Interpolative fuzzy reasoning method based on the incircle of a generalized triangular fuzzy number

Abstract

Fuzzy Rule Interpolation (FRI) is an important technique for implementing inference with sparse fuzzy rule-bases. Even if a given observation has no overlap with the antecedent of any rule from the rule-base, FRI may still conclude a conclusion. This paper introduces a new method called “Incircle FRI” for fuzzy interpolation which is based on the incircle of a triangular fuzzy number. The suggested method is defined for triangular CNF fuzzy sets, for a single antecedent universe and two surrounding rules from the rule-base. The paper also extends the suggested “Incircle FRI” to trapezoidal, and hexagonal shaped fuzzy sets by decomposing their shapes to multiple triangulars. The generated conclusion is also a CNF fuzzy set. The performance of the suggested method is evaluated based on numerical examples and a comprehensive comparison to other current FRI methods.

Keywords

Fuzzy interpolative reasoning sparse fuzzy rule-based systems incircle triangular fuzzy numbers incircle FRI method

1 Introduction

Fuzzy Rule Interpolation (FRI) techniques have been proposed to permit inference in sparse fuzzy rule based systems, where the rule antecedents are not covering all the input universes. FRI reasoning is an important concept for permitting the fuzzy rule base size reduction. It may still obtain reasonable conclusions, and the reduced (and sparse) rule-base could be considered as a fuzzy model enhancement. One way of the fuzzy rule-base reduction is omitting the rules from the rule-base which could be approximated from their adjacent rules by the FRI methods.

In recent years, several FRI methods have been presented for sparse fuzzy rule based system [1 –14].

Kóczy and Hirota [1 –3] presented a fuzzy interpolation method (KH FRI) that can infer the fuzzy interpolative reasoning result based on linear interpolation of α-cuts of the fuzzy sets. Vass et al. [4] presented a fuzzy interpolative reasoning method that modifies the α-cut of the KH FRI method based on the distance of the center points and the widths of the α-cuts to avoid abnormal conclusions. Tikk et al. [5] presented a fuzzy interpolative stabilized KH (KHstab FRI) to handle and exclude the possible abnormal conclusion, that can infer the fuzzy interpolative reasoning result the reciprocal distance as weights between all the rule antecedents and the observation for all the α-cuts.

In [6] Tikk and Baranyi compared some modified α-cuts based fuzzy interpolation method [33] and [1], in [7] authors suggest a solution for avoiding the abnormality of the conclusion. In [8] Li et al. presented a weighted fuzzy interpolative reasoning method based on the center of gravity of the trapezoidal fuzzy sets. In [9] Marsala et al. presented a fuzzy interpolative reasoning method with multiple variable rules. Qiao et al. [10] suggested a similarity transfer reasoning model to improve Kóczy and Hirota fuzzy interpolative reasoning method in sparse fuzzy rule-based systems. Shi et al. [11] pointed out that the Kóczy and Hirota fuzzy interpolative reasoning method [1] does not always lead to convex conclusions. In [12] Hsiao et al. presented an interpolative reasoning method based on the slopes of triangular fuzzy sets. In [13] Huang and Shen introduced a fuzzy interpolative reasoning method based on the representative values of fuzzy sets, and a way for preserving its position by scale and move transformation operators during the reasoning. In [14] Chang et al. presented a fuzzy interpolative reasoning method (called the CCL method) that is based on the areas of fuzzy sets. The suggested method satisfies the logical consistency with respect to the ratio of fuzziness of the corresponding fuzzy sets.

Although many methods have been proposed to deal with fuzzy interpolative reasoning, there have been some drawbacks in these methods. Some methods cannot preserve the convexity of the fuzzy interpolative reasoning result [1, 15], while other methods are limited to deal with triangular or trapezoidal membership functions [8 , 15], various methods cannot deal with the interpolation when the antecedents and the consequences of the given fuzzy rules are different kinds of membership functions [8 , 15]; the method presented in [1] gets nonconvex fuzzy interpolative reasoning results in some situations and does not properly deal with fuzzy interpolative reasoning when the fuzzy sets of the antecedents and the consequences in fuzzy rules belong to different kinds of membership functions. In order to eliminate the non-convexity drawback in [1], A modification of the KH FRI was proposed in [4] and [7] to reduces the problem of non-convex conclusions, where the conclusion is computed based on the distance of the center points and the widths of the α-cuts, instead of their lower and upper end points. The method presented in [13] can not deal with the fuzzy interpolative reasoning where observations are rectangular membership functions. In [11], Shi et al. pointed out that Koczy-and-Hirota’s fuzzy interpolative reasoning method [1] does not always lead to convex conclusions.

In this paper, a new technique of fuzzy interpolation will be introduced based on the representation of the reference point and the expression of fuzziness of a triangular-shaped fuzzy number by its incircle (inscribed circle). The proposed “Incircle FRI” follows geometrical considerations for performing fuzzy interpolation. The proposed Incircle FRI can produce a Convex and Normal Fuzzy set (CNF) for all rules and observation configurations, the proposed Incircle FRI can deal with fuzzy interpolative reasoning with different membership functions (triangular, trapezoidal, singleton and hexagonal shaped fuzzy sets), it can deal with different kinds membership functions of antecedents and consequents fuzzy rules, it can deal with the similarity propagation of the fuzziness and core between the observation and conclusion, it can handle fuzzy interpolative reasoning with logically consistent properties with respect to the ratios of fuzziness. For demonstrating the performance of the suggested FRI method, some numerical examples and comparisons with some existing FRI methods (to KH FRI [1 –3], KHstab FRI [5], VKK FRI [4], HCL FRI [12], HTY FRI [15], CCL FRI [14], and to HS FRI [13]) will be discussed briefly in the paper.

The rest of this paper is organized as follows. Section 2 introduces the background of fuzzy rule interpolative techniques, fuzzy numbers, and the suggested incircle concept for the generalized triangular fuzzy number. In Section 3, a new fuzzy interpolative reasoning method based on incircle of triangular fuzzy number is introduced. The experimental results of the proposed “Incircle FRI” with some of the existing FRI methods are presented in Section 4. Finally, the conclusion is summarised in Section 5.

2 Background of fuzzy rule interpolative techniques and fuzzy numbers

In this section, some basic concepts related to the complete and incomplete fuzzy rule bases of the interpolative reasoning techniques, the fuzzy numbers, and the suggested incircle concept of the triangular fuzzy number will be presented.

2.1 Complete and incomplete fuzzy rule bases

The FRI was originally introduced as an inference technique to handle sparse fuzzy rule bases. In case if the rule base is sparse, the antecedents of the fuzzy rules do not cover the whole input universe, the classical fuzzy reasoning methods can not produce conclusions for all the possible observations. On the other hand, the FRI techniques can perform interpolative approximate reasoning based on the existing rules even if there is no matching between the observation and the rule antecedents. The first FRI method (originally referred as “linear interpolation”, later called KH FRI) was proposed by Kóczy and Hirota [1 –3].

Later several methods have been proposed to address the sparse fuzzy rule bases, where the KH method was the baseline to many of them. Besides, various conditions were suggested to unify the FRI techniques that aimed to classify the FRI methods [16 –18]. For example, the conclusion fuzzy set must preserve a piece-wise linearity (see the benchmark set of piece-wise linearity in [32]), and also the conclusion fuzzy set must be a CNF set if all the rule antecedents and consequents are CNF sets.

Presuming that there are two linguistics variables X and Y, which are described on the universe R of real numbers, and F is a set in the fuzzy sets of R. We assume the fuzzy sets A_i in F are defined, 1 ⩽ i ⩽ n, such that: $A_{1} \underline{≺} A_{2} \dots \underline{≺} A_{i} \underline{≺} A_{i + 1} \dots \underline{≺} A_{n}$ , for a given partial ordering $\underline{≺}$ on F. We also suppose that given fuzzy sets B_i in F, 1≤i≤n, which are also ordered according to $\underline{≺}$ .

According to the definitions in [19, 20], the fuzzy functions are described by the fuzzy relations between the fuzzy sets of the inputs A_i and outputs B_i. The fuzzy rule base could be characterized and represented based on this relation. The classical reasoning methods, such as Mamdani and Sugeno [21, 22] follow that the fuzzy rule base relation requires to define all the fuzzy rule relations between the inputs and outputs.

