Lecture 10: Sensitivity Part I: General AGEC 352 Spring 2011 – February 28 R. Keeney

Exam Overview Average Score = 79 Median Score = 79 ◦ Range = ~40 Most missed question ◦ #19 – Answer is D because the line has a negative slope

Exam Overview Grading mistakes / errors in the exam ◦ If you put ‘FREE’ for number 7 or ◦ If you put False for number 14 ◦ Resubmit your exam at the end of class Other questions, see me after class ◦ Things to check  Did you miss the questions marked with X?  Are the X’s counted correctly?  Is your score = 100 – 3*(number of X’s)?

Exam Answers 1.A 2.C 3.True 4.True 5.D 6.C 7.A,B,D 8.True 9.True 10.B 11. B 12. False 13. A 14. True 15. B 16. C 17. False 18. False 19. D 20. True 21. A 22. D 23. True 24. True 25. True 26. False 27. C 28. B 29. True 30. True

Simple Problem Corner Points: (x,y) (10,0) (10,10) (0,10) Solution x = 10, y = 10, Z=20 Both constraints bind

Two Initial Questions 1)If we add a constraint, is the decision maker in the problem… ◦ Better off? Worse off? Indifferent? 2)If we remove a constraint, is the decision maker in the problem… ◦ Better off? Worse off? Indifferent? Try to find examples that produce each result to prove to ourselves the implications

Question 1: A new constraint with no impact on the objective variable  The new constraint says the sum of x and y has to be less than 100  The maximized value from the original problem was Z=x+y=20  Clearly the objective variable will be unaffected by this new constraint  DM is indifferent to this addition

Question 1: A new constraint with an adverse impact on the objective Is our solution to the initial problem still feasible? 2(10)+10 = 30 which fails the new constraint

Question 1: A new constraint generating an improvement Added constraints can have two impacts on the feasible space ◦ Leave it unchanged ◦ Shrink it Since the new constraint cannot expand the feasible space it would be impossible to find a new constraint that makes the decision maker better off.

Question 2: Removing a constraint Left for you to think through Key concept: ◦ Adding a constraint can leave the feasible space unchanged or shrink it ◦ Show that removal of a constraint can leave the feasible space unchanged or expand it Implications for the decision maker’s objective variable are going to be opposite

Simple 3 variable problem q is a new choice variable q competes with x in the constraint q is worth ½ as much as x in the objective equation Note that y is unaffected by the new choice variable q. Increasing y increases the objective variable so we can set y to 10 (its highest level) and just consider the choice between x and q.

2 Initial Questions when adding a choice variable Will the addition of a choice variable make the decision maker… ◦ Better off? Worse off? Indifferent? Will the removal of a choice variable make the decision maker… ◦ Better off? Worse off? Indifferent? Again work with a simple example to try and prove which are possible.

Impact of Adding q to the problem Optimal y = 10 is assumed but not shown Choice between x and q graphed Objective says use only x and set q=0 Adding q to the problem has no impact in this instance

Another possibility of adding q Now q enters the objective equation with a coefficient of 1.5 Graphical solution says x=0,q=10 Remember y=10 New Z = 10 + (1.5)(10) So adding q increased the objective value

Can a new decision variable be deterimental?  Not possible unless it comes with a constraint that forces its value to be a non-zero value  In this example, q was initially not worth pursuing, so the solution was to just set it to zero  What about removing a decision variable?  Beginning with the problem having x,y,q consider the impact of taking q away (returning to the initial problem)  What possibilities exist for the decision maker being better/worse/indifferent?

Review Exercises The minimization case is left to you in the handout to work out the sensitivity of the objective variable to added constraints or decision variable Look through the review handout and ask any questions on Wednesday that you do not understand Wednesday we will review shadow price sensitivity

General Sensitivity Summary Adding a constraint ◦ Worse off  Z decreases for max  Z increases for min ◦ Indifferent (Z is the same) Adding a variable ◦ Better off  Z increases for max  Z decreases for min ◦ Indifferent (Z is the same)