Advances in Mathematical Programming Models Advances in Mathematical Programming Models for Enterprise-wide Optimization Ignacio E. Grossmann Center for Advanced Process Decision-making Department of Chemical Engineering Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. EWO Seminar April 24, 2012
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Advances in Mathematical Programming ModelsAdvances in Mathematical Programming Models for Enterprise-wide Optimization
Ignacio E. GrossmannCenter for Advanced Process Decision-making
Department of Chemical EngineeringDepartment of Chemical EngineeringCarnegie Mellon UniversityPittsburgh, PA 15213, U.S.A.
EWO SeminarApril 24, 2012
Motivation for Enterprise-wide Optimization
US chemical industry:19 % of the world’s chemical output
Facing stronger international competition
US$689 billion revenues10% of US exports
Facing stronger international competitionPressure for reducing costs, inventories and ecological footprint
Major goal: Enterprise-wide Optimization
Recent research area in Process Systems Engineering:
Recent research area in Process Systems Engineering: Grossmann (2005); Varma, Reklaitis, Blau, Pekny (2007)
A major challenge: optimization models and solution methodsj g p
Enterprise-wide Optimization (EWO)
EWO involves optimizing the operations of R&D,material supply, manufacturing, distribution of a company to reduce costs, inventories, ecological footprint and to maximize profits, responsiveness .
Key element: Supply Chain
Example: petroleum industry
WellheadWellhead PumpPumpTradingTrading Transfer of Crude
Transfer of Crude
Refinery Processing
Refinery Processing
Schedule ProductsSchedule Products
Transfer of Products
Transfer of Products
TerminalLoadingTerminalLoading
3
I I t ti f l i h d li d t l
Key issues:I. Integration of planning, scheduling and control
Planning Economicsmonths, years Multiple Planning
Scheduling
Economics
Feasibility Delivery
days, weeks
ptime scales
Control
Delivery
Dynamic Performance
secs, mins
Planning LP/MILPMultiple models
Scheduling MI(N)LP
RTO MPC
4
Control RTO, MPC
II. Integration of information and models/solution methods
Strategic OptimizationModeling System
Tactical Optimization
Analytical Analytical ITIT
Strategic AnalysisStrategic Analysis
LongLong--Term Tactical Term Tactical
ScopeScope
Tactical OptimizationModeling System
Production Planning OptimizationModeling Systems
Logistics OptimizationLogistics OptimizationModeling SystemModeling System
Demand Demand Forecasting and OrderForecasting and OrderManagement SystemManagement System
o go g e act cae act caAnalysisAnalysis
ShortShort--Term Tactical Term Tactical AnalysisAnalysis
Production Scheduling Optimization Modeling Systems
Distributions Scheduling Optimization Distributions Scheduling Optimization Modeling SystemsModeling Systems
Operational Operational AnalysisAnalysis
Materials RequirementMaterials RequirementPlanning SystemsPlanning Systems
Distributions RequirementsDistributions RequirementsPlanning SystemPlanning System
Transactional ITTransactional IT
Enterprise ResourceEnterprise ResourcePlanning SystemPlanning System
E l DE l D
5Source: Tayur, et al. [1999]Source: Tayur, et al. [1999]
External DataExternal DataManagement SystemsManagement Systems
Optimization Modeling Framework:Mathematical Programmingg g
fZ )(i Obj i f i
yx htsyxfZ
0),(..),(min
Objective function
Constraints
mnRyxg
100, )(
Constraints
mn yRx 1,0,
MINLP: Mixed-integer Nonlinear Programming Problem
6
fZ )(i
Linear/Nonlinear Programming (LP/NLP)
xgx hts
xfZ
0)(0)( ..
