**Meet the editor**

Prof Elmer P. Dadios finished his doctoral degree at Loughborough University, United Kingdom in 1996. He was a recipient of the Philippines Department of Science and Technology (DOST) 50 Men and Women of Science and Technology; DOST Scholar Achievers); The National Research Council of the Philippines Basic Research Achievement Award; The National Academy of Science

and Technology (NAST) Outstanding Scientific Paper Award; The De La Salle University Miguel Febres Cordero Research Award. Currently, Dr. Dadios is a University Fellow of the De La Salle University and holds the University's highest faculty rank of Full Professor 10. He is the president of the NEURONEMECH Inc.

He has been a consultant for Robotics and Automation in the Philippine government and private corporations. He is the founder of the IEEE Computational Intelligence Society - Philippines Chapter. He is the Founder and President of the Mechatronics and Robotics Society of the Philippines.

Contents

**Preface IX**

**Part 1 Hybrid Fuzzy Logic Algorithms 1**

**Fuzzy Set Model and Data Analysis 3**

Chapter 4 **Standard Fuzzy Sets and some Many-Valued Logics 75**

**System in Supply Chain Management Evaluation 115**

**Multivalued Knowledge-Base 25**

Chapter 3 **Resolution Principle and Fuzzy Logic 55**

Chapter 5 **Parametric Type-2 Fuzzy Logic Systems 97**  Arturo Tellez, Heron Molina, Luis Villa,

Chapter 6 **Application of Adaptive Neuro Fuzzy Inference**

**Algorithms in Wavelet Domain 127**  Heydy Castillejos and Volodymyr Ponomaryov

Chapter 8 **Fuzzy Logic Approach for QoS Routing Analysis 149**

**Part 2 Techniques and Implementation 147**

Adrian Shehu and Arianit Maraj

Elsa Rubio and Ildar Batyrshin

Chapter 1 **Ambiguity and Social Judgment:**

Kazuhisa Takemura

Chapter 2 **From Fuzzy Datalog to**

Agnes Achs

Hashim Habiballa

Jorma K. Mattila

Thoedtida Thipparat

Chapter 7 **Fuzzy Image Segmentation**

### Contents

### **Preface** XI

	- **Part 2 Techniques and Implementation 147**

### Preface

*Algorithm* is used to define the notion of decidability. It is a set of rules that precisely defines a sequence of operations. This is essential for computers to process information. Computer programs contain algorithms that detail specific instructions a computer should perform to carry out a specified task. The traditional computer program performs specific instructions sequentially, and uses crisp values of information which do not support uncertainties. Thus, when a problem is getting harder and becoming more complex, alternative algorithms are required in order to obtain accurate solutions. To this date, the quest of discovering new algorithms is in a race. The fuzzy logic algorithm is one of very strong contender s in this race because fuzzy logic exhibits reasoning power similar to how humans reason out. Fuzzy logic is able to process incomplete data and provide approximate solutions to problems other methods find difficult to solve. Fuzzy logic was first proposed by Lotfi A. Zadeh of the University of California at Berkeley in 1965. This is based on the idea that humans do not think in terms of crisp numbers, but rather in terms of concepts. The degree of membership of an object in a concept may be partial, with an object being partially related with many concepts. By characterizing the idea of partial membership in concepts, fuzzy logic is better able to convert natural language control strategies in a form usable by machines.

This book presents Algorithms, Techniques, and Implementations of fuzzy logic. It is categorized into two sections, namely:

In section one, there are seven chapters that focus on hybrid fuzzy logic algorithms and methodology:

	- Application of Adaptive Neuro Fuzzy Inference System in Supply Chain Management Evaluation
	- Fuzzy Image Segmentation Algorithms in Wavelet Domain

In section two, there are seven chapters that focus on fuzzy logic modeling and implementations, particularly:

The contributions to this book clearly reveal the fuzzy logic models, techniques, and implementation which are very important for the development of new technologies. I hope the readers of this book will find it a unique and significant source of knowledge and reference for the years to come.

> **Elmer P. Dadios**  University Fellow and Full Professor, Department of Manufacturing Engineering and Management, De La Salle University, Philippines

## **Part 1**

**Hybrid Fuzzy Logic Algorithms**

**1**

*Japan*

Kazuhisa Takemura *Waseda University,*

**Ambiguity and Social Judgment:**

**Fuzzy Set Model and Data Analysis**

Comparative judgment is essential in human social lives. Comparative judgment is a type of human judgment procedure, in which the evaluator is asked which alternative is preferred (e.g., "Do you prefer Brand A to Brand B?" or "How do you estimate the probability of choosing Brand A over Brand B when you compare the two brands? "). This type of judgment is distinguished from absolute judgment, in which the evaluator is asked to assess the attractiveness of an object (e.g., "How much do you like this brand on

The ambiguity of social judgment has been conceptualized by the fuzzy set theory. The fuzzy set theory provides a formal framework for the presentation of the ambiguity. Fuzzy sets were defined by Zadeh(1965) who also outlined how they could be used to characterize complex systems and decision processes ( Zadeh, 1973). Zadeh argues that the capacity of humans to manipulate fuzzy concepts should be viewed as a major asset, not a liability. The complexities in the real world often defy precise measurement and fuzzy logic defines concepts and its techniques provide a mathematical method able to deal with thought processes which are often too imprecise and ambiguous to deal with by classical

This chapter introduces a model of ambiguous comparative judgment (Takemura,2007) and provides a method of data analysis for the model, and then shows some examples of the data analysis of social judgments. Comparative judgments in social situations often involve ambiguity with regard to confidence, and people may be unable to make judgments without some confidence intervals. To measure the ambiguity (or vagueness) of human judgment, the fuzzy rating method has been proposed and developed (Hesketh, Pryor, Gleitzman, & Hesketh, 1988). In fuzzy rating, respondents select a representative rating point on a scale and indicate higher or lower rating points, depending on the relative ambiguity of their judgment. For example, fuzzy rating would be useful for perceived temperature, with the evaluator indicating a representative value and lower and upper values. This rating scale allows for asymmetries and overcomes the problem, identified by Smithson (1987), of researchers arbitrarily deciding the most representative value from a range of scores. By making certain simplifying assumptions (which is not uncommon in fuzzy set theory), the rating can be viewed as an L-R fuzzy number, thereby making the use of fuzzy set

**1. Introduction**

a scale of 0 to 100?").

mathematical techniques.

## **Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis**

Kazuhisa Takemura *Waseda University, Japan*

### **1. Introduction**

Comparative judgment is essential in human social lives. Comparative judgment is a type of human judgment procedure, in which the evaluator is asked which alternative is preferred (e.g., "Do you prefer Brand A to Brand B?" or "How do you estimate the probability of choosing Brand A over Brand B when you compare the two brands? "). This type of judgment is distinguished from absolute judgment, in which the evaluator is asked to assess the attractiveness of an object (e.g., "How much do you like this brand on a scale of 0 to 100?").

The ambiguity of social judgment has been conceptualized by the fuzzy set theory. The fuzzy set theory provides a formal framework for the presentation of the ambiguity. Fuzzy sets were defined by Zadeh(1965) who also outlined how they could be used to characterize complex systems and decision processes ( Zadeh, 1973). Zadeh argues that the capacity of humans to manipulate fuzzy concepts should be viewed as a major asset, not a liability. The complexities in the real world often defy precise measurement and fuzzy logic defines concepts and its techniques provide a mathematical method able to deal with thought processes which are often too imprecise and ambiguous to deal with by classical mathematical techniques.

This chapter introduces a model of ambiguous comparative judgment (Takemura,2007) and provides a method of data analysis for the model, and then shows some examples of the data analysis of social judgments. Comparative judgments in social situations often involve ambiguity with regard to confidence, and people may be unable to make judgments without some confidence intervals. To measure the ambiguity (or vagueness) of human judgment, the fuzzy rating method has been proposed and developed (Hesketh, Pryor, Gleitzman, & Hesketh, 1988). In fuzzy rating, respondents select a representative rating point on a scale and indicate higher or lower rating points, depending on the relative ambiguity of their judgment. For example, fuzzy rating would be useful for perceived temperature, with the evaluator indicating a representative value and lower and upper values. This rating scale allows for asymmetries and overcomes the problem, identified by Smithson (1987), of researchers arbitrarily deciding the most representative value from a range of scores. By making certain simplifying assumptions (which is not uncommon in fuzzy set theory), the rating can be viewed as an L-R fuzzy number, thereby making the use of fuzzy set

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 5

applied for the analysis of the former model, and a fuzzy logistic regression model

*X1 =* (*X11, X12,…,X1n*), *X2* = (*X21, X22,…,X2n*),…, *Xm =* (*Xm1, Xm2,…,Xmn*), where *Xij* (*i = 1.m; j = 1.,n*) is the value of alternative *Xi* on dimension *j.* Note that the components of *Xi* may be

The relational structure < X, > is a weak order if, and only if, for all *Xa, Xb, Xc*, the

However, the weak order relation is not always assumed in this paper. That is, transitivity

As a classical preference relation is a subset of X × X , is a classical set often viewed as a

Note that "iff" is short for "if and only if" and {0,1} is called the valuation set. If the valuation set is allowed to be the real interval [0,1], is called a fuzzy preference relation.

*µa*: X × X → [0,1].

Ambiguous preference relations are defined as a fuzzy set of *X* ×*X* × *S*, where *S* is a subset of one-dimensional real number space. *S* is interpreted as a domain of preference strength. *S*

:: X × X × *S* → [0,1].

1 iff X X 0 iff not(X X )

a b a b

β

is defined as:

 .

 *Xn* be a set of multidimensional alternatives with elements of the form

(Takemura, 2004) was proposed for the analysis of the latter model.

**2.2.1 Definition 1: Set of multidimensional alternatives**

ambiguous linguistic variables rather than crisp numbers.

Let be a binary relation on X, that is, is a subset of X × X*.*

1. Connectedness (Comparability): *Xa, Xb* or *Xb Xa,* 2. Transitivity: If *Xa Xb* and *Xb Xc*, then *Xa Xc.*

**2.2.3 Definition 3: Fuzzy preference relation**

characteristic function *c* from X × X to {0,1} such that:

That is, the membership function *µa* is defined as:

**2.2.4 Definition 4: Ambiguous preference relation**

or connectedness may be violated in the preference relations.

*c*(Xj Xk) =

may be bounded, for example, *S* = [0,1]. The membership function *µ*

*µ*β

**2.2.2 Definition 2: Classic preference relation**

**2.2 Assumptions of the model**

following two axioms are satisfied.

Let X *= X1*

× *X2* × *….* ×

theoretical operations possible (Hesketh et al., 1988; Takemura, 2000). Lastly, numerical illustrations of psychological experiments are provided to examine the ambiguous comparative judgment model (Takemura, 2007) using the proposed data analysis.

### **2. Model of ambiguous comparative judgment**

### **2.1 Overview of ambiguous comparative judgment and the judgment model**

Social psychological theory and research have demonstrated that comparative evaluation has a crucial role in the cognitive processes and structures that underlie people's judgments, decisions, and behaviors(e.g.,Mussweiler,2003). People comparison processes are almost ubiquitous in human social cognition. For example, people tend to compare their performance of others in situations that are ambiguous (Festinger,1954). It is also obvious that they are critical in forming personal evaluations, and making purchase decisions (Kühberger,,.Schulte-Mecklenbeck, & Ranyard, 2011; Takemura,2011).

The ambiguity or vagueness is inherent in people's comparative social judgment. Traditionally, psychological and philosophical theories implicitly had assumed the ambiguity of thought processes ( Smithson, 1987, 1989). For example, Wittgenstein (1953) pointed out that lay categories were better characterized by a " family resemblance" model which assumed vague boundaries of concepts rather than a classical set-theoretic model. Rosch (1975) and Rosch & Mervice(1975) also suggested vagueness of lay categories in her prototype model and reinterpret-ed the family resemblance model. Moreover, the social judgment theory (Sherif & Hovland,1961) and the information integration theory (Anderson,1988) for describing judgment and decision making assumed that people evaluate the objects using natural languages which were inherently ambiguous. However, psychological theories did not explicitly treat the ambiguity in social judgment with the exception of using random error of judgment.

Takemura (2007) proposed fuzzy set models that explain ambiguous comparative judgment in social situations. Because ambiguous comparative judgment may not always hold transitivity and comparability properties, the models assume parameters based on biased responses that may not hold transitivity and comparability properties. The models consist of two types of fuzzy set components for ambiguous comparative judgment. The first is a fuzzy theoretical extension of the additive difference model for preference, which is used to explain ambiguous preference strength and does not always assume judgment scale boundaries, such as a willing to pay (WTP) measure. The second type of model is a fuzzy logistic model of the additive difference preference, which is used to explain ambiguous preference in which preference strength is bounded, such as a probability measure (e.g., a certain interval within a bounded interval from 0 to 100%).

Because judgment of a bounded scale, such as a probability judgment, causes a methodological problem when fuzzy linear regression is used, a fuzzy logistic function to prevent this problem was proposed. In both models, multi-attribute weighting parameters and all attribute values are assumed to be asymmetric fuzzy L-R numbers. For each model, A method of parameter estimation using fuzzy regression analysis was proposed. That is, a fuzzy linear regression model using the least squares method (Takemura, 1999, 2005) was applied for the analysis of the former model, and a fuzzy logistic regression model (Takemura, 2004) was proposed for the analysis of the latter model.

### **2.2 Assumptions of the model**

4 Fuzzy Logic – Algorithms, Techniques and Implementations

theoretical operations possible (Hesketh et al., 1988; Takemura, 2000). Lastly, numerical illustrations of psychological experiments are provided to examine the ambiguous

Social psychological theory and research have demonstrated that comparative evaluation has a crucial role in the cognitive processes and structures that underlie people's judgments, decisions, and behaviors(e.g.,Mussweiler,2003). People comparison processes are almost ubiquitous in human social cognition. For example, people tend to compare their performance of others in situations that are ambiguous (Festinger,1954). It is also obvious that they are critical in forming personal evaluations, and making purchase decisions

The ambiguity or vagueness is inherent in people's comparative social judgment. Traditionally, psychological and philosophical theories implicitly had assumed the ambiguity of thought processes ( Smithson, 1987, 1989). For example, Wittgenstein (1953) pointed out that lay categories were better characterized by a " family resemblance" model which assumed vague boundaries of concepts rather than a classical set-theoretic model. Rosch (1975) and Rosch & Mervice(1975) also suggested vagueness of lay categories in her prototype model and reinterpret-ed the family resemblance model. Moreover, the social judgment theory (Sherif & Hovland,1961) and the information integration theory (Anderson,1988) for describing judgment and decision making assumed that people evaluate the objects using natural languages which were inherently ambiguous. However, psychological theories did not explicitly treat the ambiguity in social judgment with the

Takemura (2007) proposed fuzzy set models that explain ambiguous comparative judgment in social situations. Because ambiguous comparative judgment may not always hold transitivity and comparability properties, the models assume parameters based on biased responses that may not hold transitivity and comparability properties. The models consist of two types of fuzzy set components for ambiguous comparative judgment. The first is a fuzzy theoretical extension of the additive difference model for preference, which is used to explain ambiguous preference strength and does not always assume judgment scale boundaries, such as a willing to pay (WTP) measure. The second type of model is a fuzzy logistic model of the additive difference preference, which is used to explain ambiguous preference in which preference strength is bounded, such as a probability measure (e.g., a

Because judgment of a bounded scale, such as a probability judgment, causes a methodological problem when fuzzy linear regression is used, a fuzzy logistic function to prevent this problem was proposed. In both models, multi-attribute weighting parameters and all attribute values are assumed to be asymmetric fuzzy L-R numbers. For each model, A method of parameter estimation using fuzzy regression analysis was proposed. That is, a fuzzy linear regression model using the least squares method (Takemura, 1999, 2005) was

comparative judgment model (Takemura, 2007) using the proposed data analysis.

