There are $5$ questions in total, $($a$)$ through $($e$).$

$($a$)-($c$)$ are problem solving questions, while $($d$)$&$($e$)$ are conceptual questions.

You don$$t need Excel for the conceptual questions.

$$ For question $($b$),$ neatly organize your Excel spreadsheet and submit the file on

Canvas.

$($c$),($d$),$ and $($e$)$ are independent questions.

A chemical company wishes to mix two elements $($E and F$)$ to create two alloys $($X and

Y$).$ The costs of the elements are as shown in the following table.

Element Cost per ton

E $$3\mathrm{.}00$

F $$4\mathrm{.}00$

To maintain the required quality for each alloy, it is necessary to specify certain maximum

or minimum percentages of the elements. These are as shown in the following table.

Alloy Specifications Selling price per ton

X No more than $30\%$ of E

At least $40\%$ of F $$5\mathrm{.}50$

Y No more than $50\%$ of E

At least $10\%$ of F $$4\mathrm{.}00$

The usage of each element is limited to $5,000$ tons, and the total usage of both elements

is limited to $10,000$ tons. Further, due to the relatively uncertain demand for alloy Y$,$ the

company would like to ensure that Y constitutes at least $30\%$ of the total quantity of the

two alloys produced. Determine the mix of the two elements that will maximize profit

under these conditions.

a$)(35$ pts$)$ Formulate the problem.

b$)(15$ pts$)$ Solve the problem using Excel solver.

c$)(15$ pts$)$ What happens to the optimal profit when the allowed usage of element F

increases from $5,000$ to $6,000$ tons? Use sensitivity report to answer this question.

d$)(15$ pts$)$ What happens to the shadow price when the RHS of the total allowed usage

constraint decreases beyond the allowable decrease? Why? Explain using the

definition of shadow price. Provide proper explanation utilizing the meaning of shadow

price.

e$)(20$ pts$)$ Suppose the objective coefficient of one of the decision variables changes

within its allowed increase$/$decrease limit$.$ Can this have an impact on the shadow

prices? Why or why not?