Exam 7: Linear Programming

Introduce slack variables as necessary and write the initial simplex tableau for the problem. -Maximize $z=2 x_{1}+x_{2}$ Subject to: $x_{1}+x_{2} \leq 71$ $3 x_{1}+x_{2} \leq 175$ $x_{1} \geq 0, x_{2} \geq 0$

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 $\$ 1.54$ to make each dog and $\$ 1.32$ for each dinosaur. What is the company's minimum cost?

Find the transpose of the matrix. -

Use the two-stage method to solve. - Minimize w=18+11 subject to: 2+2=14 6+2\geq36 2+4\leq24 \geq0,\geq0

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 $\$ 20$ and on an SST ring is $\$ 50$ ?

Provide an appropriate response. -Consider a linear program with an objective function for profit. Thinking of isoprofit lines, if the objective function is evaluated at the corner points of polygon $A B C D$ , and $p(A)=10, p(B)=20$ , and $p(C)=5$ , is it safe to assume that $p(D)$ is not the corner point at which the profit is maximized?

Use the two-stage method to solve. -Find $x_{1} \geq 0$ and $x_{2} \geq 0$ such that +\leq25 +2\geq15 2+\leq30 And $\mathrm{z}=4 \mathrm{x}_{1}+2 \mathrm{x}_{2}$ is maximized.

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

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 $270 \mathrm{lb}$ of flour, 260 dozen eggs, $210 \mathrm{lb}$ of sugar. The profit on a batch of sweet rolls is $\$ 7.00$ and on a batch of donuts is $\$ 15.00$ .

Find the pivot in the tableau. -

Find the value(s) of the function on the given feasible region. -Find the maximum and minimum of $z=6 x+6 y$ .

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 4 cents, pill #2 costs 5 cents, and pill #3 costs 8 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin A?

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

Use duality to solve the problem. -Minimize $w=4 y_{1}+2 y_{2}$ Subject to: $3 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 22$ $\mathrm{y}_{1}+4 \mathrm{y}_{2} \geq 26$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

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.

Find the value(s) of the function, subject to the system of inequalities. -Find the maximum and minimum of $Z=19 x+12 y$ subject to: $0 \leq x \leq 10,0 \leq y \leq 5,3 x+2 y \geq 6$ .

Use graphical methods to solve the linear programming problem. -Maximize $\mathrm{z}=8 \mathrm{x}+12 \mathrm{y}$ Subject to: $40 x+80 y \leq 560$ $6 x+8 y \leq 72$ $x \geq 0$ $y \geq 0$

Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. - $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{s}_{2}=23$ $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}-\mathrm{s}_{2}=23$ $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{s}_{3}=14$ $\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{s}_{3}=14$

Introduce slack variables as necessary and write the initial simplex tableau for the problem. -Maximize $\mathrm{z}=\mathrm{x}_{1}+2 \mathrm{x}_{2}$ Subject to: $x_{1}+x_{2} \leq 20$ $3 x_{1}+2 x_{2} \leq 40$ $2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 60$ $x_{1} \geq 0, x_{2}>0$

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