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Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of Newfoundlandspkenny@engr.mun.ca
ENGI 5708 Design of Civil Engineering Systems
Lecture 10: Sensitivity Analysis
2 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Lecture 10 Objective
to understand parameters influencing sensitivity of LP problems
3 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Motivation for Sensitivity AnalysisModel Idealization
Abstraction of reality, linearization• Relationship between variables, constraints, coefficients• Scope extent, system hierarchy
Quantifiable ≠ fact or importanceSubjectivity
UncertaintyNatural variability or volatility
• Regulations, economics, resourcesMechanisms
• Poor or incomplete understanding of processesData
• Bias, error or sample size
4 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
What is Sensitivity Analysis?
Tool for Decision MakingHeuristic analysis• Trial and error• What if scenario analysis
Assess importance of possible events (i.e. change in parameter value) and probable outcomes from these changes or variability
5 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Why Conduct Sensitivity Analysis?Rank Assessment
Screening tool• Identify key elements and parameters• Increase or decrease model complexity• Focus effort and resources• Model advancement or refinement
Test optimal solution robustness• Identify critical parameters• Establish thresholds (upper/lower bounds)
Contingency planning• Impact to optimal solution or decision making?• Set conditions for changes in strategy
6 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Sensitivity Analysis
Constraint Equation Coefficient (amn)Right-Hand Side Constraints (Cm)Objective Function Coefficient (kn)Add New Decision VariablesAdd Constraint Equations
( )1 1
min or maxN N
n n nn n
Z f x k x= =
= = ⇒∑ ∑Objective Function orMerit Function (Non-$)
( )1 1 1
, ,M M N
m n mn n mm m n
g x a x C= = =
= ≤ = ≥∑ ∑∑Constraint Equations
7 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Constraint Equation CoefficientPossible Variation
Volume, production rate or yield of a process or resourceExample 6-01
• Clay volume, blending time or storage capacity per unit volume of product
ImpactConstraint equation slopeFeasible regionBasic feasible solutions
Feasible Region
Example 6-01
A B
C
D
EF
8 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Constraint Equation Coefficient (cont.)
Example 6-01Double HYDITblending time
Feasible Region
1 2 1 25 5 50 10 5 50x x x x+ ≤ ⇒ + ≤
Example 6-01
9 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Constraint Equation Coefficient (cont.)
Example 6-01Double FILIT blending time
Feasible Region
1 2 1 25 5 50 5 10 50x x x x+ ≤ ⇒ + ≤
Example 6-01
10 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side ConstraintPossible Variation
Maximum capacity, resource availability, usage or timeExample 6-01
• Total clay volume, blending time or storage capacity
ImpactConstraint equation shiftFeasible regionBasic feasible solutions
Feasible Region
Example 6-01
A B
C
D
EF
11 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Example 6-01Increase total available blending timeto 70 hrs
Feasible Region
1 2 1 25 5 50 5 5 70x x x x+ ≤ ⇒ + ≤
Example 6-01
12 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Example 6-01Decrease total available blending time to 30 hrs
Feasible Region
1 2 1 25 5 50 5 5 30x x x x+ ≤ ⇒ + ≤
Example 6-01
13 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Binding ConstraintsIf binding then RHS limits value of obj.function
Example 6-01
Feasible Region
1 22 4 28x x+ ≤
1 25 5 50x x+ ≤
1 8x ≤
2 6x ≤
1 2, 0x x ≥
1 2140 160Z x x= +
Unique Optimal Solutionx1 = 6; x2 = 4
A B
C
D
EF
14 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Binding ConstraintsReduced costcoefficients
1 23 1 22 52s s s= − +
1 12 1 22 54x s s= − +
1 14 1 22 52s s s= + −
1 21 1 22 56x s s= + −
1 21480 10 24Z s s= − −
{ }3 2 1 4, , ,DB s x x s= Example 6-01
Feasible Region
Unique Optimal Solutionx1 = 6; x2 = 4
A B
C
D
Clay VolumeBlending Time
EF
15 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Non-Binding ConstraintsLook at basic variables
1 23 1 22 52s s s= − +
1 12 1 22 54x s s= − +
1 14 1 22 52s s s= + −
1 21 1 22 56x s s= + −
1 21480 10 24Z s s= − −
{ }3 2 1 4, , ,DB s x x s= Example 6-01
Feasible Region
Unique Optimal Solutionx1 = 6; x2 = 4
A B
C
D
HYDIT StorageFILIT Storage Increasing storage capacity has no effect on strategy
EF
16 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Shadow Prices or Dual Resource PricesReduced costcoefficientsTolerable rangeon variability?• ll constraint line
thru adjacent vertex
1 21480 10 24Z s s= − −
Example 6-01
Feasible Region
Unique Optimal Solutionx1 = 6; x2 = 4
A B
C
D
Clay VolumeBlending Time
EF
G
H
17 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)
Shadow Prices or Dual Resource Prices
Resource Current RHS
Optimal Usage
Lower Range
Allowable Decrease
Upper Range
Allowable Increase
Shadow Price
Wabash Red Clay 28 m3 28 m3
24 m3
(C ≡
8,2)4 m3
32 m3
(G ≡
4,6)4 m3 $10
Blending Time 50 hr 50 hr
40 hr(E ≡
2,6)10 hr
55 hr(H ≡
8,3)5 hr $24
HYDIT Curing Vat Capacity
8 m3 6 m3 6 m3
(D ≡
6,4)2 m3 ∞ Unlimited $0
FILIT Curing Vat Capacity
6 m3 4 m3 4 m3
(D ≡
6,4)2 m3 ∞ Unlimited $0
1 22 4 28x x+ ≤
1 25 5 50x x+ ≤1 8x ≤
2 6x ≤Clay VolumeBlending Time
HYDIT StorageFILIT Storage
18 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Right-Hand Side Constraint (cont.)Shadow Prices or Dual Resource Shadow price implications
If unit price of clay ≤ $10/t then purchaseIf supplier failed to deliver clay then for this range the compensation price is $10/t⇑ Vat capacity provides no improved profits
Resource Optimal Usage
Allowable Decrease
Allowable Increase
Shadow Price
Wabash Red Clay 28 m3 4 m3 4 m3 $10
Blending Time 50 hr 10 hr 5 hr $24
HYDIT Curing Vat Capacity
6 m3 2 m3 Unlimited $0
FILIT Curing Vat Capacity 4 m3 2 m3 Unlimited $0
19 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Objective Function
Possible VariationUnit cost or profitUnit gain or loss
ImpactObjective function slopeOptimal solution
Feasible Region
Increasing FILIT Profit
Increasing HYDIT Profit
1 2 3
4
1
23
4
20 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
Reading ListPike, R.W. (2001). Optimization for Engineering Systems. http://www.mpri.lsu.edu/bookindex.html
Arsham (2007). Linear Programming. http://home.ubalt.edu/ntsbarsh/opre640a/partVIII.htm#rplp
Arsham (2007). Linear Programming. http://home.ubalt.edu/ntsbarsh/Business-stat/opre/PartVII.htm#rintrodu
21 ENGI 5708 Civil Engineering Systems – Lecture 10© 2008 S. Kenny, Ph.D., P.Eng.
References
ReVelle, C.S., E.E. Whitlatch, Jr. and J.R. Wright (2004). Civil and Environmental Systems Engineering 2nd Edition, Pearson Prentice Hall ISBN 0-13-047822-9
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