qingpeng (qp) zhang [email protected]. sensitivity analysis is concerned with how changes...
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5.1 A Graphical Introduction to Sensitivity Analysis
Sensitivity analysis is concerned with how changes in an linear programming’s parameters affect the optimal solution.
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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$2
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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.
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Example: Giapetto problem
Optimal solution=(60, 180)=180
Constraint/Objective Slope
Finishing constraint -2
Carpentry constraint -1.5
Objective function -1
Basic variableBasic solution
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Changes of Parameters
Change objective function coefficient Change right-hand side of constraint Other change options Shadow price The Importance of sensitivity analysis
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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?
max 𝑧=¿ 3𝑥1+2𝑥2 ¿
𝑐1
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Change Objective Function Coefficient
𝑧=𝑐1𝑥1+2𝑥2
?
?
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Change Objective Function Coefficient
If
then
Slope is steeperB->C
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Change Objective Function Coefficient
Slope is steeper
New optimal solution:(40, 20)
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Change Objective Function Coefficient
If
then
Slope is flatterB->A
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Change Objective Function Coefficient
Slope is steeper
New optimal solution:(0, 80)z=
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Changes of Parameters
Change objective function coefficient Change right-hand side of constraint Other change options Shadow price The Importance of sensitivity analysis
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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.
𝑏1
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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.
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Change RHS
As long as the binding point (B) of finishing and carpentry constraints is feasible, optimal solution will occur at the binding point.
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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)
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Changes of Parameters
Change objective function coefficient Change right-hand side of constraint Other change options Shadow price The Importance of sensitivity analysis
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Other change options
(weekly profit) s.t. (finishing constraint)
(carpentry constraint) (demand constraint) (sign restriction)
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Other change options
(weekly profit) s.t. (finishing constraint)
(carpentry constraint) (demand constraint) (sign restriction)
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Changes of Parameters
Change objective function coefficient Change right-hand side of constraint Other change options Shadow price The Importance of sensitivity analysis
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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).
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Shadow Prices – Example
Finishing constraint Basic variable: 100 Current value
100+Δ New optimal solution
(20+Δ, 60-Δ) z=3+2=180+ Δ Current basis is optimal
one increase in finishing hours increase optimal z-value by $1The shadow price for the finishing constraint is $1
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Changes of Parameters
Change objective function coefficient Change right-hand side of constraint Other change options Shadow price The Importance of sensitivity analysis
![Page 25: QingPeng (QP) Zhang qpzhang@email.arizona.edu. Sensitivity analysis is concerned with how changes in an linear programming’s parameters affect the optimal](https://reader036.vdocument.in/reader036/viewer/2022062417/551aa83d550346e0158b5d71/html5/thumbnails/25.jpg)
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
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The Importance of Sensitivity Analysis
Deal with the uncertainty about LP parameters• Example:• The weekly demand for
soldiers is 40.• Optimal solution B• If the weekly demand is
uncertain. • As long as the demand is
at least 20, B is still the optimal solution.