**2.1 Overview of ambiguous comparative judgment and the judgment model**

(Kühberger,,.Schulte-Mecklenbeck, & Ranyard, 2011; Takemura,2011).

**2. Model of ambiguous comparative judgment**

exception of using random error of judgment.

certain interval within a bounded interval from 0 to 100%).

### **2.2.1 Definition 1: Set of multidimensional alternatives**

Let X *= X1*× *X2* × *….* × *Xn* be a set of multidimensional alternatives with elements of the form *X1 =* (*X11, X12,…,X1n*), *X2* = (*X21, X22,…,X2n*),…, *Xm =* (*Xm1, Xm2,…,Xmn*), where *Xij* (*i = 1.m; j = 1.,n*) is the value of alternative *Xi* on dimension *j.* Note that the components of *Xi* may be ambiguous linguistic variables rather than crisp numbers.

### **2.2.2 Definition 2: Classic preference relation**

Let be a binary relation on X, that is, is a subset of X × X*.*

The relational structure < X, > is a weak order if, and only if, for all *Xa, Xb, Xc*, the following two axioms are satisfied.

However, the weak order relation is not always assumed in this paper. That is, transitivity or connectedness may be violated in the preference relations.

### **2.2.3 Definition 3: Fuzzy preference relation**

As a classical preference relation is a subset of X × X , is a classical set often viewed as a characteristic function *c* from X × X to {0,1} such that:

$$c(\mathsf{X}\_{\mathsf{j}} \succ \mathsf{X}\_{\mathsf{k}}) = \begin{cases} 1 & \text{iff} \\ 0 & \text{iff} \end{cases} \qquad \begin{array}{c} \mathsf{X}\_{\mathsf{a}} \succ \mathsf{X}\_{\mathsf{b}} \\ \text{not}(\mathsf{X}\_{\mathsf{a}} \succ \mathsf{X}\_{\mathsf{b}}) \cdot \mathsf{X}\_{\mathsf{c}} \end{array}$$

Note that "iff" is short for "if and only if" and {0,1} is called the valuation set. If the valuation set is allowed to be the real interval [0,1], is called a fuzzy preference relation. That is, the membership function *µa* is defined as:

$$
\mu\_a \colon \mathcal{X} \times \mathcal{X} \to [0, 1].
$$

### **2.2.4 Definition 4: Ambiguous preference relation**

Ambiguous preference relations are defined as a fuzzy set of *X* ×*X* × *S*, where *S* is a subset of one-dimensional real number space. *S* is interpreted as a domain of preference strength. *S* may be bounded, for example, *S* = [0,1]. The membership function *µ*βis defined as:

$$
\mu\_{\land} \dots \mathbb{X} \times \mathbb{X} \times \mathbb{S} \to [0, 1].
$$

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 7

*abn* ⊗ *(Xan*○-*-Xbn*

where *log*, ○÷ ,⊗ ,○+, and ○- are logarithmic, division, product , additive, and difference

The second model of the equation (2) is the model for [0,1]. However, the model could apply to not only the interval [0,1] but also any finite interval [a,b](a<b). Therefore, the model of

Non-comparability and intransitivity properties are explained if a threshold of comparative judgment is assumed, if intransitivity is indicated by the necessity measure of fuzzy comparative relation resulting from the existence of the threshold, and if a necessity

(1

difference model and the logistic regression model, respectively. Assuming the above relation of (3) or (4), it is clear that intransitivity and non-comparability hold in the

Traditional approaches to the measurement of social judgment have involved methods such as the semantic differential, the Likert scale, or the Thurstone scale. Although insights into the ambiguous nature of social judgment were identified early in the development of measurement of social judgment, the subsequent methods used failed to capture this ambiguity, no doubt because traditional mathematics was not well developed for dealing

In order to measure the vagueness of human judgment, the fuzzy rating method has recently been proposed and developed (Hesketh et al.,1988; Takemura,1996). In the fuzzy rating method, respondents select a representative rating point on a scale and indicate lower or upper rating points if they wish depending upon the relative vagueness of their judgment (see Figure 2). For example, the fuzzy rating method would be useful for measuring perceived temperature indicating the representative value and the lower or upper values. This rating scale allows for asymmetries, and overcomes the problem, identified by Smithson (1987), of researchers arbitrarily deciding most representative value from a range of scores. By making certain simplifying assumptions ( not uncommon within fuzzy set theory), the rating can be viewed as a L-R fuzzy number, hence making possible the use of

**3. Fuzzy data analysis for the ambiguous comparative judgment model**

*)* is a necessity measure, and *θ, and Pθ* are threshold parameters for the additive

*Xa Xb iff Nes ( v(Xa Xb)*

)(2)

>θ

○-*p(Xa Xb) )*

> *P*θ

*)* (3)

*)* (4)

 *1nXb1)ype correction and drawing figures.mments and*

)=

A

operations based on the extension principle for the fuzzy set, respectively.

the equation (2)is considered to be a special case for the finite interval model.

measure for fuzzy relation does not always lead to comparability. That is,

*Xa Xb iff Nes( p(Xa Xb)* ○÷

l

or

where *Nes (*

・

comparative judgment.

**3.1 Fuzzy rating data and fuzzy set**

fuzzy set theoretic operations).

with vagueness of judgment (Hesketh et al.,1988).

*og ( p(Xa Xb)* ○÷

*Aab0*○+ *Aab1* ⊗

(1

*Xa1*○-*Xb1*

(

○-*p(Xj Xk)*

)○+…○+

**2.2.7 Explaining non-comparability and intransitivity**

Ambiguous preference relation is interpreted as a fuzzified version of a classical characteristic function *c*(*Xa Xb*).

Therefore, the ambiguous preference relation for *Xa Xb* is represented as the fuzzy set *v(Xa Xb)*. For simplicity, *v(Xa Xb)* will be assumed to be an asymmetrical L-R fuzzy number (see Figure 1).

Fig. 1. Example of Ambiguous Preference Relation

### **2.2.5 Additive difference model of ambiguous comparative judgement**

The ambiguous preference relation *v(Xa Xb)* for *Xa Xb* is represented as the following additive difference model using L-R fuzzy numbers:

*v(Xa Xb)*= *Aab0*○+ *Aab1* ⊗ (*Xa1*○-*Xb1*)○+…○+Aabn ⊗ (*Xan*○-*-Xbn*)(1)

where ⊗ , ○+, and ○-are the product, additive, and difference operation based on the extension principle for the fuzzy set, respectively.

The parameter *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. The parameter *Ajk*0 would be a fuzzy variable and larger than *Aab0* if *Xa* were more salient than *Xb*. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes are assumed to be crisp numbers.

#### **2.2.6 Logistic model of ambiguous comparative judgement**

Let an ambiguous preference relation that is bounded (e.g., fuzzy probability in [0,1]) be *p*(*Xa Xb*) for *Xa Xb*. *p*(*Xa Xb*) and be represented as the following logistic model using L-R fuzzy numbers:

l*og ( p(Xa Xb)* ○÷ (1 ○-*p(Xj Xk)*)= *1nXb1)ype correction and drawing figures.mments and Aab0*○+ *Aab1* ⊗ (*Xa1*○-*Xb1*) ○+…○+A*abn* ⊗ *(Xan*○-*-Xbn*)(2)

where *log*, ○÷ ,⊗ ,○+, and ○- are logarithmic, division, product , additive, and difference operations based on the extension principle for the fuzzy set, respectively.

The second model of the equation (2) is the model for [0,1]. However, the model could apply to not only the interval [0,1] but also any finite interval [a,b](a<b). Therefore, the model of the equation (2)is considered to be a special case for the finite interval model.

### **2.2.7 Explaining non-comparability and intransitivity**

Non-comparability and intransitivity properties are explained if a threshold of comparative judgment is assumed, if intransitivity is indicated by the necessity measure of fuzzy comparative relation resulting from the existence of the threshold, and if a necessity measure for fuzzy relation does not always lead to comparability. That is,

$$\text{Xa} \succeq \text{Xb} \text{ } \text{iff} \text{ Nes} \left( v(\text{Xa} \succeq \text{Xb}) \ge \theta \right) \tag{3}$$

or

6 Fuzzy Logic – Algorithms, Techniques and Implementations

Ambiguous preference relation is interpreted as a fuzzified version of a classical

Therefore, the ambiguous preference relation for *Xa Xb* is represented as the fuzzy set *v(Xa Xb)*. For simplicity, *v(Xa Xb)* will be assumed to be an asymmetrical L-R fuzzy number

**Ambiguous preference relation**

∈*S*

**as fuzzy set :** *v(Xa Xb)*

characteristic function *c*(*Xa Xb*).

*μβ(Xa Xb)*

1

0

*Preference strength s*

**2.2.5 Additive difference model of ambiguous comparative judgement**

The ambiguous preference relation *v(Xa Xb)* for *Xa Xb* is represented as the following

*Xan*○-*-Xbn*

where ⊗ , ○+, and ○-are the product, additive, and difference operation based on the

The parameter *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. The parameter *Ajk*0 would be a fuzzy variable and larger than *Aab0* if *Xa* were more salient than *Xb*. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes

Let an ambiguous preference relation that is bounded (e.g., fuzzy probability in [0,1]) be *p*(*Xa Xb*) for *Xa Xb*. *p*(*Xa Xb*) and be represented as the following logistic model using L-

)(1)

Fig. 1. Example of Ambiguous Preference Relation

additive difference model using L-R fuzzy numbers:

)○+…○+Aabn ⊗ (

extension principle for the fuzzy set, respectively.

**2.2.6 Logistic model of ambiguous comparative judgement**

(see Figure 1).

*v(Xa Xb)*

*Aab0*○+ *Aab1* ⊗

=

(*Xa1*○-*Xb1*

are assumed to be crisp numbers.

R fuzzy numbers:

$$X\_a \succ X\_b \text{ iff } \text{Yes}(\begin{array}{c} p(X\_a \succ X\_b) \oplus \end{array} \begin{pmatrix} \text{ } I \stackrel{\frown}{\ominus} p(X\_a \succ X\_b) \end{pmatrix} \succeq P\_\theta) \tag{4}$$

where *Nes (*・*)* is a necessity measure, and *θ, and Pθ* are threshold parameters for the additive difference model and the logistic regression model, respectively. Assuming the above relation of (3) or (4), it is clear that intransitivity and non-comparability hold in the comparative judgment.

### **3. Fuzzy data analysis for the ambiguous comparative judgment model**

### **3.1 Fuzzy rating data and fuzzy set**

Traditional approaches to the measurement of social judgment have involved methods such as the semantic differential, the Likert scale, or the Thurstone scale. Although insights into the ambiguous nature of social judgment were identified early in the development of measurement of social judgment, the subsequent methods used failed to capture this ambiguity, no doubt because traditional mathematics was not well developed for dealing with vagueness of judgment (Hesketh et al.,1988).

In order to measure the vagueness of human judgment, the fuzzy rating method has recently been proposed and developed (Hesketh et al.,1988; Takemura,1996). In the fuzzy rating method, respondents select a representative rating point on a scale and indicate lower or upper rating points if they wish depending upon the relative vagueness of their judgment (see Figure 2). For example, the fuzzy rating method would be useful for measuring perceived temperature indicating the representative value and the lower or upper values. This rating scale allows for asymmetries, and overcomes the problem, identified by Smithson (1987), of researchers arbitrarily deciding most representative value from a range of scores. By making certain simplifying assumptions ( not uncommon within fuzzy set theory), the rating can be viewed as a L-R fuzzy number, hence making possible the use of fuzzy set theoretic operations).

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 9

Firstly, the convexity of the fuzzy subset is defined as follows: A fuzzy subset A ⊆ R is

Aα= {x| μA(x) ≧ α}, α∈[0,1],

Secondly, the normality of the fuzzy subset is defined as follows: A fuzzy subset A ⊆ R is

One of the most well known fuzzy numbers is the L-R fuzzy number (Dubois & Prade,1980).

where L((x - m)/u) is a increasing monotonic function, R((x - m)/v) is a decreasing

An example of the fuzzy rating scale and of the representation of the rating data using L-R fuzzy number are shown in Figure 3. Note in Figure 3 that representations of variables are

0 100

*M ij x*

*R ij x*

*x*

*R ij x*

Fuzzy rating data

*Xij* :

L-R fuzzy number (Membership function)

abbreviated as follows: *xijL* for *xi j (0lL*, *xijR* for *xi j (0l<sup>R</sup>* , *xijM* for *xi j (1lL* = *xi j (1lR*.

*L ij x*

Fig. 3. Fuzzy Rating Data and Its Representation by L-R Fuzzy Numbers

*M ij x*

*L ij x*

convex if and only if every ordinary

The L-R fuzzy number is defined as follows:

μA(x) = L((x - m)/u), - ∞ < x < m,

= R((x - m)/v), m < x < ∞,

monotonic function, u>0, and v>0.

μ

0.1

normal if and only if ∀x ∈R, max μA(x) = 1.

<sup>x</sup>

∀x ∈R:

= 1, x=m,

subset is convex( That is, in the case of a closed interval of R).

Fig. 2. Example of Fuzzy Rating

A fuzzy set A is defined as follows. Let X denote a universal set, such as X={x1,x2,....,xn}. Then, the membership function μA⊆X by which a fuzzy set A is defined has the form

$$\mathsf{u} \mathsf{A} \,\,\, :\, \mathsf{X} \to \{0, 1\} \,\, \,\, \mathsf{A}$$

where [0,1] denotes the interval of real numbers from 0 to 1, inclusive.

The concept of a fuzzy set is the foundation for analysis where fuzziness exists (Zadeh, 1965). a fuzzy set may be expressed as:

$$\begin{aligned} \mathbf{A} &= \mathfrak{\mu} \mathbf{A}(\mathbf{x}\_{\mathrm{l}}) / \chi\_{\mathbf{1}} \oplus \mathfrak{\mu} \mathbf{A}(\mathbf{x}\_{\mathrm{2}}) / \chi\_{\mathbf{2}} \oplus \dots \quad \dots \quad \oplus \mathfrak{\mu} \mathbf{A}(\mathbf{x}\_{\mathrm{n}}) / \chi\_{\mathbf{n}} \\ &= \mathfrak{\underline{\mathfrak{\mu}}} \quad \mathfrak{\mu} \mathbf{A}(\mathbf{x}\_{\mathrm{i}}) / \chi\_{\mathbf{i}} \end{aligned}$$

where μA(xi) represents the "grade of membership" of Xi in A, or the degree to which Xi satisfies the properties of the set A. It should be noted that here the symbol '"+ " does not refer to the ordinary addition.

