Exam 7: Linear Programming

Use the two-stage method to solve. -Maximize $z=5 x_{1}+4 x_{2}$ Subject to: $x_{1}+2 x_{2}=15$ $\mathrm{x}_{1}+\mathrm{x}_{2} \geq 12$ $2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 30$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0$

Graph the linear inequality. - $y<x+2$

A toy making company has at least 300 squares of felt, $700 \mathrm{oz}$ of stuffing, and $230 \mathrm{ft}$ of trim to make dogs and dinosaurs. A dog uses 1 square of felt, $4 \mathrm{oz}$ of stuffing, and $1 \mathrm{ft}$ of trim. A dinosaur uses 2 squares of felt, $3 \mathrm{oz}$ of stuffing, and $1 \mathrm{ft}$ of trim. -It costs the company $\$ 1.39$ to make each dog and $\$ 1.70$ for each dinosaur. The company wants to minimize its costs. What are the coefficients of the objective function?

The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24rings each day using up to 60 total man hours of labor. It takes 3 man hours to make one VIP ring, versus 2 man hoursto make one SST ring. -How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $\$ 60$ and on an SST ring is $\$ 20$ ?

Use the two-stage method to solve. -Maximize $z=12 x_{1}+15 x_{2}$ Subject to: $x_{1}+x_{2}=10$ $3 x_{1}+3 x_{2} \leq 36$ $4 \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 18$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0$

A manufacturer of wooden chairs and tables must decide in advance how many of each item will be made in a givenweek. Use the table to find the system of inequalities that describes the manufacturer's weekly production. -Use $x$ for the number of chairs and $y$ for the number of tables made per week. The number of work-hours available for construction and finishing is fixed.

Solve the problem. -Two foods are to be mixed. Food I contains 10 grams of carbohydrates per unit. Food II contains 5 grams of carbohydrates per unit. The total grams of carbohydrates in the mixture may not exceed 130 grams. No more than 18 units of Food I and no more than 6 units of Food II are to be used in the mixture. Each food contains 20 grams of protein per unit. How many units of each food should be used to maximize the grams of protein in the mixture?

Graph the feasible region for the system of inequalities. - $x+2 y \leq 2$ $x+y \geq 0$

State the linear programming problem in mathematical terms, identifying the objective function and the constraints. -A car repair shop blends oil from two suppliers. Supplier I can supply at most 45 gal with $3.9 \%$ detergent. Supplier II can supply at most $67 \mathrm{gal}$ with $3.3 \%$ detergent. How much can be ordered from each to get at most 100 gal of oil with maximum detergent?

Find the pivot in the tableau. -

Use the simplex method to solve the linear programming problem. -Maximize $\mathrm{z}=9 \mathrm{x}_{1}+8 \mathrm{x}_{2}$ Subject to: $x_{1}+2 x_{2} \leq 2$ $3x_{1}+2 x_{2} \leq 8$ $2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 10$ With $\quad x_{1} \geq 0, x_{2} \geq 0$

Use the two-stage method to solve. -Find $x_{1} \geq 0$ and $x_{2} \geq 0$ such that $2 x_{1}+x_{2} \geq 12$ $\mathrm{x}_{1}+4 \mathrm{x}_{2} \leq 36$ And $\mathrm{z}=4 \mathrm{x}_{1}+3 \mathrm{x}_{2}$ is maximized.

A manufacturer of wooden chairs and tables must decide in advance how many of each item will be made in a givenweek. Use the table to find the system of inequalities that describes the manufacturer's weekly production. -Use $x$ -for the number of chairs and $y$ for the number of tables made per week. The number of work hours a yailable for construction and finishing is fixed.

Use the simplex method to solve the linear programming problem. -Maximize $z=x_{1}+3 x_{2}+x_{3}+2 x_{4}$ Subject to: $2 x_{1}+x_{2}+5 x_{3}+6 x_{4} \leq 25$ $5 x_{1}+3 x_{2}+4 x_{3}+x_{4} \leq 60$ With $x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, x_{4} \geq 0$

Use the two-stage method to solve. -Find $x_{1} \geq 0$ and $x_{2} \geq 0$ such that +2\geq24 +\leq40 And $\mathrm{z}=6 \mathrm{x}_{1}+5 \mathrm{x}_{2}$ is maximized.

Use the simplex method to solve the linear programming problem. -Maximize $\mathrm{z}=7 \mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}$ Subject to: $x_{1}+5 x_{2}+7 x_{3} \leq 8$ $x_{1}+4 x_{2}+11 x_{3} \leq 9$ With $\quad x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$

Use graphical methods to solve the linear programming problem. -Maximize $\mathrm{z}=2 \mathrm{x}+5 \mathrm{y}$ Subject to: $\quad 3 x+2 y \leq 6$ $-2 x+4 y \leq 8$ $x \geq 0$ $y \geq 0$

Provide an appropriate response. -What happens if an indicator other than the most negative one is chosen to solve a simplex tableau?

Provide an appropriate response. -Is it possible to have a bounded feasible region that does not optimize an objective function?

Find the value(s) of the function, subject to the system of inequalities. -Find the maximum and minimum of $Z=9 x-18 y$ subject to: $0 \leq x \leq 5,0 \leq y \leq 8,4 x+5 y \leq 30$ , and $4 x+3 y \leq 20$ .