustness of interval-valued full implication algorithms for fuzzy inference depends directly on the choice of the implication operator in the fuzzy rules.

For a given inference method, the choice of aggregation operators among multiple rules depends mainly on the fuzzy implication operators used in the logic inference process. In the face of so many implication operators, how to choose the appropriate aggregation operator for a multi-rule fuzzy inference system is an urgent problem to be studied.

The purpose of this paper is to establish matching relationships between the implication and aggregation operators used in fuzzy inference systems and to provide new solutions for the design and construction of multi-rule fuzzy inference systems. Firstly, the implication operator is classified according to its definition and monotonicity, and then the fuzzy inference process using different implication operators is classified. Second, in order to match different classes of implication operators and to fit different types of inference processes, two new aggregation operators are constructed in this paper. Then, the matching basis of aggregation operator when different implication operators are applied to the fuzzy inference system is given. It is experimentally demonstrated that the application of this paper’s method to a multi-rule fuzzy inference system based on the CRI method yields desirable results.

The rest of this paper is organized as follows. In Section 2, we review the basics; in Section 3, we classify fuzzy implication operators and inference processes; in Section 4, we propose dynamically large and dynamically small aggregation operators to establish the matching relationship between implication and aggregation operators in multi-rule fuzzy inference systems; in Section 5, we combine examples to illustrate the matching method of aggregation operators when different implication operators are used in multi-rule fuzzy inference systems; in Section 6, we further illustrate the applicability of this paper’s method through comparative In Section 6, the applicability of this paper is further illustrated by comparison experiments; the conclusion is in the last section.

2 Preliminaries

L.A. Zadeh proposed the compositional rule of inference (CRI) method in 1973 [4], which is one of the most commonly used fuzzy inference methods at present. The basic principle of the single-rule CRI algorithm is to make the following inference for a given observation A^*. $B^{*} = A^{*} \circ [A \to B]$ (1) where ∘ denotes the composition operation in the CRI method, [A → B] denotes fuzzy implication. However, in practical applications, most of the fuzzy rule bases contain many rules. The literature [8] proposes two processes for multi-rule CRI fuzzy inference algorithms, first inference and then aggregation and first aggregation and then inference, i.e., “FITA” and “FATI”.

FITA. Firstly, the composite inference results of the observations A^* and each rule R_ω are calculated separately. $B_{ω}^{*} = A^{*} \circ R_{ω}, ω = 1, 2, \dots Ω$ (2) Then aggregate all the results to obtain the B^*. $B^{*} = ⊎_{ω = 1}^{Ω} B_{ω}^{*}$ (3) where ⊎ denotes aggregation of multiple fuzzy sets, which in the actual calculation is to map a [0, 1] ^Ω space to a [0, 1] space by aggregation operator.

FATI. [8] Firstly, the implication relation of all rules is aggregated. $R = ⊎_{ω = 1}^{Ω} R_{ω}$ (4) Then use the CRI method to obtain the B^*. $B^{*} = A^{*} \circ R$ (5)

The aggregation operator is also called the aggregation function. The definition of the aggregation operator is given in the literature [12] as follows.

Definition 1. [12] A mapping A: [0, 1] ⁿ → [0, 1] is an aggregation operator if it is monotonically increasing function, i.e., A (x₁, x₂, ⋯ x_n) ⩽ A (y₁, y₂, ⋯ y_n), when (x₁, x₂, ⋯ x_n), (y₁, y₂, ⋯ y_n) ∈ [0, 1] ⁿ, x_i ⩽ y_i, where i = 1, 2, ⋯ n, and $A \underset{n - times}{\underset{︸}{(x, \dots x)}} = x$ is satisfied for every x ∈ [0, 1], $n \in ℕ$ .

Definition 2. [13] The mapping I: [0, 1] × [0, 1] → [0, 1] satisfies the following two conditions.

∃a, b ∈ [0, 1], such that I (a, b) =1;

∃c, d ∈ [0, 1], such that I (c, d) =0.

Then I is called a fuzzy implication operator.

According to the inference results, the literature [8] gives the following classification of inference processes.

Definition 3. [8] For a given rule A → B and observation A^*, suppose that the result deduced by an inference process is B^*, if there is always B ⊂ B^*, this inference process is called “extended inference “. On the contrary, suppose the result of the inference is B’, and if there is always B′ ⊂ B, this inference process is called “reduced inference”.

In order to describe the closeness of the output fuzzy set obtained by inference to the ideal output fuzzy set, this paper uses the nearness degree proposed in the literature [14] and defined by the Hamming distance.

Definition 4. [14] Let X = {x₁, x₂, ⋯ x_n}, A and B are fuzzy sets defined on the domains X, $q_{H} (A, B) = 1 - \frac{1}{n} \sum_{k = 1}^{n} | A (x_{k}) - B (x_{k}) |$ (6)

q_H (A, B) is called the nearness degree of A to B as defined by the Hamming distance.

3 Fuzzy implication classification

3.1 Fuzzy implication classification

In this paper, we classify the implication operators according to the definition and monotonicity of fuzzy implication operators.

Definition 5. The implication operator is a “type I extended implication” on the interval U, i.e., for ∀a ∈ U ⊂ [0, 1], there is I (a, b) ⩾ b, and for any two values a₁ and a₂ on the interval U, if a₁ ⩽ a₂, then I (a₁, b) ⩾ I (a₂, b).

Definition 6. The implication operator is called a “type II extended implication” on the interval U, i.e., for ∀a ∈ U ⊂ [0, 1], there is I (a, b) ⩾ b and for any two values a₁ and a₂ on the interval U, if a₁ ⩽ a₂, then I (a₁, b) ⩽ I (a₂, b).

