Engineering Optimization: Theory and Practice, Fourth Edition

2.5 Multivariable Optimization with Inequality Constraints 101

Example 2.14 A manufacturing firm producing small refrigerators has entered into
a contract to supply 50 refrigerators at the end of the first month, 50 at the end of the
second month, and 50 at the end of the third. The cost of producingxrefrigerators
in any month is given by $(x^2 + 000). The firm can produce more refrigerators in 1
any month and carry them to a subsequent month. However, it costs $20 per unit for
any refrigerator carried over from one month to the next. Assuming that there is no
initial inventory, determine the number of refrigerators to be produced in each month
to minimize the total cost.

SOLUTION Letx 1 ,x 2 , andx 3 represent the number of refrigerators produced in the
first, second, and third month, respectively. The total cost to be minimized is given by

total cost=production cost+holding cost

or

f (x 1 , x 2 , x 3 ) =(x^21 + 0001 )+(x^22 + 0001 )+(x^23 + 0001 )+ 20 (x 1 − 05 )

+ 20 (x 1 +x 2 − 001 )

=x 12 +x 22 +x^23 + 04 x 1 + 02 x 2

The constraints can be stated as

g 1 (x 1 , x 2 , x 3 )=x 1 − 05 ≥ 0

g 2 (x 1 , x 2 , x 3 )=x 1 +x 2 − 001 ≥ 0

g 3 (x 1 , x 2 , x 3 )=x 1 +x 2 +x 3 − 501 ≥ 0

The Kuhn–Tucker conditions are given by

∂f

∂xi

+λ 1

∂g 1

∂xi

+λ 2

∂g 2

∂xi

+λ 3

∂g 3

∂xi

= 0 , i= 1 , 2 , 3

that is,

2 x 1 + 04 +λ 1 +λ 2 +λ 3 = 0 (E 1 )

2 x 2 + 02 +λ 2 +λ 3 = 0 (E 2 )

2 x 3 +λ 3 = 0 (E 3 )

λjgj= 0 , j= 1 , 2 , 3

that is,

λ 1 (x 1 − 05 )= 0 (E 4 )

λ 2 (x 1 +x 2 − 001 )= 0 (E 5 )

λ 3 (x 1 +x 2 +x 3 − 501 )= 0 (E 6 )

gj≥ 0 , j= 1 , 2 , 3