An Atanassov’s intuitionistic fuzzy reasoning model

Abstract

The task of the standard Mamdani fuzzy logic controller is to find a crisp control action from the fuzzy rule-base and from a set of crisp inputs. In this paper we modify this controller in order to work with Atanassov’s intuitionistic fuzzy sets and to activate a set of rules having the same conclusion. Usually, the inference rules used in a fuzzy logic controller are given by a domain expert; in our system, these rules are automatically induced as fuzzy association rules starting from a training set. The fuzzy confidence value associated with each rule is used to obtain the fuzzy set inferred by our system.

Keywords

Fuzzy logic controller Atanassov’s intuitionistic fuzzy sets t-norm implication association rules

1 Introduction

The knowledge base of a rule-based system may contain imprecisions which appear in the description of the rules given by the expert. Because such an inference can not be made by the methods which use multiple valued logic, Zadeh gave a theory of approximate reasoning [38] that is the deduction of imprecise conclusions from a set of imprecise premises. The theory of approximate reasoning is based on fuzzy logic inference processes. An important part of fuzzy reasoning is represented by Fuzzy Logic Control (FLC), which is very useful when the needed models are not known or when they are too complex for analysis with conventional quantitative techniques.

In a fuzzy logic controller the expert’s knowledge is of the form $\begin{matrix} IF (a set of conditions are satisfied) \\ THEN (a set of consequences are inferred) \end{matrix}$ where the antecedents and the consequences of the rules are associated with fuzzy concepts. The task of a FLC system is to find a crisp control action from the rule-base and from the crisp inputs. Because the inputs and the outputs of rule-based systems are fuzzy sets, we have to fuzzify the crisp inputs and to defuzzify the fuzzy outputs. A standard FLC system consists of four major parts (see Fig. 1): Fuzzification interface, Fuzzy rule-base, Fuzzy inference engine and Defuzzification interface.

In this paper we modify the standard Mamdani controller in order to work with Atanassov’s intuitionistic fuzzy sets. An Atanassov’s intuitionistic fuzzy set is a natural generalization of a fuzzy set used for better modeling imperfect knowledge and is characterized by two functions representing the degree of membership and the degree of non-membership.

We obtain an Atanassov’s Intuitionistic Fuzzy Logic Controller (IFLC), a model for fuzzy reasoning based on Atanassov’s intuitionistic fuzzy sets.

There are a lot of papers studying the approximate reasoning based on Atanassov’s Intuitionistic Fuzzy Sets (IFSs) or Interval-Valued Fuzzy Sets (IVFSs) which consider the mathematical equivalence between IFS and IVFS. In the paper [14], authors present two ways of inference: first is based on interval-valued inclusion grade indicator of the observation in the premise and second is based on interval-valued similarity measure. Using the idea of interval-valued fuzzy relations, in the paper [15] it is described a method of inference based on IVFS while in the paper [33] a completely new interval-valued fuzzy reasoning is introduced. This inference process uses interval-valued restricted equivalence functions to increase the relevance of the rules in which the equivalence of the interval membership degrees of the patterns and the ideal membership degrees is greater. The papers [7 , 37] perform an extension of a fuzzy expert system to an Atanassov’s intuitionistic fuzzy expert system in order to incorporate in the expert system the uncertainty modeled by Atanassov’s intuitionistic fuzzy sets. Our system features and differences from those mentioned above are highlighted below.

Usually, the inference rules used by a reasoning system are provided by a domain expert. In our case, these rules are induced as fuzzy association rules using a training set and the system activates only a set of rules having the same conclusion (or consequence).

The proposed IFLC system works as follows:

generates using data mining techniques, from training data, the rules used by inference engine;

computes the firing level of each rule, corresponding to input data;

computes the matching value of the input data with each set of rules having the same conclusion; this value is computed taking into account the fuzzy confidence value of the rule;

selects the best set of rules used in the inference process;

computes the fuzzy set that represents the conclusion inferred and its corresponding crisp value obtained by defuzzification process.

The rest of paper is organized as follows: Section 2 presents the basic concepts with references to our approach. In Section 3 it is described the proposed IFLC system. The Section 4 contains an example about presented system and the last section discusses conclusions and future works.

