Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=3 x+4 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ & x \geq 0, y \geq 0 \end{array} $$

Wayland Company manufactures two models of its twin-size futons, standard and deluxe, in two locations, I and II. The maximum output at location I is 600 /week, whereas the maximum output at location II is \(400 /\) week. The profit per futon for standard and deluxe models manufactured at location I is \(\$ 30\) and \(\$ 20\), respectively; the profit per futon for standard and deluxe models manufactured at location II is \(\$ 34\) and \(\$ 18\), respectively. For a certain week, the company has received an order for 600 standard models and 300 deluxe models. If prior commitments dictate that the number of deluxe models manufactured at location II not exceed the number of standard models manufactured there by more than 50 , find how many of each model should be manufactured at each location so as to satisfy the order and at the same time maximize Wayland's profit.

A company manufactures products \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each product is processed in three departments: I, II, and III. The total available labor-hours per week for departments I, II, and III are 900,1080 , and 840 , respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows: $$ \begin{array}{lccc} \hline & \text { Product A } & \text { Product B } & \text { Product C } \\ \hline \text { Dept. I } & 2 & 1 & 2 \\ \hline \text { Dept. II } & 3 & 1 & 2 \\ \hline \text { Dept. III } & 2 & 2 & 1 \\ \hline \text { Profit } & \$ 18 & \$ 12 & \$ 15 \\ \hline \end{array} $$ How many units of each product should the company produce in order to maximize its profit? What is the largest profit the company can realize? Are there any resources left over?

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=4 x+6 y \\ \text { subject to } & 3 x+y \leq 24 \\ & 2 x+y \leq 18 \\ & x+3 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Minimize } & C=x-2 y+z \\ \text { subject to } & x-2 y+3 z \leq 10 \\ 2 x+y-2 z & \leq 15 \\ & 2 x+y+3 z \leq 20 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

You are given the final simplex tableau for the dual problem. Give the solution to the primal problem and the solution to the associated dual problem. Problem: Minimize \(\quad C=8 x+12 y\) subject to $\begin{aligned} x+3 y & \geq 2 \\ 2 x+2 y & \geq 3 \\ x \geq 0, y & \geq 0 \end{aligned}$ $$ \begin{array}{rrrrr|c} u & v & x & y & P & \text { Constant } \\ \hline 0 & 1 & \frac{3}{4} & -\frac{1}{4} & 0 & 3 \\ 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 2 \\ \hline 0 & 0 & \frac{5}{4} & \frac{1}{4} & 1 & 13 \end{array} $$