Geometrical Solution Of
Linear Programming Problem

Operations Research

Question: Find a geometrical interpretation and solution of the following LP problem:

Maximize  z = 3x₁ + 5x₂
subject to restrictions
x₁ + 2x₂ ≤ 2000   (time restraint)
x₁ + x₂  ≤ 1500   (plastic restraint)
     x₂  ≤ 600    (dress restraint)
and      x₁  ≥ 0, x₂ ≥ 0 (non-negative restrictions)
  
Answer:

Here is the graphical solution,

① Graph inequality constraints

   Consider two mutually perpendicular lines Ox₁ and Ox₂. Any point in the positive quadrant satisfies non-negative restrictions i.e. x₁  ≥ 0, x₂ ≥ 0

   Plot a line representing x₁ + 2x₂ = 2000

   Clearly any point below line x₁ + 2x₂ = 2000 satisfies x₁ + 2x₂ ≤ 2000. This region is coloured blue here.



   Similarly plot other lines to represent x₁ + x₂  ≤ 1500 and x₂  ≤ 600




② Find the feasible region or solution space.

As shown in figure any point in the shaded area is a feasible solution to the given LPP.