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Optimization Techniques for Minimizing Costs of Parachute Design

This guide provides a non-computer approach to optimizing functions, specifically applied to minimizing the cost of parachute design. It involves deriving the first derivative ( f'(x) ) of a cost function, setting it to zero, and solving for critical points. It includes a practical example focusing on parachute cost minimization, considering variables like total weight, impact speed limit, and chute dimensions. You will learn how to formulate and tackle optimization problems within the context of constraints and objectives to achieve efficient designs.

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Optimization Techniques for Minimizing Costs of Parachute Design

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  1. Part Four Optimization

  2. Motivation Given a function f(x), find an x0 where f(x) is the maximum or minimum.Noncomputer method:derive the exact formula of the first derivative f’(x) using calculus and then solve the equation f’(x) = 0. Question: How about find (x1, x2, x3, x4, x5) such that the following function is minimized?

  3. Example: Optimization of Parachute Cost For each chute: A = 2πr2l = √2rc = kcA m = Mt/n Cost = c0 + c1l + c2A2 Task: Given the total weight Mt, the impact speed limit vc, and the initial height z0, determine the size (r) and the number of chutes (n) such that the total costn(c0 + c1l + c2A2) is the minimum while the impact speed of the chutes when reaching the ground is less than vc. MinimizeC = n(c0 + c1l + c2A2) Subject to v≤ v0 and n ≥ 1 where A = 2πr2l = √2r c = kcA m = Mt/n t = root [ z0 – gmt/c + (gm2/c2)(1-e-(c/m)t)] v = (gm/c)(1-e-(c/m)t)

  4. Mathematical Background Optimization or mathematical programming problem Find x, which minimizes or maximizes f(x) Subject to di(x) ≤ ai i = 1, 2, …, m ci(x) = bi i = 1, 2, …, p where x: an n-dimensional design vector f(x): the objective function di(x): inequality constraints ci(x): equality constraints • If f(x) and the constraints are linear, it is linear programming. • If f(x) is quadratic and the constraints are linear, it is quadratic programming. • Otherwise, it is non-linear programming.

  5. Overall Structure

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