Exam 7: Linear Programming

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 $C$ . -Pi 11 #1 costs 14 cents, pill #2 costs 10 cents, and pill #3 costs 8 cents. Larry wants to minimize cost. What are the coefficients of the objective function?

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

Graph the linear inequality. - $2 x+4 y \geq-8$

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 $350 \mathrm{lb}$ of flour, 290 dozen eggs, $340 \mathrm{lb}$ of sugar. The profit on a batch of sweet rolls is $\$ 116.00$ and on a batch of donuts is $\$ 100.00$ .

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 $\$ 1.70$ to make each dog and $\$ 1.87$ for each dinosaur. The company wants to minimize its costs. What are the coefficients of the constraint inequality for stuffing?

Use the simplex method to solve the linear programming problem. -Maximize $z=2 x_{1}+5 x_{2}+3 x_{3}$ Subject to: $2 x_{1}+x_{2}+3 x_{3} \leq 9$ $4 x_{1}+3 x_{2}+5 x_{3} \leq 12$ With $\quad x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$

Write the word or phrase that best completes each statement or answers thequestion. -In an unbounded region, will there always be a solution?

Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. -

Graph the feasible region for the system of inequalities. - $4 y+x \geq-2$ $y+2 x \leq 10$ $4 y \leq 10 x+40$ $\mathrm{y} \geq 0$

Convert the objective function into a maximization function. -Minimize $\mathrm{w}=2 \mathrm{y}_{1}+\mathrm{y}_{2}+3 \mathrm{y}_{3}$ Subject to: $3 \mathrm{y}_{1}+2 \mathrm{y}_{2}+\mathrm{y}_{3} \geq 56$ $\mathrm{y} 2+\mathrm{y} 3 \geq 24$ $2 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 30$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0$

Graph the feasible region for the system of inequalities. - $2 x+y \geq 4$ $\mathrm{x}-1 \geq 0$

Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. - $15 x_{1}+3 x_{2} \leq 47$ $12 \mathrm{x}_{1}+6 \mathrm{x}_{2} \geq 63$

A manufacturing company wants to maximize profits on products $A, B$ , and $C$ . The profit margin is $\$ 3$ for $A, \$ 6$ for $B$ , and $\$ 15$ for $C$ . The production requirements and departmental capacities are as follows: -What is the constraint for the painting department?

Use duality to solve the problem. -Minimize $\mathrm{w}=\mathrm{y} 1+4 \mathrm{y} 2$ Subject to: $4 \mathrm{y}_{1}+3 \mathrm{y}_{2} \geq 75$ $3 \mathrm{y}_{1}+5 \mathrm{y}_{2} \geq 93$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

Find the transpose of the matrix. -

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.

Use the two-stage method to solve. -Maximize $z=4 x_{1}+6 x_{2}$ Subject to: $2 x_{1}+x_{2}=12$ $2 \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 20$ $2 \mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 24$ $x_{1} \geq 0, x_{2} \geq 0$

Find the value(s) of the function, subject to the system of inequalities. -Find the maximum and minimum of $P=24 x+21 y$ subject to: $0 \leq x \leq 10,0 \leq y \leq 5,3 x+2 y \geq 6$ .

Provide an appropriate response. -It is possible to have a system of linear inequalities with a feasible region that includes more than one enclosed region.

Solve the problem. -An agricultural research scientist is developing three new crop growth supplements -- A, B, and C. Each pound of each supplement contains four enzymes -- $\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$ , and $\mathrm{E}_{4}--$ in the amounts (in milligrams) shown in the table. The cost of $\mathrm{E}_{1}$ is $\$ 20 / \mathrm{mg}$ , the cost of $\mathrm{E}_{2}$ is $\$ 40 / \mathrm{mg}$ , the cost of $\mathrm{E}_{3}$ is $\$ 10 / \mathrm{mg}$ , and the cost of $\mathrm{E}_{4}$ is also $\$ 10 / \mathrm{mg}$ . The growth benefit for crops is expected to be proportional to 10 times the amount of $A$ used, 25 times the amount of $B$ used, and 60 times the amount of $C$ used. However, the total cost of the enzymes used in $A, B$ , and $C$ must be less than $\$ 5000$ for each treatment. How many pounds each of $A, B$ , and $C$ should be produced to maximize the growth effect?