Definition 7. The implication operator is called a “type I reduced implication” on the interval U, i.e., for ∀a ∈ U ⊂ [0, 1], there is I (a, b) ⩽ b and for any two values a₁ and a₂ on the interval U, if a₁ ⩽ a₂, then I (a₁, b) ⩽ I (a₂, b).

Definition 8. The implication operator is called a “type II reduced implication” on the interval U, i.e., for ∀a ∈ U ⊂ [0, 1], there is I (a, b) ⩽ b, and for any two values a₁ and a₂ on the interval U, if a₁ ⩽ a₂, then I (a₁, b) ⩾ I (a₂, b).

Under the classification method of this paper, a fuzzy implication operator can belong to multiple classes.

Definition 9. The implication operator is called an “extended implication”, if ∃U ⊂ [0, 1], such that I is a type I extended implication or a type II extended implication on the interval U, and I is neither a type I reduced implication nor a type II reduced implication on the interval V, for ∀V ⊂ [0, 1].

Definition 10. The implication operator is called a “reduced implication”, if ∃U ⊂ [0, 1], such that I is a type I reduced implication or a type II reduced implication on the interval U, and I is not a type I extended implication or a type II extended implication on the interval V, for ∀V ⊂ [0, 1].

Out of the perspective of inference results rationality and the need of CRI algorithm, the following 2 properties of fuzzy implication operator are considered in this paper.

Property 1. I (0, b) =1, ∀b ∈ [0, 1].

Property 2. I (0, b) =0, ∀b ∈ [0, 1].

It is obvious that Property 1 and Property 2 contain contradictory parts. That is, for a certain implication operator cannot satisfy both properties, nor necessarily one of them, or neither of them. It can be proved that all the extended implication operators in Table 1 satisfy Property 1 and all the reduced implication operators in Table 2 satisfy Property 2.

Table 1
Several extended fuzzy entailment operators

Number Name Equation Category

I ₁ Kleene-Dienes a → b = max(1 - a, b) Type I extended implication on [0,1]

I ₂ Goguen $a \to b = {\begin{matrix} 1, & a ⩽ b \\ b / - a, & a > b \end{matrix}$ Type I extended implication on [0,1]

I ₃ Łukasiewicz a → b = min(1, 1 - a + b) Type I extended implication on [0,1]

I ₄ Yager a → b = b^a Type II extended implication on [0,1]

I ₅ Gödel $a \to b = {\begin{matrix} 1, & a ⩽ b \\ b, & a > b \end{matrix}$ Type II extended implication on [0,1]

Number	Name	Equation	Category
I ₁	Kleene-Dienes	a → b = max(1 - a, b)	Type I extended implication on [0,1]
I ₂	Goguen	$a \to b = {\begin{matrix} 1, & a ⩽ b \\ b / - a, & a > b \end{matrix}$	Type I extended implication on [0,1]
I ₃	Łukasiewicz	a → b = min(1, 1 - a + b)	Type I extended implication on [0,1]
I ₄	Yager	a → b = b^a	Type II extended implication on [0,1]
I ₅	Gödel	$a \to b = {\begin{matrix} 1, & a ⩽ b \\ b, & a > b \end{matrix}$	Type II extended implication on [0,1]

Table 2

Several reduced fuzzy entailment operators

Number	Name	Equation	Category
I ₆	Mamdani	a → b = min(a, b)	Type I reduced implication on [0,1]
I ₇	Probabilistic	a → b = ab	Type I reduced implication on [0,1]
I ₈	Bold	a → b = max(0, a + b - 1)	Type II reduced implication on [0,1]

Definition 11. The implication operator is a “mixed implication”, if ∃U, V ⊂ [0, 1], such that I (a, b) is a type I extended implication or a type II extended implication on the interval U, and is a type I reduced implication or a type II reduced implication on the interval V.

For several common classes of general implication families in fuzzy inference, the following theorem is given in this paper.

Theorem 1. An (S, N)-implication, i.e., I (a, b) = S (N (a) , b), is a type I extended implication on the interval [0,1]. (S is a continuous T-conorms operator and N (·) is a strong negation function.)

Proof. By the definition of T-conorms, ∀a, b ∈ [0, 1], there are S (a, b) ⩾ a and S (a, b) ⩾ b. Then I (a, b) = S (N (a) , b) ⩾ b.

For ∀a₁, a₂∈ [0, 1], it may be assumed that a₁ ⩽ a₂, then N (a₁) ⩾ N (a₂). T-conorms operator satisfies the property S (x, y) = S (y, x), and when y ⩽ z, we have S (x, y) ⩽ S (x, z).

Then, I (a₁, b) = S (N (a₁) , b) ⩾ S (N (a₂) , b) = I (a₂, b).

Theorem 2. An R-implication, i.e., I (a, b) = Sup{x ∈ [0, 1] |T (a, x) ⩽ b} is a type I extended implication on the interval [0,1]. (T is a continuous T-norms operator.)

Proof. For any T-norms operator, we have T (a, x) ⩽ a and T (a, x) ⩽ x. Therefore, for any T-norms operator, when a > b, we have $\begin{matrix} I (a, b) = Sup {x \in [0, 1] | T (a, x) ⩽ b} \\ ⩾ Sup {x \in [0, 1] | (a \land x) ⩽ b} = b \end{matrix}$ and when a ⩽ b, we have $I (a, b) = Sup {x \in [0, 1] | T (a, x) ⩽ b} = 1 ⩾ b .$

For any values of b, ∀a₁, a₂ ∈ [0, 1], it may be assumed that a₁ ⩽ a₂, then for ∀x ∈ [0, 1], we have $T (a_{1}, x) ⩽ T (a_{2}, x) .$ $\begin{matrix} \forall x \in {x \in [0, 1] | T (a_{2}, x) ⩽ b}, we have \\ x \in {x \in [0, 1] | T (a_{1}, x) ⩽ b} \end{matrix}$

Therefore, Sup {x ∈ [0, 1] |T (a₂, x) ⩽ b} ⩽ Sup {x ∈ [0, 1] |T (a₁, x) ⩽ b}

Theorem 3. The S-implication, i.e., I (a, b) = S (a, b), is a type II extended implication on the interval [0,1]. (S is a continuous T-conorms operator.)

