Chemical Production Scheduling:

Transcription

1 Chemical Production Scheduling: Notation, Problem Classes, Modeling Approaches, and Solution Methods Christos T. Maravelias Department of Chemical and Biological Engineering University of Wisconsin Madison Enterprise-wide Optimization Seminar Carnegie Mellon University February 27, 2014

2 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models Time representation in material-based models Solution methods Concluding thoughts

3 What is Scheduling? It is the allocation of limited resources to tasks over time Michael Pinedo, 1998 Single-Stage Scheduling N jobs to be processed in M machines Release/due times p ij, c ij, s ij Assignment of jobs to machines Sequencing of jobs in the same machine Resources (processing units) A C Time D B Lateness D d i Changeover cost A D A D A D C B C B C B makespan Lateness B 1

4 Preliminaries Systematic scheduling practiced in manufacturing since early 20 th century First scheduling publications in the early 1950s Salveson, M.E. On a quantitative method in production planning and scheduling. Econometrica, 20(9), Extensive research in 1970s Closely related to developments in computing and algorithms Computational Complexity: Job Sequencing one of 21 NP-complete problems in (Karp, 1972) Widespread applications Airlines industry (e.g., fleet, crew scheduling); sports; transportation (e.g., vehicle routing) Government; education (e.g., class scheduling); services (e.g., service center scheduling) Chemical industries Batch process scheduling (e.g., pharma, food industry, fine chemicals) Continuous process scheduling (e.g., polymerization) Transportation and delivery of crude oil Scheduling in PSE First publications in early 1980s; focused on sequential facilities (Rippin, Reklaitis) Problems in network structures addressed in early 1990s (Pantelides et al.) Very challenging problem: Small problems can be very hard Most Open problems in MIPLIB are scheduling related Railway scheduling: 1,500 constraints, 1,083 variables, 794 binaries Production planning: 1,307 constraints, 792 variables, 240 binaries 2

5 Problem Statement Given are: a) Production facility data; e.g., unit capacities, unit connectivity, etc. b) Production recipes; i.e., mixing rules, processing times/rates, utility requirements, etc. d) Production costs; e.g., raw materials, utilities, changeover, etc. e) Material availability; e.g., deliveries (amount and date) of raw materials. f) Resource availability; e.g., maintenance schedule, resource allocation from planning, etc. g) Production targets or orders with due dates. Facility and recipe data Input from other planning functions Materials Planning Master Planning Production Distribution & Planning Transport Planning Production targets Material availability (release dates) Resource availability Scheduling Batching Assignment Sequencing Timing Shopfloor Management Facility & recipe data 3

6 Problem Statement Given are: a) Production facility data; e.g., unit capacities, unit connectivity, etc. b) Production recipes; i.e., mixing rules, processing times/rates, utility requirements, etc. d) Production costs; e.g., raw materials, utilities, changeover, etc. e) Material availability; e.g., deliveries (amount and date) of raw materials. f) Resource availability; e.g., maintenance schedule, resource allocation from planning, etc. g) Production targets or orders with due dates. Our goal is to find a least cost schedule that meets production targets subject to resource constraints. Alternative objective functions are the minimization of tardiness or lateness (minimization of backlog cost) or the minimization of earliness (minimization of inventory cost) or the maximization of profit. In the general problem, we seek to optimize our objective by making four types of decisions: a) Selection and sizing of batches to be carried out (batching) b) Assignment of batches to processing units or general resources. c) Sequencing of batches on processing units. d) Timing of batches. Task selection (batching) How many tasks/batches? What size? Demand (orders) A B C D Batches A1 A2 A3 B1 B2 C1 D1 Task-resource Assignment What resources each task requires? A1 A2 A3 B1 B2 C1 D1 U1 U2 Sequencing (for unary resources) In what sequence are batches processed? A1 A2 A3 D1 B1 Timing When do tasks start? A1 A2 D1 A3 B1 C1 B2 C1 B2 3

7 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models Time representation in material-based models Solution methods Concluding thoughts

8 Traditional Scheduling Notation α / β / γ Machine environment Single machine Parallel machines Machines in series Processing characteristics Preemption Release/due times Setup times Objective Makespan Tardiness Cost 4

9 From Machine to Production Environments. i I: j J: k K: c C: Jobs Machines Operations Work centers R i : J i : J k : C i : Routing of job i (jobshop) Machines for job i (open-shop) Machines in stage k Centers for job i R o u t i n g G e n e r a l i z a t i o n Operations: Single Multiple Routes: Common Job-specific Routes: Predetermined Free m m 1 m K m 1 m 2 m 1 m 3 m 2 m 4 m 5 R A = {m 1, m 3 } R B = {M 1, M 2 } R C = {M 2, M 4 } m 1 m 3 m 2 m 4 A B C J A = {m 1, m 3 } J C = {m 2, m 4 } m 1 m 3 (a) Single machine (b) Flow-shop (c) Job-shop (d) Open-shop Operation 1: J 1 = {m 1, m 2 } Operation 2: J 2 = {m 3, m 4, m 5 } c 1 = {m 1, m 2 }, c 2 = {m 3, m 4 }, c 3 = {m 5 } R B = {c 1, c 3 } R A = {c 1, c 2 } R C = {c 3, c 2 } m 1 m 2 m 3 m 4 m 5 C A B A C C A = {c 1, c 2 } A m 1 m 2 B m 2 C B = {c 1, c 3 } J B = {m 1, m 2 } B m 3 m 4 m 5 m 4 c 1 = {m 1, m 2 }, c 2 = {m 3, m 4 }, c 3 = {m 5 } 1 st Generalization Level: Machines per Stage 2 nd Generalization Level: Parallel machine similarity Operation Generalization (e) Machines in parallel (f) Flexible flow-shop (g) Flexible job-shop (h) Flexible open-shop Multi-stage Multi-purpose 6

