Engineering Optimization: Theory and Practice, Fourth Edition

Problems 51

1.10 A hollow circular shaft is to be designed for minimum weight to achieve a minimum
reliability of 0.99 when subjected to a random torque of(T , σT)=( 106 , 104 )lb-in.,
whereTis the mean torque andσTis the standard deviation of the torque,T. The
permissible shear stress,τ 0 , of the material is given by(τ 0 , στ 0 ) =( 50 , 000 , 5000 )psi,
whereτ 0 is the mean value andστ 0 is the standard deviation ofτ 0. The maximum
induced stress (τ) in the shaft is given by

τ=

T ro

J

whererois the outer radius andJis the polar moment of inertia of the cross section

of the shaft. The manufacturing tolerances on the inner and outer radii of the shaft are

specified as± 0 .06 in. The length of the shaft is given by 50±1 in.and the specific

weight of the material by 0. 3 ± 0 .03 lb/in^3. Formulate the optimization problem and

solve it using a graphical procedure. Assume normal distribution for all the random

variables and 3σvalues for the specified tolerances.Hints:(1) The minimum reliability

requirement of 0.99 can be expressed, equivalently, as [1.120]

z 1 = 2. 326 ≤

τ−τ 0

√

στ^2 +στ^20

(2) Iff (x 1 , x 2 ,... , xn)is a function of the random variablesx 1 , x 2 ,... , xn, the mean

value off (f )and the standard deviation off (σf)are given by

f=f (x 1 ,x 2 ,... ,xn)

σf=





∑n

i= 1

(

∂f

∂xi

∣

∣∣

∣

x 1 ,x 2 ,...,xn

) 2

σx^2 i





1 / 2

wherexiis the mean value ofxi, andσxiis the standard deviation ofxi.

1.11 Certain nonseparable optimization problems can be reduced to a separable form by
using suitable transformation of variables. For example, the product termf=x 1 x 2 can
be reduced to the separable formf=y^21 −y 22 by introducing the transformations
y 1 =^12 (x 1 +x 2 ), y 2 =^12 (x 1 −x 2 )
Suggest suitable transformations to reduce the following terms to separable form:
(a)f=x^21 x^32 , x 1 > 0 , x 2 > 0
(b)f=x 1 x^2 , x 1 > 0

1.12 In the design of a shell-and-tube heat exchanger (Fig. 1.20), it is decided to have the total
length of tubes equal to at leastα 1 [1.10]. The cost of the tube isα 2 per unit length and
the cost of the shell is given byα 3 D^2.^5 L, whereDis the diameter andLis the length of
the heat exchanger shell. The floor space occupied by the heat exchanger costsα 4 per unit
area and the cost of pumping cold fluid isα 5 L/d^5 N^2 per day, wheredis the diameter
of the tube andNis the number of tubes. The maintenance cost is given byα 6 NdL.
The thermal energy transferred to the cold fluid is given byα 7 /N^1.^2 dL^1.^4 +α 8 /d^0.^2 L.
Formulate the mathematical programming problem of minimizing the overall cost of the
heat exchanger with the constraint that the thermal energy transferred be greater than
a specified amountα 9. The expected life of the heat exchanger isα 10 years. Assume
thatαi, i= 1 , 2 ,... ,10, are known constants, and each tube occupies a cross-sectional
square of width and depth equal tod.