$7.$ Manufacture or Outsource. The Gorman Manufacturing Company must decide whether to manufacture a component part at its Milan, Michigan, plant or purchase the component part from a supplier. The resulting profit is dependent upon the demand for the product. The following payoff table shows the projected profit $($in thousands of dollars$).$ The state$-$of$-$nature probabilities are P$($s$\_1)=\mathrm{.}35,$ P$($s$\_2)=\mathrm{.}35,$ and P$($s$\_3)=\mathrm{.}30.$ a$.$ Use a decision tree to recommend a decision. b$.$ Use EVPI to determine whether Gorman should attempt to obtain a better estimate of demand. c$.$ A test market study of the potential demand for the product is expected to report either a favorable $($F$)$ or unfavorable $($U$)$ condition. The relevant conditional probabilities are as follows:

$[$ P$($F $|$ s$\_1)=\mathrm{.}10$ P$($U $|$ s$\_1)=\mathrm{.}90$; P$($F $|$ s$\_2)=\mathrm{.}40$ P$($U $|$ s$\_2)=\mathrm{.}60$; P$($F $|$ s$\_3)=\mathrm{.}60$ P$($U $|$ s$\_3)=\mathrm{.}40]$

What is the probability that the market research report will be favorable? d$.$ What is Gorman's optimal decision strategy? e$.$ What is the expected value of the market research information?