3.1 Sensitivity Analysis and Interpretation of Solution PDF

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3.1 Sensitivity Analysis and Interpretation of Solution

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3.1 Sensitivity Analysis and Interpretation of Solution Required Reading • Chapter 3, Introduction (pp. 84-138), section 3.1, 3.2, 3.3, 3.4, & 3.5 Once a problem is formulated as a linear program and solved, two fundamental issues are of concern to the manager: How to interpret the results? What is the sensitivity of model parameters subject to the uncertainties of the environment? Note that in formulating and solving linear programs, it is always assumed that the problem parameters (coefficients and resources) are well defined and known values. The sensitivity analysis or post-optimal analysis aids the decision maker in investigating the effect of changing parameters on the optimal solution of the LP. In this lesson, you will see, more specifically, how to determine the impact of changes in LP parameters on the optimal solution using, first, the graphical solution procedure, and then the computer solution.

Key Concepts Define & give examples of the following terms: • Sensitivity analysis • Dual value • Reduced cost • Sunk cost • Relevant cost

3.1.1 Sensitivity analysis, range of optimality, reduced cost, & range of feasibility Sensitivity analysis Sensitivity analysis in LP is very important for managers who must operate in a dynamic environment with imprecise estimates of the coefficients. It is used to determine how an optimal solution is affected by changes, within specified

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ranges, in the objective function coefficients and in the right-hand side (RHS) values of an LP problem. Sensitivity analysis allows managers to ask certain what-if questions about the problem’s solution. There are two ranges of interest: the range of optimality and the range of feasibility. For both ranges, the Par, Inc. problem is used again to illustrate graphical sensitivity analysis of the parameter of this LP problem.

Range of optimality A range of optimality of an objective function coefficient is, by definition, a range for which as long as the actual value of this coefficient is within that range, the current optimal solution will remain optimal. The range of optimality of an objective function coefficient, whose decision variable is positive in the optimal solution, is found by determining an interval for the objective function coefficient in which the original solution remains optimal while keeping all other data of the problem constant. The value of the objective function might, however, change in this range. Reduced cost is another quantity of importance associated with a decision variable whose value is 0 in the optimal solution. By definition, a reduced cost for a decision variable is the amount the variable's objective coefficient would have to improve (increase for maximization problems or decrease for minimization problems) before this variable could assume a positive value. Thus, the reduced cost for a decision variable with a positive value is 0. With the graphical approach, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines. This also applies to simultaneous changes in the objective coefficients. You should read pages 97 to 102 in your textbook for more details. When simultaneous changes in objective function coefficients are considered, we should apply the 100% rule, which states that these coefficients will not change the optimal solution as long as the sum of the percentages of the change divided by the corresponding maximum allowable change in the range of optimality for each coefficient does not exceed 100%.

Range of feasibility

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A range of feasibility of a right-hand-side value of a constraint is, by definition, a range for which as long as the actual value of this right-hand-side value is within that range, the dual price will remain the same. A dual price is, by definition, the amount the objective function will improve per lesson increase in the right-handside value of this constraint. The dual price reflects the value of an additional lesson of the resource if this resource cost is sunk, or the extra value over the normal cost of the resource when this resource cost is relevant. A resource cost is considered relevant if the amount paid is dependent upon the amount of the resource used by the decision variables (consequently, relevant costs are reflected in the objective function coefficients), while a resource cost is considered sunk if it must be paid regardless of the amount of the resource actually used by the decision variables (consequently, sunk resource costs are not reflected in the objective function coefficients). With the graphical approach, a dual price is determined by adding +1 to the right-hand-side value of the constraint in question and then resolving for the optimal solution in terms of the same binding constraints. The dual price is then equal to the difference in the values of the objective functions between the new and original problems. The dual price for a nonbinding constraint, the constraint in which there is positive slack or surplus when evaluated at the optimal solution, is equal to zero. For the graphical approach, a range of feasibility is determined by finding the values of a right-hand-side coefficient, such that the same two lines that determined the original optimal solution continue to determine the optimal solution for the problem. For this range as well, the 100% rule also states that simultaneous changes in right-hand-side values of several constraints will not change the dual prices of these constraints as long as the sum of the percentages of the changes divided by the corresponding maximum allowable change in the range of feasibility for each right-hand-side value does not exceed 100%. You should read pages 102 to 105 in your textbook for more details.

Alternatively, computer software packages, such as Microsoft Excel, that solve linear programming problems can generate all the relevant information about the optimal solution discussed previously: 1. The optimal value of the objective function;

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2. Information about the decision variables: their values and their reduced costs; 3. Information about the constraints: amount of slack or surplus and dual prices; 4. Ranges of optimality for objective function coefficients; 5. Ranges of feasibility for right-hand-side values. You should read pages 105 to 112 in your textbook for more details.

Learn Activity 1 Think about what you have read and answer the following questions: • Why is sensitivity analysis so important for decision-making? • Explain how to perform sensitivity analysis with the graphical approach? • List the different steps of the process of formulating an LP model? Answers to these questions are not provided but you are encouraged to share your answers on the discussion forum to get feedback and input from other students....