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

2024
Views:

101

Pages:

20
Author:

podtalniclov

Introduction to Sensitivity Analysis   In the previous chapter we discussed: • objective function value • values of the decision variables • reduced costs • slack/surplus In this chapter we will discuss: • changes in the coefficients of the objective function • changes in the right-hand side value of a constraint Introduction to Sensitivity Analysis    Sensitivity analysis (or post-optimality analysis) is used to determine how the optimal solution is affected by changes, within specified ranges, in: • the objective function coefficients • the right-hand side (RHS) values Sensitivity analysis is important to a manager who must operate in a dynamic environment with imprecise estimates of the coefficients. Sensitivity analysis allows a manager to ask certain what-if questions about the problem. Graphical Sensitivity Analysis  For LP problems with two decision variables, graphical solution methods can be used to perform sensitivity analysis on • the objective function coefficients, and • the right-hand-side values for the constraints. Example 1  LP Formulation Max 5x1 + 7x2 s.t. x1 < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1, x2 > 0 Example 1  Graphical Solution x2 x1 + x2 < 8 8 Max 5x1 + 7x2 7 6 x1 < 6 5 Optimal Solution: x1 = 5, x2 = 3 4 3 2x1 + 3x2 < 19 2 1 1 2 3 4 5 6 7 8 9 10 x1 7 Objective Function Coefficients    Let us consider how changes in the objective function coefficients might affect the optimal solution. The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. Managers should focus on those objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range. 8 Example 1  Changing Slope of Objective Function x2 Coincides with x1 + x2 < 8 constraint line 8 7 6 5 Objective function line for 5x1 + 7x2 5 Coincides with 2x1 + 3x2 < 19 4 Feasible Region 3 2 1 constraint line 4 3 1 2 1 2 3 4 5 6 7 8 9 10 x1 9 Range of Optimality   Graphically, 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. Slope of an objective function line, Max c1x1 + c2x2, is -c1/c2, and the slope of a constraint, a1x1 + a2x2 = b, is -a1/a2. 10 Example 1  Range of Optimality for c1 The slope of the objective function line is -c1/c2. The slope of the first binding constraint, x1 + x2 = 8, is -1 and the slope of the second binding constraint, x1 + 3x2 = 19, is -2/3. Find the range of values for c1 (with c2 staying 7) such that the objective function line slope lies between that of the two binding constraints: -1 < -c1/7 < -2/3 Multiplying through by -7 (and reversing the inequalities): 14/3 < c1 < 7 11 Example 1  Range of Optimality for c1 Would a change in c1 from 5 to 7 (with c2 unchanged) cause a change in the optimal solution? The answer is ‘no’ because when c1 = 7, the condition 14/3 < c1 < 7 is satisfied. 12 Example 1  Range of Optimality for c2 Find the range of values for c2 ( with c1 staying 5) such that the objective function line slope lies between that of the two binding constraints: -1 < -5/c2 < -2/3 Multiplying by -1: Inverting, 1 > 5/c2 > 2/3 1 < c2/5 < 3/2 Multiplying by 5: 5 < c2 < 15/2 13 Example 1  Range of Optimality for c2 Would a change in c2 from 7 to 6 (with c1 unchanged) cause a change in the optimal solution? The answer is ‘no’ because when c2 = 6, the condition 5 < c2 < 15/2 is satisfied. 14 Simultaneous Changes     The range of optimality for objective function coefficients is only applicable for changes made to one coefficient at a time. All other coefficients are assumed to be fixed at their initial values. If two or more coefficients are changed simultaneously, further analysis is usually necessary. However, when solving two-variable problems graphically, the analysis is fairly easy. 15 Simultaneous Changes   Simply compute the slope of the objective function (-Cx1/Cx2 ) for the new coefficient values. If this ratio is > the lower limit on the slope of the objective function and < the upper limit, then the changes made will not cause a change in the optimal solution. 16 Example 1  Simultaneous Changes in c1 and c2 Would simultaneously changing c1 from 5 to 7 and changing c2 from 7 to 6 cause a change in the optimal solution? (Recall that these changes individually did not cause the optimal solution to change.) Recall that the objective function line slope must lie between that of the two binding constraints: -1 < -c1/c2 < -2/3 The answer is ‘yes’ the optimal solution changes because -7/6 does not satisfy the above condition. 17 Right-Hand Sides     Let us consider how a change in the right-hand side for a constraint might affect the feasible region and perhaps cause a change in the optimal solution. The change in the value of the optimal solution per unit increase in the right-hand side is called the dual value. The range of feasibility is the range over which the dual value is applicable. As the RHS increases, other constraints will become binding and limit the change in the value of the objective function. 18 Dual Value     Graphically, a dual value is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints. The dual value is equal to the difference in the values of the objective functions between the new and original problems. The dual value for a nonbinding constraint is 0. A negative dual value indicates that the objective function will not improve if the RHS is increased. 19 Example 1  Dual Values Constraint 1: Since x1 < 6 is not a binding constraint, its dual price is 0. Constraint 2: Change the RHS value of the second constraint to 20 and resolve for the optimal point determined by the last two constraints: 2x1 + 3x2 = 20 and x1 + x2 = 8. The solution is x1 = 4, x2 = 4, z = 48. Hence, the dual price = znew - zold = 48 - 46 = 2. 20 Example 1  Dual Values Constraint 3: Change the RHS value of the third constraint to 9 and resolve for the optimal point determined by the last two constraints: 2x1 + 3x2 = 19 and x1 + x2 = 9. The solution is: x1 = 8, x2 = 1, z = 47. The dual price is znew - zold = 47 - 46 = 1. 21 Range of Feasibility   The range of feasibility for a change in the right hand side value is the range of values for this coefficient in which the original dual value remains constant. Graphically, the 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. 22