Review Questions
539
interpretation, pp. 15-21, in Progress in Engineering Optimization-1981 , R. W. Mayne
and K. M. Ragsdell, Eds., ASME, New York, 1981.
8.31
M. Avriel, R. Dembo, and U. Passey, Solution of generalized geometric programs, International Journal for Numerical Methods in Engineering, Vol. 9, pp. 149-168, 1975.
8.32
Computational aspects of geometric programming: 1. Introduction and basic notation,
pp. 115-120 (A. B. Templeman), 2. Polynomial programming, pp. 121-145 (J. Bradley),
3. Some primal and dual algorithms for posynomial and signomial geometric programs,
pp. 147-160 ( J. G. Ecker, W. Gochet, and Y. Smeers), 4. Computational experiments in
geometric programming, pp. 161-173 ( R. S. Dembo and M. J. Rijckaert), Engineering
Optimization, Vol. 3, No. 3, 1978.
8.33
A. J. Morris, Structural optimization by geometric programming, International Journal
of Solids and Structures, Vol. 8, pp. 847-864, 1972.
8.34
A. J. Morris, The optimisation of statically indeterminate structures by means of approximate geometric programming, pp. 6.1-6.17 in Proceedings of the 2nd Symposium on
Structural Optimization, AGARD Conference Proceedings 123 , Milan, 1973.
8.35
A. B. Templeman and S. K. Winterbottom, Structural design applications of geometric programming, pp. 5.1-5.15 in Proceedings of the 2nd Symposium on Structural Optimization,
AGARD Conference Proceedings 123 , Milan, 1973.
8.36
A. B. Templeman, Structural design for minimum cost using the method of geometric programming, Proceedings of the Institute of Civil Engineers, London, Vol. 46,
pp. 459-472, 1970.
8.37
J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, 5th ed., McGraw-Hill,
New York, 1989.
REVIEW QUESTIONS
8.1
State whether each of the following functions is a polynomial, posynomial, or both.
(a) f = 4 Š x 12 + 6x 1x 2 + 3x22
(b) f = 4 + 2x 12 + 5x 1x 2 + x22
2
Š4
+ 5x1 Š1
(c) f = 4 + 2x 1Š1
x 2 + 3x2
8.2
3
x2
Answer true or false:
(a) The optimum values of the design variables are to be known before finding the
optimum value of the objective function in geometric programming.
(b) _
_
j
denotes the relative contribution of the j th term to the optimum value of
the
objective function.
(c) There are as many orthogonality conditions as there are design variables in a geometric
programming problem.
(d) If f is the primal and v is the dual, f _ v.
(e) The degree of difficulty of a complementary geometric programming problem is given
by (N Š n Š 1), where n denotes the number of design variables and N represents the
total number of terms appearing in the numerators of the rational functions involved.
(f) In a geometric programming problem, there are no restrictions on the number of
design variables and the number of posynomial terms.
8.3
8.4
How is the degree of difficulty defined for a constrained geometric programming problem?
What is arithmetic-geometric inequality?
540
Geometric Programming
8.5
What is normality condition in a geometric programming problem?
8.6
Define a complementary geometric programming problem.
PROBLEMS
Using arithmetic mean-geometric mean inequality, obtain a lower bound v for each function
[f (x) _ v, where v is a constant] in Problems 8.1-8.3.
xŠ2
xŠ 3 +
4
x3 /2
f (x) =
8.2
f (x) = 1 + x +
8.3
f (x) =
8.4
An open cylindrical vessel is to be constructed to transport 80 m 3 of grain from a
warehouse to a factory. The sheet metal used for the bottom and sides cost $80 and
$10 per
square meter, respectively. If it costs $1 for each round trip of the vessel, find the dimensions of the vessel for minimizing the transportation cost. Assume that the vessel has no
salvage upon completion of the operation.
8.5
8.6
8.7
3
1
2
+
2
8.1
3
3
1
x
+
1
x2
2
x Š3 + x + 2x
Find the solution of the problem stated in Problem 8.4 by assuming that the sides
cost
$20 per square meter, instead of $10.
Solve the problem stated in Problem 8.4 if only 10 trips are allowed for transporting the
80 m 3 of grain.
An automobile manufacturer needs to allocate a maximum sum of $2.5 × 10 6 between
the development of two different car models. The profit expected from both the
models
is given by x11 5x2.
8.8
8.9
8.10
, where x i denotes the money allocated to model i (i = 1,
2). Since
the success of each model helps the other, the amount allocated to the first model should
not exceed four times the amount allocated to the second model. Determine the amounts
to be allocated to the two models to maximize the profit expected. Hint: Minimize
Š2
Š1
the
f (X) = x x1 2x 3 + 2x1 Š1 x2 x3 + 5x 2 + 3x 1x2 Š2
inverse of the profit expected.
