Expanding Scope and Computational Challenges in Process Scheduling

Expanding Scope and Computational Challenges in Process Scheduling Pedro Castro Centro de Investigação Operacional Faculdade de Ciências Universidade de Lisboa 1749-016 Lisboa, Portugal Ignacio E. Grossmann Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213, USA Qi Zhang Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213, USA Currently at BASF, SE, Ludwigshafen, Germany January 10, 2017 Carnegie Mellon FOCAPO / CPC 2017 Tucson, Arizona

FOCAPO 2017/CPC IX January 8-12, 2017, Tucson, Arizona - FOCAPO: Christos Maravelias (Wisconsin) and John Wassick (Dow) - CPC: Erik Ydstie (Carnegie Mellon University) and Larry Megan (Praxair) FOCAPO Speakers: Chrysanthos Gounaris Ignacio Grossmann Nick Sahinidis CPC Speaker: Larry Biegler Slides talks: http://focapo-cpc.org/?page=schedule Workshops: FOCAPO -Introduction to Chemical Process Operations and Optimization CPC -Introduction to Theory and Practice of MPC Joint - Introduction to Machine Learning 2

EWO Seminars: http://egon.cheme.cmu.edu/ewo/seminars.html Spring 2017 March 10: Julia and Pyomo: Software for the 21 st Century Qi Chen, Braulio Brunaud March 31: Expanding Scope and Computational Challenges in Process Scheduling Pedro Castro, Ignacio Grossmann April 7: Supply Chain Optimization at Amazon Russell Allgor April 21: Flexible Regression Methods for Big Data Simon Sheather 3

Scheduling Key in Enterprise-wide Optimization (EWO) EWO involves optimizing the operations of R&D, material supply, manufacturing, distribution of a company to reduce costs and inventories, and to maximize profits, asset utilization, responsiveness. Carnegie Mellon 2

Key issues: Carnegie Mellon Integration of planning, scheduling and control Multiple time scales Planning months, years Economics Scheduling days, weeks Feasibility Delivery Control secs, mins Dynamic Performance Multiple models Planning LP/MILP Scheduling MI(N)LP Control RTO, MPC 3

References Carnegie Mellon Reklaitis, G. V. Review of Scheduling of Process Operation. AIChE Symp. Ser. 78, 119-133 (1978). Mauderli. A. M.: Rippin. D. W. T. Production Planning and Scheduling for Mu1tipurpose Batch Chemical Plants. Comp. Chem. Eng. 3, 199 (1979). Floudas, C.A.; Lin, X. Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review. Comp. and Chem. Eng., 28, 2109 2129 (2004). Shah, N., Single- and multisite planning and scheduling: Current status and future challenges, Proceedings of FOCAPO-98 75 90 (1998). Kallrath, J. Planning and scheduling in the process industry, OR Spectrum, 24, 219-250 (2002). Maravelias C., C. Sung, Integration of production planning and scheduling: Overview, challenges and opportunities, Comp. Chem. Eng., 33, 1919 1930(2009). Baldea, M., I. Harjunkoski., Integrated production scheduling and process control: A systematic review Comp. Chem. Eng., 71, 377-390 (2014). Dias, L.S., M. Ierapetritou., Integration of scheduling and control under uncertainties: Review and challenges, Chem. Eng. Res. Design, 116, 98-113 (2016). Wassick, J. (2009), Enterprise-wide optimization in an integrated chemical complex, Comp. Chem. Eng., 33, 1950 1963. Grossmann, I.E., Advances in Mathematical Programming Models for Enterprise-Wide Optimization, Comp. Chem. Eng., 47, 2-18 (2012). Harjunkoski, I., Maravelias, C.T., Bongers, P., Castro, P., Engell, S., Grossmann, I.E., Hooker, J., Mendez, C., Sand, G. and Wassick, J., Scope for Industrial Applications of Production Scheduling Models and Solution Methods, Comp. Chem. Eng., 62, 161-193 (2014).

Carnegie Mellon Outline presentation 1. Scheduling: Basics and new applications a) Brief review state-art-scheduling b) Beyond conventional scheduling problems Heat integration, pipeline scheduling, blending 2. Demand side management: New area for scheduling a) Multiscale design/scheduling models b) Application robust optimization cryogenic energy storage 3. Integration of Planning and Scheduling: Largely unsolved problem a) Discussion of approaches b) Use of TSP constraints for changeovers c) Decomposition schemes: Bi-level and Lagrangean 5

Basic concepts Production recipe Sequence of tasks with known duration/processing rate Product I1 Filling Duration=40 min 48 o C Cp=43.9 MJ/K 94 o C 95 o C 45.9 MJ/K 113 o C 110 o C 45.8 MJ/K 93 o C K1 Heating (C1) Duration=20 min K2 Neutralization Duration=180 min K3 Heating (C2) Duration=40 min K4 Evaporation Duration=65 min K5 Cooling (H1) Duration=25 min K6 Washing Duration=85 min 94 o C 45.5 MJ/K 99 o C 97 o C 45.8 MJ/K 107 o C 44.8 MJ/K 65 o C Heating (C3) Filtration Heating (C4) Cooling (H2) Discharge K7 K8 K9 K10 K11 K12 Duration=30 min Duration=25 min Duration=20 min Duration=30 min Duration=120 min Need to consider multiple materials? No: Identity is preserved sequential facility Yes: Material-based network facility Production environment Illustrated for sequential but also applies to network facility January 10, 2017 Planning & Scheduling 8

Time representation Discrete time uniform slot size (time units) δ 2 3 4... T -3 T -2 T -1 1 t= T time points Precedence General,, ft 1 ft 2 ft 3 ft 4... ft T -2 ft T -1 ft T time of each time point is known a priori 0 H Continuous time slot 1 time slot 2 slot T -2 slot T -1 2 3 T -2 T -1 1 event points t= T T 1 T 2 T 3 T T -2 T T -1 T T,, duration of order starting time of order Immediate,, timing variables to be determined by optimization Single time grid for all resources Multiple time grids, GDP facilitates modeling of equipment availability constraint January 10, 2017 Planning & Scheduling 9

Discrete vs. continuous-time (Castro 08) Multistage, multiproduct batch plant, earliness minimization Discrete-time Reducing data accuracy ( ) makes model easier to solve One way to reduce complexity while generating good solutions T Binary variables Total variables Constraints RMIP MIP CPUs Nodes 29 710 2103 1433 Infeasible Infeasible 0.27-57 1535 4272 2777 207 207 0.47 0 142 3978 10795 6857 192 192 20.0 0 283 8034 21619 13625 184 184 54.7 0 Continuous-time More complex models, can handle just a few event points ( 10) T Binary variables Total variables Constraints RMIP MIP CPUs Nodes 5 440 511 873 154.17 184 1748 328357 CPLEX 11.1, Intel Core2 Duo T9300 @2.5 GHz Was 45,520 s with CPLEX 10.2, Pentium 4 @3.4 GHz January 10, 2017 Planning & Scheduling 10

State-Task Network (STN) (Kondili, Pantelides & Sargent 93) Material balances (multiperiod),,,,,,,,,, Material state availability Production Consumption,,, Equipment allocation constraints,, 1 1,,,, Processing time Batch size Raw material supply & product demand Assigns start of task to unit time Fewer & tighter constraints (Shah, Pantelides & Sargent 93),, 1, Process representation model Complex recipes, multiple processing routes, shared intermediates, recycles Different treatment of material states and equipment units One of most important papers in PSE 622 citations (ISI) #4 of all time Comp. Chem. Eng. January 10, 2017 Planning & Scheduling 11

Resource-Task Network (RTN) (Pantelides 94) Generalization of STN Tasks Rectangles Resources (states, units, etc.) Circles Structural parameters Link tasks & resources May be difficult to find CC 1 PW EN H h _C 1 Cast_G g _CC 1 Duration=154 min H h H h _C 1 H h Casting task δ t = 10 11 12 13 θ= 0 1 2 3 Hour= 15:30 16:30 17:30 18:30 RTN mathematical model Very simple & tight (discrete-time) Few sets of constraints Magic is in excess resource balances! 1 0 1 0 1 0 7 0 7 + 1 + 7 1 7 1 + 7 +7 7 6.3 +6.3 7 + 1 + 1 0 January 10, 2017 Planning & Scheduling 12

RTN similar to UOPSS (Kelly, 2005) Example: fruit juice processing plant (Zyngier, 2016) Continuous multiproduct plant 3 juice types (water + grape, grape pear, grape pear apple) 2 package types (bottle, carton) Process flow diagram does not provide all information UOPSS shows operating modes for blender & packaging lines RTN equivalent: tasks consuming same equipment resource January 10, 2017 Planning & Scheduling 13

Scheduling roadmap (adapted from Harjunkoski et al. 14) Network Describe Process as STN/RTN? Discrete time 1 2 3 4 5 6 7 Continuous time 2 3 4 1 5 Single time grid Continuous time 1 2 3 4 STN/RTN based Models Gather Info Plant topology & Production recipe +? Production Environment 1 2 3 4 Unit specific Mathematical Model Key Aspect: Time Representation Sequential Standard Network? Precedence i i i i Multiple time grids 1 2 3 4 1 2 3 4 Continuous time Models Use GDP to Derive Difficult Constraints E.g. time dependent pricing & availability of resources 14

Beyond conventional scheduling problems: 1) Heat integration 2) Pipeline Scheduling 3) Blending 15

Integrating scheduling & heat integration Heat integration model derived from GDP, Linking timing constraints, Classical general precedence model Timing, temperature driving force & bounds on energy transfer hot task h Heat integration cold task c hot task h Heat hot task h Heat integration integration cold task c cold task c hot task h Heat No overlap h integration cold task c c,,, 0,,,,,,,, 0,,,,,,,, 0,,,,,,,, 0,,,,,, 0 0,,,,,,, 0, January 10, 2017 Planning & Scheduling 16

Tradeoff makespan vs. utility consumption Vegetable oil refinery (Castro et al. 15) 26 streams 890 min, 26.2% Energy savings 15.5 % 37.7 % Problem/Stages 2 3 18 streams 29 s 927 s 26 streams 463 s 202,652 s 33 streams 171,971 s - January 10, 2017 Planning & Scheduling 17

RTN vs. GDP for pipeline scheduling Input node (Refinery R1) Output node (Depot D1) Dual purpose node DP1 Input R2, Output D2 Depot D3 P 1 _ls s 1 RTN pipeline segment model Product centric, FIFO policy F_P 1 Fill_P1 Rate=Whatever P 2 _bv... P P _bv Pipeline Volume Continuous interaction Discrete interaction Switch Fill A_P1_P? Dur.=Instantaneous P 1 _av G P 1 _ip Switch Empty B_P?_P1 Dur.=Instantaneous Switch Fill A/B_P?_P1 Dur.=Instantaneous Inside Pipeline Segment S s Minimum Volume M_P 1 Move_P1 Rate=Whatever Fill & Empty_P1 Rate=Whatever FE_P 1 N_P 1 Do Nothing_P1 Rate=Whatever Switch Empty A_P?_P1 Dur.=Instantaneous Pipeline Volume P1 P2 P3 P4 P5 P6 I5 P 1 _bv Switch Fill B_P1_P? Dur.=Instantaneous Switch Empty A/B_P1_P? Dur.=Instantaneous E_P 1 Empty_P1 Rate=Whatever Valve _S s I4 Empty batch I3 (Castro 10) Segment S1 Segment S2 Segment S3 (Mostafaei & Castro 17) P 1 _ls s GDP modular approach Batch centric, fewer time slots Exclusive disjunctions,,,,,,,,,,, Inclusive disjunctions I2,,,,,, 0,,,,,,,, I1,,,,,,,,, 0,,,,,,, 0,, January 10, 2017 Planning & Scheduling 18

Integrated batching & scheduling GDP model can be extended to other configurations January 10, 2017 Planning & Scheduling 19

Blending in petroleum refineries Crude oil Lee, Pinto, Grossmann & Park ( 96) Refined products Li, Karimi & Srinivasan ( 10) Kolodziej, Grossmann, Furman & Sawaya ( 13) Batch blending (MINLP) Continuous blending (MILP) Material from upstream processes Supply tanks Blending tanks Product tanks Tank contents used to fulfill product orders January 10, 2017 Planning & Scheduling 20

Alternative formulations Process networks Tank volumes, compositions, stream flows Source based Disaggregated volume & flow variables, split fractions,,,,, 1,, 1 3,, 1 2 3 4 5 1 2 3 4,, Bilinear terms (non-convex) 2 2 4,,, 1 2 3 4 6 5 6,,,,,,,,,,,,,,,, Smaller size, fewer bilinear terms but worse performance! Total flows and compositions Problem Variables Equations Bilinear DICOPT BARON terms Feasible? CPUs 6T-3P-2Q-029 103 202 64 No 3.13 8T-3P-2Q-146 223 617 256 No 1227 8T-4P-2Q-480 313 879 376 No 302 8T-4P-2Q-531 273 732 358 No 97.4 8T-3P-2Q-718 223 603 244 Yes 3.56 8T-3P-2Q-721 223 623 256 No 265 8T-4P-2Q-852 305 859 376 No 231,,,,,,,,,,,,,,,,,,,,,, Individual flows and split fractions BARON Bilinear DICOPT Variables Equations CPUs terms Feasible? 0.33 219 294 64 No 91.3 689 965 480 Yes 453 941 1383 720 No 43.3 878 1218 684 No 3.97 672 939 456 Yes 21.4 689 971 480 Yes 134 933 1363 720 Yes January 10, 2017 Planning & Scheduling 21

Global optimization of bilinear MINLPs 2-stage MILP-NLP strategy MILP relaxation Bilinear envelopes (McCormick 76) Integration with spatial B&B Piecewise McCormick (Bergamini et al. 05) Recommended for 2,,9 Multiparametric disaggregation (Kolodziej, Castro & Grossmann 13) 10, 100, 1000, Standalone procedures, guarantee global optimality as Local solution of reduced NLP Fix binary variables Using values from MILP relaxation The tighter the relaxation ( ), the most likely to get feasible or global optimal solution Bilinear term, Domain of divided into partitions 1, Partition dependent bounds for,,,,,,,,,,,, 1/ /,,,...,,... Single active partition,,, January 10, 2017 Planning & Scheduling 22

Insights from crude oil blending (Castro 16) Advantages of discrete-time Simpler model Tighter MILP-LP relaxation Easier to account for time-varying inventory costs Better for cost minimization Discrete-time Continuous-time Problem Slots Solution (k$) Solution (k$) Slots P1 97 209.585 210.537 8 P2 81 319.140 320.496 8 P3 97 284.781 287.000 8 P4 121 319.875 333.331 7 Major surprise! Zero MINLP-MILP gap from bilinear envelopes! Better than BARON & GloMIQO Advantages of continuous-time More accurate model Fewer slots to represent schedule nonlinear blending constraints Better for gross maximization Discrete-time Continuous-time Slots Solution (k$) Solution (k$) Slots 97 7983 7985 4 81 10240 10246 7 49 8542 8574 8 121 13258 13258 7 Approach Cost [$] Gap CPUs Cost [$] Gap CPUs McCormick P1 209585 0.0000% 72.6 P3 284781 0.0000% 346 GloMIQO 209585 0.0001% 1557 284781 11.1% 3600 BARON 209585 0.0001% 305 397208 112% 3600 McCormick P2 319140 0.0000% 662 P4 322300 7.6% 3600 GloMIQO 319252 10.9% 3600 No sol. 17.6% 3600 BARON 319140 38.5% 3600 324746 37.9% 3600 January 10, 2017 Planning & Scheduling 23

Carnegie Mellon Time-sensitive pricing motivates the active management of electricity demand demand side management (DSM) Price [$/MWh] Hourly electricity prices in 2013 Source: PJM Interconnection LLC Time [h] Electricity prices change on an hourly basis (more frequently in the real-time market) Challenge, but also opportunity for electricity consumers Chemical plants are large electricity consumers high potential cost savings 24

Carnegie Mellon Strategic planning models have to incorporate long-term and short-term decisions for demand side management Industrial Case Study: Mitra, Grossmann, Pinto, Arora (2014) Uncertain demand Air separation plant Air feed GO2 GN2 On-site customers Electricity LO2 LN2 LAr Storage Off-site customers Given: Power-intensive plant Product demands for each season Seasonal electricity prices on an hourly basis Upgrade options for existing equipment New equipment options Additional storage tanks Determine: Production / inventory levels Mode of operation Product purchases Upgrades for equipment Purchase of new equipment Purchase of new tanks for each season on an hourly basis 25

The operational model is based on a surrogate representation in the product space 1 Carnegie Mellon Disjunction of feasible regions, reformulated with convex hull: Feasible region: projection in product space Modes: different ways of operating a plant Mass balances: differences for products with and without inventory Inventory balance Demand satisfaction Energy consumption: requires correlation with production levels for each mode + Inventory and transition cost 1. Zhang et al. (2016). Optimization & Engineering, 17, 289-332. 26

Carnegie Mellon Transient plant behavior is captured with logic constraints 1,2 Minimum uptime: 48 hours Off Minimum downtime: 24 hours Ramp up transition After 6 hrs Production mode Link between state and transitional variables State diagram for transitions: nodes: states (modes) = different ways of operating a plant arcs = allowed transitions (including constraints, e.g. min. up-/downtime) / / Enforce minimum stay in a mode Coupling between transitions Forbidden transitions 1. Mitra et al. (2012). Computers & ChemE, 38, 171-184. 2. Zhang et al. (2016). Computers & ChemE, 84, 382-393. Rate of change constraint 27

A multiscale time representation based on the seasonal behavior of electricity prices is applied 1 Carnegie Mellon Year 1, spring: Investment decisions Year 2, spring: Investment decisions Spring Summer Fall Winter 250.00 250.00 250.00 250.00 200.00 150.00 100.00 200.00 200.00 200.00 150.00 150.00 150.00 Mo Tu We Th Fr Sa Su 100.00 Mo Tu Su 100.00 Mo Tu Su Mo Tu 100.00 Su 50.00 50.00 50.00 50.00 0.00 1 25 49 73 97 121 145 0.00 1 25 49 73 97 121 145 0.00 1 25 49 73 97 121 145 0.00 1 25 49 73 97 121 145 Spring Summer Fall Winter Horizon: 10 years, each year has 4 seasons (spring, summer, fall, winter) Each season is represented by one week on an hourly basis Each representative week is repeated in a cyclic manner (13 weeks reduced to 1) Connection between periods: Only through investment (design) decisions 1. Mitra et al. (2014). Computers & ChemE, 65, 89-101. 28

Retrofitting an air separation plant Carnegie Mellon Superstructure Air Separation Plant Liquid Nitrogen LIN 1.Tank LIN 2.Tank? Existing equipment Option A Liquid Argon LAR 1.Tank LAR 2.Tank? Option B? (upgrade) Liquid Oxygen LOX 1.Tank LOX 2.Tank? Additional Equipment Gaseous Oxygen Gaseous Nitrogen Pipelines Time Spring - Investment decisions: (yes/no) - Option B for existing equipment? - Additional equipment? - Additional Tanks? Fall - Investment decisions: (yes/no) - Option B for existing equipment? - Additional equipment? - Additional Tanks? Spring Summer Fall Winter The resulting MILP has 191,861 constraints and 161,293 variables (18,826 binary.) Solution time: 38.5 minutes (GAMS 23.6.2, GUROBI 4.0.0, Intel i7 (2.93GHz) with GB RAM)

Investments increase flexibility help realizing savings. Carnegie Mellon Power consumption Price in $/MWh Remarks on case study Power consumption 1 25 49 73 97 121 145 Hour of a typical week in the summer season Power consumption w/ investment Power consumption w/o investment Summer prices in $/MWh LN2 inventory profile 200 150 100 50 0 Annualized costs: $5,700k/yr Annualized savings: $400k/yr Buy new liquefier in the first time period (annualized investment costs: $300k/a) Buy additional LN2 storage tank ($25k/a) Don t upgrade existing equipment ($200k/a) equipment: 97%. Inventory level 1 25 49 73 97 121 145 Hour of a typical week in the summer season outage level LN2 w/ investment 2 tanks capacity 1 tank capacity LN2 w/o investment Source: CAPD analysis; Mitra, S., I.E. Grossmann, J.M. Pinto and Nikhil Arora, "Integration of strategic and operational decision- making for continuous power-intensive processes, submitted to ESCAPE, London, Juni 2012 30

Comparison of seasonal schedules Carnegie Mellon Spring Summer Power consump on Price in $/MWh 200 150 100 50 Power consump on Price in $/MWh 200 150 100 50 0 1 25 49 73 97 121 145 Hour of a typical week in the spring season Power consump on w/ investment: spring Power consump on w/o investment: spring Spring 0 1 25 49 73 97 121 145 Hour of a typical week in the summer season Power consump on w/ investment: summer Power consump on w/o investment: summer Summer Fall Winter Power consump on Price in $/MWh 200 150 100 50 Power consump on Price in $/MWh 200 150 100 50 1 25 49 73 97 121 145 Hour of a typical week in the fall season Power consump on w/ investment: fall Power consump on w/o investment: fall Fall 0 0 1 25 49 73 97 121 145 Hour of a typical week in the winter season Power consump on w/ investment: winter Power consump on w/o investment: winter Winter 31

Carnegie Mellon Industrial case study: Integrated Air Separation Unit -Cryogenic Energy Storage (CES) participates in two electricity markets Zhang, Heuberger, Grossmann, Pinto, Sundramoorthy (2015) GO2, GN2 Vented gas Gas demand Air ASU LO2, LN2 Driox Purchased liquid LO2, LN2, LAr LO2, LN2 Liquid inventory Liquid demand CES inventory For internal use Electricity generation Purchased electricity Sold electricity Provided reserve Electric energy market Operating reserve market Uncertainty in reserve demand 32

Carnegie Mellon Adjustable Affine Robust Optimization ensures feasible schedule for provision of operating reserve capacity Multistage formulation: first stage: base plant operation, reserve capacity recourse: liquid produced (linear with reserve demand) Large-scale MILP: 53,000 constraints, 55,000 continuous variables, 2,500 binaries CPLEX 12.5, 10 min CPU-time (1% gap) 0.8 0.1 CES Inventory 0.6 0.4 0.2 0.05 0-0.05 In and Out Flows 0-0.1 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 Liquid Flow into CES Tank Converted to Power for Internal Use Converted to Power to be Sold Committed Reserve Capacity CES Inventory Spinning Reserve Price Electricity Price Time [h] 33

Approaches to Planning and Scheduling Carnegie Mellon Decomposition Sequential Hierarchical Approach Simultaneous Planning and Scheduling Detailed scheduling over the entire horizon Planning months, years Planning Scheduling days, weeks Challenges: Challenges: Scheduling Different models / different time scales Mismatches between the levels Very Large Scale Problem Solution times quickly intractable Goal: Planning model that integrates major aspects of scheduling 34

Approaches to Integrating Scheduling at Planning Level Carnegie Mellon Extensive review: Maravelias, Sung (2009) - Relaxation/Aggregation of detailed scheduling model Erdirik, Wassick, Grossmann (2006, 2007, 2008) Single stage multiproduct batch/continuous with sequence dependent changeovers - Projection of scheduling model onto Planning level decisions Sung, Maravelias (2007, 2009) General MILP STN model for multiproduct batch scheduling - Iterative decomposition of Planning and Scheduling Models - Bilevel decomposition - Lagrangean decomposition 35

MILP Planning Models Multiple Stage Batch/Continuous Carnegie Mellon Erdirik, Grossmann (2006) I Relaxation/Aggregation of detailed scheduling model Scheduling model Continuous time domain representation Based on time slots Sequence dependent change-over times handled rigorously Incorporates mass balances and intermediate storage II. Replace the detailed timing constraints by: Model A. (Relaxed Planning Model) Constraints that underestimate the sequence dependent changeover times Weak upper bounds (Optimistic Profit) Model B. (Detailed Planning Model) Sequencing constraints for accounting for transitions rigorously (Traveling salesman constraints) Tight upper bounds (Realistic estimate Profit) 36

Proposed Model B (Detailed Planning) Carnegie Mellon Sequence dependent changeovers: Sequence dependent changeovers within each time period: 1. Generate a cyclic schedule where total transition time is minimized. KEY VARIABLE: ZP ' ii mt :becomes 1 if product i is after product i on unit m at time period t, zero otherwise P1, P2, P3, P4, P5 P1 ZP P1, P2, M, T = 1 P5 P4 P2 KEY VARIABLE: ii mt P4 P3 ZP P2, P3, M, T = 1 2. Break the cycle at the pair with the maximum transition time to obtain the sequence. ZZP ' :becomes 1 if the link between products i and i is to be broken, zero otherwise P1 P4 P2 P4? ZZP P4, P3, M, T P3 37

Changeovers within each period Carnegie Mellon According to the location of the link to be broken: P1 P2, P3, P4, P5, P1 ZZP P1, P2, M, T = 1 P4 P2 P3, P4, P5, P1, P2 ZZP P2, P3, M, T = 1 P4, P5, P1, P2, P3 ZZP P3, P4, M, T = 1 The sequence with the minimum total transition time is the optimal sequence within time period t. P5, P1, P2, P3, P4 ZZP P4, P5, M, T = 1 P4 P3 P1, P2, P3, P4, P5 ZZP P5, P1, M, T = 1 YP ZP i, m, t imt i' ii' mt YPimt ' ZPiimt ' i ', m, t i [ ] YP imt YP ' ',, i i i mt ZPiimt i m t Generate the cycle and break the cycle to find the optimum sequence where transition times are minimized. YP imt ZP i,, i m, t i, m, t ZP iimt,,, YP i ', mt, 1 i, i ' i, m, t ZP YP YP i, m, t i iimt,,, imt,, i ', mt, i ' i i' ZZP ii' mt 1 m, t ZZPii' mt ZPii' mt i, i ', m, t Having determining the sequence, we can determine the total transition time within each week. 38

Changeovers within each period Carnegie Mellon 1) generate the cycle P1 P5 P2 2) break the cycle to obtain the sequence P 4, P 5 P5, P1 P1, P 2 P 2, P 3 P4 P5 P1 P2 P3 P4 P3 ZZP P4, P3, M, T =1 P 3, P 4 TRNP m, t P4, P5 P5, P1 P1, P2 P2, P3 P3, P4 P3, P4 Total transition time within period t on unit m Transition time required to change the operation from P1 to P2 TRNP ZP ZZP m, t mt, ii, ' ii, ', mt, ii, ' ii, ', mt, i i' i i' 39

Multiperiod Refinery Planning Problem Fractionation index model for CDU Given: refinery configuration Alattas, Palou, Grossmann (2012) Carnegie Mellon Time horizon with N time periods Inventories and changeovers of M crudes Determine What crude oil to process and in which time period? The quantities of these crude oils to process? The sequence of processing the crudes? 40

Multiperiod MINLP Model Carnegie Mellon Max Profit= Product sales minus the costs of product inventory, crude oil, unit operation and net transition times. s.t. Performance CDU (FI Model) each crude, each time period Mass balances, inventories each crude, each time period Sequencing constraints (Traveling Salesman, Erdirik, Grossmann (2008)) 0-1 variables to assign crude in period t 0-1 variables to indicate position of crude in sequence 0-1 variables to indicate where cycle is broken Continuous variables flows, inventories, cut temperatures 41

Carnegie Mellon Example: 5 crudes, 4 weeks Produce fuel gas, regular gasoline, premium gasoline, distillate, fuel oil and treated residue Optimal solution ($1000 s) Profit 2369.0 Sales 22327.9 Crude oil cost 16267.5 Other feedstock 44.6 Inventory cost 126.3 Operating cost 3246.5 Transition cost 274.0 MINLP model: 13,680 variables (900 0-1), 15,047 constraints Nonlinear variables: 28% GAMS/DICOPT 23.3.3 (CONOPT/CPLEX): 37 seconds (94% NLP, 6% MIP) 42

Multisite Planning and Scheduling Multi-Scale Optimization Challenge (Spatial, Temporal) Calfa, Agrawal, Grossmann, Wassick (2013) Carnegie Mellon Raw Materials Plants Final Products Customers Demand Demand Demand Demand Production Production Production Production Month 1 Month 2 Month 3 Month 4 Time Multi-period integrated planning and scheduling of a network of multiproduct batch plants located in multiple sites 43

Bilevel Decomposition Algorithm Carnegie Mellon Includes TSP constraints Integer cuts are added to ULP to generate new schedules and avoid infeasible ones to be passed to the LLS problem 44

Lagrangean Decomposition Carnegie Mellon ULP problem can become expensive to solve for large industrial cases Temporal Lagrangean Decomposition (TLD) can be applied to ULP problem: each time period becomes a subproblem : Inventory levels, assignments (changeovers across periods) 45

Hybrid BD-LD Decomposition Carnegie Mellon Multipliers are updated using the Subgradient Method Lagrangean subproblems are solved in parallel using GAMS grid computing capabilities * Maximum 30 LD iterations allowed * http: //interfaces.gams-software.com/doku.php?id=the_gams_grid_computing_facility 46

Carnegie Mellon Computational Results: Problem Sizes 4 weeks 6 weeks 12 weeks Ex. 1 2 3 Problem Disc. Vars. Cont. Vars. Const. NZ Elems. Nodes Time [s] ULP 528 925 1,412 4,537 5,015 0.992 LLS 507 1,039 1,726 5,049 29 0.180 FS 936 1,201 2,924 9,113 94,929 44.981 ULP 6,328 52,783 43,169 145,009 57 2.343 LLS 4,412 53,047 45,378 145,831 0 1.623 FS 128,400 95,563 437,649 3,998,885 57,536 12,228.943 ULP 119,397 834,195 590,810 2,206,546 0 4,070.48 LLS 228,701 898,119 1,140,007 6,836,510 0 452.53 FS 6,726,779 3,138,985 22,895,121 648,785,966 Not enough RAM to solve problem FS in Example 3 47

Concluding remarks Carnegie Mellon 1. Scheduling: Variety of powerful approaches available a) STN & RTN discrete/continuous-time models have reached maturity b) GDP facilitates formulation of complex constraints, widening the scope c) Increased emphasis on nonlinear models (MINLP) 2. Demand side management: Link with electric power: new application area for scheduling a) Large-scale MILP models can yield significant $ savings b) Application robust optimization cryogenic energy storage 3. Integration of Planning and Scheduling: Remains major unsolved problem a) Not a single approach has emerged as winner b) Showed effectiveness of TSP constraints for changeovers c) Showed need for decomposition schemes: Bi-level and Lagrangean

Research Challenges Carnegie Mellon -The modeling challenge: Integration of planning, scheduling, control models for the various components of the supply chain, including nonlinear process models. - The multi-scale optimization challenge: Coordinated optimization of models over geographically distributed sites, and over the long-term (years), medium-term (months) and short-term (days, min) decisions. - The uncertainty challenge: Anticipating impact of uncertainties in a meaningful way. - Algorithmic and computational challenges: Effectively solving large scale MIP models including nonconvex problems in terms of efficient algorithms, and modern computer architectures. 49