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Mixed-integer Models for Planning and Scheduling - Ignacio E. Grossmann

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This document discusses optimization models for planning and scheduling in chemical processes. It provides an overview of mixed-integer models and their applications. Specifically, it describes models for single and multi-stage batch processes with parallel units and due dates. This includes mixed-integer linear programming (MILP) models to optimize scheduling and assignment for minimizing makespan and costs while meeting due dates. The document also discusses generalized disjunctive programming and constraint programming approaches for more complex scheduling problems.

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1 Mixed-integer Models for Planning and Scheduling Ignacio E. Grossmann Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213 University of Alicante January 16-17, 2018
2 1. Introduction 2. Modeling 3. Solution strategies General Plan Short Course Goals 1. Gain an appreciation for optimization-based planning/scheduling 2. Learn general guidelines for modeling and solving these problems
3 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 . Enterprise-wide Optimization (EWO) Key issue: - Integration of information, modeling and solution methods
4 Source: Tayur, et al. [1999] Enterprise Resource Planning System Materials Requirement Planning Systems Distributions Requirements Planning System Transactional IT External Data Management Systems Strategic Optimization Modeling System Tactical Optimization Modeling System Production Planning Optimization Modeling Systems Logistics Optimization Modeling System Production Scheduling Optimization Modeling Systems Distributions Scheduling Optimization Modeling Systems Analytical IT Demand Forecasting and Order Management System Strategic Analysis Long-Term Tactical Analysis Short-Term Tactical Analysis Operational Analysis Scope Methods/Tools for Enterprise-wide Optimization
5 Enterprise-wide Optimization • The supply chain is large, complex, and highly dynamic • Optimization can have very large financial payout WellheadWellhead PumpPump TradingTrading Transfer of Crude Transfer of Crude Refinery Optimization Refinery Optimization Schedule Products Schedule Products Transfer of Products Transfer of Products Terminal Loading Terminal Loading Petroleum industry Dennis Houston (2003)
6 • Pharmaceutical process (Shah, 2003)  Primary production has five synthesis stages  Two secondary manufacturing sites  Global market Lifecycle Management 0.5 - 2 yrs 1 - 2 yrs 1.5 - 3.5 yrs 2.5 - 4 yrs 0.5-2 yrs Discovery Market 2-5 yrs Submission& Approval 10-20 yrs Phase 3Phase 2a/bPhase 1 Pre- clinical Development Targets Hits Leads Candidate R&D Pharmaceutical industry Pharmaceutical supply chain (Gardner et al , 2003)
7 -The modeling challenge: Planning, scheduling, control models for the various components of the supply chain, including nonlinear process models? Research Challenges - 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: How to effectively anticipate effect of uncertainties ? - Algorithmic and computational challenges: How to effectively solve the various models including nonconvex problems in terms of efficient algorithms, and in terms of modern computer architectures?
8 Planning and scheduling Given existing plant/supply chain and product demands over time, Determine production & inventory targets, time activities Objectives: Economics (Profit, Cost) Feasibility (Just-in-time) No clear cut distinction between Planning and Scheduling Planning: Usually driven by economics, longer term Scheduling: Usually driven by feasibility, short term Long-term goal: Integration of both
9 Overview of Discrete-Continuous Optimization Models a. Linear and mixed-integer linear programming b. Nonlinear and mixed-integer nonlinear programming c. Logic-based optimization: Generalized disjunctive programming Constraint programming Lagrangean decomposition Outline Examples of Models Single stage, parallel units wo/w due dates. Multistage, parallel units, due dates. Aggregated Multistage Zero-Wait Discrete time MILP short-term batch scheduling (State Task Network) Continuous time MILP short-term batch scheduling (State Task Network) MINLP multistage cyclic scheduling model Refinery scheduling and blending, Multiperiod refinery planning MILP Supply Chain Optimization Model NLP multiperiod production planning model with process models Supply Chain with Stochastic Inventory Management
10 References Mendez, C.A., J. Cerda, I.E. Grossmann, I. Harjunkoski, M. Fahl, “State-of-the-art Review of Optimization Methods for Short-term Scheduling of Batch Processes,” Comp. & Chemical Engineering 30, 913-946 (2006). 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). Kallrath, J. “Planning and scheduling in the process industry,” OR Spectrum, 24, 219-250 (2002). Pinto, J.M. & Grossmann, I.E., “Assignments and sequencing models of the scheduling of process systems,” Annals of Operations Research, 81, 433 – 466 (1998). Shah, N., “Single- and multisite planning and scheduling: Current status and future challenges,” Proceedings of FOCAPO-98 75 – 90 (1998). Mauderli. A. M.: Rippin. D. W. T. Production Planning and Scheduling for Mu1tip;;pose Batch Chemical Plants. Comp. Chem.Eng. 3, 199 (1979). Reklaitis, G. V. Review of Scheduling of Process Operation. AIChE Symp. Ser. 78, 119-133 (1978). 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,” Computers and Chemical Engineering, 62, 161-193 (2014).
11 ROAD-MAP FOR BATCH SCHEDULING (1) TASK TOPOLOGY: - Single Stage (single unit or parallel units) - Multiple Stage (multiproduct or multipurpose) - Network (2) EQUIPMENT ASSIGNMENT - Fixed - Variable (3) EQUIPMENT CONNECTIVITY - Partial - Full (4) INVENTORY STORAGE POLICIES - Unlimited intermediate storage (UIS) - Non-intermediate storage (NIS) - Finite intermediate storage (FIS): Dedicated or shared storage units - Zero wait (ZW) (5) MATERIAL TRANSFER - Instantaneous (neglected) -Time consuming (no-resource, pipes, vessels) A B C 1 2 3S1 S2Heat Reaction1 Separation Reaction 3 S3 S5 S4 S7 S6 Reaction2 1h 1h 3h 2h 2h 90% 10% 40% 60%70% 30%
12 ROAD-MAP BATCH SCHEDULING (6) BATCH SIZE: - Fixed - Variable (mixing and splitting operations) (7) BATCH PROCESSING TIME - Fixed - Variable (unit / batch size dependent) (8) DEMAND PATTERNS - Due dates (single or multiple product demands) - Scheduling horizon (fixed, minimum/maximum requirements) (9) CHANGEOVERS - None - Unit dependent - Sequence dependent (product or product/unit dependent) (10) RESOURCE CONSTRAINTS - None (only equipment) - Discrete (manpower) - Continuous (utilities) 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon i i’changeoveri’
13 ROAD-MAP BATCH SCHEDULING (11) TIME CONSTRAINTS - None - Non-working periods - Maintenance - Shifts (12) COSTS - Equipment - Utilities (fixed or time dependent) - Inventory - Changeovers (13) Degree of certainty - Deterministic - Stochastic
14 ROAD-MAP FOR OPTIMIZATION APPROACHES (A) TIME DOMAIN REPRESENTATION - Discrete time - Continuous time (B) EVENT REPRESENTATION DISCRETE TIME - Global time intervals CONTINUOUS TIME - Time slots - Unit-specific direct precedence - Global direct precedence - Global general precedence - Global time points - Unit- specific time event (C) MATERIAL BALANCES - Lots (Order or batch oriented) - Network flow equations (STN or RTN problem representation) (D) OBJECTIVE FUNCTION - Makespan - Earliness/ Tardiness - Profit - Inventory - Cost 7 6 5 4 3 2 1 9 8 19 18 11 10 2 2 21 20 25 24 23 TIME TASK TIME EVENTS TASK TIME TASK
15 Optimization approaches Mixed-integer (non)linear programming (MILP/MINLP) Disjunctive programming/logic optimization Constraint logic programming Combinatorial optimization (traveling salesman) Local search methods (simulated annealing, genetic algorithms) Decomposition techniques: Lagrangean decomposition
16 Mathematical Programming min f(x, y) Cost s.t. h(x, y) = 0 Process equations g(x, y)  0 Specifications x X Continuous variables y {0,1} Discrete variables Continuous optimization Linear programming: LP Nonlinear programming: NLP Discrete optimization Mixed-integer linear programming: MILP Mixed-integer nonlinear programming: MINLP
17 Generalized Disjunctive Programming (GDP)     Ω ,0)( 0)( )(min 1 falsetrue,Y Rc,Rx trueY Kk γc xg Y Jj xs.t. r xfcZ jk k n jkk jk jk k k k                      • Raman and Grossmann (1994) (Extension Balas, 1979) Objective Function Common Constraints Continuous Variables Boolean Variables Logic Propositions OR operator Disjunction Fixed Charges Constraints Multiple Terms / Disjunctions
18 Constraint Programming Key Features  Variables: Integer, Boolean, Continuous  Constraints • Algebraic constraints: x1 + x2 ≤ 5 • Logical constraints: [a.end ≤ b.start]  [b.end ≤ a.start] • Global constraints: alldifferent(x) • Meta-constraints: iS (xi < 5) = 3  State Objective Function (optional)  Write an algorithmic search for finding values of the variables satisfying the constraints • Implicit enumeration • Constraint propagation – domain reduction
19 Problems Multistage, parallel, due dates MILP Multistage Aggregate Zero-Wait IP/LP Multipurpose, resource constraints: State-Task-Network (Resource Task Network) Discrete time MILP Continuous time MILP Single stage, parallel MILP Single stage parallel, due dates MILP Batch Continuous Multistage, single, cyclic schedule MINLP Multiperiod production planning NLP Refinery scheduling/blending Iterative MILP Multiperiod Supply Chain MILP Stochastic Inventory Magmt. MINLP
20 Given N tasks (i) and M units (m) Tasks (Jobs/Orders) = batches fixed lot size, fixed processing times pim Fi ={m | units m can perform task i} N ={ (i1,i2) cannot share same unit} Find how to assign tasks j to machine m to minimize completion time MS Parallel units (single stage) Unit m Task 1 Task 2 . Task N Processing Time Machine m TimeTime
21 xim assign task i unit m (0-1) Min Makespan Makespan is largest time for every unit m Every task i to one unit m MILP Optimization Model }1,0{,0 1 min      im m im im i im xMS ix mMSxpst MS
22 Parallel Units (Single Stage) with Due Dates (Jain and Grossmann, 2001) Given: N jobs/orders (release dates, processing times, due dates) M units (cost different for each unit) Find schedule that minimizes cost and meets all due dates Job 1 Job 2 Job n Time Unit 1 Unit m Time Release date Due date Processing time ri di pim
23 MILP Optimization Model itasktimestartts otherwise munittoitaskif x iim      0 1 Mmrdpx Iix Iixpdts rtsts xC iiii Ii imim Mm im Mm imimii ii Ii Mm imim               }{min}{max 1 .. min Cost processing Earliest start Latest start Assign to only one unit Redundant constraint
24 Sequencing tasks in each unit trueareyORythentruexANDxIf unitgivenonitaskbeforeitaskifyLet iiiimiim ii ''' ' '1 imiiii ptststhenyIf  '' 1 MmiiIiixxyy miimiiii  ,',',1''' ',',)1( '' iiIiiyMxptsts iiim Mm imii   Big-M Constraint 0},1,0{},1,0{ '  iiiim tsyx MILP CPLEX 6.5 3 tasks, 2 units 0.04 sec 12 tasks, 3 units 926 sec 20 tasks, 5 units 18,000 sec
25 Short Term Scheduling Multistage Plants Stage I II III IV ¬ Individual customer orders 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon (ICI) Pinto, Grossmann (1995)
26 Given: ¬ Product demands and deadlines ¬ Topology of the plant ¬ Processing times of orders in units ¬ Set-up times Determine: ¬ Assignment of orders to units ¬ Timing of operations of units Objective Minimize Earliness Deliver as close as possible to deadline Just-in-time Short Term Scheduling Problem
27 ¬ All data are deterministic and fixed over the time horizon of interest. ¬ Set-up times are only equipment dependent. ¬ Batch sizes are fixed parameters. ¬ Each order must be processed by exactly one unit per stage (No splitting). ¬ No resource constraints considered ¬ Unlimited intermediate storage between stages Parameters di due date of order i SUj set-up time in unit j Tij processing time of order i in unit j Assumptions
28 Tsjk starting time in unit j during time slot k Tejk finishing time in unit j during time slot k Tsiiil starting time for order i in stage l Teiiil finishing time for order i in stage l Wijkl 0-1 variable that assigns order i stage l to unit j slot k . . . Order 1 2 NO 1 2 3 1 2 1 2 Unit 2 Unit NU time Time Slot Set-up Processing time Unit 1 . . . Representation and Variables
29 MILP Model Assignment Wijkl kSLj  j   1 i,l Si Wijkl lSi  i   1 j, k SLj Every order i, stage l Every unit j, slot k Tejk  Tsjk 1 j, k SLj  K j   iiilil LSliTsiTei   ,1 TeiiLi  di i Due date Unit slots Order stages Timing   j i Sl jijijkljkjk SLkjSUpWTsTe i   ,   i j SLk jijijklilil SliSUpWTsiTei j    , End/Start Unit End/Start Order
30 MILP Model Time Matching jkilijkl TsiTsiWIf 1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1 Big-M Other j i Sl lijk i Sl ijkl SLkjWW ii     ,,1 Objective: Minimize Earliness iiL i iTeihMax  Maximize end times order last stage L symmetry breaking
31 Reformulation Time matching constraints :   ijjkilijkl SlSLkjiTsTsiWM  ,,,1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1 i j k ijkl j k jkijklil SliTsWTsi   , jjk i l ijkljk SLkjTs   , ijijklijkl SlSLkjiWU  ,,,.0  jjkjk SLkjyU  ,. ¬ Fewer constraints ¬ Tighter LP relaxation Alternative formulation: • Multiply and perform exact linearization • Disaggregated variables and apply assignment constraints )( jkilijkl TsTsW  ijkl Reformulation
32 Heuristic 1. Determine optimal sequence for each unit independently Apply single machine scheduling method 2. Enforce order in unit slots, but not requiring use of all slots 1 5 2 1 3 5 2 4 6 empty slots Unit j Optimal sequence for all 6 slots If orders 1, 5, 2 assigned unit j Actual sequence 1, 5, 2 Preordering has effect of fixing many Wiljk to zero
33 STEP 1. Minimize In-Process time (residence time) Determine assignments Overestimation of set-up times MILP - Zero gap - Interior point method Algorithm (with preordering) A A B C A A B C LP -Feasibility guaranteed from step 1 -Bound on optimum easily determined - STEP 2. Minimize Earliness Fixed assignment Eliminate unnecessary set-up times
