Engineering Optimization: Theory and Practice, Fourth Edition

666 Stochastic Programming

Time per unit (min) for product: Stage capacity

A B C (mins/day)

Stage Mean

Standard

deviation Mean

Standard

deviation Mean

Standard

deviation Mean

Standard

deviation

1 4 1 8 3 4 4 1720 172

2 12 2 0 0 8 2 1840 276

3 4 2 16 4 0 0 1680 336

The profit per unit is also a random variable with the following data:

Profit($)

Product Mean Standard deviation

A 6 2

B 4 1

C 10 3

Assuming that all amounts produced are absorbed by the market, determine the daily

number of units to be manufactured of each product for the following cases.

(a)The objective is to maximize the expected profit.

(b)The objective is to maximize the standard deviation of the profit.

(c)The objective is to maximize the sum of expected profit and the standard deviation

of the profit.

Assume that all the random variables follow normal distribution and the constraints have

to be satisfied with a probability of 0.95.

11.18 In a belt-and-pulley drive, the belt embraces the shorter pulley 165◦and runs over it

at a mean speed of 1700 m/min with a standard deviation of 51 m/min. The density of

the belt has a mean value of 1 g/cm^3 and a standard deviation of 0.05 g/cm^3. The mean

and standard deviations of the permissible stress in the belt are 25 and 2.5 kgf/cm^2 ,

respectively. The coefficient of friction (μ) between the belt and the pulley is given

byμ= 0 .25 andσμ= 0 .05. Assuming a coefficient of variation of 0.02 for the belt

dimensions, find the width and thickness of the belt to maximize the mean horsepower

transmitted. The minimum permissible values for the width and the thickness of the belt

are 10.0 and 0.5 cm, respectively. Assume that all the random variables follow normal

distribution and the constraints have to be satisfied with a minimum probability of 0.95.

Hint:Horsepower transmitted=(T 1 −T 2 )ν/75, whereT 1 andT 2 are the tensions on

the tight side and slack sides of the belt in kgfandνis the linear velocity of the belt

in m/s:

T 1 =Tmax−Tc=Tmax−

wν^2

g

and

T 1

T 2

=eμθ

whereTmaxis the maximum permissible tension,Tcthe centrifugal tension,wthe weight

of the belt per meter length,gthe acceleration due to gravity in m/s, andθthe angle of

contact between the belt and the pulley.