This paper is an extension of [28]; in order to prove the performance of the proposed IFLC system compared to the classic FLC system, we extended the sample data used in the Section 4, while as defuzzification method we employed the Center-of-Gravity technique.

2 Basic concepts

2.1 Atanassov’s intuitionistic fuzzy sets

The knowledge base of a rule-based system may contain imperfect information which is inherent in the description of the rules given by the expert. This information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory or deficient in some other ways. In these cases, some difficulties appear: the difficulty of representing the deduction rules expressed, generally, by means of natural language and the difficulty of utilization of these rules when the observed facts do not match the condition expressed in the premise of the rule, but are not too different from them. Nowadays, we can handle much of this information’s imperfection using fuzzy logic [12]. A membership function of a classical fuzzy set assigns to each element from the universe of discourse a number from the unit interval to indicate the membership degree to the set under consideration. The non-membership degree is just natural defined as the complement to 1 of the membership degree. However, a human being who expresses the membership degree of a given element in a fuzzy set very often does not express corresponding degree of non membership as the complement to 1. This reflects a well known psychological fact that the linguistic negation not always identifies with logical negation. Thus Atanassov [5] introduced the concept of an intuitionistic fuzzy set characterized by two functions expressing the degree of membership and the degree of non membership, respectively. Atanassov’s intuitionistic fuzzy sets (see [5 , 9]), can be viewed as a tool that may help better model imperfect information, especially under imperfectly defined facts and imprecise knowledge.

Definition 1. [5, 6] Let X denote a universe of discourse. Then an Atanassov’s intuitionistic fuzzy set A in X is a set of ordered triples $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ where μ_A, ν_A : X → [0, 1] are functions such that $0 \leq μ_{A} (x) + ν_{A} (x) \leq 1, \forall x \in X .$

For each x the numbers μ_A (x) and ν_A (x) represent the degree of membership and the degree of non-membership of the element x ∈ X to A, respectively. For each element x ∈ X we can compute the, so-called, Atanassov’s intuitionistic fuzzy index (or hesitation [35]) of x in A defined as follows: $π_{A} (x) = 1 - μ_{A} (x) - ν_{A} (x) .$

This notion was extended in [16], obtaining a generalized Atanassov’s intuitionistic fuzzy index associated with a strong negation. Of course, a fuzzy set is a particular case of the Atanassov’s intuitionistic fuzzy set with ν_A (x) =1 - μ_A (x). The working with Atanassov’s intuitionistic fuzzy sets instead of fuzzy sets imply the adding of another degree of freedom (ν_A or π_A) to μ_A. The Atanassov’s intuitionistic fuzzy sets offer a new possibility to represent imperfect knowledge and, therefore, to describe in a more adequate way many real problems. Such problems appear when we face with human opinions involving two or more answer of the type [34]: “Yes”, “Not”, “I do not know”, “I am not sure”, etc.

Voting can be a good example of such a situation, as human voters may be divided into three groups of those who: "Vote for", "Vote against", "Abstain or Give invalid votes".

This third group is of great interest (from the point of view of customer behavior analysis, for instance) because people from this undecided group after proper enhancement (e.g. different marketing activities) can finally become sure, i. e. become persons voting for or against (customers wishing to buy products advertised).

The majority of applications operate with Atanassov’s intuitionistic fuzzy numbers which was proposed by Burillo et al. in [17], the literature being rich in various definitions (see, for example, [11 , 32]).

Definition 2. [20] An Atanassov’s intuitionistic fuzzy subset $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in ℝ}$ of the real line is called Atanassov’s intuitionistic fuzzy number if:

A is if-normal: there exists $x_{0} \in ℝ$ such that μ_A (x₀) =1 and ν_A (x₀) =0,

A is if-convex: its membership function is fuzzy convex μ_A (λx₁ + (1 - λ) x₂) ≥ min (μ_A (x₁) , μ_A (x₂)) $\forall x_{1}, x_{2} \in ℝ, λ \in [0, 1]$ and its non-membership function is fuzzy concave ν_A (λx₁ + (1 - λ) x₂) ≤ max (ν_A (x₁) , ν_A (x₂)) $\forall x_{1}, x_{2} \in ℝ, λ \in [0, 1],$

μ_A and ν_A are continuous by parts.