34 STBS- CAPD Software Visual Basic Interface (Valli, 1998) Example: 25 units, 8 orders Solves in few seconds!
35 Assignments of orders to units
36 Optimal Sequencing Multistage Plants ZW policy M Stages NB Batches Given NB Batches with processing times pij and sequence dependent changeover times τimj Find optimal sequence that minimizes Cycle Time Key observation: can define sequences in terms of slacks Birewar, Grossmann (1990)
37 5 2 3 1 5 3 A B (a) Sequence A - B 5 2 3 1 5 3 A B (b) Slack times with zero clean-up times SL = 1 SL = 0 SL = 1 AB1 AB2 AB3 (c) Slack times with clean-up times SL = 0 SL = 0 SL = 1 AB1 AB2 AB3 5 2 3 1 5 3 A B Slack Determination 2 2 1 2
38 1 2 3 4 5 Asymmetric Traveling Salesman Problem ATSP 1 0 il if batchi followed by batchl y otherwise    
39 Cycle time stage 1 1 1,...il l y i NB  1 1,...il i y l NB  Only one out Only one in il NB i NB l il NB i i ySLpCT     1 1 1 1 1
40 Subtours may occur 1 2 3 4 , 1 ,il i Q i Q Let B set batches Q subset of batches y Q B Q         Above Q = {1,2}, Q = {3,4}
41 1 1,...il l y i NB  1 1,...il i y l NB  1 ,il i Q i Q y Q B Q         0,1ily  Integer Program ATSP il NB i NB l il NB i i ySLpCT     1 1 1 1 1min
42 Aggregation Model Consider ni batches for product i, i=1, NP Let NPRim number batches product i followed by product m 1 1 1 1 1 min NP NP NP i i im im i i m CT n p SL NPR        1,...im i m NPR n i NP  1,...im m i NPR n m NP  1 1,ii iNPR n i NP   0,1,2,..iNPR  Can be solved as LP treating NPRi  0 (as continuous) CYCLE http://newton.cheme.cmu.edu/interfaces
43 A B C D 4 3 3 3 2 5 Example 4 products, 20 batches nA = 7 nB = 5 nC = 3 nD = 5 Solution from LP Eg NPRBA = 3 NPRAA = 4
44 3 A B C D 4 3 3 3 3 A B C D 3 3 3 3 B D 2 2 + B -> D -> B -> D -> B A 4 B -> D -> B -> D -> B A -> A -> A-> A-> A + C -> D -> B -> A -> C -> D -> B -> A -> C -> D B -> D -> B -> D -> B A -> A -> A-> A-> A Loop Breaking Procedure
45 Multipurpose Plants Given: • the time horizon • the available units and storage tanks, and their capacities • the available utilities (steam, cooling water) • the production recipe (mass balance coefficients, utility requirements) • the prices of raw materials and final products Consider: Arbitrary topology Flows (no fixed batch size) Different transfer/Intermediate Storage Policies Resource Constraints Determine • the sequence and the timing of tasks taking place in each unit • the batch size of tasks (i.e. the processing time and the allocated resources) • the amount of raw materials purchased and final products sold
46 State Task Network (STN) (Kondili, Pantelides, Sargent, 1993) S1 S2Heat Reaction1 Separation Reaction 3 S3 S5 S4 S7 S6 Reaction2 1h 1h 3h 2h 2h 0 1 2 3 4 5 6 Time (h) Reactor 1 Reactor 2 Reactor 3 Column Reaction 2 Reaction 3 Separation Reaction 1Heating Note: Related Resource Task Network (RTN)
47 Discrete-time STN Model (1) Variables: Wijt = 1 if unit j starts processing task i at the beginning of time period t; 0 otherwise. Bijt = Amount of material which starts undergoing task i in unit j at the beginning of period t. Sst = Amount of material stored in state s, at the beginning of period t. Uut = Demand of utility u over time interval t. pij = 3 Task i starts at t=2 in unit j Wij2 = 1, Bij2  0 Wijt = 0, Bijt = 0, t2 2 3 4 5 T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) Allocation Constraints: tjW jIi ijt ,1    tjIiWMW jijtij Ii pt tt jti j ij ,,11 ' 1 ' ''    
48 iijijtijtijijt KjtiVWBVW  ,,maxmin tsCST sst ,0  Capacity limitations: Objective Function:      H t utut us HsHs H t st R st s H t st D st s UCSCRCDCZ 1 1,1, 11 max Batch Units Storage capacity tsDRBBSTST stst Kj ijt Ti is Kj ptij Ti isstst isi is s ,,1        Material balances: Inventories Produced Consumed Purchased/Sold tuBWU i i p ijtuiijtui Kjt ut ,)( 1 0          tuUU utut ,0 max  Availability of utilities: Linear function batch size Discrete-time STN Model (2)
49 Reformulation Previous MILP is expensive to solve tjW jIi ijt ,1    tjIiWMW jijtij Ii pt tt jti j ij ,,11 ' 1 ' ''     Culprit: big-M allocation constraints tjW i j pt tt tij Ii ,1 1 ˆ ˆ    Solution: Replace by constraint below (Shah, 1992) Fewer and tighter ! 2 hr 3 hr Unit j Tasks iIj t
50 Reaction 2 Reaction 1 Heating Reaction 3 Separation Product 1 Feed A Hot A Int BC Feed B Feed C Impure C Int AB Product 2 Classical Kondili Example MILP 72 0-1 variables 179 continuous variables 250 constraints STN 2003 CPLEX 7.5: 0.45 sec, 22 nodes, IBM-T40 1 2 3 4 5 6 7 8 9 10 Heating Reaction 1 Reaction 2 Reaction 3 Separation Heater Reactor 1 Reactor 2 Reactor 1 Reactor 2 Reactor 1 Reactor 2 Still 52 20 52 80 50 56 80 50 50 80 50 80 50 130 Optimal Schedule 1987 Kondili’s B&B: 908 sec, 1466 nodes, Vax-8600 1992 Shah’s B&B: 119 sec, 419 nodes, SUN Sparc STN http://newton.cheme.cmu.edu/interfaces
51 • Main idea  Treat the production recipe as a set of (tasks) that transform a set of resources into another set • Tasks, set I  e.g. Fill, Heat, Mix, Store, React, etc.  Represented as a rectangle • Resources, set R  e.g. equipment, material, utilities, equipment state, material location  Represented as a circle Compact Equations: Resource balances Resource Task Network Pantelides (1994) Castro et al. (2001, 2004)
52 ReactionA B Reactor C 1.0 0.7 0.3 Pack_D11 Rate=0-5.5714 Int1 Line3 PD11 S1 S2 T1_R1 Duration= 2 h T1_R4 Duration= 2 h R1 R4 T2_R4 Duration= 2 h T2_R1 Duration= 2 h S4 T3_R3 Duration= 4 h T4_R2 Duration= 2 h S5 S6S3 0.6 0.4 0.6 0.4 R2 T4_R3 Duration= 2 h R3 T3_R2 Duration= 4 h Legend: Discrete interaction: Continuous interaction: Examples of RTNs
53 • Model entities  Variables  Ni,t- assigns the start of task i to time point t (binary variable)  i,t- amount of material processed by task i at time point t (nonnegative continuous variable)  Rr,t- excess amount of resource r at time point t (nonnegative continuous variable)  Parameters (related to RTN structure)  Amount of resource r consumed/produced by task i at a time  relative to the start of the task – i,r,- usually linked to equipment resources and N – i,r,- usually linked to material resources and  – + sign indicates production, - indicates consumption Discrete-time RTN formulation
54 • Derive the RTN and model parameters for the following production recipe  Consider a reaction task i, that lasts 5 hours. It converts material A to B. It is carried out in a reactor. It uses 0.25 kg/s of steam per t of material being processed during the first hour. Then it used 2 kg/s of cooling water per t of material being processed until the end of the operation  Assume time intervals of one hour: =1 h Example RTN Discrete Model
55 Resource balances Resource limits Capacity constraints MILP Model TtRrNRRR tr Ii tiirtiirttrtrtr i     ,)( , 0 ,,,,,,11,1 0 ,     TtRrRR r,tr,t  ,0 max TtIiVNVN r Rr rititir Rr riti EQ i EQ i    ,max ,,,, min ,,,  
56 • Nonzero parameters  Reactor: i,R,0=-1; i,R,5=1  Materials: i,A,0=-1; i,B,5=1  Utilities: i,S,0=-0.25; i,S,1=0.25; i,CW,1=-2; i,CW,5=2 Reaction Duration=5 h A B RS CW 0.25 2 RTN model and parameters
57 Time Representations T1 T2 T3 1 2 32 0 1 2 3 4 5 6 7 8 t (hr) 3 2T1 T2 T3 Event-Based Representation 2 hr 1.5 hr 3 hr Interval = 0.5 hr Discrete Time Representation T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) • Fixed time points • Large size • Constant processing times Continuous Time Representation II Continuous Time Representation I 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 • Small size • Variable proc. Times • Unknown time points niWsHTTs innin  ,)1( Postulate no of points Iterative scheme for optimal niWsHTTs innin  ,)1(
58 Continuous-time STN Model I (Maravelias, Grossmann, 2003) 1. Common mixed continuous-time representation  Fewer time points 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 • If produce ZW-state finish at time point • Else finish at or before time point 2. Unit-task decoupling and New Assignment Constraints  Fewer binaries (Ierapetritou, Floudas, 1998) 4. Addition of New Valid Inequalities  Tighter LP Relaxation 3. Utility Constraints RI1 RI2 RII 0 2 4 6 8 t (hr) 0 2 4 6 8 t (hr) CWMAX CW R2-40 R2-46.4 R1-40 R1-49.6 R1-40 R4-72 R3-40 R3-60.8 (kg/min) 40 20
59 Continuous-time STN Model II Zsjn = 1 if a task in I(j) starts in unit j at time point n Zpjn = 1 if a task in I(j) is being processed in unit j at time point n Zfjn = 1 if a task in I(j) finishes in unit j, at or before time point n Wsin = 1 if task i starts at time point n Wpin = 1 if task i is being processed at time point n Wfin = 1 if task i finishes at or before time point n Assignment Constraints jnjn ZpZs  If a task is assigned to start in unit j at time point n, then equipment j at time point n is not processing any other task Logic Condition: njWsZs in jIi jn   , )( njWfZf in jIi jn   , )( njZfZsZp nn jn nn jnjn    , ' ' ' ' 0 2 3 … n-1 n 0 2 3 … n-1 n Reaction Reaction 1 Reaction 2Reaction Reaction 1 Reaction 2 Task-Unit Decoupling: Define new tasks njWfWs jIi nn inin    ,1)( )( ' '' iWfWs n in n in   njWs jIi in  ,1 )( njWf jIi in  ,1 )( iWfi  00 iWsN  0|| Integer assignment constraints
60 A WsA2 WfA4 Tn=Ts 0 2 6 8 10 n 1 2 3 4 5 D 0 6 0 0 0 Tf 0 8 8 8 8 A Tn Time that corresponds to time point n (i.e. start of period n; finish of period n-1) Tsin Start time of task i that starts at time point n Tfin Finish time of task i that starts at time point n Din Duration of task i that starts at time point n Timing Constraints Continuous-time STN Model III niTTs nin  , ni TfTf D Ws DTsTf BsD Ws inin in in ininin iniiin in                             ,,0 1  ni TfTf Wf ZWiTTf ZWiTTf Wf inin in nin nin in                       ,, , , 1 niBsWsD iniiniin  , niWsHDTsTf inininin  ,)1( niWsHDTsTf inininin  ,)1( niTTs nin  , niWfHTTf innin  ,)1(1 niZWiWfHTTf innin  ),()1(1 niWsHTfTf ininin   ,1 niDTfTf ininin   ,1 MIP Constraints (big-M)
61 S3 40% 60% 75% S1 S2 S4 BO A,S3,4=5 BO A,S4,4=15 BsA2=BfA4=20 BI A,S1,2=8 BI A,S2,2=12 25% Batch Size Constraints and Mass Balances niWsBBsWsB in MAX iinin MIN i  , niWfBBfWfB in MAX iinin MIN i  , )(,, iSIsniBsB inis I isn   )(,, iSOsniBfB inis O isn   1, )()( 1,     nsBBSSSS sIi I isn sOi O isnnssnsn nsCS ssn  , Utility Constraints nriBsWsR inirsinir I irn  ,, nriBfWfR inirsinir O irn  ,, nrRRRR i I irn i O irnrnrn    ,11 nrRR MAX rrn  , RI1 RI2 RII 0 2 4 6 8 t (hr) 0 2 4 6 8 t (hr) CWMAX CW R2-40 R2-46.4 R1-40 R1-49.6 R1-40 R4-72 R3-40 R3-60.8 (kg/min) 40 20 WsA2, BsA,2 WfA4, BfA2 A CW Continuous-time STN Model IV
62 Continuous-time STN Model V Addition of new valid inequalities  Tighter LP relaxation DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H    )( jIi n in jHD DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H     )( ' ' , jIi n nn in njTHD      )( ' ' ' ,)( jIi nini nn ini njTBfWf  Smaller Bs’ Lower Production Lower Profit  STN http://newton.cheme.cmu.edu/interfaces
63 MILP Formulation Novel Tightening Constraints Utility Constraints Novel Assignment Constraints njWfWs jIi nn inin    ,1)( )( ' '' iWfWs n in n in   niBsWsD iniiniin  , niWsHDTsTf inininin  ,)1( niWsHDTsTf inininin  ,)1( niTTs nin  , niWfHTTf innin  ,)1(1 niZWiWfHTTf innin  ),()1(1 niWsBBsWsB in MAX iinin MIN i  , niWfBBfWfB in MAX iinin MIN i  , niBfBpBpBs inininin   ,11 )(,, iSIsniBsB inis I isn   )(,, iSOsniBfB inis O isn   1, )()( 1,     nsSSSPBBSS snsn sIi I isn sOi O isnnssn nriBsWsR inirsinir I irn  ,, nriBfWfR inirsinir O irn  ,, nrRRRR i I irn i O irnrnrn    ,11    )( ' ' , jIi n nn in njTHD    )( ' ' ' ,)( jIi niin nn iin njTvdBffdWf Finish time of task i Mass balances  s ssnSSZ max
64 Example T1 T2 T3 T4 T5 T7 T8 T6 T9 T10 F1 S1 S2 INT1 S3 P1 P2 WS P3 F2 S5 INT2 S4 ADD S6 0.95 0.05 0.1 0.9 0.5 0.5 0.98 0.02 U2 U1 U3 U4 U5 U6 Unlimited Storage Finite Storage No Intermediate Storage Zero-Wait Cooling Water Low Pressure Steam High Pressure Steam
65 Results U1 U2 U3 U4 U5 U6 0 2 4 6 8 10 12 t (hr) T1 T1 T1 T2T2 T2 T3 T3 T3 T4 T5 T4 T7 T8 T9 T10 T5 0 10 20 30 40 50 0 2 4 6 8 10 12 CW-MAX HPS-MAX CW LPS Z = $13,000 3067 Constraints 180 Binary Variables 1587 Continuous Variables LP Relaxation 19,500 CPU s (CPLEX 7.5) 62.8 Nodes 2,107
66 |T| Binary variables Total variables Constraints RMIP MIP CPUs Nodes 5 440 511 873 154.17 184 1748 328357 Multistage multiproduct plant: Total earliness minimization. CPLEX 11.1 on Intel Core2 Duo T9300 @2.5 GHz, Windows Vista |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 Discrete-time Continuous-time Was 45,520 s with CPLEX 10.2 on Pentium 4, 3.4 GHz Progress in Discrete time makes Continuous less Pressing Castro (2008)
67 Continuous multistage plants ••• Stage 1 Stage 2 Stage M Product 1 2 NP • • • • • • • • • • • • • • • Cyclic schedules (constant demand rates, infinite horizon) Intermediate storage Given : N Products Transition times (sequence dependent) Demand rates Determine : PLANNING Amount of products to be produced Inventory levels SCHEDULING Cyclic production schedule Sequencing Lengths of production Cycle time Objective : Maximize Profit = + Sales of products - inventory costs - transition costs Pinto, Grossmann (1996)
68 Optimal length of cycle determined largely by transition and inventory costs 2001000 0 100 200 300 400 500 Length cycle (hrs) $/hr sales of products profit transition costs inventory costs Critical to model properly inventory levels and transition times Optimal Trade-offs
69 a) Assignments of products to slots and vice versa iy k ik  1 ky i ik  1 Assumptions : 1. Each product is processed in the same sequence at each stage 2. Each stage consists of a single production line 3. The schedule is cyclic time Stage 1 Stage 2 Stage M k = 1 k = 2 k = NP k = 1 k = 2 k = NP k = 1 k = 2 k = NP ••• ••• ••• ••• Transition Processing Time Slot Binary variables for assignments yik = 1 product i assigned to slot k 0 otherwise Basic ideas : a. NP products b. NP time slots at each stage MINLP Model (1)
70 b) Definition of transition variables kjiyyz jkikijk   ,11 c) Processing rates, mass balances and amounts produced mkiTpppWp ikmimikm   1...111   MmkiWpWp ikmimikm  mkTpW kmkmkm   treated as continuous MILP model MINLP Model (2) d) Timing constraints mkiyUTpp ik T imikm  0 mkTppTp i ikmkm   mkTsTeTp kmkmkm  mNPkzTeTs i j ijkijmkmmk    1...111   i j ijij zTs 1111  1...11   MmkTsTs kmkm 1...11   MmkTeTe kmkm mzTpTc k i j ijkijmkm              Cycle time Processing time Transitions ijkz
