MIXIMAZATION PROBLEM BY USING THE SIMPLEX METHOD IN MATHEMATICS

MIXIMAZATION PROBLEM BY USING THE SIMPLEX METHOD

1.     Write the inequality in the standard form. I.e. all inequality should be in the form of ≤.

2.     Introduce the slack variables u, v, w….. Their no. should be same to the no. of constraints.

3.     Add the slack variables to each constraints and replace the sign ≤ by =

4.     Make the initial simplex tableau: For this we write the coefficient of variables of the system of equations in the table, the last row of which contains the coefficient of variables of the objective function by changing their sign.

5.     Choose the pivot column: For this we choose the column of the number in the last row which is the least negative number (neglect the positive). If two or more least negative no. is same select any one of them. Also if there are no negative no. in the last row then stop!

6.     Choose the pivot row: For this see the positive elements (neglect the negative) in the pivot column and take the ratio of the constant to each positive element of the pivot column, of the same row. The row with least ratio is the pivot row. If two or more such ratio is same choose any one of them.

7.     The pivot element is the element common to the both pivot column and pivot row. Encircle the pivot element.

8.     Make the simplex table by making the pivot element one and other element of the pivot column zero, by using the row equivalent method.

9.     In this table, if the numbers of the last row are all positive then stop!. If not, continue the above process to choose the pivot element in the new table. This process will be continued until all the numbers in the last row are positive.

10.                        Read the solution: In the final simplex table (the table in which the elements of the last row are all positive), the number common to the last row and the last column is the maximum value of the objective function.

11.                        Find the value of the variables: The value of the variable will be zero if the number common to the variable column and the last row is non zero. If the variable column contains 1 in a row and zero elsewhere, then the value of that variable is the number in the last column of that row. In other cases the value of variable is found by making equations.

12.                        NOTE: In the minimizing problem by simplex method, it all the constraints are given in the form of ≤ then don’t change them into ≥. Only multiply the objective function by – and it will automatically change the minimum problem to maximum and apply the above method to find the maximum value. In this case, the minimum value will be the same to the maximum value but in the opposite sign. If the sign is ≤ then multiply the objective – and do the numerical as same as maximize.