Reference no: EM131842070

From Introduction to Operations Research 10 Ed., Hillier, Frederick. Consider the following problem.

Minimize W = 5y1 + 4y2,
subject to
4y1 + 3y2 >= 4
2y1 +y2 >= 3
y1 +2y2 >= 1
y1 +y2 >= 2
and
y1 >=0, y2>= 0.

Because this primal problem has more functional constraints than variables, suppose that the simplex method has been applied directly to its dual problem. If we let x5 and x6 denote the slack variables for this dual problem, the resulting final simplex tableau is

Basic Var    Eq.    Z    x1 x2   x3   x4   x5   x6 Right Side

Z                 (0)      1     3    0     2     0     1     1         9

x2                (1)      0     1    1    -1    0     1     -1        1

x4                (2)      0      2     0     3     1   -1     2        3

For each of the following independent changes in the original primal model, you now are to conduct sensitivity analysis by directly investigating the effect on the dual problem and then inferring the complementary effect on the primal problem.

For each change, apply the procedure for sensitivity analysis summarized at the end of Sec.to the dual problem (do not reoptimize), and then give your conclusions as to whether the current basic solution for the primal problem still is feasible and whether it still is optimal. Then check your conclusions by a direct graphical analysis of the primal problem.

Summary of Procedure for Sensitivity Analysis

1. Revision of model: Make the desired change or changes in the model to be investigated next.

2. Revision of final tableau: Use the fundamental insight (as summarized by the formulas on the bottom of Table) to determine the resulting changes in the final simplex tableau.

3. Conversion to proper form from Gaussian elimination: Convert this tableau to the proper form for identifying and evaluating the current basic solution by applying (as necessary) Gaussian elimination.

4. Feasibility test: Test this solution for feasibility by checking whether all its basic variable values in the right-side column of the tableau still are nonnegative.

5. Optimality test: Test this solution for optimality (if feasible) by checking whether all its nonbasic variable coefficients in row 0 of the tableau still are nonnegative.

6. Reoptimization: If this solution fails either test, the new optimal solution can be obtained (if desired) by using the current tableau as the initial simplex tableau (and making any necessary conversions) for the simplex method or dual simplex method.