Sensitivity and Uncertainty Analysis in Sensitivity and Uncertainty Analysis in OptimizationOptimization--Driven ModelsDriven Models

David RheinheimerDavid RheinheimerUC Davis UC Davis ––

drheinheimer@ucdavis.edudrheinheimer@ucdavis.edu

Dr. Jay LundDr. Jay LundUC Davis UC Davis ––

jrlund@ucdavis.edujrlund@ucdavis.edu

2008 California Water and Environment Modeling Forum 2008 California Water and Environment Modeling Forum

February 28, 2008February 28, 2008

OutlineOutline

•

Review uncertainty and senstivity

•

Optimization-driven models–

Linear programming

–

LP sensitivity analysis

•

Automated sensitivity screening for LP–

Development of automated process

–

Application to CALSIM

Uncertainty vs. sensitivityUncertainty vs. sensitivity

Loucks and Beek (2005)

Uncertainty: sources of uncertainty Uncertainty: sources of uncertainty in modelsin models

Loucks and Beek (2005)

Optimization modelsOptimization models•

Optimization models help decide the best use of resources under constraints

•

Also…used for simulation

•

Numerous modeling methods

•

Each method includes:–

Decision variables–

Objective function–

Constraints

Mathematical programming:Linear programmingNon-linear programmingInteger programmingDynamic programming

Heuristic methods

Optimization modelsOptimization modelsGeneral optimization models:

Linear Programming (subset of general optimization models):

1

n

j jj

c x=∑

1

for all 1, 2,3, ,n

ij j ij

a x b i m=

≤ =∑ …

0 for all 1, 2,3, ,jx j n≥ = …

Maximize: (minimize)

Subject to:

( )f X

( ) for all 1, 2,3, ,i ig X b i m≤ = …

Maximize: (minimize)

Subject to:

Simple LP problemSimple LP problem

x1

≥

0

x2

≥

0

z = c1

x1

+ c2

x2

x1

≤

b3

a21

x1

+ a22

x2

≤

b2

a11

x1

+ a12

x2

≤

b1

x2

x1

Optimal solution = z* = (x1

*, x2

*)

Feasible region

Complex LP problemsComplex LP problems•

CALSIM–

Optimization-simulation model for SWP/CVP planning

–

Mixed Linear Integer Programming–

876 months, >300 nodes, >900 arcs,>2000 constraints, layers and sub-layers

•

CALVIN–

CA-wide economic optimization model for water distribution

•

Many others (TMDL, well-placement, etc.)

Uncertainty in optimizationUncertainty in optimization

•

Goal: map uncertainty in input to uncertainty in output

•

Approach 1: Integrate uncertainty into model

•

Approach 2: Uncertainty analysis–

Monte carlo simulations

–

Difficult to impossible for all inputs/outputs in large models

ex: x1

≤

b3 P[x1 ≤ b3] ≤ p3

Uncertainty in optimizationUncertainty in optimization

Need to focus on “important”

parameters:

1.

identify major input parameters2.

develop input uncertainty ranges

3.

perform uncertainty-weighted sensitivity analysis

4.

focus on more sensitive uncertain parameters

Sensitivity analysis in optimizationSensitivity analysis in optimization

•

How do model outputs respond to changes in inputs?

•

Common method: change one input at a timevery time consuming

•

LP solvers provide sensitivity outputs

a21

x1

+ a22

x2

≤

b2

range of basis

Sensitivity analysis in LP modelsSensitivity analysis in LP modelsx2

x1

z = c1

x1

+ c2

x2

Feasible region

lagrange multiplier:b + Δb z + Δz

New objective function value

Standard LP solver outputs

slack variable

IndexIndex--based sensitivity screeningbased sensitivity screening

•

Indices based on LP outputs

can help screen LP parameters to scrutinize

–

Lagrange Multiplier Index (LMI)–

Slack Variable Index (SVI)

–

Range of Basis Index (RBI)

IndexIndex--based sensitivity screeningbased sensitivity screening

General process for each index:1.

Specify parameter uncertainty range

2.

Calculate index values3.

Rank and assess results

i,min i i,maxb b b≤ ≤ i,min i i,maxc c c≤ ≤or

IndexIndex--based sensitivity screeningbased sensitivity screeningLagrange multiplier index (LMI)

Slack variable index (SVI)

Range of basis index (RBI)

2i,max i,min

i i

b bLMI L

−= ⋅

( )

( )

[ constraints]

[ constraints]

i i i,mini

i i,min

i i,max ii

i,max i

S b bSVI

b b

S b bSVI

b b

− −= ≤

−

− −= ≥

−

Large LMI --

high sensitivity and/or uncertainty in constraint.

Negative SVI --

non-binding constraint could potentially be binding, changing the optimal solution.

( )i max mini

max i,min

r c cRBI

c c− −

=−

Negative RBI --

uncertain cost coefficient could potentially change optimal solution

Implementation of indicesImplementation of indices

Goal:Processor

to computes/sort index values

Uncertainty ranges

Sensitivity index processor

Formatted LP output parameters Sorted

sensitivity indices

ExampleExampleLP sensitivity output

User-specified uncertainty range

Sensitivity index output

Next stop: CalSimNext stop: CalSim--IIII•

CalSim-II: optimization-simulation model for SWP and CVP planning

•

many nodes and arcs: traditional analysis approaches unrealistic

Subject to: physical/legal constraints

DWR (2003)

Previous CalSimPrevious CalSim--II workII work

DWR (2005)

Major Rim Flows Lagrange Multiplier Index

0

10000002000000

30000004000000

50000006000000

70000008000000

9000000

Octobe

rNov

embe

rDec

embe

rJa

nuary

Februa

ryMarc

h

April

May

June July

Augus

tSep

tembe

rMonth (WY1922)

Inde

x Va

lue Trinity Lake

Shasta Lake

Lake Oroville

Folsom Lake

Lake Oroville Lagrange Multiplier Index

02000000

40000006000000

800000010000000

1200000014000000

1600000018000000

Octobe

rNov

embe

rDec

embe

rJa

nuary

Februa

ryMarc

h

April

MayJu

ne July

Augus

tSep

tembe

r

Month (WY1922)

Inde

x Va

lue

+/- 5%

+/- 10%

ConclusionsConclusions•

Comprehensive uncertainty analysis impossible for large optimization models

•

Senstivitity analysis is very possible for LP models

•

Can combine parameter uncertainty ranges with sensitivity analysis to screen parameters for uncertainty reduction

•

Further work needed:–

Develop CalSim-II ranges of uncertainty–

Apply this approach to CalSim-II–

Explore alternative screening approaches

Thank you!Thank you!

Questions??Questions??

ReferencesReferencesDWR (2000). CALSIM: Water Resource Simulation Model Manual

[Draft]. Sacramento.DWR (2003). CalSim II Simulation of Historical SWP/CVP Operations.

Sacramento, California Department of Water Resources Bay-Delta Office.

DWR (2005). CalSim-II Model Sensitivity Analysis Study. Sacramento, California Department of Water Resources Bay-Delta Office.

Loucks, J. R. and E. Beek (2005). Model Sensitivity and Uncertainty Analysis. Water Resources Systems Planning and Management: An Introduction to Methods, Models and Applications. Paris, UNESCO Publishing.