10 Network Production Environment Discrete manufacturing A job (e.g., circuit chip) moves through operations consisting of parallel machines Each job is not split into multiple jobs; jobs are not merged Chemical production: tasks involve fluids Fluids coming from different batches can be mixed into a vessel; fluids of different types can be mixed to be converted to a new fluid; output of a batch (stored in a vessel) can be used in multiple downstream batches. No mixing/splitting restrictions may be added (e.g., quality control) Sequential processing Materials cannot be mixed/split/recycled Problem defined in terms of: Batches, stages, and units Problems similar to discrete manufacturing Network processing Materials can be mixed/split/recycled Amounts of materials must be monitored Problem defined in terms of: Materials, tasks, resources Problem is different from discrete manufacturing Batches RM1 RM2 40% Int2 40% A Int3 60% 60% 80% ImB 20% 10% 90% B Int1 Fermentation Centrifugation Drying RM3 Mendez et al., (2006), Computers & Chemical Engineering 7

11 Basic Insights - I Sequential-looking process Tasks: T1 in U1: F I (2 h); T2 in U2: I P (3 h) Capacities: β min /β max (kg): U1: 20/40, U2: 20/40, V: 0/40v Network-looking process Tasks: T0/U0: F I (2 h); T1/U1: I P1 (3 h) T2/U2: I P2 (4 h); T3/U3: I P3 (2 h) Capacities: β min /β max (kg): All units should run 20 kg batches No special restrictions for intermediate I Each batch of intermediate I should be consumed by a single downstream batch F U1-T1 I V U2-T2 P U1 V U2 T1/20 T1/20 Two batches of T1 mix into one Store/20 batch of T2 20 T2/ (h) F U0 U1-T1 P1 U0-T0 I U2-T2 P2 U3-T3 P3 T0/20 T0/20 T0/20 Batches Stage 1 Stage 2 U1 T1/40 One batch of T1 splits 20 into two batches of T2 V Store/ U2 T2/20 T2/ (h) U1 T1/20 U2 U3 T2/20 T3/ (h) Sequential plant structure does NOT imply sequential processing Materials handling restrictions determine type of processing 8

12 Basic Insights - II Sequential or network? Tasks: T0/U1: F1 A (2 h); T1/U1: F2 I (2 h); T2/U2: I (90%) + A (10%) P1 (4 h); T3/U3: I (80%) + A (20%) P2 (2 h) Capacities: β min /β max (kg): U1: 16/20; U2: 20/20; U3: 20/20 Sequential or network? Tasks: T1/U1: F I (60%) + U (40%) (2 h) T2/U2: I P1 (3 h); T3/U3: U F (50%) + W (50%) (2 h) Capacities: β min /β max (kg): U1: 20/20; U2: 12/12; U3: 16/16 A batch of intermediate I should be consumed by a single downstream batch of T2 or T3 Each batch of T2 or T3 requires additive A A can be stored and used in multiple batches A batch of intermediate I should be consumed by a single downstream batch of T2 Byproduct U should be treated when enough material is accumulated F1 U1-T0 A V1 F U1-T1 I U2-T2 P1 F2 U1-T1 I U2-T2 U3-T3 P1 P2 U V1 U3-T3 W U1 V1 U2 T0/20 T1/18 T1/ Store 20Store 18Store 14 2 T2/20 4 U3 T3/ (h) U1 U2 T1/20 T1/ T2/12 T2/12 8 V1 Store 8 U3 8 T3/ (h) We have materials mixing (A+I) We do not have batch mixing for I We have multiple output materials (I+U) We do not have batch splitting for I 9

13 Problem Classes α / β / γ α Production environment Material handling constraints are key (not facility structure): sequential, network, hybrid β Processing characteristics Typical characteristics: setups, changeovers, release/due times, etc. Chemical production characteristics: storage constraints, material transfer constraints, utilities, etc. γ Objective functions min Cost, min Lateness, min Tardiness, max Throughput, etc. Production Environment - α Sequential multistage (Sms) Sequential multipurpose (Smp) Network (N) Hybrid (H) Objective Function - γ min Makespan (M max ) min Cost (C tot ) min Max Lateness (L max ) min Weighted Lateness (Σw i L i ) Problem Class e.g. Sms/s, u, r/m max Processing characteristics - β - Setups (s) & changeovers (c) Utilities (u) Storage restrictions (st) Material transfers (mt) Unit connectivity constraints (uc) Release/due times (r)... Selection of one Multiple selections possible Maravelias (2011), AIChE Journal 10

14 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models Time representation in material-based models Solution methods Concluding thoughts

15 From Problems to Models Math Programming Scheduling Models in the PSE literature ( ): For sequential processes we developed batch-based approaches Track batches; do not account for material amounts Fixed number and size of batches: only assignment and sequencing decisions; no batching Task selection (batching) Demand (orders) A B C D Batches A1 A2 A3 B1 B2 C1 D1 Task-resource Assignment A3 B1 A2 A1 B2 C1 D1 For network processes we developed material-based approaches We model amounts of material (material balances) We make batching, assignment and sequencing/timing decisions RM1 RM2 RM3 40% Int2 40% 60% 60% A Int3 80% Int1 ImB 20% 10% 90% B U1 U2 A1 D1 Sequencing A2 A3 B1 C1 B2 Other common assumptions No storage constraints No utility requirements Also account for: Storage constraints Utility requirements Transfer operations Blending 11

