L19_Sensitivity analysis - Changes affecting Optimality

# L19_Sensitivity analysis - Changes affecting Optimality -...

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Addition of a new constraint The addition of a new constraint to an existing model can lead to one of two cases: 1. The new constraint is redundant , meaning that it is satisfied by the current optimal solution and hence can be dropped altogether from the model altogether. 2. The current solution violates the new constraint in which case the dual simplex method can be used to recover feasibility.

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We introduce this constraint in the final simplex tableau as an additional row where the usual additional (slack or surplus) variable is taken as the basic variable for this new row. Because this new row will probably have non- zero coefficients for some of the other basic variables, the conversion to proper form for Gaussian elimination is done and then reoptimization is done in the usual way. The following example illustrates this.
Example: Consider the LPP Maximize 3 2 1 2 x x x z + - = subject to 0 , , 20 10 2 60 3 3 2 1 3 2 1 3 2 1 3 2 1 - + + - + + x x x x x x x x x x x x Applying the Simplex method, the optimal tableau is given in the next slide. Now a new constraint 1 2 3 3 2 28 x x x - + solution. is added. Find the new optimal

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x2 0 0 1 - 3/2 0 - 1/2 1/2 5 x1 0 1 0 1/2 0 1/2 1/2 15 z 1 0 0 3/2 0 3/2 1/2 25 Basic z x1 x2 x3 s1 s2 s3 Sol s1 0 0 0 1 1 - 1 - 2 10 s4 0 3 -2 1 0 0 0 1 28 s4 0 0 0 0 z 1 0 0 0 0 3/7 2/7 3/7 22 x2 0 0 1 0 0 4/7 5/7 -3/7 8 x1 0 1 0 0 0 1/7 3/7 1/7 14 s1 0 0 0 0 1 - 12/7 -15/7 2/7 8 x3 0 0 0 1 0 5/7 1/7 -2/7 2 0 0 0 -7/2 0 - 5/2 -1/2 1 -7
Changes affecting Optimality 1. Changes in the objective coefficients c j The changes in the objective coefficients, c j , affect " z j - c j " and hence the optimality. We recompute the z-Row by replacing c j with c' j . We note that C B will also change to C' B . In the previous problem, we replace the objective

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## This note was uploaded on 02/23/2011 for the course MATH 112 taught by Professor Ritadubey during the Spring '11 term at Amity University.

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L19_Sensitivity analysis - Changes affecting Optimality -...

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