Write the question from Operation Research assignment of MB0032 for SMU MBA. The question is “Describe the Penalty Cost method (Big M Method) for solving L.P.P.” From the SMU MBA MB002 assignment you already have gotten classification of Operations Research models.
Consider a L.P.P. when at least one of the constraints is of the type ≥ or =. While expressing in the standard form, add a non negative variable to each of such constraints. These variables are called artificial variables. Their addition causes violation of the corresponding constraints, since they are added to only one side of an equation, the new system is equivalent to the old system of constraints if and only if the artificial variables are zero. To guarantee such assignments in the optimal solution, artificial variables are incorporated into the objective function with large positive coefficients in a minimization program or very large negative coefficients in a maximization program. These coefficients are denoted by ± M.
Whenever artificial variables are part of the initial solution Xo, the last row of simplex table will contain the penalty cost M. The following modifications are made in the simplex method to minimize the error of incorporating the penalty cost in the object function. This method is called Big M-method to penalty cost method.
1. The last row of the simplex table is decomposed into two rows, the first of which involves those terms not containing M, while the second involves those containing M.
2. The step 1 of the simplex method is applied to the last row created in the above modification and followed by steps 2,3 and 4 until this row contains no negative elements. Then step 1 of simplex algorithm is applied to those elements next to the last row that are positioned over zero in the last row.
3. Whenever an artificial variable ceases to be basic, it is removed from the first column of the table as a result of step 4, it is also deleted from the top row of the table as is the entire column under it.
4. The last row is removed from the table whenever it contains all zones.
5. If non-zero artificial variables are present in the final basic set, then the program has no solution. In contrast, zero valued artificial variables in the final solution may exist when one or more of the original constraints equations are redundant.
Consider a L.P.P. when at least one of the constraints is of the type ≥ or =. While expressing in the standard form, add a non negative variable to each of such constraints. These variables are called artificial variables. Their addition causes violation of the corresponding constraints, since they are added to only one side of an equation, the new system is equivalent to the old system of constraints if and only if the artificial variables are zero. To guarantee such assignments in the optimal solution, artificial variables are incorporated into the objective function with large positive coefficients in a minimization program or very large negative coefficients in a maximization program. These coefficients are denoted by ± M.
Whenever artificial variables are part of the initial solution Xo, the last row of simplex table will contain the penalty cost M. The following modifications are made in the simplex method to minimize the error of incorporating the penalty cost in the object function. This method is called Big M-method to penalty cost method.
1. The last row of the simplex table is decomposed into two rows, the first of which involves those terms not containing M, while the second involves those containing M.
2. The step 1 of the simplex method is applied to the last row created in the above modification and followed by steps 2,3 and 4 until this row contains no negative elements. Then step 1 of simplex algorithm is applied to those elements next to the last row that are positioned over zero in the last row.
3. Whenever an artificial variable ceases to be basic, it is removed from the first column of the table as a result of step 4, it is also deleted from the top row of the table as is the entire column under it.
4. The last row is removed from the table whenever it contains all zones.
5. If non-zero artificial variables are present in the final basic set, then the program has no solution. In contrast, zero valued artificial variables in the final solution may exist when one or more of the original constraints equations are redundant.
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