Because the operating principle of the proposed system is independent of the type of fuzzy numbers used, for an easy implementation we will work with trapezoidal Atanassov’s intuitionistic fuzzy numbers.

Definition 3. An trapezoidal Atanassov’s intuitionistic fuzzy number $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in ℝ}$ is given by $μ_{A} (x) = {\begin{matrix} 0 ifx < a_{1} \\ ∥ \frac{x - a_{1}}{a_{2} - a_{1}} ∥ if a_{1} \leq x \leq a_{2} \\ ∥ 1 if a_{2} \leq x \leq a_{3} \\ ∥ \frac{x - a_{4}}{a_{3} - a_{4}} if a_{3} \leq x \leq a_{4} \\ ∥ 0 for a_{4} < x \end{matrix}$

$ν_{A} (x) = {\begin{matrix} 1 ifx < b_{1} \\ ∥ \frac{x - b_{2}}{b_{1} - b_{2}} ∥ if b_{1} \leq x \leq b_{2} \\ ∥ 0 if b_{2} \leq x \leq b_{3} \\ ∥ \frac{x - b_{3}}{b_{4} - b_{3}} if b_{3} \leq x \leq b_{4} \\ ∥ 1 if b_{4} < x \end{matrix}$

where $b_{1}, a_{1}, b_{2}, a_{2}, a_{3}, b_{3}, a_{4}, b_{4} \in ℝ$ and b₁ ≤ a₁ ≤ b₂ ≤ a₂ ≤ a₃ ≤ b₃ ≤ a₄ ≤ b₄.

By convention, we ignore the situations where the denominator is equal to zero.

A such number will be denoted by $A = (b_{1}, a_{1}, b_{2}, a_{2}, a_{3}, b_{3}, a_{4}, b_{4}) .$

The rule $if X is A then Y is B$ is represented by its conditional possibility distribution [38, 39] π_Y/X defined by $π_{Y / X} (v, u) = μ_{A} (u) \to μ_{B} (v), \forall u \in U, \forall v \in V$ where → is an implication operator, μ_A and μ_B are the membership functions of the fuzzy sets A and B, respectively, U is the domain of X and V is the domain of Y.

Definition 4. A fuzzy implication is a function I : [0, 1] ² → [0, 1] satisfying the following conditions:

I1: If x ≤ z then I (x, y) ≥ I (z, y) for all x, y, z ∈ [0, 1]

I2: If y ≤ z then I (x, y) ≤ I (x, z) for all x, y, z ∈ [0, 1]

I3: I (0, y) =1 (falsity implies anything)

for all y ∈ [0, 1]

I4: I (x, 1) =1 (anything implies tautology)

for all x ∈ [0, 1]

I5: I(1,0)=0 (Booleanity).

The following properties for fuzzy implication are required by different authors and they have their significance in various applications:

I6: I (1, x) = x (tautology cannot justify anything) for all x ∈ [0, 1]

I7: I (x, I (y, z)) = I (y, I (x, z)) (exchange principle) for all x, y, z ∈ [0, 1]

I8: x ≤ y if and only if I (x, y) =1 (implication defines ordering) for all x, y ∈ [0, 1]

I9: I (x, 0) = N (x) for all x ∈ [0, 1] is a strong negation

I10: I (x, y) ≥ y for all x, y ∈ [0, 1]

I11: I (x, x) =1 (identity principle) for all x ∈ [0, 1]

I12: I (x, y) = I (N (y) , N (x)) for all x, y ∈ [0, 1] and a strong negation N

I13: I is a continuous function.

Czogala and Leski [21] analyzing a set of eight implications (Kleene-Dienes, Reichenbach, Lukasiewicz, Godel, Rescher-Gaines, Goguen, Zadeh, Fodor) concluded that the Lukasiewicz implication $I_{L} (x, y) = min (1 - x + y, 1)$ satisfy all set of previous properties and, therefore, it is one of the most important implications. The implication from fuzzy set theory was extended to Atanassov’s intuitionistic fuzzy and interval-valued fuzzy set theory. For instance, the Lukasiewicz implication in this case is (see [19],[10]) $I_{L}^{e} (x, y) = (\min (1, x_{2} + y_{1}), \max (0, x_{1} + y_{2} - 1))$ where x = (x₁, x₂) and y = (y₁, y₂). Because we decompose an Atanassov’s intuitionistic fuzzy rule in two standard fuzzy rules, one characterized by membership function and the other by non-membership function, we will work with Lukasiewicz implication I_L.