)(min
nRxxg
0)(
LP Codes: Very large-scale models LP Codes:CPLEX, XPRESS, GUROBI, XA
MILP Codes:Great Progress over last decade despite NP-hardPlanning/Scheduling: Lin, Floudas (2004)
CPLEX, XPRESS, GUROBI, XA
MINLP Codes:
g g , ( )Mendez, Cerdá , Grossmann, Harjunkoski (2006)Pochet, Wolsey (2006)
MINLP Codes:DICOPT (GAMS) Duran and Grossmann (1986)a-ECP Westerlund and Petersson (1996)MINOPT Schweiger and Floudas (1998)MINLP BB (AMPL)Fl h d L ff (1999)
New codes over last decadeLeveraging progress in MILP/NLP
MINLP-BB (AMPL)Fletcher and Leyffer (1999)SBB (GAMS) Bussieck (2000)Bonmin (COIN-OR) Bonami et al (2006)FilMINT Linderoth and Leyffer (2006)
Issues:ConvergenceNonconvexitiesScalability
88
yff ( )BARON Sahinidis et al. (1998)Couenne Belotti, Margot (2008)
GlobalOptimization
Scalability
Modeling systems g yMathematical Programming
GAMS (Meeraus et al, 1997)
AMPL (Fourer et al., 1995)
AIMSS (Bisschop et al. 2000)
1 Al b i d li t ti d l1. Algebraic modeling systems => pure equation models
2. Indexing capability => large-scale problems
3. Automatic differentiation => no derivatives by user3. Automatic differentiation no derivatives by user
4. Automatic interface with LP/MILP/NLP/MINLP solvers
Constraint Programming OPL (ILOG), CHIP (Cosytech), Eclipse Have greatly facilitatedHave greatly facilitated development and
implementation of Math Programming models
99
min kZ c f (x ) Generalized Disjunctive Programming (GDP)
Chemical engineers, Operations Research, Industrial Engineering
Larry Biegler (ChE)Nicola Secomandi (OR)John Hooker (OR)
L hi h U i it K t S h i b (I d E )Lehigh University: Katya Scheinberg (Ind. Eng)Larry Snyder (Ind. Eng.)Jeff Linderoth (Ind. Eng.)
16
Projects and case studies with partner companies:“Enterprise-wide Optimization for Process Industries”
ABB: Optimal Design of Supply Chain for Electric MotorsABB: Optimal Design of Supply Chain for Electric Motors Contact: Iiro Harjunkoski Ignacio Grossmann, Analia Rodriguez
Air Liquide: Optimal Coordination of Production and Distribution of Industrial GasesContact: Jean Andre, Jeffrey Arbogast Ignacio Grossmann, Vijay Gupta, Pablo Marchetti
Air Products: Design of Resilient Supply Chain Networks for Chemicals and GasesContact: James Hutton Larry Snyder, Katya Scheinbergy y , y g
Braskem: Optimal production and scheduling of polymer productionContact: Rita Majewski, Wiley Bucey Ignacio Grossmann, Pablo Marchetti
Cognizant: Optimization of gas pipelinesContact: Phani Sistu Larry Biegler, Ajit Gopalakrishnan
Dow: Multisite Planning and Scheduling Multiproduct Batch Processes Contact: John Wassick Ignacio Grossmann, Bruno Calfa
Dow: Batch Scheduling and Dynamic Optimization Contact: John Wassick Larry Biegler, Yisu Nie
Ecopetrol: Nonlinear programming for refinery optimizationContact: Sandra Milena Montagut Larry Biegler, Yi-dong Lang
ExxonMobil: Global optimization of multiperiod blending networksExxonMobil: Global optimization of multiperiod blending networksContact: Shiva Kameswaran, Kevin Furman Ignacio Grossmann, Scott Kolodziej
ExxonMobil: Design and planning of oil and gasfields with fiscal constraintsContact: Bora Tarhan Ignacio Grossmann, Vijay Gupta
G&S Construction: Modeling & Optimization of Advanced Power PlantsContact: Daeho Ko and Dongha Lim Larry Biegler, Yi-dong Lang
Praxair: Capacity Planning of Power Intensive Networks with Changing Electricity PricesContact: Jose Pinto Ignacio Grossmann, Sumit Mitra
UNILEVER: Scheduling of ice cream productionContact: Peter Bongers Ignacio Grossmann, Martijn van Elzakker
BP*: Refinery Planning with Process ModelsContact: Ignasi Palou-Rivera Ignacio Grossmann, Abdul Alattas
17
Contact: Ignasi Palou Rivera Ignacio Grossmann, Abdul AlattasPPG*: Planning and Scheduling for Glass Production
NLP required for process modelsMINLP required for cyclic scheduling, stochastic inventory MIDO for integration of control
20
inventory, MIDO for integration of control
Nonlinear CDU Models in Refinery Planning Optimization
Typical Refinery Configuration (Adapted from Aronofsky, 1978)
Alattas, Palou-Rivera, Grossmann (2010)
butaneFuel gas
Prem.SR Fuel gas
Cat Ref
Crude1,…
Gasoline
Reg.Gasoline
SR Naphtha
SR Gasoline
Distillateblending
CDUDistillate
SR DistillateProduct Blending
blending
Gas oil
Cat Crack
Crude2,…. Fuel Oil
SR GO
Hydrotreatment
blending
Treated Residuum
SR Residuum
21
1000
1200
Li
He
Res
Refinery Planning Models
LP planning models
Fixed yield model 1200
R0
200
400
600
800
0% 10% 20% 30% 40% 50% 60% 70% 80% 90%
C d V l %
TBP
(ºF)
Fuel Gas
Naphtha
ight Distillate
eavy Distillate
siduum B
ottomFixed yield modelSwing cuts model
200
400
600
800
1000
1200
TBP
(ºF)
Fuel Gas
Naphtha
Light Distillat
Heavy D
istillat
Residuum
Botto
Crude Volume %
Nonlinear FI Model (Fractionating Index)
FI Model is crude independent
00% 10% 20% 30% 40% 50% 60% 70% 80% 90%
Crude Volume %
e te om
FI Model is crude independent FI values are characteristic of the column FI values are readily calculated and updated from refinery data
Avoids more complex, nonlinear modeling equations Avoids more complex, nonlinear modeling equations Generates cut point temperature settings for the CDU Adds few additional equations to the planning model
22
Planning Model Example Results
Crude1 Louisiana Sweet Lightest
Crude2 Texas Sweet
Crude3 Louisiana Sour
Comparison of nonlinear fractionation index (FI) with the
Crude3 Louisiana Sour
Crude4 Texas Sour Heaviest
p f ( )fixed yield (FY) and swing cut (SC) models
Economics: maximum profit
Model Case1 Case2 Case3
FI yields highest profit
FI 245 249 247
SC 195 195 191
FY 51 62 59
23
Model statistics LP vs NLP FI model larger number of equations and variables Impact on solution time ~30% nonlinear variables
• Major Decisions (Network + Inventory)k b f C d h i l i i b Network: number of DCs and their locations, assignments between
retailers and DCs (single sourcing), shipping amounts Inventory: number of replenishment, reorder point, order quantity,
safety stock
• Objective: (Minimize Cost)
Total cost = DC installation cost + transportation cost + fixed order cost+ working inventory cost + safety stock cost+ working inventory cost + safety stock cost
26Trade-off: Transportation vs inventory costs
supplierDC i ll i
INLP Model Formulation
retailersupplier
DC
DC – retailer transportation
DC installation cost
EOQ
Safety StockXj Yij
Assignm
Supplier RetailersDistribution Centers
ments
N INLP27
Nonconvex INLP: 1. Variables Yij can be relaxed as continuous2. Problem reformulated as MINLP3. Solved by Lagrangean Decomposition (by distribution centers)
Computational Results
Each instance has the same number of potential DCs as the retailers
• Suboptimal solution in 3 out of 6 cases with BARON for 10 hour limit.Large optimality gaps
28
Large optimality gaps
The multi-scale optimization challenge
Temporal integration long-term, medium-term and p g g ,short-term Bassett, Pekny, Reklaitis (1993), Gupta, Maranas (1999), Jackson, Grossmann (2003), Stefansson, Shah, Jenssen (2006), Erdirik-Dogan, Grossmann (2006), Maravelias, Sung (2009), Li and Ierapetritou (2009), Verderame , Floudas (2010)
Spatial integration geographically distributed sites G t M (2000) T i ki Sh h P t lid (2001)Gupta, Maranas (2000), Tsiakis, Shah, Pantelides (2001), Jackson, Grossmann (2003), Terrazas, Trotter, Grossmann (2011)
Decomposition is key: Benders, Lagrangean, bilevel
29
Multi-site planning and scheduling involves differenttemporal and spatial scales Terrazas, Grossmann (2011)
• Allowed integrality gap is 0.1% • MIP Solvers: CPLEX 11.2.1, XPRESS (version: Aug 13 2009 for GAMS)
37
• Machine: Intel Centrino Duo, 2 Ghz
- The uncertainty challenge:
Sh i i b t ti i tiShort term uncertainties: robust optimizationComputation time comparable to deterministic models
Long term uncertainties: stochastic programmingComputation time one to two orders of magnitude larger than deterministic modelsthan deterministic models
38
Risk ManagementGlobal Sourcing Project withGlobal Sourcing Project with Uncertainties
Gi
You, Wassick, Grossmann (2009)
• GivenInitial inventoryInventory holding cost and throughput cost
Inventory holding cost and throughput costTransport times of all the transport linksUncertain production reliability and demands 25,000 shipping
links/modesp y
• DetermineInventory levels, transportation and sale amounts
• Objective: Minimize CostTwo-stage stochastic MILP model
Page 39
Two-stage stochastic MILP model1000 scenarios (Monte Carlo sampling)
Risk Management
MILP P bl SiMILP Problem Size
DeterministicStochastic
ProgrammingCase Study 1 Deterministic
ModelProgramming
Model1,000 scenarios
# of Constraints 62 187 52 684 187# of Constraints 62,187 52,684,187# of Cont. Var. 89,014 75,356,014# of Disc. Var. 7 7
Impossible to solve directlytakes 5 days by using standard L-shaped Bendersy y g ponly 20 hours with multi-cut version Benders30 min if using 50 parallel CPUs and multi-cut version
Page 40
Risk Management
Simulation Results to Assess Benefits Stochastic Model
Stochastic Planner vs Deterministic Planner
11
12Stochastic SolnDeterministic SolnAverage
5 70±0 03%
Stochastic Planner vs Deterministic Planner
10
11
)
5.70±0.03%cost saving
9
Cos
t ($M
M)
7
8
C
61 10 19 28 37 46 55 64 73 82 91 100
i
Page 41
Iterations
Off h ilfi ld h i l i d t i t
Optimal Development Planning under Uncertainty
Decisions: N b d it f TLP/FPSO f iliti
facilities
Offshore oilfield having several reservoirs under uncertaintyMaximize the expected net present value (ENPV) of the project
Tarhan, Grossmann (2009)
Number and capacity of TLP/FPSO facilities Installation schedule for facilities Number of sub-sea/TLP wells to drill Oil production profile over time Oil production profile over time
TLP FPSO
ReservoirsReservoirswells
Uncertainty:
Initial productivity per well
42
Size of reservoirs Water breakthrough time for reservoirs
Non-linear Reservoir Model
Initial oil
U t i d
Initial oil production Assumption: All wells in the same reservoir are identical.
Rat
e (k
bd)
Unconstrained Maximum Oil Production
Oil
and
Wat
er
Water Rate
Sing
le W
ell
Tank Cumulative Oil (MBO)
Size of the reservoirUncertainty is represented by discrete distributions functions
43
Uncertainty is represented by discrete distributions functions
Decision Dependent Scenario TreesEndogeneous uncertainty: size field
Invest in F in year 1 Invest in F
Assumption: Uncertainty in a field resolved as soon as WP installed at field
Invest in F in year 5
g y
H
ves
Size of F: M L
Invest in F
Scenario tree
HSize of F: LM
4444
Scenario tree – Not unique: Depends on timing of investment at uncertain fields– Central to defining a Stochastic Programming Model
Multi-stage Stochastic Nonconvex MINLP
M i i P b bilit i ht d f NPV t i t iMaximize.. Probability weighted average of NPV over uncertainty scenariossubject to
Equations about economics of the model Surface constraints Surface constraints Non-linear equations related to reservoir performance Logic constraints relating decisions
if there is a TLP available, a TLP well can be drilled
Every scenario,
time period
Non-anticipativity constraintsNon-anticipativity prevents a decision being taken now from using information that will only become available in the future
Every pair scenarios,
time periodDisjunctions (conditional constraints)
Problem size MINLP increases ti ll ith b f ti i d
Decomposition algorithm:L l i &
time period
exponentially with number of time periodsand scenarios
Lagrangean relaxation & Branch and Bound
45
Multistage Stochastic Programming Approach
RS: Reservoir sizeOne reservoir, 10 years, 8 scenarios
Solution proposes building 2 small FPSO’s in the first year and then add new facilities / drill wells (recourse action) depending on the positive or negative outcomes.
RS: Reservoir size
Multistage Stochastic Programming ApproachOne reservoir, 10 years, 8 scenarios
Solution proposes building 2 small FPSO’s in the first year and then add new facilities / drill wells (recourse action) depending on the positive or negative outcomes.
Distribution of Net Present Value
8
10
4
6t P
rese
nt V
alue
($ x
10
9 )
0
2
1 2 3 4 5 6 7 8 9
Net
-2
Scenarios
Deterministic Mean Value = $4.38 x 109
Multistage Stoch Progr = $4.92 x 109 => 12% higher and more robustg g g
No co ve N : 00 d sc ete va s, 970 co t va s, 8090 Co st a ts
Economics vs. performance?
Multiobjective Optimization Approach
49
Optimal Design of Responsive Process Supply ChainsObjective: design supply chain polystyrene resisns under responsive and economic criteria You, Grossmann (2008)
50
Possible Plant SiteSupplier Location
Distribution CenterCustomer Location
Production Network of Polystyrene Resins
Example
Production Network of Polystyrene ResinsThree types of plants:
1. Enterprise-wide Optimization area of great industrial interestG t i i t f ff ti l i l l h iGreat economic impact for effectively managing complex supply chains
2. Key components: Planning and SchedulingM d li h ll
3 C t ti l h ll li i
Modeling challenge:Multi-scale modeling (temporal and spatial integration )