The Marketing Book 5th Edition

216 The Marketing Book

4 The fuzzy mathematical method is easier to
perform than the traditional method, once the
membership of assessment facets is defined.

Some of the main characteristics, advantages,
limitations and applications of simulation
and fuzzy sets in marketing are presented in
Table 9.8.

Fuzzy decision trees

Inductive decision trees were first introduced
in 1963 with the Concept Learning System
Framework. Since then, they have continued to
be developed and applied. The structure of a
decision tree starts with a root decision node,
from which all branches originate. A branch is a
series of nodes where decisions are made at
each node, enabling progression through
(down) the tree. A progression stops at a leaf
node, where a decision classification is given.
As with many data analysis techniques
(e.g. traditional regression models), decision
trees have been developed within a fuzzy
environment. For example, the well-known
decision tree method ID3 was developed to
include fuzzy entropy measures. The fuzzy
decision tree method was introduced by Yuan
and Shaw (1995) to take account of cognitive
uncertainty, i.e. vagueness and ambiguity.
Central to any method within a fuzzy
environment is the defining of the required
membership functions. Incorporating a fuzzy
aspect (using membership functions) enables
the judgements to be made with linguistic
scales.

Summary of fuzzy decision tree

method

In this section a brief description of the func-
tions used in the fuzzy decision tree method are
exposited. A fuzzy set A in a universe of
discourseUis characterized by a membership
functionA, which take values in the interval
[0, 1]. For all U, the intersection ABof
two fuzzy sets is given by AB= min(A(u),
A(u)).

A membership function (x) of a fuzzy

variableYdefined on Xcan be viewed as a

possibility distribution of YonX, i.e. (x) =

(x), for all xX. The possibilistic measure of

ambiguity –E(Y) – is defined as:

E(Y)=g( )=

n

i=1

( *i– *i+1) 1n[i],

where *={ * 1 , * 2 ,.. ., *n} is the permutation

of the possibility distribution ={ (x 1 ),

(x 2 ),.. ., (xn)}, sorted so that *i≥ *i+1for

i= 1,.. ., n, and *n+1=0.

The ambiguity of attribute Ais then:

E(A)=

1

m

m

i=1

E(A(ui)),

where E(A(ui)) = g(Ts (ui)/max

15 j 5 s

(Tj(ui))),

with Tj the linguistic scales used within an

attribute.

The fuzzy subsethood S(A, B) measures the

degree to which Ais a subset of B(see Kosko,

1986) and is given by:

S(A,B)=

uU

min(A(u),B(u))

uU

A(u)

Given fuzzy evidence E, the possibility of

classifying an object to Class Cican be defined

as:

=(CiE)=

S(E,Ci)

max

j

S(E,Cj)

where S(E,Ci) represents the degree of truth for

the classification rule. Knowing a single piece

of evidence (i.e. a fuzzy value from an attrib-

ute), the classification ambiguity based on this

fuzzy evidence is defined as:

G(E)=g( (CE))

The classification ambiguity with fuzzy parti-

tioningP= {E 1 ,.. ., Ek} on the fuzzy evidence F,

denoted as G(P|F), is the weighted average of