Question

A manager must decide how many machines of a certain type to buy. The machines will be used to manufacture a new gear for which there is increase demand. The manager has narrowed the decision to two alternatives; buy one machine or buy two. If only one machine is purchased and demand is more than it can handle, a second machine can be purchased at a later time. however, the cost per machine would be lower if the two machines were purchased at the same time.
The estimated probability of low demands is .30, and the estimated probability of high demand is .70.
The net present value associated with the purchase of two machines initially is $75,000 if demand is low and $130,000 if demand is high.
The net present value for one machine and low demand is $90,000 . If demand is high, there are three options. One option is to do nothing, which would have a net present value of $90,000 . A second option is subcontract; that would have a net present value of $110,000 . The third option is to purchase a second machine. This option would have a net present value of $100,000.
How many machines should the manager purchase initially ? Use a decision tree to analyze this problem.

Answer 1

We can look at this problem as follows:

If the decision today is to buy 2, then the expected payoff is the weighted average of 130000 and 75000 weighted by the probabilities of high- and low- demand. This comes to 113500.

If the decision today is to buy 1, then there is a future decision in case of high demand (sub-contract or buy 2nd machine) the average payoff of which is 100000 (average of 90000, 110000 and 100000). In case of low demand there's no decision required in future. The average of this would be 0.7*100000 + 0.3*90000 = 97000.

Since the expected payoff from buying 2 machines today (113500) is higher the manager should buy two machines today.

					Do nothing	90000
		High demand (0.7)	100000 (avg of 90000, 110000 and 100000)	Tomorrow	Sub-contract	110000
	Buy 1				Buy second machine	100000
	97000 (0.7*100000 + 0.3*90000)	Low demand (0.3)	90000
Today
		High demand (0.7 x 130000)	91000
	Buy 2
	113500 (91000 + 22500)	Low demand (0.3 x 75000)	22500