Other notions used in this paper are given by the following definitions.

Definition 5. A function T : [0, 1] ² → [0, 1] is a triangular norm (in short, t - norm) iff it is commutative, associative, non-decreasing and T (x, 1) = x, ∀ x ∈ [0, 1].

Definition 6. A linguistic variable is defined by a quin-tuple (x, T (x) , U, G, M) where: x is the name of variable, T (x) is the set of terms (set of linguistic values of x), U is the universe of discourse, G is a syntactic rule for generating the names of values of x and M is a semantic rule with each value its meaning.

2.2 Fuzzy association rules

Mining of association rules represents one of the most important task in data mining.

Association rules are used to represent and identify dependencies between items in a database [40]. These are expressions of the type X → Y, where X and Y are sets of items and X∩ Y ≠ ∅. This means that if all the items in X exist in a transaction then all the items in Y with a high probability are also in the transaction, and X and Y should not have any common items [3]. Knowledge of this type of relationship can enable proactive decision making to proceed from the inferred data.

Professor Mamdani did a pioneering job by investigating the use of fuzzy logic for interpreting the human derived control rules. Conceptually, fuzzy association rules follow the same scheme proposed by Mamdani in regression/control. In the described case it is used their meaning as descriptive rules of information, hence having a descriptive rule set that are a particular chunk of information.

The task of discovering association rules was first introduced in [2]. Initially, association rules mining focused on market basket data which stores items purchased on a per-transaction basis. A typical example of an association rule has the following statement: 86% of customers who purchase bread also purchase butter.

Several extensions to the classical relational model have been proposed to support quantitative data [4]. Fuzzy association rules can handle both quantitative and categorical data and are expressed in linguistic terms, which are more natural and understandable for human beings.

An example of fuzzy association rules is the following: $age (X, young) \land income (X, high) cars (X, many)$ (1)Here, the notions of young, high and many can be modeled as fuzzy sets.

The basic problem of finding fuzzy association rules was introduced in [30]. Let DB = {t₁, …, t_n} be a database characterized by a set I = {i₁, …, i_m} of categorical or quantitative attributes (items). For each attribute i_k, (k = 1, …, m), we will consider n (k) associated fuzzy sets. Let $F_{i_{k}} = {F_{i_{k}}^{1}, \dots, F_{i_{k}}^{n (k)}}$ be the set of all these fuzzy sets.

For an attribute i_k and a fuzzy set $F_{i_{k}}^{j}$ , the membership function, denoted $μ_{F_{i_{k}}^{j}}$ , is defined as: $μ_{F_{i_{k}}^{j}} dom (i_{k}) \to [0, 1], : k = \bar{1, m}, j = \bar{1, n (k)}$ (2)Typically, a domain expert provides the set of fuzzy sets associated with attributes, along with their corresponding membership functions.

For a specified transaction t_j ∈ DB and an attribute i_k, we denote with t_j [i_k] the value of attribute i_k in transaction t_j.

Example 1. Let us consider DB = {t₁, t₂, t₃, t₄, t₅} the database sample with quantitative attributes presented in Table 1. As it can be seen, the set of attributes is I = {Age, :Income, :Cars}.

For the quantitative attribute Age we can take into consideration the following three fuzzy sets: young, middle - aged and old.

Thus, we have F_Age = {young, :middle - aged, :old}. Similarly, let F_Income = {low, :middle, :high} the set of fuzzy sets associated with attribute Income.

For the last attribute Cars we consider the following set of fuzzy sets F_Cars = {few, :many}.

Definition 7. We call fuzzy itemset the tuple 〈XF_X〉,where X ⊆ I is a set of attributes and F_X is a set of fuzzy sets associated with attributes from X.

For example, the tuple 〈{Age, Income}{young,high}〉 represents a fuzzy itemset for the database given in Example 1.

