Engineering Economic Analysis

Graphical Solutions 247

FIGURE 8-t Graph ofPW of benefitsversus

PW of cost.

$30

$27

$24

--------------------------Alternative 2

$6

$3

o $3 $6 $9 $12 $15 $18$21 $24 $27$30

PresentWorthof Cost

Alternative 2

PW of cost- $20

PW of benefits =$28(PjF, 6%,1) = 28(0.9434)=$26.42

I.

I

,.

The alternatives are plotted in Figure 8-1, which looks very simple yet tells us a great deal about

the situation.''

On a graph of PW of benefits versus PW of cost (for convenience,we willcall it a benefit...,

cost graph), there will be a line where NPW O.Where the scales used on the two axes are

identical, as in this case, the resulting line will be at a 45° angle. If unequal scales are used~the

line will be at some other angle.

For the chosen interest rate (6% in this example), thisNPW= 0 line divides the graph iIlto

areas of desirable and undesirable alternatives.To the left (or above) the line is "de~irable,"while

to the right (or below) the line is "undesirable." We see that to the left of the liIle,PW of bynefits

exceeds thePW of cost or, we could say, NPW is positive. To the right of the lU1e,in the shaded

area, PW of benefits is less than PW of cost; thus, NPW is negative.

In this example,both alternativesare to the left of the NPW=0 line. Therefore, both

alternatives will have a rate of return greater than the 6% interest rate used in COIlstructingthe

graph. In fact, other rate of return lines could also be computed and plotted on the graphfor this

special case ofa one-year analysis period:We must emphasize ai the outs~t that the additioIlal

rate of return lines shownin Figure 8-2 can be plotted only for this special situation. For:analysis

period~ greater than one y~ar,the NPW.. 0 line is the only line that caIl be accurately cir"aWn.

The graphicalresultS in Figure 8-2 agree with the calculations.made in Example 7-6, Where the

=' ~rates,ofretQrnJor'the'two alternatives were"50%' tind 40%, respectively.:: ~

Figure 8..2showsthat1;beslope of a line on the graph represents a p~!cular rate ofrettirnfor

thisspecialcase?fa oIle-rearan~ysispe~od.Bet:veenthe2rigin apdAlt. 1,theslqp~!~We~~ am

a'=5Q%~Tat~DfY'eti.Ibi':wIfile fr15m 'the o~n.gihToKft~ 2: the sTope represeIlts a'"40% raie~fretlllJJ. ..

I'

l

'

..

il

:1

Ii

.,"'"....

<.,