MITESD_77S10_soln04 (1)

# MITESD_77S10_soln04 (1) - 16.888/ESD 77 Multidisciplinary...

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16.888/ESD 77 Multidisciplinary System Design± Optimization:± Assignment 4 Part a) Solution± Part A1 (a) Increasing mutation rate results in increased population density. (b) The required number of bits: n nbits = f ( n , Δ x d , ) = ln x UB x LB ln d = 17 i = 1 ⎣⎝ Δ x ⎠⎦ (c) Using the same formula as above, we get: nbits = 2. (d) The binary to decimal conversion can be written as: d = (binary to decimal conversion) + x min Here x =1 and therefore to get a decimal number 3, we should look for a binary string min that represent decimal number 2. The corresponding binary string is 10 (i.e., 1.2 2 1 + 0.2 0 = 2 ). (e) The correct answer is (b) – B&C. ± A is not possible because it’s starting bit is 1, but both parents starting bit is 0. D is not± possible since it’s 4 th bit is 1, but that of both parents is 0. ± F 100 (f) p = 1 = = 0.10 . F 1 5 F 1000 i i = (g) In this population of size 5, only F 2 has a better fitness than F 1. Hence, assuming week dominance, F 2 would be selected in 4 out of 5 instances. The only time it would lose when it is compared to F 2 . Therefore the probability that F 1 would be selected is 4/5 = 0.80. (h) A 16 bit variable can have 2 16 possible values and a 4 bit variable can have 2 4 possible± values. ± Hence the total number of possible population members = 2 16 . 2 4 =2 20 . ± 1

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Part A2 This problem has 5 continuous variables (i.e., the c/s dimensions of I beams and supporting column) and 3 integer variables (i.e., number of beams and material identifiers). Therefore, we can use heuristic algorithms like Simulated Annealing, Genetic Algorithm; or Branch and Bound Algorithms, or perform explicit enumeration on integer variables while optimizing for 5 continuous variables (this would mean 4*4*4 = 64 gradient-based optimization runs for each possible inter parameter settings). This problem also brings out the role of constraints and bounding characteristics in MDO problem formulations. The first step would be to use some “physical” bounds on the cross-sectional dimensions. For example, one cannot produce an I-beam of thickness 0 or 1 mm or it is impractical for the height of I-beam to go over say, 5 m in an over-bridge construction. In addition, for it to be an I-beam, impose two geometrical constraints:
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MITESD_77S10_soln04 (1) - 16.888/ESD 77 Multidisciplinary...

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