Figure 1 demonstrates the antecedent space of the complete fuzzy rule bases having two dimensionals antecedents. It describes the observations (x₁) and (x₂) are matching with the fuzzy rules 1,2,4 and 5.

Fig. 1

Complete fuzzy rule base [23].

Thus, the conclusion could be computed based on one of the classical fuzzy reasoning methods, like Zadeh- Mamdani “max-min” Compositional Rule of Inference (CRI).

The sparse fuzzy rule-base (incomplete rule-base) is a rule-base in which the union of the antecedents Ai of the fuzzy rules (Ri): “if X is Ai then Y is Bi” are not covering the antecedent universe. There is input A* such that it is not overlapping with any of the rule antecedents. The aim of a fuzzy interpolation method is to provide the corresponding conclusion to such an observation A* too. Figure 2 describes the issue, where the observations (x1.1, x2.1) and (x1.2, x2.2) refer to the two-dimensional inputs, which does not overlap with any of the rule antecedents. This case with classical fuzzy reasoning no conclusion can be obtained.

Fig. 2

The incomplete (sparse) fuzzy rule base with a not overlapping observation [23].

2.2 Fuzzy numbers

A fuzzy set (A) defined on a universe of discourse X which holds total ordering, is a fuzzy number, i.e. a CNF set, if it is normal, its height is equal to one, and convex. It has a membership grade of any elements between two other elements greater than, or equal to the minimum membership degree of these two boundary elements. A convex fuzzy set can be defined by ∀x, y ∈ U, ∀ λ ∈ [0, 1] : μA (λ . x + (1 - λ . y)) ⩾ min(μA (x) , μA (y)). The support of a fuzzy set (A) is the set of all elements in the universe of discourse X with a membership degree μA (x) is greater than zero. Supp (A) : x ∈ U, μA (x) > 0.

The height of a fuzzy set is the maximum membership degree of all the elements of the universe, and it can be defined by Height (A) : max(x) ∈ U (μA (x)). A fuzzy set is said to be normal if at least one element of the universe has a membership degree equal to 1, ∃x ∈ U, μA (x) : Height (A) = 1. The α-cut and the strong α-cut of a fuzzy set is the crisp subset of the universe where the membership degrees are greater (strong α-cut), or greater, or equal (α-cut) than a specified α value. The α-cut can be represented by Aα : x ∈ U, μA (x) ⩾ α, α ∈ [0, 1] and Aα : x ∈ U, μA (x) > α, α ∈ [0, 1]. The kernel of a fuzzy set is the crisp subset of the universe where the membership degrees are equal to 1. Kernel (A) : x ∈ U, μA (x) = 1.

In case of a convex fuzzy set A on R_n, all of its α–cuts A_α are convex sets for all α ∈ (0, 1], i.e. its α–cuts are intervals. In most cases particular types of the fuzzy numbers such as trapezoidal and triangular are used for real-life applications. A fuzzy number (A) is called a generalized trapezoidal fuzzy number if its membership function is given as follows: $μ_{A} (X) = {\begin{matrix} \begin{matrix} \frac{H (X - a_{1})}{(a_{2} - a_{1})} & , & a_{1} ⩽ X ⩽ a_{2} \end{matrix} \\ \begin{matrix} H & , & a_{2} ⩽ X ⩽ a_{3} \end{matrix} \\ \begin{matrix} \frac{H (X - a_{4})}{(a_{3} - a_{4})} & , & a_{3} ⩽ X ⩽ a_{4} \end{matrix} \\ \begin{matrix} 0 & , & \begin{matrix} \begin{matrix} \begin{matrix} X < a_{1} & or \end{matrix} \end{matrix} & X > a_{4} \end{matrix} \end{matrix} \end{matrix}$ where, 0 < H ⩽ 1. The generalized trapezoidal fuzzy number A is denoted by A = (a₁, a₂, a₃, a₄ ; H) and has the shape of a trapezoid. A triangular fuzzy number is a particular case of generalized trapezoidal fuzzy number, having triangle–shaped membership function. Precisely, A = (a₁, a₂, a₃, a₄ ; H) called a triangular fuzzy number if a₂ = a₃ as shown in Fig. 3.

Fig. 3

Generalized trapezoidal and triangular fuzzy numbers.

2.3 Incircle of a triangular fuzzy number

The incircle of a triangular fuzzy number can be considered as an incircle of a triangle. The incircle of a triangle is that circle which touches all three main sides (AB, BC, and AC) of the triangle, and the points of tangency of the incircle of ΔABC (i.e. T_A, T_B, T_C) with its sides as shown in Fig. 4.

Fig. 4

The Incircle CIR(I) with Gergonne Point in a triangle ABC.

Consequently, that follows the Ceva^squos theorem directly (see theorem 1), where cevians of particular importance in the general triangle (medians, angle bisectors, etc.) are synchronous, and the fact that two tangents to a circle from a point outside the circle are equal. Furthermore, the common point of a triangle ABC that called a particular point of the triangle could be defined as the point of intersection of the cevians AT_A, BT_B and CT_C. This point is called the Gergonne Point (GP) of the triangle [24], which is the concurrence point for the cevians from the vertices to the points of tangency on the opposite sides of the triangle as shown in Fig. 5.

Fig. 5

Trilinear Coordinates α β γ of point P.

Theorem 1. The three cevians joining the vertices of a triangle to the point of tangency of the opposite sides with the incircle are concurrent [25].

By using Ceva’s Theorem, the following result could be got directly by the given the existence of a triangle with vertices A, B and C. The Trilinear Coordinates (TCs) of point P, which related to triangle ABC are three ordered numbers. Each corresponding to the distance from P to one of the sideline. TCs are generally referred via α:β:γ as shown in Fig. 5. If point P has TCs α:β:γ, then the Cartesian Coordinates of P are calculated as follows: $\begin{matrix} p = (\frac{1}{α . | S_{2 (a 2 a 3)} | + β . | S_{3 (a 1 a 3)} | + γ . | S_{1 (a 1 a 2)} |}) . \\ (α . | S_{2 (a 2 a 3)} | . A + β . | S_{3 (a 1 a 3)} | . B + γ . | S_{1 (a 1 a 2)} | . C) \end{matrix}$ (1)

TCs of the GP are given as: $\begin{matrix} \frac{| S_{3 (a 1 a 3)} | . | S_{1 (a 1 a 2)} |}{| S_{3 (a 1 a 3)} | + | S_{1 (a 1 a 2)} | - | S_{2 (a 2 a 3)} |} : \\ \frac{| S_{2 (a 2 a 3)} | . | S_{1 (a 1 a 2)} |}{| S_{2 (a 2 a 3)} | + | S_{1 (a 1 a 2)} | - | S_{3 (a 1 a 3)} |} : \\ \frac{| S_{2 (a 2 a 3)} | . | S_{3 (a 1 a 3)} |}{| S_{2 (a 2 a 3)} | + | S_{3 (a 1 a 3)} | - | S_{1 (a 1 a 2)} |} \end{matrix}$ (2)

Based on the properties of the incircle triangular fuzzy number, some notations for the fuzzy rules and for the observation fuzzy sets could be determined. It will be used to perform the approximate conclusion. The “center” of the triangle could be denoted by the Gergonne Point (GP), and the main sides of the triangular are indicated as SD₁, SD₂ and SD₃. Besides, the length of tangents and the vertices of the triangle can be determined to its incircle which denotes by PS₁, PS₂ and PS₃ as shown in Fig. 6. In the followings these sides PS₁, PS₂ and PS₃ will be referred as “fuzziness sides”.

Fig. 6

Notations of triangular fuzzy number.

Assuming that the triangular fuzzy set is A = (a₁, a₂, a₃; H). Hence, a triangle with the coordinates of vertices could be got as following: A = (a₁,0), B = (a2, H) and C = (a3,0) with H. If H is equal to 1, then the fuzzy number is normal. The trapezoidal fuzzy set can be represented by two triangular fuzzy numbers AL = (a₁, a₂, Mp; H) and AR = (Mp, a₃, a₄; H). In the following, some notations required for calculating the approximating conclusion of the proposed “Incircle FRI” are presented.

Supposing that there are a single dimensional antecedent space and two adjacent fuzzy rules to the observation (like in the original KH FRI) and triangular shaped fuzzy observation and rule antecedents.