16 Modeling Attributes Key modeling entities Batches/tasks Material amounts Both Scheduling decisions Task number & size (batching) Task-unit assignment Sequencing/timing I. Key Modeling Elements Tasks Materials Tasks & materials Sequencing/timing Sequential, no batching Single-unit II. Scheduling Decisions Unit-batch assignment Sequencing/timing Sequential, no batching Single-stage, multistage, multi-purpose Batching (no & size) Unit-batch assignment Sequencing/timing 1. Network processes 2. Network & sequential processes Sequential with batching 12

17 Modeling Attributes Key modeling entities Batches/tasks Material amounts Both Scheduling decisions Task number & size (batching) Task-unit assignment Sequencing/timing Modeling of time (four types of decisions) 1. Selection between precedence based and time-grid-based approaches 2. Selection of (i) type of precedence relationship (local vs. global) (ii) type of time grid (common vs. unit specific) Precedence-based Time-grid-based Local precedence relations (L) B Y BA = 1 Y AC = 1 Y CD = 1 A C Y BC = 0 Y AD = 0 Y BD = 0 Global precedence relations (G) B Y BA = 1 Y AC = 1 Y CD = 1 A Y BC = 1 Y BD = 1 C Y AD = 1 D D Unit-specific time grids (U) U1 U2 T 2 T 3 T 4 A D C T 3 T 4 T 5 T 6 Common (global) time grid (C) U1 U2 A D C B T 3 T 4 T 5 T 6 T 7 B 12

18 Modeling Attributes Key modeling entities Batches/tasks Material amounts Both Scheduling decisions Task number & size (batching) Task-unit assignment Sequencing/timing Modeling of time (four types of decisions) 1. Selection between precedence based and time-grid-based approaches 2. Selection of (i) type of precedence relationship (local vs. global) (ii) type of time grid (common vs. unit specific) 3. Specific representation assumptions Precedence-based Local precedence relations (L) B Y BA = 1 Y AC = 1 Y CD = 1 A C Y BC = 0 Y AD = 0 D (LG1) Use start times (Ts i ) to enforce precedences; Ts i defined as any time between the end of j i changeover and the actual start of i. Ts C TsA + τ A + σ ACY + M 1 Y ) A ( AC C AC (LG2) Use end times (Te i ) to enforce precedences; Te i defined as the end of i (i.e., before the i j changeover). Te C A TeA + σ ACYAC + τ C + M ( 1 YAC ) C (LG3) τ i : processing time of task i; σ ij : i j changeover time; M: sufficiently large number Ts A Ts C Te A Te C 12

19 Modeling Attributes Key modeling entities Batches/tasks Material amounts Both Scheduling decisions Task number & size (batching) Task-unit assignment Sequencing/timing Modeling of time (four types of decisions) 1. Selection between precedence based and time-grid-based approaches 2. Selection of (i) type of precedence relationship (local vs. global) (ii) type of time grid (common vs. unit specific) 3. Specific representation assumptions 4. Selection between discrete- and continuous-time Levels III. Modeling of time 1. Precedence vs. time grid Precedence-based Time-grid-based 2. Type of precedence/grid Local Global Unit-specific Common 3. Specific assumptions L1 L2 G1 G2 U1 U2... C1 C Time representation Discrete vs. Continuous 12

20 The Universe of Modeling Approaches I. Key Modeling Elements Tasks Materials Tasks & materials Sequencing/timing Sequential, no batching Single-unit II. Scheduling Decisions Unit-batch assignment Sequencing/timing Sequential, no batching Single-stage, multistage, multi-purpose Batching (no & size) Unit-batch assignment Sequencing/timing 1. Network processes 2. Network & sequential processes Sequential with batching Scheduling Decisions Levels 1. Precedence vs. time grid 2. Type of precedence/grid 3. Specific assumptions 4. Time representation III. Modeling of time Precedence-based Time-grid-based Local Global Unit-specific Common L1 L2 G1 G2 U1 U2... C1 C2... Discrete vs. Continuous Batching, Assignment & Sequencing/timing Assignment & Sequencing/timing Precedence -based Time-grid -based Local Global Unit-specific Common Sequencing/timing L1 L2 G1 G2 U1 U2 C1 C2 L1D L1C C1D C1C C2C Tasks Modeling of Time Tasks & Materials Materials Key Modeling Elements 13

21 The Universe of Modeling Approaches Batching, Assignment & Sequencing/timing Assignment & Sequencing/timing Sequencing/timing Scheduling Decisions Precedence -based Time-grid -based Local Global Unit-specific Common L1 L2 G1 G2 U1 U2 C1 C2 STN (Kondili et al., 1993) Decisions: Batching, assignment & timing Modeling element: Materials Modeling of time: 1. Time-grid-based 2. Common grid 3. Tasks time pts 4. Discrete-time L1D L1C C1D C1C C2C Tasks Modeling of Time Tasks & Materials Materials Key Modeling Elements 13

22 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models for: Sequential environments under storage and utility constraints General production environment Time representation in material-based models Solution methods Concluding thoughts

23 PSE Scheduling Literature: Batch-based formulations Exploit sequential process structure Batch-centric approach Batches are assigned to units Precedence constraints for batches in the same unit Batching problem is solved prior to scheduling Other common assumptions: No storage constraints No utility constraints Task-unit assignment A3 B2 A2 B1 C1 A1 D U1 U2 A1 D1 A2 Sequencing A3 B1 C1 B2 Material-based formulations Handle network environments (mixing, splitting, recycle) Consider wide range of processing constraints Storage & utility constraints Consider batching and scheduling simultaneously Material-based approach Tasks consume/produce materials & resources Can we formulate general models for problems in sequential environments? 14