71 e) Inventory levels for intermediates   1...1,min 1   MmkI0TsTsTpI1 kmkmkmkmkmkm  I2km  km  km1km1 max 0,Tekm  Tskm1  I1km k m  1... M 1 I3km  km1km1 min Tekm1  Tekm,Tpkm1  I 2km k m  1... M 1 0  I1km  Imaxkm k m  1... M 1 0  I2km  Imaxkm k m  1... M 1 0  I3km  Imaxkm k m  1... M 1 Imaxkm  Ipikm i  k m Ipikm  Uim I yik  0 i k m  1... M Tskm Tekm Tpkm Tskm+1 Tekm+1 Tpkm+1 Tskm Tekm Tpkm Tskm+1 Tekm+1 Tpkm+1 Tpkm Tpkm+1 Tskm+1 - Tskm Tekm+1 - Tekm time time Inventory level Inventory level stage m stage m+1 I0km I1km I2km I3kmI0km I1km I3km I2km MINLP Model (3) Inventories: Case 1 Case 2
72 f) Demand constraints iTcdWp i k ikM  MINLP Model (4) Note: Linear g) Objective function : Maximize Profit Profit  pi WpikM Tck  i  INCOME  Ctrij zijk Tck  j  i  TRANSITION COST  Cinvim Ipikm Tcm  k  i  INTERMEDIATE INVENTORY COST  1 2 Cinvfi piM  WpikM Tc     k  i  TppikM FINAL INVENTORY COST Note: Nonlinear Non-differentiable MINLP
73 min {f1(x), f2(x)} = - max {-f1(x), -f2(x)} = f1(x) - max {0, f1(x)-f2(x)} Handling Non-differentiabilities Option 1: Smooth Approximation   f (x)2  2 2  f(x) 2   max 0, f(x) Replace by Balakrishna, Biegler (1992) 0    f (x)  U1 1 y  0    U2y Option 2: Mixed-integer approach Raman, Grossmann (1991)   max 0, f(x) Replace by    f(x) for f(x) > 0, and   0 for f(x) < 0  y = 1,  = f(x) for f(x) > 0, and y=0,  = 0 for f(x) < 0 Option 2 is superior due to errors introduced with option 1
74 time (h) time (h) Intermediate storage (ton) Stages 1-2 Intermediate storage (ton) Stages 2-3 10 20 10 20 A C B E HF D G stage 1 stage 2 stage 3 cycle time = 675 hrs Optimal solution Profit = $6609/h - 47% improvement MINLP model 448 binary 0-1 variables, 2050 continuous variables, 3010 constraints Example 3 stages, 8 products, 3 DICOPT (CONOPT/CPLEX): 38.2 secs MULTISTAGE http://newton.cheme.cmu.edu/interfaces
75 Refinery Scheduling and Blending crude-oil marine vessels storage tanks charging tanks crude dist. units other prod. units comp. stock tanks blend headers finished product tanks shipping points crude-oil blending gasoline blending product delivery fractionation and reaction processes crude-oil unloading STANDARD REFINERY SYSTEM CRUDE OIL UNLOADING AND BLENDING PRODUCTION UNIT SCHEDULING PRODUCT SCHEDULING AND BLENDING gasoline can yield 60-70% of refinery’s profit !! •Production logistics •Production quality (scheduling) (blending) Multiple product demands Inventory and pumping constraints Resource allocation Timing of operations Logistic and operating rules Variable product recipes Product specifications Complex correlations for product properties • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
76 Problem statement i’ i’’ B2 blenders product tanks component tanks min/max product specifications - prmin p,k1 ≤ prp,k1,t ≤ prmax p,k1 - prmin p,k2 ≤ prp,k2,t ≤ prmax p,k2 … - prmin p,kn ≤ prp,kn,t ≤ prmax p,kn component properties - pri’’,k1 - pri’’,k2 … -- pri’’,kn - pri,kn p i i’ i’’ B3 B1 p’ p’ p’’ GIVEN: •Scheduling horizon •Components, Products •Storage tanks, Blend headers •Component properties, stocks and supplies •Product specifications, stocks and demands •Min/Max flowrates and concentrations •Correlations for predicting product properties •Operating rules THE GOAL IS TO DETERMINE: •Allocation of resources •Inventory levels in tanks •Component concentrations in each product •Volume of each product •Pumping rates •Production and storage tasks timing MAXIMIZE PRODUCTION PROFIT REQUIRES SIMULTANEOUS SCHEDULING AND BLENDING • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
77 Proposed optimization approach fi’’,t fi’,t fi,t FI i’’,p,t FI i’,p,t FI i,p,t FP p,t i’ i’’ B2 blenders product tanks component tanks p i i’ i’’ B3 B1 p’ p’ p’’ • Linear approximations for product properties 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon Product Due Dates D1 D2 D3 D4 time DISCRETE TIME FORMULATION CONTINUOUS TIME FORMULATION • MILP multiperiod optimization method • Discrete or continuous time domain representation • Discrete decisions for resource allocation and operating rules • Iterative procedure for improving predictions • Integrated production logistics and quality specifications • Variable recipes and min/max component concentrations • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results Product Due Dates D1 D2 D3 D4 time T1 T2 T3 T4 T5 T6 T7 T8 T9 SLOTS Sub-interval
78 Product property prediction 86 87 88 89 90 91 92 0 0.2 0.4 0.6 0.8 1 Component 'A' volume fraction property Non-linear correlation linear volum. average linear volum. average + bias Property value Comp A: 92.1 Comp. B: 86.1 Correction factor ‘bias’ = 0.24 BLEND 40% COMPONENT ‘A’ 60% COMPONENT ‘B’ tkpvprpr I tpi i kitkp ,,,,,,,   tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFbiasFpr FprFbiasFpr P tptkp P tpkp I tpi i ki P tptkp P tpp,k ,,,,,, max , ,,,,,,, min    •Volumetric average tpiFFv I tpi P tp I tpi ,,,,,,,  •Non-linear flowrate-composition matching Linear approximation for non-linear product properties Volumetric product property correlation (Linear) • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
79 Iterative procedure Generate initial guess recipes Compute non-linear properties for all products and time slots Compute correction factors 'bias' Solve MILP model using correction factors Compute non-linear properties for all products and time slots Fix product recipes for products on-spec All products on-spec or iteration limit reached Solution for the Scheduling and Blending problem component volume fractions in blends component volume fractions in blends NO YES • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
80 Integrated scheduling and blending model Discrete time domain representation Product Due Dates D1 D2 D3 D4 time tnA B t p tp  , tpFF P tp i I tpi ,,,,  tpiFrcpFFrcp P tppi I tpi P tppi ,,, max ,,,, min ,  tpAserateFAserate tpttp P tptpttp ,)()( , max ,, min  tiFefinvV ttp I tpitii I ti , ', ',,,   tiVVV i I tii ,max , min  tpddFinvV td pd tt p tpp P tp , ' ',,    tpVVV p P tpp ,max , min  tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFprFbiasFprFpr P tpkp I tpipk I tpi i ki P tpp,k ,,, max ,,,,,,,, min   dpddF dd dp dt P tp ,' ' ,,             t p I tpi i i P tpp FcFpMax ,,,          i t I tii p t P tpp t p I tpi i i P tpp VspVspFcFpMax ,,,,, ALLOCATION COMPOSITION CONCENTRATION, MATERIAL BALANCE AND STORAGE CAPACITY VOLUMETRIC FLOW-RATE, MATERIAL BALANCE AND STORAGE CAPACITY SPECIFICATION PRODUCT DEMANDS OBJECTIVE FUNCTION MAXIMIZE PROFIT • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results C O M P O N E N T S P R O D U C T S
81 Integrated scheduling and blending model Continuous time domain representation Product Due DatesD1 D2 D3 D4 time T1 T2 T3 T4 T5 T6 T7 T8 T9 tpSErateFAhrateSErate ttp P tptppttp ,)()1()( max ,, minmin  tpAhrateF tpp P tp ,, max ,  tiFEfiniV ttp I tpitii I ti , ', ',,,   tiFSfinvV ttp I tpitii I ti ,' ', ',,,   tiVVV i I tii ,' max , min  p dd pd dt p tpp P tp dpddFinvV ,' ' ',,    tpVV P tpp ,' , min  tpAlSE tpptt ,, min  tAhSE p tptt   , tSE tt   1 dt TtdS  1 dt TtdE  d p tp B t p tp TttdAnA    )1,(,,1, FLOWRATES STORAGE CAPACITY MATERIAL BALANCE DURATION SEQUENCING SUB-INTERVALS BOUNDS ALLOCATION tnA B t p tp  , tpFF P tp i I tpi ,,,,  tpiFrcpFFrcp P tppi I tpi P tppi ,,, max ,,,, min ,  ALLOCATION COMPOSITION CONCENTRATION tiVVV i I tii ,max , min  tpddFinvV td pd tt p tpp P tp , ' ',,    tpVVV p P tpp ,max , min  tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFprFbiasFprFpr P tpkp I tpipk I tpi i ki P tpp,k ,,, max ,,,,,,,, min   STORAGE CAPACITY SPECIFICATION MATERIAL BALANCE OBJECTIVE FUNCTION MAXIMIZE PROFIT • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results C O M P O N E N T S P R O D U C T S S L O T S SLOTS
82 Treatment of infeasible solutions Penalty for preferred recipe deviations Penalty for min/max specification deviations Penalty for component shortages tpiFDFrcp tpi R tpi P tpip ,,,,,,,   tpiFDFrcp tpi R tpi P tpip ,,,,,,,       t p i R tpi R ip R tpi R ip DpltyDpltyPenalty ,,,, tkpFprDFprop I tpi i ki S tpk P tpkp ,,,,,,,, min ,    tkpFprDFprop I tpi i ki S tpk P tpkp ,,,,,,,, max ,        t p k S tpk S pk S tpk S pk DpltyDpltyPenalty ,,,,,, tiSFeprodiniV ti ttp I tpitii I ti ,, ', ',,,     t i ti SH i SpltyPenalty , Relax some hard constraints that can generate an infeasible solution • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
83 Examples and numerical results 12 STORAGE TANKS 3 BLEND HEADERS 8-DAY TIME HORIZON 6 DUE DATES • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results i’ i’’ B2 blenders product tanks component tanks G2 C3 C4 C5 B3 B1 G1 G3 C2 C7 C6 C1 C8 C9 0 Day 1 8 days Day 3 Day 4 Day 5 Day 7 Day 8 0 Discrete time formulation 0 Continuous time formulation SLOTS
84 Examples and numerical results COMPONENT C1 C2 C3 C4 C5 C6 C7 C8 C9 COST($/BBL) 24.00 20.00 26.00 23.00 24.00 50.00 50.00 50.00 50.00 PROD.RATE (MBBL/DAY) 15.00 33.00 20.00 14.00 18.00 10.00 0.00 0.00 0.00 INITIALSTOCK (MBBL) 48.00 20.00 75.00 22.00 30.00 54.00 12.00 20.00 15.00 MIN.STOCK (MBBL) 5.0 5.0 5.0 5.0 5.0 5.0 0.0 0.0 0.0 MAX.STOCK (MBBL) 100.00 250.00 250.00 100.00 100.00 100.00 100.00 100.00 100.00 PROPERTY P1 93.00 104.00 104.90 94.80 87.40 118.00 87.30 95.20 93.30 P2 92.10 91.90 91.90 81.50 86.10 100.00 79.50 85.80 81.90 P3 0.7069 0.8692 0.6167 0.6731 0.6540 0.7460 0.7460 0.8187 0.7339 P4 3.60 1.00 100.00 94.90 91.50 15.00 0.00 1.30 34.30 P5 16.30 4.50 100.00 97.10 95.50 100.00 0.00 6.00 57.10 P6 94.30 93.50 100.00 100.00 100.00 100.00 0.00 93.90 95.90 P7 35.00 22.70 351.10 117.10 93.00 31.30 63.30 16.00 52.40 P8 0.007 0.00 0.00 0.009 0.0002 0.05 0.0063 0.1805 0.057 P9 0.00 88.60 0.00 2.30 0.20 0.00 43.98 65.30 21.30 P10 0.00 0.1 61.30 48.90 36.00 0.00 1.04 0.60 33.30 P11 0.00 3.30 0.00 1.10 0.10 0.00 3.33 0.90 0.80 P12 0.00 0.00 0.00 0.00 0.00 15.40 0.00 0.00 0.00 9 Components 12 properties Min/Max stock Initial stock Flowrates Cost • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
85 Examples and numerical results 8 Linear and 4 non- linear properties Min/Max component concentrations Min/Max storage capacities Multiple product demands PRODUCT G1 G2 G3 PRICE ($/BBL) 31.00 31.00 31.00 REQUIREMENT (MBBL) MIN MAX LIFT MIN MAX LIFT MIN MAX LIFT DAY 1 (MBBL) 5.00 45.00 10.00 5.00 50.00 12.00 5.00 50.00 10.00 DAY 3 (MBBL) 5.00 50.00 25.00 DAY 4 (MBBL) 5.00 45.00 25.00 5.00 50.00 23.00 DAY 5 (MBBL) DAY 7 (MBBL) 5.00 45.00 30.00 DAY 8 (MBBL) 5.00 45.00 10.00 5.00 50.00 22.00 INVENTORY (MBBL) 5.00 150.00 5.00 150.00 5.00 150.00 RATE (MBBL/DAY) 5.00 45.00 5.00 50.00 5.00 50.00 RECIPE (%) MIN MAX MIN MAX MIN MAX C1 0.00 22.00 0.00 25.00 0.00 25.00 C2 0.00 20.00 0.00 24.00 0.00 24.00 C3 2.00 10.00 0.00 10.00 0.00 10.00 C4 0.00 6.00 0.00 23.00 0.00 23.00 C5C 0.00 25.00 0.00 25.00 0.00 25.00 C6 0.00 10.00 0.00 10.00 0.00 10.00 C7 0.00 100.00 0.00 0.00 0.00 0.00 C8 0.00 100.00 0.00 0.00 0.00 0.00 C9 0.00 100.00 0.00 0.00 0.00 0.00 SPECIFICATIONS MIN MAX MIN MAX MIN MAX P1 95.00 98.00 98.00 P2 85.00 88.00 88.00 P3 0.72 0.775 0.72 0.775 0.72 0.775 P4 20.00 50.00 20.00 48.00 22.00 50.00 P5 46.00 71.00 46.00 71.00 46.00 71.00 P6 85.00 85.00 85.00 P7 45.00 60.00 45.00 60.00 60.00 90.00 P8 0.015 0.015 0.008 P9 42.00 42.00 42.00 P10 18.00 18.00 18.00 P11 1.00 1.00 1.00 P12 2.70 2.70 2.70 3 Products Price • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
86 Ex. 1: Blending problem (fixed scheduling) INITIAL RECIPE C2 20 C3 3.598 C4 6 C6 7.246 C7 C8 16.156 C1 22 C5 25 ITERATION 1 C1 22 C2 20 C3 2 C4 4.938 C5 25 C6 10 C7 5.181 C8 1.114 C9 9.767 ITERATION 2 C1 22 C2 20 C3 2 C4 4.847 C5 25 C6 10 C7 5.198 C8 0.958 C9 9.997 C1 C2 C3 C4 C5 C6 C7 C8 C9 Convergence to the preferred recipe for product G1 MIN. SPEC. INITIAL RECIPE ITERATION 1 ITERATION 2 MAX SPEC. BLEND COST ($/BBL) 29.30 29.97 29.99 QUALITY Value Value approx. Value Approx. P1 95.00 97.891 97.898 97.7737 97.893 97.8928 P2 85.00 88.417 88.470 88.0493 88.438 88.4335 P3 0.72 0.7418 0.7325 0.7324 0.775 P4 20.00 34.455 35.418 35.409 50.00 P5 46.00 46.00 50.80 50.833 71.00 P6 85.00 96.460 91.797 91.780 P7 45.00 60.00 60.00 60.00 60.00 P8 0.0378 0.0152 0.0150 0.0150 0.0150 0.015 P9 28.458 22.974 22.923 42.00 P10 14.256 15.974 16.005 18.00 P11 0.8964 1.00 1.00 1.00 P12 1.1223 1.5684 1.5488 1.5687 1.5684 2.70 Objective: Find product recipes that satisfy all specifications at minimum cost • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
87 Ex. 2: Blending and scheduling (limited production) G1 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime Objective: Make scheduling and blending decisions that maximize profit Min/max production for each time interval Product specifications Operating conditions Demands at specific due dates Discrete time formulation Continuous time formulation Profit: $1,611.21 Operates blenders at full capacity for 2.67 days less than discrete time G2 G3 G1 G2 G3 Mbbl Mbbl EVOLUTION OF COMPONENT STOCKS 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime
88 Ex. 3: Blending and scheduling (flexible production) 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime Objective: Maximize profit of the refinery Product specifications Operating conditions Demands at specific due dates Profit: $2,448.05 (increased by 52%) Operates blenders at full capacity for 2.6 days less than discrete time Increase production by 36% Discrete time formulation Continuous time formulation G1 G2 G3 G1 G2 G3 Mbbl Mbbl EVOLUTION OF COMPONENT STOCKS
89 Computational results Modest number of 0-1 variables Very low CPU time requirements EXAMPLE DISCRETIME CONTINUOUSTIME Binary, Continuous, Constraints Objective CPU Binary, Continuous, Constraints Objective CPU 2 9,757,679 1,611.21 0.26 9,841,832 1,611.21 0.26 3 18,757,679 2,448.05 0.23 18,841,832 2,448.05 0.26 4 9,757,679 1,234.49 0.23 9,841,832 1,234.49 0.26 • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results Seconds on Pentium IV (2.4 GHz) with GAMS 21.2 / CPLEX 8.1
Multiperiod Refinery Planning Model • LP planning models (PIMS) – Fixed yield model – Swing cuts model • NLP planning models – Aggregate distillation model – Fractionation index (FI) model 90 Alattas, Palou-Rivera, Grossmann (2012) Goal: Develop nonlinear multiperiod planning
FI Model (Fractionating Index) – 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 – Generates cut point temperature settings for the CDU – Adds few additional equations to the planning model 91
92 Multiperiod Refinery Planning Problem • Time horizon with N time periods • Inventories and changeovers of M crudes • Given: refinery configuration 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?