Proof. The proof of Theorem 3 is obvious by the definition of T-conorms.

Theorem 4. The T-implication, i.e., I (a, b) = T (a, b), is a type I reduced implication on the interval [0,1]. (T is a continuous T-norms operator.)

Proof. By the definition of T-norms, the proof of Theorem 4 is obvious.

Based on the above definition of fuzzy entailment and its classification, this paper will continue to classify the inference process.

3.2 Fuzzy inference process classification

In this paper, based on Definition 3, the inference process is classified in more detail by combining the fuzzy implication operators selected in the inference process. It is worth noting that for fixed composition operators, such as Sup-min composition (denoted by the symbol ∨ -∧), the results of fuzzy inference in the CRI method depend mainly on the use of the implication operator. In view of this, the following definitions and theorems are given in this paper.

Definition 12. The inference process is “type I extended inference”, i.e., the fuzzy implication operator chosen in the inference process is the type I extended implication on the interval [0,1]. The inference process is “type II extended inference”, i.e., the fuzzy implication operator chosen in the inference process is the type II extended implication on the interval [0,1]. The inference process is “type III extended inference”, i.e., the fuzzy implication operator selected in the inference process is extended implication.

Theorem 5. Suppose that the fuzzy sets used in fuzzy reasoning systems are all normal fuzzy sets. For the CRI method, using the Sup-min composition operator, type I extended inference, type II extended inference and type III extended inference are extended inference [8].

Proof. Suppose a rule is given that if x is A, then y is B. The observation result is A^* and the corresponding inference result is B^*. Let the domain X = {x₁, x₂, ⋯ x_n}, Y = {y₁, y₂, ⋯ y_m}, $A, A^{*} \in F (X)$ , B, $B^{*} \in F (Y)$ .

(1) Consider the type I extended inference process, according to the CRI method Equations (1)–(3), and using the Sup-min composition operator, we have $\begin{matrix} B^{*} (y_{j}) = \underset{i}{\lor} {A^{*} (x_{i}) \land I [A (x_{i}), B (y_{j})]} \\ 1 pt 1 pt ⩾ \underset{i}{\lor} {A^{*} (x_{i}) \land B (y_{j})} \end{matrix}$ (7)

A^* is a normal fuzzy set, so there exists A^* (x_k) =1. Further, there exists A^* (x_k) ⩾ B (y_j), at this time A^* (x_k) ∧ B (y_j) = B (y_j), then in equation (7), we have $B^{*} (y_{j}) ⩾ \underset{i}{\lor} {A^{*} (x_{i}) \land B (y_{j})} ⩾ A^{*} (x_{k}) \land B (y_{j}) = B (y_{j})$

Further, B ⊂ B^*.

Therefore, according to the classification of fuzzy inference processes in Definition 3, type I extended inference is extended inference.

(2), (3) are proved in the same way as (1), and type II and type III extended inference are also extended inference.

Definition 13. The inference process is “type I reduced inference”, i.e., the fuzzy implication operator chosen in the inference process is a type I reduced implication on the interval [0,1]. The inference process is “type II reduced inference”, i.e., the fuzzy implication operator chosen in the inference process is a type II reduced implication on the interval [0,1]. The inference process is “type III reduced inference”, i.e., the fuzzy implication operator chosen in the inference process is a reduced implication.

Theorem 6. Suppose that the fuzzy sets used in fuzzy reasoning systems are all normal fuzzy sets. For the CRI method, using the Sup-min composition operator, Type I reduced inference, Type II reduced inference and Type III reduced inference are all reduced inference [8].

Based on the previous analysis, the proof of Theorem 8 is obvious.

Definition 14. The inference process is “mixed inference”, i.e., the fuzzy implication operator chosen in the inference process is a mixed implication.

Theorem 7. Suppose that the fuzzy sets used in fuzzy reasoning systems are all normal fuzzy sets. For the CRI method, using Sup-min composition operators, the mixed inference process is neither extended nor reduced inference [8].

Proof. By the proof (1) of Theorem 5, in CRI inference methods, using Sup-min composition operators, and using mixed implication operators, there are I [A (x_i) , B (y_j)] ⩾ B (y_j) and I [A (x_p) , B (y_q)] ⩾ B (y_q).

Thus, B^* (y_j) ⩾ B (y_j) and B^* (y_j) ⩾ B (y_q).

Therefore, the mixed inference process is neither extended nor reduced inference.

In this paper, we next propose the corresponding aggregation operator selection methods for each of the above inference processes.

4 Aggregation operator and its usage

4.1 Dynamic maximum aggregation operator and dynamic minimum aggregation operator

There is a considerable literature discussing aggregation operators and their properties to support a good choice of aggregation approaches in some specific problem domains [17–21]. In order to make aggregation operators in multi-rule fuzzy inference systems better match different types of implication operators and satisfy different types of inference processes, two new aggregation operators are proposed in this paper.

In this paper, we found through preliminary experiments that using min, max and mean as aggregation operators in a multi-rule fuzzy inference system all have significant drawbacks. The disadvantage of the min and max aggregation operators is that the minimum or maximum value is chosen as the aggregation result, ignoring a large amount of useful information. The disadvantage of the mean aggregation operator is that the average of all the information is chosen as the aggregation result, making some special information not obvious enough, such as the maximum and minimum values. From the perspective of compensating for the shortcomings of the above three aggregation operators, this paper proposes two new aggregation operators, which dynamically take the large aggregation operator, which is denoted by the symbol “δ_max” and the dynamic minimum aggregation operator, which is denoted by the symbol “δ_min”. It can not only reflect the specificity of extreme values, but also make use of more observations of variables, which are defined as follows.