Definition 8. A fuzzy association rule is an implication of the following form: $if X is F_{X} then Y is F_{Y}$ where X = {x₁, …, x_p} and Y = {y₁, …, y_q} are disjoint subset of I, and F_X = {a₁, …, a_p} and F_Y = {b₁, …, b_q} are fuzzy sets related to attributes from X and Y respectively. More exactly, a_i ∈ F_{x
_i}, (i = 1, …, p), and b_i ∈ F_{y
_i}, (i = 1, …, q). We denote this rule with: $〈 X F_{X} 〉 \Rightarrow 〈 Y F_{Y} 〉$

The intuitive signification of this fuzzy association rule, 〈X,F_X〉⇒〈YF_Y〉, is: "if a transaction (tuple) satisfies the property X ∈ F_X, then it will also satisfy the property Y ∈ F_Y with a high probability".

An example of a fuzzy association rule is the following: $" if Age is young and Income is high then Cars is many "$ Here, X = {Age, Income}, Y = {Cars}, F_X = {young, high}, F_Y = {many} and the rule can be represented as:

$\begin{matrix} 〈 {Age, Income} {young, high} 〉 \\ \Rightarrow 〈 {Cars} {many} 〉 \end{matrix}$

In order to express the quality of a fuzzy association rule two quality measures, fuzzy support and fuzzy confidence, have been proposed in [30].

Definition 9. The fuzzy support value of fuzzy itemset 〈 XF_X〉 in DB is: ${FS}_{X F_{X}} = \frac{\sum_{t_{i} \in DB} \prod_{x_{j} \in X} α_{a_{j}} (t_{i} [x_{j}])}{| DB |}$ (3)where $α_{a_{j}} (t_{i} [x_{j}]) = {\begin{matrix} μ_{a_{j}} (t_{i}), & if μ_{a_{j}} (t_{i} [x_{j}]) \geq ω \\ 0, & otherwise \end{matrix}$ (4)

and ω is a user specified minimum threshold for the membership function. Thus, the values of membership functions less than this minimum threshold, ω, are ignored.

Definition 10. Let 〈X,F_X〉⇒〈Y,F_Y〉 be a fuzzy association rule. The fuzzy support value of the rule is defined as fuzzy support value of the itemset 〈{X,Y}{F_X,F_Y}〉: ${FS}_{〈 X F_{X} 〉〈 Y F_{Y}} 〉 = {FS}_{〈 {X, Y} {F_{X}, F_{Y}} 〉}$

Definition 11. Let 〈X,F_X〉⇒〈Y,F_Y〉, a fuzzy association rule. The fuzzy confidence value of the rule is defined as: ${FC}_{〈 X F_{X} 〉 \Rightarrow 〈 Y F_{Y}} 〉 = \frac{FS 〈 Z, F_{Z} 〉}{FS 〈 X, F_{X} 〉}$ where Z = {X, Y} and F_Z = {F_X, F_Y}.

A fuzzy association rule is considered as interesting if it has enough support and high confidence value.

3 Proposed atanassov’s intuitionistic fuzzy system

According to the structure of a FLC, an IFLC requires the following operations: fuzzification, reasoning and defuzzification.

3.1 Fuzzification and firing levels

A fuzzification operator transforms crisp data into fuzzy sets. For instance, x₀ ∈ U is fuzzified into ${\bar{x}}_{0}$ according to the relations: $μ_{{\bar{x}}_{0}} (x) = {\begin{matrix} 1 & if x = x_{0} \\ 0 & otherwise \end{matrix}$ (5)

$ν_{{\bar{x}}_{0}} (x) = {\begin{matrix} 0 & if x = x_{0} \\ 1 & otherwise \end{matrix}$ (6)

The μ-firing level and ν-firing level of an Atanassov’s intuitionistic fuzzy set A and a crisp value x₀ as input are μ_A (x₀) and ν_A (x₀), respectively.

3.2 Reasoning

In order to perform reasoning a set of rules are necessary. Typically rules for fuzzy logic controllers appear in if-then form and are obtained from the knowledge of experts and operators. As a result, the rules are limited, subjective and inaccurate.

