Engineering Optimization: Theory and Practice, Fourth Edition

560 Dynamic Programming

the minimum ofF (s 5 , x 4 ) for any specified value of, s 5 , is given by

∂F

∂x 4

= 0. 6 −

( 11. 5596 )( 1. 3500 )

(s 5 x 4 − 1. 3500 )^2

= 0 or x 4 ∗=

6. 44

s 5

(E 20 )

f 4 ∗(s 5 )= 0. 6 x∗ 4 +

11. 5596

s 5 − 1. 3500 /x 4 ∗

=

3. 864

s 5

+

16. 492

s 5

=

20. 356

s 5

(E 21 )

Since the value ofs 5 is specified as 0.5 in., the minimum weight of the structure can

be calculated from Eq. (E 21 ) as

f 4 ∗(s 5 = 0. 5 )=

20. 356

0. 5

= 40 .712 lb (E 22 )

Once the optimum value of the objective function is found, the optimum values of the

design variables can be found with the help of Eqs. (E 20 ), (E 15 ), (E 9 ), and (E 5 ) as

x∗ 4 = 21 .88 in^2

s 4 =s 5 −

1. 3500

x 4 ∗

= 0. 5 − 0. 105 = 0 .395 in.

x∗ 3 =

4. 2445

s 4

= 01 .73 in^2

s 3 =s 4 −

1. 5625

x 3 ∗

= 0. 3950 − 0. 1456 = 0 .2494 in.

x∗ 2 =

1. 6124

s 3

= 6. 4 7 in^2

s 2 =s 3 −

0. 6750

x 2 ∗

= 0. 2494 − 0. 1042 = 0 .1452 in.

x∗ 1 =

1. 5625

s 2

= 01 .76 in^2

9.6 Example Illustrating the Tabular Method of Solution

Example 9.3 Design the most economical reinforced cement concrete (RCC) water

tank (Fig. 9.6a) to store 100,000 liters of water. The structural system consists of a

tank, four columns each 10 m high, and a foundation to transfer all loads safely to the

ground [9.10]. The design involves the selection of the most appropriate types of tank,

columns, and foundation among the seven types of tanks, three types of columns, and

three types of foundations available. The data on the various types of tanks, columns,

and foundations are given in Tables 9.1, 9.2, and 9.3, respectively.

SOLUTION The structural system can be represented as a multistage decision pro-

cess as shown in Fig. 9.13. The decision variablesx 1 ,x 2 , andx 3 represent the type of