Notation 1: The main sides of the triangular fuzzy set A, SD_1(a1a2), SD_2(a2a3), SD_3(a1a3). H denotes the height of the fuzzy set (A). The main sides of a triangular fuzzy set (A) can be calculated by Equation (3). $\begin{matrix} | {SD}_{1} | = \sqrt{(a_{1} - a_{2})^{2} + H^{2}}, \\ | {SD}_{2} | = \sqrt{(a_{2} - a_{3})^{2} + H^{2}}, \\ | {SD}_{3} | = a_{3} - a_{1} \end{matrix}$ (3)

Notation 2: Applying Equations (2) and (3). The TCs and GP of a triangle shaped fuzzy set A, which has vertices a₁, a₂ and a₃ could be found by:

$\begin{matrix} α = \frac{(a 3 - a 1) . \sqrt{(a 1 - a 2)^{2} + H^{2}}}{(a 3 - a 1) + \sqrt{(a 1 - a 2)^{2} + H^{2}} - \sqrt{(a 2 - a 3)^{2} + H^{2}}} \\ β = \frac{\sqrt{(a 2 - a 3)^{2} + H^{2}} . \sqrt{(a 1 - a 2)^{2} + H^{2}}}{\sqrt{(a 2 - a 3)^{2} + H^{2}} + \sqrt{(a 1 - a 2)^{2} + H^{2}} - (a 3 - a 1)} \\ γ = \frac{(a 3 - a 1) . \sqrt{(a 2 - a 3)^{2} + H^{2}}}{\sqrt{(a 2 - a 3)^{2} + H^{2}} + (a 3 - a 1) - \sqrt{(a 1 - a 2)^{2} + H^{2}}} \end{matrix}$

Notation 3: Using Equation (1), the Cartesian Coordinates of GP for triangle fuzzy set A can be calculated as follows: $\begin{matrix} {GP}_{A} = (X_{A}, μ_{A}) = \\ (\frac{a_{1} . α | {SD}_{2} | + a_{2} . β | {SD}_{3} | + a_{3} . γ | {SD}_{1} |}{α | {SD}_{2} | + β | {SD}_{3} | + γ | {SD}_{1} |}, \\ \frac{β | {SD}_{3} |}{α | {SD}_{2} | + β | {SD}_{3} | + γ | {SD}_{1} |}) \end{matrix}$ (4)

Regarding the height property of the fuzzy set, a fuzzy set (A) is normal if there is at least one element on the universe of discourse has a membership degree equal to 1, ∃_x ∈ U, μA (x) : Height (A) = 1.

Notation 4: Fuzziness sides PS₁, PS₂ and PS₃ will be used to determine the left and right fuzziness length of the antecedent, consequent and observation fuzzy sets, which can be calculated as follows: $\begin{matrix} A_{(PS 1)} = \frac{({SD}_{1} + {SD}_{3} - {SD}_{2})}{2} \\ A_{(PS 2)} = \frac{({SD}_{1} + {SD}_{2} - {SD}_{3})}{2} \\ A_{(PS 3)} = \frac{({SD}_{2} + {SD}_{3} - {SD}_{1})}{2} \end{matrix}$ (5)

3 A new fuzzy interpolative reasoning method based on incircle of triangular fuzzy number

In this section, a new fuzzy interpolative reasoning method will be presented for sparse fuzzy rule based systems, which is based on the incircle of the triangular fuzzy set of the fuzzy rules and observation. To facilitate the discussion in our approach, the representative values of the approximating conclusion fuzzy set will be determined by the Gergonne Points (GP.x), left fuzziness (PS₁) and right fuzziness (PS₃). In the followings, the proposed “Incircle FRI” will be discussed in details.

3.1 Single antecedent variable with triangular fuzzy sets

The triangular membership function is a particular case of the fuzzy sets, which are vastly used in fuzzy rule-based systems because of its simplicity. A triangular membership function can be described by vertices (a₁, a₂, a₃; H), where a₁ refers to the left side of the support, a₂ denotes the center point and a₃ refers to the right side of the support, and H denotes the height of the fuzzy set.

Figure 7 describes the suggested reference point of the fuzzy set, which is the Gergonne Point (GP_x.A) of fuzzy set A, the main left, right and base sides of triangle ABC, which indicate by SD₁, SD₂ and SD₃, respectively. The left PS₁ and right PS₃ fuzziness sides of triangular membership function can be described, where PS₁ refers to the left fuzziness and PS₃ refers to the right fuzziness.

Fig. 7

The main notations of the triangular fuzzy number represented by GP.x, SD1, SD3, SD3, PS1, PS2 and PS3.

For simplicity, in the initial version of the suggested reasoning method, the two adjacent fuzzy rules A₁ ⇒ B₁ and A₂ ⇒ B₂ to the observation will be taken into consideration from the rule-base only. A₁, A₂, B₁ and B₂ denote the fuzzy sets of the antecedents and consequents, respectively. Considering that the observation fuzzy set A^* occurs between the fuzzy sets A₁ and A₂. The conclusion B^* fuzzy set denotes the fuzzy interpolative reasoning result as shown in Fig. 8. The scheme of the fuzzy interpolative reasoning of the two fuzzy-rules and observation using triangular fuzzy sets can be defined as follows: $\begin{matrix} Rule 1 : If X is A_{1} then Y is B_{1} \\ Rule 2 : If X is A_{2} then Y is B_{2} \\ Observation : X is A^{*} . \\ - - - - - - - - - - - - - - - - - - - - - - - - \\ Conclusion : Y = (B^{*}) \end{matrix}$ where Rule₁ and Rule₂ are two adjacent fuzzy rules, A_i = (a_i,1, a_i,2, a_i,3) , B_i = (b_i,1, b_i,2, b_i,3) , 1 ⩽ i ⩽ 2, the observation A^* = (a₁, a₂, a₃) , andB^* = (b₁, b₂, b₃). The suggested fuzzy interpolative reasoning method “Incircle FRI” is built upon the following steps:

Fig. 8

Fuzzy interpolative reasoning using triangular membership functions.

Step 1: To perform interpolation for single antecedent fuzzy rules, the two closest fuzzy rules (m) [26], Rule₁, Rule₂,..., and Rule_m must be determine. Which are the nearest to the observation $A_{i} \underline{≺} A^{*} \underline{≺} A_{i + 1}$ as shown in Fig. 8. The closest fuzzy rules can be determined by the distances using Equation (6): $D = d (A_{i}, A^{*}) = d ({GP}_{x} . A_{i}, {GP}_{x} . A^{*})$ (6)

Step 2: Suppose the two adjacent fuzzy rules A₁ and A₂ are the left and right antecedent fuzzy sets to the observation fuzzy set A^*. Therefore, based on [26], the weight between the adjacent triangular fuzzy rules, Rule₁ and observation A^* can be determined as follows: $W_{i} = 1 - \frac{| {GP}_{x} . A^{*} - {GP}_{x} . A_{i} |}{{GP}_{x} . A_{2} - {GP}_{x} . A_{1}}$ (7)

First, GP . _X refers to the Gergonne Point of the fuzzy sets. It could be calculated by Equation (4). Second, W_i denotes the weight of Rule_i, 0 ⩽ W_i ⩽ 1, 1 ⩽ i ⩽ m, where i = 1, 2 that represents the individual fuzzy rules as given in Fig. 8 holding the property of W₁ + W₂ = 1.

Step 3: Calculate the Gergonne Point GP_x of the fuzzy interpolative reasoning result B^*, shown as follows: ${GP}_{x} . B^{*} = \sum_{i = 1}^{2} W_{i} \times {GP}_{x} . B_{i}$ (8)

The required Gergonne Points GP_x . B₁ and GP_x . B₂ can be calculated by Equation (4).

Step 4: The fuzziness sides of the B^* fuzzy set can be calculated by Equation (9): ${PS}_{M} (B^{*}) = {\begin{matrix} {PS}_{M} (A^{*}) \times \sum_{i = 1}^{2} W_{i} \times \frac{{PS}_{M} (B_{i})}{{PS}_{M} (A_{i})}, \\ \begin{matrix} if \begin{matrix} \end{matrix} {\begin{matrix} \exists \end{matrix}}_{i} {PS}_{M} (A_{i}) > 0 \end{matrix} \\ {PS}_{M} (A^{*}), \\ \begin{matrix} if \begin{matrix} \end{matrix} \forall_{i} {PS}_{M} (A_{i}) = 0 \end{matrix} \end{matrix}}$ (9)

Where M ∈ [PS₁, PS₃], PS₁ refers to the left fuzziness side and PS₃ denotes the right fuzziness side of triangular fuzzy set. If one of the antecedents fuzzy sets A₁ and A₂, the left PS₁ or right PS₃ are great than zero, the top part of the Equation (9) is implemented to conclude the left and right fuzziness sides PS₁ and PS₃ of the fuzzy set B^*. Otherwise, if both antecedents fuzzy sets A₁ and A₂ are found, the left PS₁ and right PS₃ are zero. I.e. in case of singleton fuzzy sets, the bottom part of the Equation (9) could be used.