Definition 15. The dynamic maximum aggregation operator “δ_max” is defined by the equation (8). The dynamic minimum aggregation operator “δ_min” is defined by the equation (9).

$\begin{matrix} δ_{max} (x_{1}, x_{2}, \dots x_{n}) = \\ \frac{max (x_{1}, x_{2}, \dots x_{n})}{1 + max (x_{1}, x_{2}, \dots x_{n}) - min (x_{1}, x_{2}, \dots x_{n})} \end{matrix}$ (8)

$\begin{matrix} δ_{min} (x_{1}, x_{2}, \dots x_{n}) = \\ 1 pt \frac{min (x_{1}, x_{2}, \dots x_{n})}{1 + max (x_{1}, x_{2}, \dots x_{n}) - min (x_{1}, x_{2}, \dots x_{n})} \end{matrix}$ (9)

The numerator of equation (8) is the maximum operation, and the numerator of equation (9) is the minimum operation, which makes the aggregation result reflect the specialness of extreme values to some extent. The denominators of both (8) and (9) are 1 + max(x₁, x₂, ⋯ x_n) - min(x₁, x₂, ⋯ x_n), making the dynamic maximum and dynamic minimum aggregation operators essentially a compromise operation, like the mean.

In the following, we prove that the dynamic maximum aggregation operator and the dynamic minimum aggregation operator satisfy the definition of the aggregation operator.

Proof. (1) The dynamic maximum aggregation operator and the dynamic minimum aggregation operator satisfy monotonicity.

∀ (x₁, x₂, ⋯ x_n), (y₁, y₂, ⋯ y_n) ∈ [0, 1] ⁿ, and x_i ⩽ y_i, i = 1, ⋯ n.

Let x_max = max(x₁, x₂, ⋯ x_n), x_min = min(x₁, x₂, ⋯ x_n), y_max = max(y₁, y₂, ⋯ y_n), y_min = min(y₁, y₂, ⋯ y_n). Then we have $\begin{matrix} δ_{max} (x_{1}, x_{2}, \dots x_{n}) \\ = \frac{max (x_{1}, x_{2}, \dots x_{n})}{1 + max (x_{1}, x_{2}, \dots x_{n}) - min (x_{1}, x_{2}, \dots x_{n})} \\ = \frac{x_{max}}{1 + x_{max} - x_{min}} \end{matrix}$ $\begin{matrix} δ_{max} (y_{1}, y_{2}, \dots y_{n}) \\ = \frac{max (y_{1}, y_{2}, \dots y_{n})}{1 + max (y_{1}, y_{2}, \dots y_{n}) - min (y_{1}, y_{2}, \dots y_{n})} \\ = \frac{y_{max}}{1 + y_{max} - y_{min}} \end{matrix}$ $\begin{matrix} δ_{max} (y_{1}, y_{2}, \dots y_{n}) - δ_{max} (x_{1}, x_{2}, \dots x_{n}) \\ = \frac{(1 - x_{min}) y_{max} - (1 - y_{min}) x_{max}}{(1 + y_{max} - y_{min}) (1 + x_{max} - x_{min})} ⩾ 0 \end{matrix}$

Thus, δ_max (x₁, x₂, ⋯ x_n) ⩽ δ_max (y₁, y₂, ⋯ y_n). δ_min (x₁, x₂, ⋯ x_n) in the same way.

(2) The dynamic maximum aggregation operator and the dynamic minimum aggregation operator satisfy the idempotency. $\begin{matrix} δ_{max} (x, x, \dots x) \\ 1 pt 1 pt 1 pt = \frac{max (x_{1}, x_{2}, \dots x_{n})}{1 + max (x_{1}, x_{2}, \dots x_{n}) - min (x_{1}, x_{2}, \dots x_{n})} \\ 1 pt 1 pt 1 pt = \frac{x}{1 + x - x} = x \end{matrix}$

δ_min (x, x, ⋯ x) in the same way.

For the two aggregation operators proposed above, this paper gives the matching method and theoretical basis for applying them to a multi-rule fuzzy inference system with different types of implication operators.

4.2 Basic requirements for fuzzy inference

The basic requirement of single-rule fuzzy inference is that given a rule A → B, if the system observation is A^* = A, the inference result should be B, also known as reducibility. This property has been studied by many authors, e.g., in the literature [19–24]. The basic requirements of fuzzy inference should also be satisfied in the case of having multiple rules in the rule base. However, it can be shown that many fuzzy inference methods satisfy reducibility in the single rule or simple case and do not have reducibility when they have multiple rules, including the CRI method proposed in the literature [4], the full implication triple I algorithm proposed in the literature [25], and some similarity-based fuzzy inference algorithms proposed in the literature [26–28]. To this end, this paper proposes two basic requirements for multi-rule fuzzy inference in terms of the reasonableness of the inference results.

Results non-0 principle. In order to infer useful information, the result of fuzzy inference based on multiple rules, its membership degree should not be all 0 on all elements. When the value of membership degree on all elements is equal to 0, it means that the result obtained by inference is empty information.

Results non-1 principle. In order to infer meaningful information, the result of fuzzy inference based on multiple rules, its membership degree should not be all 1 on all elements. When the value of membership degree on all elements is equal to 1, it means that the result obtained by inference is meaningless information.

The meaning of the above two principles is that the final result of fuzzy inference should contain rich and specific information. Next, this paper will propose the selection method of aggregation operator based on these two principles.

4.3 Aggregation operator and its usage

For the three fuzzy inference processes proposed in Section 3 and the two aggregation operators proposed in Section 4, the following provides a solution for the design and construction of multi-rule fuzzy inference systems by establishing a matching method between the implication and aggregation operators used in different inference processes.

