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1. Graphic Method:

The Graphic Method is a graphical approach used to solve LPPs with two decision variables. Here are the steps:

  • Plot the constraints on a graph to determine the feasible region.
  • Identify the corner points (vertices) of the feasible region.
  • Evaluate the objective function at each corner point.
  • Select the corner point that optimizes the objective function.

This method is straightforward and intuitive but is limited to problems with two variables due to graphical representation constraints.

2. Simplex Method:

The Simplex Method is a widely used algorithm to solve LPPs with any number of decision variables. Here’s an outline of the steps:

  1. Convert to Standard Form: Convert the LPP into standard form by introducing slack, surplus, and artificial variables to convert inequalities into equations.
  2. Initialization: Start with an initial feasible solution, often at the origin (0,0), and calculate the values of the objective function.
  3. Iteration:
    • Choose a pivot column: Select the most negative coefficient in the objective function.
    • Choose a pivot row: Determine the row with the smallest non-negative ratio of the constant term to the coefficient of the pivot column.
    • Perform row operations: Update the tableau by using the pivot element to make the pivot column a unit vector and zero out other entries in the pivot column.
    • Repeat the process until an optimal solution is reached or no further improvement is possible.
  4. Optimality Test: Check if the current solution is optimal. If all coefficients in the objective function are non-negative, the solution is optimal; otherwise, continue the iteration.
  5. Solution Interpretation: Interpret the final solution obtained from the simplex algorithm in the context of the problem.

3. Big M Method:

The Big M Method is an extension of the Simplex Method that handles problems with artificial variables. It involves the following steps:

  1. Introduce Artificial Variables: Add artificial variables to each constraint in the LPP to convert inequalities into equations.
  2. Adjust the Objective Function: Modify the objective function to include a penalty term for each artificial variable.
  3. Apply the Simplex Method: Apply the Simplex Method to the modified LPP, treating artificial variables like regular decision variables.
  4. Optimality Test: Check if the solution satisfies all the optimality conditions. If the artificial variables are still present in the final solution, it indicates that the original problem is infeasible or unbounded.
  5. Post-Optimal Analysis: Perform post-optimal analysis to remove artificial variables from the final solution and interpret the optimal solution.

These methods provide systematic approaches to solving LPPs and can handle a wide range of problems efficiently. The choice of method depends on the problem’s complexity, the number of decision variables, and the presence of constraints.