Step 5: Now, based on the results of steps 3 and 4, the representative values of the conclusion B^* fuzzy set will be determined. The Gergonne Point GP is determined by Equation (8). The left fuzziness[GP_X.B^* - B^*PS₁] and the right fuzziness side [GP_X.B^* + B^*PS₃] are defined by Equation (9). Finally, the fuzzy interpolative reasoning conclusion B^* for the triangular fuzzy set can be collected as: $\begin{matrix} B_{1}^{*} = {GP}_{x} . B^{*} - B_{(ps 1)}^{*} \\ B_{2}^{*} = {GP}_{x} . B^{*} \\ B_{3}^{*} = {GP}_{x} . B^{*} + B_{(ps 3)}^{*} \end{matrix}$

In general, the result of the proposed “Incircle FRI” is satisfied with logically consistent properties and with respect to the ratios of fuzziness sides based on the two-fuzzy-rules interpolative reasoning technique, which obtained by Equation (9). The top equation of (9) is used to infer the fuzziness sides of the interpolated conclusion fuzzy set B^* if there exists a fuzzy rule whose fuzziness sides of the antecedent part is larger than zero. Otherwise, the bottom equation of (9) is used when the fuzziness sides of the antecedent part of the given fuzzy rules are zero. That means the larger the fuzziness of the membership function of a fuzzy set is the more fuzziness the fuzzy set has. Therefore, the ratio R(A, B) of the Left-Right fuzziness sides (A) fuzzy set to the Left-Right fuzziness sides (B) fuzzy set is defined in [14] as follows: $RF . {PS}_{m} (A, B) = \frac{RF . {PS}_{m} (B)}{RF . {PS}_{m} (A)}$ (10) where RF.PS_m (A) > 0, m ∈ the left RF.PS₁ and right RF.PS₃ fuzziness sides of the membership function A and B fuzzy sets, respectively. Thus, if there are two fuzzy rules A₁ ⇒ B₁, A₂ ⇒ B₂ and observation A^*, as shown in Fig. 8. The fuzziness sides ratios of the logically consistent properties will be satisfied with the fuzzy interpolative reasoning result as follows:

Property 1: Min (RF . PS_m (A₁, B₁) , RF . PS_m (A₂, B₂)) ⩽ RF . PS_m (A^*, B^*) ⩽ Max (RF . PS_m (A₁, B₁) , RF . PS_m (A₂, B₂)), where m ∈ [ps₁, ps₃]. Then RF.PS_m (A^*, B^*) can be calculated by Equation (11) as follows: $\begin{matrix} RF . {PS}_{m} (A^{*}, B^{*}) = \\ ((1 - W 2) \times RF . {PS}_{m} (A_{1}, B_{1})) + \\ (W 2 \times RF . {PS}_{m} (A_{2}, B_{2})) \end{matrix}$ (11) where the weight W_i between antecedents and observation could be defined by Equation (7).

Property 2: If RF . PS_m (A₁, B₁) = RF . PS_m (A₂, B₂) = K, then RF . PS_m (A^*, B^*) = K, where K ≥0 and m ∈ [ps₁, ps₃] . Min (RF . PS_m (A₁, B₁) , RF . PS_m (A₂, B₂)) = Max (RF . PS_m (A₁, B₁) , RF . PS_m (A₂, B₂)) = K and RF.PS_m (A^*, B^*) = K.

It is evident that Property 1 and Property 2 are logically consistent with regard to the left and right ratios of fuzziness (RF . PS₁ and RF . PS₃) based on the two fuzzy rules. Based on (9), the weight W_i of RF.PS_m(A_i,B_i) contributing to RF.PS_m(A^*, B^*) is determined by the distance of the Gergonne Points between A_i and A^*. That is, the closer the Gergonne Point of A^* to the Gergonne Point of A_i, the larger the weight of RF.PS_m (A_i, B_i), where i = 1, 2 and m ∈ PS₁, PS₃.

3.2 Single antecedent variable with trapezoidal fuzzy sets

The incircle concept of triangular fuzzy numbers can be extended to trapezoidal fuzzy set. A trapezoidal fuzzy set can be represented through two triangular fuzzy sets AL = (a₁, a₂, Mp; H) and AR = (Mp, a₃, a₄; H). Thus, the notations in Equations (3), (4) and (5) will be used to calculate AL and AR separately.

Figure 9 describes left and right Gergonne Points of the trapezoidal fuzzy set that are denoted by GP_x.A_L and GP_x.A_R. The main sides of the left triangle AL are described by SDL₁, SDL₂ and SDL₃, respectively. And the fuzziness sides of AL can be described by the left PS₁A_L and the right PS₃A_L. For the right triangle AR, the main sides are described by SDR₁, SDR₂ and SDR₃, respectively. Besides, the fuzziness sides of AR can be represented by the left PS₁A_R and the right PS₃A_R. The fuzziness sides PS₁A_L and PS₃A_R will be used to describe the left fuzziness and the right fuzziness of the trapezoidal membership function to determine the conclusion.

Fig. 9

The main notations of the trapezoidal fuzzy number represented by notations of two triangular fuzzy sets AL and AR.

An example for the suggested fuzzy interpolative reasoning using trapezoidal fuzzy sets is shown in Fig. 10. The main steps of the suggested incircle fuzzy interpolative reasoning method with trapezoidal fuzzy sets (represented by two AL and AR triangular fuzzy sets) could be determined as follows:

Fig. 10

Fuzzy interpolative reasoning using trapezoidal membership functions.

Step 1: To perform the interpolation for single antecedent fuzzy rules, the two closest fuzzy rules (m) [26], Rule₁, Rule₂,..., and Rule_m must be defined, which are close to the observation $A_{i} \underline{≺} A^{*} \underline{≺} A_{i + 1}$ as shown in Fig. 10.

Thus, the closest fuzzy rules can be determined by using the average of two Gergonne Points GP.xA_L and GP.xA_R of the trapezoidal fuzzy sets via AVG.GP_X = (GP_X.A_L and GP_X.A_R)/2. Then, distances can be computed using Equation (12): $\begin{matrix} D = d (A_{i}, A^{*}) \\ = d (AVG . {GP}_{x} . A_{i}, AVG . {GP}_{x} . A^{*}) \end{matrix}$ (12)

Step 2: The two adjacent fuzzy rules A₁ and A₂ are the left and the right antecedent fuzzy sets to the observation fuzzy set A^*. Hence, the weight of trapezoidal fuzzy rules neighbouring and observation can be determined by Equation (13): $\begin{matrix} {WL}_{i} = 1 - \frac{| {GP}_{x} . A^{*} L - {GP}_{x} . {AL}_{i} |}{{GP}_{x} . A_{2} L - {GP}_{x} . A_{1} L} \\ {WR}_{i} = 1 - \frac{| {GP}_{x} . A^{*} R - {GP}_{x} . {AR}_{i} |}{{GP}_{x} . A_{2} R - {GP}_{x} . A_{1} R} \end{matrix}$ (13) where WL_i and WR_i denote the weight of Rule_i, 0 ⩽ WL_i ⩽ 1, 0 ⩽ WR_i ⩽ 1. And i = 1, 2, it represents the individual fuzzy rules as given in Fig. 10 holding the property of WL1 + WL2 = 1 and WR1 + WR2 = 1.

Step 3: Calculate the two Gergonne Points GP_x.AL and GP_x.AR of the fuzzy interpolative reasoning result B^*, which can be defined by Equation (14): $\begin{matrix} {GP}_{x} . B^{*} L = \sum_{i = 1}^{2} {WL}_{i} \times {GP}_{x} . {BL}_{i} \\ {GP}_{x} . B^{*} R = \sum_{i = 1}^{2} {WR}_{i} \times {GP}_{x} . {BR}_{i} \end{matrix}$ (14)

Regarding the main point (MP) of the trapezoidal fuzzy set can be calculated by MP.B^* = AVG.GPX.B₁ + (((AVG.GPX.A*- AVG.GPX.A₁) × (AVG.GPX.B₂- AVG.GPX.B₁))/(AVG.GPX.A₂- AVG.GPX.A₁))).

