Lecture 10:
Sensitivity Part I: General
AGEC 352
Fall 2011 – October 15
R. Keeney
Simple Problem
max Z  x  y
s.t.
x  10; y  10
x, y  0
Solution
x = 10, y = 10, Z=20
Both constraints
bind
y
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Payoff: 1.0 x + 1.0 y = 20.0
: 0.0 x + 1.0 y = 10.0
Corner
Points:
(x,y)
(10,0)
(10,10)
(0,10)
: 1.0 x + 0.0 y = 10.0
0
1
2
3
4
5
6
7
8
9
Optimal Decisions(x,y): (10.0, 10.0)
: 1.0x + 0.0y <= 10.0
: 0.0x + 1.0y <= 10.0
10 11 12 13 14 15 16 17 18 19 20
x
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
max Z  x  y

s.t.
x  y  100
x  10; y  10

x, y  0


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
max Z  x  y problem still feasible?
 2(10)+10 = 30 which fails
s.t.
the new constraint
2 x  y  20

x  10; y  10
x, y  0
y
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Payoff: 1.0 x + 1.0 y = 20.0
: 0.0 x + 1.0 y = 10.0
: 1.0 x + 0.0 y = 10.0
: 2.0 x + 1.0 y = 20.0
0
1
2
3
4
5
6
7
8
9
Optimal Decisions(x,y): (10.0, 10.0)
: 1.0x + 0.0y <= 10.0
: 0.0x + 1.0y <= 10.0
: 2.0x + 1.0y <= 20.0
10 11 12 13 14 15 16 17 18 19 20
x
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
max Z  x  y  .5q
s.t.
x  q  10; y  10
x, y , q  0
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

q
12
11
10
Payoff: 1.0 x + 0.5 q = 10.0
9

8
7
6
5
: 1.0 x + 1.0 q = 10.0
4

3
2
1
0
0
1
2
3
4
5
6
7
Optimal Decisions(x,q): (10.0, 0.0)
: 1.0x + 1.0q <= 10.0
8
9
10 11 12
x

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

q
12
11
10
Payoff: 1.0 x + 1.5 q = 15.0

9
8
7
6
5

: 1.0 x + 1.0 q = 10.0
4
3
2

1
0
0
1
2
3
4
5
6
7
Optimal Decisions(x,q): ( 0.0, 10.0)
: 1.0x + 1.0q <= 10.0
8
9
10 11 12
x

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)