Question $3$

The BB Pasta Company has traditionally packaged their pasta tomato sauce product in cylindrical glass jars for $50$ years. Management believes that the cylindrical container is important for the product's image. But consumer tastes can change, and the new marketing managers think that younger people regard the round jars as old$-$fashioned and unappealing. The Marketing Department wants to analyze whether or not to package the tomato sauce in rectangular cartons that will save on transportation costs by eliminating the empty spaces in the shipping boxes. It is also believed that the change can also improve the company's market by uplifting the product's image. However, previous study has shown that a small segment of the existing market would stop buying the product if the packaging were to be changed from the glass jar to carton.

The following payoff table has been established for the present net worth of retaining the old glass jar versus using the new rectangular carton.

$\backslash$table$[[\backslash$table$[[$National Market Response$],[$to New Packaging$]],$Action$],[$Retain Old Packaging,Use New Packaging,$],[$Moderate $($M$),$OMR $0,-2,000,000,],[$Strong $($S$),0,850,000,],[,0,3,$ XXX$,$X$00,]]$

As prior probabilities for the new packaging response events, the Marketing Department calculated the following estimates: $P(W)=0\mathrm{.}30$;$P(M)=0\mathrm{.}45$;$P(S)=0\mathrm{.}25.$ The new packaging is to be test marketed for six months in a selected city. Three outcomes are possible: decreased sales $($D$),$ unchanged sales $($U$),$ and increased sales $($I$).$ Historical experience with other products shows the following conditional result probabilities:

$P(D|W)=0\mathrm{.}7,P(D|M)=0\mathrm{.}2,P(D|S)=0$

$P(U|W)=0\mathrm{.}3,P(U|M)=0\mathrm{.}4,P(U|S)=0\mathrm{.}1$

$P(I|W)=0,P(I|M)=0\mathrm{.}4,P(I|S)=0\mathrm{.}9$

$($a$)$ Construct the decision tree for this decision problem, taking into account that it has already decided that the new packaging will be test marketed in the selected city.

$(4$ marks$)$

$($b$)$ Fold back the decision tree to find the optimal strategy for BB Pasta.

$(1$ mark$)$

NOTE: The payoff for the strong market response to new packaging is indicated by $3,xxx,x00($OMR$).$ You must replace the $xxx,x$ by adding the last $3$ digits of the student ID numbers of both team members, and dividing by $2.$ For example, if the ID numbers of the team members are $188686$ and $180177,$ the average of the last $3$ digits is $431\mathrm{.}5,$ and the payoff will be $3,431,500$OMR. If the average is$,$ for example, $212\mathrm{.}0,$ the payoff becomes $3,212,000$OMR.