Lecture 10: Sensitivity Part I: General AGEC 352 Fall 2011 – October 15 R. Keeney

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… 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… 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