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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.