In our system, these rules are automatically induced as fuzzy association rules starting from a training set; for this we can use any algorithm for mining fuzzy association rules (see [18], [22], [26], [27]. We consider that the training set is described as a set of transactions DB = {t₁, …, t_n} characterized by a set I = {i₁, …, i_m} of attributes. These attributes are represented by the input and output variables of fuzzy logic controller. For each attribute (variable) i_k, k = 1, …, m, we consider n (k) linguistic values represented as fuzzy sets.

We generate only rules with input attributes in premise and output attributes in conclusion. The generated rules has the following form: $\begin{matrix} R_{i} : & if X_{1} is A_{i}^{1} and \dot{.} . and X_{r} is A_{i}^{r} \\ then Y is C_{i} : ({FS}_{i}, {FC}_{i}) \end{matrix}$ (7)where X_j, j ∈ {1, 2, . . . , r} represent the input variables, Y is an output variable, $A_{i}^{j}, j \in {1, 2, . . ., r}$ and C_i are linguistic values associated with X_j and Y respectively, and (FS_i, FC_i) are the fuzzy support and the fuzzy confidence of the rule.

For a given rule R_i the input data x = {x₁, …, x_r} generates the μ-firing level μ_i, the ν-firing level ν_i and the Atanassov’s intuitionistic fuzzy index π_i, computed as $\begin{matrix} μ_{i} = T (μ_{i}^{1}, . . ., μ_{i}^{r}), \\ ν_{i} = T (ν_{i}^{1}, . . ., ν_{i}^{r}), \\ π_{i} = T (π_{i}^{1}, . . ., π_{i}^{r}) \end{matrix}$ (8)where T is a t-norm while $μ_{i}^{j}$ is the μ-firing level, $ν_{i}^{j}$ is the ν-firing level and $π_{i}^{j}$ is the Atanassov’s intuitionistic fuzzy index, respectively, for $A_{i}^{j}, j \in {1, 2, . . ., r}$ .

We partition the generated rules in subsets with rules having the same conclusion: $R (C) = {\begin{matrix} R_{1} : & if X_{1}^{1} is A_{1}^{1} and \dot{.} . and X_{1}^{r_{1}} is A_{1}^{r_{1}} \\ then Y is C : ({FS}_{1}, {FC}_{1}) \\ \dots \\ R_{P} : & if X_{P}^{1} is A_{P}^{1} and \dot{.} . and X_{P}^{r_{P}} is A_{P}^{r_{P}} \\ then Y is C : ({FS}_{P}, {FC}_{P}) \end{matrix}$ For each subset R (C) = {R₁, …, R_P} we compute the matching value of input data x as follows: $MR (C) = \frac{\sum_{i = 1}^{P} α_{i} * {FC}_{i}}{\sum_{i = 1}^{P} {FC}_{i}}$ (9)where α_i is the firing level of the rule R_i, computed as $α_{i} = (1 - π_{i}) μ_{i} + π_{i} ν_{i} .$ (10)In order to compute the output of IFLC for the input x, the system selects the subset of rules R (C) having the maximum matching measure. This subset will be identified as a single rule having the conclusion C and the firing levels l_μ and l_ν, where: $l_{μ} = \frac{\sum_{i = 1}^{P} μ_{i} * {FC}_{i}}{\sum_{i = 1}^{P} {FC}_{i}}$ (11)and $l_{ν} = \frac{\sum_{i = 1}^{P} ν_{i} * {FC}_{i}}{\sum_{i = 1}^{P} {FC}_{i}} .$ (12)Using the Lukasiewicz implication, we get two conclusions:

the μ-conclusion C^μ, which is the traditional output of the rule computed using the membership function and the firing level l_μ $μ_{C^{μ}} (v) = I_{L} (l_{μ}, μ_{C} (v)), \forall v \in V .$

the ν-conclusion C^ν, which is the output of the rule computed using the non-membership function and the firing level l_ν $μ_{C^{ν}} (v) = I_{L} (l_{ν}, μ_{C} (v)), \forall v \in V .$

The crisp value y₀ associated to a inferred conclusion C′ is obtained by defuzzification method.

