Engineering Optimization: Theory and Practice, Fourth Edition

Problems 665

11.13 The range (R)of a projectile is given by

R=

V 02

g

sin 2φ

whereV 0 is the initial velocity of the projectile,gthe acceleration due to gravity, andφ

the angle from the horizontal as shown in Fig. 11.6. If the mean and standard deviations

ofV 0 andφare given byV 0 =100 ft/s,σV 0 = 0 ft/s, 1 φ= 30 ◦, andσφ= 3 ◦, find

the first-order mean and standard deviation of the rangeR, assuming thatV 0 andφ

are statistically independent. Evaluate also the second-order mean range. Assume that

g= 32 .2 ft/s^2.

11.14 Maximizef= 4 x 1 + 2 x 2 + 3 x 3 +c 4 x 4

subject to

x 1 +x 3 +x 4 ≤ 24

3 x 1 +x 2 + 2 x 3 + 4 x 4 ≤ 48

2 x 1 + 2 x 2 + 3 x 3 + 2 x 4 ≤ 36

xi≥ 0 , i=1 to 4

wherec 4 is a discrete random variable that can take values of 4, 5, 6, or 7 with probabil-

ities of 0.1, 0.2, 0.3, and 0.4, respectively. Using the simplex method, find the solution

that maximizes the expected value off.

11.15 Find the solution of Problem 11.14 if the objective is to maximize the variance off.

11.16 A manufacturing firm can produce 1, 2, or 3 units of a product in a month, but the
demand is uncertain. The demand is a discrete random variable that can take a value of
1, 2, or 3 with probabilities 0.2, 0.2, and 0.6, respectively. If the unit cost of production
is $400, unit revenue is $1000, and unit cost of unfulfilled demand is $0, determine the
output that maximizes the expected total profit.

11.17 A factory manufactures productsA, B, andC. Each of these products is processed
through three different production stages. The times required to manufacture 1 unit of
each of the three products at different stages and the daily capacity of the stages are
probabilistic with means and standard deviations as indicated below.

Figure 11.6 Range of a projectile.