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1. Big-M Method of LPP With Two Solved Examples

2. Introduction The Big-M method is an extension of the Simplex method. It is used to solve Linear Programming Problems (LPP) with artificial variables. It assigns a very large penalty 'M' to artificial variables in the objective function.

3. Formula & Terms Objective Function (Maximization): Z = c1x1 + c2x2 + ... + cnxn + M(A1 + A2 + ...) Terms: - c1, c2... : Coefficients of decision variables - x1, x2... : Decision variables - M : A very large positive number (penalty) - A1, A2... : Artificial variables added to convert constraints to equalities

4. Example 1 Maximize Z = 3x1 + 2x2 Subject to: x1 + x2 ≥ 4 x1 + 2x2 ≤ 6 x1, x2 ≥ 0 Solution: - Introduce artificial variable A1 for ≥ constraint. - New objective: Max Z = 3x1 + 2x2 - M(A1) - Solve using Simplex table until optimal solution is reached.

5. Example 2 Minimize Z = 5x1 + 4x2 Subject to: x1 + x2 = 6 x1 + 2x2 ≥ 8 x1, x2 ≥ 0 Solution: - Introduce artificial variable A2 for ≥ constraint. - New objective: Min Z = 5x1 + 4x2 + M(A2) - Solve using Simplex table, remove artificial variable in final solution.

6. Applications - Used in solving LPP with ≥ or = constraints - Industrial production planning - Resource allocation problems - Transportation and assignment problems

7. Conclusion The Big-M method is a powerful technique to handle artificial variables. It allows the Simplex method to work with ≥ and = constraints. Choosing a very large M ensures artificial variables are removed from the final solution.

8. References 1. Operations Research by Kanti Swarup 2. Introduction to Linear Programming by Taha 3. Lecture Notes & Research Papers on LPP