Question

Consider the following Linear Programming Problem (LPP):

$\text{Maximize}z=3x_1+2x_2$

$\text{Subject to:}$
$x_1\le 4$
$x_2\le 6$
$3x_1+2x_2\le 18$
$x_1\ge 0,x_2\ge 0$

• A. The LPP has multiple optimal solution
• B. The LPP is unbounded
• C. The LPP is infeasible
• D. The LPP has a unique optimal solution

Answer 1

The correct option is A The LPP has multiple optimal solution

Linear Programming Problem (LPP)

$\text{Maximize,}z=3x_1+2x_2$

$\text{Constraints:}$

$x_1\le 4...\left(i\right)$

$x_2\le \mathrm{6...}\left(ii\right)$

$3x_1+2x_2\le 18...\left(iii\right)$

$x_1\le 0,x_2\le \mathrm{0...}\left(iv\right)$

$\text{Using graphical method}$


Because objective functions have slope same as constraint (iii) i.e. objective function is parallel to constraint. Therefore the LPP has multiple optimal solutions.

$\text{For example at Point B,}$

$\text{Maximum,}z=3x_1+2x_2$

$=3\left(4\right)+2\left(3\right)=18$

$\text{and at point C,}$

$\text{Maximum, z = 3(2) + 2(6) = 18}$