24 General Models for Sequential Environments 1) Simultaneous batching, assignment and sequencing 1 Variable number of batches variable batchsizes variable processing times Introduce new selection variables; if a batch is selected then it is assigned and sequenced Task selection (batching) Demand (orders) Batches A B B1 B2 C C1 D A1 A2 A3 D1 Task-unit assignment A3 B2 A2 B1 C1 A1 D1 U1 U2 A1 D1 Sequencing A2 A3 B1 C1 B2 2) Simultaneous batching, assignment, sequencing + general storage constraints 2 Consider timing (waiting in units & tanks) and capacity (tank number & size) constraints Storage tanks modeled as additional resources 3) Simultaneous batching, assignment, sequencing + storage and utility constraints 3 Storage tanks modeled as resources Adopt common (discrete) time grid to monitor utilities Express resource balance constraints for units, tanks, and utitilties 1 Prasad & Maravelias, Computers and Chemical Engineering, 32 (6), Sundaramoorthy & Maravelias, Industrial and Engineering Chemistry Research, 47 (17), Sundaramoorthy & Maravelias, Industrial and Engineering Chemistry Research, 48 (13),

25 Facilities with Multiple Environments No methods available for facilities with combined environments But such facilities are common! Feedstocks ADJUNCT MALT Brew House Batch Fermentation Semi-continuous Network processing D. W. Continuous Liter 500 BPM ¼ STD 700 BPM DILUTION ½ STD 600 BPM COOKER 1 MASHING 1 LAUTER 1 BOILING 1 WHIRLPOOL 1 ½ PREMIER 600 BPM ½ T.A., ¼ BOH. 750 BPM Liter 350 BPM No batch mixing Batch splitting allowed LATA 1200 LPM ½ T.A EXP 750 BPM BOILING 2 WHIRLPOOL 2 Sequential processing Aging Filtering Storage Bottling Can we formulate models applicable to all/combined environments? 16

26 Unified Framework Network M 30% 60% Feed1 intermediate 40% Mixer U1 Feed 2 Feed 3 R1 70% Reactor U2 R2 Reactor U3 P1 P2 R3 Reactor U4/U5 R4 Reactor U4/U5 Sequential Intermediate 1 Intermediate 2 S1 Separator U6/U7 S2 Separator U6/U7 P3 (5 orders) P4 (7 orders) Traditional process representation S1 60% {U1} S3 T1 40% Material-based S4 30% T2 70% {U2, U3} S2 {U2, U3} T3 Batch-based S5 S6 U4 U5 U6 U7 Stage 1 Stage 2 S4 70% S1 {U1} 60% S3 30% S5 T1 T2 40% S2 Sundaramoorthy & Maravelias, AIChE, 2011 {U2, U3} {U2, U3} T3 S6 {U4,U5} T4 {U4,U5} T6 S7 S9 {U6,U7} T5 {U6,U7} T7 S8 S10 Network task Network material Hybrid material Sequential task Sequential material 17

27 Industrial Application TANQUES DE GOBIERNO Processing Stages: Fermentation Filtering Storage Bottling Production Type: Batch, continuous, and semi-continuous Planning Horizon: 6 weeks Production Environment: Fermentation Filtering Storage Bottling Litro 500 BPM FERMENTACIÓN ¼ STD D. W. 700 BPM DILUTION TANQUES DE GOBIERNO I TANQUES DE GOBIERNO II ½ STD 600 BPM ½ PREMIER 600 BPM ½ T.A., ¼ BOH. 750 BPM Litro 350 BPM 8 processing units (lines) 22 products families 25 product subfamilies 162 products REPOSO LATA 1200 LPM ½ T.A EXP TANQUE REPOSO 750 BPM Product families: Products (for bottling) are grouped into families Changeover costs/times between families; setup costs/times between products Products belong to subfamilies Product family major changeover Product family Kopanos et al.,iecr, 2011 Product (item) minor changeover (sequence independent) 18

28 Industrial Application: Executed Schedule Executed Schedule 19

29 Industrial Application: Integrated Approach No memory across planning periods Minimize setup time & costs 20

30 Industrial Application Schedule Found Using Integrated Framework 21

31 Industrial Application Comparison with Implemented Solution 22

32 Remarks Sequential environments Simultaneous batching, assignment, sequencing Storage policies, shared storage vessels, general resource constraints Network environments* Non-simultaneous and multiple material transfers 1 Resource-constrained material transfers and changeover activities 2... Hybrid environments 3 * Different handling constraints for different materials; e.g., batch integrity maintained for major product but byproduct is mixed Combined environments 3 * Upstream sequential followed by downstream network, followed by continuous processing All of the above 4 * * Material-based models 1 Gimenez et al., Computers and Chemical Engineering, 33 (9), Gimenez et al., Computers and Chemical Engineering, 33 (10), Sundaramoorthy & Maravelias, AIChE J., 57(3), Velez & Maravelias, Industrial & Engineering Chemistry Research, 52(9),

33 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models Time representation in material-based models Solution methods Concluding thoughts

34 Material-Based Model Problem Statement Given are a set of tasks i I, processing units j J, materials (states) k K, and resources r R A processing unit j can be used to carry out tasks i I j. Task i in unit j has processing time τ ij and variable batchsize in [β j min β j max ] Each task can consume/produce multiple materials; conversion coefficient ρ ik Material k can be produced/consumed by multiple tasks; is stored in a dedicated tank Task i requires ψ ir units of resource r during its execution Generalizations for continuous processing, variable processing times, release and due dates, variable utility consumption, etc. Tasks trigger changes in: (1) unit utilization, (2) material inventories, (3) resource utilization/availability Tasks are mapped onto one or more time grids Changes in unit, inventories, resources models through balances over time 1 Sundaramoorthy & Maravelias, AIChE J.,

