Chapter 4.14, Problem 4P

Explanation of Solution

Formulation of linear programming:

Given,

Maximize, ${\text{z=|2x}}_{\text{1}}{\text{-3x}}_{\text{2}}\text{|}$

Subject to the constraints,

$\begin{array}{l}{\text{4x}}_{\text{1}}{\text{+x}}_{\text{2}}\le \text{4}\\ {\text{2x}}_{\text{1}}{\text{-x}}_{\text{2}}\le \text{0}\text{.5}\\ {\text{x}}_{\text{1}}{\text{,x}}_{\text{2}}\ge \text{0}\end{array}$

Consider that,

Maximize,

$z=z\text{'}+z\text{'}\text{'}$

As given in the problem,

${\text{z=|2x}}_{\text{1}}{\text{-3x}}_{\text{2}}\text{|}$

So, $2x_1\text{-}3x_2=z\text{'}\text{-}z\text{'}\text{'}$, all the variables are non-negative.

First, note that in any basic feasible solution, the variables both $z\text{'}$ and $z\text{'}\text{'}$ cannot be positive