5.2 MULTI-OBJECTIVE FUZZY PATTERNRECOGNITION MODEL (MOFPR)Ifone assumes that a decision making problem is to identify an optimum value fromn alternatives in which each one has m objectives, the values of mobjectives in n alternatives can form an objective value matrix asfollows: where denotes the value ofobjective i in the alternative j (i = 1,2, …, m; j = 1,2,…

,n).There existdifferences between values and units of m objectives in matrix X.Furthermore, there are positive and negative correlations between the optimumvalue and its evaluation objectives. Hence it is necessary to normalize theelements of a matrix X.

Ifthe optimum value and a particular factor are positively correlated, i.e. thebigger the factor value, the larger the membership degree to the optimum, thenormalizing formula is defined as:Alternatively,the normalizing formula for the negatively correlated factor is defined as:                                                                                    In formulae (3)and (4), denotes the absolute ora relative optimum value for objective i; and denotes thecorresponding minimum value. After normalizing, the matrix X becomes anormalized matrix R in which thevalues are within the interval 0,1.Inmatrix R, if  = 1, the alternative j is the optimum and if  = 0, the alternative j is the worst, according to theobjective i only. Supposing that there is an ideal optimum alternative in which all objective membership degreesto the optimum are equal to 1, denoted by , the worst alternativeis expressed as Inthis case, the decision-making problem becomes a fuzzy pattern recognitionproblem, i.e. evaluating to what membership degree each alternative in matrix R belongs to the ideal optimum.

Becausedifferent objectives have different contributions in the process of evaluatingan alternative, different weights should be given to m objectives. Theweighting vector is denoted by subject to arestriction,In matrix R, alternative j can be expressed as The distance ofalternative j to the w worst alternative can be described asThe distance ofalternative j to the worst alternative can be described asIn equations(5.7) and (5.8), p is a distanceparameter. When p = 1 and p = 2, the distances are calledHamming 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.Ifthe 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 betweenalternative j and the optimum or the worst. The weighted distance to theoptimum of alternative,j can bedescribed asSimilarly, theweighted distance of alternative j to the worst can be described asInorder to solve optimal membership degree uj, an objectivefunction is established as follows:=   +                                                                                                (5.

11)Using the condition,   Amulti-objective fuzzy pattern recognition model can be obtained:According tothis model the bigger the uj, the better the alternative j.5.3 MOFPR MODEL TO EVALUATE THEGROUND WATER   VULNERABILITY USING THEDRASTIC SYSTEMAquifervulnerability and its evaluation have an intrinsic property, i.e. fuzziness.

Bythe DRASTIC system, this fuzziness is taken into account by dividing the valuesof each affecting factor into ranges, and then assigning a rating to eachrange. However, it should be noted that if a factor value can be measurednumerically, the fuzziness should be described continuously rather than in themanner of ranges that are also difficult to be determined. The membershipdegree of ‘”vulnerability” can just describe the fuzzinesscontinuously and efficiently. For example, factor D is divided into seven rangeswhich, in the DRASTIC system, are assigned seven ratings respectively, butusing the MOFPR model, the membership degree decreases continuously from 1 to 0calculated by equation (5.

4), i.e.                                                                  = 

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