35 Time Grids Early Discrete 1,2,3,4 : horizon η divided into uniform periods of known length δ Processing times are approximated Large-scale MIP models Continuous 5,6,7 : horizon η divided into periods of unknown (variable) length Exact processing times Smaller MIP models Continuous time models have been studied extensively since ,9,10,11,12,13, unit-specific time grids; wide range of constraints, etc. 1 Kondili et al., Computers and Chemical Engineering, 17, Shah et al., Computers and Chemical Engineering, 17, Pantelides, 2 nd Conference on Foundations of Computer Aided Process Operations, Bassett et al., AIChE J., 42(12), Zhang & Sargent, Computers and Chemical Engineering, Schilling & Pantelides, Computers and Chemical Engineering, 20, Mockus & Reklaitis, Industrial and Engineering Chemistry Research, 38, Ierapetritou & Floudas, Industrial and Engineering Chemistry Research, 37, Castro et al., Industrial and Engineering Chemistry Research, 40(9), Maravelias & Grossmann, Industrial and Engineering Chemistry Research, 24, Sundaramoorthy & Karimi, Chemical Engineering Science, 60, Janak & Floudas, Computers and Chemical Engineering, 32, Gimenez et al., Computers and Chemical Engineering, 22,

36 Discrete Time: Modeling Advantages Inventory (kg) Inventory (kg) S kn t (hr) Discrete: IC k = nn νν kk δδss kkkk S kn δ parameter S kn variable linear T 1 variable S kn variable t (hr) Continuous: IC k = nn νν kk TT nn SS kkkk bilinear A. Inventory cost, IC k (ν k : unit cost [$/(kg hr)]) Real Profile ζ rn Availability: 30 kw during and hr 20 kw during hr t (hr) Calculation of time-varying resource availability θ rn For δ=0.5: θ rn = 30, n = 1-4, 10-12; θ rn = 20, n = t (hr) Resource constraint: RR rrrr θθ rrrr paramter C. Time-varying resource availability Deliveries: 20 kg of A 30 kg of t=0.5 t=1.75 hr A B Real timeline t (hr) C Calculation of shipments ξ kn (δ = 1): Deliveries/orders moved to next/previous point if needed A B Order: 25 kg of t = h Discrete timeline t (hr) ξ A1 = 20 ξ B2 = 30 C ξ C5 = -25 Material balance: SS kkkk = SS kk,nn 1 + ii,jj ρρ iiii BB iiii,nn ττiiii + ii,jj ρρ iiii BB iiiinn + ξξ kkkk E. Modeling of release and due times Resource usage (kw) Resource usage (kw) R rn δ parameter R rn variable t (hr) Discrete: UC r = nn σσ rr δδrr rrrr R rn T n variable R rn variable linear t (hr) Continuous: UC r = nn σσ rr TT nn RR rrrr bilinear B. Utility cost, UC r (σ r : unit cost [$/(kw hr)]) Real Profile σ rn Pricing: $0.04/kWh during and hr $0.03/kWh during hr t (hr) Calculation of time-varying resource price σ rn For δ=0.5: σ rn = 0.04, n = 1-5, 9-12; σ rn = 0.03, n = t (hr) UC r = nn σσ rrrr δδrr rrrr linear D. Time-varying resource pricing ττ RR rr,nn+1 = RR rr,nn + ii,jj iiii ss=0 (φφ iiiiii XX iiii,nn ss + ψψ iiiiii BB iiii,nn ss ) Recipe of task i = R in unit j = U: F ψ RF0 =-1 Load B = 10 kg of input F Remove product P (75%) after 4 hr Electricity consumption (kw/kg): 0-1 h: 10; 1-3 h: 5; 3-4 h: 10 kw/kg ψ RP4 = Coefficient ψψ RR,EE,ss (kw/kg) Electricity consumption (kw) 0 (ψψ RR,EE,ss BB RR,UU,nn ss ) -10 s = F. Variable resource consumption during task P Velez & Maravelias, Ann Rev Chemical & Biomolecular Engineering,

37 Computational Comparison Continuous vs. Discrete Averaged over instances Adding Features to Discrete Averaged over 18 instances CPU time (s) Optimality gap or Quality (%) CPU time (s) Optimality gap or Quality (%) Objective.Processing Time Discrete (time) Continuous (time) Discrete (gap) Continuous (gap) Discrete (quality) Continuous (quality) Objectives: S: Sales without orders P: Profit with orders Processing Times: CPT: fixed processing times VPT1/2: variable processing times weak/strong dependence on batch size Sundaramoorthy & Maravelias, Industrial and Engineering Chemistry Research, 50, Objective.Processing Time.Features Features: U: utilities SET: setups INT: intermediate due dates HC: holding and backlog costs 27

38 Outline Introduction Problem classes Classification of modeling approaches New MIP scheduling models Time representation in material-based models Solution methods Tightening methods Reformulation Concluding thoughts

39 Major Types of Scheduling Constraints 1. A unit can process at most one task at a time X ijt = 1 if task i starts in unit j at time t Basic difference between discrete- and continuous-time formulations nn ii IIjj nn nn ττ iiii +1 XX iiiiiii 1, jj, nn 2. Batch-sizes are within the unit capacity B ijt = batch-size of task i in unit j starting at time t If a task is carried out then its batch-size is within a lower and upper bound ββ jj mmmmmm XX iiiiii BB iiiiii ββ jj mmmmmm XX iiiiii, ii, jj JJ ii, nn 3. Material balance constraints S st = inventory of material s at time t SS kkkk = SS kk,nn 1 + ii IIkk + ρρ iiii jj JJii BB iiii,nn ττiiii + ii IIkk ρρ iiii jj JJii BB iiiiii + ξξ kkkk γγ kk, kk,n 28

