Engineering Optimization: Theory and Practice, Fourth Edition

Problems 667

11.19 An article is to be restocked every three months in a year. The quarterly demandUis
random and its law of probability in any of the quarters is as given below:

U Probability mass function,PU(u)

0 0.2

1 0.3

2 0.4

3 0.1

> 3 0.0

The cost of stocking an article for a unit of time is 4, and when the stock is exhausted,

there is a scarcity charge of 12. The orders that are not satisfied are lost, in other words,

are not carried forward to the next period. Further, the stock cannot exceed three articles,

owing to the restrictions on space. Find the annual policy of restocking the article so as

to minimize the expected value of the sum of the cost of stocking and of the scarcity

charge.

11.20 A close-coiled helical spring, made up of a circular wire of diameterd, is to be designed
to carry a compressive loadP. The permissible shear stress isσmaxand the permissible
deflection isδmax. The number of active turns of the spring isnand the solid height of
the spring has to be greater thanh. Formulate the problem of minimizing the volume
of the material so as to satisfy the constraints with a minimum probability ofp. Take
the mean diameter of the coils (D)and the diameter of the wire (d)as design variables.
Assumed, D, P,σmax,δmax, h, and the shear modulus of the material,G, to be normally
distributed random variables. The coefficient of variation ofdandDisk. The maximum
shear stress,σ, induced in the spring is given by

σ=

8 PDK

π d^3

whereKis the Wahl’s stress factor defined by

K=

4 D−d

4 (D−d)

+

0. 615 d

D

and the deflection (δ) by

δ=

8 P D^3 n

Gd^4

Formulate the optimization problem for the following data:

G=N ( 840 , 000 , 84 , 000 )kgf/cm^2 , δmax=N ( 2 , 0. 1 )cm,

σmax=N ( 3000 , 150 )kgf/cm^2 ,

P=N ( 12 , 3 )kgf, n= 8 , h=N ( 2. 0 , 0. 4 )cm, k= 0. 05 ,

p= 0. 99

11.21 Solve Problem 11.20 using a graphical technique.