Example 3.1 Assume that you have been asked to look into purchasing some storage tanks for your company,
and for the purchase of these tanks, you are given a budget of $1680. After some research, you
find two tank manufacturers that meet your requirements. From Manufacturer A, you can pur-
chase 16-ft
3
-capacity tanks that cost $120 each. Moreover, the type of tank requires a floor
space of 7.5 ft
2. Manufacturer B makes 24-ft
3
-capacity tanks that cost $240 each and that
require a floor space of 10 ft
2
. The tanks will be placed in a section of a lab that has 90 ft
2
of
floor space available for storage. You are looking for the greatest storage capacity within the
budgetary and floor-space limitations. How many of each tank must you purchase?
First, we need to define theobjective function, which is the function that we will attempt
to minimize or maximize. In this example, we want to maximize storage capacity. We can rep-
resent this requirement mathematically as
(3.1)
subject to the following constraints:
(3.2)
(3.3)
(3.4)
(3.5)
In Equation (3.1),Zis the objective function, while the variablesx 1 andx 2 are calledde-
sign variables, and represent the number of 16-ft
3
-capacity tanks and the number of 24-ft
3- capacity tanks, respectively. The limitations imposed by the inequalities in Equations (3.2)–(3.5)
are referred to as a set ofconstraints. Although there are specific techniques that deal with solv-
ing linear programming problems (the objective function and constraints are linear), we will
solve this problem graphically to illustrate some additional concepts.
Let us first review how you would plot the regions given by the inequalities. For example, to
plot the region as given by the linear inequality 120x 1 240 x 2 1680, we must first plot the line
120 x 1 240 x 2 1680 and then determine which side of the line represents the region. For
example, after plotting the line 120x 1 240 x 2 1680, we can test pointsx 1 0 andx 2 0 to
see if they fall inside the inequality region; because substitution of these points into the inequal-
ity satisfies the inequality, that is, (120)(0)(240)(0)1680, the shaded region represents the
given inequality (see Figure 3.2(a)). Note that if we were to substitute a set of points outside the
region, such asx 1 15 andx 2 0, into the inequality, we would find that the inequality is not
satisfied. The inequalities in Equations (3.2)–(3.5) are plotted in Figure 3.2(b).
The shaded region shown in Figure 3.2(b) is called afeasible solution region. Every point
within this region satisfies the constraints. However, our goal is to maximize the objective func-
tion given by Equation (3.1). Therefore, we need to move the objective function over the fea-
sible region and determine where its value is maximized. It can be shown that the maximum
value of the objective function will occur at one of the corner points of the feasible region. By
evaluating the objective function at the corner points of the feasible region, we see that the
maximum value occurs atx 1 8 andx 2 3. This evaluation is shown in Table 3.1.
Thus, we should purchase eight of the 16-ft
3
-capacity tanks from Manufacturer A and
three of the 24-ft
3
-capacity tanks from Manufacturer B to maximize the storage capacity within
the given constraints.
x 2 0
x 1 0
7.5x 1 10 x 2 90
120 x 1 240 x 2 1680
maximize Z 16 x 1 24 x 2
3.1 Engineering Design Process 47
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