Question $3.(50$ points$)$ A winery produces three types of wine: Merlot, Chardonnay, and Cabernet. Each barrel of Merlot requires $0\mathrm{.}5$ units of grapes, $0\mathrm{.}3$ units of oak, and $0\mathrm{.}1$ units of yeast; each barrel of Chardonnay requires $0\mathrm{.}6$ units of grapes, $0\mathrm{.}4$ units of oak, and $0\mathrm{.}2$ units of yeast; and each barrel of Cabernet requires $0\mathrm{.}7$ units of grapes, $0\mathrm{.}5$ units of oak, and $0\mathrm{.}3$ units of yeast. The winery has $250$ units of grapes, $100$ units of oak, and $60$ units of yeast available each month.

According to a contract with a local distributor, the winery is required to supply at least $120$ barrels of Merlot, $90$ barrels of Chardonnay, and $80$ barrels of Cabernet each month. Due to storage limitations at the distributor, the winery cannot send more than $180$ barrels of Merlot, $110$ barrels of Chardonnay, and $90$ barrels of Cabernet per month.

The winery makes a profit of $$15$ per barrel of Merlot, $$12$ per barrel of Chardonnay, and $$18$ per barrel of Cabernet.

You are asked to determine the optimal product mix that maximizes the monthly total profit. You will formulate a linear programming model. You will NOT solve this problem by Excel Solver.

SHOW ALL YOUR WORK ONA PIECE OF PAPER.

a$)(12$ points$)$ What are the decision variables? How many decision variables are there? Clearly explain the decision variables in English and provide the mathematical notation used in the algebraic formulation of the decision problem.

b$)(8$ points$)$ What is the objective function? Clearly explain the objective function in English and provide the mathematical formulation.

c$)(25$ points$)$ What are the constraints of the problem? Clearly explain each constraint in English and provide the mathematical formulation.

d$)(5$ point$)$ Are there any sign or type restrictions in your formulation? Why or why not? Clearly explain the sign and type restrictions $($if any$).$