40 Motivating Example Capacities: U1: kg U2: kg U3: kg {U2,U3} S1 {U1} T1 S2 T2 S3 T3 S4 Demand: 90 kg S3 and 25 kg S4 # of Batches Min. Cost T1 T2 T3 ($) LP-Relaxation No tightening With tightening* Optimal solution Calculation of some bounds for specific models using auxiliary LPs & MIPs Burkard & Hatzl, 2006; Janak & Floudas, 2008 Can we generalize to all networks and models and calculate bounds fast? 1. Demand number of batches: T2 must produce 90 kg in at least 2 batches T3 has to produce 25 kg, but the minimum capacity is 35 kg, so T3 must produce 35 kg in 1 batch {U2,U3} S1 {U1} S2 T2: S3 T1 T3: S4 2.Number of batches intermediate demand: 125 kg of S2 are needed {U2,U3} S1 {U1} S2 T2: S3 T1 125 T3: S4 3.Intermediate demand number of batches: T1 has to produce at least 125 kg; Capacity of U1 is kg, so 3 batches are required {U2,U3} S1 {U1} S2 T2: S3 T1: T3: S4 29

41 Basic Propagation Basic parameters: ω k : minimum required amount of material k Calculated once µ i is known for all tasks consuming material k For final products, ω k is the customer demand ωω kk = max 0, ii IIkk ρρ 0 iiii μμ ii ξξ kk μ = max{20/0.2,50/0.8} μ i : minimum cumulative production of task i μ = 100 Calculated once ω k is known for all materials produced by task I # of Batches Capacity: 30-40kg μμ ii = max kk KK ii + + ωω kk ρρ iiii µ i = 50 µ i * = U1 U Tightening of µ i If units have min and max capacities, some values of μ i are not feasible μ i is feasible if, for some m, jj JJii αα jj mm ββ jj mmmmmm μμ ii jj JJii αα jj mm ββ jj mmaaaa ω = 0.5*50+30 where αα mm jj is the number of batches in unit j for range m, ββ mmmmmm jj /ββ mmmmmm jj is the minimum/maximum capacity of unit j, and J i is the set of units for task i Otherwise, increase µ i to the nearest attainable amount µ * i µ i ω = 55 50% T1: 50 T2: 30 # of Batches 50% Capacity: U kg, U kg µ i = 55 µ i * = % 80%

42 Algorithm for General Networks 1. Demand: 15kg S4, 20kg S5, and 50kg S6 50% T2 15 S4 S1 S3 50% T1 20 S5 25% {U1,U2} T3 50 S6 75% S2 {U3} 2. T3 only produces S6: T2 produces both S4 and S5 50% T2: S4 S1 S3 50% T1 20 S5 25% {U1,U2} T3: S6 75% S2 {U3} 3. T2 and T3 both need S3 50% T2: S4 S1 S3 50% T S5 25% {U1,U2} T3: S6 75% S2 {U3} 4. T1 produces S3 50% T2: S4 S1 S3 50% T1: S5 25% {U1,U2} T3: S6 75% S2 {U3} U1 U # of Batches Capacity: U kg, U kg µ i = 55 µ i * =

43 Remarks Challenges A material can be produced by many tasks A task can be carried out in many units (of unequal capacity) Recycle streams Algorithm depends on network structure General Networks No Loops Loops No Recycle Materials Recycle Materials Single Loop Nested Loops T1 S1 S2 T2 T1 S1 S2 T2 T1 S1 T2 S1 T1 S2 T2 Use tear streams; iterate until bounds converge Use combination of backward and forward propagation for nested loops 32

44 The Algorithm General networks Recycle loops Recycle materials Nested loops 1. Set ν is =0 s S T, i I T I s + ; t s =0 s S T ; and S i C ={s:s S T S i + } 2. Set S NC =S, I NC =I, I A =, and S A =S F YES Is S RC =? NO 8. Set S RC =S A S R 9. Choose an s S RC and label it s* NO Stop: Infeasible YES Is t s 1 s S T? 7. t s =t s +1 s S T s.t. ν is >ρ is μ i i I T I s + 3. Calculate ω s s S A Is S A S R =? 4. Calculate ν is s S A, i I s + using LP i NO YES Is ν is ρ is μ i s S T, i I T I s +? YES NO 5. Set S NC =S NC \S A, S i C =S i C {s:s S i + S A }, S A =, and I A ={i:i I NC, S i C =S i + } 11. Calculate ψ is* and ν is* i I S* + and remove s* from S RC 10. Set μ ir = i I s* SL, ω s R = s S s* SL I RNC =I s* SL, S RNC =S s* SL, I RA =, and S RA ={s*} YES Is S RNC = & I RNC =? NO 12. Calculate ω sr s S RA (eqn. 10 or 14). Set S RNC =S RNC \ S RA, S RA =, and I RA ={i:i I RNC, S i - S RNC = } 13. Calculate μ ir i I RA (eqn. 11 or 16). Set I RNC =I RNC \I RA, I RA =, and S RA ={s:s S RNC, I s + I RNC = } 6. Calculate μ i and μ ĩ i I A. Set I NC =I NC \I A. I A = and S A ={s:s S NC, I s - I NC = } Stop YES Is I A = & S A =? NO Is S RA or S RNC =? NO 14. Set S RA ={s:s S R S RNC, (I s + I s* SL )\I RNC } and I RA ={i:i I RNC, (S i - S s* SL )\S RNC, S i - S s* SL >1}. Calculate ξ s s S RA YES

