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Linear Programming: Sensitivity Analysis and Interpretation of Solution

Learning Objectives. Understand the usefulness of sensitivity analysisPerform

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Linear Programming: Sensitivity Analysis and Interpretation of Solution

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    1. Linear Programming: Sensitivity Analysis and Interpretation of Solution

    2. Learning Objectives Understand the usefulness of sensitivity analysis Perform “one-way” sensitivity analyses for changes in objective function coefficients and right-hand-side constants Interpret dual prices and reduced costs Interpret LINDO sensitivity analysis tables

    3. Usefulness of Sensitivity Analysis When solving an LP model we assume that all relevant factors are known with certainty. However, such certainty rarely exists, so sensitivity analysis helps assess the robustness of a solution

    4. Usefulness of Sensitivity Analysis (cont.) Sensitivity analysis helps answer questions about how the optimal solution changes given various changes of inputs Objective function coefficients Right hand side constants Addition or deletion of decision variables

    5. Approaches to Sensitivity Analysis Change the data and re-solve the model (sometimes this is the only practical approach) Perform ad-hoc “one-way” sensitivity analyses (by hand or using LINDO) Ranges of Optimality (for objective function coefficients and dual prices ) Dual Prices Reduced Costs

    6. Sensitivity Analysis Areas Objective function coefficients (shifting slope of objective function) RHS constants (enlarging, shrinking feasible region) Changing the optimal solution by adding a “non-basic” variable (reduced costs)

    7. Blue Ridge Hot Tubs

    8. Sensitivity of Objective Function Coefficients

    11. Changes in Objective Function Coefficients “Allowable Increase” and “Allowable Decrease” columns for the decision variables/changing cells indicate the amounts by which an objective function coefficient can change without changing the optimal solution, assuming all other coefficients remain constant.

    12. Sensitivity of Objective Function Coefficients

    13. Check for Alternate Optimal Solutions The signal: values of zero (0) in the “Allowable Increase” or “Allowable Decrease” columns for the decision variables

    14. Blue Ridge Hot Tubs

    16. Sensitivity of RHS Constants

    17. Changes in Constraint RHS Values The dual price of a constraint indicates the amount by which the objective function value changes (improves or worsens) given a unit change (loosening or tightening) in the RHS value of the constraint, assuming all other coefficients remain constant. Loosen Tighten < increase RHS decrease RHS > decrease RHS increase RHS MAX MIN Improves increases Z decreases Z Worsens decreases Z increases Z

    18. Changes in RHS Values Dual prices are valid only within RHS changes falling within the values in “Allowable Increase” and “Allowable Decrease” columns

    19. Key Points About Dual Prices The dual prices of resources equate the marginal value of the resources consumed with the marginal benefit of the entities being produced Resources in excess supply have a dual price (or marginal value) of zero

    20. Another Use of Dual Prices Suppose a new Hot Tub (the Typhoon-Lagoon) is being considered. It generates a marginal profit of $320 and requires: 1 pump (dual price = $200) 8 hours of labor (dual price = $16.67) 13 feet of tubing (dual price = $0) Q: Would it be profitable to produce any? A: $320 - $200*1 - $16.67*8 - $0*13 = -$13.33 No!

    21. Blue Ridge Hot Tubs, Revisited

    22. Reduced Costs The reduced cost for each product equals its per-unit marginal profit minus the per-unit value of the resources it consumes (priced at their dual prices) OR the amount by which the objective function is worsened by a one unit inclusion of a (currently) non-basic variable into the basis

    23. Reduced Costs (cont.) OR the amount by which the objective function coefficient must be changed by in order to allow that decision variable to become non-zero

    24. Reduced Costs (cont.) Basically, products whose marginal profits are less than the marginal value of the goods required for their production will not be produced in an optimal solution

    25. Review Understand the usefulness of sensitivity analysis Perform “one-way” sensitivity analyses for changes in objective function coefficients and right-hand-side constants Interpret dual prices and reduced costs Interpret LINDO sensitivity analysis tables

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