Exam 7: Linear Programming

Solve the problem. -An athlete uses three different exercise regimens for increasing cardiovascular fitness. An hour of regimen A rates a 3 in effectiveness, an hour of regimen $B$ rates a 4 in effectiveness, and an hour of regimen $C$ rates a 5 in effectiveness. In any given week, the athlete can train at most 5 hours by regimen $A$ . In any given week the combined hours of regimens $A$ and $C$ may not exceed 9. In any given week the combined hours of regimens B and C may not exceed 10. How many hours of each regimen should the athlete perform in a week in order to maximize the increase in fitness?

Use the simplex method to solve the linear programming problem. -Maximize $z=x_{1}+2 x_{2}+4 x_{3}+6 x_{4}$ Subject to: $x_{1}+2 x_{2}+3 x_{3}+x_{4} \leq 100$ $3 x_{1}+x_{2}+2 x_{3}+x_{4} \leq 75$ With $\quad x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, x_{4} \geq 0$

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

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

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 constraint for the finishing department?

Write the word or phrase that best completes each statement or answers thequestion. -Explain why the graphing method is not satisfactory for solving a linear programming problem with 3 variables.

Find the value(s) of the function, subject to the system of inequalities. -Find the minimum of $Z=16 x+15 y+19$ subject to: $x \geq 0, y \geq 0, x+y \geq 1$ .

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

Provide an appropriate response. -The feasible region of a set of two inequalities must always be unbounded.

Convert the constraints into linear equations by using slack variables. -Maximize $\mathrm{z}=4 \mathrm{x}_{1}+12 \mathrm{x}_{2}+7 \mathrm{x}_{3}$ Subject to: $2 x_{1}+4 x_{2}+x_{3} \leq 12$ +6+\leq24 \geq0,\geq0,\geq0

Find the pivot in the tableau. -

Solve the problem. -An electronics store stocks VCRs, stereo systems, and television sets. They have limited storage space and can stock a total of at most 80 of these three machines. They know from past experience that they should stock twice as many VCRs as stereo systems and at least 20 television sets. If each VCR sells for $\$ 400$ , each stereo system sells for $\$ 1900$ , and each television set sells for $\$ 600$ , how many of each should be stocked and sold for maximum revenues?

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. - $3 x-2 y \geq-6$ $x - 1 < 0$

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 $\mathrm{w}=6 \mathrm{x}_{1}+3 \mathrm{x}_{2}$ Subject to: $3 x_{1}+2 x_{2} \geq 24$ $2 x_{1}+5 x_{2} \geq 38$ $\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0$

Provide an appropriate response. -Is it possible that the feasible region of a linear program include more than one distinct area?

Convert the constraints into linear equations by using slack variables. -Maximize $z=3 x_{1}+5 x_{2}$ Subject to: $3 x_{1}+3 x_{2} \leq 30$ $x_{1}+4 x_{2} \leq 40$ $x_{1} \geq 0, x_{2} \geq 0$

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

Convert the objective function into a maximization function. -Minimize $\mathrm{w}=\mathrm{y}_{1}+3 \mathrm{y}_{2}+\mathrm{y}_{3}+4 \mathrm{y}_{4}$ Subject to: $\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}+\mathrm{y}_{4} \geq 27$ $2 \mathrm{y}_{1}+2 \mathrm{y}_{2}+\mathrm{y}_{3}+2 \mathrm{y}_{4} \geq 53$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0, \mathrm{y}_{4} \geq 0$

Use duality to solve the problem. -Minimize $w=y_{1}+3 y_{2}+2 y_{3}$ Subject to: $\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3} \geq 50$ $2 \mathrm{y}_{1}+\mathrm{y}_{2} \geq 25$ $\mathrm{y}_{1} \geq 0, \mathrm{y}_{2} \geq 0, \mathrm{y}_{3} \geq 0$