Optimization Problems

Find [Graphics:../Images/Bhatti_Nonlinear_gr_1.gif] to     Minimize [Graphics:../Images/Bhatti_Nonlinear_gr_2.gif]

Subject to [Graphics:../Images/Bhatti_Nonlinear_gr_3.gif] and [Graphics:../Images/Bhatti_Nonlinear_gr_4.gif].

Building design example

To save energy costs for heating and cooling an architect is considering designing a partially buried rectangular building. The total floor space needed is [Graphics:../Images/Bhatti_Nonlinear_gr_5.gif]. Lot size limits the building plan dimension to [Graphics:../Images/Bhatti_Nonlinear_gr_6.gif]. It has already been decided that the ratio between the plan dimensions must be equal to the golden ratio (1.618) and that each story must be [Graphics:../Images/Bhatti_Nonlinear_gr_7.gif] high. The heating and cooling costs are estimated at [Graphics:../Images/Bhatti_Nonlinear_gr_8.gif]of the exposed surface area of the building. The owner has specified that the annual energy costs should not exceed [Graphics:../Images/Bhatti_Nonlinear_gr_9.gif]. Determining building dimensions to minimize cost of excavation.

Optimization variables

n = Number of stories
d = Depth of building below ground
h = Height of building above ground
= Length of building in plan
w = Width of building in plan

Objective function

Minimize excavation cost

[Graphics:../Images/Bhatti_Nonlinear_gr_11.gif]

Constraints

Number of stories related to the building height

[Graphics:../Images/Bhatti_Nonlinear_gr_12.gif]

Plan dimensions

[Graphics:../Images/Bhatti_Nonlinear_gr_13.gif]    

Floor space requirement

[Graphics:../Images/Bhatti_Nonlinear_gr_14.gif]

Lot size

[Graphics:../Images/Bhatti_Nonlinear_gr_15.gif] [Graphics:../Images/Bhatti_Nonlinear_gr_16.gif]

Energy cost

[Graphics:../Images/Bhatti_Nonlinear_gr_17.gif]

Physical limitations

[Graphics:../Images/Bhatti_Nonlinear_gr_18.gif]

The complete optimization problem can be stated as follows.

Find [Graphics:../Images/Bhatti_Nonlinear_gr_19.gif] in order to

Minimize [Graphics:../Images/Bhatti_Nonlinear_gr_20.gif]

Subject to [Graphics:../Images/Bhatti_Nonlinear_gr_21.gif]