μA is called a membership function, or a possibility function. The Xi values are drawn from a global set of all possible values, X. Grade of membership take values between 0 and 1. The membership function has a value of 0 when the properties of the fuzzy set are not at all satisfied, and 1 when the properties of fuzzy set are completely satisfied.

Hesketh et al.(1988) pointed out that fuzzy rating data can be represented as fuzzy sets by making certain implifying assumptions, which are not uncommon within fuzzy set theory. According to Hesketh et al.(1988), those assumptions are:

Making those assumptions, fuzzy rating data in this study can be expressed as a fuzzy number which is a kind of fuzzy set. The concept of the fuzzy number can be defined from the concept of the fuzzy subset(Kaufman & Gupta,1985). The properties of fuzzy numbers are the convexity and the normality of a fuzzy subset.

Firstly, the convexity of the fuzzy subset is defined as follows: A fuzzy subset A ⊆ R is convex if and only if every ordinary

$$A\_{\mathfrak{a}} = \{ \mathbf{x} \mid \mathfrak{p}A \left( \mathbf{x} \right) \ge \mathfrak{a} \}, \mathfrak{a} \in [0, 1],$$

subset is convex( That is, in the case of a closed interval of R).

Secondly, the normality of the fuzzy subset is defined as follows: A fuzzy subset A ⊆ R is normal if and only if

∀x ∈R, max μA(x) = 1.

<sup>x</sup>

One of the most well known fuzzy numbers is the L-R fuzzy number (Dubois & Prade,1980).

The L-R fuzzy number is defined as follows:

∀x ∈R:

8 Fuzzy Logic – Algorithms, Techniques and Implementations

A fuzzy set A is defined as follows. Let X denote a universal set, such as X={x1,x2,....,xn}.

The concept of a fuzzy set is the foundation for analysis where fuzziness exists (Zadeh,

where μA(xi) represents the "grade of membership" of Xi in A, or the degree to which Xi satisfies the properties of the set A. It should be noted that here the symbol '"+ " does not

μA is called a membership function, or a possibility function. The Xi values are drawn from a global set of all possible values, X. Grade of membership take values between 0 and 1. The membership function has a value of 0 when the properties of the fuzzy set are not at all

Hesketh et al.(1988) pointed out that fuzzy rating data can be represented as fuzzy sets by making certain implifying assumptions, which are not uncommon within fuzzy set theory.

3. The fuzzy membership function takes its maximum value, one, at the point on the

4. The extent of the fuzzy support is represented by the horizontal lines to either side

5. The fuzzy membership function tapers uniformly from its value of one at the representative point to a value of zero beyond the fuzzy support or the left and right

Making those assumptions, fuzzy rating data in this study can be expressed as a fuzzy number which is a kind of fuzzy set. The concept of the fuzzy number can be defined from the concept of the fuzzy subset(Kaufman & Gupta,1985). The properties of fuzzy numbers

extensions. The membership value of the lower point and the upper point is 0.

Then, the membership function μA⊆X by which a fuzzy set A is defined has the form

where [0,1] denotes the interval of real numbers from 0 to 1, inclusive.

satisfied, and 1 when the properties of fuzzy set are completely satisfied.

1) Low ambiguity

2)High ambiguity

A = μA(x1)/x1 ⊕ μA(x2)/x2 ⊕ ... ⊕ μA(xn)/xn

According to Hesketh et al.(1988), those assumptions are:

2. The global set X is represented along the horizontal axis.

fuzzy support represented by the representative point.

1. The fuzzy set has a convex membership function.

are the convexity and the normality of a fuzzy subset.

Fig. 2. Example of Fuzzy Rating

1965). a fuzzy set may be expressed as:

μA :X→[0, 1],

= Σ μA(xi)/xi,

refer to the ordinary addition.

of evaluated point.

n

i=1

$$\begin{cases} \mu \text{A(x)} = \text{L}((\text{x} - \text{m})/\text{u}), \text{ } \cdot \text{ } \infty \text{ } \text{x} \le \text{m}, \\\\ \text{ } = \text{1}, \text{ } \text{x} \text{=} \text{m}, \\\\ \text{ } = \text{R}((\text{x} - \text{m})/\text{v}), \text{ } \text{m} \le \text{x} \le \text{w}, \end{cases}$$

where L((x - m)/u) is a increasing monotonic function, R((x - m)/v) is a decreasing monotonic function, u>0, and v>0.

An example of the fuzzy rating scale and of the representation of the rating data using L-R fuzzy number are shown in Figure 3. Note in Figure 3 that representations of variables are abbreviated as follows: *xijL* for *xi j (0lL*, *xijR* for *xi j (0l<sup>R</sup>* , *xijM* for *xi j (1lL* = *xi j (1lR*.

Fig. 3. Fuzzy Rating Data and Its Representation by L-R Fuzzy Numbers

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 11

 = ( ) *R k x*0 α

fuzzy coefficient for the *j*-th attribute and the α-level set of fuzzy input data *Xjk*, () ()

j jk a x α α is defined in the same manner, respectively. ( )

assumed to be 1 (a crisp number) for the purpose of estimation for the fuzzy bias parameter

To define the dissimilarity between the predicted and observed values of the dependent

L 2 <sup>k</sup> z ) <sup>α</sup> +( ( ) R <sup>k</sup> y <sup>α</sup> - ( )

The definition in Equation (12) can be applied to interval data as well as to L-R fuzzy numbers. That is, Equation (12) represents the sum of squares for the distance between

To generalize, a dissimilarity indicator representing the square of the distance for L-R fuzzy

In the case of a triangular fuzzy number with *wj =* 1, the above equation is approximately

The proposed method is to estimate fuzzy coefficients using minimization of the sum of *Dk*

R 2 k 1 z ) +( ( ) R k 0 y - ( )

<sup>N</sup> <sup>2</sup> k k 1 Min D =

L 2 k j z ) <sup>α</sup> +( ( ) R <sup>k</sup> <sup>j</sup> y <sup>α</sup> - ( )

=1 (11)

L

<sup>k</sup> z ) <sup>α</sup> (12)

k j z ) <sup>α</sup> ) (13)

k 0 z ) (14)

*2*

0k x <sup>α</sup> and ( )

L R j jk a x α α ,

R 0k x <sup>α</sup> are

j jk a x α α is a product between the lower value of the α-level

R 2

R 2

R 2

(15)

j(h) <sup>1</sup> a 0, ≥ ∈j J (16)

j(h) j(h) <sup>2</sup> a 0, a 0, ≤ ≥∈j J (17)

j(h) <sup>3</sup> a 0, ≤ ∈j J (18)

j(h) j(h) −+ ≥ aa0 (19)

( ) *L k x*0 α

L <sup>k</sup> y <sup>α</sup> - ( )

L L

variable, the following indicator *Dk* ( ) <sup>α</sup> *<sup>2</sup>*was adopted:

n

j=0

L k 0 y - ( )

 wj(( ( ) L <sup>k</sup> <sup>j</sup> y <sup>α</sup> - ( )

> L 2 k 0 z ) +( ( ) L k 1 y - ( )

Objective function:

Subject to: <sup>L</sup>

L R

R

L R

where α*j = jh/n, j = 0,...,n*, *h* is an equal interval, and *wj* is a weight for the *j*-th level.

 *Dk* ( ) <sup>α</sup> *<sup>2</sup>*=( ( )

numbers can be written as follows:

Dk2=

Dk2 =( ( )

In the above Equation (9), () ()

R R

() () R L

interval data.

represented as:

respecting *k*. That is,

*A0.*

j jk a x α α , or () ()

### **3.2 Analysis of the additive difference type model**

The set of fuzzy input-output data for the *k*-th observation is defined as:

$$\left(\mathbf{Y}\_{\text{abk}}; \mathbf{X}\_{\text{a1k}'} \; \mathbf{X}\_{\text{a2k}'}..., \mathbf{X}\_{\text{ank}} \; \mathbf{X}\_{\text{b1k}'} \mathbf{X}\_{\text{b2k}'}..., \mathbf{X}\_{\text{bnk};}\right) \tag{5}$$

where *Yabk* indicates the *k*-th observation's ambiguous preference for the *a-*th alternative (a) over the *b-*th alternative (b), which represented by fuzzy L-R numbers, and *Xajk* and *Xbjk* are the *j*-th attribute values of the alternatives (a and b) for observation *k*.

Let *Xabjk* be *Xajk* - *Xbjk*, where - is a difference operator based on the fuzzy extension principle, and denote *Xk.* as the abbreviation of *Xabk* in the following section. Therefore, a set of fuzzy input-output data for the *i*-th observation is re-written as:

$$(\text{\textquotedblleft}\_{\text{k}}; \text{\textquotedblright}\_{\text{1k}}, \text{\textquotedblleft}\_{\text{2k}}, \dots, \text{\textquotedblleft}\_{\text{nk}}\text{\textquotedblright}\_{\text{\textquotedblleft}}), k = 1, 2, \dots, N\text{\textquotedblright}\tag{6}$$

where *Yk* is a fuzzy dependent variable, and *Xjk* is a fuzzy independent variable represented by L-R fuzzy numbers. For simplicity, assume that *Yk* and *Xjk* are positive for any membership value, α ∈ (0,1).

The fuzzy linear regression model (where both input and output data are fuzzy numbers) is represented as follows:

$$\overline{\mathbf{Y}}\_{\mathbf{k}} = \mathbf{A}\_0 \oplus \mathbf{A}\_1 \otimes \mathbf{X}\_{1\mathbf{k}} \oplus \dots \oplus \mathbf{A}\_n \otimes \mathbf{X}\_{n\mathbf{k}} \tag{7}$$

where is a fuzzy estimated variable, A*j*(*j = 1,…,n*) is a fuzzy regression parameter represented by an L-R fuzzy number, ⊗ is an additive operator, and ⊕ is the product operator based on the extension principle.

It should be noted that although the explicit form of the membership function of Yk cannot be directly obtained, the α-level set of Yk can be obtained from Nguyen's theorem (Nguyen, 1978).

Let ( ) L <sup>k</sup> z <sup>α</sup> be a lower value of Yk , and ( ) R <sup>k</sup> z <sup>α</sup> be an upper value of Yk .

Then,

$$\mathbf{Z}\_{\mathbf{k}} = \left[ \mathbf{z}\_{\mathbf{k}(a)}^{\mathrm{L}}, \mathbf{z}\_{\mathbf{k}(a)}^{\mathrm{R}} \right], \quad \alpha \in \{0, 1\} \tag{8}$$

Where

$$\mathbf{z}\_{\mathbf{k}(\alpha)}^{\mathcal{L}} = \sum\_{\mathbf{j}=0}^{n} \left[ \min \left( \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathcal{L}} \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathcal{R}} \right) \right] \tag{9}$$

$$\mathbf{z}\_{\mathbf{k}(\alpha)}^{\mathbb{R}} = \sum\_{\mathbf{j}=0}^{n} \left[ \max \left( \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathbb{R}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathbb{L}} \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathbb{R}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathbb{R}} \right) \right] \tag{10}$$

10 Fuzzy Logic – Algorithms, Techniques and Implementations

where *Yabk* indicates the *k*-th observation's ambiguous preference for the *a-*th alternative (a) over the *b-*th alternative (b), which represented by fuzzy L-R numbers, and *Xajk* and *Xbjk* are

Let *Xabjk* be *Xajk* - *Xbjk*, where - is a difference operator based on the fuzzy extension principle, and denote *Xk.* as the abbreviation of *Xabk* in the following section. Therefore, a set

where *Yk* is a fuzzy dependent variable, and *Xjk* is a fuzzy independent variable represented by L-R fuzzy numbers. For simplicity, assume that *Yk* and *Xjk* are positive for any

The fuzzy linear regression model (where both input and output data are fuzzy numbers) is

where is a fuzzy estimated variable, A*j*(*j = 1,…,n*) is a fuzzy regression parameter represented by an L-R fuzzy number, ⊗ is an additive operator, and ⊕ is the product

It should be noted that although the explicit form of the membership function of Yk cannot be directly obtained, the α-level set of Yk can be obtained from Nguyen's theorem (Nguyen,

> ( ) { ( ) () () () () } <sup>n</sup> <sup>L</sup> LL LR k j jk j jk

> ( ) { ( ) () () () () } <sup>n</sup> <sup>R</sup> RL RR k j jk j jk

z min a x ,a x α α <sup>α</sup> <sup>α</sup> <sup>α</sup>

z max a x ,a x <sup>α</sup> αααα

R

j 0

=

j 0

=

( ) Y ;X , X , ,X X ,X , ,X abk a1k a2k ank; b1k b2 k bnk; (5)

( ) Y ;X ,X , ,X k 1k 2k nk , *k=1,2,….,N* (6)

YAAX A X <sup>k</sup> = ⊕ ⊗ ⊕⊕ ⊗ 0 1 1k n nk (7)

<sup>k</sup> z <sup>α</sup> be an upper value of Yk .

() () ( ] L R Z z ,z , 0,1 <sup>k</sup> k k α α <sup>=</sup> α ∈ (8)

<sup>=</sup> (9)

<sup>=</sup> (10)

**3.2 Analysis of the additive difference type model**

membership value, α ∈ (0,1).

operator based on the extension principle.

<sup>k</sup> z <sup>α</sup> be a lower value of Yk , and ( )

represented as follows:

1978).

Then,

Where

Let ( ) L

The set of fuzzy input-output data for the *k*-th observation is defined as:

the *j*-th attribute values of the alternatives (a and b) for observation *k*.

of fuzzy input-output data for the *i*-th observation is re-written as:

$$\mathfrak{X}\_{0k(a)}^L = \mathfrak{X}\_{0k(a)}^\mathbb{R} = 1 \tag{11}$$

In the above Equation (9), () () L L j jk a x α α is a product between the lower value of the α-level fuzzy coefficient for the *j*-th attribute and the α-level set of fuzzy input data *Xjk*, () () L R j jk a x α α , () () R L j jk a x α α , or () () R R j jk a x α α is defined in the same manner, respectively. ( ) L 0k x <sup>α</sup> and ( ) R 0k x <sup>α</sup> are assumed to be 1 (a crisp number) for the purpose of estimation for the fuzzy bias parameter *A0.*

To define the dissimilarity between the predicted and observed values of the dependent variable, the following indicator *Dk* ( ) <sup>α</sup> *<sup>2</sup>*was adopted:

$$D\_{\mathbf{k}}\ (a) \ \mathbf{\color{red}{2}} = (\mathbf{y}\_{\mathbf{k}\langle a\rangle}^{\mathbf{L}} \cdot \mathbf{z}\_{\mathbf{k}\langle a\rangle}^{\mathbf{L}})^2 + (\mathbf{y}\_{\mathbf{k}\langle a\rangle}^{\mathbf{R}} \cdot \mathbf{z}\_{\mathbf{k}\langle a\rangle}^{\mathbf{R}})^2 \tag{12}$$

The definition in Equation (12) can be applied to interval data as well as to L-R fuzzy numbers. That is, Equation (12) represents the sum of squares for the distance between interval data.

