"A manager is trying to decide whether to build a small, medium, or large facility. Demand can be low, average, or high, with the estimated probabilities being 0.25, 0.40, and 0.35, respectively.
A small facility is expected to earn an after-tax net present value of just $18,000 if demand is low. If demand is average, the small facility is expected to earn $75,000; it can be increased to medium size to earn a net present value of $60,000. If demand is high, the small facility is expected to earn $75,000 and can be expanded to medium size to earn $60,000 or to large size to earn $125,000.
A medium-sized facility is expected to lose an estimated $25,000 if demand is low and earn $140,000 if demand is average. If demand is high, the medium-sized facility is expected to earn a net present value of $150,000; it can be expanded to a large size for a net payoff of $145,000.
If a large facility is built and demand is high, earnings are expected to be $220,000. If demand is average for the large facility, the present value is expected to be $125,000; if demand is low, the facility is expected to lose $60,000."

Draw a decision tree for the three options described in the question above. What should management do to achieve the highest expected payoff?

To solve this problem, we'll construct a decision tree to visualize the different options and their associated payoffs based on the probabilities of demand being low, average, or high.

Let's denote the options as follows:

- S: Small facility

- M: Medium facility

- L: Large facility

Here's the decision tree:

```

/ Low (0.25)

/

S ----<---- Average (0.40)

/ \ \

Low / \ Average \ High

/ \ \

$18,000 $75,000 $75,000

(S) (S) / \

/ \

$60,000 (M) $125,000 (L)

(M) (L)

/ Low (0.25)

/

M ----<---- Average (0.40)

/ \ \

Low / \ Average \ High

/ \ \

-$25,000 $140,000 $150,000

(M) (M) / \

/ \

$145,000 (L) $220,000 (L)

(L) (L)

/ Low (0.25)

/

L ----<---- Average (0.40)

/ \ \

Low / \ Average \ High

/ \ \

-$60,000 $125,000 $220,000

(L) (L) (L)

```

To achieve the highest expected payoff, management should choose the option with the highest expected value.

Based on the decision tree and the probabilities given:

- Expected payoff for a Small facility: $ 0.25 \times 18000 + 0.40 \times 75000 + 0.35 \times 75000 = 62250 $

- Expected payoff for a Medium facility: $ 0.25 \times (-25000) + 0.40 \times 140000 + 0.35 \times 150000 = 111750 $

- Expected payoff for a Large facility: $ 0.25 \times (-60000) + 0.40 \times 125000 + 0.35 \times 220000 = 101500 $

Thus, management should choose to build a Large facility to achieve the highest expected payoff.