Big-M_Method_Maths_Projectszdczsczs.pptx

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1. Big-M Method of LPP Maths Mini Project Group Members: [Add Names] Guide: [Teacher Name]

2. Introduction • Linear Programming optimizes an objective function under constraints. • Big-M Method modifies the simplex method to handle ≥ and = constraints. • Introduces artificial variables and assigns large penalty (M) to remove them.

3. Need for Big-M Method • Handles constraints of type ≥ or =. • Adds slack, surplus & artificial variables. • Uses a large penalty M to push artificial variables out of solution.

4. Steps of Big-M Method 1. Convert constraints to standard form. 2. Add artificial variables for ≥ and = constraints. 3. Modify objective function (Max: subtract M*A, Min: add M*A). 4. Solve using Simplex method. 5. Ensure artificial variables are zero in final solution.

5. Example Problem Maximize: Z = 3x₁ + 2x₂ Subject to: x₁ + x₂ ≥ 4 x₁ + 2x₂ ≤ 6 x₁, x₂ ≥ 0

6. Standard Form Conversion x₁ + x₂ - s₁ + A₁ = 4 x₁ + 2x₂ + s₂ = 6 Modified Objective: Z = 3x₁ + 2x₂ - M*A₁

7. Initial Simplex Table Show initial table with A₁ in basis & Z-row with -M.

8. Iterations Perform simplex iterations until A₁ leaves basis. Highlight pivot steps.

9. Final Solution x₁ = [value], x₂ = [value], Z = [value] A₁ = 0 → feasible solution found.

10. Applications • Operations Research: Resource allocation. • Manufacturing: Production planning. • Transportation: Logistics cost optimization. • Finance: Portfolio optimization.

11. Advantages & Limitations Advantages: • Handles any type of constraint. • Extends simplex method. Limitations: • Requires very large M. • More computational steps.

12. Conclusion • Big-M Method is a robust way to handle ≥ and = constraints. • Ensures feasible solutions by penalizing artificial variables. • Widely used in real-world optimization problems.