4.3.1 Extended inference

For the extended inference process based on the CRI method, the following necessary conditions are given in this paper.

Necessary condition 1. Assume that there are Ω (Ω ≥ 2) rules in the rule base of fuzzy inference system. Using CRI method with Sup-min composition operator, if the corresponding inference process is extended inference, it is necessary to use the δ_min operator for aggregation in order to obtain useful inference results.

Proof. (1) First consider the FITA method, for the ωth rule (1 ⩽ ω ⩽ Ω), then we have $B_{ω}^{*} (y_{j}) = \underset{i}{\lor} {A^{*} (x_{i}) \land I [A_{ω} (x_{i}), B_{ω} (y_{j})]}$ (10) $B^{*} (y_{j}) = \oplus_{ω = 1}^{Ω} B_{ω}^{*} (y_{j})$ (11) where ⊕ denotes the aggregation operator.

If there exists a rule ω′: A_ω′ → B_ω′, 1 ⩽ ω′ ⩽ Ω, such that A^*∩ A_ω′ = Ø, i.e., there is no overlap between A^* and A_ω′, as shown in Fig. 1.

When A^* (x_k) =0, there is

Fig. 1

No overlap between rule and observation.

$A^{*} (x_{k}) \land I [A_{ω^{'}} (x_{k}), B_{ω^{'}} (y_{j})] = 0 .$ When A^* (x_k) >0, there is $A_{ω^{'}} (x_{k}) = 0$ as the marked dot in Fig. 1, at which time A^* (x_k) ∧ I [A_ω′ (x_k) , B_ω′ (y_j)] = A^* (x_k) ∧ I [0, B_ω′ (y_j)].

During the extended type inference, if the implication operator I used satisfies property 1, i.e., $I [0, B_{ω^{'}} (y_{j})] = 1,$ then for any j, 1 ⩽ j ⩽ m, in equation (10), we have $B_{ω}^{*} (y_{j}) = \underset{i}{\lor} {A^{*} (x_{i}) \land I [0, B_{ω^{'}} (y_{j})]} = 1$

At this point the value of membership degree is equal to 1 on all elements in $B_{ω'}^{*}$ and the inference result obtained using rule ω′ is meaningless information.

If the δ_max aggregation operator is chosen, then in equation (11), there is $\begin{matrix} B^{*} (y_{j}) = δ_{max} \\ 1 pt 1 pt = 1 pt \frac{1}{1 + 1 - min (B_{1}^{*} (y_{j}), B_{2}^{*} (y_{j}), \dots, B_{Ω}^{*} (y_{j}))} ⩾ 0.5 \end{matrix}$ If the δ_min aggregation operator is chosen, then in equation (11), there is $\begin{matrix} B^{*} (y_{j}) = δ_{min} [B_{1}^{*} (y_{j}), B_{2}^{*} (y_{j}), \dots, B_{Ω}^{*} (y_{j})] \\ 1 pt 1 pt = 1 pt \frac{min (B_{1}^{*} (y_{j}), B_{2}^{*} (y_{j}), \dots, B_{Ω}^{*} (y_{j}))}{1 + 1 - min (B_{1}^{*} (y_{j}), B_{2}^{*} (y_{j}), \dots, B_{Ω}^{*} (y_{j}))} \end{matrix}$

The rule ω′ is very different from the observed result, and the influence of the rule ω′ on the inference result should be reduced in order to inference reasonably. If δ_max is used as the aggregation operator, the membership degree of the inference result on all elements in B^* is greater than or equal to 0.5, which is obviously unsatisfactory. If δ_min is used as the aggregation operator, there is no such effect.

(2) Now consider FATI, if given a system observation A^*, then according to the Equations (4)–(5), $B^{*} (y_{j}) = \underset{i}{\lor} {A^{*} (x_{i}) \land \oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})]}$ (12)

If δ_max is used as the aggregation operator, assume that the still existing ω′, 1 ⩽ ω′ ⩽ Ω, makes A^*∩ A_ω′ = Ø, as described in (1) above. If the implication operator I used satisfies Property 1, then for any j, 1 ⩽ j ⩽ m, in equation (12), we have $\begin{matrix} \oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})] ⩾ 0.5 \\ B^{*} (y_{j}) = \underset{i}{\lor} {A^{*} (x_{i}) \land \oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})]} ⩾ 0.5 \end{matrix}$

Therefore, similar to FITA, δ_min needs to be used as the aggregation operator.

Overall, in the extended inference, using the CRI method with the Sup-min composition operator, it is necessary to use the δ_min operator for aggregation.

4.3.2 Reduced inference

For the reduced inference process based on the CRI method, the following necessary conditions are given in this paper.

Necessary condition 2. Assume that there are Ω (Ω ≥ 2) rules in the rule base of fuzzy inference system. Using CRI method with Sup-min composition operator, if the corresponding inference process is reduced inference, it is necessary to use the δ_max operator for aggregation in order to obtain useful inference results.

Proof. (1) First consider the FITA method, if given a system observation A′, then $B_{ω}^{'} s (y_{j}) = \underset{i}{\lor} {A^{'} (x_{i}) \land I [A_{ω} (x_{i}), B_{ω} (y_{j})]}$ (13) $B^{'} (y_{j}) = \oplus_{ω = 1}^{Ω} B_{ω}^{'} (y_{j})$ (14)

If there exists a rule ω′: A_ω′ → B_ω′, 1 ⩽ ω′ ⩽ Ω, such that A′∩ A_ω′ = Ø. When A′ (x_k) =0, there is $A^{'} (x_{k}) \land I [A_{ω^{'}} (x_{k}), B_{ω^{'}} (y_{j})] = 0 .$ When A′ (x_k) >0, we have A_ω′ (x_k) =0, then $A^{'} (x_{k}) \land I [A_{ω^{'}} (x_{k}), B_{ω^{'}} (y_{j})] = A^{'} (x_{k}) \land I [0, B_{ω^{'}} (y_{j})] .$

During the reduction type inference, if the implication operator I used satisfies Property 2, I [0, B_ω′ (y_j)] =0, then for any j, 1 ⩽ j ⩽ m, in equation (13), we have $B_{ω'}^{*} (y_{j}) = \underset{i}{\lor} {A^{'} (x_{i}) \land I [0, B_{ω} (y_{j})]} = 0$

If the δ_min aggregation operator is chosen, then in equation (14), there is

The membership degree of the aggregation result on all elements in B′ is equal to 0, and the inference results in null information. For reasonable inference, it is necessary to use δ_max as the aggregation operator in this case.

