3 Simplex Algorithm – Special cases There are four special cases arise in the use of the simplex method.DegeneracyAlternative optimalUnbounded solutioninfeasible solution
4 Degeneracy ( no improvement in objective) Degeneracy: It is situation when the solution of the problem degenerates.Degenerate Solution: A Solution of the problem is said to be degenerate solution if value or values of basic variable(s) become zeroIt occurs due to redundant constraints.
5 Degeneracy – Special cases (cont.) This is in itself not a problem, but making simplex iterations form a degenerate solution, give rise to cycling, meaning that after a certain number of iterations without improvement in objective value the method may turn back to the point where it started.
7 Degeneracy – Special cases (cont.) The solution:Step 1. write inequalities in equation formLet S1 and S2 be the slack variablesX1 + 4X2 + s1= 8X1 + 2X2 + s2= 4X1, X2 ,s1,s2≥ 0Let X1=0, X2=0f=0, S1=8, S2=4
8 Degeneracy – Special cases (cont.) X2 will enter the basis because of having minimum objective function coefficient.Initial TableauEntering VariableLeaving VariableBasisX1X2S1S2RHSRatio1488/4=224/2=2f-3-9S1 and S2 tie for leaving variable( with same minimum ratio 2) so we can take any one of them arbitrary.
9 Degeneracy – Special cases (cont.) Tableau 1BasisX1X2S1S2RHSRatio1/4128-1/20 minf-3/49/418Here basic variable S2 is 0 resulting in a degenerate basic solution.
10 Degeneracy – Special cases (cont.) Tableau 2Same objective function no change and improvement ( cycle)BasisX1X2S1S2RHS11/2-1/22-1f3/218Same objective
11 Degeneracy – Special cases (cont.) This is redundant constraint because its presence or absence does not affect, neither on optimal point nor on feasible region.Feasible Region
12 Alternative optimalIf the f-row value for one or more non basic variables is 0 in the optimal tableau, alternate optimal solution exists.When the objective function is parallel to a binding constraint, objective function will assume same optimal value. So this is a situation when the value of optimal objective function remains the same.We have infinite number of such points
14 Alternative optima – Special cases (cont.) The solutionMax 2x1+ 4x2Let S1and S2 be the slack variablesx1 + 2x2 + s1= 5x1 + x2 + s2 = 4x1, x2, s1, s2 ≥0Initial solutionLet x1 = 0, x2 = 0,f=0, s1= 5, s2 = 4
15 Alternative optima – Special cases (cont.) Initial TableauEntering VariableLeaving VariableBasisX1X2S1S2RHSRatio1255/2=2.544/1=4f-2-4
16 Alternative optima – Special cases (cont.) Optimal solution is 10 when x2=5/2, x1=0.How do we know that alternative optimal exist ?BasisX1X2S1S2RHSRatio1/215/25-1/23/23 minf210
17 Alternative optima – Special cases (cont.) Entering VariableBy looking at f-row coefficient of the nonbasic variable.The coefficient for x1 is 0, which indicates that x1 can enter the basic solution without changing the value of f. Optimal sol. f= x1=0, x2=5/2Leaving VariableBasisX1X2S1S2RHSRatio1/215/25-1/23/23f210
18 Alternative optima – Special cases (cont.) The second alternative optima is:The new optimal solution is 10 when x1=3, x2=1BasisX1X2S1S2RHS1-123f10
19 Any point on BC represents an alternate optimum with f=10 As the objective function line is parallel to line BC so the optimal solution lies on every point on line BCObjective Function
20 Alternative optimal – Special cases (cont.) In practice alternate optimals are useful as they allow us to choose from many solutions experiencing deterioration in the objective value.
21 Unbounded SolutionWhen determining the leaving variable of any tableau, if there is no positive ratio (all the entries in the pivot column are negative and zeros), then the solution is unbounded.