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Advancesensit
1.
Slides Prepared by
JOHN S. LOUCKS ST. EDWARD’ S UNIVERSITY © 2005 Thomson/South-Western Slide 1
2.
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation of Solution Introduction to Sensitivity Analysis Graphical Sensitivity Analysis Sensitivity Analysis: Computer Solution Simultaneous Changes © 2005 Thomson/South-Western Slide 2
3.
Standard Computer Output Software
packages such as The Management Scientist and Microsoft Excel provide the following LP information: Information about the objective function: • its optimal value • coefficient ranges (ranges of optimality) Information about the decision variables: • their optimal values • their reduced costs Information about the constraints: • the amount of slack or surplus • the dual prices • right-hand side ranges (ranges of feasibility) © 2005 Thomson/South-Western Slide 3
4.
Standard Computer Output
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 © 2005 Thomson/South-Western Slide 4
5.
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 the manager who must operate in a dynamic environment with imprecise estimates of the coefficients. Sensitivity analysis allows him to ask certain what-if questions about the problem. © 2005 Thomson/South-Western Slide 5
6.
Objective Function Coefficients
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. © 2005 Thomson/South-Western Slide 6
7.
Right-Hand Sides
The improvement in the value of the optimal solution per unit increase in the right-hand side is called the shadow price. The range of feasibility is the range over which the shadow price is applicable. As the RHS increases, other constraints will become binding and limit the change in the value of the objective function. © 2005 Thomson/South-Western Slide 7
8.
Example : Olympic
Bike Co. Olympic Bike is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from special aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional. The number of pounds of each alloy needed per frame is summarized on the next slide. © 2005 Thomson/South-Western Slide 8
9.
Example 2: Olympic
Bike Co. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. Aluminum Alloy Steel Alloy Deluxe 2 3 Professional 4 2 How many Deluxe and Professional frames should Olympic produce each week? © 2005 Thomson/South-Western Slide 9
10.
Example : Olympic
Bike Co. Model Formulation • Verbal Statement of the Objective Function Maximize total weekly profit. • Verbal Statement of the Constraints Total weekly usage of aluminum alloy < 100 pounds. Total weekly usage of steel alloy < 80 pounds. • Definition of the Decision Variables x1 = number of Deluxe frames produced weekly. x2 = number of Professional frames produced weekly. © 2005 Thomson/South-Western Slide 10
11.
Example : Olympic
Bike Co. Model Formulation (continued) Max 10x1 + 15x2 (Total Weekly Profit) s.t. 2x1 + 4x2 < 100 (Aluminum Available) 3x1 + 2x2 < 80 (Steel Available) x1, x2 > 0 © 2005 Thomson/South-Western Slide 11
12.
Example : Olympic
Bike Co. Partial Spreadsheet: Problem Data A B C D 1 Material Requirements Amount 2 Material Deluxe Profess. Available 3 Aluminum 2 4 100 4 Steel 3 2 80 © 2005 Thomson/South-Western Slide 12
13.
Example : Olympic
Bike Co. Partial Spreadsheet Showing Solution A B C D 6 Decision Variables 7 Deluxe Professional 8 Bikes Made 15 17.500 9 10 Maximized Total Profit 412.500 11 12 Constraints Amount Used Amount Avail. 13 Aluminum 100 <= 100 14 Steel 80 <= 80 © 2005 Thomson/South-Western Slide 13
14.
Example : Olympic
Bike Co. Optimal Solution According to the output: x1 (Deluxe frames) = 15 x2 (Professional frames) = 17.5 Objective function value = $412.50 © 2005 Thomson/South-Western Slide 14
15.
Example : Olympic
Bike Co. Range of Optimality Question Suppose the profit on deluxe frames is increased to $20. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $20? © 2005 Thomson/South-Western Slide 15
16.
Example : Olympic
Bike Co. Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$8 Deluxe 15 0 10 12.5 2.5 $C$8 Profess. 17.500 0.000 15 5 8.333333333 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $B$13 Aluminum 100 3.125 100 60 46.66666667 $B$14 Steel 80 1.25 80 70 30 © 2005 Thomson/South-Western Slide 16
17.
Example : Olympic
Bike Co. Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x1 is between 7.5 and 22.5. Since 20 is within this range, the optimal solution will not change. The optimal profit will change: 20x1 + 15x2 = 20(15) + 15(17.5) = $562.50. © 2005 Thomson/South-Western Slide 17
18.
Example : Olympic
Bike Co. Range of Optimality Question If the unit profit on deluxe frames were $6 instead of $10, would the optimal solution change? © 2005 Thomson/South-Western Slide 18
19.
Example : Olympic
Bike Co. Range of Optimality Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$8 Deluxe 15 0 10 12.5 2.5 $C$8 Profess. 17.500 0.000 15 5 8.33333333 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $B$13 Aluminum 100 3.125 100 60 46.66666667 $B$14 Steel 80 1.25 80 70 30 © 2005 Thomson/South-Western Slide 19
20.
Example : Olympic
Bike Co. Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x1 is between 7.5 and 22.5. Since 6 is outside this range, the optimal solution would change. © 2005 Thomson/South-Western Slide 20
21.
End of Chapter
3 © 2005 Thomson/South-Western Slide 21
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