Exam 7: Linear Programming

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 $\$ 1.24$ to make each dog and $\$ 1.76$ for each dinosaur. The company wants to minimize its costs. What are the coefficients of the constraint inequality for felt?

Graph the feasible region for the system of inequalities. - $2 y+x \geq-2$ $\mathrm{y}+3 \mathrm{x} \leq 9$ $y \leq 0$ $x \geq 0$

Find the value(s) of the function, subject to the system of inequalities. -Find the maximum and minimum of $P=9 x-12 y$ subject to: $0 \leq x \leq 5,0 \leq y \leq 8,4 x+5 y \leq 30$ , and $4 x+3 y \leq 20$ .

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 $\$ 1.26$ to make each dog and $\$ 1.87$ for each dinosaur. The company wants to minimize its costs. What are the coefficients of the dual objective function?

State the linear programming problem in mathematical terms, identifying the objective function and the constraints. -A breed of cattle needs at least 10 protein and 8 fat units per day. Feed type I provides 6 protein and 2 fat units at $\$ 3 / \mathrm{bag}$ . Feed ty pe II provides 2 protein and 5 fat units at $\$ 2 / \mathrm{bag}$ . Which mixture fills the needs at minimum cost?

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 $\$ 1.03$ to make each dog and $\$ 1.48$ for each dinosaur. The company wants to minimize its costs. What are the coefficients of the constraint inequality for trim?

Graph the linear inequality. - $y \geq 5$

State the linear programming problem in mathematical terms, identifying the objective function and the constraints. -A firm makes products $A$ and B. Product A takes 3 hours each on machine $L$ and machine M; product B takes 3 hours on $L$ and 2 hours on M. Machine $L$ can be used for 13 hours and M for 8 hours. Profit on product $A$ is $\$ 7$ and $\$ 10$ on B. Maximize profit.

Provide an appropriate response. -If a system of inequalities includes $x \leq 1$ , then the feasibility region is restricted to what?

Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill \#1 contains 4 units of $A$ and 3 of B. Pill \#2 contains 1 unit of A, 2 of B, and 4 of C. Pill \#3 contains 10 units of A, 1 of B, and 5 of $C$ . -Pill #1 costs 8 cents, pill #2 costs 2 cents, and pill #3 costs 10 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin $\mathrm{C}$ ?

Convert the objective function into a maximization function. -Minimize $w=y_{1}+2 y_{2}+3 y_{3}$ Subject to: $4 y_{1}+3 y_{2}+y_{3} \geq 16$ $\mathrm{y}_{1}+2 \mathrm{y}_{2}+4 \mathrm{y}_{3} \geq 40$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0$

Graph the feasible region for the system of inequalities. - $3 y+x \geq-6$ $y+2 x \leq 8$ $y \leq 0$ $x \geq 0$

Use the simplex method to solve the linear programming problem. -Maximize $\mathrm{z}=4 \mathrm{x}_{1}+2 \mathrm{x}_{2}$ Subject to: $2 x_{1}+3 x_{2} \leq 6$ $x_{1}+3 x_{2} \leq 4$ $2 \mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 8$ With $\quad x_{1} \geq 0, x_{2} \geq 0$

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 $\$ 1.16$ to make each dog and $\$ 1.84$ for each dinosaur. What is the company's minimum cost?

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 $\$ 1.46$ to make each dog and $\$ 1.63$ for each dinosaur. The company wants to minimize its costs. How much will it cost if the supply of trim is reduced to at least $100 \mathrm{ft}$ ?

Introduce slack variables as necessary and write the initial simplex tableau for the problem. -Maximize $\mathrm{z}=4 \mathrm{x}_{1}+\mathrm{x}_{2}$ Subject to: 2+5\leq5 3+3\leq10 \geq0,\geq0

Use duality to solve the problem. -Minimize $\mathrm{w}=5 \mathrm{y} 1+2 \mathrm{y} 2$ Subject to: $\mathrm{y}_{1}+\mathrm{y}_{2} \geq 19.5$ $2 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 24$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. - $2 x_{1}+5 x_{2} \leq 7$ $x_{1}+3 x_{2} \geq 5$

A bakery makes sweet rolls and donuts. A batch of sweet rolls requires $3 \mathrm{lb}$ of flour, 1 dozen eggs, and $2 \mathrm{lb}$ of sugar. Abatch of donuts requires $5 \mathrm{lb}$ of flour, 3 dozen eggs, and $2 \mathrm{lb}$ of sugar. Set up an initial simplex tableau to maximizeprofit. -The bakery has $700 \mathrm{lb}$ of flour, 540 dozen eggs, $460 \mathrm{lb}$ of sugar. The profit on a batch of sweet rolls is $\$ 94.00$ and on a batch of donuts is $\$ 142.00$ .

Solve the problem. -A political mailing will have several pages on the economy, the military, and the environment. The total number of these pages in the booklet should not be greater than 110, including at most 20 pages on the economy. For the target group that will receive the booklet, market research suggests that there will be a positive impact proportional to 5 times the number of pages on the economy, a positive impact proportional to 3 times the number of pages on the military, and a positive impact proportional to 2 times the number of pages on the environment. The editor insists that the number of pages on the environment be at least twice as many as the number on the military. Find the number of pages that should be devoted to the economy, the military, and the environment. (Hint: Write the constraint involving environment and military pages in the form $\leq 0$ .)