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 boundvfor each function
[f (x)≥v, wherevis a constant] in Problems 8.1–8.3.8.1 f (x)=
x−^2
3+
2
3x−^3 +
4
3x^3 /^28.2 f (x)= 1 +x+1
x
+1
x^2
8.3 f (x)=^12 x−^3 +x^2 + 2 x8.4 An open cylindrical vessel is to be constructed to transport 80 m^3 of grain from a ware-
house 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 dimen-
sions of the vessel for minimizing the transportation cost. Assume that the vessel has no
salvage upon completion of the operation.
8.5 Find the solution of the problem stated in Problem 8.4 by assuming that the sides cost
$20 per square meter, instead of $10.
8.6 Solve the problem stated in Problem 8.4 if only 10 trips are allowed for transporting the
80 m^3 of grain.
8.7 An automobile manufacturer needs to allocate a maximum sum of $2. 5 × 106 between
the development of two different car models. The profit expected from both the models
is given byx^11.^5 x 2 , wherexidenotes the money allocated to modeli(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 the
inverse of the profit expected.
8.8 Write the dual of the heat exchanger design problem stated in Problem 1.12.
8.9 Minimize the following function:
f (X)=x 1 x 2 x− 32 + 2 x− 11 x− 21 x 3 + 5 x 2 + 3 x 1 x 2 −^28.10 Minimize the following function:
f (X)=^12 x^21 +x 2 +^32 x− 11 x− 218.11 Minimizef (X)=^20 x 2 x 3 x^44 +^20 x 12 x 3 −^1 +^5 x 2 x^23
subject to
5 x 2 −^5 x 3 −^1 ≤ 1
10 x 1 −^1 x 23 x 4 −^1 ≤ 1
xi> 0 , i=1 to 4