JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III QUANTITATIVE ANALYSIS FOR BUSINESS DECISIONS Previous Question Paper (August/September - 2021 - R18)

The graphical method is a technique used to solve linear programming problems (LPP) involving two decision variables. The steps are as follows:

1. Formulate the objective function and the constraints as linear equations or inequalities.
2. Treat each inequality constraint as an equation to plot lines on a Cartesian plane (x1, x2).
3. Identify the feasible region, which is the area satisfying all constraints simultaneously, including non-negativity restrictions (x1, x2 ≥ 0).
4. Determine the corner points (vertices) of the feasible region by solving the intersection of the boundary lines.
5. Evaluate the objective function at each corner point.
6. The optimal solution is the corner point that yields the maximum or minimum value of the objective function, depending on the problem type.

Given: Max Z = 4x₁ + 3x₂, subject to 4x₁ + 3x₂ ≤ 24, x₁ ≤ 4.5, x₂ ≤ 6, x₁, x₂ ≥ 0.

Plotting the constraints:

1. 4x₁ + 3x₂ = 24: Intercepts are (6, 0) and (0, 8).
2. x₁ = 4.5: A vertical line.
3. x₂ = 6: A horizontal line.

The feasible region is a polygon with vertices determined by the intersections:

• (0, 0)
• (4.5, 0)
• (4.5, 2) [Intersection of x₁=4.5 and 4x₁+3x₂=24: 4(4.5)+3x₂=24 => 18+3x₂=24 => x₂=2]
• (1.5, 6) [Intersection of x₂=6 and 4x₁+3x₂=24: 4x₁+3(6)=24 => 4x₁=6 => x₁=1.5]
• (0, 6)

Evaluating Z = 4x₁ + 3x₂ at vertices:

• (0, 0): Z = 0
• (4.5, 0): Z = 4(4.5) + 0 = 18
• (4.5, 2): Z = 4(4.5) + 3(2) = 18 + 6 = 24
• (1.5, 6): Z = 4(1.5) + 3(6) = 6 + 18 = 24
• (0, 6): Z = 4(0) + 3(6) = 18

The maximum value of Z is 24. Since the objective function is parallel to the constraint 4x₁ + 3x₂ = 24, there are multiple optimal solutions along the line segment connecting (4.5, 2) and (1.5, 6).