A firm must decide whether to build a small, medium, or large stamping plant. A consultant's report indicates that there is a $\mathrm{.}20$ probability that demand will be low and a $\mathrm{.}80$ probability that demand will be high. If the firm builds a small facility and demand turns out to be low, the net profit will be $$42$ million. If the demand turns out to be high, the firm can either subcontract and realize a net profit of $$42$ million or expand the facility for a net profit of $$48$ million. A medium$-$sized facility could be built: if the demand turns out to be low, its net profit is estimated at $$22$ million; if demand turns out to be high, the firm could do nothing and realize a net profit of $$46$ million, or it could expand and realize a profit of $$50$ million. If the firm builds a large facility and demand is high, there will be a net profit of $$72$ million, whereas a low demand will result in a cost of $$20$ million.

a$)$ Construct a payout table to help determine which course of action maximizes expected profits.

b$)$ Using standard deviation as a measure, which option is riskiest?

c$)$ Would the decision suggested in a$)$ be affected if the probability of low demand is greater than originally assumed? Explain.

d$)$ Given that some of the cost, revenue, and probability data may not be reliable, what course of action would you suggest for the firm?