JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD MCA II OPERATIONS RESEARCH Previous Question Paper (August - 2017 - R09)

To solve the LPP by the graphical method, we analyze the constraints:

1. Constraint 1: 3x + 2y ≤ 50. Intercepts are (50/3, 0) and (0, 25).
2. Constraint 2: 5x + 3y ≤ 10. Intercepts are (2, 0) and (0, 10/3).
3. Constraint 3: x + y ≥ 0 and x, y ≥ 0 are non-negativity constraints.

Comparing the constraints 3x + 2y ≤ 50 and 5x + 3y ≤ 10, the feasible region is entirely bounded by the line 5x + 3y ≤ 10 and the axes in the first quadrant, because 5x + 3y ≤ 10 is more restrictive than 3x + 2y ≤ 50. The vertices of the feasible region are (0, 0), (2, 0), and (0, 10/3).

Evaluating the objective function z = 3x + 5y at the vertices:

1. At (0, 0): z = 3(0) + 5(0) = 0
2. At (2, 0): z = 3(2) + 5(0) = 6
3. At (0, 10/3): z = 3(0) + 5(10/3) = 50/3 ≈ 16.67

The maximum value is 16.67 at x = 0, y = 3.33.

The phases of an Operations Research study are as follows:

1. Definition of the Problem: Clearly identifying the objectives, the constraints, and the decision variables involved in the system.
2. Construction of a Model: Developing a mathematical representation (model) that relates the variables to the objective function and constraints.
3. Solution of the Model: Utilizing mathematical techniques and algorithms (such as Simplex, Graphical method, etc.) to determine the optimal values for the decision variables.
4. Validation of the Model: Testing the model to ensure it accurately represents the real-world system and produces reliable results.
5. Implementation and Control: Translating the mathematical solution into operational instructions and establishing a monitoring system to ensure the solution remains valid as conditions change.