Chapter 6, Problem 11RP

Explanation of Solution

Finding range of values that makes the current basis remains optimal:

• The given Linear Programming (LP) is,

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

$\text{s}\text{.t}\text{.}{\text{2x}}_{\text{1}}{\text{+x}}_{\text{2}}\le \text{8}$        (1)

${\text{4x}}_{\text{1}}{\text{+x}}_{\text{2}}\le \text{10}$        (2)

${\text{x}}_{\text{1}}{\text{, x}}_{\text{2}}\ge \text{0}$

• Substitute the value of “${\text{x}}_{\text{1}}$” as “0” in Equation(1),

$\begin{array}{c}\text{2}\times {\text{0+x}}_{\text{2}}\text{=8}\\ {\text{x}}_{\text{2}}\text{=8}\end{array}$

• Hence the point “${\text{(x}}_1,{\text{x}}_2)$” is “(0,8)”.
• Substitute the value of “${\text{x}}_2$” as “0” in Equation(1),

$\begin{array}{c}{\text{2x}}_1\text{+0=8}\\ {\text{2x}}_1\text{=8}\\ {\text{x}}_1=\frac{8}{2}\\ {\text{x}}_1=4\end{array}$

• Hence the point “${\text{(x}}_1,{\text{x}}_2)$” is “(4,0)”