(2) Now consider FATI, if given a system observation A′, then $B^{'} (y_{j}) = \underset{i}{\lor} {A^{'} (x_{i}) \land \oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})]}$ (15)

If δ_min is used as the aggregation operator, assume that the still existing ω′, as described in (1) above. If the implication operator I used satisfies Property 2, $\oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})] = 0$ for any j, 1 ⩽ j ⩽ m, in equation (15), we have $B^{'} (y_{j}) = \underset{i}{\lor} {A^{'} (x_{i}) \land 0]} = 0$

Therefore, similar to FITA, δ_max needs to be used as the aggregation operator.

For all the reduced inference, there is $I [A_{ω} (x_{i}), B_{ω} (y_{j})] ⩽ B_{ω} (y_{j})$ In the FITA method, there is $B_{ω}^{'} (y_{j}) \leq B_{ω} (y_{j}) .$ Whenever B_ω (y_j) =0, there is $B_{ω}^{'} (y_{j}) = 0$ if δ_min is used as the aggregation operator, and there is no such effect if δ_max is used as the aggregation operator. In the FATI method, whenever B_ω (y_j) =0, using δ_min as the aggregation operator gives $\oplus_{ω = 1}^{Ω} I [A_{ω} (x_{i}), B_{ω} (y_{j})] = 0$

Thus, we have B′ (y_j) =0. If we use δ_max as the aggregation operator, there is no such effect.

Overall, it is necessary to use the δ_min operator for aggregation in reduced inference using the CRI method with the Sup-min composition operator.

4.3.3 Mixed inference

For the mixed inference process based on CRI method, the following necessary conditions are given in this paper.

Necessary condition 3. Assume that there are Ω (Ω≥2) rules in the rule base of fuzzy inference system. Using the CRI method with the Sup-min composition operator, if the corresponding inference process is a hybrid inference, it is necessary to use the δ_min operator for aggregation when the number of close to 1 in the aggregated elements is greater than the number of close to 0 in order to obtain the desired inference result. On the contrary, it is necessary to use the δ_max operator for aggregation.

Proof. Consider the FITA method, if given a system observation A′. If there exists rule ω′, there is no overlap between A′ and A_ω′.The following two cases may occur in the mixed type inference.

(1) If the implication operator I used is one of the two types of extended implication on the interval U, when A_ω′ (x_k) ∈ U, there is $I [A_{ω^{'}} (x_{k}), B_{ω} (y_{j})] ⩾ B_{ω} (y_{j}) .$ Whenever B_ω (y_j) =1, we have $B_{ω}^{'} (y_{j}) = 1 .$ If δ_max is used as the aggregation operator, then $B^{'} (y_{j}) = 0.5 .$

If we use δ_min as the aggregation operator, there is no such effect.

(2) If the implication operator I used is one of the two types of reduced implication on the interval V, when A_ω′ (x_k) ∈ V, there is $I [A_{ω^{'}} (x_{k}), B_{ω} (y_{j})] ⩽ B_{ω} (y_{j}) .$ Whenever B_ω (y_j) =0, we have $B' (y_{j}) = 0 .$ If we use δ_min as the aggregation operator, then $B^{'} (y_{j}) = 0 .$

If we use δ_max as the aggregation operator, there is no such effect.

From the above analysis, it is clear that we cannot stipulate that mixed inference must use the δ_max operator for aggregation or must use the δ_min operator for aggregation.

In the rule base of fuzzy inference system, there are usually fewer rules that are the same as or close to the observed value, and more rules that are different from the observed value. Thus, if the inference result $B_{ω}^{'}$ , ω = 1, 2, ⋯ Ω, obtained using the mixed implication, has more membership degree close to 1 on element y_j than close to 0, it means that the rules that differ more from the observation have inference result membership degree closer to 1 on element y_j. To reduce the influence of such rules, it is necessary to use the δ_min operator for aggregation. Conversely, it is necessary to use the δ_max operator for aggregation.

For multiple rules and a given observation, the aggregation operator selection method in a multi-rule fuzzy inference system can be interpreted according to the principle of non-1 results as follows, if the inference result obtained using one rule is all 1, it is necessary to take a smaller value after aggregation in order to reduce the influence of this rule on the final inference result. In other words, it is necessary to use the δ_min aggregation operator. In addition, according to the principle of non-0 results, the aggregation operator selection method in the multi-rule fuzzy inference system can be interpreted as follows, if the inference result obtained by using one rule is all 0, a larger value is needed after aggregation in order to retain more information. In other words, it is necessary to use the δ_max aggregation operator.

5 Numerical examples

This section will illustrate the use of the aggregation operator proposed in this paper with numerical examples and show how the matching of the implication and aggregation operators affects the inference results, which is consistent with the findings in Section 4.

Suppose there are five rules in the rule base of fuzzy inference system, A_ω → B_ω, ω = 1, 2, ⋯ , 5, where $A_{ω} \in F (X)$ and B_ω $\in F (Y)$ are the fuzzy sets defined on the domain X and Y, respectively, and the membership function defining these fuzzy sets are shown in (16) and (17).

The image of the membership function is shown in Fig. 2.

