5.2 the values of m objectives in

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:

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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.

                                                                  =