3.3 Defuzzification

The μ-conclusion and the ν-conclusion inferred using the set of rules R (C), where the conclusion C is the trapezoidal Atanassov’s intuitionistic fuzzy number (b₁, a₁, b₂, a₂, a₃, b₃, a₄, b₄), are given by the following expressions: $μ_{C^{μ}} (x) = {\begin{matrix} 1 - l_{μ} ifx \leq a_{1} ∥ \\ \frac{x - a_{1}}{a_{2} - a_{1}} + 1 - l_{μ} \\ ∥ ifx \in [a_{1}, l_{μ} (a_{2} - a_{1}) + a_{1}] ∥ \\ 1 ifx \in [l_{μ} (a_{2} - a_{1}) + a_{1}, \\ ∥ l_{μ} (a_{3} - a_{4}) + a_{4}] ∥ \\ \frac{x - a_{4}}{a_{3} - a_{4}} + 1 - l_{μ} \\ ∥ ifx \in [l_{μ} (a_{3} - a_{4}) + a_{4}, a_{4}] \\ ∥ 1 - l_{μ} if a_{4} \leq x \end{matrix}$ $μ_{C^{ν}} (x) = {\begin{matrix} 1 ifx \leq l_{ν} (b_{1} - b_{2}) + b_{2} ∥ \\ \frac{x - b_{2}}{b_{1} - b_{2}} + 1 - l_{ν} ∥ \\ ∥ ifx \in [l_{ν} (b_{1} - b_{2}) + b_{2}, b_{2}] \\ ∥ 1 - l_{ν} ifx \in [b_{2}, b_{3}] \\ ∥ \frac{x - b_{3}}{b_{4} - b_{3}} + 1 - l_{ν} \\ ∥ ifx \in [b_{3}, l_{ν} (b_{4} - b_{3}) + b_{3}] ∥ \\ 1 ifx \geq l_{ν} (b_{4} - b - 3) + b_{3} \end{matrix}$ where l_μ and l_ν are the firing levels defined in the previous section.

The conclusions C^μ and C^ν are defuzzified into the numbers y^μ and y^ν using the Center-of-Gravity technique, that for a discrete fuzzy set C gives:

$z_{0} = \frac{\sum_{j = 1}^{N} z_{j} μ_{C} (z_{j})}{\sum_{j = 1}^{N} μ_{C} (z_{j})}$ The defuzzified output corresponding to conclusion C of the set R (C) is computed as a linear combination $y = (1 - π) y^{μ} + π y^{ν},$ where π is the Atanassov’s intuitionistic fuzzy index associated with the set R (C), computed as $π = \frac{\sum_{i = 1}^{P} π_{i} * {FC}_{i}}{\sum_{i = 1}^{P} {FC}_{i}} .$

For π = 0 the IFLC system reduces to the system from [29].

4 A case study

In order to show how the proposed system works, we consider an example inspired from [1] concerning washing machines. We transform this example into an IFLC system with two inputs and one output, which are linguistic variables. The input variables are Degree-of-Dirt (DD) and Type-of-Dirt (TD); the output variable is Washing-Time (WT). We consider the universes of discourse [0, 100] for the input variables and [0, 60] for the output variable.

For the input variable DD we can take into consideration the following three linguistic terms (see Fig. 4): $F_{DD} = {Small, Medium, Large}$ defined as the following trapezoidal Atanassov’s intuitionistic fuzzy numbers: $\begin{matrix} Small = (0, 0, 0, 0, 20, 30, 40, 50) \\ Medium = (10, 20, 30, 40, 60, 70, 80, 90) \\ Large = (50, 60, 70, 80, 100, 100, 100, 100) \end{matrix}$

Similarly, let

F_TD = {VeryNotGreasy, NotGreasy,

Medium, Greasy, VeryGreasy}

be the set of linguistic terms associated with the input variable TD defined by the following trapezoidal Atanassov’s intuitionistic fuzzy numbers (see Fig. 5): $\begin{matrix} VeryNotGreasy = (0, 0, 0, 0, 10, 20, 30, 40) \\ NotGreasy = (5, 10, 15, 20, 30, 35, 40, 45) \\ Medium = (10, 20, 30, 40, 60, 70, 80, 90) \\ Greasy = (55, 60, 65, 70, 80, 85, 90, 95) \\ VeryGreasy = (60, 70, 80, 90, 100, 100, 100, 100) . \end{matrix}$

