fundamental-engineering-optimization-methods.pdf

Establishing the feasible region this is done by

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1. Establishing the feasible region. This is done by plotting the constraint boundaries. 2. Plotting the level curves (or contours) of the cost function and identifying the minimum. The graphical method is normally implemented in a computational software package such as Matlab © and Mathematica ©. Both packages include functions that aid the plotting and visualization of cost function contours and constraint boundaries. Code for Matlab implementation of graphical optimization examples considered in this chapter is provided in the Appedix. Learning Objectives: The learning goals in this chapter are: 1. Recognize the usefulness and applicability of the graphical method. 2. Learn how to apply graphical optimization techniques to problems of low dimensions. 3.1 Functional Minimization in One-Dimension Graphical function minimization in one-dimension is performed by computing and plotting the function values at a set of discrete points and identifying its minimum value on the plot. We assume that the feasible region for the problem is a closed interval: ܵ ൌ ሾݔ ǡ ݔ ሿǢ then, the procedure can be summarized as follows: 1. Define a grid over the feasible region: let W ݔ ൌ ݔ ൅ ݇ߜǡ ݇ ൌ Ͳǡͳǡʹǡ ǥ Z where ߜݔ defines the granularity of the grid. 2. Compute and compare the function values over the grid points to find the minimum. An illustrative example for one-dimensional minimization is provided below. Example 3.1: Graphical function minimization in one-dimension Let the problem be defined as: Minimize ݁ subject to ݔ ൑ ͳ ² Then, to find a solution, we define a grid over the feasible region as follows: OHW ൌ ͲǤͲͳǡ ݔ ൌ െͳǡ െͲǤͻͻǡ ǥ ǡ െͲǤͲͳǡͲǡͲǤͲͳǡǥ ǡͲǤͻͻǡͳ ² Then, ݂ሺݔሻ ൌ ݁ ିଵ ǡ ݁ ି଴Ǥଽଽ ǡ ǥ ǡ ݁ ି଴Ǥ଴ଵ ǡ ͳǡ ݁ ଴Ǥ଴ଵ ǡ ǥ ǡ ݁ ଴Ǥଽଽ ǡ ݁ ² By comparison, ݂ ௠௜௡ ൌ ݁ ିଵ DW ݔ ൌ െͳǤ
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 36 Graphical Optimization 3.2 Graphical Optimization in Two-Dimensions Graphical optimization is most useful for optimization problems involving functions of two variables. Graphical function minimization in two-dimensions is performed by plotting the contours of the objective function along with the constraint boundaries on a two-dimensional grid. In Matlab ©, the grid points can be generated with the help of ‘meshgrid’ function. Mathematica © also provide similar capabilities. In the following we discuss three examples of applying graphical method in engineering design optimization problems where each problem contains two optimization variables. Example 3.2: Hollow cylindrical cantilever beam design (Arora, p. 85) We consider the minimum-weight design of a cantilever beam of length L, with hollow circular cross- section (outer radius ܴ ³ inner radius ܴ ) subjected to a point load P. The maximum bending moment on the beam is given as PL,
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