Step 4: The fuzziness sides of the B^* fuzzy set can be calculated by Equations (15) and (16):

${PS}_{M} (B^{*} L) = {\begin{matrix} {PS}_{M} (A^{*} L) \times \sum_{i = 1}^{2} {WL}_{i} \times \frac{{PS}_{M} ({BL}_{i})}{{PS}_{M} ({AL}_{i})}, \\ \begin{matrix} if \begin{matrix} \end{matrix} {\begin{matrix} \exists \end{matrix}}_{i} {PS}_{M} ({AL}_{i}) > 0 \end{matrix} \\ {PS}_{M} (A^{*} L), \\ \begin{matrix} if \begin{matrix} \end{matrix} \forall_{i} {PS}_{M} ({AL}_{i}) = 0 \end{matrix} \end{matrix}}$ (15) ${PS}_{M} (B^{*} R) = {\begin{matrix} {PS}_{M} (A^{*} R) \times \sum_{i = 1}^{2} {WR}_{i} \times \frac{{PS}_{M} ({BR}_{i})}{{PS}_{M} ({AR}_{i})}, \\ \begin{matrix} if \begin{matrix} \end{matrix} {\begin{matrix} \exists \end{matrix}}_{i} {PS}_{M} ({AR}_{i}) > 0 \end{matrix} \\ {PS}_{M} (A^{*} R), \\ \begin{matrix} if \begin{matrix} \end{matrix} \forall_{i} {PS}_{M} ({AR}_{i}) = 0 \end{matrix} \end{matrix}}$ (16)

M ∈ [PS₁, PS₃], PS₁ refers to the left fuzziness side and PS₃ denotes the right fuzziness side for the two triangles fuzzy sets AL and AR.

The top part of Equations (15) and (16) is performed to conclude the left fuzziness side PS₁ and the right fuzziness side PS₃ of the fuzziness of B^* fuzzy set. If one of the antecedents A₁ and A₂ fuzzy sets exits, PS₁ or PS₃ is bigger than zero, if both PS₁ or PS₃ of A₁ and A₂ fuzzy sets are zero, the bottom part of Equations (15) and (16) could be used.

Step 5: Based on the results of steps 3 and 4, the following notations are determined: First, the left and right Gergonne Points GP_x.B^*L and GP_x.B^*R. Second, the left fuzziness PS₁ and right fuzziness PS₃ of the conclusion B^* fuzzy set. Finally, the FRI conclusion B^* for the trapezoidal fuzzy set can be determined based on the following three cases:

The first case: The results of AL and AR triangular fuzzy sets have the same values, then the conclusion can be determined as follows:

$B^{*} = [{GP}_{x} . B^{*} L, {GP}_{x} . B^{*} L, {GP}_{x} . B^{*} R, {GP}_{x} . B^{*} R]$ (17)

The second case: The result of the left triangular AL has the same values, and the right triangular AR has the same values too, but both left and right values are not the same. The conclusion values will be defined by GP_x1x2 = (GP_x.B^*L + GP_x.B^*R)/2 as follows: $B^{*} = [{GP}_{x 1 x 2}, {GP}_{x 1 x 2}, {GP}_{x 1 x 2}, {GP}_{x 1 x 2}]$ (18)

The third case: Because we are working with two triangular separately, in case the Left triangular has the same values, we will use B₁^*, but in case the Right one has the same values, we will use B₂^*, then, the conclusion can be determined as follows:

$\begin{matrix} B_{1}^{*} = & [{GP}_{x .} B^{*} L, {GP}_{x .} B^{*} L, {GP}_{x .} B^{*} R, \\ {GP}_{x .} B^{*} R + B^{*} {Rps}_{3}] \\ B_{2}^{*} = & [{GP}_{x .} B^{*} L - B^{*} {Lps}_{1}, {GP}_{x .} B^{*} L, \\ {GP}_{x .} B^{*} R, {GP}_{x .} B^{*} R] \end{matrix}$ (19)

The fourth case: When all values results of the left and the right triangles are different. The conclusion can be determined as follows: $\begin{matrix} B_{1}^{*} = {GP}_{x} . B^{*} L - B^{*} L_{(PS 1)} \\ B_{2}^{*} = MP . B^{*} - B^{*} L_{(PS 3)} \\ B_{3}^{*} = MP . B^{*} - B^{*} R_{(PS 1)} \\ B_{4}^{*} = {GP}_{x} . B^{*} R + B^{*} R_{(PS 3)} \end{matrix}$ (20)

Because B^*1≤B^*2≤B^*3≤B^*4, we can see that the proposed method can preserve the convexity of the fuzzy interpolative reasoning result with trapezoidal fuzzy set, add to that the value of the left point (B^*₁) is smaller than or equal to the values of the reference point (GPX. B^*L and GPX. B^*R), which are also smaller than or equal to the value of the right point (B^*₄).

3.3 Single antecedent variable with hexagonal fuzzy sets

The incircle concept of triangular and trapezoidal fuzzy numbers can be further extended to hexagonal, or any complex polygonal fuzzy membership functions. In the followings the extension of the suggested incircle fuzzy interpolation method to hexagonal fuzzy sets will be discussed.

A Hexagonal fuzzy set can be represented by two triangular fuzzy sets AL = (a₁, a₂, a₃) and AR = (a₄, a₅, a₆). Figure 11 describes the representative values of a hexagonal fuzzy set, which is denoted by (a1, a2, a3, a4, a5, a6). The Gergonne Points (left and right) denotes by a3 and a4, respectively. a1 and a6 are the left and the right side of the support points, respectively, and a2 and a5 are the intermediate points. Thus, the Equations (3), (4) and (5) can be used to calculate AL and AR separately.

Fig. 11

The main notations of the hexagonal fuzzy number represented by notations of two triangular fuzzy sets AL and AR.

An example of the suggested incircle fuzzy interpolative reasoning using hexagonal fuzzy sets is shown in Fig. 12. The main steps of the suggested incircle fuzzy interpolative reasoning method with hexagonal fuzzy sets (represented by two AL and AR triangular fuzzy sets) are the followings:

Fig. 12

Fuzzy interpolative reasoning using hexagonal membership functions.

Step 1: To perform interpolation for single antecedent fuzzy rules. The two closest fuzzy rules (m) [26], Rule₁, Rule₂,..., and Rule_m must be determine, which are the nearest to the observation $A_{i} \underline{≺} A^{*} \underline{≺} A_{i + 1}$ as shown in Fig. 12.

Step 2: The adjacent fuzzy rules A₁ and A₂ are the left and right antecedent fuzzy sets to the observation fuzzy set A^*. Hence, the weight of hexagonal fuzzy rules neighbouring and observation can be determined by Equation (13).

Step 3: Compute the two Gergonne Point GP_x.AL and GP_x.AR of the fuzzy interpolative reasoning result B^*, according to Equation (14).

Step 4: The fuzziness sides of the B^* fuzzy set can be calculated using the same Equations (15) and (16).

Step 5: Based on the results of steps 3 and 4, the Gergonne Point of the conclusion a3 and a4 will be determined based on the results of the intermediate points a2 and a5, which are denoted by GP_x.B^*L and GP_x.B^*R. The left support a1 and right support a6 of the conclusion B^* fuzzy set will be determined based on the left fuzziness PS₁A_L and right fuzziness PS₃A_R. Finally, the conclusion B^* of the incircle fuzzy interpolative reasoning for the hexagonal fuzzy set can be determined as follows: $\begin{matrix} B_{1}^{*} = {GP}_{x} . B^{*} L - B^{*} L_{(PS 1)} \\ B_{2}^{*} = {GP}_{x} . B^{*} L \\ B_{3}^{*} = {GP}_{x} . B^{*} L + B^{*} L_{(PS 3)} \\ B_{4}^{*} = {GP}_{x} . B^{*} R - B^{*} R_{(PS 1)} \\ B_{5}^{*} = {GP}_{x} . B^{*} R \\ B_{6}^{*} = {GP}_{x} . B^{*} R + B^{*} R_{(PS 3)} \end{matrix}$ (21)

4 Comparison of the proposed incircle FRI method with some other FRI methods

In this section, we discuss the performance of the proposed “Incircle FRI” method, some numerical examples in [13 , 27–29] will be compared with the results of KH FRI [1 –3], KHstab FRI [5], VKK FRI [4], CCL FRI [14], HS FRI [13], HTY FRI [15], and HCL FRI [12]. The KH FRI, the KHstab FRI, and the VKK FRI methods were tested by using the Matlab FRI toolbox [30, 31]. In addition, we present a comparison summary of the seven FRI methods and Incircle FRI based on five evaluation criteria (i.e., CNF property”, “different membership functions,” “different kinds membership functions of the antecedents and the consequents fuzzy rules”, “the similarity propagation of the fuzziness and core between the observation and conclusion”, and “logically consistent with respect to the ratios of fuzziness”).

