Chapter 5 Sensitivity Analysis QingPeng QP Zhang qpzhangemailarizonaedu 51 A Graphical Introduction to Sensitivity Analysis Sensitivity analysis is concerned with how changes in an linear programmings ID: 620116
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Slide1
SIE 340Chapter 5. Sensitivity Analysis
QingPeng
(QP) Zhang
qpzhang@email.arizona.eduSlide2
5.1 A Graphical Introduction to Sensitivity AnalysisSensitivity analysis
is concerned with how changes in an linear programming’s
parameters
affect the
optimal solution
.Slide3
Example: Giapetto problem
Weekly profit (revenue - costs)
=
number of soldiers produced each week
= number of trains produced each week.
Profit generated by each soldier
$3
Profit generated by each train
$2Slide4
Example: Giapetto problem
(weekly profit)
s.t.
(finishing constraint)
(carpentry constraint)
(demand constraint)
(sign restriction)
= number of soldiers produced each week = number of trains produced each week.
Slide5
Example: Giapetto problem
Optimal solution
=(60, 180)
=180
Constraint/Objective
Slope
Finishing
constraint-2
Carpentry constraint-1.5
Objective function-1
Basic variableBasic solutionSlide6
Changes of Parameters
Change objective function coefficient
Change right-hand side of
constraint
Other change options
Shadow priceThe Importance of sensitivity analysisSlide7
Change Objective Function Coefficient
How would
changes
in the problem’s
objective
function coefficients
or the constraint’s right-hand sides change this optimal solution?
Slide8
Change Objective Function Coefficient
?
?Slide9
Change Objective Function Coefficient
If
then
Slope is steeper
B->C
Slide10
Change Objective Function Coefficient
Slope is steeper
New optimal solution:
(40, 20)
Slide11
Change Objective Function Coefficient
If
then
Slope is flatter
B-
>A
Slide12
Change Objective Function Coefficient
Slope is steeper
New optimal solution:
(
0
, 80
)
z=
Slide13
Changes of Parameters
Change objective function coefficient
Change right-hand side of
constraint
Other change options
Shadow priceThe Importance of sensitivity analysisSlide14
Change RHS
(weekly profit)
s.t.
(finishing constraint)
(carpentry constraint)
(demand constraint)
(sign restriction) = number of soldiers produced each week
= number of trains produced each week.
Slide15
Change RHS
is the number of finishing hours.
Change in b1 shifts the finishing constraint parallel to its current position.
Current optimal point (B) is where the carpentry and finishing constraints are binding.
Slide16
Change RHS
As long as the binding point (B) of finishing and carpentry constraints is feasible, optimal solution will occur at the binding point
.Slide17
Change RHS
If
>120,
>40 at the
binding point
.
If
<80, <0 at the binding point.So, in order to keep the basic solution, we need:
(
z is changed)
(
demand constraint
)
(sign restriction)
Slide18
Changes of Parameters
Change objective function coefficient
Change right-hand side of
constraint
Other change options
Shadow price
The Importance of sensitivity analysisSlide19
Other change options
(weekly profit)
s.t.
(finishing constraint)
(carpentry constraint)
(demand constraint)
(sign restriction)
Slide20
Other change options
(weekly profit)
s.t.
(finishing constraint)
(carpentry constraint)
(demand constraint)
(sign restriction
)
Slide21
Changes of Parameters
Change objective function coefficient
Change right-hand side of
constraint
Other change options
Shadow price
The Importance of sensitivity analysisSlide22
Shadow Prices
To determine how a constraint’s
rhs
changes the optimal z-value.
The
shadow price for the
ith constraint of an LP is the amount by which the optimal z-value is improved (increased in a max problem or decreased in a min problem).Slide23
Shadow Prices – Example
Finishing constraint
Basic variable: 100
Current value
100+
Δ
New optimal solution
(20+Δ, 60-Δ)z=3+2=180+ ΔC
urrent basis is optimalone increase in finishing hours increase optimal z-value by $1
The shadow price for the finishing constraint is $1
Slide24
Changes of Parameters
Change objective function coefficient
Change right-hand side of
constraint
Other change options
Shadow price
The Importance of sensitivity analysisSlide25
The Importance of Sensitivity Analysis
If LP parameters change, whether we have to solve the problem again?
In previous example:
sensitivity
analysis shows it is
unnecessary as long as:
z is changed Slide26
The Importance of Sensitivity Analysis
Deal with the uncertainty about LP parameters
Example:
The weekly demand
for soldiers
is 40
.
Optimal solution BIf the weekly demand is uncertain. As long as the demand is at least 20, B is still the optimal solution.