A dam is being considered to reduce river flooding. But if a dam is built, what height should it be? Increasing the dam’s height will (1) reduce a flood’s probability, (2) reduce the damage when floods occur, and (3) cost more. Which dam height minimizes the expected total annual cost? The state

Question

A dam is being considered to reduce river flooding. But if a dam is built, what height should it be? Increasing the dam’s height will (1) reduce a flood’s probability, (2) reduce the damage when floods occur, and (3) cost more. Which dam height minimizes the expected total annual cost? The state uses an interest rate of 5% for flood protection projects, and all the dams should last 50 years.

Dam Height (ft)	First Cost
No dam	$0
20	700,000
30	800,000
40	900,000

Accepted Answer

The easiest way to solve this problem is to choose the dam height with the lowest equivalent uniform annual cost (EUAC). Calculating the EUAC of the first cost requiresmultiplying the first cost by (A/P, 5%, 50). For example, for the dam 20 ft high, this is 700,000(A/P, 5%, 50) = $38,344.

Calculating the annual expected flood damage cost for each alternative is simplified because the term for the P(no flood) is zero, because the damages for no flood are $0. Thus we need to calculate only the term for flooding. This is done by multiplying the P(flood) times the damages if a flood happens. For example, the expected annual flood damage cost with no levee is 0.25 × $800,000, or $200,000.

Then the EUAC of the first cost and the expected annual flood damage are added together to find the total EUAC for each height. The 30 ft dam is somewhat cheaper than the 40 ft dam.

Dam Height (ft)	EUAC of First Cost	Annual P (flood)> Height	Damages if Flood Occurs	Expected Annual Flood Damages	Total Expected EUAC
No dam	$0	0.25 ×	$800,000 =	$200,000	$200,000
20	38,344	0.05 ×	500,000 =	25,000	63.344
30	43,821	0.01 ×	300,000 =	3000	46.821
40	49,299	0.002 ×	200,000 =	400	49.699

Question 10.9: A dam is being considered to reduce river flooding. But if a...

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