Example X1 [13 , 27–29]: This example uses only triangular fuzzy sets in the fuzzy rules and for the observation A^* = [7 , 9]. Table 1 and Fig. 13 describe all the attribute values and the results of fuzzy interpolative reasoning methods. The conclusion fuzzy set of the proposed Incircle FRI is represented by triangular membership function as B^* = (GP_x.B^*-PS₁, GP_x.B^*, GP_x.B^*+PS₃). Based on Equations (4), (7), and (8) the Gergonne Point value was calculated by the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂. Where GP_x.B^* = 5.42. Also, the left fuzziness PS₁ = 0.472 and the right fuzziness PS₃ = 1.737 are obtained based on Equations (5), (7), and (9). Consequently, the proposed Incircle FRI gets the interpolated consequence as triangular fuzzy set B^* = [4.95 5.42 7.16]. Figure 13 describes the results of various FRI methods, in which, the KH [1 –3], KHstab [5], and VKK [4] FRI methods generated an abnormal (non-convex) fuzzy sets. While the FRI methods CCL [14], HS [13], HTY [15], HCL [12], and the proposed Incircle FRI have CNF conclusion.

Fig. 13

A comparison of FRI results of Example X1 for several methods.

Based on Equations (10) and (11) and Table 1, the logically consistent properties and the respect to ratios of the fuzziness two adjacent Rules (A₁ ⇒ B₁) and (A₂ ⇒ B₂), the left ratio fuzziness RF.PS ₁ (A₁, B₁) = 0.4 and the ratio RF.PS ₁ (A₂, B₂) = 0.5, and the left ratio fuzziness RF.PS ₁ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, the HCL FRI, the HTY FRI and HS FRI methods are 0.44, 0.44, 0.22, 0.66 and 0.43, therefore, the proposed Incircle FRI, CCL FRI and HS FRI satisfy property 1, where MIN (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂, B₂)) = 0.4≤RF.PS ₁ (A^*, B^*) = [Incircle(0.44), CCL(0.44), HS(0.43)]≤MAX (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂, B₂)) = 0.5. The right ratio fuzziness of Rules ((A₁ ⇒ B₁ and A₂ ⇒ B₂) RF.PS ₃ (A₁, B₁) = RF.PS ₃ (A₂, B₂) = 2, and the right ratio fuzziness RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, the HCL FRI, the HTY FRI, HS FRI, KHstab FRI, and KH FRI methods are 2, 2, 0.8, 0.88, 1.12, 2, and 2, respectively. Thus, the right ratio fuzziness RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, KHstab FRI, and KH FRI satisfy Property 2, where MIN (RF.PS ₃ (A₁, B₁), RF.PS ₃ (A₂, B₂)) = MAX (RF.PS ₃ (A₁, B₁), RF.PS ₃ (A₂, B₂)) = RF.PS ₃ (A^*,B^*) = 2.

Table 1

Fuzzy interpolative reasoning results of Example X1

Attribute Values	Methods	Results of Fuzzy Interpolative reasoning
	KH FRI [1–3]	B^* = (6.36 5.38 7.38)
A₁ = [0 5 6]	KHstab FRI [5]	B^* = (6.36 5.38 7.38)
A₂ = [11 13 14]	VKK FRI [4]	B^* = (6.15 5.38 7.84)
B₁ = [0 2 4]	CCL FRI [14]	B^* = (4.94 5.38 7.38)
B₂ = [10 11 13]	HS FRI [13]	B^* = (5.83 6.26 7.38)
A^* = [7 8 9]	HTY FRI [15]	B^* = (5.76 6.42 7.30)
	HCL FRI [12]	B^* = (6.36 6.58 7.38)
The Incircle FRI		B* = (4.95 5.42 7.16)

Therefore, the fuzzy interpolative reasoning result of the proposed Incircle FRI is logically consistent in terms of Property 1 and Property 2.

Example X2 [13 , 28]: This example is the same example as X1, which uses triangular fuzzy sets only, but the observation is a singleton fuzzy set A^* = [8 , 8]. Table 2 and Fig. 14 give all attribute values and once again generate non-convex fuzzy conclusion. The HCL FRI [12] produce non-convex fuzzy set. Despite it is a non-triangular fuzzy set, the FRI methods CCL FRI [14], HS FRI [13], HTY FRI [15], and proposed Incircle FRI generate normal singleton conclusion. The Incircle FRI produces a singleton fuzzy set represented by the Gergonne Point GP_x.B^* = 5.42, the left fuzziness PS₁ = 0 and the right fuzziness PS₃ = 0, then B^* = [5.42, 5.42, 5.42]. In this example the proposed Incircle FRI performs better than the KH, KHstab, VKK and HCL FRI.

Table 2

Fuzzy interpolative reasoning results of Example X2

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (7.27 5.38 6.25)
A₁ = [0 5 6]	KHstab FRI [5]	B^* = (7.27 5.38 6.25)
A₂ = [11 13 14]	VKK FRI [4]	B^* = (7.00 5.38 7.00)
B₁ = [0 2 4]	CCL FRI [14]	B^* = (5.38 5.38 5.38)
B₂ = [10 11 13]	HS FRI [13]	B^* = (6.49 6.49 6.49)
A^* = [8 8 8]	HTY FRI [15]	B^* = (6.49 6.49 6.49)
	HCL FRI [12]	B^* = (7.27 –6.25)
The Incircle FRI		B* = (5.42 5.42 5.42)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Fig. 14

A comparison of FRI results of Example X2 for several methods.

Based on Equations (10) and (11) and Table 2, the left ratio fuzziness and the right ratio fuzziness of this example have the same results in Example X1. The results of the proposed Incircle FRI, CCL FRI satisfy property 1 for the left ratio fuzziness RF.PS ₁ (A^*, B^*), where MIN (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂, B₂)) = 0.4≤RF.PS ₁ (A^*, B^*) = [Incircle(0.44), CCL(0.44)]≤MAX (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂, B₂)) = 0.5. While, the right ratio fuzziness RF.PS ₃ (A₁, B₁) = RF.PS ₃ (A₂, B₂) = 2, and the results of the right ratio fuzziness RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI and CCL FRI satisfy Property 2, which are 2. Therefore, the fuzzy interpolative reasoning result of the proposed Incircle FRI is logically consistent in terms of Property 1 and Property 2 for the left and right ratio fuzziness, respectively.

Example X3 [13 , 28]: In this example the antecedents of the fuzzy rules are singleton fuzzy sets. The observation is a triangular fuzzy set A^* = [5 , 8]. Table 3 and Fig. 15 describe all the results and attributes values of this example. The result of the proposed Incircle FRI is computed using Equations (4), (7), and (8), which produce a triangular fuzzy set, in which the value of the Gergonne Point GP_x.B^* is calculated based on the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂, where GP_x.B^* = 6.37. According to Equations (5), (7), and (9), the left fuzziness PS₁ = 1.089 and the right fuzziness PS₃ = 1.910 are determined. Finally, the conclusion of the proposed Incircle FRI is a CNF triangular fuzzy set B^* = [5.28, 6.37, 8.28]. Meanwhile, the KH FRI [1 –3], the KHstab FRI [5], the HCL FRI [12], the CCL FRI [14], and the HS FRI [13] give a very reasonable conclusion. In contrast, in this case the VKK FRI [4] and the HTY FRI [15] could not generate a usable conclusion.