93 MINLP Model 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
94 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)
95 Simultaneous Cyclic Scheduling and Control of a Multiproduct CSTR Reactor Given is a CSTR reactor N products Lower bounds demand rates Dynamic model for reactions Determine cyclic schedule Cycle time Sequence Amounts to produce Lenghts transitions and their dynamic profile Objective: Maximize total profit Terrazas, Flores , Grossmann (2008) MIDO Optimization model
96 Scheduling and Control Formulation
97 Scheduling Formulation (Pinto’s cyclic scheduling model)
98 Simultaneous Approach for solving Dynamic Optimization Problems Problem • Few discrete decisions • Can exhibit instability due to: • Open loop models • Choice of control laws • Can add stability constraints • Requires simultaneous approach • Large, nonlinear NLP sub-problem • NLP expense >> MILP expense
99   K 1k iki0 z(t)zz(t) k x x x x element i k = 1 k = 2 Differential variables Continuous ´ ´ xxxx   K 1k ik(t)uu(t) k Algebraic and Manipulated variables Discontinuous ´ ´ Simultaneous Formulation Collocation on Finite Elements to tfCollocation pts · · ·· · · · · · · · · Polynomials · Mesh points hi ´ ´ ´ ´´ ´´ ´´ ´ ´ ´
100 Control Formulation
101 Optimal sequence ADECB Cycle time = 124.8 h Profit = $7889/h Results
102 Technology 1 Technology I Technology 1 Technology I DMK lpt PUjpt Q PL jkpt Q WH klpt Suppliers Plants j=1,…,J Warehouses k=1,…,K Markets l=1,…,L Wijpt INVkpt CPL ijt CWH kt Model = Plant location problem (Current et al.,1990) plus Long range planning of chemical processes (Sahinidis et al., 1989) • Three-echelon supply chain • Different technologies available at plants • Multi-period model Multiperiod MILP formulation supply chain Guillen, Grossmann (2008)
103 Notation
104 1. Mass balances Plants Warehouses Markets 2. Capacity Expansion Plants Plants Binary variable (1 if technology i is expanded in plant j in period t) Multiperiod MILP formulation (I)
105 Transport links Multiperiod MILP formulation (II) Binary variable (1 if warehouse k is expanded in period t) Warehouses 3. Capacity Expansion Warehouses Binary variable (1 if there is a transport link between plant j and warehouse k in period t) Binary variable (1 if there is a transport link between warehouse k and market l in period t) 4. Transportation links
106 5. Objective function Summation of discounted cash flows Net Earnings Fixed cost Multiperiod MILP formulation (III)
107 Case study (I) Problem : • Redesign a petrochemical SC to fulfill future forecasted demand Case study Existing plant Potential plant
108 One step oxidation of ethylene Cyanation / oxidation of ehtylene Acetaldehyde Acrylonitrile Ethylene Ammoxidation of propylene Propylene Phenol HCN HCl H2SO4 O2 NH4 Hydration of propylene Reaction of benzene and propylene Isopropanol Oxidation of cumene Benzene Cumene Acetone 0.67 0.38 1.35 1 0.61 By-product 0.05 0.83 1.20 0.76 H2SO4 NaOH Others 0.010.010.01 1 Active carbon 0.01 1 0.43 0.15 1 1 0.900.17 0.83 0.83 0.40 1 O2 Technologies in each Plant Site
109 Multiperiod MILP Models: • Number of 0-1 variables: 450 • Number of continuous variables: 4801 • Number of equations: 4682 • CPU* time: 0.33 seconds *Solved with GAMS 21.4 / CPLEX 9.0 (Pentium 1.66GHz) Potential Supply Chain Horizon: 3 yrs
110 NPV = $132 million Optimal Solution
111 Multiperiod Production Planning LP Model i jt jt jt jt j J t T j J t T it ijt jt jt jt jt i I j JM t T j J t T j J t T Max PROFIT S P W V SF                             TtJMjJjIiWW iitijijijt  ,',,' TtJMjIiQW iitijt  ,, TtJj dSd aPa U jtjt L jt U jtjt L jt        , , 1 , j j j t ijt jt jt ijt jt i O i I V W P V W S j J t T           TtJjSdSF jt U jt jt  , TtJjSFSF U jtjt  ,0 TtJjVV U jtjt  , , , , 0jt jt it jtS P W V  Mass balance process Capacity Mass balance chemicals Shortfalls Purchases Sales Limit inventory Fixed price purchases What if prices not fixed but given by contracts?
112 . Discount after d jt amount. TtJRjPPCOST d jt d jt d jt d jt d jt  ,2211  TtJRj P P y P P y d jt d jt d jt d jt d jt d jt d jt d jt                              , 00 0 2 1 2 2 1 1  TtJRjPPP d jt d jt d jt  ,21 Disjunctive model TtJRjPPCOST d jt d jt d jt d jt d jt  ,2211  TtJRjPPP d jt d jt d jt  ,21 TtJRjPPP d jt d jt d jt  ,12111 TtJRjyP d jt d jt d jt  ,0 111  TtJRjyP d jt d jt d jt  ,212  TtJRjUyP d jt d jt d jt  ,0 22 TtJRjyyy d jt d jt d jt  ,21 }1,0{, 21 d jt d jt yy MILP model (Convex hull) Price drops after amount 
113 Bulk discount TtJRj P PCOST y P PCOST y b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt                              , 0 2 2 1 1     Disjunctive Model TtJRjPPCOST b jt b jt b jt b jt b jt  ,2211  TtJRjPPP b jt b jt b jt  ,21 TtJRjyP b jt b jt b jt  ,0 11  TtJRjyUPy b jt b jt b jt b jt b jt  ,222  TtJRjyyy b jt b jt b jt  ,21 }1,0{, 21 b jt b jt yy MILP Model (Convex Hull) Price for total amount drops after amount 
114 Fixed-duration contracts TtJRj P P P PCOST PCOST PCOST y P P PCOST PCOST y P PCOST y l jt l tj l jt l tj l jt l jt l tj l jt l tj l tj l jt l tj l jt l jt l jt l jt l jt l tj l jt l jt l tj l jt l tj l jt l jt l jt l jt l jt l jt l jt l jt l jt l jt                                                                              , 3 2, 3 1, 3 2, 3 2, 1, 3 1, 3 3 2 1, 2 1, 2 1, 2 2 1 1 1             Disjunctive Model TtJRjPCOST LCp T lp tj lp j l jt p t     ,   TtJRjPP LCp T lp tj l jt p t     ,   LCpTTTTtJRjyUPy p t plp j l j lp tj lp j lp j  ,,,   TJRjyy l j LCp lp j   , }1,0{lp jy  , }3,2,1{LC MILP Model Price gradually drops if amount  purchased fixed number of months
115 Case 0-1 variables Cont. variables Constraints CPU time [s] Time periods Solution [105 $] 1 0 12,606 13,416 0.18 10 18,085.95 2 6,160 40,606 46,002 0.95 10 22,073.06 Computational Results 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 9 10 Time periods Qauntities[kton] fixed Ethy fixed Acet fixed Naph discount Ethy discount Acet discount Naph bulk Ethy bulk Acet bulk Naph length Ethy length Acet length Naph Purchases raw materials 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 PROFIT REVENUES PURCHASES OPERATION STORAGE*100 SHORTFALLS 1E5$ Without Contracts With Contracts Comparison without/with contracts
116 Computational Results ILOG CPLEX Dec 1, 2008 22.9.2 LNX 7311.8080 LX3 x86/Linux Cplex 11.2.0, GAMS Link 34 MIP Presolve eliminated 44425 rows and 38571 columns. Reduced MIP has 909 rows, 1367 columns, and 3267 nonzeros. Reduced MIP has 270 binaries, 0 generals, 0 SOSs, and 0 indicators. Implied bound cuts applied: 3 Flow cuts applied: 40 Gomory fractional cuts applied: 15 MIP Solution: 22073.060039 (959 iterations, 21 nodes) MODEL STATISTICS BLOCKS OF EQUATIONS 47 SINGLE EQUATIONS 46,002 BLOCKS OF VARIABLES 30 SINGLE VARIABLES 40,606 NON ZERO ELEMENTS 85,033 DISCRETE VARIABLES 6,160 S O L V E S U M M A R Y MODEL MULT OBJECTIVE NPV TYPE MIP DIRECTION MAXIMIZE SOLVER CPLEX FROM LINE 878 **** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 22073.0600 RESOURCE USAGE, LIMIT 0.470 10000.000 ITERATION COUNT, LIMIT 1437 1000000
117 Decomposable MILP Problems A D1 D3 D2 Complicating Constraints max 1,.. { , 1,.. , 0} T i i i i i c x st Ax b D x d i n x X x x i n x        x1 x2 x3 Lagrangean decomposition complicating constraints D1 D3 D2 Complicating Variables A x1 x2 x3y 1,.. max 1,.. 0, 0, 1,.. T T i i i n i i i i a y c x st Ay D x d i n y x i n          Benders decomposition complicating variables Note: can reformulate by defining 1i iy y  and apply Lagrangean decomposition Complicating constraints
118  MILP optimization problems can often be modeled as problems with complicating constraints.  The complicating constraints are added to the objective function (i.e. dualized) with a penalty term (Lagrangean multiplier) proportional to the amount of violation of the dualized constraints.  The Lagrangean problem is easier to solve (eg. can be decomposed) than the original problem and provides an upper bound to a maximization problem. Lagrangean Relaxation (Fisher, 1985)
119 n Zx eDx bAx cxZ     max bAx Assume that is complicating constraint n LR Zx eDx AxbucxuZ    )(max)( 0where u Lagrange multipliers (IP) Assume integers only Easily extended cont. vars. Lagrangean Relaxation (Fisher, 1985)
120 0uwhere ( )LRZ u Z n Zx eDx bAx cxZ     max n LR Zx eDx AxbucxuZ    )(max)( bAx  ZuZLR )( 0)(  Axb 0u This is a relaxation of original problem because: i) removing the constraint relaxes the original feasible space, ii) always holds as in the original space since and Lagrange multiplier is always . Lagrangean Relaxation Yields Upper Bound Complicating Constraint  Lagrangean Relaxation
121 n LR Zx eDx AxbucxuZ    )(max)(Relaxed problem: min ( ) 0 D LRZ Z u u   Lagrangean dual: )( 1uZLR )( 2uZLR )( 3uZLR Z n Zx eDx bAx cxZ     maxOriginal problem: DZdual gap Lagrangean Relaxation
122   n xu D Zx eDx AxbucxZ      )(maxmin 00 min ( ) 0 D LRZ Z u u   n LR Zx eDx AxbucxuZ    )(max)(Relaxed problem: Lagrangean dual: Combined Relaxed and Lagrangean Dual Problems: Graphical Interpretation
123   n xu D Zx eDx AxbucxZ      )(maxmin 00  n ZxeDxx  ,  n ZxbxAx  , Optimization of Lagrange multipliers (dual) can be interpreted as optimizing the primal objective function on the intersection of the convex hull of non- complicating constraints set and the LP relaxation of the relaxed constraints set . 0 ),( max       x ZxeDxConvx bAx cxZ n D Nice Proof Frangioni (2005) Graphical Interpretation
124  eDxx  Conv n ZxeDxx  ,  bAxx  cx ZLP ZD Z 0 ),( max       x ZxeDxConvx bAx cxZ n D dual gap Graphical Interpretation
125 Lagrangean relaxation yields a bound at least as tight as LP relaxation  eDxx  Conv  n ZxeDxx  ,  bAxx  cx ZLP ZD Z ( ) ( )D LR LPZ P Z Z u Z   Theorem
126  Lagrangean Decomposition is a special case of Lagrangean Relaxation.  Define variables for each set of constraints, add constraints equating different variables (new complicating constraints) to the objective function with some penalty terms. n Zx eDx bAx cxZ     max n n Zy Zx yx eDy bAx cxZ         max New complicating constraints n n LD Zy Zx eDy bAx xyvcxvZ        )(max)( Dualize x = y Lagrangean Decomposition (Guignard & Kim, 1987)
127 n n LD Zy Zx eDy bAx xyvcxvZ        )(max)( n LD Zx bAx xvcvZ    )(max)(1 n LD Zy eDy vyvZ    max)(2 Subproblem 1 Subproblem 2  )()(min 21 0 vZvZZ LDLD v LD   Lagrangean dual Lagrangean Decomposition (Guignard & Kim, 1987)
128  Lagrangean decomposition is different from other possible relaxations because every constraint in the original problem appears in one of the subproblems. Subproblem 1 Subproblem 2 Graphically: The optimization of Lagrangean multipliers can be interpreted as optimizing the primal objective function on the intersection of the convex hulls of constraint sets. Notes
129 Z Graphical Interpretation Subproblem 1  eDxx   bAxx  cx ZLP ZLR Conv n ZxeDxx  , Conv n ZxbxAx  , ZLD Subproblem 2 Note: ZLR, ZLD refer to dual solutions
130  The bound predicted by “Lagrangean decomposition” is at least as tight as the one provided by “Lagrangean relaxation” (Guignard and Kim, 1987)  For a maximization problem LPLRLD ZZZPZ )( Solution of Dual Problem ZLR or ZLD u or  minimum Piecewise linear => Non-differentiable Theorem
131 1,... max{ ( ) , } max { ( )}k k x k K cx u b Ax Dx d x X cx u b Ax         Assuming Dx  d is a bounded polyhedron (polytope) with extreme points 1,2...k x k K , then How to iterate on multipliers u? 0 01,.. min max{ ( )} min{ ( ), 1,.. }k k k k u uk K cx u b Ax cx u b Ax k K           => Dual problem Cutting plane approach 1 min . . ( ), 1,.. 0, k k ns t cx u b Ax k K u R          Kn = no. extreme points iteration n subgradient Note: xk generated from max{cx + uk(b-Ax) subproblems
132 Update formula for multipliers (Fisher, 1985) 21 ( )( ) / [0,2] k k LB k k k k LD k u u Z Z b Ax b Ax where          Subgradient ( )k k s b Ax  Steepest descent search 1k k k u u s   Subgradient Optimization Approach Note: Can also use bundle methods for nondifferentiable optimization Lemarechal, Nemirovski, Nesterov (1995)
133 Solution of Langrangean Decomposition Select MaxI, ε, ak Set UB = +, LB= -  Solve (RP’) to find v0 | ZLD - ZLB |<ε? or k=MaxI? Solve (P) with fixed binaries or use heuristics: Obtain ZLB Solve (P1) and (P2): Obtain ZLD k = k+1 Update uk For k = 1..K Return ZLB & Current Solution YES NO Iterative search in multilpliers of dual Notes: Heuristic due to dual gap Obtaining Lower Bound might be tricky Remarks 1. Methods can be extended to NLP, MINLP 2. Size of dual gap depends greatly on how problems are decomposed 3. From experience gap often decreases with problem size. Upper Bound Lower Bound