To generalize, a dissimilarity indicator representing the square of the distance for L-R fuzzy numbers can be written as follows:

$$\mathbf{Dk}^2 = \sum\_{\mathbf{j}=0}^n \mathbf{w} \mathbf{j} ((\mathbf{y}\_{\mathbf{k}(\neq \mathbf{j})}^\mathcal{L} \mathbf{z}\_{\mathbf{k}(\neq \mathbf{j})}^\mathcal{L})^2 + (\mathbf{y}\_{\mathbf{k}(\neq \mathbf{j})}^\mathcal{R} \mathbf{z}\_{\mathbf{k}(\neq \mathbf{j})}^\mathcal{R})^2) \tag{13}$$

where α*j = jh/n, j = 0,...,n*, *h* is an equal interval, and *wj* is a weight for the *j*-th level.

In the case of a triangular fuzzy number with *wj =* 1, the above equation is approximately represented as:

$$\mathbf{Dk}^2 = (\mathbf{y}\_{\mathbf{k}(0)}^\mathcal{L} - \mathbf{z}\_{\mathbf{k}(0)}^\mathcal{L})^2 + (\mathbf{y}\_{\mathbf{k}(1)}^\mathcal{L} - \mathbf{z}\_{\mathbf{k}(1)}^\mathcal{R})^2 + (\mathbf{y}\_{\mathbf{k}(0)}^\mathcal{R} - \mathbf{z}\_{\mathbf{k}(0)}^\mathcal{R})^2 \tag{14}$$

The proposed method is to estimate fuzzy coefficients using minimization of the sum of *Dk 2* respecting *k*. That is,

$$\text{Objective function: Min } \sum\_{\mathbf{k}=1}^{N} \mathbf{D}\_{\mathbf{k}} \stackrel{\ast}{}^{\mathbf{2}} \tag{15}$$

$$\text{Subject to: } \operatorname{a}^{\mathcal{L}}\_{j(h)} \ge 0, \text{ j} \in \mathcal{J}\_1 \tag{16}$$

$$\mathbf{a}\_{\mathbf{j}(\mathbf{h})}^{\mathcal{L}} \le \mathbf{0}, \; \mathbf{a}\_{\mathbf{j}(\mathbf{h})}^{\mathcal{R}} \ge \mathbf{0}, \; \mathbf{j} \in \mathcal{J}\_2 \tag{17}$$

$$\mathbf{a}\_{j(h)}^{\mathbb{R}} \le \mathbf{0}, \ j \in \mathbb{J}\_3 \tag{18}$$

$$-\mathbf{a}\_{j(h)}^{\mathcal{L}} + \mathbf{a}\_{j(h)}^{\mathcal{R}} \ge 0 \tag{19}$$

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 13

highest value (such as 1). The present study develops the concept of logistic regression for the crisp numbers, and then proposes the fuzzy version of logistic regression analysis for

where *Pabk* indicates the *k*-th observation's ambiguous preference for the *a-*th alternative (a) over the *b-*th alternative (b), which is represented by fuzzy L-R numbers, and *Xajk* and *Xbjk*

Let *Xabjk* be *Xajk* ○- *Xbjk*, where ○- is a difference operator based on the fuzzy extension principle, and denote *Xk.* as the abbreviation of *Xabk* in the following section. Therefore, a set

where *Pk* is a fuzzy dependent variable, and *Xjk* is a fuzzy independent variable represented by L-R fuzzy numbers. For simplicity, I assume that *Pk* and *Xjk* are positive for any

The fuzzy logic regression model (where both input and output data are fuzzy numbers) is

 *log*(*Pk* ○÷ (1 ○-*Pk*)) = ⊗ ⊕ ⊗ ⊕⊕ ⊗ AX AX A X 0 i0 1 i1 m im (24)

where *log*(*Pk*○÷ (1○-*Pk*)) is the estimated fuzzy log odds, ○÷ is the division operator, ○- is the difference operator, ⊗ is the product operator, and ⊕ is the additive operator based on the

It should be noted that although the explicit form of the membership function of *log*(*Pk*○÷ (1○-*Pk*)) cannot be directly obtained, the α -level set of *log*(*Pk*○÷ (1○-*Pk*)) can be

bound. Then, the α level set of the fuzzy dependent variable *Pk* can be represented as

[*log*(*Pk*○÷ (1○-*Pk*))]<sup>α</sup><sup>=</sup>

() () () () LL RR kk kk [min(log(P /(1 P )),log(P /(1 P ))) αα αα − −

() () () ()

LL RR max(lo kk kk g(P /(1 P )),log(P /(1 P )))] αα αα − − (25)

kP <sup>α</sup> be the lower bound of the dependent fuzzy variable and ( )

Therefore, the α level set of the left term in Equation (24) is as follows:

( ) P ;X ,X , ,X ;X ,X , ,X abk a1k a2k ank b1k b2k bnk (22)

( ) P ;X ,X , ,X k 1k 2k nk , k=1,2,….,N (23)

R

kP <sup>α</sup> be the upper

The set of fuzzy input-output data for the *k-*th observation is defined as:

are the *j*-th attribute values of the alternatives (a and b) for observation *k*.

of fuzzy input-output data for the *i*-th observation is re-written as:

fuzzy input and output data.

membership value, α ∈ (0,1).

() () <sup>α</sup> ( ] L R <sup>k</sup> k k P P ,P , 0,1 α α = α <sup>∈</sup> .

extension principle for the fuzzy set, respectively.

obtained using Nguyen's theorem (Nguyen, 1978).

represented as follows:

Let ( ) L

Where { } <sup>123</sup> j∈ =∪∪ 0,....,n J J J , 12 23 31 J J ,J J ,J J , ∩ = ϕ ∩ = ϕ ∩ = ϕ

$$\mathbf{z}\_{\mathbf{k}}^{\mathcal{L}}(a) = \sum\_{\mathbf{j}\_a \mathbf{l}\_1} \mathbf{a}\_{\mathbf{j}(a)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(a)}^{\mathcal{L}} + \sum\_{\mathbf{j}\_a \mathbf{l}\_2 \mathbf{l}\_3} \mathbf{a}\_{\mathbf{j}(a)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(a)}^{\mathcal{R}} \tag{20}$$

$$\mathbf{z}\_{\mathbf{k}}^{\mathbb{R}}(\boldsymbol{\alpha}) = \sum\_{\mathbf{j}\_{\mathbf{k}} \mathbf{J}\_{1} \mathbf{J}\_{12}} \mathbf{a}\_{\mathbf{j}(\boldsymbol{\alpha})}^{\mathbb{R}} \mathbf{x}\_{\mathbf{jk}(\boldsymbol{\alpha})}^{\mathbb{R}} + \sum\_{\mathbf{j}\_{\mathbf{k}} \mathbf{J}\_{3}} \mathbf{a}\_{\mathbf{j}(\boldsymbol{\alpha})}^{\mathbb{R}} \mathbf{x}\_{\mathbf{jk}(\boldsymbol{\alpha})}^{\mathbb{L}} \tag{21}$$

The estimated coefficients can be derived through quadratic programming. The proposed fuzzy least squares method is also shown in Figure 4.

Fig. 4. Fuzzy Least Squares Regressions Analysis for Fuzzy Input and Output Data

### **3.3 Analysis of the logistic type model**

Although the fuzzy linear regression analysis in the fuzzy additive difference model can give satisfactory results, these fuzzy regression analyses may fail to interpret psychological judgment data that have bounds on a psychological scale. For example, a perceived purchase probability has [0,1] interval and cannot be greater than 1 or less than 0. For such data, these fuzzy regression analyses may predict the values that are greater than 1 or less than 0. It may happen that the predicted values are greater than the highest bound or less than the lowest bound, and this causes a significant problem if the predicted values are used in a subsequent analysis. Therefore, the present study also attempted to solve this problem by setting predicted values to be greater than the lowest value (such as 0) or less than the

12 Fuzzy Logic – Algorithms, Techniques and Implementations

() () () () () 1 2 3 L LL LR k j jk j jk j J jJJ z ax ax ∈ ∈

( ) () () () () 1, 12 3 R R R R L k j jk j jk jJ J j J z ax ax ∈ ∈

The estimated coefficients can be derived through quadratic programming. The proposed

*z <sup>L</sup> z*

*j* +

*M y*

*M z*

*L y*

*n*

*j*

11

<sup>2</sup>

Fig. 4. Fuzzy Least Squares Regressions Analysis for Fuzzy Input and Output Data

Although the fuzzy linear regression analysis in the fuzzy additive difference model can give satisfactory results, these fuzzy regression analyses may fail to interpret psychological judgment data that have bounds on a psychological scale. For example, a perceived purchase probability has [0,1] interval and cannot be greater than 1 or less than 0. For such data, these fuzzy regression analyses may predict the values that are greater than 1 or less than 0. It may happen that the predicted values are greater than the highest bound or less than the lowest bound, and this causes a significant problem if the predicted values are used in a subsequent analysis. Therefore, the present study also attempted to solve this problem by setting predicted values to be greater than the lowest value (such as 0) or less than the

==

*N*

*k*

α αα αα = + (20)

α αα αα = + (21)

μ

*R*

*<sup>j</sup>* + <sup>2</sup>

*j*

*Y k*

*R y*

2

Where { } <sup>123</sup> j∈ =∪∪ 0,....,n J J J , 12 23 31 J J ,J J ,J J , ∩ = ϕ ∩ = ϕ ∩ = ϕ

fuzzy least squares method is also shown in Figure 4.

0.1

→

*Min*

**3.3 Analysis of the logistic type model**

highest value (such as 1). The present study develops the concept of logistic regression for the crisp numbers, and then proposes the fuzzy version of logistic regression analysis for fuzzy input and output data.

The set of fuzzy input-output data for the *k-*th observation is defined as:

$$\left(\mathbf{P}\_{\rm abk}; \mathbf{X}\_{\rm a1k}, \mathbf{X}\_{\rm a2k}, \dots; \mathbf{X}\_{\rm ank}; \mathbf{X}\_{\rm b1k}, \mathbf{X}\_{\rm b2k}, \dots; \mathbf{X}\_{\rm bnk}\right) \tag{22}$$

where *Pabk* indicates the *k*-th observation's ambiguous preference for the *a-*th alternative (a) over the *b-*th alternative (b), which is represented by fuzzy L-R numbers, and *Xajk* and *Xbjk* are the *j*-th attribute values of the alternatives (a and b) for observation *k*.

Let *Xabjk* be *Xajk* ○- *Xbjk*, where ○- is a difference operator based on the fuzzy extension principle, and denote *Xk.* as the abbreviation of *Xabk* in the following section. Therefore, a set of fuzzy input-output data for the *i*-th observation is re-written as:

$$\left(\mathbb{P}\_{\mathbf{k}}; \mathbb{X}\_{1\mathbf{k}}, \mathbb{X}\_{2\mathbf{k}}, \dots, \mathbb{X}\_{n\mathbf{k}}\right), \mathbf{k} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{N} \tag{23}$$

where *Pk* is a fuzzy dependent variable, and *Xjk* is a fuzzy independent variable represented by L-R fuzzy numbers. For simplicity, I assume that *Pk* and *Xjk* are positive for any membership value, α ∈ (0,1).

The fuzzy logic regression model (where both input and output data are fuzzy numbers) is represented as follows:

$$\overline{\log(P\_k \oplus (1 \odot P\_k))} = \mathbf{A}\_0 \oplus \mathbf{X}\_{i0} \oplus \mathbf{A}\_1 \otimes \mathbf{X}\_{i1} \oplus \dots \oplus \mathbf{A}\_{\mathfrak{m}} \otimes \mathbf{X}\_{\text{im}} \tag{24}$$

where *log*(*Pk*○÷ (1○-*Pk*)) is the estimated fuzzy log odds, ○÷ is the division operator, ○- is the difference operator, ⊗ is the product operator, and ⊕ is the additive operator based on the extension principle for the fuzzy set, respectively.

It should be noted that although the explicit form of the membership function of *log*(*Pk*○÷ (1○-*Pk*)) cannot be directly obtained, the α -level set of *log*(*Pk*○÷ (1○-*Pk*)) can be obtained using Nguyen's theorem (Nguyen, 1978).

Let ( ) L kP <sup>α</sup> be the lower bound of the dependent fuzzy variable and ( ) R kP <sup>α</sup> be the upper bound. Then, the α level set of the fuzzy dependent variable *Pk* can be represented as () () <sup>α</sup> ( ] L R <sup>k</sup> k k P P ,P , 0,1 α α = α <sup>∈</sup> .

Therefore, the α level set of the left term in Equation (24) is as follows:

$$[\log(P\_k \ominus)(1 \ominus P\_k)]\_{\circ} -$$

$$\begin{aligned} \text{[min}(\overline{\log(\mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{L}} / (1 - \mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{L}}))}) & \overline{\log(\mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{R}} / (1 - \mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{R}}))}) \\\\ \max(\overline{\log(\mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{L}} / (1 - \mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{L}}))}, \overline{\log(\mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{R}} / (1 - \mathcal{P}\_{\mathbf{k}(a)}^{\mathcal{R}}))})) \end{aligned} \tag{25}$$

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 15

The participant also rated the desirability of the attribute information for each computer using a fuzzy rating method. The fuzzy rating scale of desirability ranged from 0 point to 100 points. (see Figure 6). That is, the participant answered the lower value, the

0 100

100

The fuzzy coefficients were obtained by fuzzy linear regression analysis using the least squares under constraints, as shown in Tables 1 and 2. The dependent variable of Table 1 was the same as that in Table 2. However, the independent variables in Table 1 are objective values measured by crisp numbers, whereas in Table 2 the independent variables are fuzzy rating values measured by an L-R fuzzy number. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. The parameter *Ajk*0 would be a fuzzy variable and larger than *Aab0* if *Xa* were more salient than *Xb*. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes are assumed to be crisp numbers as explained before. According to Tables 1 and 2, the preference strength concerning comparative judgment was influenced most by whether the target computer was new or used. The impact of the hard disks' attributes was smaller than that of the new-used dimension.

The participant was a 43-year-old adult. The participant rated the ambiguous probability of preferring a certain computer (DELL brand) out of seven different computers. Three types of attribute information (hard disk: 100 or 60 GB; memory: 2.80 or 2.40 GHz; new or used product) were manipulated in the same manner as in the previous judgment task.. That is, the participant answered the lower value, the representative value , and upper value for the probability that superior alternative is preferred to inferior alternative. The participant used the fuzzy rating method to provide representative, lower, and upper values of probabilities

representative value , and upper value for each attribute value.

