Linear Programming Simplex Method Computational Problems Breaking Ties

Linear Programming – Simplex Method: Computational Problems Breaking Ties in Selection of Non-Basic Variable – if tie for non-basic variable with largest relative profit ( ), arbitrarily select incoming variable. Ties in Minimum Ratio Rule (Degeneracy) – if more than one basic variable have same minimum ratio, select either variable to leave the basis. This will result in a basic variable taking on a value of 0. When this occurs, the solution is referred to as a degenerate basic feasible solution. When this occurs, you may transition through more than one simplex tableau with the same objective (Z) value.

Linear Programming – Simplex Method: Computational Problems Unbounded Solutions – if when performing the minimum ratio rule, none of the ratios are positive, then the solution is unbounded (e. g Max Z = or Min = ). See pages 44 -46 for good examples.

Simplex Method – Finding an Initial Basic Feasible Solution Min Z = -3 x 1 + x 2 + x 3 s. t. x 1 – 2 x 2 + x 3 <= 11 -4 x 1 + x 2 +2 x 3 >= 3 2 x 1 - x 3 = -1 x 1, x 2, x 3 >= 0 Standard Form: (-) Max Z = 3 x 1 - x 2 - x 3 s. t. x 1 – 2 x 2 + x 3 + x 4 = 11 -4 x 1 + x 2 +2 x 3 -x 5 = 3 -2 x 1 + x 3 = 1 x 1, x 2, x 3, x 4, x 5 >= 0

Simplex Method – Finding an Initial Basic Feasible Solution (-) Max Z = 3 x 1 - x 2 - x 3 s. t. x 1 – 2 x 2 + x 3 + x 4 = 11 -4 x 1 + x 2 +2 x 3 -x 5 = 3 -2 x 1 + x 3 = 1 x 1, x 2, x 3, x 4, x 5 >= 0 Only x 4 is basic. Introduce artificial variables. s. t. x 1 – 2 x 2 + x 3 + x 4 = 11 -4 x 1 + x 2 +2 x 3 -x 5 + x 6 =3 -2 x 1 + x 3 + x 7 = 1 x 1, x 2, x 3, x 4, x 5, x 6, x 7 >= 0

Simplex Method – Solve Using Big-M Method Let M be an arbitrarily large number, then: (-) Max Z = 3 x 1 - x 2 - x 3 + 0 x 4 + 0 x 5 – Mx 6 – Mx 7 s. t. x 1 – 2 x 2 + x 3 + x 4 = 11 -4 x 1 + x 2 +2 x 3 -x 5 + x 6 =3 -2 x 1 + x 3 + x 7 = 1 x 1, x 2, x 3, x 4, x 5, x 6, x 7 >= 0 Note: If the simplex algorithm terminates with one of the artificial variables as a basic variable, then the original problem has no feasible solution.