Minimize
the following
function:
Write the dual
of the heat
exchanger design problem stated in Problem 1.12.
Minimize the following function: f (X) =
21x
12
+ x2 +
1
x Š1
Š1 x2
23
2
4
Minimize f (X) = 20x 2x 3x 4 + 20x 1 x3 Š1
8.11
subject to
5 x3Š1 _ 1
Š
10x1Š 1x 3Š1 4 _ 1
5x2
x
xi > 0,
i2
=
1 to
4
+ 5x 2x3
2
Problems
Minimize f (X) = x1Š
8.12
2
+
22x
1x
4
541
3
subject to
3
4
2
x Š2
1 x2 +
3
Š2
8 x 2x3
i = 1, 2, 3
xi > 0,
Minimize f (X) = x1Š
8.13
_1
3/2
3x
+ x1
2
Š1
x3
subject to
2
1 Š2 3
x Š1 2 + 2
x3 _ 1
x
x
x3 > 0
x11 > 0, 1 x2 > 0,
Minimize f = x1Š
8.14
x2Š
x32 Š
1
2
subject to
x1 + x 22 + x 3 _ 1
3
xi > 0,
8.15
i = 1, 2, 3
+ c a 2x2 + · · · + c
Prove that the function y = c e 1 1 ax
1
a convex function with respect to x 1, x22e, . . . , x n.ne
a
x
n
n
, c
is
i
_ 0, i = 1, 2, . . . , n,
8.16
Prove that f = ln x is a concave function for positive values of x.
8.17
The problem of minimum weight design of a helical torsional spring subject to a
stress
constraint can be expressed as [8.27]
Minimize f (d, D) =
2E
d6 +
14,680M
Dd 2
2Q
4
subject to
14.5M
_max
d 2 .885D 0 115
.
_1
where d is the wire diameter, D the mean coil diameter,
the density, E is
Young's
modulus, the angular deflection in degrees, M the torsional moment, and Q the number
of inactive
forN-m,
the
following
data:turns.
E = 20Solve
× 1010this problem
= 15 geometric
× 10 7 Pa, programming
= 20 , Q = approach
2, M = 0.3
Pa, _maxusing
4
and
= 7.7 × 10 N/m 3.
8.18
Solve the machining economics problem given by Eqs. (E 2) and (E 4) of Example 8.7 for
the given data.
8.19
8.20
Solve the machining economics problem given by Eqs. (E 2), (E 4), and (E 6) of
Example
8.7 for the given data.
Determine the degree of difficulty of the problem stated in Example 8.8.
542
Geometric Programming
Figure 8.5
8.21
Floor consisting of a plate with supporting beams [8.36].
A rectangular area of dimensions A and B is to be covered by steel plates with supporting
beams as shown in Fig. 8.5. The problem of minimum cost design of the floor subject to a
constraint on the maximum deflection of the floor under a specified uniformly distributed
live load can be stated as [8.36]
Minimize f (X) = cost of plates + cost of beams
=k
f
. ABt + k b. Ak 1nZ2
/3
(1)
subject to
56.25WB4
EA
t Š3
n +
Š4
, 4 .69WBA 3 nŠ
Ek2
ZŠ1
4/3
_1
(2)
where W is the live load on the floor per unit area, k f and k b are the unit costs of plates
and beams, respectively, . the weight density of steel, t the thickness of plates, n
4/3
the
1
/32
number of beams, k Z
the cross-sectional area of each beam, k 2Z
the area moment
of inertia of each beam, k 1 and k 2 are constants, Z the section modulus of each
beam,
and E the elastic modulus of steel. The two terms on the left side of Eq. (2)
denote the
contributions of steel plates and beams to the deflection of the floor. By assuming the data
8.22
as A = 10 m, B = 50 m, W = 1000 kg f/m 2, k b = $0.05/ kg f, k f = $0.06/ kg f, . = 7850
kg f/m 3, E = 2.1 × 10 5 MN/m 2, k 1 = 0.78, and k 2 = 1.95, determine the solution of
the
problem (i.e., the values of t _, n _, and Z _).
8.23
Solve the zero-degree-of-difficulty bearing problem given by Eqs. (E
Example 8.12.
8)
and (E 9) of
8.24
Solve the one-degree-of-difficulty bearing problem given by Eqs. (E 12) and (E 13) of
Example 8.12.
The problem of minimum volume design of a statically determinate truss consisting of n
members (bars) with m unsupported nodes
and subject
Minimize
f = to q load
l x conditions can be stated
i
as follows [8.14]:
i=1
i
n
(1)