Fig. 6. Example of a Fuzzy Desirability Rating.

1) Low ambiguity

2)High ambiguity 0

**<IP_ADDRESS> Analysis and results**

**4.1.2 Example of the logistic model <IP_ADDRESS> Participant and procedure**

(see Figure 7 ).

Let ( ) L <sup>k</sup> z <sup>α</sup> be a lower value of [*log*(*Pk*○÷ (1○-*Pk*))]α, and ( ) R <sup>k</sup> z <sup>α</sup> be an upper value of [*log*(*Pk*○÷ (1○-*Pk*))]<sup>α</sup>

where

$$\mathbf{z}\_{\mathbf{k}(\alpha)}^{\mathcal{L}} = \sum\_{\mathbf{j}=0}^{n} \left[ \min \left( \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathcal{L}} \mathbf{a}\_{\mathbf{j}(\alpha)}^{\mathcal{L}} \mathbf{x}\_{\mathbf{jk}(\alpha)}^{\mathcal{R}} \right) \right] \tag{26}$$

$$\mathbf{z}\_{\mathbf{k}\left(\alpha\right)}^{\mathrm{R}} = \sum\_{\mathbf{j}=0}^{\mathrm{n}} \left[ \max \left( \mathbf{a}\_{\mathbf{j}\left(\alpha\right)}^{\mathrm{R}} \mathbf{x}\_{\mathbf{jk}\left(\alpha\right)}^{\mathrm{L}} \mathbf{a}\_{\mathbf{j}\left(\alpha\right)}^{\mathrm{R}} \mathbf{x}\_{\mathbf{jk}\left(\alpha\right)}^{\mathrm{R}} \right) \right] \tag{27}$$

$$\mathcal{X}\_{0k}^{L}(\alpha) = \mathcal{X}\_{0k}^{R}(\alpha) = \mathbf{1} \tag{28}$$

In the above Equation (26), is a product between the lower value of the �-level fuzzy coefficient for the *j*-th attribute and the α-level set of fuzzy input data *Xjk*, , or is defined in the same manner, respectively. and are assumed to be 1 (a crisp number) for the purpose of estimation for the fuzzy bias parameter *A0*. The parameter estimation method is basically the same as the fuzzy logistic regression method and a more concrete procedure is described in Takemura (2004).

### **4. Numerical example of the data analysis method**

To demonstrate the appropriateness of the proposed data analysis methods, the detail numerical examples are shown for the individual level analysis (Takemura,2007) and group level analysis (Takemura, Matsumoto, Matsuyama, & Kobayashi, 2011) of ambiguous comparative judgments.

### **4.1. Individual level analysis of ambiguous comparative model**

### **4.1.1 Example of additive difference model**

### **<IP_ADDRESS> Participant and procedure**

The participant was a 43-year-old faculty member of Waseda University. The participant rated differences in WTP for two different computers (DELL brand) with three types of attribute information (hard disk: 100 or 60 GB; memory: 2.80 or 2.40 GHz; new or used product). The participant compared a certain alternative with seven different alternatives. The participant provided representative values and lower and upper WTP values using a fuzzy rating method. (see Figure 5)

The participant was asked the amount of money he would be willing to pay to upgrade the inferior from inferior alternative to superior alternative using fuzzy rating method. That is, the participant answered the lower value, the representative value, and upper value for the amount of money he would be willing to pay.

$$\begin{array}{cccc} \text{Lower Value} & \text{Representative Value} & \text{Upper Value} \\\\ \text{(} & ) \text{ Yen} & \text{(} & ) \text{ Yen} & \text{(} & ) \text{ Yen} \end{array}$$

Fig. 5. Example of a Fuzzy Rating in WTP Task.

The participant also rated the desirability of the attribute information for each computer using a fuzzy rating method. The fuzzy rating scale of desirability ranged from 0 point to 100 points. (see Figure 6). That is, the participant answered the lower value, the representative value , and upper value for each attribute value.

Fig. 6. Example of a Fuzzy Desirability Rating.

### **<IP_ADDRESS> Analysis and results**

14 Fuzzy Logic – Algorithms, Techniques and Implementations

( ) { ( ) () () () () } <sup>n</sup> <sup>L</sup> LL LR k j jk j jk

( ) { ( ) () () () () } <sup>n</sup> <sup>R</sup> RL RR k j jk j jk

In the above Equation (26), is a product between the lower value of the �-level fuzzy coefficient for the *j*-th attribute and the α-level set of fuzzy input data *Xjk*, , or is defined in the same manner, respectively. and are assumed to be 1 (a crisp number) for the purpose of estimation for the fuzzy bias parameter *A0*. The parameter estimation method is basically the same as the fuzzy logistic regression method and a more concrete procedure is described in

To demonstrate the appropriateness of the proposed data analysis methods, the detail numerical examples are shown for the individual level analysis (Takemura,2007) and group level analysis (Takemura, Matsumoto, Matsuyama, & Kobayashi, 2011) of ambiguous

The participant was a 43-year-old faculty member of Waseda University. The participant rated differences in WTP for two different computers (DELL brand) with three types of attribute information (hard disk: 100 or 60 GB; memory: 2.80 or 2.40 GHz; new or used product). The participant compared a certain alternative with seven different alternatives. The participant provided representative values and lower and upper WTP values using a

The participant was asked the amount of money he would be willing to pay to upgrade the inferior from inferior alternative to superior alternative using fuzzy rating method. That is, the participant answered the lower value, the representative value, and upper value for the

> Lower Value Representative Value Upper Value ( ) Yen ( ) Yen ( ) Yen

z min a x ,a x α α <sup>α</sup> <sup>α</sup> <sup>α</sup>

z max a x ,a x <sup>α</sup> αααα

( ) *L k x*0 α = ( ) *R k x*0 α

R

<sup>=</sup> (26)

<sup>=</sup> (27)

<sup>k</sup> z <sup>α</sup> be an upper value of [*log*(*Pk*○÷

=1 (28)

<sup>k</sup> z <sup>α</sup> be a lower value of [*log*(*Pk*○÷ (1○-*Pk*))]α, and ( )

j 0

=

j 0

=

**4. Numerical example of the data analysis method**

**4.1. Individual level analysis of ambiguous comparative model**

**4.1.1 Example of additive difference model**

amount of money he would be willing to pay.

Fig. 5. Example of a Fuzzy Rating in WTP Task.

**<IP_ADDRESS> Participant and procedure**

fuzzy rating method. (see Figure 5)

Let ( ) L

where

(1○-*Pk*))]<sup>α</sup>

Takemura (2004).

comparative judgments.

The fuzzy coefficients were obtained by fuzzy linear regression analysis using the least squares under constraints, as shown in Tables 1 and 2. The dependent variable of Table 1 was the same as that in Table 2. However, the independent variables in Table 1 are objective values measured by crisp numbers, whereas in Table 2 the independent variables are fuzzy rating values measured by an L-R fuzzy number. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. The parameter *Ajk*0 would be a fuzzy variable and larger than *Aab0* if *Xa* were more salient than *Xb*. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes are assumed to be crisp numbers as explained before. According to Tables 1 and 2, the preference strength concerning comparative judgment was influenced most by whether the target computer was new or used. The impact of the hard disks' attributes was smaller than that of the new-used dimension.

### **4.1.2 Example of the logistic model**

### **<IP_ADDRESS> Participant and procedure**

The participant was a 43-year-old adult. The participant rated the ambiguous probability of preferring a certain computer (DELL brand) out of seven different computers. Three types of attribute information (hard disk: 100 or 60 GB; memory: 2.80 or 2.40 GHz; new or used product) were manipulated in the same manner as in the previous judgment task.. That is, the participant answered the lower value, the representative value , and upper value for the probability that superior alternative is preferred to inferior alternative. The participant used the fuzzy rating method to provide representative, lower, and upper values of probabilities (see Figure 7 ).

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 17

0% 100%

100% 0%

The fuzzy coefficients were obtained by fuzzy linear regression analysis using least squares under constraints, as shown in Tables 3 and 4. However, in Table 3 the independent variables are objective values measured by crisp numbers, whereas in Table 4 the independent variables are fuzzy rating values measured by an L-R fuzzy number. The parameter *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. According to Tables 3 and 4, the bounded preference strength was influenced most by whether the target computer was new or used. Interestingly, the impact of the attribute for memory was slightly greater than was the case

Attribute Value

Hard Disk(M) Representative 0.000

Fuzzy Memory(L) Lower 1.781 Coefficient Memory(M) Representative 1.781 Memory(R) Upper 1.881

(R) Upper 1.443

Hard Disk (L) Lower 0.000

Hard Disk (R) Upper 0.009

New or Used(R) Lower 1.791 New or Used(M) Representative 2.097 New or Used(L) Upper 2.777

(L) Lower 0.847

(M) Representative 1.201

1) Low ambiguity

2)High ambiguity

Fig. 7. Example of Fuzzy Probability Rating.

Note: The independent variables are crisp numbers.

Table 3. Coefficients of Fuzzy Logistic Regression Analysis

*Ajk* <sup>0</sup>

*A*jk0

*A*jk0

**<IP_ADDRESS> Analysis and results**

in Tables 1 and 2.

Note: The independent variables are crisp numbers.

Table 1. Coefficients of Fuzzy Regression Analysis

Note: The independent variables are fuzzy L-R numbers.

Table 2. Coefficients of Fuzzy Regression Analysis

Fig. 7. Example of Fuzzy Probability Rating.

### **<IP_ADDRESS> Analysis and results**

16 Fuzzy Logic – Algorithms, Techniques and Implementations

Hard Disk(L) Lower 78.5

Fuzzy Memory(L) Lower 0.0 Coefficient Memory(M) Representative 0.0

(R) Upper 33 111.2

Hard Disk(L) Lower 33.9

Fuzzy Memory(L) Lower 0.0 Coefficient Memory(M) Representative 0.0

(R) Upper 48 004.0

Hard Disk (M) Representative 85.7 Hard Disk (R) Upper 986.8

Memory(R) Upper 0.0 New or Used(R) Lower 22 332.5 New or Used (M) Representative 22 332.5 New or Used(L) Upper 22 332.5

Hard Disk (M) Representative 33.9 Hard Disk (R) Upper 33.9

Memory(R) Upper 0.0 New or Used(R) Lower 446.1 New or Used(M) Representative 446.1 New or Used(L) Upper 446.1

(L) Lower 36 082.1

(M) Representative 36 082.1

(L) Lower 25 450.8

(M) Representative 29 420.1

Attribute Value

Note: The independent variables are crisp numbers. Table 1. Coefficients of Fuzzy Regression Analysis

*Ajk*<sup>0</sup>

*A*jk0

*A*jk0

Note: The independent variables are fuzzy L-R numbers. Table 2. Coefficients of Fuzzy Regression Analysis

*Ajk* <sup>0</sup>

*A*jk0

*Ajk* <sup>0</sup>

Attribute Value

The fuzzy coefficients were obtained by fuzzy linear regression analysis using least squares under constraints, as shown in Tables 3 and 4. However, in Table 3 the independent variables are objective values measured by crisp numbers, whereas in Table 4 the independent variables are fuzzy rating values measured by an L-R fuzzy number. The parameter *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. According to Tables 3 and 4, the bounded preference strength was influenced most by whether the target computer was new or used. Interestingly, the impact of the attribute for memory was slightly greater than was the case in Tables 1 and 2.

Note: The independent variables are crisp numbers.

Table 3. Coefficients of Fuzzy Logistic Regression Analysis

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 19

Then, please estimate the amount of money you would be willing to pay to upgrade the inferior alternative from inferior alternative to superior alternative using fuzzy rating method. That is, the participants answered the lower value, the representative value, and

The fuzzy coefficients were obtained by fuzzy linear regression analysis using the least squares under constraints, as shown in Tables 5 for the digital camera data and Table 6 for mobile phone data. The independent variables in Table5 and Table 6 are objective values measured by crisp numbers. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables.According to Tables 5, the preference strength concerning comparative judgment was influenced most by whether the target digital camera was 2.5 or 5.0 inches. The impact of the memory's attribute was smaller than those of display size and weight dimensions. According to Tables 6, the preference strength concerning comparative judgment was influenced most by whether the target mobile phone was 2.8 or 3.0 inches. The impact of the pixel number's attribute was smaller than those of display size and weight dimensions. The participants also rated the desirability of the attribute information for each computer using a fuzzy rating method. The fuzzy rating scale of desirability ranged from 0 point to 100 points. (see Figure 6). That is, the participant answered the lower value, the representative value ,

Which alternative do you prefer ? Please circle the superior alternative.

upper value for the amount of money you would be willing to pay.

Minimum: 2,000 yen ----- Maximum: 10, 000 yen

 **Brand A Brand B**  Weight: 130g Weight: 160g Memory: 25MB Memory: 50MB Display: 50 inches Display: 25 inches

**Difference**

Fig. 8. Example of Fuzzy WTP Rating

and upper value for each attribute value.

**<IP_ADDRESS> Analysis and results**

Representative Value: 5,000 yen

**Question:**

Note: The independent variables are fuzzy L-R numbers.

Table 4. Coefficients of Fuzzy Logistic Regression Analysis

### **4.2 Group level analysis of ambiguous comparative model**

### **4.2.1 Example of additive difference model**

### **<IP_ADDRESS> Participants and procedure**

The participant s were 100 undergraduate university students (68 female and 32 male students) enrolled in an economic

psychology class at Waseda University. They were recruited for an experiment investigating "consumer preference ".

Their average age was 21.3 years old. The participants rated differences in WTP for two different digital cameras with three types of attribute information (weight: 130 gram or1 60 gram; memory: 25 or 50 MB; display size:2.5 or 5.0 inches). The participants compared a certain alternative with seven different alternatives. The participants also rated differences in WTP for two different mobile phones with three types of attribute information (weight: 123 gram or132 gram; pixel number:3,200,000 or 5,070,000 pixels; display size:2.8 or 3.0 inches). The participants compared a certain alternative with seven different mobile phones. The participant provided representative values and lower and upper WTP values using a fuzzy rating method. The participants were asked the amount of money he would be willing to pay to upgrade the inferior from inferior alternative to superior alternative using fuzzy rating method. That is, the participants answered the lower value, the representative value , and upper value for the amount of money he would be willing to pay. An example of fuzzy WTP rating is illustrated in the Figure 8.