Fig. 2

Membership functions of fuzzy sets, (a) premise, (b) conclusion.

$\begin{matrix} \begin{matrix} μ_{A 1} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00] \\ μ_{A 2} (x) = [0.00, 0.25, 0.50, 0.75, 1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00] \\ μ_{A 3} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.50, 1.00, 0.50, 0.00, 0.00, 0.00, 0.00] \\ μ_{A 4} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.50, 1.00, 0.50, 0.00, 0.00] \\ μ_{A 5} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.50, 1.00, 1.00] \end{matrix} \end{matrix}$ (16)

$\begin{matrix} μ_{B 1} (x) = [0.00, 0.50, 1.00, 0.50, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00] \\ μ_{B 2} (x) = [0.00, 0.25, 0.50, 0.75, 1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00] \\ μ_{B 3} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.50, 1.00, 0.50, 0.00, 0.00, 0.00, 0.00] \\ μ_{B 4} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.50, 1.00, 0.50, 0.00, 0.00] \\ μ_{B 5} (x) = [1.00, 0.75, 0.50, 0.25, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.50, 1.00, 1.00, 1.00] \end{matrix}$ (17)

First, the type I extended implication on the interval [0,1] is used, such as the Kleene-Dienes implication discussed in Section 3. Second, the type II extended implication on the interval [0,1] is used, such as the Yager implication discussed in Section 3. Then, the type I extended implication on the interval [0,1] is used, such as the Mamdani implication discussed in Section 3. Again, the type II reduced implication on the interval [0,1] is used, such as the Bold implication discussed in Section 3.

The experiments are based on the CRI method, and the FITA process is chosen to use the Sup-min composition operator. Set observation value A^* = A₅, then the ideal inference result should be B^* = B₅. The results obtained by using δ_min aggregation operator and δ_max aggregation operator, respectively, are shown in Fig. 3.

Fig. 3

Comparison of inference results, (a) type I extended inference, (b) type II extended inference, (c) type I reduced inference, (d) type II reduced inference.

It can be found that rules A₁ and A₂ differ from the observed result A₅ by a large amount and are completely non-overlapping, as discussed in Section 4. In order to inference reasonably, the influence of rules A₁ and A₂ on the inference results should be reduced as much as possible.

If the δ_max is chosen as the aggregation operator during the two types of extended inference, then the membership degree of inference result B^* greater than or equal to 0.5 on every element, and the inference yields meaningless information. To reduce this effect, the δ_min must be used as the aggregation operator.

If the δ_min is chosen as the aggregation operator during the two types of reduced inference, then the membership degree of inference result B^* is equal to 0 on every element, and the inference yields empty information. To reduce this effect, the δ_max must be used as the aggregation operator.

Finally, for the mixed inference, use the mixed implication operator, such as the Rescher implication operator discussed in Section 3.

Repeat the above experiments. The inference results obtained from the calculation are shown in (18).

$\begin{matrix} μ_{B^{*} 1} (x) = [1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00] \\ μ_{B^{*} 2} (x) = [1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00] \\ μ_{B^{*} 3} (x) = [1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00] \\ μ_{B^{*} 4} (x) = [1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 1.00] \\ μ_{B^{*} 5} (x) = [0.00, 0.0, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.50, 1.00, 1.00, 1.00] \end{matrix}$ (18)

According to the necessary condition 3, it is necessary to use the δ_min aggregation operator at this time, and the final inference result obtained from the calculation is shown in (19). According to the Equation (6), there is $q_{H} (B_{5}, B^{*}) = 0.99 .$

If the δ_max aggregation operator is used, the final inference result obtained from the calculation is shown in (20). According to the Equation (6), there is $q_{H 2} (B_{5}, B^{*}) = 0.63 .$

$\begin{matrix} μ_{B^{*}} (y) = [0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, \\ 0.00, 0.00, 0.00, 0.33, 1.00, 1.00, 1.00] \end{matrix}$ (19)

$\begin{matrix} μ_{B^{*}} (y) = [0.50, 0.50, 0.50, 0.50, 0.50, 0.50, 0.50, \\ 0.50, 0.50, 0.50, 0.67, 1.00, 1.00, 1.00] \end{matrix}$ (20)

Since q_H1 (B₅, B^*) > q_H2 (B₅, B^*), it can be seen that the aggregation operator selection method proposed in this paper also has good applicability when using mixed fuzzy implication operators in the inference process.

It is worth noting that since the common max aggregation operator and min aggregation operator are special case of the aggregation operator in this paper, the aggregation operator selection basis proposed in this paper is also suitable for t maximum and minimum aggregation operators.

In the next section, we will analyze the superiority of various aggregation operators in aggregating multiple rules or their results through comparative experiments.

6 Comparison experiments

The max, min, and weighted average aggregation operators mentioned in the previous section are applied to aggregate rules or their results in multiple fuzzy reasoning. Among them, the weighted average operator is proposed for specific problems, i.e., for rules with different importance. Since each rule is given the same importance in the CRI method, the weights of each rule are set to the same value in this section, at which point the weighted average aggregation operator is equivalent to the mean aggregation operator. In order to compare the effects of different aggregation operators on the obtained inference results, this section adopts a comparison experiment using the CRI method. The method employed here is as follows,

setting different observations;

examining the inference results and calculating their closeness to the corresponding ideal inference results.

For the type I extended inference, type II extended inference, type I reduced inference, type II reduced inference and mixed inference processes proposed in this paper, and I₁, I₄, I₆ and I₈ are chosen as the implication operators, and A^* = A₁, A₂, A₃, A₄, A₅ are set respectively, and the corresponding ideal inference results are B^* = B₁, B₂, B₃, B₄, B₅ The experimental results are shown in Figs. 4–7, and the nearness degree is shown in Tables 4–7.