For the output variable WT we consider the following set of linguistic terms:

F_WT = {VeryShort, Short, Medium,

Long, VeryLong}

with the membership and the non-membership functions defined as trapezoidal Atanassov’s intuitionistic fuzzy numbers (see Fig. 6): $\begin{matrix} VeryShort = (0, 0, 0, 0, 5, 7.5, 10, 20) \\ Short = (2.5, 5, 7.5, 10, 15, 17.5, 20, 25) \\ Medium = (10, 15, 20, 25, 35, 40, 45, 50) \\ Long = (30, 40, 42.5, 45, 50, 52.5, 55, 57.5) \\ VeryLong = (40, 50, 52.5, 55, 60, 60, 60, 60) . \end{matrix}$

In order to extract the rules used by the inference engine of IFLC, we use a modified implementation of Fuzzy Apriori-T algorithm [18]. This algorithm runs on a training dataset of the form (DD, TD, WT) and keep only rules with support and confidence greater than or equal to the minimum support threshold and minimum confidence threshold respectively.

A fragment from training dataset is presented in Table 2.

Table 3 contains the fuzzy association rules obtained applying the Fuzzy Apriori-T algorithm on training dataset.

Partitioning this set of rules, we get the following subsets having the same conclusion: $\begin{matrix} R (VeryLong) = {R_{1}}, \\ R (Long) = {R_{2}, R_{3}, R_{4}}, \\ R (Medium) = {R_{5}, R_{6}, R_{7}}, \\ R (Short) = {R_{8}}, \\ R (VeryShort) = {R_{9}} . \end{matrix}$ Now, we have inference rules for our IFLC system. Let consider that we want to compute the output for the following input data x = (16, 86). Using the t-norm Product T (x, y) = xy, we compute for every rule R_i the values: the μ-firing level μ_i, the ν-firing level ν_i, the Atanassov’s intuitionistic fuzzy index π_i and the firing index α_i.

After this step, for each partition of rules we compute the matching value of the input data x using the formula (9) (see Table 4) and select the rules set having the maximum matching measure, R (Long).

For the set R (Long), the conclusions Long^μ and Long^ν are defuzzified into y^μ = 30.86302 and y^ν = 28.62719, respectively. Because the Atanassov’s intuitionistic fuzzy index for the set R (Long) is π = 0.05126, the conclusion inferred by the IFLC system is y = 30.74841. The response surfaces given by IFLC system is presented in Fig. 7. In order to evaluate the performance of the IFLC system we made a comparison with classical FLC system [29] for various inputs, which highlighted the performance of the proposed algorithm compared to the classic version. For sample inputs/outputs presented in the Table 5, the dispersions of the results are $D_{FLC}^{2} = 0.921554379$ for FLC system and $D_{IFLC}^{2} = 0.852100401$ for IFLC system. Because $D_{IFLC}^{2} < D_{FLC}^{2}$ it means that the Atanassov’s intuitionistic fuzzy system is more robust than the classical. Other statistics highlight this property. For instance, the amplitude (difference between extreme values) is 2.67875 for FLC system and 2.54143 for IFLC system; the coefficient of variation (standard deviation relative to the arithmetic mean) is 327.438368 * 10^-4 for FLC system and 314.863087 * 10^-4 for IFLC system.

5 Conclusion and future works

This paper presents a fuzzy controller model of Mamdani type working with Atanassov’s intuitionistic fuzzy sets; thus this controller is an extension of the model from [36],[29] and [28]. While the standard Mamdani controller activates a set of rules with different conclusions, our model activates a set of rules having the same conclusion; thus we obtain a fuzzy set as output, which can be defuzzified in order to obtain a crisp value. Moreover, the rules used by Fuzzy Inference Engine are generated using Data Mining techniques and taking into account the rule fuzzy confidence. In the future we intend to extend this version in order to work with crisp data, intervals and/or linguistic terms as inputs, and to work with Atanassov’s intuitionistic implication instead of dividing the Atanassov’s intuitionistic fuzzy rule into two fuzzy rules.

Acknowledgments

This work was partially supported by the strategic grant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007-2013.

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