### **Question:**

18 Fuzzy Logic – Algorithms, Techniques and Implementations

Hard Disk (M) Representative 0.000 Hard Disk (R) Upper 0.000

New or Used(R) Lower 0.043 New or Used(M) Representative 0.043 New or Used(L) Upper 0.043

(M) Representative 1.806

Hard Disk(L) Lower 0.000

Fuzzy Memory(L) Lower 0.008 Coefficient Memory(M) Representative 0.008 Memory(R) Upper 0.008

(L) Lower 1.806

(R) Upper 1.806

The participant s were 100 undergraduate university students (68 female and 32 male

psychology class at Waseda University. They were recruited for an experiment investigating

Their average age was 21.3 years old. The participants rated differences in WTP for two different digital cameras with three types of attribute information (weight: 130 gram or1 60 gram; memory: 25 or 50 MB; display size:2.5 or 5.0 inches). The participants compared a certain alternative with seven different alternatives. The participants also rated differences in WTP for two different mobile phones with three types of attribute information (weight: 123 gram or132 gram; pixel number:3,200,000 or 5,070,000 pixels; display size:2.8 or 3.0 inches). The participants compared a certain alternative with seven different mobile phones. The participant provided representative values and lower and upper WTP values using a fuzzy rating method. The participants were asked the amount of money he would be willing to pay to upgrade the inferior from inferior alternative to superior alternative using fuzzy rating method. That is, the participants answered the lower value, the representative value , and upper value for the amount of money he would be willing to pay. An example of fuzzy

Attribute Value

Note: The independent variables are fuzzy L-R numbers.

**4.2.1 Example of additive difference model**

**<IP_ADDRESS> Participants and procedure**

students) enrolled in an economic

WTP rating is illustrated in the Figure 8.

"consumer preference ".

Table 4. Coefficients of Fuzzy Logistic Regression Analysis

*Ajk* <sup>0</sup>

Ajk0

*Ajk* <sup>0</sup>

**4.2 Group level analysis of ambiguous comparative model**

Which alternative do you prefer ? Please circle the superior alternative.

Then, please estimate the amount of money you would be willing to pay to upgrade the inferior alternative from inferior alternative to superior alternative using fuzzy rating method. That is, the participants answered the lower value, the representative value, and upper value for the amount of money you would be willing to pay.

Fig. 8. Example of Fuzzy WTP Rating

### **<IP_ADDRESS> Analysis and results**

The fuzzy coefficients were obtained by fuzzy linear regression analysis using the least squares under constraints, as shown in Tables 5 for the digital camera data and Table 6 for mobile phone data. The independent variables in Table5 and Table 6 are objective values measured by crisp numbers. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables.According to Tables 5, the preference strength concerning comparative judgment was influenced most by whether the target digital camera was 2.5 or 5.0 inches. The impact of the memory's attribute was smaller than those of display size and weight dimensions. According to Tables 6, the preference strength concerning comparative judgment was influenced most by whether the target mobile phone was 2.8 or 3.0 inches. The impact of the pixel number's attribute was smaller than those of display size and weight dimensions. The participants also rated the desirability of the attribute information for each computer using a fuzzy rating method. The fuzzy rating scale of desirability ranged from 0 point to 100 points. (see Figure 6). That is, the participant answered the lower value, the representative value , and upper value for each attribute value.

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 21

The participant s were 100 undergraduate university students (68 female and 32 male students). Their average age was 21.3 years old. The participants rated the ambiguous probability of preferring a certain digital camera out of seven different digital cameras. The three types of attribute information (weight: 130 gram or1 60 gram; memory: 25 or 50 MB; display size:2.5 or 5.0 inches) were manipulated in the same manner as in the previous individual judgment task. They also rated the ambiguous probability of preferring a certain mobile phone out of seven different mobile phones. The three types of attribute information (weight: 123 gram or132 gram; pixel number:3,200,000 or 5,070,000 pixels; display size:2.8 or 3.0 inches) were manipulated in the same manner in the previous judgment task. The participant provided representative values and lower and upper values of probabilities. That is, the participants answered the lower value, the representative value , and upper value for the probability that superior alternative is preferred to inferior alternative. The participants used the fuzzy rating method to provide representative, lower, and upper

The fuzzy coefficients were obtained by fuzzy logistic regression analysis using the least squares under constraints, as shown in Tables 7 for the digital camera data and Table 8 for mobile phone data. The independent variables in Table 7 and Table 8 are objective values measured by crisp numbers. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. According to Tables 7, the bounded preference strength was influenced most by whether the target digital camera was 2.5 or 5.0 inches. The impact of the memory's attribute was smaller than those of display size and weight dimensions. According to Tables 8, the bounded preference strength t was influenced most by whether the target mobile phone was 2.8 or 3.0 inches. The impact of the weight's attribute was smaller than those of display

Weight(L) Lower 0.035

Fuzzy Memory(L) Lower 0.003 Coefficient Memory(M) Representative 0.003 Memory(R) Upper 0.003

(R) Upper 1.072

Table 7. Coefficients of Fuzzy Logistic Regression Analysis for Digital Camera Data

Weight (M) Representative 0.038 Weight (R) Upper 0.054

Display Size(R) Lower 2.625 Display Size (M) Representative 2.625 Display Size(L) Upper 2.625

> (L) Lower -0.122 (M) Representative 0.459

**4.2.2 Example of the logistic model <IP_ADDRESS> Participants and procedure**

values of probabilities (see Figure 7 ).

size and pixel number dimensions.

Note: The independent variables are crisp numbers.

Attribute Value

*Ajk* <sup>0</sup>

Ajk0 Ajk0

**<IP_ADDRESS> Analysis and results**

Note: The independent variables are crisp numbers.

Table 5. Coefficients of Fuzzy Regression Analysis for Digital Camera Data

Note: The independent variables are crisp numbers.

Table 6. Coefficients of Fuzzy Regression Analysis for Mobile Phone Data

### **4.2.2 Example of the logistic model**

### **<IP_ADDRESS> Participants and procedure**

20 Fuzzy Logic – Algorithms, Techniques and Implementations

Weight (M) Representative 48.57 Weight (R) Upper 68.33

Memory(R) Upper 14.62 Display Size(R) Lower 223.10 Display Size (M) Representative 4791.98 Display Size(L) Upper 4791.98

(L) Lower 11361.25

(M) Representative 11361.25

Weight(L) Lower 48.57

Fuzzy Memory(L) Lower 8.29 Coefficient Memory(M) Representative 8.29

(R) Upper 15447.54

Weight(L) Lower 28.84

Fuzzy Pixel Number(L) Lower -12.12 Coefficient Pixel Number(M) Representative 28.55

(R) Upper 12569.35

Table 6. Coefficients of Fuzzy Regression Analysis for Mobile Phone Data

Weight (M) Representative 28.84 Weight (R) Upper 53.44

Pixel Number(R) Upper 28.55 Display Size(R) Lower -233.73 Display Size(M) Representative 190.29 Display Size(L) Upper 190.29

(L) Lower 7758.98

(M) Representative 8234.94

Table 5. Coefficients of Fuzzy Regression Analysis for Digital Camera Data

Attribute Value

*Ajk* <sup>0</sup>

*Ajk*<sup>0</sup>

*A*jk0

Attribute Value

Note: The independent variables are crisp numbers.

*Ajk* <sup>0</sup>

*A*jk0 *A*jk0

Note: The independent variables are crisp numbers.

The participant s were 100 undergraduate university students (68 female and 32 male students). Their average age was 21.3 years old. The participants rated the ambiguous probability of preferring a certain digital camera out of seven different digital cameras. The three types of attribute information (weight: 130 gram or1 60 gram; memory: 25 or 50 MB; display size:2.5 or 5.0 inches) were manipulated in the same manner as in the previous individual judgment task. They also rated the ambiguous probability of preferring a certain mobile phone out of seven different mobile phones. The three types of attribute information (weight: 123 gram or132 gram; pixel number:3,200,000 or 5,070,000 pixels; display size:2.8 or 3.0 inches) were manipulated in the same manner in the previous judgment task. The participant provided representative values and lower and upper values of probabilities. That is, the participants answered the lower value, the representative value , and upper value for the probability that superior alternative is preferred to inferior alternative. The participants used the fuzzy rating method to provide representative, lower, and upper values of probabilities (see Figure 7 ).

### **<IP_ADDRESS> Analysis and results**

The fuzzy coefficients were obtained by fuzzy logistic regression analysis using the least squares under constraints, as shown in Tables 7 for the digital camera data and Table 8 for mobile phone data. The independent variables in Table 7 and Table 8 are objective values measured by crisp numbers. The parameter of *Ajk*0 involves a response bias owing to presentation order, context effects, and the scale parameter of the dependent variables. According to Tables 7, the bounded preference strength was influenced most by whether the target digital camera was 2.5 or 5.0 inches. The impact of the memory's attribute was smaller than those of display size and weight dimensions. According to Tables 8, the bounded preference strength t was influenced most by whether the target mobile phone was 2.8 or 3.0 inches. The impact of the weight's attribute was smaller than those of display size and pixel number dimensions.

Note: The independent variables are crisp numbers.

Table 7. Coefficients of Fuzzy Logistic Regression Analysis for Digital Camera Data

Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis 23

marketing research, risk perception research, and human judgment and decision-making research. Empirical research using possibilistic analysis and least squares analysis will be

Results of these applications to psychological study indicated that the parameter estimated in the proposed analysis was meaningful for social judgment study. This study has a methodological restriction on statistical inferences for fuzzy parameters. Therefore, we plan further work on the fuzzy theoretic analysis of social judgment directed toward the statistical study of fuzzy regression analysis and fuzzy logistic regression analysis such as

This work was supported in part by Grants in Aids for Grant-in-Aid for Scientific Research on Priority Area, The Ministry of Education, Culture, Sports, Science and Technology(MEXT). I thank Matsumoto,T., Matsuyama,S.,and Kobayashi,M.. for their

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**6. Acknowledgment**

**7. References**

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*Analysis and Application,* 64, 369–380.

Psychology: General, 104, 192-233.

categories. *Cognitive Psychology,* 7, 573-603.

input-output data. *Fuzzy Sets and Systems,* 47, 173–181.

consequences*. Psychological Review,* 110, 472–489.

Academic Press.

Note: The independent variables are crisp numbers.

Table 8. Coefficients of Fuzzy Logistic Regression Analysis for Mobile Phone Data

### **5. Conclusion**

This chapter introduce fuzzy set models for ambiguous comparative judgments, which do not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that is used to explain ambiguous preference strength. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and can be reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes are assumed to be crisp numbers. The second type of model was a fuzzy logistic model for explaining ambiguous preference in which preference strength is bounded, such as a probability measure.

In both models, multi-attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L-R numbers. For each model, parameter estimation method using fuzzy regression analysis was introduced. Numerical examples for comparison were also demonstrated. As the objective of the numerical examples was to demonstrate that the proposed estimation might be viable, further empiric studies will be needed. Moreover, because the two models require different evaluation methods, comparisons of the psychological effects of the two methods must be studied further.

In this chapter, the least squares method was used for data analyses of the two models. However, the possibilistic linear regression analysis (Sakawa & Yano, 1992) and the possibilistic logistic regression analysis (Takemura, 2004) could also be used in the data analysis of the additive difference type model and the logistic type model, respectively. The proposed models and the analyses for ambiguous comparative judgments will be applied to marketing research, risk perception research, and human judgment and decision-making research. Empirical research using possibilistic analysis and least squares analysis will be needed to examine the validity of these models.

Results of these applications to psychological study indicated that the parameter estimated in the proposed analysis was meaningful for social judgment study. This study has a methodological restriction on statistical inferences for fuzzy parameters. Therefore, we plan further work on the fuzzy theoretic analysis of social judgment directed toward the statistical study of fuzzy regression analysis and fuzzy logistic regression analysis such as statistical tests of parameters, outlier detection, and step-wise variable selection.

### **6. Acknowledgment**

22 Fuzzy Logic – Algorithms, Techniques and Implementations

Weight (M) Representative 0.002 Weight (R) Upper 0.009

Pixel Number(R) Upper 0.024 Display Size(R) Lower 0.161 Display Size(M) Representative 0.165 Display Size(L) Upper 0.232

(L) Lower -0.871

(M) Representative 0.030

Weight(L) Lower 0.002

Fuzzy Pixel Number(L) Lower 0.012 Coefficient Pixel Number(M) Representative 0.017

(R) Upper 0.887

This chapter introduce fuzzy set models for ambiguous comparative judgments, which do not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that is used to explain ambiguous preference strength. This model can be reduced to the Fuzzy Utility Difference Model (Nakamura, 1992) if multi-attribute weighting parameters are assumed to be crisp numbers, and can be reduced to the Additive Difference Model (Tversky, 1969) if multi-attribute weighting parameters and the values of multi-attributes are assumed to be crisp numbers. The second type of model was a fuzzy logistic model for explaining ambiguous preference in which preference strength is bounded, such as a probability

In both models, multi-attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L-R numbers. For each model, parameter estimation method using fuzzy regression analysis was introduced. Numerical examples for comparison were also demonstrated. As the objective of the numerical examples was to demonstrate that the proposed estimation might be viable, further empiric studies will be needed. Moreover, because the two models require different evaluation methods, comparisons of the

In this chapter, the least squares method was used for data analyses of the two models. However, the possibilistic linear regression analysis (Sakawa & Yano, 1992) and the possibilistic logistic regression analysis (Takemura, 2004) could also be used in the data analysis of the additive difference type model and the logistic type model, respectively. The proposed models and the analyses for ambiguous comparative judgments will be applied to

psychological effects of the two methods must be studied further.

Table 8. Coefficients of Fuzzy Logistic Regression Analysis for Mobile Phone Data

Attribute Value

*Ajk* <sup>0</sup> *Ajk* <sup>0</sup>

*Ajk*<sup>0</sup>

Note: The independent variables are crisp numbers.

**5. Conclusion**

measure.

This work was supported in part by Grants in Aids for Grant-in-Aid for Scientific Research on Priority Area, The Ministry of Education, Culture, Sports, Science and Technology(MEXT). I thank Matsumoto,T., Matsuyama,S.,and Kobayashi,M.. for their assistance, and the editor and the reviewers for their valuable comments.

### **7. References**

**2**

Agnes Achs

 *Hungary*

**From Fuzzy Datalog to**

*University of Pecs Faculty of Engineering,*

**Multivalued Knowledge-Base**

Despite the fact that people have very different and ambiguous concepts and knowledge, they are able to talk to one another. How does human mind work? How can people give answers to questions? Modelling human conversation and knowledge demands to deal with

Human knowledge consists of static and dynamic knowledge chunks. The static ones include the so called lexical knowledge or the ability to sense similarities between facts and between predicates. Through dynamic attainments one can make deductions or one can give answers to a question. There are several and very different approaches to make a model of human knowledge, but one of the most common and widespread fields of research is based

Fuzzy sets theory, proposed by Zadeh (1965), is a realistic and practical means to describe the world that we live in. The method has successfully been applied in various fields, among others in decision making, logic programming, and approximate reasoning. In the last decade, a number of papers have dealt with that subject, e.g. (Formato et al 2000, Sessa 2002, Medina et al 2004, Straccia et al 2009). They deal with different aspects of modelling and handling uncertainty. (Straccia 2008) gives a detailed overview of this topic with widespread references. Our investigations have begun independently of these works, and have run parallel to them. Of course there are some similar features, but our model differs from the

As a generalization of fuzzy sets, intuitionistic fuzzy sets were presented by Atanassov (Atanassov 1983), and have allowed people to deal with uncertainty and information in a much broader perspective. Another well-known generalization of an ordinary fuzzy set is the interval-valued fuzzy set, which was first introduced by Zadeh (Zadeh 1975). These generalizations make descriptions and models of the world more realistic, and practical.