Table 3

Fuzzy interpolative reasoning results of Example X3

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (5.33 6.33 9.00)
A₁ = [3 3 3]	KHstab FRI [5]	B^* = (5.33 6.33 9.00)
A₂ = [12 12 12]	VKK FRI [4]	B^* = (–0.00 -)
B₁ = [4 4 4]	CCL FRI [14]	B^* = (5.33 6.33 8.33)
B₂ = [10 11 13]	HS FRI [13]	B^* = (5.71 6.28 8.16)
A^* = [5 6 8]	HTY FRI [15]	B^* = (-)
	HCL FRI [12]	B^* = (5.33 6.55 9.00)
The Incircle FRI		B* = (5.28 6.37 8.28)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Fig. 15

A comparison of FRI results of Example X3 for several methods.

Example X4 [13 , 28]: In this example only trapezoidal fuzzy sets are used for the fuzzy rules and for the observation A^* = [6 , 10]. Table 4 and Fig. 16 describe all the attribute values and the results of FRI methods. The conclusion of the proposed Incircle FRI is a trapezoidal membership function B₁^* = [GPx.B^*_L-TR_LPS₁, GPx.B^*_L, GPx.B^*_R, GPx.B^*_R +TR_RPS₃]. Based on Equations (4), (13), and (14) the Gergonne Points values are calculated by the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂, where GP_x.B^*_L = 4.54 and GP_x.B^*_R = 7.47.

The left fuzziness of the left triangular TR_LPS₁ = 0 and the right fuzziness of the right triangular TR_RPS₃ = 1.05 could be obtained based on Equations (5), (13), (15), and (16). Consequently, the Incircle FRI gets conclusion as trapezoidal fuzzy set B^* = [4.54 4.54 7.47 8.53] according to Equation (19). Figure 16 shows the results of the FRI methods. The HCL FRI [12] is unable to generate a conclusion in this case. The KH FRI [1 –3], the KHstab FRI [5], the VKK FRI [4], and the HTY FRI [15] generate an abnormal trapezoidal fuzzy set. In contrast, the CCL FRI [14], the HS FRI [13], and the proposed Incircle FRI generate CNF trapezoidal fuzzy sets.

Fig. 16

A comparison of FRI results of Example X4 for several methods.

Concerning ratios of the fuzziness of the two Rules (A₁ ⇒ B₁) and (A₂ ⇒ B₂) obtained by Equations (10) and (11) and Table 4, the left ratio fuzziness RF.PS ₁ (A₁, B₁) = 0.5, RF.PS ₁ (A₂, B₂) = 1, therefore, the left ratio fuzziness RF.PS ₁ (A^*, B^*) of the proposed Incircle FRI and CCL FRI are 0.64 and 0.69, therefore, the proposed Incircle FRI and CCL FRI methods satisfy property 1, where MIN (RF.PS ₁ (A, B))≤RF.PS ₁ (A^*, B^*)≤MAX (RF.PS ₁ the (A, B)) is equal 0.5≤[Incircle (0.64), CCL (0.69)]≤1. For the right ratio fuzziness, the ratio RF.PS ₃ (A₁, B₁) = RF.PS ₃ (A₂, B₂) = 1, and RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, HTY FRI, HS FRI, CCL FRI, KHstab FRI, and KH FRI methods are 1, 1.62, 0.71, 1, 1, and 1, respectively. Hence, the right ratio fuzziness RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, KHstab FRI, and KH FRI satisfy Property 2. Therefore, the fuzzy interpolative reasoning results of the proposed method are logically consistent in terms of Property 1 and Property 2.

Table 4

Fuzzy interpolative reasoning results of Example X4

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (5.45 4.25 7.50 8.5)
A₁ = [0 4 5 6]	KHstab FRI [5]	B^* = (5.45 4.25 7.50 8.5)
A₂ = [11 12 13 14]	VKK FRI [4]	B^* = (5.32 4.38 7.38 8.68)
B₁ = [0 2 3 4]	CCL FRI [14]	B^* = (4.25 4.25 7.5 8.5)
B₂ = [10 11 12 13]	HS FRI [13]	B^* = (5.23 5.23 7.61 8.32)
A^* = [6 6 9 10]	HTY FRI [15]	B^* = (4.98 7.44 6.44 8.06)
	HCL FRI [12]	B^* = (-)
The Incircle FRI		B* = (4.54 4.54 7.47 8.53)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Example X5 [14, 29]: In this example the rule antecedents are triangular fuzzy sets, the rule consequents are trapezoidal fuzzy sets. The observation is a triangular fuzzy set A^* = [7 , 9]. Table 5 and Fig. 17 show all the results of the FRI methods and attributes values of this example.

Table 5

Fuzzy interpolative reasoning results of Example X5

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (6.36 5.38 6.38 7.38)
A₁ = [0 5 6]	KHstab FRI [5]	B^* = (6.36 5.38 6.38 7.38)
A₂ = [11 13 14]	VKK FRI [4]	B^* = (6.16 5.38 6.38 7.84)
B₁ = [0 2 3 4]	CCL FRI [14]	B^* = (4.94 5.38 6.38 7.38)
B₂ = [10 11 12 13]	HS FRI [13]	B^* = (5.93 6.36 6.80 7.47)
A^* = [7 8 9]	HTY FRI [15]	B^* = (5.87 6.20 7.20 7.41)
	HCL FRI [12]	B^* = (-)
The Incircle FRI		B* = (5.0 5.9 5.9 7.3)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Fig. 17

A comparison of FRI results of Example X5 for several methods.

Based on Equations (4), (5), (13), (14), (15), and (16) the conclusion of the Incircle FRI can be calculated by the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂. The center values GP_x.B^* _L = 5.331, GP_x.B^* _R = 6.435, and MP.B^* = 5.9, the fuzziness values of the left triangular are TR_LPS₁ = 0.3233 and TR_LPS₃ = 0.0_, the fuzziness values of the right triangular are TR_RPS₁ = 0.0 and TR_RPS₃ = 0.898. Therefore, the Incircle FRI produced a trapezoidal fuzzy conclusion B^* = [5.0 5.9 5.9 7.3] that is calculated according to Equation (20). This case the HCL FRI is unable to generate any conclusion. The conclusion of the KH FRI [1 –3], the KHstab FRI [5], and the VKK FRI [4] are not convex and normal.

The CCL FRI [14], the HS FRI [13], the HTY FRI [15], and the Incircle FRI generate CNF trapezoidal conclusion.

Based on by Equations (10) and (11) and Table 5, the ratios of fuzziness the two adjacent Rules (A₁ ⇒ B₁, A₂ ⇒ B₂), and the ratios of fuzziness of the observation and conclusion (A^* ⇒ B^*) of the FRI methods. The left ratio fuzziness of RF.PS ₁ (A₁, B₁) = 0.4, RF.PS ₁ (A₂,B₂) = 0.5, and the left ratio fuzziness RF.PS ₁ (A^*, B^*) of the proposed Incircle FRI, the HS FRI, the CCL FRI and HTY FRI are 0.44, 0.43, 0.44, and 0.33, respectively, therefore, the proposed Incircle FRI, the HS FRI and the CCL FRI satisfy property 1, where MIN (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂,B₂))≤RF.PS ₁ (A^*, B^*)≤MAX (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂,B₂)), which is equal 0.4≤(0.44, 0.44, 0.43)≤0.5. In the right ratio fuzziness, the ratio RF.PS ₃ (A₁, B₁) = RF.PS ₃ (A₂, B₂) = 1, and RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, the HS FRI, the CCL FRI, HTY FRI, KHstab FRI, and KH FRI are 1, 0.67, 1, 0.21, 1, and 1. Therefore, the right ratio fuzziness RF.PS ₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, KHstab FRI, and KH FRI satisfy Property 2. Hence, the fuzzy interpolative reasoning results of the proposed method are logically consistent in terms of Property 1 and Property 2.

Example X6 [14]: In this example antecedents of the fuzzy rules are trapezoidal fuzzy sets. The consequent fuzzy sets are triangular. The observation is a trapezoidal fuzzy set A^* = [6 , 10]. Table 6 and Fig. 18 show all the results of the FRI methods and attributes values to this example. The representative values of the proposed Incircle FRI are determined by the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂ using Equations (4), (5), (13), (14), (15), and (16). This case, the proposed Incircle FRI produced a trapezoidal fuzzy conclusion B^* = [4.37 4.37 6.66 8.56] that is calculated according to Equation (19). The KH FRI [1 –3], the KHstab FRI [5], and the HTY FRI [15] can not generate convex and normal fuzzy conclusion. Meanwhile, the CCL [14], the HS [13], and the VKK FRI [4] have a CNF trapezoidal fuzzy conclusion.