134 Multisite planning with sequence dependent changeovers is an integrated problem Month 1 Month 2 Month n -1 Month n Multi-site Network Market 3 Market 2 Market 1 A D B C Continuous Processes Production Cycle Given: • Network of production sites and markets • Capacities in each plant for producing different products (e.g. polymers) • Forecasted demands • All costs and product prices Determine: • Product assignment to sites • Production and shipment volumes • Inventory levels • Sequence of production in each site Objective: Maximize profit Profit = Sales - Operating costs - Inventory costs - Distribution costs- Changeover costs Production Site 1 Production Site 2 Production Site n-1 Production Site n
135 The solution coordinates sites and markets across time periods A B A BSite 1 Site 2 A 20 20 100 100 50 Site 1 Site 2 Market 1 Market 2 Site 1 Site 2 Market 1 Market 2 10 10 20 100 20 60 40 40 Month 1 Month 2 Changeover time Production time Inventory Distribution Sales Month 1 Month 2 Changeovers Inventories Decrease Capacity Increase Cost
136 Mathematical Model  Objective: Maximize Profit subject to  Market constraints • Balance of sales vs. shipments to markets  Production constraints • Capacity constraints: Limited capacity at each production sites • Inventory constraints: Penalties for inventory over or under target  Links across periods: a) Carry over inventories from last month b) Changeover to first product in next month • Time Balances: Task should not take longer than available time • Sequencing Constraints: Traveling Salesman Problem Constraints Source of complexity of the model: TSP constraints
137 Mathematical Model (MILP) sk t i ski t m smi t si t si t si t xfzzz finvinvp , 1 ,, ,,,, 1 ,          m s t tsmtsmtmtmtsts fgxcxcprofit ,,,,, 2 , 2 , 1 , 1 m,tfx s tsmtm   0,,, 2 Links across time periods Objective Function            i s t si t si t i k s t ski t ski t ski t ski t m s smi t smi t t i m s si t si t si t mi t mi t tpCOP zzzzzzCTR f penpsl profit ,, ,,,,,,,, ,,,, ,,,,, ] )([   Market Constraints  s smi t mi t fsl ,,, tsfxJxJxJ m tsmtstststststs ,0,,1, 1 , 3 , 1 , 2 1, 1 , 1   Sites Variables Market Variables Shipments Shipments Market Variables Site Variables Site Variables Site Variables Shipments
138 Mathematical Model (MILP) si t si t si t si t si t si t yCAPp tpCAPp ,,, ,,,   sisi t si t si t sisi t QUOTAinvpen invQUOTApen ,,, ,,,     t s t i si t i ski t ski t ski t k kis t HTRTtp zzzzzztrt     , ,,,,,,,  Capacity Constraints Inventory Time Balance Sequencing             ik k sk t si t sii t sk t sii t sii t si t ski t ski t i k ski t i ski t sk t k ski t si t yyz yz zy zzz zz zy zy ,,,, ,,, ,,, ,,,, ,, ,,, ,,, 1 1 si t k ski t i si t i si t k ski t si t i ski t sk t xlzzz xl xf zzxl zzxf ,,, , , ,,, ,,, 1 1           tsyBxA tstststs ,e ts,,,, 1 ,  Market Variables Product assignment Variables
139 Decomposition strategies are required Multisite production planning with sequence dependent changeovers integrates functions but involves large-scale optimization problem Computational expense Performance (Economic Profit) Problem size More sites, markets, products and time periods Decomposition strategies are required Options: Benders, Lagrangean, Two - Level. Problem can become intractable 3 markets 3 sites 3 time periods 3 products 6 markets 3 sites 6 products 6 markets 6 sites 6 products 6 time periods 6 time periods Solved in 1 s Solved in 780 s Not solved to optimality in 10,000 s 982 cont. vars. 36 discrete vars. 1,042 constraints 5,437 cont. vars. 126 discrete vars. 4,984 constraints 10,621 cont. vars. 252 discrete vars. 9463 constraints
140 • Temporal and Spatial Lagrangean decompositions for multi-scale problem Two types of Lagrangean decomposition Simultaneous solution of sites, markets across time periods Production Sites Markets Time period 1 Time period 2 Production Sites Markets Time period 1 Time period 2 Time periods Solved Indepen- dently Time periods Solved Indepen- dently Sites solved independentlyProduction Sites Markets Time period 1 Time period 2 Markets solved independently Spatial Decomposition Temporal Decomposition Full space solution Which is better ?
141 Example 1: 3 sites, 3 markets, 3 months & 3 products Price [$/ton] Operating Cost [$/ton] Inventory Cost [$/ton] Distrib. Cost [$/ton] Product A 1.20 0.010 0.010x10-2 0.2 Product B 1.00 0.008 0.008x10-2 0.2 Product C 1.00 0.015 0.015x10-2 0.2 A B C A - 100 | 0.2 200 | 0.4 B 100 | 0.2 - 100 | 0.2 C 200 | 0.4 100 | 0.4 - Changeover Costs | Time [$] | month fractionsPrices and Costs Month 1 Month 3 Market 3 Market 2 Market 1 Production Site 1 Production Site 2 Production Site 3 Month 2 Forecast for Market 3 0 50 100 150 200 250 300 350 400 450 500 Month 1 Month 2 Month 3 Tons Product A Product B Product C
142 B Example 1 - Output B C BSite 1 Site 2 0.2 Site 1 Site 2 Market 1 Market 2 Month 1 Month 2 Month 2 Site 3 Market3 Site 3 Month 3 0.780.02 0.59 0.390.02 A 0.51 0.470.02 A A 0.182 0.7980.02 0.133 A 0.8470.02 Profit: $ 2.576 million
143 10 20 30 40 50 60 70 80 90 100 2 2.5 3 3.5 x 10 4 Iteration ObjectiveFunction[$] Temporal vs. Spatial Decomposition Temporal Lagrangean Subproblem Feasible Solution (Temporal Decomposition) Spatial Lagrangean Subproblem Feasible Solution (Spatial Decomposition) Example 2 Full space Temporal Spatial Upper Bound 28,820 27,970 31,026 Best Feasible Solution 27,373 26,838 27,322 Gap [%] 5 4 13 CPU time [seconds] 10,000 + 804 5489 Temporal Spatial Sum of LP multipliers 171 1176 Size: 6 Sites, 6 Market, 6 Months, 6 Products (10,621 cont. vars., 252 discrete vars., 9463 constraints) Objectives: Compare computational performances of full space model, temporal and spatial decomposition Upper Bounds Lower Bounds Temporal is superior Selection criterion predicts better performance of the temporal decomposition Numerical results confirm expected results Temporal decomposition obtains better result than full space solution Observations: STvTw  
144 Time Safety Stock Reorder Point Order placed Lead Time InventoryLevel Replenishment • Inventory System under Demand Uncertainty  Total Inventory = Working Inventory (WI) + Safety Stock (SS)  Estimate WI with Economic Order Quantity (EOQ) model Stochastic Inventory System
145 Time InventoryLevel Average Inventory (Q/2) Order quantity (Q)  F = Fixed ordering cost for each replenishment  h = Unit inventory holding cost Replenishment Constant Demand Rate = D Economic Order Quantity Model
146 Total Cost 2 * 2     Q h Q D F Q FD h Order Cost Curve Order quantity Q Annual Cost Optimal Order Quantity (Q*) Minimum Total Cost Economic Order Quantity (EOQ) Economic Order Quantity Model
147 Time Inventory Level Lead Time Order Quantity (Q) Reorder Point (r)  When inventory level falls to r, order a quantity of Q  Reorder Point (r) = Demand over Lead Time Order placed Replenishment (Q,r) Inventory Policy
148 Reorder Point (r) Time InventoryLevel Lead Time Place order Receive order Safety Stock Reorder Point = Expected Demand over Lead Time + Safety Stock Stochastic Inventory = Working Inventory (EOQ) + Safety Stock Stochastic Inventory Model
149 Safety Stock (Service Level) Lead time = L Safety Stock Level
150* GD Eppen, “Effect of centralization in a multi-location newsboy problem”, Management Science, 1979, 25(5), 498 • Single retailer: Retailer Retailer Retailer Retailer Retailer • Centralized system:  All retailers share common inventory  Integrated demand • Decentralized system:  Each retailer maintains its own inventory  Demand at each retailer is Risk-Pooling Effect*
151 • Given: A potential supply chain  Including fixed suppliers, retailers and potential DC locations  Each retailer has uncertain demand, using (Q, r) policy  Assume all DCs have identical lead time L (lumped to one supplier) Suppliers RetailersDistribution Centers Problem Statement
152 • Objective: (Minimize Cost)  Total cost = DC installation cost + transportation cost + fixed order cost + working inventory cost + safety stock cost • Major Decisions (Network + Inventory)  Network: number of DCs and their locations, assignments between retailers and DCs (single sourcing), shipping amounts  Inventory: number of replenishment, reorder point, order quantity, neglect inventories in retailers retailer supplier DC Supplier RetailersDistribution Centers Problem Statement
153 Annual EOQ cost at a DC: ordering cost transportation cost Working inventory cost v(x)= g + ax EOQ cost
154 The optimal number of orders is: The optimal annual EOQ cost: Annual working inventory cost at a DC: ordering cost transportation cost inventory cost Convex Function of n Working Inventory cost
155 • Demand at retailer i ~ N(i,  i) • Centralized system (risk-pooling) • Expected annual cost of safety stock at a DC is: where za is the standard normal deviate for which Reorder Point (ROP) Time InventoryLevel Lead Time Safety Stock Cost for DCs
156 retailer supplier DC Other Parameters and Variables
157 retailer supplier DC DC – retailer transportation Safety Stock EOQ DC installation cost Supplier RetailersDistribution Centers Assignments Nonconvex INLP INLP Model Formulation
158 • Small Scale Example  A supply chain includes 3 potential DCs and 6 retailers (pervious slide)  Different weights for transportation (β) and inventory (θ) β = 0.01, θ = 0.01 β = 0.1, θ = 0.01 β = 0.01, θ = 0.1 • Model Size for Large Scale Problem  INLP model for 150 potential DCs and 150 retailers has 22,650 binary variables and 22,650 constraints – need effective algorithm to solve it … Illustrative Example
159 Non-convex MINLP Avoid unbounded gradient • Variables Yij can be relaxed as continuous variables (MINLP)  Local or global optimal solution always have all Yij at integer  If h=0, it reduces to an “uncapacitated facility location” problem  NLP relaxation is very effective (usually return integer solutions) Z1j Z2j Model Properties
160 • MINLP Heuristic Method  Solve the convex relaxation (MILP), using secant for convex envelope  Use optimal value of X and Y variables as initial point, solve the reformulated problem with an MINLP solver (BARON, Dicopt, etc.) Convex Relaxation Algorithm 1 – MINLP Heuristic
161 Supplier RetailersDistribution Centers • Lagrangean Relaxation (LR) and Decomposition  LR: dualizing the single sourcing constraint:  Spatial Decomposition: decompose the problem for each potential DC j  Implicit constraint: at least one DC should be installed,  Use a special case of LR subproblem that Xj=1 decompose by DC j Algorithm 2 - Lagrangean Relaxation
162 Solve the reduced NLP to get UB No Initialization Convergence or iteration limit Yes Stop Solve the | j | LR subproblem for Xj=1, set Vj as the optimal obj. fun. value Update subgradient Update LB, adjust step length parameter, fix all the X variables NoYes LR Algorithm Flowchart
163 • Case 1: 33 retailers and DCs  Each instance has the same number of potential DCs as the retailers 300 600 900 1200 1500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 θ ObjectiveFunctionValue DICOPT, SBB, BARON, Lagrangean 0.01 0.1 1 10 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 θ CPUTime(s) BARON (global) DICOPT (Algorithm 1) SBB (Algorithm 1) Lagrangean (Algorithm 2) 300 600 900 1200 1500 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 β ObjectiveFunctionValue DICOPT, SBB BARON, Lagrangean 0.1 1 10 100 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 β CPUTime(s) BARON (global) DICOPT (Algorithm 1) SBB (Algorithm 1) Lagrangean (Algorithm 2) Computational Results
