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Leveraging Big Data for Robust Process Operation Under Uncertainty
Fengqi You
Joint work with graduate student Chao Ning
Process-Energy-Environmental Systems Engineering (PEESE)School of Chemical and Biomolecular Engineering
Cornell University, Ithaca, New York
www.peese.org
• Let Uncertainty Data “Speak” in Math Programming• Data-driven stochastic programming [Jiang, 13]• Data-driven chance constraint programming [Guan, 15]• Distributionally robust optimization [Delage & Ye, 10]• Data-driven static/adaptive robust optimization
• When Big Data “meets” Robust Optimization• Data-driven static robust optimization [Bertsimas et al., 13]• Adaptive/adjustable robust optimization better balances
conservativeness and has high computational tractability
• Objective: A novel data-driven adaptive robust optimization framework – fill the knowledge gap
Data-Driven Decision Making under Uncertainty
2
• Two main components• Decisions: All the decisions are made “here-and-now”• Uncertainty set: Often constructed based on a priori
and relatively simple assumptions about uncertainty
• Drawback: Solution could be overly conservative
Background: Static Robust Optimization
3
0min max ,
s.t. , 0 ,U
i
f
f U i
x u
x u
x u u
Uncertainty setDecisions
• “Wait-and-see” decisions made after uncertainty is revealed• Well represents sequential decision-making problems• Less conservative than Static Robust Optimization• Recourse decisions address feasibility issues
Two-Stage Adaptive Robust Optimization (ARO)
4
( , )min max min
. . ,
, :
T T
U
s t
x y x uu
x
y
c x b y
Ax d x S
x u y S Wy h Tx Mu “wait-and-see” decisions
“here-and-now” decisions
“here-and-now” decisions
Uncertainty “wait-and-see” decisions
• Linear decision rule
• After replacing y with the linear decision rule, we can solve it as a Static Robust Optimization
• Less adaptive but computationally inexpensive
Decision Rules for ARO
5
y q Qu qQ
“here-and-now” decisions
An affine function of uncertainty
Example When Affine Decision Rule Fails to Work
6
ARO with affine decision rule (AARC)
ARO with general decision rule
Worst-case profit 0 6,600
“over-conservatism by refusing to open any of
the facilities”
Location-Transportation Problem
,max min max
. . , ,
, ,
0, , ,
0 , 0,1 ,
i i i i ij ijUx v yi i i j
ij ji
ij ij
ij
i i
i
c x k v y
s t y j U
y x i U
y i j U
x Mv iv i
• Ardestani-Jaafari, Amir, and Erick Delage. “Linearized RobustCounterparts of Two-stage Robust Optimization Problem withApplications in Operations Management.” 2016
Demand Uncertainty
5.9 5.6 4.95.6 5.9 4.9
ˆ , 0 1, ,j j j j j jj
U j
0.6130,000
icM100,000ik
The parameters of costs
Uncertainty set
ˆ20,000 18,000 2j j
Example: 2 facilities and 3 customers
• Linear decision rule
• After replacing y with the linear decision rule, we can solve it as a Static Robust Optimization
• Less adaptive but computationally inexpensive
• Generalized decision rule (This work)
• Fully adaptive• Challenging to solve
Decision Rules for ARO
7
y y u
y q Qu
A general function of uncertainty, determined
by optimization
“here-and-now” decisions
An affine function of uncertainty
“wait-and-see” decisionsy
min
s.t.
T
y
y
b y
y S
Wy h Tx Mu
• Box Uncertainty Set• Soyster (1973)
• Pros: Tractable• Cons: Very conservative
• Budgeted Uncertainty Set• Bertsimas and Sim (2003)
• Pros: Control conservatism• Cons: Most suitable for independent and symmetric uncertainty
Uncertainty Sets – “Heart” of Robust Optimization
8
budget , 1 1, , i i i i i i ii
U u u u u z z z i
1
box , L Ui i i iU u u u u i
• The “bridge” between data and uncertainty set
• Dirichlet Process (DP) Mixture Model [Blei & Jordan, 06]• A powerful Bayesian nonparametric model• Ability to adjust its complexity to that of data
Data-Driven Uncertainty Set for ARO
9
0 0
1 2
(1, )
, , ( )
( )
~
~
~
~i
k
k
i
i i i l
Beta
F F
l Mult
o l p o
1 kkkF
“Stick Breaking”
Data Sample
1 11
2 21
Pr new observation Dataset
Predictive PosteriorVariationalinference
Dirichlet Process Mixture Model
Uncertainty Set
Features of DP Mixture Model
10
• Dirichlet Process (DP) mixture model [Blei & Jordan, 06]• Model data with complicated characteristics (e.g. multimode)• Handle data outliers, asymmetry, and correlation
• Why DP mixture model is better?
