The student engineering society is building a snack cart to raise money. Members must decide what capacity the cart should be able to serve. To serve 100 customers per hour costs $10,310, and to serve 150 customers per hour costs $13,400. The 50% increase in capacity is less than 50% of $10,310

Question

The student engineering society is building a snack cart to raise money. Members must decide what capacity the cart should be able to serve. To serve 100 customers per hour costs $10,310, and to serve 150 customers per hour costs $13,400. The 50% increase in capacity is less than 50% of $10,310 because of economies of scale; but the increase in net revenue will be less than 50%, since the cart will not always be serving 150 customers per hour. The estimated net annual income for the lower capacity is $3300, and for the higher capacity it is $4000. After 5 years the cart is expected to have no salvage value. The engineering society is unsure of what interest rate to use to decide on the capacity. Make a recommendation.

Accepted Answer

Since we do not know the interest rate, the easiest way to analyze the problem is to graph the PW of each alternative versus the interest rate, as in Figure 8–1. The Excel function¹

[latex]=-cost +PV(interest rate,life,-annual benefit)[/latex]

is used to graph these two equations:

[latex]PW_{Low}=-\$10,310+\$3300(P/A,i,5)[/latex]

[latex]PW_{high}=-\$13,400+\$4000(P/A,i,5)[/latex]

5-BUTTON SOLUTION

H	G	F	E	D	C	B	A
Answer	Solve for	FV	PV	PMT	n	i	Alternative	1
18.04%	i	0	10,310	3300	5		Low	2
15.03%	i	0	13,400	4000	5		High	3
4.30%	i	0	3,090	700	5		High - low	4

We want to maximize the PW, so the choice between these alternatives is defined by where their curves intersect (the incremental rate of return of 4.3%), not by where those curves intersect the PW = 0 axis. Those internal rates of return of 15 and 18% are irrelevant to the choice between the lowand high-capacity carts.

Figure 8–1 also shows the main advantage of the graphical approach for realworld decision making. We can easily see that for interest rates between 0% and about 8%, there is little difference between the choices. In this region we are better off focusing our decision making on risk, benefits, or costs that we could not quantify, as well as on uncertainties in the data—since the PWs of our two alternatives are nearly the same. These topics are discussed in later chapters. Until then it is clearer if we analyze problems as though all numbers are known precisely.

From Figure 8–1, we see that if the interest rate is below 4.3%, the high-capacity cart has a higher PW. From 4.3 to 18% the lower-capacity cart has a higher PW. Above 18%, the third alternative of doing nothing is a better choice than building a low-capacity cart because the PW < 0. This can be summarized in a choice table:

Best Choice	Interest Rate
High capacity	0% ≤ i ≤ 4.3%
Low capacity	4.3% ≤ i ≤ 18.0%
Do nothing	18.0% ≤ i

This example has been defined in terms of a student engineering society. But very similar problems are faced by a civil engineer sizing the weighing station for trucks carrying fill to a new earth dam, by an industrial engineer designing a package-handling station, and by a mechanical engineer sizing energy conservation equipment.

¹ The cost term in the equation has a minus sign because it is a cost and we are calculating a present worth. The annual benefit term has a minus sign because of the sign convention for the PV function.

Question 8.1: The student engineering society is building a snack cart to ...

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