Exam 7: Linear Programming

Graph the feasible region for the system of inequalities. - $3 x+2 y \leq-6$ $x-1 \geq 0$

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

State the dual problem. Use $\mathrm{y}_{1}, \mathrm{y}_{2}$ , and $\mathrm{y}_{3}$ as the variables. Given: $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0$ , and $\mathrm{y}_{3} \geq 0$ . -Minimize $\mathrm{w}=2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}$ Subject to: $x_{1}+3 x_{2} +2 x_{3}\geq 34$ $2 x_{1}+4 x_{2} +3 x_{3}\geq 59$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} , \mathrm{x}_{3}\geq 0$

Use the two-stage method to solve. -Minimize $w=5 x_{1}+2 x_{2}$ Subject to: $x_{1}+x_{2}=10$ $3 x_{1}+x_{2} \geq 18$ $\mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 12$ $x_{1} \geq 0, x_{2} \geq 0$

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 maximum profit if the capacity of the finishing department changes to 33,000 hours?

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 maximum profit if the profit margin on A changes to $\$ 7.00$ ?

Graph the linear inequality. - $-3 x-5 y \leq 15$

Find the pivot in the tableau. -

Provide an appropriate response. -Explain why a different slack variable must be used for each constraint when converting constraints to linear equations.

Use the two-stage method to solve. -Find $x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0$ such that $x_{1}+x_{2}+x_{3} \leq 125$ $x_{1}+x_{2}+x_{3} \geq 75$ And $z=2 x_{1}+5 x_{2}+3 x_{3}$ is maximized.

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 10 cents, and pill #3 costs 1 cent. How many of each pill must Larry take to minimize his cost?

Provide an appropriate response. -Explain the result if the simplex tableau is solved using a quotient other than the smallest non-negative quotient.

Find the pivot in the tableau. -

Provide an appropriate response. -A negative number in the rightmost column of a simplex tableau tells you that you have made what kind of error when pivoting?

Convert the objective function into a maximization function. -Minimize $w=2 y 1+4 y 2+3 y 3$ Subject to: $\mathrm{y}_{1}+\mathrm{y}_{2} \geq 14$ $2 \mathrm{y}_{1}+3 \mathrm{y}_{2}+\mathrm{y}_{3} \geq 18$ $\mathrm{y}_{1}+2 \mathrm{y}_{2}+\mathrm{y}_{3} \geq 10$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0$

Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. - $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3} \leq 41$ $x_{1}+x_{2}+x_{3} \geq 23$ $\mathrm{x}_{1}+\mathrm{x}_{2} \geq 14$

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 are the constants in the model?

Convert the constraints into linear equations by using slack variables. -Maximize $z=1.1 x_{1}+2.6 x_{2}$ Subject to: $1.9 x_{1}+1.5 x_{2} \leq 35$ $1)6 x_{1}+1.5 x_{2} \leq 35$ $x_{1} \geq 0, x_{2} \geq 0$

Use graphical methods to solve the linear programming problem. -Minimize $z=4 x+5 y$ Subject to: $\quad 2 x-4 y \leq 10$ $2 x+y \geq 15$ $x \geq 0$ $y \geq 0$

Use the indicated entry as the pivot and perform the pivoting once. -