Table 6

Fuzzy interpolative reasoning results of Example X6

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (5.45 4.25 6.50 8.50)
A₁ = [0 4 5 6]	KHstab FRI [5]	B^* = (5.45 4.25 6.50 8.50)
A₂ = [11 12 13 14]	VKK FRI [4]	B^* = (5.31 5.38 5.38 8.68)
B₁ = [0 2 4]	CCL FRI [14]	B^* = (5.38 5.38 5.38 7.38)
B₂ = [10 11 13]	HS FRI [13]	B^* = (5.46 5.46 5.46 8.55)
A^* = [6 6 9 10]	HTY FRI [15]	B^* = (5.07 7.26 5.26 8.15)
	HCL FRI [12]	B^* = (-)
The Incircle FRI		B* = (4.37 4.37 6.66 8.56)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Fig. 18

A comparison of FRI results of Example X6 for several methods.

Based on Equations (10) and (11) and Table 6, the left ratio fuzziness of the Rule1 (A₁ ⇒ B₁) RF.PS ₁ (A₁, B₁) = 0.5 and the ratio of the Rule2 (A₂ ⇒ B₂) RF.PS ₁ (A₂, B₂) = 1. Thus, the left ratio fuzziness RF.PS ₁ (A^*, B^*) of the proposed Incircle FRI and CCL FRI are 0.64 and 0.69, therefore, the proposed Incircle FRI and CCL FRI satisfy property 1, where MIN (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂, B₂)) = 0.5≤RF.PS ₁ (A^*, B^*) [Incircle(0.64), CCL(0.69)]≤MAX (RF.PS ₁ (A₁, B₁), RF.PS ₁ (A₂,B₂)) = 1.

For the right ratio fuzziness, the ratio RF.PS₃ (A₁, B₁) = RF.PS₃ (A₂, B₂) = 2, and the right ratio fuzziness RF.PS₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, the HTY FRI, HS FRI, the VKK FRI, the KHstab FRI, and the KH FRI methods are 2, 2, 2.8, 3.1, 3.3, 2, and 2, respectively. Thus, the right ratio fuzziness RF.PS₃ (A^*, B^*) of the proposed Incircle FRI, CCL FRI, the KHstab FRI, and the KH FRI satisfy Property 2, where RF.PS₃ (A₁, B₁) = RF.PS₃ (A₂, B₂) = RF.PS₃ (A^*, B^*) = 2. Therefore, the fuzzy interpolative reasoning result of the proposed method is logically consistent in terms of Property 1 and Property 2.

Example X7 [13 , 29]: In this example the two adjacent fuzzy rules and observation are hexagonal fuzzy sets. Table 7 and Fig. 19 show all the results of the FRI methods and attributes values to this example with the observation describes by hexagonal fuzzy set A^* = [6, 6.5, 7, 9, 10, 10.5]. The representative values of the Incircle FRI are determined by the fuzzy interpolation of the fuzzy sets A₁, A₂, B₁, and B₂ using Equations (4), (5), (13), (14), (15), and (16) that produce a hexagonal fuzzy conclusion B^* = [5.59 6.02 6.55 8.63 9.52 10.13] according to Equation (21), where GP_x.B^* _L = 6.017, GP_x.B^* _R = 9.52, the fuzziness sides of the left triangular are TR_LPS₁ = 0.424 and TR_LPS₃ = 0.531, the fuzziness sides of the right triangular are TR_RPS₁ = 0.898 and TR_RPS₃ = 0.601. The KH FRI [1 –3], the VKK FRI [4], The CCL FRI [14], and the HS FRI [13] generate CNF hexagonal fuzzy conclusion. In contrast, the KHstab FRI [5] can not create a CNF conclusion and the HTY FRI [15] and HCL FRI [12] have no conclusion at all.

Table 7

Fuzzy interpolative reasoning results of Example X7

Attribute values	Methods	Results of fuzzy interpolative reasoning
	KH FRI [1–3]	B^* = (5.73 5.74 5.75 5.89 7.25 8.56)
A₁ = [0 1 3 4 5 5.5]	KHstab FRI [5]	B^* = (5.73 5.87 6.00 5.89 7.25 8.56)
A₂ = [11 11.5 12 13 13.5 14]	VKK FRI [4]	B^* = (5.60 5.69 5.71 6.26 7.46 8.68)
B₁ = [0 0.5 1 3 4 4.5]	CCL FRI [14]	B^* = (5.2 5.4 5.7 8.6 9.7 10.2)
B₂ = [10.5 11 12 13 13.5 14]	HS FRI [13]	B^* = (5.47 5.79 6.08 8.42 9.23 9.70)
A^* = [6 6.5 7 9 10 10.5]	HTY FRI [15]	B^* = (-)
	HCL FRI [12]	B^* = (-)
The Incircle FRI		B* = (5.59 6.02 6.55 8.63 9.52 10.13)

Note: the sign (-) indicates no clear evidence for the method to handle the case in the example.

Fig. 19

A comparison of FRI results of Example X7 for several methods.

Table 8 presents a summary of evaluation for the proposed Incircle FRI method compared with the current methods (i.e., KH FRI [1 –3], KHstab FRI [5], VKK FRI [4], HCL FRI [12], HTY FRI [15], CCL FRI [14], and HS FRI [13]) according to the criteria (i.e., CNF property”, “different membership functions,” “different kinds membership functions of the antecedents and the consequents fuzzy rules, the similarity propagation of the fuzziness and core between the observation and conclusion, and “logically consistent with respect to the ratios of fuzziness (property 1 and property 2)”). From Table 8, we can see that the Incircle FRI methods satisfy with these five evaluation criteria, where the sign (√) indicates the technique is satisfied with all criteria for all selected examples, while a sign (-) shows the method has a problem in most examples, and the sign (x) indicates the technique does not satisfy with all examples.

Table 8

Description of the evaluation criteria of the Incircle FRI with existing methods

Criteria	Methods
	KH	KHstab	VKK	HCL	HTY	CCL	HS	Incircle
	FRI	FRI	FRI	FRI	FRI	FRI	FRI	FRI
	[1 –3]	[5]	[4]	[12]	[15]	[14]	[13]
CNF property	x	x	x	–	–	√	√	√
Handle different Membership Functions	√	√	√	x	–	√	√	√
Different membership functions of the ANT and the CON	√	√	√	x	–	√	√	√
Propagation of the fuzziness between observation and conclusion	x	x	x	x	x	–	–	√
Logically consistent with respect to the ratios of fuzziness	–	–	x	x	x	√	–	√

5 Conclusions

In this paper, a new fuzzy interpolative reasoning method called “Incircle FRI”, is introduced, which is defined for triangular CNF fuzzy sets, for a single antecedent universe and two surrounding rules from the rule-base. The suggested “Incircle FRI” is based on the incircle of a triangular fuzzy number, the Gergonne Point (GP) as a reference point of the triangular fuzzy set, and the “fuzziness sides”, i.e. the distances of the endpoints of the support, and the core from the incircle InTouch points (noted by PS₁, PS₂ and PS₃ in this paper). The proposed method calculates the conclusion by holding the same rate of distances among the observation and the two rule antecedents, and the conclusion and the two corresponding rule consequents with the Gergonne Points (for the reference point of the conclusion), and with the “fuzziness sides” (for the shape of the conclusion). The paper also extends the suggested “Incircle FRI” to trapezoidal, and hexagonal shaped fuzzy sets by decomposing their shapes to multiple triangulars. The generated conclusion is also a CNF fuzzy set. The performance of the suggested “Incircle FRI” is discussed based on numerical examples, and a comprehensive comparison to other FRI methods, namely with the KH method [1 –3], the KHstab method [5], the VKK method [4], the HCL method [12], the HTY method [15], the CCL method [14], and the HS method [13]. From the experimental results and Table 8, we can see that the proposed method is considered one of the best current FRI methods. Consequently, the proposed method provides a useful method as a fuzzy interpolation in dispersed rules-based systems.

Footnotes

Acknowledgments

The described study was carried out as part of the EFOP-3.6.1-16-00011 Younger and Renewing University - Innovative Knowledge City - institutional development of the University of Miskolc aiming at intelligent specialization project implemented in the framework of the Szechenyi 2020 program. The realization of this project is supported by the European Union, cofinanced by the European Social Fund.

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