In the beginning, our knowledge-base model was based on the concept of fuzzy logic, later on it was extended to intuitionistic and interval-valued logic. In this model, the static part is a background knowledge module, while the dynamic part consists of a Datalog based deduction mechanism. To develop this mechanism, it was necessary to generalize the Datalog language

and to extend it into fuzzy and intuitionistic direction. (Achs 1995, 2007, 2010).

**1. Introduction**

on fuzzy logic.

uncertainty and deductions.

others detailed in literature.

## **From Fuzzy Datalog to Multivalued Knowledge-Base**

Agnes Achs *University of Pecs Faculty of Engineering, Hungary*

### **1. Introduction**

24 Fuzzy Logic – Algorithms, Techniques and Implementations

Sherif,M.,& Hovland,C,I. (1961). *Social judgment: Assimilation and contrast effects in communication and attitude change.* New Haven: Yale University Press. Smithson, M. (1987). *Fuzzy set analysis for the behavioral and social sciences*. New York:

Takemura, K. (1999). A fuzzy linear regression analysis for fuzzy input-output data using

Takemura, K. (2000). Vagueness in human judgment and decision making. In Z. Q. Liu & S.

Takemura, K. (2004). Fuzzy logistic regression analysis for fuzzy input and output data.

Takemura, K. (2005). Fuzzy least squares regression analysis for social judgment study.

Takemura, K. (2007). Ambiguous comparative judgment: Fuzzy set model and data analysis.

Takemura, K. ,Matsumoto,T.,Matsuyama,S.,& Kobayashi,M., (2011). Analysis of consumer's

Takemura,K. (1996). *Psychology of decision making.* Tokyo:Fukumura Syuppan. (in Japanese). Takemura,K. (2011) Model of multi-attribute decision making and good decision.

Zadeh,A. (1973). Outline of a new approach to the analysis of complex systems and decision processes, *IEEE Transactions on Systems, Man and Cybernetics, SMC* 3(1), 28-44.

the least squares method under linear constraints and its application to fuzzy rating

Miyamoto (Eds), *Soft Computing for Human Centered Machines* (pp. 249–281). Tokyo:

Proceedings of the joint 2nd International Conference on Soft Computing and Intelligent Systems and the 5th International Symposium on Advanced Intelligent

ambiguous comparative judgment. *Discussion Paper, Department of Psychology,*

Smithson,M.(1989) Ignorance and uncertainty. New York: Springer-Verlag-

data. *Journal of Advanced Computational Intelligence,* 3, 36–40.

*Journal of Advanced Computational Intelligence,* 9, 461–466.

Operations Research,56(10),583-590 (In Japanese)

Zadeh,A. (1965). Fuzzy sets, *Information and Control*, 8, 338-353.

Tversky, A. (1969). Intransitivity of preferences. *Psychological Review,* 76, 31–48. Wittgenstein,L. (1953). *Philosophical investigations.* New York:MacMillan.

Systems 2004 (WE8-5), Yokohama, Japan.

*Japanese Psychology Research,* 49, 148–156.

Springer-Verlag.

Springer Verlag.

*Waseda University*.

Despite the fact that people have very different and ambiguous concepts and knowledge, they are able to talk to one another. How does human mind work? How can people give answers to questions? Modelling human conversation and knowledge demands to deal with uncertainty and deductions.

Human knowledge consists of static and dynamic knowledge chunks. The static ones include the so called lexical knowledge or the ability to sense similarities between facts and between predicates. Through dynamic attainments one can make deductions or one can give answers to a question. There are several and very different approaches to make a model of human knowledge, but one of the most common and widespread fields of research is based on fuzzy logic.

Fuzzy sets theory, proposed by Zadeh (1965), is a realistic and practical means to describe the world that we live in. The method has successfully been applied in various fields, among others in decision making, logic programming, and approximate reasoning. In the last decade, a number of papers have dealt with that subject, e.g. (Formato et al 2000, Sessa 2002, Medina et al 2004, Straccia et al 2009). They deal with different aspects of modelling and handling uncertainty. (Straccia 2008) gives a detailed overview of this topic with widespread references. Our investigations have begun independently of these works, and have run parallel to them. Of course there are some similar features, but our model differs from the others detailed in literature.

As a generalization of fuzzy sets, intuitionistic fuzzy sets were presented by Atanassov (Atanassov 1983), and have allowed people to deal with uncertainty and information in a much broader perspective. Another well-known generalization of an ordinary fuzzy set is the interval-valued fuzzy set, which was first introduced by Zadeh (Zadeh 1975). These generalizations make descriptions and models of the world more realistic, and practical.

In the beginning, our knowledge-base model was based on the concept of fuzzy logic, later on it was extended to intuitionistic and interval-valued logic. In this model, the static part is a background knowledge module, while the dynamic part consists of a Datalog based deduction mechanism. To develop this mechanism, it was necessary to generalize the Datalog language and to extend it into fuzzy and intuitionistic direction. (Achs 1995, 2007, 2010).

From Fuzzy Datalog to Multivalued Knowledge-Base 27

this model it is impossible to make any true fact false and still have a model consistent with

An interpretation assigns truth or falsehood to every possible instance of the program's predicates. An interpretation is a model, if it makes the rules true, no matter what assignment of values from the domain is made for the variables in each rule. Although there are infinite many implications, it is proved that it is enough to consider only the Herbrand

The Herbrand universe of a program *P* (denoted by *HP*) is the set of all possible ground terms constructed by using constants and function symbols occurring in *P*. The Herbrand base of *P* (*BP*) is the set of all possible ground atoms whose predicate symbols occur in *P* and

In general, a term is a variable, a constant or a complex term of the form *f(t1, …, tn)*, where *f* is a function symbol and *t1, …, tn* are terms. An atom is a formula of the form *p(t)*, where *p* is a predicate symbol of a finite arity (say *n*) and *t* is a sequence of terms of length *n* (arguments). A literal is either an atom (positive literal) or its negation (negative literal). A term, atom or literal is ground if it is free of variables. As in fuzzy extension, we did not deal with function symbols, so in our case the ground terms are the constants of the program.

In the case of Datalog programs there are several equivalent approaches to define the semantics of the program. In fuzzy extension we mainly rely on the fixed-point base aspect. The above concepts are detailed in classical works such as (Ceri et al 1990, Loyd 1990,

In fuzzy Datalog (fDATALOG) the facts can be completed with an uncertainty level, the rules with an uncertainty level and an implication operator. With the use of this operator and these levels deductions can be made. As in classical cases, logical correctness is extremely important as well, i.e., the consequence must be a model of the program. This means that for each rule of the program, the truth-value of the fuzzy implication following

β

≥ *0),*

 *A1,…,An (n*

where *A* is an atom (the head of the rule), *A1,…,An* are literals (the body of the rule); *I* is an

For getting a finite result, all the rules in the program must be safe. An fDATALOG rule is safe if all variables occurring in the head also occur in the body, and all variables occurring in a negative literal also occur in a positive one. An fDATALOG program is a finite set of

∈ (0,1] (the level of the rule).

*; I*, where r is a formula of the form

the rule has to be at least as large as the given uncertainty level.

*A* ←

**2.1.1 Syntax and semantics of fuzzy datalog**

**Definition 1**. An fDATALOG rule is a triplet *r;*

β

More precisely, the notion of fuzzy rule is the following:

interpretation defined on the Herbrand universe and the Herbrand base.

the database.

Ullman 1988).

**2.1 Fuzzy Datalog**

implication operator and

safe fDATALOG rules.

whose arguments are elements of *HP*.

In many frameworks, in order to answer a query, we have to compute the whole intended model by a bottom-up fixed-point computation and then answer with the evaluation of the query in this model. This always requires computing a whole model, even if not all the facts and rules are required to determine answer. Therefore a possible top-down like evaluation algorithm has been developed for our model. This algorithm is not a pure top-down one but the combination of top down and bottom up evaluations. Our aim is to improve this algorithm and perhaps to develop a pure top down evaluation based on fuzzy or multivalued unification algorithm. There are fuzzy unification algorithms described for example in (Alsinet et al 1998, Formato et al 2000, Virtanen 1994), but they are inappropriate for evaluating our knowledge-base.

However, the concept of (Julian-Iranzo et al 2009, 2010) is similar but not identical with one of our former ideas about evaluating of special fuzzy Datalog programs (Achs 2006). Reading these papers has led to the assumption that this former idea may be the base of a top-down-like evaluation strategy in special multivalued cases as well. Based on this idea, a multivalued unification algorithm was developed and used for to determine the conclusion of a multivalued knowledge-base.

In this chapter this possible model for handling uncertain information will be provided. This model is based on the multivalued extensions of Datalog. Starting from fuzzy Datalog, the concept of intuitionistic Datalog and bipolar Datalog will be described. This will be the first pillar of the knowledge-base. The second one deals with the similarities of facts and concepts. These similarities are handled with proximity relations. The third component connects the first two with each other. In the final part of the paper, an evaluating algorithm is presented. It is discussed in general, but in special cases it is based on fuzzy, or multivalued unification, which is also mentioned.

### **2. Extensions of datalog**

When one builds a knowledge-base, it is very important to deal with a database management system. It is based on the relational data model developed by Codd in 1970. This model is a very useful one, but it can not handle every problem. For example, the standard query language for relational databases (SQL) is not Turing-complete, in particular it lacks recursion and therefore concepts like transitive closure of a relation can not be expressed in SQL. Along with other problems this is why different extensions of the relational data model or the development of other kinds of models are necessary. A more complete one is the world of deductive databases. A deductive database consists of facts and rules, and a query is answered by building chains of deductions. Therefore the term of deductive database highlights the ability to use a logic programming style for expressing deductions concerning the contents of a database. One of the best known deductive database query languages is Datalog.

As any deductive database, a Datalog program consists of facts and rules, which can be regarded as first order logic formulas. Using these rules, new facts can be inferred from the program's facts so that the consequence of a program will be logically correct. This means that evaluating the program, the result is a model of the formulas belonging to the rules. On the other hand, it is also important that this model will contain only those true facts, which are the consequences of the program; that is, the minimality of this model is expected, i.e. in 26 Fuzzy Logic – Algorithms, Techniques and Implementations

In many frameworks, in order to answer a query, we have to compute the whole intended model by a bottom-up fixed-point computation and then answer with the evaluation of the query in this model. This always requires computing a whole model, even if not all the facts and rules are required to determine answer. Therefore a possible top-down like evaluation algorithm has been developed for our model. This algorithm is not a pure top-down one but the combination of top down and bottom up evaluations. Our aim is to improve this algorithm and perhaps to develop a pure top down evaluation based on fuzzy or multivalued unification algorithm. There are fuzzy unification algorithms described for example in (Alsinet et al 1998, Formato et al 2000, Virtanen 1994), but they are inappropriate

However, the concept of (Julian-Iranzo et al 2009, 2010) is similar but not identical with one of our former ideas about evaluating of special fuzzy Datalog programs (Achs 2006). Reading these papers has led to the assumption that this former idea may be the base of a top-down-like evaluation strategy in special multivalued cases as well. Based on this idea, a multivalued unification algorithm was developed and used for to determine the conclusion

In this chapter this possible model for handling uncertain information will be provided. This model is based on the multivalued extensions of Datalog. Starting from fuzzy Datalog, the concept of intuitionistic Datalog and bipolar Datalog will be described. This will be the first pillar of the knowledge-base. The second one deals with the similarities of facts and concepts. These similarities are handled with proximity relations. The third component connects the first two with each other. In the final part of the paper, an evaluating algorithm is presented. It is discussed in general, but in special cases it is based on fuzzy, or

When one builds a knowledge-base, it is very important to deal with a database management system. It is based on the relational data model developed by Codd in 1970. This model is a very useful one, but it can not handle every problem. For example, the standard query language for relational databases (SQL) is not Turing-complete, in particular it lacks recursion and therefore concepts like transitive closure of a relation can not be expressed in SQL. Along with other problems this is why different extensions of the relational data model or the development of other kinds of models are necessary. A more complete one is the world of deductive databases. A deductive database consists of facts and rules, and a query is answered by building chains of deductions. Therefore the term of deductive database highlights the ability to use a logic programming style for expressing deductions concerning the contents of a database. One of the best known deductive

As any deductive database, a Datalog program consists of facts and rules, which can be regarded as first order logic formulas. Using these rules, new facts can be inferred from the program's facts so that the consequence of a program will be logically correct. This means that evaluating the program, the result is a model of the formulas belonging to the rules. On the other hand, it is also important that this model will contain only those true facts, which are the consequences of the program; that is, the minimality of this model is expected, i.e. in

for evaluating our knowledge-base.

of a multivalued knowledge-base.

**2. Extensions of datalog**

database query languages is Datalog.

multivalued unification, which is also mentioned.

this model it is impossible to make any true fact false and still have a model consistent with the database.

An interpretation assigns truth or falsehood to every possible instance of the program's predicates. An interpretation is a model, if it makes the rules true, no matter what assignment of values from the domain is made for the variables in each rule. Although there are infinite many implications, it is proved that it is enough to consider only the Herbrand interpretation defined on the Herbrand universe and the Herbrand base.

The Herbrand universe of a program *P* (denoted by *HP*) is the set of all possible ground terms constructed by using constants and function symbols occurring in *P*. The Herbrand base of *P* (*BP*) is the set of all possible ground atoms whose predicate symbols occur in *P* and whose arguments are elements of *HP*.

In general, a term is a variable, a constant or a complex term of the form *f(t1, …, tn)*, where *f* is a function symbol and *t1, …, tn* are terms. An atom is a formula of the form *p(t)*, where *p* is a predicate symbol of a finite arity (say *n*) and *t* is a sequence of terms of length *n* (arguments). A literal is either an atom (positive literal) or its negation (negative literal). A term, atom or literal is ground if it is free of variables. As in fuzzy extension, we did not deal with function symbols, so in our case the ground terms are the constants of the program.

In the case of Datalog programs there are several equivalent approaches to define the semantics of the program. In fuzzy extension we mainly rely on the fixed-point base aspect. The above concepts are detailed in classical works such as (Ceri et al 1990, Loyd 1990, Ullman 1988).

### **2.1 Fuzzy Datalog**

In fuzzy Datalog (fDATALOG) the facts can be completed with an uncertainty level, the rules with an uncertainty level and an implication operator. With the use of this operator and these levels deductions can be made. As in classical cases, logical correctness is extremely important as well, i.e., the consequence must be a model of the program. This means that for each rule of the program, the truth-value of the fuzzy implication following the rule has to be at least as large as the given uncertainty level.

### **2.1.1 Syntax and semantics of fuzzy datalog**

More precisely, the notion of fuzzy rule is the following:

**Definition 1**. An fDATALOG rule is a triplet *r;* β*; I*, where r is a formula of the form

$$A \gets A\_1, \ldots, A\_n \qquad \quad (n \ge 0),$$

where *A* is an atom (the head of the rule), *A1,…,An* are literals (the body of the rule); *I* is an implication operator and β∈ (0,1] (the level of the rule).

