University:
California State University, Northridge
Course:
MGT 360 | Management and Organizational Behavior
Academic year:

2024
Views:

98

Pages:

17
Author:

Britney Bird

Limitations of Classical Sensitivity Analysis    Classical sensitivity analysis provided by most computer packages does have its limitations. It is rarely the case that one solves a model once and makes recommendations. More often, a series of models is solved using a variety of input data sets before a final plan is adopted. Limitations of Classical Sensitivity Analysis  Simultaneous Changes The range analysis for objective function coefficients and the constraint right-hand sides is only applicable for changes in a single coefficient. We should resolve the model with the new coefficient values. Limitations of Classical Sensitivity Analysis  Changes in Constraint Coefficients Classical sensitivity analysis provides no information about changes resulting from a change in a coefficient of a variable in a constraint. We must simply change the coefficient and rerun the model. Limitations of Classical Sensitivity Analysis  Non-intuitive Dual Values Constraints with variables naturally on both the lefthand and right-hand sides often lead to dual values that have a non-intuitive explanation. This is often the case with constraints that involve percentages. The model should be resolved with the new percentages. Example 3  Consider the following linear program: Min s.t. 6x1 + 9x2 ($ cost) x1 + 2x2 < 8 10x1 + 7.5x2 > 30 x2 > 2 x1, x2 > 0 Example 3  Computer Output OBJECTIVE FUNCTION VALUE = 27.000 Variable x1 x2 Value 1.500 2.000 Reduced Cost 0.000 0.000 Constraint 1 2 3 Slack/Surplus 2.500 0.000 0.000 Dual Value 0.000 -0.600 -4.500 Example 3  Computer Output (continued) OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value x1 0.000 6.000 x2 4.500 9.000 Upper Limit 12.000 No Limit RIGHTHAND SIDE RANGES Constraint Lower Limit Current Value 1 5.500 8.000 2 15.000 30.000 3 0.000 2.000 Upper Limit No Limit 55.000 4.000 Example 3  Optimal Solution According to the output: x1 = 1.5 x2 = 2.0 Objective function value = 27.00 Example 3  Range of Optimality Question: Suppose the unit cost of x1 is decreased to $4. Is the current solution still optimal? What is the value of the objective function when this unit cost is decreased to $4? Example 3  Computer Output OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value x1 0.000 6.000 x2 4.500 9.000 Upper Limit 12.000 No Limit RIGHTHAND SIDE RANGES Constraint Lower Limit Current Value 1 5.500 8.000 2 15.000 30.000 3 0.000 2.000 Upper Limit No Limit 55.000 4.000 Example 3  Range of Optimality Answer: The output states that the solution remains optimal as long as the objective function coefficient of x1 is between 0 and 12. Because 4 is within this range, the optimal solution will not change. However, the optimal total cost will be affected: 6x1 + 9x2 = 4(1.5) + 9(2.0) = $24.00. Example 3  Range of Optimality Question: How much can the unit cost of x2 be decreased without concern for the optimal solution changing? Example 3  Computer Output OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value x1 0.000 6.000 x2 4.500 9.000 Upper Limit 12.000 No Limit RIGHTHAND SIDE RANGES Constraint Lower Limit Current Value 1 5.500 8.000 2 15.000 30.000 3 0.000 2.000 Upper Limit No Limit 55.000 4.000 Example 3  Range of Optimality Answer: The output states that the solution remains optimal as long as the objective function coefficient of x2 does not fall below 4.5. Example 3  Range of Feasibility Question: If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution? Example 3  Computer Output OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value x1 0.000 6.000 x2 4.500 9.000 Upper Limit 12.000 No Limit RIGHTHAND SIDE RANGES Constraint Lower Limit Current Value 1 5.500 8.000 2 15.000 30.000 3 0.000 2.000 Upper Limit No Limit 55.000 4.000 Example 3  Range of Feasibility Answer: A dual value represents the improvement in the objective function value per unit increase in the righthand side. A negative dual value indicates a deterioration (negative improvement) in the objective, which in this problem means an increase in total cost because we're minimizing. Since the righthand side remains within the range of feasibility, there is no change in the optimal solution. However, the objective function value increases by $4.50.