Engineering Optimization: Theory and Practice, Fourth Edition

586 Dynamic Programming

Figure 9.26 Pipe network.

For the segmentsBitoCjandCitoDj

To nodej

From nodei 1 2 3

1 8 12 19

2 9 11 13

3 7 15 14

Find the solution using dynamic programming.

9.8 Consider the problem of controlling a chemical reactor. The desired concentration of

material leaving the reactor is 0.8 and the initial concentration is 0.2. The concentration

at any timet,x(t ), is given by

dx

dt

=

1 −x

1 +x

u(t )

whereu(t )is a design variable (control function).

Findu(t )which minimizes

f=

∫T

0

{[x(t )− 0 .8]^2 +u^2 (t )}dt

subject to

0 ≤u(t )≤ 1

Choose a grid and solveu(t )numerically using dynamic programming.

9.9 It is proposed to build thermal stations at three different sites. The total budget available

is 3 units (1 unit=$10 million) and the feasible levels of investment on any thermal

station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different

investments is given below: