QUESTION $3[20$MARKS$]$ Bamba$-$Zonke Limited has had a hard time for the last couple of years in the market in which they operate. The management must now decide whether a new factory should be built and how the factory should be$.$ The managing director suggests that the company should first carry out market research on the success or otherwise of the product to be produced. The following information is available: P $($favorable selling conditions$)=0\mathrm{.}55$ P $($unfavorable selling conditions$)=0\mathrm{.}55$ P $($market research favorable$|$ favorable selling conditions$)=0\mathrm{.}75$ P $($market research $|$ unfavorable selling conditions$)=0\mathrm{.}8$ A large factory with favorable selling conditions is estimated to realize a profit of $$300$ million while it will make a loss of $$200$ million under unfavorable selling conditions. In addition, it is estimated that a small factory will make a profit of $$200$ million if selling conditions are favorable and a loss of $$120$ million under unfavorable selling conditions. The cost of carrying out market research will amount to $$10$ million and it is not taken into account in the calculation of the profit or loss. i$.$ What amount will Bamba$-$Zonke be willing to pay for perfect information? $(10$ marks$)$ ii$.$ Analyse the decision problem using a decision tree diagram. $(10$ marks