Problem $3(35$ points$)$

The Moon Coffee Shop blends coffee on the premises for its customers. It sells three basic blends in $1-$pound bags, Special, Moon Dark, and Moon Regular. It uses four different types of coffee to produce the blends $-$ Brazilian, Mocha, Columbian, and Mild. The shop used the following blend recipe requirements.

The cost of Brazilian coffee is $\backslash (\backslash$$ $3\mathrm{.}00\backslash )$ per pound, the cost of Mocha is $\backslash (\backslash$$ $3\mathrm{.}75\backslash )$ per pound, the cost of Columbian is $\backslash (\backslash$$ $3\mathrm{.}90\backslash )$ per pound, and the cost of Mild is $\backslash (\backslash$$ $2\mathrm{.}70\backslash )$ per pound. The shop has $110$ pounds of Brazilian coffee, $70$ pounds of Mocha, $80$ pounds of Columbian, and $150$ pounds of Mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize the profit $($i$.$e$.,$ revenue $-$ cost$).$

$($a$)$ Formulate algebraically the Linear Programming $($LP$)$ model for this problem. Define the decision variables, objective function, and constraints. $(19$ points$)$

$($b$)$ Formulate this same linear programming problem on a spreadsheet and SOLVE using Excel Solver $($Provide the corresponding "Excel Spreadsheet" and the "Answer Report"$).$ Include "managerial statements" that communicate the results of the analyses $($i$.$e$.$ describe verbally the results$).(16$ points$)$