• Parameter space has infinite dimensions
• Infer the number of components from data
DP mixture model
• Finite number of parameters
• Specify the number of components a priori
Parametric mixture models
6 parameters
Variational Inference for DDANRO Uncertainty Set
11
,,
i i
i i
v
Variationalinference
Inference results
Uncertainty dataq is variational distribution
Update kq
Update
Update q
,k kq η H
1
1
ELBO ELBOELBOt t
t
q qtol
q
Yes
No
Parameters in uncertainty sets
1
1
iji
iji i j j
vv v
1
1 dimi
ii i
s
u
1, , NU u u
, , ,i i i is μ Ψ
Evidence lower bound
Update iq l
1
1 1 1
, , , , ,N M M
i k k ki k k
q q l q q q
l β η H η H
,i iμ Ψ
Example 1: Data-driven uncertainty set for ARO
12
Box type uncertainty set
Budgeted uncertainty set Data-driven uncertainty set
Uncertainty data0 20 40 60 80 100 120 140 160 180
0
50
100
150
200
u1
u 2
Outliers
Uncertainty data
Outliers
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
u1
u 2
Uncertainty Set
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
u1
u 2
Uncertainty Set
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
u1
u 2
Uncertainty Set
128 Olin 128 Olin
Uncertainty Sets under Different Parameters
13
Budget based Uncertainty Set Data Driven Uncertainty Set
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
250
u1
u 2
Uncertainty set (=0.5)Uncertainty set (=1.0)Uncertainty set (=1.5)
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
250
u1u 2
Uncertainty set (=0.50)Uncertainty set (=0.93)Uncertainty set (=0.98)
Data-Driven Adaptive Nested Robust Optimization
14
*
1/21
:
, 1, i
i i i i ii
U s
u u μ Ψ z z z
( , )1, ,
1/21
min max max min
. . ,
, 1,
, :
i
T T
i m U
i i i i i i
s t
U s
x y x uu
x
y
c x b y
Ax d x S
u u μ Ψ z z z
x u y S Wy h Tx Mu
Uncertainty set using l1 and l∞ norms
DDANRO1∩∞
• Size depends on data • Multi-level (min-max-
max-min) optimization
Model Features
• Adaptive to uncertainty• Less conservative • Captures the nature of
uncertainty data
Advantages
component iChallenge: How to solve the multi-level optimization problem?
Tailored Column & Constraint Gen. Algorithm
15
min
. . , ,
, ,
T
T l
ll
l
s tl L
l L
l L
x y
c x
Ax db y
Tx Wy h Mu
x S y S
Master problem
Sub-problems
max min
. .
i
Ti U
Q
s t
yu
y
x b y
Wy h Tx Muy S
First-stage decisions
Optimality or feasibility cuts
• Multi-level optimization to single-level SIP
Example 2: ARO under correlated uncertainties
16
1 2 1 2
1 2
1 1 1
2 2 2
min 3 5 max min 6y 10
. . 100 , 0, 1, 2
U
i i
x x y
s t x xx y ux y ux y i
x yu
Uncertainties
Uncertainty set is constructeddirectly from data.
Motivating Example 2
17
ARO with box uncertainty set
ARO with budgeteduncertainty set
Data-driven ARO with l1and l∞ norms based set
Min. obj. 824.8 732.3 620.3First-stagedecisions
1
2
20.379.7
xx
1
2
32.267.8
xx
1
2
41.458.6
xx
35 40 45 50 55 60 65 70 75 80 8520
30
40
50
60
70
80
90
100
u1
u 2
Data-driven uncertainty setBox based uncertainty setBudgeted based uncertainty set
35 40 45 50 55 60 65 70 75 80 8520
30
40
50
60
70
80
90
100
u1
u 2
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%500
550
600
650
700
750
800
850
Data Coverage of Uncertainty Set
Obj
ectiv
e Fu
nctio
n V
alue
ARO with budgeted setThe proposed DDANRO
• Uncertain parameters from historical data• Demands of 4 products (correlated uncertainty)• Processing times of 3 reactions (with outliers)• Asymmetry, multimode and correlated data
18
Objective• Maximize profit
• Assignment constraint• Time constraint• Batch size constraint• Mass balance constraint• Storage constraint• Demand constraint
Constraints
Application 1: Batch Process Scheduling
19
Affected by outliers in processing time data
Static robustoptimization
box uncertainty
set
ARO budgeted
uncertainty set
DDANRO
• DDANRO yields the highest profit ($46,597)
• Reduces conservatism of ARO solution in the presence of outlier-corrupted data.
Data-Driven Robust Batch Scheduling Results
Application 2: Process Network Planning
20
Process Network
• 38 processes• 28 chemicals
Objective• Maximize NPV
• Supply (10)• Demand (16)
Constraints• Expansion constraint• Investment constraint• Mass balance constraint• Capacity constraint• Demand constraint• Supply constraint
Uncertainty
Data-Driven Robust Process Network Planning
21
Static robust optimization w/ box uncertainty
ARO with budget based uncertainty
(Гd=3, Гs=2)
DDANRO(Φd=3, Φs=2)
Max. NPV(m.u.) 761.79 799.03 857.38
Computational Results for Application 2
22
Int. Variables Cont. Var. Constraints Total CPU (s)Original ARO 152 681 945
466.4Master (last iter.) 152 7,450 9,748Subproblem 112 13,033 38,067
23
http://you.cbe.cornell.edu
Fengqi YouRoxanne E. and Michael J. Zak Professor
Cornell University
318 Olin Hall, Ithaca, New York [email protected] (email)
www.peese.org