Exam 7: Linear Programming

Use the two-stage method to solve. -Minimize $w=x_{1}+2 x_{2}$ Subject to: $2 \mathrm{x}_{1}+\mathrm{x}_{2}=18$ $2 x_{1}+2 x_{2} \geq 17$ $\mathrm{x}_{1}+\mathrm{x}_{2} \leq 9$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0$

Solve the problem. -The manager of a concert hall estimates that 600 people attend each classical concert, that 1000 people attend each jazz concert, and that 300 people attend each rock concert. In any given month, the total of the number of classical concerts and the number of jazz concerts may not exceed 9 and the number of rock concerts must be no more than 8 . Furthermore there should be twice as many rock concerts as classical concerts in any given month. How many of each type of concert should there be in a month to maximize attendance?

Use the two-stage method to solve. -Maximize $z=18 x_{1}+12 x_{2}$ Subject to: $2 x_{1}+3x_{2}=24$ $2 x_{1}+ x_{2} \leq 45$ $2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \geq 20$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0$

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 $\$ 1.67$ to make each dog and $\$ 1.96$ for each dinosaur. The company wants to minimize its costs. What are the constants in the corresponding dual model?

Use graphical methods to solve the linear programming problem. -Minimize $z=2 x+4 y$ Subject to: $x+2 y \geq 10$ $3 x+y \geq 10$ $x \geq 0$ $y \geq 0$

Use duality to solve the problem. -Minimize $\mathrm{w}=4 \mathrm{y} 1+2 \mathrm{y} 2$ Subject to: $3 \mathrm{y}_{1}+2 \mathrm{y}_{2} \geq 60$ $4 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 40$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

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 11 cents, and pill #3 costs 6 cents. Larry wants to minimize cost. What are the constants in the problem?

Convert the constraints into linear equations by using slack variables. -Maximize $z=2 x_{1}+8 x_{2}$ Subject to: $x_{1}+6 x_{2} \leq 15$ $2 x_{1}+9 x_{2} \leq 25$ $x_{1} \geq 0, x_{2} \geq 0$

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.

Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. - $\mathrm{x}_{1}+\mathrm{x}_{2} \leq 20$ $2 \mathrm{x}_{1}+4 \mathrm{x}_{2} \geq 24$

Find the transpose of the matrix. -

Use graphical methods to solve the linear programming problem. -Maximize $z = 6x + 7y$ Subject to: $2 x+3 y \leq 12$ $2 x+ y \leq 8$ $x \geq 0$ $y \geq 0$

Find the value(s) of the function on the given feasible region. -Find the maximum and minimum of $z=20 x+5 y$ .

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

Use the simplex method to solve the linear programming problem. -Maximize $z=4 x_{1}+2 x_{2}$ Subject to: $x_{1}+5 x_{2} \leq 14$ $2 x_{1}+3 x_{2} \leq 12$ With $\quad x_{1} \geq 0, x_{2} \geq 0$

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 $\$ 40$ and on an SST ring is $\$ 30$ ?

Introduce slack variables as necessary and write the initial simplex tableau for the problem. -Maximize $z=2 x_{1}+5 x_{2}$ Subject to: $5 x_{1}+10 x_{2} \leq 113$ $10 x_{1}+15 x_{2} \leq 129$ $x_{1} \geq 0, \quad x_{2} \geq 0$

Solve the problem. -A toy manufacturer makes three types of toys. Toy A requires 12 minutes to assemble, 12 minutes to paint, 4 minutes to package, and sells for $\$ 20$ . Toy B requires 20 minutes to assemble, 12 minutes to paint, 8 minutes to package, and sells for $\$ 25$ . Toy $\mathrm{C}$ requires 24 minutes to assemble, 20 minutes to paint, 16 minutes to package, and sells for $\$ 30$ . The manufacturer has available 800 work-minutes for assembly, 600 work-minutes for painting, and 720 work-minutes for packaging each day. They must make at least 10 toys of type A and 15 of type $B$ each day. How many of each type of toy should be produced each day to maximize revenues?

Use duality to solve the problem. -Minimize $w=4 y_{1}+4 y_{2}$ Subject to: $5 y_{1}+10 y_{2} \geq 100$ $10 \mathrm{y}_{1}+20 \mathrm{y}_{2} \geq 150$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$

Provide an appropriate response. -Does a linear program with at least three constraints always have a closed feasible region?