Q2Soft Computing
Question
Q.2. Let A be a fuzzy set that tells about a student as shown in the figure, where the linguistic variable P represents a Pass student, F stands for a Fair student, G represents a Good student, and VG represents a Very Good student. Triangular membership functions are shown with peaks approximately: P peak 0.6 at x=60, F peak 0.4 at x=70, VG and V peaks approximately 0.2 at x=80 and x=90, over the universe of discourse x=50 to 100. Calculate the defuzzified value of the fuzzy set A with weighted average method and center of sums.
Answer
Reading the given triangular membership plot, the peak (representative) points of each linguistic term and their corresponding membership heights are approximately: P at x=60 with mu=0.6, F at x=70 with mu=0.4, and VG and V at x=80 and x=90 respectively, each with mu=0.2 (these are the illustrative values read from the described figure; in an actual examination, the candidate reads these directly off the printed graph).
Weighted Average Method
The weighted average method defuzzifies by taking the sum of each peak value multiplied by its membership degree, divided by the sum of the membership degrees: x = sum(mu_i x_i) / sum(mu_i).
Center of Sums Method
The center of sums method is similar to the weighted average but uses the area (or, for simple triangular approximations without overlap correction, again the peak value times height as a simplified proxy for area) contributed by each fuzzy set's membership function, summed over all the sets, without subtracting overlapping regions counted twice (unlike the center of gravity method, which would require integrating the true union membership function). Using the same peak-height values as areas (a common simplification when only peak heights, not full triangle base widths, are given), the center of sums calculation gives the same numerical result of x* = 70 in this specific case, since both methods reduce to the same weighted-average-style computation when only representative peak points (rather than full membership function shapes) are used as the basis of calculation.
It should be noted that the exact numerical answer depends on the precise peak coordinates and heights read directly from the given figure in the original examination paper; the method demonstrated here, computing a mu-weighted average of the representative x-values for each linguistic term, is the standard, generally applicable procedure for both weighted average and (in this simplified peak-only form) center of sums defuzzification, and should be applied using the specific numerical values visible on the actual printed graph.