Engineering Optimization: Theory and Practice, Fourth Edition

114 Classical Optimization Techniques

d

d

h

h

D

P

(a)

(b)

Indentation or crater

of diameter d and depth h

Spherical (ball)

indenter of

diameterD

Figure 2.13 Brinell hardness test.

2.60 A manufacturer produces small refrigerators at a cost of $60 per unit and sells them to

a retailer in a lot consisting of a minimum of 100 units. The selling price is set at $80

per unit if the retailer buys 100 units at a time. If the retailer buys more than 100 units

at a time, the manufacturer agrees to reduce the price of all refrigerators by 10 cents for

each unit bought over 100 units. Determine the number of units to be sold to the retailer

to maximize the profit of the manufacturer.

2.61 Consider the following problem:

Minimizef=(x 1 − 2 )^2 +(x 2 − 1 )^2

subject to

2 ≥x 1 +x 2

x 2 ≥x 12

Using Kuhn–Tucker conditions, find which of the following vectors are local minima:

X 1 =

{

1. 5

0. 5

}

, X 2 =

{

1

1

}

, X 3 =

{

2

0

}

2.62 Using Kuhn–Tucker conditions, find the value(s) ofβfor which the pointx∗ 1 = 1 , x 2 ∗= 2

will be optimal to the problem:

Maximizef (x 1 , x 2 )= 2 x 1 +βx 2

subject to

g 1 (x 1 , x 2 )=x^21 +x 22 − 5 ≤ 0

g 2 (x 1 , x 2 )=x 1 −x 2 − 2 ≤ 0

Verify your result using a graphical procedure.