MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO 16.888/ESD.77J Multidisciplinary System Design Optimization (MSDO) Spring 2010 Assignment 4 Instructors: Prof. Olivier de Weck Prof. Karen Willcox Dr. Anas Alfaris Dr. Douglas Allaire TAs: Andrew March Kaushik Sinha Issued: Lecture 13 Due: Lecture 17 You are expected to solve Part (a) individually and Part (b) in your project team. Each person must submit their own Part (a) but you should submit Part (b) as a group. Please indicate the name(s) of your teammate(s). Topics: genetic algorithms, mixed-integer optimization, scaling Part (a) Part A1 The following questions refer to a Genetic Algorithm as defined in Lecture 10: a) Which of the following will most increase population diversity? a. Increasing mutation rate b. Changing the crossover location c. Increasing the amount of elitism d. All of the above e. None of the above b) In a binary GA, how many bits are needed to represent numbers between 1 and 10 with a resolution of 0.0001? a. 3 b. 4 c. 16 d. 17 e. 18 c) In a binary GA, what is the minimum number of bits required to represent a discrete design variable that can have values, 1, 2, 3 or 4? Page 1 of 6MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO a. 1 b. 2 c. 3 d. 4 d) For a discrete variable that can have the values, 1, 2, 3, or 4, in the binary representation of this variable with the minimum number of bits, what would the representation of 3 be? a. 1 b. 01 c. 10 d. 111 e. 010 f. 0110 e) Given the parents 01001101 and 01100100, which of the following are possible single point crossovers? A: 11001101 B: 01000100 C: 01001100 D: 01111101 a. A & B b. B & C c. A & D d. A & C e. A, B, C & D f) In a population, the fitness function values of each member are: F1=100, F2=800, F3=1, F4=90, F5=9. Using roulette wheel selection, what is the approximate probability the member with F1 will be chosen as a parent? a. 0.01 b. 0.10 c. 0.80 d. 0.90 e. 0.99 g) In a population, the fitness function values of each member are: F1=100, F2=800, F3=1, F4=90, F5=9. Now, using the fraction of the population a member dominates as the selection criterion, what is the approximate probability the member with F1 will be chosen as a parent? a. 1 b. 0.90 c. 0.80 d. 0.50 e. 0.10 Page 2 of 6MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO h) In a binary encoded GA with two design variables, one with 4 bits, and one with 16 bits, what is the total number of possible population members? a. b. c. d. 218 220 224 232 Part A2 Your objective is to design the cheapest possible bridge to span a highway. The total span of the bridge (across both halves of the highway) must be L=30 meters, and it must support its own weight and a load q=33x104 N/m along its span (a total load of about 1x106 N plus its own weight). I-beams Middle Support The bridge span will be supported by between one and four I-beams. In the figure above, the I-beams would be parallel to each other going into the page, for example it could be one I-beam in the middle of the bridge, or one I-beam on both sides of the bridge, etc. and this will be represented by the design variable, nIbeams. The shape of the I-beams will be represented by three continuous design variables, the height, h, flange width, b, and thickness, t. The middle support will be rectangular (when viewed from above), and will have two design variables, the width, w, and depth, d. b w dh t Where, ρIbeams, is the density of the material used for the I-beams, the mass of the I-beams can be computed using, M Ibeams =[2bt +(h − 2t)t]LρIbeamsnIbeams , and where, ρSupport , is the density of the material used for the support, and H=5m is the height of the bridge above the ground, the mass of the middle support can be computed using, M Support = wdHρSupport . Page 3 of 6MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO There is a constraint that the stress the I-beam is less than the material failure stress for the I-beam, σFailure-Ibeams. Note: g is the gravitational constant (9.81 m/s2). q⎜⎛ L ⎟⎞2 + M Ibeams ⎜⎛ L ⎟⎞g σ =⎝ 2 ⎠ ⎝ 4 ⎠ ⎛⎜ h ⎞⎟ ≤σIbeams Failure−Ibeams8IIbeamnIbeams ⎝ 2 ⎠ Where, IIbeam, is the moment of inertia for the I-beam given by, (h − 2t)3 t ⎡t3b ⎛ h t ⎞2 ⎤ IIbeam = + 2⎢+ tb⎜ −⎟⎥ . 12 ⎢⎣ 12 ⎝ 2 2 ⎠ ⎥⎦ In addition that the shear stress in the I-beams is less than the material failure stress: τ Ibeams = M Ibeamsg + qL ≤σ Failure−Ibeams .4[2bt +(h − 2t)t]nIbeams For the middle-support there are two constraints, the column cannot buckle and the stress must be less than the material failure stress. Buckling is based on a requirement that the applied load is less than a critical load, PApplied = M Ibeams 2 g + qL ≤ PCrit Where the critical load is a function of the lowest moment of inertia of the support and the modulus of elasticity of the support material, Esupport, ⎧ w3d wd 3 ⎫π 2 ESupport min⎨⎩ 12 ,12 ⎬⎭PCrit = 2.4H The stress requirement is that the applied stress is less than the support material failure stress, PAppliedσ Support = wd ≤σ Failure−Support The bridge span (I-beams) can be made from Al 6061, A36 Steel, A514 Steel, or Titanium; however, the support can be made from Al 6061, A36 Steel, A514 Steel, or Concrete. The reason for the difference is that concrete cannot be loaded in tension. The material properties and prices are listed in the Table: Material Density (kg/m3) Modulus of Elasticity (GPa=109N/m2) Failure Stress (MPa=106N/m2) Cost ($/kg) Al 6061 2700 70 270 2.05 A36 Steel 7850 210 250 0.62 A514 Steel 7900 210 700 0.90 Titanium 4500 120 760 16.00 Concrete 2400 31 70 0.04 Your objective is to find the dimensions of the I-beams, number of I-beams, and material type for the I-beams, as well as the dimensions of the support and material type for the support to minimize cost of the bridge. Where cIbeams, and cSupport are the cost per kilogram of the materials used for the I-beams and Support, the total bridge cost is: C = cIbeamsM Ibeams + M SupportcSupport Page 4 of 6MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO a) Please explain what optimization algorithm you chose to find the cheapest possible bridge and why. b) What are the design variables and cost for the cheapest bridge? Part A3 Consider the 8-dimensional function: f (x) = 12 (0.0001x12 + 0.001x22 + 0.01x32 + 0.1x42 + x52 +10x62