engi 5708 design of civil engineering...

Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of [email protected]

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


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


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


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


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


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


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


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


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


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



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



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



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


A B

C

D

Clay VolumeBlending Time

EF



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


A B

C

D

HYDIT StorageFILIT Storage Increasing storage capacity has no effect on strategy

EF



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


A B

C

D

Clay VolumeBlending Time

EF

G

H



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


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


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


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


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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