Though choice theory based on utility maximisation is commonly applied in understanding individual preferences, insights from other approaches to decision theory state that human decision-making process is more dependent on few
normative rules and hence heuristics play an important role in the decision process.
Moreover, the assumption of rationality in choice theory implies that decisions are based on a consistent line of reasoning which could have their basis on some normative rules (Klein and Methlie, 1990). The fuzzy logic (FL) theory is one technique which allows for heuristics in decision-making.
The FL theory is based on the fuzzy set theory. Different from the classical set theory where an element can either belong to a set or not, the fuzzy set theory allows an element to belong to multiple sets with varying levels of membership.
Hence, while classical logic, based on reasoning with precise propositions, is a two-valued logic where the truth value of a statement can either be TRUE or FALSE, the truth value of a statement under fuzzy logic, which applies approximate reasoning to imprecise propositions, can assume varying values (Chen and Pham, 2001).
Let us consider a set ‘OLD’ and an element x = ‘66 years’, the difference between classical and fuzzy sets can thus be given in the following manner:
Assuming the following rule pertaining to age classification:
IF x ≥ 70 THEN OLD
The membership of x = ‘66 years’ under the classical and fuzzy sets can be graphically represented as follows:
Figure 4.3 Classical and Fuzzy set representations OLD x
OLD x
Fuzzy Set Representation Classical Set Representation
The above figure reveals that under the assumed rule, the element x does not belong to the set OLD with the classical set representation; however, in case of fuzzy set, the element belongs to the set with some level of membership.
In relation to the theory of fuzzy sets and the example given, it is evident that
‘membership function’ plays a crucial role in the FL theory.
Membership of an element is given by the degree to which the element belongs to a set while the function mapping the degrees of potential membership is termed as the membership function (Kasabov, 1996). Recalling the example of age, the membership function can be mathematically denoted as follows under both the set theories:
Let denote the level of membership for the element x in set A. Under classicalA set theory,
( ) 1
A x
if x70 ( ) 0
A x
if x70
Under fuzzy set theory, the membership of the element can be given as:
( ) : [0,1]
A x U
This equation implies that set A is a fuzzy subset of the universal set U and the membership level of the element x in set A can take the value from 0 to 1, where 0 means that the element completely does not belong to the set while 1 implies the contrary (Kasabov, 1996).
Generating subjective membership functions for different age categories, the membership of x = 66 in set ‘OLD’ can be given as follows:
Figure 4.4 Graphical Representation of Membership functions
In this case, from the above figure, it can be seen that x = 66 belongs to the set
‘OLD’ with 80% of membership. The concept of membership and membership functions thus forms a crucial component in the FL theory. Though FL is an expert decision system where many system components such as membership function and rules are specified by an expert, several methods for specifying the membership function as well as fuzzy rules can be identified. The methods of specifying these can take any of the following forms:
1. Eliciting an expert’s opinion: this can either take the form of enquiring from an expert of the knowledge domain or where public surveys are concerned, eliciting required information from the respondent
2. Understanding the physical environment: the system can be designed based on the designer’s understanding of the physical meaning surrounding the technique used. In this case, the designer serves as an expert of the system 3. Employing machine learning: through the use of neural networks, genetic
algorithms or other methods of machine learning, rules and membership functions can be derived from the data (Kasabov, 1996)
Besides the membership functions, as necessary in the case of classical logic, rules form an integral part of the FL system. The rules in the FL system are termed as fuzzy rules as they pertain to the vagueness associated with the variables, which normally assume a linguistic form. As in classical logic, the rules follow an IF-THEN structure with the only striking characteristic that the associated variables are fuzzy representations. While the effect of operators associated with the rules base, such as AND, OR, NOT are similar to that of Boolean logic, FL rules need to be consistent (such that rules are not contradictory), complete (all possible rules are formed) and concise (there is absence of redundancy) (Chen and Pham, 2001).
Besides fuzzy rules, the FL system comprise of two other main components: the input variables and the output variable. These three components thus form the fuzzy inference system (FIS) which maps the input and output variables. To illustrate, the FIS for four inputs (view, noise, sunlight and charge), can be represented as follows:
Figure 4.5 The fuzzy inference system
In this case, the four input variables as well as the output variable (choice) are linked through the fuzzy rules. In order to map the relation between the numeric input and output in FIS, several steps are conducted. As an important characteristic of FL is its ability to deal with vague and imprecise data, linguistic characterisation forms an important aspect of the FIS which is carried out through the linguistic
membership function associated with each of the variables. The conversion of numeric input values to a linguistic category is termed ‘fuzzification’. The fuzzification process detects the membership function associated with each input variable and the degree to which the input variable matches the conditional element in the rule is computed. For each of the rule incorporated in the FIS, the associated values in the output membership function is computed using minimum (which truncates the membership function) or product (which scales the membership function) method. This process is called implication. Combining the effect of all rules on the output membership function forms the aggregation process while defuzzification computes a single numeric output from the aggregate membership function (Kasabov, 1996; MATLAB Handbook, R2007a).
Based on the form of the output variable and the type of fuzzy rule, two main types of fuzzy systems exist within the Fuzzy Logic Toolbox in MATLAB. These are the Zadeh-Mamdani and the Takagi-Sugeno fuzzy rules. The Zadeh-Mamdani fuzzy rule takes the following form:
IF x is A THEN y is B
Where x is the input variable, y is the output variable while A and B are fuzzy sets.
In case of the Takagi-Sugeno system, the output variable is a mathematical function and thus the rule takes the following form:
IF x is A and y is B THEN z is f (x, y)
Where x and y are the input variables, z is the output variable, while A and B are fuzzy sets (Kasabov, 1996).
It can be thus noted that in case of the Mamdani method, the output variable consists of membership functions which allows for linguistic representation in the rule base. In case of the Takagi-Sugeno method however, the output variable is a numeric computation.
The overview of the FL theory and the method of FIS have revealed that this method can be especially important in analysing linguistic data and heuristic based system. The output variable in the FIS can be specified over a range of 0-1 such that it reflects the choice associated with each rule. The specific applicability of this analytical method in relation to the research focus of this thesis stems from the examination and performance of the FL system to analyse each of the different representation data. Based on the characteristics of the FL theory and system, it can be hypothesised that data based on linguistic representation will be better analysed using this method. Moreover, the requirements for the FIS as well as the number of rules required for the system can vary based on the method of attribute input and representation method used during the survey. As a heuristic based system, this analytical method can also provide some insight on the decision-making process of individuals by examining the rules required in the system for optimum output.
While this method of analysis is commonly applied in physical systems where accurate observations can be made and more general rules developed, increasing applications are found in social surveys. The following section provides an overview of the FL application related to choice experiment as well as noise annoyance.