164 • Case 2: 88 ~150 retailers  Each instance has the same number of potential DCs as the retailers No. Retailers β θ Algorithm 1 Algorithm 2 DICOPT (CONOPT) SBB (CONOPT) BARON (global optimum) BARON+CONOPT Lagrangean Obj. fun. Time (s) Obj. fun. Time (s) Obj. fun. Time (s) Obj. fun. Time (s) 33 0.001 0.1 398.64 0.22 398.64 0.20 398.64 53.31 398.64 15.8 33 0.001 0.5 580.46 0.23 580.46 0.22 580.46 74.27 580.46 17.9 33 0.001 1.0 728.21 0.17 728.21 0.17 728.21 39.14 728.21 15.85 33 0.001 5.0 1460.40 0.18 1460.40 0.20 1460.40 93.22 1460.40 37.28 33 0.003 0.1 770.26 0.23 770.26 0.27 734.60 75.67 734.60 42.79 88 0.001 0.1 935.02 20.89 935.02 20.94 867.55* > 10 hr 867.55 356.1 88 0.001 0.5 1386.28 42.16 1386.28 38.50 1295.02* > 10 hr 1230.99 322.54 88 0.005 0.1 2297.74 42.11 2297.74 42.16 2297.80* > 10 hr 2284.06 840.28 88 0.005 0.5 3082.19 42.35 3082.19 42.38 3022.67* > 10 hr 2918.3 934.85 150 0.001 0.5 2205.37 229.57 2205.37 228.31 1847.93* > 10 hr 1847.93 659.1 150 0.005 0.1 3689.71 397.88 3689.71 398.34 3689.71* > 10 hr 3689.71 3061.2 * Suboptimal solution obtained with BARON for 10 hour limit. Computational Results
165 No. Retailers β θ Lagrangean Relaxation (Algorithm 2) BARON (global optimum) Upper Bound Lower Bound Gap Iter. Time (s) Upper Bound Lower Bound Gap 88 0.001 0.1 867.55 867.54 0.001 % 21 356.1 867.55* 837.68 3.566 % 88 0.001 0.5 1230.99 1223.46 0.615 % 24 322.54 1295.02* 1165.15 11.146 % 88 0.005 0.1 2284.06 2280.74 0.146 % 55 840.28 2297.80* 2075.51 10.710 % 88 0.005 0.5 2918.3 2903.38 0.514 % 51 934.85 3022.67* 2417.06 25.056 % 150 0.001 0.5 1847.93 1847.25 0.037 % 13 659.1 1847.93* 1674.08 10.385 % 150 0.005 0.1 3689.71 3648.4 1.132 % 53 3061.2 3689.71* 3290.18 12.143 % • Case 2: 88 ~150 retailers  Each instance has the same number of potential DCs as the retailers * Suboptimal solution obtained with BARON for 10 hour limit. Computational Results
Carnegie Mellon 166 Van Hentenryck (1988), Puget (1994), Hooker (2000) Variables: Continuous, integer, boolean Constraints: Algebraic h(x) ≤ 0 Disjunctions [ A1x ≤ b1 ]  A2x ≤ b2 ] Conditional If g(x) ≤ 0 then r(x) ≤ 0 Unusual Operators: All different all different(x1, x2, ...xn) Constraint (Logic) Programming 
Carnegie Mellon 167 Constraint Programming 1. Declarative language with high level operators (OPL-ILOG) 2. Tree search: implicit enumeration Depth first search Lower bound: partial solution Upper bound: feasible solution 3.Constraint propagation: Domain Reduction Reduction of bounds/discrete values: a) Tighten bounds linear/monotonic functions b)"edge-finding" for jobshop scheduling Job i Job j Earliest Latest Earliest Latest Earliest Latest Earliest Latest {Job i before Job j} OR {Job j before Job i} {Job j before Job i}
Carnegie Mellon 168 Constraint Programming Example Problem Find the order at which jobs A, B and C are sequenced on machine M Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7
Carnegie Mellon 169 A1 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A2 A3 B1 B2 B3 B1 B2 B3 B1 B2 B3 C1 C2 C3 C 3 ABC C2 ACB
Carnegie Mellon 170 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A1 A2 A3 B1 B2 B3 B1 B2 B3 B1 B2 B3 C1C2C3 C3 C2 C1 ABC ACB BAC CAB BCA CBA
Carnegie Mellon 171 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st A1 A2 A3 B2 B3 B1 B3 B1 B2 C1C2C3 C3 C2 C1 ABC ACB BAC CAB BCA CBA
Carnegie Mellon 172 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st C must be 2nd B2 B3 C2C3 A1 ABC ACB
Carnegie Mellon 173 Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st C must be 2nd A{1,2,3} B{1,2,3} C{1,2,3} B3 C2 A1 ACB
Carnegie Mellon 174 Constraint vs. Mathematical Programming Constraint Programming  Fast algorithms for special problems Computationally effective for highly constrained, feasibility and machine sequencing problems Not effective for optimization problems with complex structure and many feasible solutions Mathematical Programming  Intelligent search strategy but computationally expensive for large problems  Does not exploit special structure Computationally effective for optimization problems with many feasible solutions Not effective for feasibility problems and machine sequencing problems MAIN IDEA Decompose problem into two parts  Use MP for high-level optimization decisions  Use CP for low-level sequencing decisions
Carnegie Mellon 175 Hybrid MILP/CP Models Two classes of scheduling problems: Multistage batch plants Continuous time State Task Network •Combine Constraint Programming (CP) with Mixed Integer Linear Programming (MILP) to overcome combinatorial explosion •Two subproblems: assignment (MILP) and sequencing (CP)
Carnegie Mellon 176 Dvyx vyxGts xfM   ,, 0),,(.. )(min:)2( pmn T Rvyx avCyBxA aCvByAxts xcM    ,}1,0{,}1,0{ .. min:)1( MILP/CP Hybrid Models MILP: CP: Complicating rows Complicating variables (not in objective function) Hybrid: MILP (optimality) CP (feasibility) Jain, Grossmann (2001) The picture can't be displayed. Dvyx Rvyx vyxG xx aCvByAxts xcM pmn T      ,, ,}1,0{,}1,0{ 0),,( .. min:)3(
Carnegie Mellon 177 Theorem: If the MILP/CP decomposition method is applied to solve problem (M3) with the above cuts, the method converges to the optimal solution or proves infeasibility in a finite number of iterations. Decomposition Hybrid Model MILP CP No-good Cuts (weak) Balas & Jeroslow (1972)
Carnegie Mellon 178 Job 1 Job 2 Job n Time Unit 1 Unit m Time Release date Due date Processing time Scheduling of parallel units (Single Stage) (Jain and Grossmann, 2001) Given: n jobs/orders (release dates, processing times, due dates) m units (cost different for each unit) Find schedule that minimizes cost and meets all due dates
Carnegie Mellon 179 Decomposition Strategy Job 1 Job 2 Job 3 Machine 1 Machine 2 Machine 3 MILP CP Job 4 Job 5 Job 6 Assignment Sequencing
Carnegie Mellon 180 Hybrid Optimization Model Mmrdpx Iix Iixpdts rtsts xC iiii Ii imim Mm im Mm imimii ii Ii Mm imim             }{min}{max 1 .. min Assignment tasks to units: Mixed-integer linear programming     otherwise munittoitaskif xim 0 1
Carnegie Mellon 181 MmIimzthenxif iim  ,)()1( IiDdurationiDstartiIiMz MmIitsx Iitrequiresi pdurationi Iipdstarti Iirstarti i iim z z zi i i i i       .,.; ,0},1,0{ . . . Sequencing of tasks in each unit Global constraint (implicit) Constraint Programming
Carnegie Mellon 182 Decomposition Strategy Separate problem into assignment and sequencing problems Assign jobs Sequence jobs Feasible? Add cuts Stop YesNo Fix assignments Analyze solution Find the minimum assignment (production) cost MILP Find a feasible sequence for the chosen assignment CP
Carnegie Mellon 183 Cuts No sequence can be found in machine 2 in the assignment of is infeasible and can be cut out The cuts also exclude large number of supersets CP-1 CP-2 CP-3 due date Cut out this assignment OK OK Inf. A B C ED F 3 2 1 (yD2 + yE2 ≤ 1)   k m m k k im k m m k Ii im IBxiI MmBx k m    ,1 1Cuts for infeasible assignments:
Carnegie Mellon 184 Results Test Problems MILP: CPLEX 6.5 CP: ILOG Scheduler 4.4 SUN Ultra 60 Tasks Units MILP(s) CP(s) Hybrid (s) 3 2 0.04 0.01 7 3 0.31 0.04 12 3 926.3 3.84 15 5 1784.7 553.5 20 5 18,142.7 * 68,853.5 * 0.02 0.6 4.5 2.6 14.6 * Suboptimal Order magnitude reductions!
Carnegie Mellon 185 Effect of new data Release dates, due dates and durations with arbitrary rational numbers (Harjunkoski et al., 2002) 2 machines 3 machines 3 machines 5 machines 5 machines 3 jobs 7 jobs 12 jobs 15 jobs 20 jobs Problem Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 MILP 0.04 0.04 0.31 0.27 926.3 199.9 1784.7 73.3 18142.7 102672.3 2M, 3J 3M, 7J 3M, 12J 5M, 15J 5M, 20J Problem Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Hybrid 0.00 0.01 0.51 0.02 5.36 0.03 0.64 0.92 36.63 4.79 CPU times from Jain and Grossmann (2001) (integer numbers) CLP 0 0.02 0.04 0.14 3.84 0.38 553.5 9.28 68853.5 2673.9 Hybrid 0.02 0.01 0.52 0.02 4.18 0.02 2.25 0.04 14.13 0.41

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Mixed-integer Models for Planning and Scheduling - Ignacio E. Grossmann

  • 1. 1 Mixed-integer Models for Planning and Scheduling Ignacio E. Grossmann Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA 15213 University of Alicante January 16-17, 2018
  • 2. 2 1. Introduction 2. Modeling 3. Solution strategies General Plan Short Course Goals 1. Gain an appreciation for optimization-based planning/scheduling 2. Learn general guidelines for modeling and solving these problems
  • 3. 3 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 . Enterprise-wide Optimization (EWO) Key issue: - Integration of information, modeling and solution methods
  • 4. 4 Source: Tayur, et al. [1999] Enterprise Resource Planning System Materials Requirement Planning Systems Distributions Requirements Planning System Transactional IT External Data Management Systems Strategic Optimization Modeling System Tactical Optimization Modeling System Production Planning Optimization Modeling Systems Logistics Optimization Modeling System Production Scheduling Optimization Modeling Systems Distributions Scheduling Optimization Modeling Systems Analytical IT Demand Forecasting and Order Management System Strategic Analysis Long-Term Tactical Analysis Short-Term Tactical Analysis Operational Analysis Scope Methods/Tools for Enterprise-wide Optimization
  • 5. 5 Enterprise-wide Optimization • The supply chain is large, complex, and highly dynamic • Optimization can have very large financial payout WellheadWellhead PumpPump TradingTrading Transfer of Crude Transfer of Crude Refinery Optimization Refinery Optimization Schedule Products Schedule Products Transfer of Products Transfer of Products Terminal Loading Terminal Loading Petroleum industry Dennis Houston (2003)
  • 6. 6 • Pharmaceutical process (Shah, 2003)  Primary production has five synthesis stages  Two secondary manufacturing sites  Global market Lifecycle Management 0.5 - 2 yrs 1 - 2 yrs 1.5 - 3.5 yrs 2.5 - 4 yrs 0.5-2 yrs Discovery Market 2-5 yrs Submission& Approval 10-20 yrs Phase 3Phase 2a/bPhase 1 Pre- clinical Development Targets Hits Leads Candidate R&D Pharmaceutical industry Pharmaceutical supply chain (Gardner et al , 2003)
  • 7. 7 -The modeling challenge: Planning, scheduling, control models for the various components of the supply chain, including nonlinear process models? Research Challenges - 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: How to effectively anticipate effect of uncertainties ? - Algorithmic and computational challenges: How to effectively solve the various models including nonconvex problems in terms of efficient algorithms, and in terms of modern computer architectures?
  • 8. 8 Planning and scheduling Given existing plant/supply chain and product demands over time, Determine production & inventory targets, time activities Objectives: Economics (Profit, Cost) Feasibility (Just-in-time) No clear cut distinction between Planning and Scheduling Planning: Usually driven by economics, longer term Scheduling: Usually driven by feasibility, short term Long-term goal: Integration of both
  • 9. 9 Overview of Discrete-Continuous Optimization Models a. Linear and mixed-integer linear programming b. Nonlinear and mixed-integer nonlinear programming c. Logic-based optimization: Generalized disjunctive programming Constraint programming Lagrangean decomposition Outline Examples of Models Single stage, parallel units wo/w due dates. Multistage, parallel units, due dates. Aggregated Multistage Zero-Wait Discrete time MILP short-term batch scheduling (State Task Network) Continuous time MILP short-term batch scheduling (State Task Network) MINLP multistage cyclic scheduling model Refinery scheduling and blending, Multiperiod refinery planning MILP Supply Chain Optimization Model NLP multiperiod production planning model with process models Supply Chain with Stochastic Inventory Management
  • 10. 10 References Mendez, C.A., J. Cerda, I.E. Grossmann, I. Harjunkoski, M. Fahl, “State-of-the-art Review of Optimization Methods for Short-term Scheduling of Batch Processes,” Comp. & Chemical Engineering 30, 913-946 (2006). 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). Kallrath, J. “Planning and scheduling in the process industry,” OR Spectrum, 24, 219-250 (2002). Pinto, J.M. & Grossmann, I.E., “Assignments and sequencing models of the scheduling of process systems,” Annals of Operations Research, 81, 433 – 466 (1998). Shah, N., “Single- and multisite planning and scheduling: Current status and future challenges,” Proceedings of FOCAPO-98 75 – 90 (1998). Mauderli. A. M.: Rippin. D. W. T. Production Planning and Scheduling for Mu1tip;;pose Batch Chemical Plants. Comp. Chem.Eng. 3, 199 (1979). Reklaitis, G. V. Review of Scheduling of Process Operation. AIChE Symp. Ser. 78, 119-133 (1978). 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,” Computers and Chemical Engineering, 62, 161-193 (2014).
  • 11. 11 ROAD-MAP FOR BATCH SCHEDULING (1) TASK TOPOLOGY: - Single Stage (single unit or parallel units) - Multiple Stage (multiproduct or multipurpose) - Network (2) EQUIPMENT ASSIGNMENT - Fixed - Variable (3) EQUIPMENT CONNECTIVITY - Partial - Full (4) INVENTORY STORAGE POLICIES - Unlimited intermediate storage (UIS) - Non-intermediate storage (NIS) - Finite intermediate storage (FIS): Dedicated or shared storage units - Zero wait (ZW) (5) MATERIAL TRANSFER - Instantaneous (neglected) -Time consuming (no-resource, pipes, vessels) A B C 1 2 3S1 S2Heat Reaction1 Separation Reaction 3 S3 S5 S4 S7 S6 Reaction2 1h 1h 3h 2h 2h 90% 10% 40% 60%70% 30%
  • 12. 12 ROAD-MAP BATCH SCHEDULING (6) BATCH SIZE: - Fixed - Variable (mixing and splitting operations) (7) BATCH PROCESSING TIME - Fixed - Variable (unit / batch size dependent) (8) DEMAND PATTERNS - Due dates (single or multiple product demands) - Scheduling horizon (fixed, minimum/maximum requirements) (9) CHANGEOVERS - None - Unit dependent - Sequence dependent (product or product/unit dependent) (10) RESOURCE CONSTRAINTS - None (only equipment) - Discrete (manpower) - Continuous (utilities) 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon i i’changeoveri’
  • 13. 13 ROAD-MAP BATCH SCHEDULING (11) TIME CONSTRAINTS - None - Non-working periods - Maintenance - Shifts (12) COSTS - Equipment - Utilities (fixed or time dependent) - Inventory - Changeovers (13) Degree of certainty - Deterministic - Stochastic
  • 14. 14 ROAD-MAP FOR OPTIMIZATION APPROACHES (A) TIME DOMAIN REPRESENTATION - Discrete time - Continuous time (B) EVENT REPRESENTATION DISCRETE TIME - Global time intervals CONTINUOUS TIME - Time slots - Unit-specific direct precedence - Global direct precedence - Global general precedence - Global time points - Unit- specific time event (C) MATERIAL BALANCES - Lots (Order or batch oriented) - Network flow equations (STN or RTN problem representation) (D) OBJECTIVE FUNCTION - Makespan - Earliness/ Tardiness - Profit - Inventory - Cost 7 6 5 4 3 2 1 9 8 19 18 11 10 2 2 21 20 25 24 23 TIME TASK TIME EVENTS TASK TIME TASK
  • 15. 15 Optimization approaches Mixed-integer (non)linear programming (MILP/MINLP) Disjunctive programming/logic optimization Constraint logic programming Combinatorial optimization (traveling salesman) Local search methods (simulated annealing, genetic algorithms) Decomposition techniques: Lagrangean decomposition
  • 16. 16 Mathematical Programming min f(x, y) Cost s.t. h(x, y) = 0 Process equations g(x, y)  0 Specifications x X Continuous variables y {0,1} Discrete variables Continuous optimization Linear programming: LP Nonlinear programming: NLP Discrete optimization Mixed-integer linear programming: MILP Mixed-integer nonlinear programming: MINLP
  • 17. 17 Generalized Disjunctive Programming (GDP)     Ω ,0)( 0)( )(min 1 falsetrue,Y Rc,Rx trueY Kk γc xg Y Jj xs.t. r xfcZ jk k n jkk jk jk k k k                      • Raman and Grossmann (1994) (Extension Balas, 1979) Objective Function Common Constraints Continuous Variables Boolean Variables Logic Propositions OR operator Disjunction Fixed Charges Constraints Multiple Terms / Disjunctions
  • 18. 18 Constraint Programming Key Features  Variables: Integer, Boolean, Continuous  Constraints • Algebraic constraints: x1 + x2 ≤ 5 • Logical constraints: [a.end ≤ b.start]  [b.end ≤ a.start] • Global constraints: alldifferent(x) • Meta-constraints: iS (xi < 5) = 3  State Objective Function (optional)  Write an algorithmic search for finding values of the variables satisfying the constraints • Implicit enumeration • Constraint propagation – domain reduction
  • 19. 19 Problems Multistage, parallel, due dates MILP Multistage Aggregate Zero-Wait IP/LP Multipurpose, resource constraints: State-Task-Network (Resource Task Network) Discrete time MILP Continuous time MILP Single stage, parallel MILP Single stage parallel, due dates MILP Batch Continuous Multistage, single, cyclic schedule MINLP Multiperiod production planning NLP Refinery scheduling/blending Iterative MILP Multiperiod Supply Chain MILP Stochastic Inventory Magmt. MINLP
  • 20. 20 Given N tasks (i) and M units (m) Tasks (Jobs/Orders) = batches fixed lot size, fixed processing times pim Fi ={m | units m can perform task i} N ={ (i1,i2) cannot share same unit} Find how to assign tasks j to machine m to minimize completion time MS Parallel units (single stage) Unit m Task 1 Task 2 . Task N Processing Time Machine m TimeTime
  • 21. 21 xim assign task i unit m (0-1) Min Makespan Makespan is largest time for every unit m Every task i to one unit m MILP Optimization Model }1,0{,0 1 min      im m im im i im xMS ix mMSxpst MS
  • 22. 22 Parallel Units (Single Stage) with Due Dates (Jain and Grossmann, 2001) Given: N jobs/orders (release dates, processing times, due dates) M units (cost different for each unit) Find schedule that minimizes cost and meets all due dates Job 1 Job 2 Job n Time Unit 1 Unit m Time Release date Due date Processing time ri di pim
  • 23. 23 MILP Optimization Model itasktimestartts otherwise munittoitaskif x iim      0 1 Mmrdpx Iix Iixpdts rtsts xC iiii Ii imim Mm im Mm imimii ii Ii Mm imim               }{min}{max 1 .. min Cost processing Earliest start Latest start Assign to only one unit Redundant constraint
  • 24. 24 Sequencing tasks in each unit trueareyORythentruexANDxIf unitgivenonitaskbeforeitaskifyLet iiiimiim ii ''' ' '1 imiiii ptststhenyIf  '' 1 MmiiIiixxyy miimiiii  ,',',1''' ',',)1( '' iiIiiyMxptsts iiim Mm imii   Big-M Constraint 0},1,0{},1,0{ '  iiiim tsyx MILP CPLEX 6.5 3 tasks, 2 units 0.04 sec 12 tasks, 3 units 926 sec 20 tasks, 5 units 18,000 sec
  • 25. 25 Short Term Scheduling Multistage Plants Stage I II III IV ¬ Individual customer orders 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon (ICI) Pinto, Grossmann (1995)
  • 26. 26 Given: ¬ Product demands and deadlines ¬ Topology of the plant ¬ Processing times of orders in units ¬ Set-up times Determine: ¬ Assignment of orders to units ¬ Timing of operations of units Objective Minimize Earliness Deliver as close as possible to deadline Just-in-time Short Term Scheduling Problem
  • 27. 27 ¬ All data are deterministic and fixed over the time horizon of interest. ¬ Set-up times are only equipment dependent. ¬ Batch sizes are fixed parameters. ¬ Each order must be processed by exactly one unit per stage (No splitting). ¬ No resource constraints considered ¬ Unlimited intermediate storage between stages Parameters di due date of order i SUj set-up time in unit j Tij processing time of order i in unit j Assumptions
  • 28. 28 Tsjk starting time in unit j during time slot k Tejk finishing time in unit j during time slot k Tsiiil starting time for order i in stage l Teiiil finishing time for order i in stage l Wijkl 0-1 variable that assigns order i stage l to unit j slot k . . . Order 1 2 NO 1 2 3 1 2 1 2 Unit 2 Unit NU time Time Slot Set-up Processing time Unit 1 . . . Representation and Variables
  • 29. 29 MILP Model Assignment Wijkl kSLj  j   1 i,l Si Wijkl lSi  i   1 j, k SLj Every order i, stage l Every unit j, slot k Tejk  Tsjk 1 j, k SLj  K j   iiilil LSliTsiTei   ,1 TeiiLi  di i Due date Unit slots Order stages Timing   j i Sl jijijkljkjk SLkjSUpWTsTe i   ,   i j SLk jijijklilil SliSUpWTsiTei j    , End/Start Unit End/Start Order
  • 30. 30 MILP Model Time Matching jkilijkl TsiTsiWIf 1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1 Big-M Other j i Sl lijk i Sl ijkl SLkjWW ii     ,,1 Objective: Minimize Earliness iiL i iTeihMax  Maximize end times order last stage L symmetry breaking
  • 31. 31 Reformulation Time matching constraints :   ijjkilijkl SlSLkjiTsTsiWM  ,,,1   ijjkilijkl SlSLkjiTsTsiWM  ,,,1 i j k ijkl j k jkijklil SliTsWTsi   , jjk i l ijkljk SLkjTs   , ijijklijkl SlSLkjiWU  ,,,.0  jjkjk SLkjyU  ,. ¬ Fewer constraints ¬ Tighter LP relaxation Alternative formulation: • Multiply and perform exact linearization • Disaggregated variables and apply assignment constraints )( jkilijkl TsTsW  ijkl Reformulation
  • 32. 32 Heuristic 1. Determine optimal sequence for each unit independently Apply single machine scheduling method 2. Enforce order in unit slots, but not requiring use of all slots 1 5 2 1 3 5 2 4 6 empty slots Unit j Optimal sequence for all 6 slots If orders 1, 5, 2 assigned unit j Actual sequence 1, 5, 2 Preordering has effect of fixing many Wiljk to zero
  • 33. 33 STEP 1. Minimize In-Process time (residence time) Determine assignments Overestimation of set-up times MILP - Zero gap - Interior point method Algorithm (with preordering) A A B C A A B C LP -Feasibility guaranteed from step 1 -Bound on optimum easily determined - STEP 2. Minimize Earliness Fixed assignment Eliminate unnecessary set-up times
  • 34. 34 STBS- CAPD Software Visual Basic Interface (Valli, 1998) Example: 25 units, 8 orders Solves in few seconds!
  • 36. 36 Optimal Sequencing Multistage Plants ZW policy M Stages NB Batches Given NB Batches with processing times pij and sequence dependent changeover times τimj Find optimal sequence that minimizes Cycle Time Key observation: can define sequences in terms of slacks Birewar, Grossmann (1990)
  • 37. 37 5 2 3 1 5 3 A B (a) Sequence A - B 5 2 3 1 5 3 A B (b) Slack times with zero clean-up times SL = 1 SL = 0 SL = 1 AB1 AB2 AB3 (c) Slack times with clean-up times SL = 0 SL = 0 SL = 1 AB1 AB2 AB3 5 2 3 1 5 3 A B Slack Determination 2 2 1 2
  • 38. 38 1 2 3 4 5 Asymmetric Traveling Salesman Problem ATSP 1 0 il if batchi followed by batchl y otherwise    
  • 39. 39 Cycle time stage 1 1 1,...il l y i NB  1 1,...il i y l NB  Only one out Only one in il NB i NB l il NB i i ySLpCT     1 1 1 1 1
  • 40. 40 Subtours may occur 1 2 3 4 , 1 ,il i Q i Q Let B set batches Q subset of batches y Q B Q         Above Q = {1,2}, Q = {3,4}
  • 41. 41 1 1,...il l y i NB  1 1,...il i y l NB  1 ,il i Q i Q y Q B Q         0,1ily  Integer Program ATSP il NB i NB l il NB i i ySLpCT     1 1 1 1 1min
  • 42. 42 Aggregation Model Consider ni batches for product i, i=1, NP Let NPRim number batches product i followed by product m 1 1 1 1 1 min NP NP NP i i im im i i m CT n p SL NPR        1,...im i m NPR n i NP  1,...im m i NPR n m NP  1 1,ii iNPR n i NP   0,1,2,..iNPR  Can be solved as LP treating NPRi  0 (as continuous) CYCLE http://newton.cheme.cmu.edu/interfaces
  • 43. 43 A B C D 4 3 3 3 2 5 Example 4 products, 20 batches nA = 7 nB = 5 nC = 3 nD = 5 Solution from LP Eg NPRBA = 3 NPRAA = 4
  • 44. 44 3 A B C D 4 3 3 3 3 A B C D 3 3 3 3 B D 2 2 + B -> D -> B -> D -> B A 4 B -> D -> B -> D -> B A -> A -> A-> A-> A + C -> D -> B -> A -> C -> D -> B -> A -> C -> D B -> D -> B -> D -> B A -> A -> A-> A-> A Loop Breaking Procedure
  • 45. 45 Multipurpose Plants Given: • the time horizon • the available units and storage tanks, and their capacities • the available utilities (steam, cooling water) • the production recipe (mass balance coefficients, utility requirements) • the prices of raw materials and final products Consider: Arbitrary topology Flows (no fixed batch size) Different transfer/Intermediate Storage Policies Resource Constraints Determine • the sequence and the timing of tasks taking place in each unit • the batch size of tasks (i.e. the processing time and the allocated resources) • the amount of raw materials purchased and final products sold
  • 46. 46 State Task Network (STN) (Kondili, Pantelides, Sargent, 1993) S1 S2Heat Reaction1 Separation Reaction 3 S3 S5 S4 S7 S6 Reaction2 1h 1h 3h 2h 2h 0 1 2 3 4 5 6 Time (h) Reactor 1 Reactor 2 Reactor 3 Column Reaction 2 Reaction 3 Separation Reaction 1Heating Note: Related Resource Task Network (RTN)
  • 47. 47 Discrete-time STN Model (1) Variables: Wijt = 1 if unit j starts processing task i at the beginning of time period t; 0 otherwise. Bijt = Amount of material which starts undergoing task i in unit j at the beginning of period t. Sst = Amount of material stored in state s, at the beginning of period t. Uut = Demand of utility u over time interval t. pij = 3 Task i starts at t=2 in unit j Wij2 = 1, Bij2  0 Wijt = 0, Bijt = 0, t2 2 3 4 5 T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) Allocation Constraints: tjW jIi ijt ,1    tjIiWMW jijtij Ii pt tt jti j ij ,,11 ' 1 ' ''    
  • 48. 48 iijijtijtijijt KjtiVWBVW  ,,maxmin tsCST sst ,0  Capacity limitations: Objective Function:      H t utut us HsHs H t st R st s H t st D st s UCSCRCDCZ 1 1,1, 11 max Batch Units Storage capacity tsDRBBSTST stst Kj ijt Ti is Kj ptij Ti isstst isi is s ,,1        Material balances: Inventories Produced Consumed Purchased/Sold tuBWU i i p ijtuiijtui Kjt ut ,)( 1 0          tuUU utut ,0 max  Availability of utilities: Linear function batch size Discrete-time STN Model (2)
  • 49. 49 Reformulation Previous MILP is expensive to solve tjW jIi ijt ,1    tjIiWMW jijtij Ii pt tt jti j ij ,,11 ' 1 ' ''     Culprit: big-M allocation constraints tjW i j pt tt tij Ii ,1 1 ˆ ˆ    Solution: Replace by constraint below (Shah, 1992) Fewer and tighter ! 2 hr 3 hr Unit j Tasks iIj t
  • 50. 50 Reaction 2 Reaction 1 Heating Reaction 3 Separation Product 1 Feed A Hot A Int BC Feed B Feed C Impure C Int AB Product 2 Classical Kondili Example MILP 72 0-1 variables 179 continuous variables 250 constraints STN 2003 CPLEX 7.5: 0.45 sec, 22 nodes, IBM-T40 1 2 3 4 5 6 7 8 9 10 Heating Reaction 1 Reaction 2 Reaction 3 Separation Heater Reactor 1 Reactor 2 Reactor 1 Reactor 2 Reactor 1 Reactor 2 Still 52 20 52 80 50 56 80 50 50 80 50 80 50 130 Optimal Schedule 1987 Kondili’s B&B: 908 sec, 1466 nodes, Vax-8600 1992 Shah’s B&B: 119 sec, 419 nodes, SUN Sparc STN http://newton.cheme.cmu.edu/interfaces
