MA 404 If the fuzzy set "intensity about 7" is defined as 0.1 17 + 0.6 1 + + 0.8 5 6 7 + (eq.94-4) determine the fuzzy membership of I on the universe of accelerations, A. Question 5 [4 marks [A Total of 24 marks for Question (a) The fuzzy sets A, B, and C are all defined on the universe X = following membership functions: 2r HB(x)=2³, μc(x) = (0,5) with the 1 HA(x)= = 1+5(x-5)²' 2+5 Find the intervals along the z axis corresponding to the A-cut sets for each of the fuzzy sets A, B, and C for the following values of A: (i) = 0.2 (ii) A=1 5 marks (b) Determine the crisp A-cut relations for A = 0.1j, for j = 0, 1, for the following fuzzy relation matrix R: [0.2 0.7 0.4 1 1 0.9 0.5 0.1 R= 0 0.8 1 0.6 0.2 0.5 1 0.3 (cy-Q5-1) 5 marks (e) Two companies bid for a contract. A committee has to review the estimates of thosecompanies and give reports to its chairperson. The reviewed reports are eval uated on a nondimensional scale and assigned a weighted score that is represented bya fuzzy membership function, as illustrated by the two fuzzy sets, B₁ and B₂, in Figure Q5(c). The chairperson is interested in the lowest bid, as well as a metricto measure the combined "best" score. For the logical union of the membershipfunc tions shown, we want to find the defuzzified quantity. For each of the following methods calculate the defuzzified value, 2*, if possible. (i) Centroid method () Weighted average method (iii) Mean max membership " [10 marks] 0.7 0.5 A 3 4 (3) 1 By 05 6 2 0 1 Figure Q5(c) 2 3 4 5 6 (b) Question 4 (a) Given the continuous, noninteractive fuzzy sets A and B on universes X and Y. using notation for continuous fuzzy variables, 1-0.1|1 = for x € [0, +10] B = {[0.2y|} for y € [0, +5] (eq.Q4-1) (eq.Q4-2) i. Construct a fuzzy relation R for the Cartesian product of A and B. [4 marks] ii. Use max-min composition to find B', given the fuzzy singleton A' = 13. [4 marks] Hint: You can solve this problem graphically, or you can approximate the continuous fuzzy variables as discrete variables and use matrix operations. In any case, sketch the solution. (b) In the statistical characterization of fractured reservoirs, the goal is to classify the geology according to different kinds of fractures, which are mainly tectonic and re- gional fractures. The purpose of this classification is to do critical simulation based on well data, seismic data, and fracture pattern. After pattern recognition(using Cauchy-Euler detection algorithms or other methods) and classification of the frac- ture images derived from the outcrops of fractured reservoirs, a geological engineer can get different patterns corresponding to different fracture morphologies. Suppose the engineer obtains five images (I1,..., Is) from five different outcrops of fractured reservoirs, and their percentage values corresponding to three kinds of fractures (tectonic fracture, regional fracture, and other fracture), as given below: Table Q4 1: را 15 Tectonic fracture 0.6 0.6 0.3 0.5 0.2 Regional fracture 0.3 0.1 0.2 0.2 0.6 Other fracture 0.1 0.3 0.5 0.3 0.2 i. Develop a similarity relation using the cosine amplitude method. [4 marks] [4 marks] ii. Develop a similarity relation using the max-min method. iii. Since the similarity relation will be a tolerance relation, find the associated equivalence relation. [4 marks] = (c) Relating earthquake intensity to ground acceleration is an imprecise science. Sup- pose we have a universe of earthquake intensities (on the Mercalli scale), I (5,6,7,8,9), and a universe of accelerations, A {0.2, 0.4, 0.6, 0.8, 1.0, 1.2), in gravity. The following fuzzy relation, R. exists on the Cartesian space 1 x A: 0.2 0.4 0.6 0.8 1.0 1.2 5 [0.75 I 0.85 0.5 0.2 0 6 0.5 0.8 1 0.7 0.3 0 R 7 0.1 0.5 0.8 1 0.7 0.1 (eq.Q4-3) 8 0 0.2 0.5 0.85 1 0.6 9 0 0 0.2 0.5 0.9 1