$22.$ A firm must decide whether to construct a small, medium, or large stamping plant. A consultant's report indicates a $\mathrm{.}20$ probability that demand will be low and an $\mathrm{.}80$ probability that demand will be high.

If the firm builds a small facility and demand turns out to be low, the net present value will be $$42$ million. If demand turns out to be high, the firm can either subcontract and realize the net present value of $$42$ million or expand greatly for a net present value of $$48$ million.

The firm could build a medium$-$size facility as a hedge: if demand turns out to be low, its net present value is estimated at $$22$ million; if demand turns out to be high, the firm could do nothing and realize a net present value of $$46$ million, or it could expand and realize a net present value of $$50$ million.

If the firm builds a large facility and demand is low, the net present value will be $-$$$20$ million, whereas high demand will result in a net present value of $$72$ million.

a$.$ Analyze the problem using a decision tree.

b$.$ What is the maximin alternative?

c$.$ Compute EVPI and interpret it$.$

d$.$ Perform sensitivity analysis on P $($high$)$