$1.$ The following table summarizes the key facts about two products, A and

B$,$ and the resources, Q$,$ R$,$ and S$,$ required to produce them.

The company needs to decide how many units of product A and product B to produce

for maximizing profit.

a$.$ Formulate the linear programming model algebraically.

b$.$ Formulate and solve a linear programming model for this problem on a

spreadsheet.

$2.$ The Oak Works is a family$-$owned business that makes handcrafted dining

room tables and chairs. They obtain the oak from a local tree farm, which ships them

$2,500$ pounds of oak each month. Each table uses $50$ pounds of oak while each chair

uses $25$ pounds of oak. The family builds all the furniture itself and has $480$ hours of

labor available each month. Each table or chair requires six hours of labor. Each table

nets Oak Works $$400$ in profit, while each chair nets $$100$ in profit. Since chairs are

often sold with the tables, they want to produce at least twice as many chairs as tables.

The Oak Works would like to decide how many tables and chairs to produce so as to

maximize profit.

a$.$ Formulate the linear programming model algebraically.

b$.$ Formulate and solve a linear programming model for this problem on a

spreadsheet.

$3.$Weenies and Buns is a food processing plant that manufactures buns and

frankfurters for hot dogs. They grind their own flour for the buns at a maximum rate of

$200$ pounds per week. Each bun requires $0\mathrm{.}1$ pound of flour. They currently have a

contract with Pigland, Inc., which specifies that a delivery of $800$ pounds of pork product

is delivered every Monday. Each frankfurter requires $1/4$ pound of pork product. All the

other ingredients in the buns and frankfurters are in plentiful supply. Finally, the labor

force at Weenies and Buns consists of five employees working full time $(40$ hours per

week each$).$ Each bun requires two minutes of labor, and each frankfurter requires three

minutes of labor. Each bun yields a profit of $$0\mathrm{.}20,$ and each frankfurter yields a profit of

$$0\mathrm{.}40.$

Weenies and Buns would like to know how many buns and how many frankfurters they

should produce each week so as to achieve the highest possible profit.

a$.$ Formulate the linear programming model algebraically.

b$.$ Formulate and solve a linear programming model for this problem on a

spreadsheet.