5.2 MULTI-OBJECTIVE FUZZY PATTERN

RECOGNITION MODEL (MOFPR)

If

one assumes that a decision making problem is to identify an optimum value from

n alternatives in which each one has m objectives, the values of m

objectives in n alternatives can form an objective value matrix as

follows:

where denotes the value of

objective i in the alternative j (i = 1,2, …, m;

j = 1,2,…,n).

There exist

differences between values and units of m objectives in matrix X.

Furthermore, there are positive and negative correlations between the optimum

value and its evaluation objectives. Hence it is necessary to normalize the

elements of a matrix X.

If

the optimum value and a particular factor are positively correlated, i.e. the

bigger the factor value, the larger the membership degree to the optimum, the

normalizing formula is defined as:

Alternatively,

the normalizing formula for the negatively correlated factor is defined as:

In formulae (3)

and (4), denotes the absolute or

a relative optimum value for objective i; and denotes the

corresponding minimum value. After normalizing, the matrix X becomes a

normalized matrix R in which the

values are within the interval 0,1.

In

matrix R, if = 1, the alternative j is the optimum and if = 0, the alternative j is the worst, according to the

objective i only. Supposing that

there is an ideal optimum alternative in which all objective membership degrees

to the optimum are equal to 1, denoted by , the worst alternative

is expressed as

In

this case, the decision-making problem becomes a fuzzy pattern recognition

problem, i.e. evaluating to what membership degree each alternative in matrix R belongs to the ideal optimum.

Because

different objectives have different contributions in the process of evaluating

an alternative, different weights should be given to m objectives. The

weighting vector is denoted by subject to a

restriction,

In matrix R, alternative j can be expressed as

The distance of

alternative j to the w worst alternative can be described as

The distance of

alternative j to the worst alternative can be described as

In equations

(5.7) and (5.8), p is a distance

parameter. When p = 1 and

p = 2, the distances are called

Hamming and Euclidean distances respectively, which are commonly used . It can be seen from equations (5.7) and (5.8)

that if djg = 0, then alternative j is the optimum and if djb

= 0, then alternative j is the worst.

If

the membership degree to the optimum is denoted by for alternative j ,(1

-) is its membership degree to the worst.

In the view of fuzzy sets, the membership degree may be regarded as a weight.

Thus, the equation (5.9) or (5.10) will better describe the difference between

alternative j and the optimum or the worst. The weighted distance to the

optimum of alternative,j can be

described as

Similarly, the

weighted distance of alternative j to the worst can be described as

In

order to solve optimal membership degree uj, an objective

function is established as follows:

= +

(5.11)

Using the condition,

A

multi-objective fuzzy pattern recognition model can be obtained:

According to

this model the bigger the uj, the better the alternative j.

5.3 MOFPR MODEL TO EVALUATE THE

GROUND WATER VULNERABILITY USING THE

DRASTIC SYSTEM

Aquifer

vulnerability and its evaluation have an intrinsic property, i.e. fuzziness. By

the DRASTIC system, this fuzziness is taken into account by dividing the values

of each affecting factor into ranges, and then assigning a rating to each

range. However, it should be noted that if a factor value can be measured

numerically, the fuzziness should be described continuously rather than in the

manner of ranges that are also difficult to be determined. The membership

degree of ‘”vulnerability” can just describe the fuzziness

continuously and efficiently. For example, factor D is divided into seven ranges

which, in the DRASTIC system, are assigned seven ratings respectively, but

using the MOFPR model, the membership degree decreases continuously from 1 to 0

calculated by equation (5.4), i.e.

=