Chapter 6.10, Problem 3P

Explanation of Solution

Given LP:

  ${\text{max z =5x}}_{\text{1}}{\text{+3x}}_{\text{2}}{\text{+x}}_{\text{3}}$

• The above function is subject to the following constraints,

  $\begin{array}{l}{\text{2x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}\le \text{6}\\ {\text{x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+x}}_{\text{3}}\le 7\\ {\text{x}}_{\text{1}}{\text{, x}}_{\text{2}}{\text{, x}}_{\text{3}}\ge \text{0}\end{array}$

To solve dual of LP:

• The dual of the Linear Programming (LP) is as follows,

  ${\text{min w = 6y}}_{\text{1}}{\text{+ 7y}}_{\text{2</}}$

Explanation of Solution

Given LP:

  ${\text{max z =5x}}_{\text{1}}{\text{+3x}}_{\text{2}}{\text{+x}}_{\text{3}}$

• The above function is subject to the following constraints,

  $\begin{array}{l}{\text{2x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}\le \text{6}\\ {\text{x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+x}}_{\text{3}}\le 7\\ {\text{x}}_{\text{1}}{\text{, x}}_{\text{2}}{\text{, x}}_{\text{3}}\ge \text{0}\end{array}$

To solve the max problem:

• The dual of the Linear Programming (LP) is as follows,

  ${\text{min w = 6y}}_{\text{1}}{\text{+ 7y}}_{\text{2}}$

• The above function is subject to the following constraints,

  $\begin{array}{l}{\text{2y}}_{\text{1}}{\text{+ y}}_{\text{2}}\ge \text{5}\\ {\text{y}}_{\text{1}}{\text{+ 2y}}_{\text{2}}\ge \text{3}\\ {\text{y}}_{\text{1}}{\text{+ y}}_{\text{2}}\ge \text{1}\\ {\text{y}}_{\text{1}}{\text{, y}}_{\text{2}}\ge \text{0}\end{array}$

• The user find the optimal solution for the dual LP to be “$\text{w=}\frac{\text{49}}{\text{3}}$”, “${\text{y}}_1\text{=}\frac{7}{\text{3}}$”, and “${\text{y}}_2\text{=}\frac{1}{\text{3}}$”