Fuzzy Multiple Objective Decision Making

219

16

Fuzzy Multiple

Objective Programming

in Interval Piecewise

Regression Model

Tanaka et al. (1982) introduced a fuzzy linear regression model with symmetric trian-

gular fuzzy parameters by using linear programming (LP). Since membership func-

tions of fuzzy sets are often described as possibility distributions, this approach is

usually called possibilistic regression analysis. The properties of possibilistic regres-

sion formulated by Diamond and Tanaka (1998), Tanaka (1987), Tanaka et al. (1982)

and Tanaka and Watada (1988) have been studied further. Kim et al. (1996), Moskowitz

and Kim (1993), and Tedden and Woodal (1994) discussed the degree of t of the fuzzy

linear model. The effects of outliers were also examined (Diamond and Tanaka, 1998).

Three shortcomings associated with the fuzzy regression model have been observed

(Yu et al., 2005). First, in possibility analysis, Tanaka’s methodologies were extremely

sensitive to outliers or high variabilities of data and they also ignored certain informa-

tion contained in the data (Kim et al., 1996; Tedden and Woodal, 1994). Second, in

necessity analysis, the necessity area cannot be obtained due to large variations in the

data or an inappropriate model (Tanaka et al., 1989; Tanaka and Ishibuchi, 1992).

A necessity model indicates than the assumed model is somewhat reliable.

Therefore, Tanaka and Lee (1998) proposed a measure of tness, which is the ratio

of necessity spread divided by possibility spread and averaged over the sample size.

However, the approximation model cannot be obtained if the required measure of

tness is set too high. This model must be analyzed with respect to the data property,

rather than mere addition of the terms of the polynomial. These issues show the dif-

culty of managing data with large variations by applying a polynomial or non-linear

form. Therefore, the piecewise concept to manage data with large variations was

proposed (Yu et al., 1999 and 2001).

Third, when we use LP in possibilistic regression analysis, some coefcients tend

to become crisp because of the characteristic of LP. This shortcoming can be alle-

viated by quadratic programming (QP) proposed by Tanaka and Lee (1998). They

devised an interval regression analysis based on QP (Best, 1984; Gill and Murray,

1978; Goldfarb and Indani, 1983) to obtain the possibility and necessity models

simultaneously. In their unied approach, they assumed simplicity so that the center

coefcients of the possibility regression and the necessity regression model are the

same. For a data set with crisp inputs and interval outputs, the possibility and the

necessity models can be considered at the same time.

220 Fuzzy Multiple Objective Decision Making

Fuzzy piecewise possibility and necessity regression models (Yu et al., 1999,

2001, and 2005) are employed when a function behaves differently in different parts

of the range of crisp input variables. Yu et al.’s 2001 paper requires the analyst to

set the number of change points so that the positions of change points and the fuzzy

piecewise regression model can be obtained simultaneously. The proper number of

change points is still a problem. Hence, we incorporate the concepts of measure of

tness (Tanaka and Lee, 1998), interval piecewise regression (Yu et al., 2005), and

multiple-objective technique (Ida and Gen, 1997; Lee and Li, 1993; Li and Lee,

1990; Zimmermann, 1978) to nd the measure of tness and the number of change

points, considering all objectives.

The tness measure should be as high as possible. However, due to the parsimoni-

ous rule, the number of change points should be as few as possible. Therefore, the

three objectives of minimizing the number of change points, maximizing tness,

and minimizing the objective of obtaining the regression models (Yu et al., 2005)

are formulated.

16.1 INTRODUCTION TO MEASURE OF FITNESS AND

FUZZY MULTIPLE OBJECTIVE PROGRAMMING

Tanaka and Lee (1998) dened a measure of tness to gauge the similarities of the

obtained possibility and necessity regression models. The larger the tness, the bet-

ter the model ts the data. The measure of tness can be introduced as the overlap of

the possibility and necessity models as explained below.

Assume that the input–output data (x

j

; Y

j

) are (x

j

; Y

j

) = (x

1j

, …, x

qj

; Y

j

), j = 1, …, n

where x

j

is the jth input vector and Y

j

is the corresponding interval output. The pre-

dicted possibility and necessity models are as follows:

Yaax ccxd dx

Yaax ccx

Possibility model: (x ),||

||

Necessity model: (x ),||

ji

i

q

ij i

i

q

ij i

i

q

ij

ji

i

q

ij i

i

q

ij

*

0

1

0

1

0

1

*0

1

0

1

∑∑ ∑

∑∑

=+ +++













=+ +













== =

==

aa

x

i

q

ii

j0

1

+∑

=

represents the center of possibility and necessity models and

cc

xd

dx

||

||

i

q

iij

i

q

ii

j0

1

0

1

+∑ ++∑

==

and

cc

x

||

i

q

ii

j0

1

+∑

=

represent the radii of the possi-

bility and necessity models, respectively. The measure of tness for all data

Y

φ

can

be dened as:

n

ccx

ccxd dx

1

||

||

||

Y

i

i

q

ij

i

i

q

ij i

i

q

ij

j

p

0

1

0

1

0

1

1

∑

∑∑

∑

φ=

+

+++





























=

==

=