Chapter 6: Chapter 6

Solve each linear programming problem by the simplex method. $$ \begin{array}{lc} \text { Maximize } & P=5 x+3 y \\ \text { subject to } & x+y \leq 80 \\ & 3 x \leq 90 \\ & x \geq 0, y \geq 0 \end{array} $$

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{aligned} \text { Minimize } & C=3 x+2 y \\ \text { subject to } & 2 x+3 y \geq 90 \\ & 3 x+2 y \geq 120 \\ & x \geq 0, y \geq 0 \end{aligned} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{cc} \text { Maximize } & P=3 x-4 y \\ \text { subject to } & x+3 y \leq 15 \\ & 4 x+y \leq 16 \\ & x \geq 0, y \geq 0 \end{array} $$

The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gallons of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $$\$ 300$$ and that for pipeline water is $$\$ 500$$, how much water should the manager get from each source to minimize daily water costs for the city?

Solve each linear programming problem by the method of corners. $$ \begin{array}{lr} \text { Maximize } & P=x+3 y \\ \text { subject to } & 2 x+y \leq 6 \\ x+y & \leq 4 \\ x & \leq 1 \\ x \geq 0, y & \geq 0 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=10 x+12 y \\ \text { subject to } & x+2 y \leq 12 \\ & 3 x+2 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{aligned} \text { Minimize } & C=6 x+4 y \\ \text { subject to } & 6 x+y \geq 60 \\ & 2 x+y \geq 40 \\ & x+y \geq 30 \\ & x \geq 0, y \geq 0 \end{aligned} $$

Ace Novelty manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires \(1.5 \mathrm{yd}^{2}\) of plush, \(30 \mathrm{ft}^{3}\) of stuffing, and 5 pieces of trim; each Saint Bernard requires \(2 \mathrm{yd}^{2}\) of plush, \(35 \mathrm{ft}^{3}\) of stuffing, and 8 pieces of trim. The profit for each Panda is $$\$ 10$$ and the profit for each Saint Bemard is $$\$ 15$$. If $3600 \mathrm{yd}^{2}\( of plush, \)66,000 \mathrm{ft}^{3}$ of stuffing and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize profit?

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{aligned} \text { Minimize } & C=10 x &+y \\ \text { subject to } & 4 x+y & \geq 16 \\ x+2 y & \geq 12 \\ x & \geq 2 \\ x & \geq 0, y & \geq 0 \end{aligned} $$

Solve each linear programming problem by the method of corners. $$ \begin{aligned} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ y & \leq 6 \\ x & \geq 0, y \geq 0 \end{aligned} $$