45 The Algorithm General networks Recycle loops Recycle materials Nested loops NO Stop: Infeasible YES Is t s 1 s S T? 7. t s =t s +1 s S T s.t. ν is >ρ is μ i i I T I s + Update Tear Streams 1. Set ν is =0 s S T, i I T I s + ; t s =0 s S T ; and S i C ={s:s S T S i + } 2. Set S NC =S, I NC =I, I A =, and S A =S F Backward Propagation 3. Calculate ω s s S A Is S A S R =? 4. Calculate ν is s S A, i I s + using LP i NO YES Is ν is ρ is μ i s S T, i I T I s +? NO YES Is S RC =? NO 11. Calculate ψ is* and ν is* i I S* + and remove s* from S RC Recycle Streams YES 8. Set S RC =S A S R 9. Choose an s S RC and label it s* 10. Set μ ir = i I s* SL, ω s R = s S s* SL I RNC =I s* SL, S RNC =S s* SL, I RA =, and S RA ={s*} Is S RNC = & I RNC =? NO 12. Calculate ω sr s S RA (eqn. 10 or 14). Set S RNC =S RNC \ S RA, S RA =, and I RA ={i:i I RNC, S i - S RNC = } YES 5. Set S NC =S NC \S A, S i C =S i C {s:s S i + S A }, S A =, and I A ={i:i I NC, S i C =S i + } 13. Calculate μ ir i I RA (eqn. 11 or 16). Set I RNC =I RNC \I RA, I RA =, and S RA ={s:s S RNC, I s + I RNC = } 6. Calculate μ i and μ ĩ i I A. Set I NC =I NC \I A. I A = and S A ={s:s S NC, I s - I NC = } Stop YES Is I A = & S A =? NO Is S RA or S RNC =? NO Computational requirements: avg = 0.26 sec, max = 4.3 sec 14. Set S RA ={s:s S R S RNC, (I s + I s* SL )\I RNC } and I RA ={i:i I RNC, (S i - S s* SL )\S RNC, S i - S s* SL >1}. Calculate ξ s s S RA YES

46 Tightening Constraints The algorithm calculates: ω s : μ i1 : minimum required amount of material s minimum cumulative production of task i Problem data β j max = maximum capacity of unit j ρ is+ = fraction of material s produced by task i Variable in all time-indexed formulations X ijt = 1 if task i starts in unit j at time t Tightening constraints 1A. Number of batches of task i jj JJii,nn XX iiiiii μμ 1 ii max jj JJ ii ββ jj mmmmmm 1B. Number of batches of all tasks producing material s ii IIkk +,jj JJ ii,nn XX iiiiii ωω kk 2A. Cumulative production of task i mmmmmm max ρρ ii II + iiii ββ jj kk jj JJ ii jj JJii,nn ββ jj mmmmmm XX iiiiii μμ ii ii 2B. Cumulative production of material k ii IIkk +,jj JJ ii,nn ρρ iiiiββ jj mmmmmm XX iiiiii ωω kk kk ii kk 34

47 Computational Results Discrete-time model Shah et al., 1993 (SPS) Testing library: 36 instances, 1800 CPU resource limit Four formulations F1: SPS F2: SPS + 1A&B F3: SPS + 2A&B F4: SPS + 1A&B + 2A&B Makespan minimization: 95% of problems solved to optimality by SPS Cost minimization: less than 40% of problems solved to optimality by SPS Average Time or Optimality Gap No Tightening 189 s 1.3% Tightening (1 &2) 1.4 s 2.4 s Optimal Solution Suboptimal Solution 2.1% 1.5% Problems Example 1 minimize Cost over 120 hours (δ = 1 hr S1 S5 S7 T1 T3 S2 80% S4 T2 20% 30% 70% S6 10% 90% T4 95% 5% S8 T5 S3 20% 80% T6 S9 S10 T7 S11 Without tightening: 2.22% gap after 16 hours With tightening: Solved to optimality in 5.8 sec 35

48 Computational Challenge Equivalent schedules Formed by shifting tasks earlier or later Have the same number of batches Have the same objective value In the B&B algorithm: Branching on X ijn leads to equivalent schedules There are millions of equivalent fractional solutions Bound does not improve The number of batches is a key feature Leads to schedules with different objectives U2 P1 F U1 I U3 P2 T1 T2 T3 Proc. Time (hr) U1 100 Cost ($/batch) U2 Capacity 20 (kg) Profit ($/kg) U U U U X ijn = U U U U U U ~600 more equivalent schedules 36

49 Parallel Branch-and-bound Algorithm 1. Branch on the sum of binaries for a task and unit over time 2. Solve nodes as MIPs in parallel 3. Prune node if: node is infeasible best node bound is worse than best integer solution 4. Branch again if node is not solved to optimality within time limit Integer solutions are shared among all nodes as soon as they are found Task 2 in Unit 1 XT2,U1,n = n =20 = =17 =18 X n T2,U3,n <8 =8 =9 =10 >10 Easy problems solved faster with standard MIP Branch-and-Bound Hard problems are more likely to be solved with Parallel Branch-and-Bound Goal: Branch on # of batches, but without overhead of parallel method Velez & Maravelias, Computers & Chemical Engineering, 55, Task 2 in Unit 3 Solved in parallel 37

50 Reformulation Introduce New Variable & Defining Equation N ij = number of batches of each task i executed on unit j NN iiii = nn XX iiiiii Tried various reformulations using SOS1 binaries ( bb BBiiii bbzz iiiiii = nn XX iiiiii ) Branching Alternatives No priorities High priority on N ij High priorities on X ijn Specific N ij priorities calculated from LP-relaxation at each node Various priorities with and without strong branching with SOS1 variables Summary of Results Using N ij was faster than Z ijb by a factor of ~2 Using priorities on X ijn was worse than using no priorities For cost/makespan, using or not using priorities on N ij gave similar results For profit, using priorities on N ij was faster than no priorities by 12% Using priorities on N ij gave the best results Results presented for N ij reformulation without priorities 38

