1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution n Introduction to Sensitivity Analysis n Graphical Sensitivity Analysis n Sensitivity Analysis: Computer Solution n Limitations of Classical Sensitivity Analysis

3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n In the previous chapter we discussed: objective function value objective function value values of the decision variables values of the decision variables reduced costs reduced costs slack/surplus slack/surplus n In this chapter we will discuss: changes in the coefficients of the objective function changes in the coefficients of the objective function changes in the right-hand side value of a constraint changes in the right-hand side value of a constraint Introduction to Sensitivity Analysis

4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction to Sensitivity Analysis n Performed after the objective solution is found n Performed after the objective solution is found in an LP problem n “How much can we change the problem before the optimal solution changes?” the objective function coefficients the objective function coefficients the right-hand side (RHS) values the right-hand side (RHS) values n Sensitivity analysis allows a manager to ask certain what-if questions about the problem.

5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Graphical Sensitivity Analysis n For LP problems with two decision variables, graphical solution methods can be used to perform sensitivity analysis on the objective function coefficients the objective function coefficients the right-hand-side values for the constraints. the right-hand-side values for the constraints.

6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example Pivoting the solution line around optimal solution

7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example If the objective function line pivots outside Of the available area, we have To find the new optimal solution

8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example Boundaries of area (or “range of optimality”) defined by Constraints, therefore we can use their slopes to Calculate the boundaries

9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: A Simple Maximization Problem Par Inc Solution Let s = number of standard golf bags produced d = number of deluxe golf bags produced Max 10s + 9d Subject to 7/10s + 1d <= 630 (Cutting and Dyeing) 1/2s + 5/6d <= 600 (Sewing) 1s + 2/3d <= 708 (Finishing) 1/10s + 1/4d <= 135 (Inspecting and Packing) s, d >= 0

10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Par Inc Solution n The objective function will remain optimal if n slope of line b <= slope of optimal line <= slope of line a n (the “higher” slope is the one that is more counterclockwise)

11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Par Inc Constraint Slopes n We need to find the slopes of the constraint lines n Optimal Solution at intersection of two constraints C+D: 7/10s + 1d = 630 C+D: 7/10s + 1d = 630 F: 1s + 2/3d = 708 F: 1s + 2/3d = 708 n Express in terms of D (or whatever your vertical axis is) C+D: d = -7/10s C+D: d = -7/10s F: d = -3/2s F: d = -3/2s

12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Par Inc Constraints n Slope of the optimal line, in general form n X (horizontal) term over Y (vertical) term, representing the profit (in this case) Profit of standard bag (Cs) Profit of deluxe bag (Cd) n Therefore -3/2 <= -Cs/Cd <= -7/10 -3/2 <= -Cs/Cd <= -7/10

13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If we want to find the range for standard bags.. n -3/2 <= -Cs/Cd <= -7/10 n Lower Range Sub in 9 for deluxe profit: -3/2 <= -Cs/9 Sub in 9 for deluxe profit: -3/2 <= -Cs/9 Elim. Negatives:3/2 => Cs/9 Elim. Negatives:3/2 => Cs/9 Times 9:27/2 => Cs Times 9:27/2 => Cs Decimal, switch:Cs <= 13.5 Decimal, switch:Cs <= 13.5 n Upper Range Sub in 9:-Cs/9 <= -7/10 Sub in 9:-Cs/9 <= -7/10 Elim. NegativesCs/9 => 7/10 Elim. NegativesCs/9 => 7/10 Times 9:Cs => 63/10 Times 9:Cs => 63/10 Decimal: Cs => 6.3 Decimal: Cs => 6.3

14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Range of Optimality for Standard Bag Price n 6.3 <= Cs <= 13.5

15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If we want to find the range for deluxe bags.. n -3/2 <= -Cs/Cd <= -7/10 n Flip:-2/3 <= -Cd/Cs <= -10/7 n Lower Range Sub in 10 for std profit: -2/3 <= -Cd/10 Sub in 10 for std profit: -2/3 <= -Cd/10 Elim. Negatives:2/3 => Cd/10 Elim. Negatives:2/3 => Cd/10 Times 10:20/3 => Cd Times 10:20/3 => Cd Decimal, switch:Cs <= 6.67 Decimal, switch:Cs <= 6.67 n Upper Range Sub in 10:-Cd/10 <= -10/7 Sub in 10:-Cd/10 <= -10/7 Elim. NegativesCd/10 => 10/7 Elim. NegativesCd/10 => 10/7 Times 10:Cd => 100/7 Times 10:Cd => 100/7 Decimal: Cs => Decimal: Cs => 14.29

16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Range of Optimality for Deluxe Bag Price n 6.67 <= Cd <= n So what’s the problem with this method? You can only change one value at a time, and not both! You can only change one value at a time, and not both!

17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. What if the standard bag profit is 18? The slope changes, but if it passes a certain value, the optimal value changes Objective function line, At old optimal point

18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Right hand Sides (Constraints) n What if the amounts of production hours we have available change? n What if we have 10 extra hours cutting and dyeing time? n From this:7/10s + 1d <= 630 n To this:7/10s + 1d <= 640

19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. An increase in hours available Means that the constraint line Slides away from the origin Feasible region may expand as a result

New Optimal Solution Results… n Optimal Solution now s = 527.5d = s = 527.5d = P = Change in P = P = Change in P = Change in P per hour = 43.75/10 = Change in P per hour = 43.75/10 = “for every extra hour of cutting and dyeing, profit can increase by 4.375” “for every extra hour of cutting and dyeing, profit can increase by 4.375” n This change in profit is called the dual value Marginal increase of profit when x changes by ONE point Marginal increase of profit when x changes by ONE point Before other constraints become binding Before other constraints become binding

21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution

22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution

23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution

24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Limitations of Classical Sensitivity Analysis n Can’t do simultaneous changes Can only change one variable at a time Can only change one variable at a time n Provides no info on changing constraints Ie if we can reduce production time per product Ie if we can reduce production time per product n Advanced constraints can become difficult to analyze Ie d >= 0.3s Ie d >= 0.3s

25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Electronic Communications Problem n If we have time

26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 3