Lec-9 Big M Method Minimization Problem || In Hindi & Urdu || Operation research

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(Big M Method Minimization Problem || In Hindi || Solve an LPP Example || Operation research)

The Big M method is a technique used in linear programming to handle linear programming problems with both equality and inequality constraints. It involves introducing a large positive constant (often denoted as "M") into the objective function and converting inequality constraints into equality constraints by introducing slack variables. The artificial objective function is then created by adding M multiplied by the slack variables to the original objective function, with the goal of finding a feasible solution for the original problem. The linear programming problem is then solved using standard optimization algorithms, such as the simplex method.

The Big M method is often used when dealing with constraints that cannot be easily expressed as equalities, or when the constraints have both equality and inequality components. For example, if a constraint in a linear programming problem is in the form "ax + by ≥ c", it can be converted into an equality constraint "ax + by + s = c", where "s" is the slack variable. The slack variable represents the excess or surplus of the constraint, and is constrained to be non-negative (s ≥ 0). By introducing slack variables, the original inequality constraints are converted into equality constraints, which allows for the use of standard linear programming techniques, such as the simplex method.

The artificial objective function in the Big M method is used to encourage the slack variables to be zero in the optimal solution, which means that the original inequality constraints are satisfied as equalities. The artificial variable (often denoted as R) is used to represent the artificial objective function, and its coefficient in the artificial objective function is set to one, while the coefficients of the other variables are set to zero. The goal is to minimize the value of the artificial objective function, while satisfying all the constraints, including the artificial constraints. If the optimal solution for the artificial objective function has a value of zero for the artificial variable (R), then the original problem is feasible and the solution is optimal. If the optimal solution for the artificial objective function has a positive value for the artificial variable (R), then the original problem is infeasible, and no feasible solution exists.

Once the linear programming problem is solved using the Big M method, the artificial variables and the artificial objective function are typically removed from the final solution, and the results are interpreted in terms of the original problem. The optimal solution for the variables represents the values that optimize the objective function, subject to the constraints. The values of the slack variables represent the surplus or excess of the resources or constraints, and can be used to interpret the feasibility of the solution. The choice of the value of M in the Big M method is important and should be carefully considered, as it can affect the stability and efficiency of the optimization algorithm. A suitable value of M depends on the specific problem and should be chosen judiciously.




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