MITESD_77S10_assn03 (1)


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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO 16.888/ESD.77J Multidisciplinary System Design Optimization (MSDO) Spring 2010 Assignment 3 Instructors: Prof. Olivier de Weck Prof. Karen Willcox Dr. Anas Alfaris Dr. Douglas Allaire TAs: Andrew March Kaushik Sinha Issued: Lecture 9 Due: Lecture 14 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 : sensitivity analysis, gradient based search algorithms, comparison of algorithm performance, single objective optimization of team design projects Part (a) Sensitivity Analysis Part A1 In lecture 9, numerous methods of estimating derivatives were discussed. You are going to test their accuracy on a set of given functions. You should estimate the first-derivative using (1) a first-order finite-difference, (2) a second-order central-difference, and (3) a complex step estimate. In addition you should estimate the second-derivative using (1) a second-order estimate, and (2) a complex step second-derivative estimate. Note for the complex step, the second derivative estimate is: f ′′ ( x ) = 2 2 [ f ( x ) − ℜ e ( f ( x + i ⋅ ∆ x ) )] . x Please plot the error between the analytical first-derivatives and your approximations on one plot and the error between the analytical second-derivative and your approximations on a second plot both using a log-scale (the command loglog() in M ATLAB® ). Please use step sizes from 1x10 -15 to 10 (the command logspace() in M ATLAB may help). (a) For the function f(x)=x 2 , at x=1. (b) For the function f(x)=x 3 , at x=1. (c) For the function f(x)=e x , at x=1. Page 1 of 5
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO (d) Please comment about the change in the accuracy of the central-difference and the complex step between (a) and (b). (e) Does any step size appear to estimate the first-derivative well, the second-derivative? Caution: if the error is zero it will not be shown on a log-scale so please take care in plotting your results. Part A2 This problem is to use sensitivity analysis on revenue management for a very simplified airline pricing model. We will assume that an airline has one flight per day from Boston to Atlanta and they use an airplane that seats 150 people. The airline wishes to determine three prices, p i (i=1,2,3), one for seats in each of the three fare buckets it will use. The fare buckets are designed
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This note was uploaded on 11/08/2011 for the course AERO 16.851 taught by Professor Ldavidmiller during the Fall '03 term at MIT.

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