9.7 Conversion of a Final Value Problem into an Initial Value Problem 567Figure 9.15 Conversion of a final value problem to an initial value problem: (a) final value
problem; (b) initial value problem.
an initial value problem as shown in Fig. 9.15b. This initial value problem is identical
to the one considered in Fig. 9.3 except for the stage numbers. If the stage numbers
1 , 2 ,... , nare reversed ton, n− 1 ,... ,1, Fig. 9.15bwill become identical to Fig. 9.3.
Once this is done, the solution technique described earlier can be applied for solving
the final value problem shown in Fig. 9.15a.
Example 9.4 A small machine tool manufacturing company entered into a contract
to supply 80 drilling machines at the end of the first month and 120 at the end of the
second month. The unit cost of manufacturing a drilling machine in any month is given
by $(50x+ 0. 2 x^2 ) where, xdenotes the number of drilling machines manufactured in
that month. If the company manufactures more units than needed in the first month,
there is an inventory carrying cost of $8 for each unit carried to the next month. Find
the number of drilling machines to be manufactured in each month to minimize the
total cost. Assume that the company has enough facilities to manufacture up to 200
drilling machines per month and that there is no initial inventory. Solve the problem
as a final value problem.
SOLUTION The problem can be stated as follows:
Minimizef (x 1 , x 2 ) =( 50 x 1 + 0. 2 x 12 ) +( 50 x 2 + 0. 2 x 22 ) + 8 (x 1 − 08 )subject to
x 1 ≥ 08
x 1 +x 2 = 002
x 1 ≥ 0 , x 2 ≥ 0