Why Sensitivity Analysis


So far: find an optimium solution given certain
constant parameters (costs, demand, etc)
How well do we know these parameters?


Usually not very accurately - rough estimates
Do our results remain valid?

If the parameters change ...

... how much does the objective function change?

... how much do the optimal values of the decision
variables change?
Notes 4
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1
General Optimization Problem

Minimize some cost or maximize benefit

Constraints:



£ Restrictions on supply of some resource

³ Restriction on satisfying demand for some resource

= Both supply restriction and demand requirement

Variable-type constraints
Decision variable determine the levels of some
activity
Coefficients = per unit impact of activities
Notes 4
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2
Changing Constraints

Relaxing constraints:


Optimal value same or better
Tightening constraints:

Optimal value same or worse
Original
Notes 4
Relaxed
IE 312
Tightened
3
Crude Oil Model
min 20 x1  15x2
s.t. 0.3 x1  0.4 x2  2.0 (gasoline)
0.4 x1  0.2 x2  1.5 (jet fuel)
0.2 x1  0.3 x2  0.5 (lubricant s)
x1  9
x2  6
Satisfy
Demand
Supply
Restriction
x1 , x2  2
Notes 4
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4
Solution
(LINDO)
LP OPTIMUM FOUND AT STEP
1
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
X1
X2
Notes 4
ROW
2)
3)
4)
5)
6)
7)
8)
112.5000
VALUE
2.000000
3.500000
SLACK OR SURPLUS
0.000000
0.000000
0.950000
7.000000
2.500000
0.000000
1.500000
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REDUCED COST
0.000000
0.000000
DUAL PRICES
-37.500000
0.000000
0.000000
0.000000
0.000000
-18.750000
0.000000
5
160
Sensitivity
140
120
80
60
‘Plenty’ of this crude
40
20
6
75
5.
5
5.
25
5.
5
75
4.
5
4.
25
4.
4
75
3.
5
3.
25
3.
3
75
2.
5
2.
2.
25
0
2
Cost
100
Crude Supply
Notes 4
IE 312
6
RHS Coefficients
Constraint
RHS
RHS
Type
Increase Decrease
Supply (<)
Relax
Tighten
Demand (>) Tighten
Relax
Notes 4
IE 312
7
LHS Coefficients
Constraint
LHS
LHS
Type
Increase Decrease
Supply (<)
Tighten
Relax
Demand (>)
Relax
Tighten
Notes 4
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8
New Constraints

Adding constraints tightens the feasible set

Removing constraints relaxes the feasible set

What about unmodeled constraints?
Notes 4
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9
Rate of Change
Supply
Demand
Optimal
Value
Optimal
Value
Maximize
RHS
RHS
Optimal
Value
Optimal
Value
Minimize
RHS
Notes 4
IE 312
RHS
10
Objective Function Changes
Model
Coefficient
Coefficient
Form
Increase
Decrease
Maximize Same or better Same or worse
Minimze Same or worse Same or better
Notes 4
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11
Crude Oil: Changing x1 Coefficient
140
120
100
Cost
80
60
40
20
0
0
5
10
15
20
25
30
35
Coefficient
Notes 4
IE 312
12
Rate of Change
Maximize
Minimize
Optimal
Value
Optimal
Value
Coefficient.
Coefficient.
Notes 4
IE 312
13
New Activities

Adding activities


Optimal value same or better
Removing activities

Notes 4
Optimal value same or worse
IE 312
14
Quantifying Effects

Now know the qualitative effects of







Notes 4
Changing RHS coefficients
Changing LHS coefficients
Changing objective function coefficients
Adding/deleting constraints
Adding/deleting activities
How much does the objective change?
Quantitative change
IE 312
15
Back to Crude Oil Example
min 20 x1  15x2
s.t. 0.3 x1  0.4 x2  2.0 (gasoline)
0.4 x1  0.2 x2  1.5 (jet fuel)
0.2 x1  0.3 x2  0.5 (lubricant s)
x1  9
x2  6
x1 , x2  2
Decreasing RHS
will make objective
better or no worse,
but by how much?
How much are we willing to pay to have one more barrel available?
Notes 4
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16
Answer using the Dual
max 2v1  1.5v2  0.5v3  9v4  6v5
s.t. 0.3v1  0.4v2  0.2v3  1v4  20
0.4v1  0.2v2  0.3v3  1v5  15
v1 , v2 , v3  0
v4 , v5  0
Notes 4
IE 312
17
LINDO Solution
LP OPTIMUM FOUND AT STEP
4
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
V1
V2
V3
V4
V5
Notes 4
92.50000
VALUE
20.000000
35.000000
0.000000
0.000000
0.000000
IE 312
REDUCED COST
0.000000
0.000000
0.950000
0.000000
0.000000
18
Interpretation



