It may arise in applying the simplex method:
It may arise in applying the simplex method It includes Unrestricted variables Tie for entering basic variable Tie for leaving basic variable
Unrestricted variables… :
Unrestricted variables… Variable which can assume positive , negative or zero values are unrestricted variables Usually in LP problem , It assume that all the variables should have non negative values. One or more of the variables, can have either positive, negative or zero values.
Cont…:
Cont… It requires that all the decision variables must have non negative value at each iteration. Therefore in order to convert an LP problem involving unrestricted variables into an equivalent problem having only restricted variables . We have to express each of unrestricted variables as the difference of two non negative variables.
Tie for entering basic variable…:
Tie for entering basic variable… A situation may arise at any iteration when two or more columns may have exactly the same value. In order to break this tie, the selection for key column solution can be made arbitrarily The number of iteration required to arrive at the optimal solution can be minimized by adopting the following rules- If there is a tie between two decision variables, then the selection can be made arbitrarily. If there is a tie between a decision variables and surplus variable, then select the decision variable to enter into basis first. If there is a tie between two surplus variables, then selection can be made arbitrarily.
Tie for leaving basic variable… :
Tie for leaving basic variable… It arise due to redundant constraint, i.e one or more of the constraints makes another unnecessary in the LP problems. It may occur at any iteration of the simplex method. When there is a tie in the minimum ratios, the selection is made arbitrarily. The number of iteration required to arrive at the optimal solution can be minimized by adopting the following rules- Divide the co efficient of slack variables in the simplex table where degeneracy is detected by the corresponding positive number of the key column in the row, starting from left to right. The row which contains smallest ratio comparing from left to right colomnwise becomes the key row.
Slide 7:
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