For getting a finite result, all the rules in the program must be safe. An fDATALOG rule is safe if all variables occurring in the head also occur in the body, and all variables occurring in a negative literal also occur in a positive one. An fDATALOG program is a finite set of safe fDATALOG rules.

From Fuzzy Datalog to Multivalued Knowledge-Base 29

*T1 = T(T0)*

*Tn = T(Tn-1)*  ...

It is clear that *NTP* is inflationary transformation over *L*, and if *P* is negation-free, then *NTP* is monotone as well. (A transformation *T* is inflationary if *X* ≤ *T(X)* for every *X* ∈ *L* and it is

In (Ceri et al 1990) it is shown that an inflationary transformation over a complete lattice has a fixed point and if it is monotone then it has a least fixed point (Loyd 1990). Therefore *NTP* has a fixed point, i.e. there exists an *X* ∈ *F(BP)* for which *NTP(X)* = *X*. If *P* is positive, then *X*

The fixed point of the transformation will be denoted by *lfp(NTP)*. It can be shown (Achs

α*Ai*

β

*A* the condition *I(*

α

In the case of c because of the construction of *T0 Ai* is not negative, that is *Ai* is not among the

**Proof** In the case of a positive Datalog program, the least fixed point is the least model (Ceri et al 1990, Ullman 1988). In the case of fuzzy Datalog, according to the definition of the consequence transformation, the level of the rule's head is the least value satisfying the criterion of modelness. The application of the transformations may arise only one problem. A lower level would be ordered to the same rule's head, but according to the definitions we should accept the higher value. But such a case can arise only in the case of programs

According to the above statements, the meaning of the programs can be defined by this

**Definition 3.** *lfp(NTP)* is the nondeterministic semantics of fDATALOG *P* program.

*A)* ∈ *lfp(NTP)* and *(|Ai|,*

α*body,* α*<sup>A</sup> ) = 1* ≥

**Proposition 1**. For negation-free fDATALOG program *P lfp(NTP)* is the least model.

*)* ∉ *lfp(NTP).*

α*Ai*

*body = 0*, so *I(*

containing any negation. Therefore the proposition is true.

An ordering relation can be defined over *F(BP)*. For *G; H : BP* → [*0; 1*]; *G* ≤ *H* iff *(*

ω } if ω

is a limit ordinal.

 *)* ∈ *lfp(NTP), 1*≤ *i* ≤ *n*.

α*body,* α*<sup>A</sup> )* ≥ β

, namely *lfp(NTP)* is a model.

∀*d* ∈ *BP)*

is realized.

 *=* least upper bound of { *Tn | n* <

*G(d)* ≤ *H(d)*. It easily can be seen that *L = (F(BP),* ≤ *)* is a complete lattice.

is the least fixed point. (That is for any *Z=T(Z)* : *X* ≤ *Z*.)

**Proof** In *ground(P)* there are rules in the next forms:

*; I);* ∃*i : (|Ai|,*

α

In the case of a, b because of the construction of

Moreover, the next proposition is true as well.

1995) that this fixed point is a model of *P*.

**Theorem 1**. *lfp(NTP)* is a model of *P*.

β*; I); (A,* α

β

*= 0*, therefore

and let

a/ *(A* ←*;*

facts, so

fixed point:

β*; I).* b/ *(A* ← *A1, ..., An;*

c/ *(A* ← *A1, ..., An;*

α*Ai* *T*ω

monotone if *T(X)* ≤ *T(Y)* if *X* ≤ *Y*).

There is a special type of rule, called fact. A fact has the form *A* ←*;* β*; I*. From now on, the facts are referred as (*A,*β), because according to implication *I*, the level of *A* easily can be computed and in the case of the implication operators detailed in this chapter it is β.

For defining the meaning of a program, we need again the concepts of Herbrand universe and Herbrand base, but this time they are based on fuzzy logic. Now a ground instance of a rule *r;* β*; I* in *P* is a rule obtained from *r* by replacing every variable in *r* with a constant of *HP*. The set of all ground instances of *r;* β*; I* is denoted by *ground(r);* β*; I*. The ground instance of *P* is *ground(P) =* ∪ *(r; I;* β*)*∈*P (ground(r); I;* β*)*.

An interpretation of a program *P* is a fuzzy set of the program's Herbrand base, *BP*, i.e. it is: ∪ *<sup>A</sup>*∈*BP (A;* α*A).* An interpretation is a model of *P* if for each *(A* ← *A1,…,An;* β*; I)* ∈ *ground(P)*

$$I(\alpha\_{A1\_{A\gets\mathcal{A}}An}, \alpha\_{\mathcal{A}}) \ge \beta.$$

A model *M* is least if for any model *N*, *M* ≤ *N*. A model *M* is minimal if there is not any model *N*, where *N* ≤ *M*.

To be short α *A1*∧*...*∧ *An* will be denoted as αbody and αA as αhead.

In the extensions of Datalog several implication operators are used, but all cases are restricted to min-max conjunction and disjunction, and to the complement to 1 as negation. So: α*A*∧*B = min(*α*A,* α*B),* α*A*∨*B = max(*α*A,* α*B)* and α¬*A = 1* − α*A*.

The semantics of fDATALOG is defined as the fixed points of consequence transformations. Depending on evaluating sequences two semantics can be defined: a deterministic and a nondeterministic one. Further on only the nondeterministic semantics will be discussed, the deterministic one is detailed in (Achs 2010). It was proved that the two semantics are equivalent in the case of negation- and function-free fDatalog programs, but they differ if the program has any negation. In this case merely the nondeterministic semantics is applicable. The nondeterministic transformation is as follows:

**Definition 2.** Let *BP* be the Herbrand base of the program *P*, and let *F(BP)* denote the set of all fuzzy sets over *BP*. The consequence transformation *NTP* : *F(BP)* → *F(BP)* is defined as

$$NT\_F(\mathcal{X}) = \{ (\mathcal{A}, \mathcal{\alpha}\_{\mathcal{A}}) \} \cup \mathcal{X} \tag{1}$$

where

$$(\mathcal{A} \gets \mathcal{A}\_{\mathcal{l}}, \dots, \mathcal{A}\_{\mathcal{n}}; \mathcal{\mathcal{J}} \mathcal{I}) \in \operatorname{ground}(\mathcal{P}), \ (\lvert A\_{i} \rvert, \lVert \mathcal{A}\_{\mathcal{l}i} \rangle \in \mathcal{X}, \ (\mathsf{1} \leq \mathsf{i} \leq \mathsf{n}); \mathcal{I})$$

α*A = max(0, min{*γ *| I(*α*body,* γ*)* ≥ β*}).*

It can be proved that this transformation has a fixed point. To prove it, let us define the powers of a transformation:

For any *T : F(BP)* → *F(BP)* transformation let

$$T\_0 = \{ \cup \{ (\mathbf{A}, \alpha\_{\mathbf{A}}) \} \mid (\mathbf{A} \leftarrow \text{; I}; \mathfrak{f}) \in \text{ground}(\mathbf{P}), \alpha\_{\mathbf{A}} = \max(0, \min\{ \gamma \mid \mathbf{I}(1, \gamma) \ge \beta \}) \mid \cup \{ (\mathbf{A}, \bullet) \mid \exists \ (\mathbf{A}, \bullet) \in \text{ground}(\mathbf{P}) \}$$

and let

28 Fuzzy Logic – Algorithms, Techniques and Implementations

For defining the meaning of a program, we need again the concepts of Herbrand universe and Herbrand base, but this time they are based on fuzzy logic. Now a ground instance of a

An interpretation of a program *P* is a fuzzy set of the program's Herbrand base, *BP*, i.e. it is:

A model *M* is least if for any model *N*, *M* ≤ *N*. A model *M* is minimal if there is not any

In the extensions of Datalog several implication operators are used, but all cases are restricted to min-max conjunction and disjunction, and to the complement to 1 as negation.

The semantics of fDATALOG is defined as the fixed points of consequence transformations. Depending on evaluating sequences two semantics can be defined: a deterministic and a nondeterministic one. Further on only the nondeterministic semantics will be discussed, the deterministic one is detailed in (Achs 2010). It was proved that the two semantics are equivalent in the case of negation- and function-free fDatalog programs, but they differ if the program has any negation. In this case merely the nondeterministic semantics is

**Definition 2.** Let *BP* be the Herbrand base of the program *P*, and let *F(BP)* denote the set of all fuzzy sets over *BP*. The consequence transformation *NTP* : *F(BP)* → *F(BP)* is defined as

> α*A )}* ∪

 *ground(P), (|Ai|,*

γ *| I(*α*body,* γ*)* ≥ β*}).*

It can be proved that this transformation has a fixed point. To prove it, let us define the

 *ground(P),*

α*A1 ,…,* α*An).*

α

*body* = *min(*

α*Ai)* ∈ *X, (1* ≤ *i* ≤ *n);*

*A= max(0, min{*

 *ground(P)}*

γ *| I(1,* γ*)* ≥ β*}) }* ∪

*NTP(X) = {(A,*

*A = max(0, min{*

α

*; I* in *P* is a rule obtained from *r* by replacing every variable in *r* with a constant of

*; I* is denoted by *ground(r);*

β

β

β

 *X ,* (1)

), because according to implication *I*, the level of *A* easily can be

*; I*. From now on, the

β.

*; I*. The ground instance

*; I)* ∈ *ground(P)*

There is a special type of rule, called fact. A fact has the form *A* ←*;*

*P (ground(r); I;*

*I(*α *A1*∧*...*∧ *An ,*α*A)* ≥ β*.*

α*A,* α*B)* and α¬*A = 1* − α*A*.

applicable. The nondeterministic transformation is as follows:

 *A1,…,An;*

*Ai*, if *Ai* is negative) and

For any *T : F(BP)* → *F(BP)* transformation let

β*; I )* ∈

α

← *; I;* β*)* ∈

> ∃ *(B* ← *...*¬ *A...; I;* β*)* ∈

*{(A, 0) |*

 *An* will be denoted as αbody and αA as αhead.

computed and in the case of the implication operators detailed in this chapter it is

β

β*)*.

*A).* An interpretation is a model of *P* if for each *(A* ← *A1,…,An;*

β

β*)*∈

*HP*. The set of all ground instances of *r;*

facts are referred as (*A,*

of *P* is *ground(P) =* ∪ *(r; I;*

α

model *N*, where *N* ≤ *M*.

α *A1*∧*...*∧

> α*A,* α*B),* α*A*∨*B = max(*

> > *(A* ←

rule *r;* β

∪ *<sup>A</sup>*∈*BP (A;*

To be short

So: α*A*∧*B = min(*

where

literal, and

¬

*T0 = {* ∪*{(A,*α*A)} | (A*

powers of a transformation:

$$T\_1 = T(T\_0)$$

$$T\_n = T(T\_{n-1})$$

$$\dots$$

$$T\_{\alpha} \text{=least upper bound of } \{ T\_n \mid n \le \alpha \} \text{ if } \alpha \text{ is a limit ordinal.}$$

An ordering relation can be defined over *F(BP)*. For *G; H : BP* → [*0; 1*]; *G* ≤ *H* iff *(*∀*d* ∈ *BP) G(d)* ≤ *H(d)*. It easily can be seen that *L = (F(BP),* ≤ *)* is a complete lattice.

It is clear that *NTP* is inflationary transformation over *L*, and if *P* is negation-free, then *NTP* is monotone as well. (A transformation *T* is inflationary if *X* ≤ *T(X)* for every *X* ∈ *L* and it is monotone if *T(X)* ≤ *T(Y)* if *X* ≤ *Y*).

In (Ceri et al 1990) it is shown that an inflationary transformation over a complete lattice has a fixed point and if it is monotone then it has a least fixed point (Loyd 1990). Therefore *NTP* has a fixed point, i.e. there exists an *X* ∈ *F(BP)* for which *NTP(X)* = *X*. If *P* is positive, then *X* is the least fixed point. (That is for any *Z=T(Z)* : *X* ≤ *Z*.)

The fixed point of the transformation will be denoted by *lfp(NTP)*. It can be shown (Achs 1995) that this fixed point is a model of *P*.

**Theorem 1**. *lfp(NTP)* is a model of *P*.

**Proof** In *ground(P)* there are rules in the next forms:

a/ *(A* ←*;* β*; I).* b/ *(A* ← *A1, ..., An;* β*; I); (A,* α*A)* ∈ *lfp(NTP)* and *(|Ai|,* α*Ai )* ∈ *lfp(NTP), 1*≤ *i* ≤ *n*. c/ *(A* ← *A1, ..., An;* β*; I);* ∃*i : (|Ai|,* α*Ai )* ∉ *lfp(NTP).*

In the case of a, b because of the construction of α*A* the condition *I(*α*body,* α*<sup>A</sup> )* ≥ β is realized. In the case of c because of the construction of *T0 Ai* is not negative, that is *Ai* is not among the facts, so α*Ai = 0*, therefore α*body = 0*, so *I(*α*body,* α*<sup>A</sup> ) = 1* ≥ β, namely *lfp(NTP)* is a model.

Moreover, the next proposition is true as well.

**Proposition 1**. For negation-free fDATALOG program *P lfp(NTP)* is the least model.

**Proof** In the case of a positive Datalog program, the least fixed point is the least model (Ceri et al 1990, Ullman 1988). In the case of fuzzy Datalog, according to the definition of the consequence transformation, the level of the rule's head is the least value satisfying the criterion of modelness. The application of the transformations may arise only one problem. A lower level would be ordered to the same rule's head, but according to the definitions we should accept the higher value. But such a case can arise only in the case of programs containing any negation. Therefore the proposition is true.

According to the above statements, the meaning of the programs can be defined by this fixed point:

**Definition 3.** *lfp(NTP)* is the nondeterministic semantics of fDATALOG *P* program.

From Fuzzy Datalog to Multivalued Knowledge-Base 31

As the next examples show, there some problems would arise if the program had any

*M1 = {(p(a), 0.7) } and M2 = {(q(b), 1) }.*

*lfp(NTP) = {(r(a), 0.8), (p(a), 0.6), (q(a), 0.5)},*

*lfp(NTP) = {(r(a), 0.8), (p(a), 0.5), (q(a), 0.5)}.*  According to the above examples, in the case of programs containing negation there are problems with the model's minimality. However, the nondeterministic semantics – *lfp(NTP)* – is minimal under certain conditions. These conditions are referred to as stratification. Stratification gives an evaluating sequence in which the literals are evaluated before

To stratify a program, it is necessary to define the concept of dependency graph. This is a directed graph, whose nodes are the predicates of *P*. There is an arc from predicate *p* to predicate *q* if there is a rule whose body contains *p* or ¬*p* and whose head predicate is *q*. A program is recursive, if its dependency graph has one or more cycles. A program is stratified if whenever there is a rule with head predicate *p* and a negated body literal ¬q,

The stratification of a program *P* is a partition of the predicate symbols of *P* into subsets