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

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

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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.

9. 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.

10. 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

11. 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

12. 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?

13. 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

14. 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