MIME 6740 Homework 7 April 25, 2003

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Due May, 2, 2003

Write a computer program for nonlinear constrained optimization using the extended interior penalty function method. Use the program that you wrote in homework 4 for unconstrained minimization.

Solve the following problem using your problem:

We want to construct a cofferdam to block a stream temporarily in a construction site. The dam configuration is shown below. Each cycle consists of a steel cylinder filled with soil joined with metal plates. The dam is defined by design variables x1, x2, x3 shown in the figure. We are given the dam length, l, and the design life, t (in days). All dimensions are in meters.

We want to determine the values of the design variables to minimize the cost of the design.

The cost consists of the fill cost, Cf, the steel cost, Cs, and the flood cost, Cfl.

The fill cost is: Cf = 0.21lx1 x2

The steel cost is: Cs = 2.28[2lx1+2(l/x3)x1x2]

There is always a chance of a flood. The chance reduces with the dam height increasing. The following empirical formula is used for the expected cost of a flood:

Cfl = 40,000t/x4

x4 is an intermediate design variable related to the height through the following equation:

x4+33.30.8x1.

The following constraints should also be considered:

No slipping on the river bottom: 1.0425(height)  thickness or 1.0425x1  x2

No excessive stresses:

(height)(cycle length)2875 or x1x32875

All costs are in dollars. The length, l, is 300 meters. The design life is 365 days.

Your homework should include the optimization problem formulation, a brief description of the algorithm, a printout out of the program and the output and a discussion of the results. The discussion should include but not be limited to the following information: what constraints are active at the optimum, what design variables and constraints are more important (they affect the optimum solution). You should also explain what you did to validate the solution.