Quantitative Techniques: Theory and Problems

Multiple Solution Problems

Problem 2.22

Solve graphically the following linear programming problem:

Maximise x₁ + x₂

subject to:

–2x₁ + x₂ ≤ 1

x₁ ≤ 2

x₁ + x₂ ≤ 3

x₁, x₂ ≥ 0

Solution

Step 1: Finding the vertex of each constraint by treating the constraint of inequality nature as equality.

Constraint (i) in limiting form –2x₁ + x₂ = 1

When x₁ = 0      x₂ = 1

When x₂ = 0      x₁ = –1/2

Thus the vertices are (0, 1) and (–1/2, 0).

Constraint (iii) in limiting form x₁ + x₂ = 3

When x₁ = 0      x₂ = 3

When x₂ = 0      x₁ = 3

Thus the vertices are (0, 3) and (3, 0).

Step 2: Plotting the co-ordinates of 1^st constraint on the graph and joining them by a straight line, and shading the feasible region. Similarly drawing a straight line and shading feasible ...