Problem $2.$ Management of the Albert Frank Co$.$ has established goals for the market share it wants each of the company's two new products to capture in their respective markets.

Goal $1$: Market share of product $1$ is at least $15\%$

Goal $2$: Market share of product $2$ is at least $10\%$

Three advertising campaigns are being planned to try to achieve these market shares. Campaign $1$ targets directly on the first product. Campaign $2$ targets the second product. Campaign $3$ promotes both products.

Letting x$1,$ x$2,$ and x$3$ be the amount of money allocated $($in millions of dollars$)$ to these

respective campaigns, the resulting market share $($expressed as a percentage$)$ for the

two products are estimated to be

Market share for product $1=0\mathrm{.}5\%$x$1+0\mathrm{.}2\%$x$3$

Market share for product $2=0\mathrm{.}3\%$x$2+0\mathrm{.}2\%$x$3$

The two goals are equally weighted. Besides, a total of $$55$ million is available for the

three advertising campaigns, but management wants at least $$10$ million devoted to the

third campaign.

In this light, management wants to know how to most effectively allocate the available

money to the three campaigns.

a$.(10$ points$)$ Formulate the goal programming model in mathematic form.

b$.(10$ points$)$ Solve the model formulated in part $($a$)$ on a spreadsheet. Include your

excel spreadsheet in your homework submission.

c$.(5$ points$)$ Interpret this solution to management in its language.

d$.(5$ points$)$ Look at the sensitivity report, if the penalty for not meeting product $1$s

market share goal is doubled from its current value, without resolving the model, can you

tell whether the optimal production plan you obtained in part $($b$)$ will change? Why or

why not? Explain.

e$.(5$ points$)$ Look at the sensitivity report, please explain the meaning of the shadow

price associated with the constraint $$A total of $$55$ million is available for the three

advertising campaigns$$ in the problem context.