Chapter 4: Chapter 4

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Minimize } & C=x+2 y \\ \text { subject to } & 4 x+7 y \leq 70 \\ & 2 x+y=20 \\ & x \geq 0, y \geq 0 \end{aligned} $$

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

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

Use the method of this section to solve each linear programming problem. $$ \begin{array}{ll} \text { Maximize } & P=5 x+4 y+2 z \\ \text { subject to } & x+2 y+3 z \leq 24 \\ & x-y+z \geq 6 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=5 x+4 y \\ \text { subject to } & 3 x+5 y \leq 78 \\ & 4 x+y \leq 36 \\ & x \geq 0, y \geq 0 \end{array} $$

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

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Maximize } & P=x-2 y+z \\ \text { subject to } & 2 x+3 y+2 z \leq 12 \\ & x+2 y-3 z \geq 6 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

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

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

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{ll} \text { Minimize } & C=200 x+150 y+120 z \\ \text { subject to } & 20 x+10 y+z \geq 10 \\ & x+y+2 z \geq 20 \\ & x \geq 0, y \geq 0, z & \geq 0 \end{array} $$