Building Problem

A hypothetic building design problem in optimization with constraints proposed by Bhatti (2000, pp. 3-5). To save energy costs for heating and cooling, an architect wishes to design a cuboidal building that is partially underground. Let  be the number of stories (which therefore must be a positive integer),  the depth of the building below ground,  the height of the building above ground,  the length of the building, and  the width of the building. The floor space needed is at least , the lot size requires that , the building shape is specified so that  (the golden ratio, approximately 1.618), each story is 3.5 m high, heating and cooling costs are estimated at  for exposed surface of the building, and it has been specified that annual climate control costs should not exceed . The problem then asks to minimize the volume that must be excavated to build the building.

This is equivalent to minimizing the function

(1)

subject to the constraints

	(2)
	(3)
	(4)
	(5)
	(6)
	(7)
	(8)

There is a fairly large region of parameter spacing giving near-optimal solution (and all well within the specified precision of the problem), with  minimized near  m,  m,  m, (corresponding to  and ), giving .

REFERENCES:

Bhatti, M. A. Practical Optimization Methods with Mathematica Applications. New York: Springer-Verlag, 2000.