Fig. 4

Comparison of type I extended inference results, (a) A1, (b) A2, (c) A3, (d) A4, (e) A5.

Fig. 5

Comparison of type II extended inference results, (a) A1, (b) A2, (c) A3, (d) A4, (e) A5.

Fig. 6

Comparison of type I reduced inference results, (a) A1, (b) A2, (c) A3, (d) A4, (e) A5.

Fig. 7

Comparison of inference results for type II reduced inference, (a) A1, (b) A2, (c) A3, (d) A4, (e) A5.

Table 3

Several mixed fuzzy implication operators

Number	Name	Equation	Category
I ₉	Rescher	$a \to b = {\begin{matrix} 1, & a ⩽ b \\ 0, & a > b \end{matrix}$	Type I extended implication on [0,b]; Type I reduced implication on (b,1].
I ₁₀	Early Zadeh	a → b = (a ∧ b) ∨ (1 - a)	Type I extended implication on [0,1], when b∈[0,1-a]; Type II reduced implication on [0,0.5], Type I extended implication on [0.5,1], when b∈[1-a,1].
I ₁₁	Klir and Yuan l	$a \to b = {\begin{matrix} \begin{matrix} b & a = 1 \end{matrix} \\ \begin{matrix} 1 - a & a \neq 1, b \neq 1 \end{matrix} \\ \begin{matrix} 1 & a \neq 1, b = 1 \end{matrix} \end{matrix}$	Type I extended implication on [0,1], when b∈[0,1-a]; Type II reduced implication on [0,1], when b∈[1-a,1]; Type II extended implication on [0,1], when b = 1.

Table 4

Type I extended inference nearness degree

Nearness Degree	A ₁	A ₂	A ₃	A ₄	A ₅
δ_min	0.56	0.52	0.56	0.58	0.75
δ_max	0.41	0.40	0.42	0.41	0.51
minimum	0.50	0.46	0.50	0.50	0.64
maximum	0.25	0.25	0.25	0.25	0.25
mean	0.30	0.29	0.30	0.30	0.33

Table 5

Type II extended inference nearness degree

Nearness Degree	A ₁	A ₂	A ₃	A ₄	A ₅
δ_min	0.99	0.96	0.98	0.98	0.99
δ_max	0.57	0.65	0.58	0.58	0.63
minimum	0.99	0.95	1.00	1.00	1.00
maximum	0.14	0.28	0.14	0.14	0.25
mean	0.31	0.41	0.31	0.31	0.40

Table 6

Type I reduced inference nearness degree

Nearness Degree	A ₁	A ₂	A ₃	A ₄	A ₅
δ_min	0.85	0.71	0.85	0.85	0.75
δ_max	0.86	0.86	0.82	0.87	0.87
minimum	0.85	0.71	0.85	0.85	0.75
maximum	0.87	0.95	0.82	0.86	0.96
mean	0.88	0.79	0.86	0.88	0.81

Table 7

Type II reduced inference nearness degree

Nearness Degree	A ₁	A ₂	A ₃	A ₄	A ₅
δ_min	0.85	0.71	0.85	0.85	0.75
δ_max	0.89	0.87	0.88	0.87	0.89
minimum	0.85	0.71	0.85	0.85	0.75
maximum	0.93	0.98	0.95	0.89	1.00
mean	0.88	0.78	0.88	0.87	0.81

The experimental results of type I extended inference show that the fuzzy inference results using the dynamic minimum aggregation operator are closer to the ideal value than those of the minimum, maximum, dynamic maximum and mean aggregation operators. It shows that the dynamic minimum operator can be a good choice for aggregating multiple rules in a multiple fuzzy inference system. Analyzing its essence, it is due to the fact that the use of dynamic minimum aggregation operator can avoid the problem of missing rules caused by minimum operation and also can reduce the compromise effect caused by mean operation.

The experimental results of type II extended inference show that the fuzzy inference results using the minimum aggregation operator are closer to the ideal value than those of the maximum, dynamic maximum, and mean aggregation operators. However, the dynamic minimum aggregation operators is also close to the ideal value, and it is observed that the difference in the closeness of the inference results using the dynamic minimum aggregation operator and the minimum operator is not significant, indicating that the dynamic minimum aggregation operator can be used as a new method for aggregating multiple rules in multiple fuzzy inference systems.

In the experimental results of the two types of reduced inference, the inference results of both the minimum and dynamic minimum aggregation operators are 0. The maximum operation, although not well reduced, has better inference results than the dynamic maximum aggregation operation. This is because the membership functions of the rules and observations are triangular fuzzy numbers, and there are few intersecting parts between the rules, or even no overlap at all, which all make the membership degree of the fuzzy set has few non-0 values, leading to the great influence of 0 in the calculation process of dynamic maximum aggregation.

Setting the rules and observations as inverted triangular fuzzy numbers, i.e., fuzzy sets with more non-0 values of membership degree, the same experiment is performed. As the maximum operation is affected by the maximum value of 1, the inference results without exception, the membership degree is 1, which does not satisfy the result non-1 principle, and all the information obtained is meaningless, as shown in Fig. 8. At this time, the dynamic maximum aggregation operator shows a more desirable inference effect than the maximum aggregation operator.

Fig. 8

Comparison of inference results of inverted triangular fuzzy numbers.

7 Conclusion

First, this paper proposes a new classification of implication operators based on the definition and monotonicity of implication operators, and then classifies the fuzzy inference process. Then, the dynamic take large and dynamic take small operators are proposed as a new kind of aggregation operator. Finally, the matching method of implication operator and aggregation operator in multi-rule fuzzy inference system is given. The analysis of the relationship between implication and aggregation operators in this paper is beneficial to the design and construction of multi-rule fuzzy inference systems. In the future, we will study the problem of aggregation of multiple rules in continuous domain multi-input fuzzy inference systems and explore its applications in various fields.

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