51 Branching on N ij Example: Maximize profit over η = 120 hr (δ = 1 hr) F U1 I U2 U3 P1 P2 T1 T2 T3 Proc. Time (hr) Cost ($/batch) Capacity (kg) Profit ($/kg) LP-Relaxation: $689.7 Optimal Solution: $686 Integrality gap: $3.7 Original formulation 10 hours, 10 million nodes Bound improves from $689.7 to $689.5 Only closes ~5% of integrality gap Reformulation Route node: z LP = 689.7; N T3 = 28.5 Branching once on N T3 improves bound to $686.4 (closes gap by 89%) Branching twice improves bound to $686 Closes 100% of integrality gap in 5 nodes CPLEX solves to optimality in <1 second $686.4 NN 2 TTT = 22.6, 2 NN 2 TTT = 57, NN 2 TTT = 28 N T1 22 $689.7 NN 1 1 TTT = 22.8, NN 1 TTT = 57, NN 1 TTT = 28.5 N T3 28 N T1 23 N T $684.2 NN 3 TTT = 22.8, NN 3 TTT = 56, NN 3 TTT = 29 $ NN 4 TTT = 22, 4 = 57, NN 4 TTT = 26.5 NN TTT 5 $686 NN 5 TTT = 23, 5 = 57, NN 5 TTT = 28 NN TTT 39

52 Computational Results STN Testing library: 8 instances; 120-hr horizon ; 1-hr time step Settings: 3 hour resource limit ; default CPLEX settings Makespan Minimization Cost Minimization Profit Maximization Original Formulation 1121s Reformulation 85.1s 1.3% 1944s Original Formulation 1.6% 1.4s 0.8% Reformulation 0.7% 0.2s 7.5s Original Formulation 1.6% 2.8s 1.4% Reformulation 1.1% 1.5s 987s Instances Instances Instances Solution Time (s) Original Formulation Reformulation Instances Solution Time (s) Original Formulation Reformulation Instances Solution Time (s) Original Formulation Reformulation Instances 40

53 Computational Results RTN Testing library: 8 instances; 120-hr horizon ; 1-hr time step Settings: 3 hour resource limit ; default CPLEX settings Makespan Minimization Cost Minimization Profit Maximization Original Formulation 1.6% Original Formulation Original Formulation 1.0% 1551s 1.0% 2.5% 1.3% 1.4s 17.6% Reformulation 1.6% Reformulation Reformulation 1.5% 100s 220s 11.5s 0.6% 1.4s 340s Instances Instances Instances Solution Time (s) Original Formulation Reformulation Instances Solution Time (s) Original Formulation Reformulation Instances Solution Time (s) Original Formulation Reformulation Instances 40

54 Example 1 Example: Maximize profit over η = 240 hr; δ = 1 hr 2,372 Binary variables 6,305 Continuous variables 11,810 Constraints Original formulation: Runs out of memory after 40 hours (1.1% gap) Reformulation: Solves in 12.6 s F1 U1: U4: % F2 T1 T7 5% U2: 2-8 U3: U5: S1 S6 S4 T2 U6: 1-4 S2 90% 50% T8 IN2 10% Up to 59 batches run in a single unit 128 time points would be required in a continuous-time formulation P1 T5 S3 T4 T3 IN1 50% S6 T9 98% P2 T6 2% WS P3 T10 Papageorgiou & Pantelides, Industrial and Engineering Chemistry Research, 35,

55 Example 2 F1 F2 F3 F4 F6 F5 T20 T10 T30 T31 T60 T61 S10 S30 S31 80% S60 S61 S20 75% T11 20% 25% T62 T32 15% IN1 T21 60% 40% 50% IN2 50% IN3 35% T22 S21 85% T40 T50 S22 S40 S50 T23 T41 T51 P1 P2 P3 IN4 S70 S71 P4 T70 T71 T72 65% 95% 5% S72 Resource1 Resource2 U1 (1.2-6) U2 (1-5) U3 (1.4-7) U4 (1.4-7) U5 (1.6-8) U6 (1.2-2) U7 (1.4-7) U8 (1.6-8) No Storage 20 kg Inventory Capacity 50 kg Inventory Capacity Unlimited Storage Utility availability & cost, deliveries, and orders: Amount R1: Availability R2: Availability R1: Cost R2: Cost Cost ($/amount) Feed Delivery (variable amounts) Orders Due(variable amounts) Objective: max Profit, while filling customer orders Papageorgiou & Pantelides, Industrial and Engineering Chemistry Research, 35,

56 Example 2 Original model runs out of memory after 2.5 days (0.6% optimality gap) Reformulation is solved in 3.5 minutes Inventory B. Inventory Profile R2: Usage R2: Availability R1: Usage R1: Availability R1: Cost C. Resource R2 Profile C. Resource R1 Profile Amount U1 U2 U3 U4 U5 U6 U7 U F1 F4 F5 P1 P3 P A. Gantt Chart Cost ($/amount) 43

57 Concluding Thoughts General framework Notation; problem classes; classification of modeling approaches New scheduling models Sequential environment Simultaneous batching and scheduling with storage and utility constraints General scheduling model Problems in all/combined production environments under wide range of processing characteristics and constraints Solution methods Constraint propagation for tightening constraints Reformulation and branching methods Orders of magnitude reduction in computational requirements Can be applied to all time-indexed MIP scheduling formulations 44

58 Concluding Thoughts Where do we go next? Solution methods for nonlinear models Online scheduling Simply solving to optimality can lead to poor schedules (even in the deterministic case) What are set-point trajectories in scheduling? Do we need them? Are recursive feasibility and stability relevant/useful? Do we need terminal region/penalties? How can we generate them systematically? What can we learn from process control? Acknowledgements Modeling: Arul Sundaramoorthy Solution methods: Sara Velez Research supported by National Science Foundation: CBET

59 Questions?