Notes 4
Our cost will be reduced by $20 or $ 35,
respectively, if the demand for gasoline
or jet fuel is one unit less.
Smaller demand for lubricants has no
effect on the objective
We are not willing to pay anything for
availability of more crude!
IE 312
19
What is the Dual?



The primal is the original optimization problem
The dual is an LP defined on the same input
parameters but characterizing the sensitivities of
the primal
There is one dual variable for each main
constraint
Primal
< constraint > constraint = constraint
vi  0
vi  0
Minimize objective
Unrestricted
vi  0
Maximize objective vi  0
Unrestricted
Notes 4
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20
Interpretation

The dual variables provide implicit
prices for marginal units of the
resource modeled by the constraint



Notes 4
Zero unless active
How much we are willing to pay for more
of a resource (supply constraint)
How much we benefit from not having to
satisfy a requirement (demand constraint)
IE 312
21
What to Optimize?

Implicit marginal value (minimization
primal) or price (maximization primal) is
a
i, j
vj
All
activitiesi

Notes 4
Maximize value or minimize price!
IE 312
22
Dual Constraints

For each activity xj in a minimization
primal there is a main dual constraint
a
v  cj
i, j i
i

For a maximization primal, each xj  0
has a main dual constraint
 ai, j vi  c j
i
Notes 4
IE 312
23
Optimal Solution

If primal has optimal solution
c x  b v
j
*
j
*
i i
j


Notes 4
i
Either the primal optimal makes a main inequality
active or the corresponding dual is zero
Either a nonnegative primal variable has optimal
value xj = 0 or the corresponding dual price vj
must make the j-th dual constraint active
IE 312
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Dual of a Min Primal
b v
max
i i
i
s.t.
a
v  cj
i, j i
i
vi  0
vi  0
vi URS
Notes 4
IE 312
for all activities j
for all primal ' s i
for all primal ' s i
for all primal ' s i
25
Dual of a Max Primal
b v
min
i i
i
s.t.
a
v  cj
i, j i
i
vi  0
vi  0
vi URS
Notes 4
for all activities j
for all primal ' s i
for all primal ' s i
for all primal ' s i
IE 312
26
Top Brass
max 12 x1  9 x2
x1  1000 (brass footballs)
s.t.
x2  1500 (brass soccer balls)
x1  x2  1750 (brass plaques)
4 x1  2 x2  4800 (feet of wood)
x1 , x2  0
Notes 4
IE 312
27
Graphical Solution
4 x1  2 x2  4800
2000
x1  1000
x2  1500
1500
Optimal Solution
1000
x1  0
x1  x2  1750
500
x2  0
Notes 4
500
1000
1500
IE 312
2000
28
Dual
min
s.t.
1000v1  1500v2  1750v3  4800v4
v1  v3  4v4  12
v2  v3  2v4  9
v1 , v2 , v3 , v4  0
Notes 4
IE 312
29
Lindo Solution
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
V1
V2
V3
V4
Notes 4
17700.00
VALUE
0.000000
0.000000
6.000000
1.500000
IE 312
REDUCED COST
350.000000
400.000000
0.000000
0.000000
30
Interpretation



Notes 4
We are willing to pay up to
$6/each for additional brass Our objective is
sensitive to these
plaques
estimates!
We are willing to pay up to
$1.5/foot for more wood
We don’t need any more
brass footballs or soccer balls
IE 312
31
Formulating Duals
max
s.t.
6 x1  x2  13x3
3x1  x2  2 x3  7
5 x1  x2  6
x2  x3  2
x1  0
x2  0
Notes 4
IE 312
32
Formulating Duals
min
s.t.
7 x1  44 x3
 2 x1  4 x2  x3  15
x1  4 x2  5
5 x1  x2  3x3  11
x1  0
x2  0
Notes 4
IE 312
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