  • 51. 51 • Main idea  Treat the production recipe as a set of (tasks) that transform a set of resources into another set • Tasks, set I  e.g. Fill, Heat, Mix, Store, React, etc.  Represented as a rectangle • Resources, set R  e.g. equipment, material, utilities, equipment state, material location  Represented as a circle Compact Equations: Resource balances Resource Task Network Pantelides (1994) Castro et al. (2001, 2004)
  • 52. 52 ReactionA B Reactor C 1.0 0.7 0.3 Pack_D11 Rate=0-5.5714 Int1 Line3 PD11 S1 S2 T1_R1 Duration= 2 h T1_R4 Duration= 2 h R1 R4 T2_R4 Duration= 2 h T2_R1 Duration= 2 h S4 T3_R3 Duration= 4 h T4_R2 Duration= 2 h S5 S6S3 0.6 0.4 0.6 0.4 R2 T4_R3 Duration= 2 h R3 T3_R2 Duration= 4 h Legend: Discrete interaction: Continuous interaction: Examples of RTNs
  • 53. 53 • Model entities  Variables  Ni,t- assigns the start of task i to time point t (binary variable)  i,t- amount of material processed by task i at time point t (nonnegative continuous variable)  Rr,t- excess amount of resource r at time point t (nonnegative continuous variable)  Parameters (related to RTN structure)  Amount of resource r consumed/produced by task i at a time  relative to the start of the task – i,r,- usually linked to equipment resources and N – i,r,- usually linked to material resources and  – + sign indicates production, - indicates consumption Discrete-time RTN formulation
  • 54. 54 • Derive the RTN and model parameters for the following production recipe  Consider a reaction task i, that lasts 5 hours. It converts material A to B. It is carried out in a reactor. It uses 0.25 kg/s of steam per t of material being processed during the first hour. Then it used 2 kg/s of cooling water per t of material being processed until the end of the operation  Assume time intervals of one hour: =1 h Example RTN Discrete Model
  • 55. 55 Resource balances Resource limits Capacity constraints MILP Model TtRrNRRR tr Ii tiirtiirttrtrtr i     ,)( , 0 ,,,,,,11,1 0 ,     TtRrRR r,tr,t  ,0 max TtIiVNVN r Rr rititir Rr riti EQ i EQ i    ,max ,,,, min ,,,  
  • 56. 56 • Nonzero parameters  Reactor: i,R,0=-1; i,R,5=1  Materials: i,A,0=-1; i,B,5=1  Utilities: i,S,0=-0.25; i,S,1=0.25; i,CW,1=-2; i,CW,5=2 Reaction Duration=5 h A B RS CW 0.25 2 RTN model and parameters
  • 57. 57 Time Representations T1 T2 T3 1 2 32 0 1 2 3 4 5 6 7 8 t (hr) 3 2T1 T2 T3 Event-Based Representation 2 hr 1.5 hr 3 hr Interval = 0.5 hr Discrete Time Representation T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) • Fixed time points • Large size • Constant processing times Continuous Time Representation II Continuous Time Representation I 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 • Small size • Variable proc. Times • Unknown time points niWsHTTs innin  ,)1( Postulate no of points Iterative scheme for optimal niWsHTTs innin  ,)1(
  • 58. 58 Continuous-time STN Model I (Maravelias, Grossmann, 2003) 1. Common mixed continuous-time representation  Fewer time points 0 1 2 3 4 5 6 7 8 t (hr) T1 T2 T3 • If produce ZW-state finish at time point • Else finish at or before time point 2. Unit-task decoupling and New Assignment Constraints  Fewer binaries (Ierapetritou, Floudas, 1998) 4. Addition of New Valid Inequalities  Tighter LP Relaxation 3. Utility Constraints RI1 RI2 RII 0 2 4 6 8 t (hr) 0 2 4 6 8 t (hr) CWMAX CW R2-40 R2-46.4 R1-40 R1-49.6 R1-40 R4-72 R3-40 R3-60.8 (kg/min) 40 20
  • 59. 59 Continuous-time STN Model II Zsjn = 1 if a task in I(j) starts in unit j at time point n Zpjn = 1 if a task in I(j) is being processed in unit j at time point n Zfjn = 1 if a task in I(j) finishes in unit j, at or before time point n Wsin = 1 if task i starts at time point n Wpin = 1 if task i is being processed at time point n Wfin = 1 if task i finishes at or before time point n Assignment Constraints jnjn ZpZs  If a task is assigned to start in unit j at time point n, then equipment j at time point n is not processing any other task Logic Condition: njWsZs in jIi jn   , )( njWfZf in jIi jn   , )( njZfZsZp nn jn nn jnjn    , ' ' ' ' 0 2 3 … n-1 n 0 2 3 … n-1 n Reaction Reaction 1 Reaction 2Reaction Reaction 1 Reaction 2 Task-Unit Decoupling: Define new tasks njWfWs jIi nn inin    ,1)( )( ' '' iWfWs n in n in   njWs jIi in  ,1 )( njWf jIi in  ,1 )( iWfi  00 iWsN  0|| Integer assignment constraints
  • 60. 60 A WsA2 WfA4 Tn=Ts 0 2 6 8 10 n 1 2 3 4 5 D 0 6 0 0 0 Tf 0 8 8 8 8 A Tn Time that corresponds to time point n (i.e. start of period n; finish of period n-1) Tsin Start time of task i that starts at time point n Tfin Finish time of task i that starts at time point n Din Duration of task i that starts at time point n Timing Constraints Continuous-time STN Model III niTTs nin  , ni TfTf D Ws DTsTf BsD Ws inin in in ininin iniiin in                             ,,0 1  ni TfTf Wf ZWiTTf ZWiTTf Wf inin in nin nin in                       ,, , , 1 niBsWsD iniiniin  , niWsHDTsTf inininin  ,)1( niWsHDTsTf inininin  ,)1( niTTs nin  , niWfHTTf innin  ,)1(1 niZWiWfHTTf innin  ),()1(1 niWsHTfTf ininin   ,1 niDTfTf ininin   ,1 MIP Constraints (big-M)
  • 61. 61 S3 40% 60% 75% S1 S2 S4 BO A,S3,4=5 BO A,S4,4=15 BsA2=BfA4=20 BI A,S1,2=8 BI A,S2,2=12 25% Batch Size Constraints and Mass Balances niWsBBsWsB in MAX iinin MIN i  , niWfBBfWfB in MAX iinin MIN i  , )(,, iSIsniBsB inis I isn   )(,, iSOsniBfB inis O isn   1, )()( 1,     nsBBSSSS sIi I isn sOi O isnnssnsn nsCS ssn  , Utility Constraints nriBsWsR inirsinir I irn  ,, nriBfWfR inirsinir O irn  ,, nrRRRR i I irn i O irnrnrn    ,11 nrRR MAX rrn  , RI1 RI2 RII 0 2 4 6 8 t (hr) 0 2 4 6 8 t (hr) CWMAX CW R2-40 R2-46.4 R1-40 R1-49.6 R1-40 R4-72 R3-40 R3-60.8 (kg/min) 40 20 WsA2, BsA,2 WfA4, BfA2 A CW Continuous-time STN Model IV
  • 62. 62 Continuous-time STN Model V Addition of new valid inequalities  Tighter LP relaxation DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H    )( jIi n in jHD DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H DA1 DB1 DA2 DB2 DA3 DB3 1 2 3 H     )( ' ' , jIi n nn in njTHD      )( ' ' ' ,)( jIi nini nn ini njTBfWf  Smaller Bs’ Lower Production Lower Profit  STN http://newton.cheme.cmu.edu/interfaces
  • 63. 63 MILP Formulation Novel Tightening Constraints Utility Constraints Novel Assignment Constraints njWfWs jIi nn inin    ,1)( )( ' '' iWfWs n in n in   niBsWsD iniiniin  , niWsHDTsTf inininin  ,)1( niWsHDTsTf inininin  ,)1( niTTs nin  , niWfHTTf innin  ,)1(1 niZWiWfHTTf innin  ),()1(1 niWsBBsWsB in MAX iinin MIN i  , niWfBBfWfB in MAX iinin MIN i  , niBfBpBpBs inininin   ,11 )(,, iSIsniBsB inis I isn   )(,, iSOsniBfB inis O isn   1, )()( 1,     nsSSSPBBSS snsn sIi I isn sOi O isnnssn nriBsWsR inirsinir I irn  ,, nriBfWfR inirsinir O irn  ,, nrRRRR i I irn i O irnrnrn    ,11    )( ' ' , jIi n nn in njTHD    )( ' ' ' ,)( jIi niin nn iin njTvdBffdWf Finish time of task i Mass balances  s ssnSSZ max
  • 64. 64 Example T1 T2 T3 T4 T5 T7 T8 T6 T9 T10 F1 S1 S2 INT1 S3 P1 P2 WS P3 F2 S5 INT2 S4 ADD S6 0.95 0.05 0.1 0.9 0.5 0.5 0.98 0.02 U2 U1 U3 U4 U5 U6 Unlimited Storage Finite Storage No Intermediate Storage Zero-Wait Cooling Water Low Pressure Steam High Pressure Steam
  • 65. 65 Results U1 U2 U3 U4 U5 U6 0 2 4 6 8 10 12 t (hr) T1 T1 T1 T2T2 T2 T3 T3 T3 T4 T5 T4 T7 T8 T9 T10 T5 0 10 20 30 40 50 0 2 4 6 8 10 12 CW-MAX HPS-MAX CW LPS Z = $13,000 3067 Constraints 180 Binary Variables 1587 Continuous Variables LP Relaxation 19,500 CPU s (CPLEX 7.5) 62.8 Nodes 2,107
  • 66. 66 |T| Binary variables Total variables Constraints RMIP MIP CPUs Nodes 5 440 511 873 154.17 184 1748 328357 Multistage multiproduct plant: Total earliness minimization. CPLEX 11.1 on Intel Core2 Duo T9300 @2.5 GHz, Windows Vista |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 Discrete-time Continuous-time Was 45,520 s with CPLEX 10.2 on Pentium 4, 3.4 GHz Progress in Discrete time makes Continuous less Pressing Castro (2008)
  • 67. 67 Continuous multistage plants ••• Stage 1 Stage 2 Stage M Product 1 2 NP • • • • • • • • • • • • • • • Cyclic schedules (constant demand rates, infinite horizon) Intermediate storage Given : N Products Transition times (sequence dependent) Demand rates Determine : PLANNING Amount of products to be produced Inventory levels SCHEDULING Cyclic production schedule Sequencing Lengths of production Cycle time Objective : Maximize Profit = + Sales of products - inventory costs - transition costs Pinto, Grossmann (1996)
  • 68. 68 Optimal length of cycle determined largely by transition and inventory costs 2001000 0 100 200 300 400 500 Length cycle (hrs) $/hr sales of products profit transition costs inventory costs Critical to model properly inventory levels and transition times Optimal Trade-offs
  • 69. 69 a) Assignments of products to slots and vice versa iy k ik  1 ky i ik  1 Assumptions : 1. Each product is processed in the same sequence at each stage 2. Each stage consists of a single production line 3. The schedule is cyclic time Stage 1 Stage 2 Stage M k = 1 k = 2 k = NP k = 1 k = 2 k = NP k = 1 k = 2 k = NP ••• ••• ••• ••• Transition Processing Time Slot Binary variables for assignments yik = 1 product i assigned to slot k 0 otherwise Basic ideas : a. NP products b. NP time slots at each stage MINLP Model (1)
  • 70. 70 b) Definition of transition variables kjiyyz jkikijk   ,11 c) Processing rates, mass balances and amounts produced mkiTpppWp ikmimikm   1...111   MmkiWpWp ikmimikm  mkTpW kmkmkm   treated as continuous MILP model MINLP Model (2) d) Timing constraints mkiyUTpp ik T imikm  0 mkTppTp i ikmkm   mkTsTeTp kmkmkm  mNPkzTeTs i j ijkijmkmmk    1...111   i j ijij zTs 1111  1...11   MmkTsTs kmkm 1...11   MmkTeTe kmkm mzTpTc k i j ijkijmkm              Cycle time Processing time Transitions ijkz
  • 71. 71 e) Inventory levels for intermediates   1...1,min 1   MmkI0TsTsTpI1 kmkmkmkmkmkm  I2km  km  km1km1 max 0,Tekm  Tskm1  I1km k m  1... M 1 I3km  km1km1 min Tekm1  Tekm,Tpkm1  I 2km k m  1... M 1 0  I1km  Imaxkm k m  1... M 1 0  I2km  Imaxkm k m  1... M 1 0  I3km  Imaxkm k m  1... M 1 Imaxkm  Ipikm i  k m Ipikm  Uim I yik  0 i k m  1... M Tskm Tekm Tpkm Tskm+1 Tekm+1 Tpkm+1 Tskm Tekm Tpkm Tskm+1 Tekm+1 Tpkm+1 Tpkm Tpkm+1 Tskm+1 - Tskm Tekm+1 - Tekm time time Inventory level Inventory level stage m stage m+1 I0km I1km I2km I3kmI0km I1km I3km I2km MINLP Model (3) Inventories: Case 1 Case 2
  • 72. 72 f) Demand constraints iTcdWp i k ikM  MINLP Model (4) Note: Linear g) Objective function : Maximize Profit Profit  pi WpikM Tck  i  INCOME  Ctrij zijk Tck  j  i  TRANSITION COST  Cinvim Ipikm Tcm  k  i  INTERMEDIATE INVENTORY COST  1 2 Cinvfi piM  WpikM Tc     k  i  TppikM FINAL INVENTORY COST Note: Nonlinear Non-differentiable MINLP
  • 73. 73 min {f1(x), f2(x)} = - max {-f1(x), -f2(x)} = f1(x) - max {0, f1(x)-f2(x)} Handling Non-differentiabilities Option 1: Smooth Approximation   f (x)2  2 2  f(x) 2   max 0, f(x) Replace by Balakrishna, Biegler (1992) 0    f (x)  U1 1 y  0    U2y Option 2: Mixed-integer approach Raman, Grossmann (1991)   max 0, f(x) Replace by    f(x) for f(x) > 0, and   0 for f(x) < 0  y = 1,  = f(x) for f(x) > 0, and y=0,  = 0 for f(x) < 0 Option 2 is superior due to errors introduced with option 1
  • 74. 74 time (h) time (h) Intermediate storage (ton) Stages 1-2 Intermediate storage (ton) Stages 2-3 10 20 10 20 A C B E HF D G stage 1 stage 2 stage 3 cycle time = 675 hrs Optimal solution Profit = $6609/h - 47% improvement MINLP model 448 binary 0-1 variables, 2050 continuous variables, 3010 constraints Example 3 stages, 8 products, 3 DICOPT (CONOPT/CPLEX): 38.2 secs MULTISTAGE http://newton.cheme.cmu.edu/interfaces
  • 75. 75 Refinery Scheduling and Blending crude-oil marine vessels storage tanks charging tanks crude dist. units other prod. units comp. stock tanks blend headers finished product tanks shipping points crude-oil blending gasoline blending product delivery fractionation and reaction processes crude-oil unloading STANDARD REFINERY SYSTEM CRUDE OIL UNLOADING AND BLENDING PRODUCTION UNIT SCHEDULING PRODUCT SCHEDULING AND BLENDING gasoline can yield 60-70% of refinery’s profit !! •Production logistics •Production quality (scheduling) (blending) Multiple product demands Inventory and pumping constraints Resource allocation Timing of operations Logistic and operating rules Variable product recipes Product specifications Complex correlations for product properties • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 76. 76 Problem statement i’ i’’ B2 blenders product tanks component tanks min/max product specifications - prmin p,k1 ≤ prp,k1,t ≤ prmax p,k1 - prmin p,k2 ≤ prp,k2,t ≤ prmax p,k2 … - prmin p,kn ≤ prp,kn,t ≤ prmax p,kn component properties - pri’’,k1 - pri’’,k2 … -- pri’’,kn - pri,kn p i i’ i’’ B3 B1 p’ p’ p’’ GIVEN: •Scheduling horizon •Components, Products •Storage tanks, Blend headers •Component properties, stocks and supplies •Product specifications, stocks and demands •Min/Max flowrates and concentrations •Correlations for predicting product properties •Operating rules THE GOAL IS TO DETERMINE: •Allocation of resources •Inventory levels in tanks •Component concentrations in each product •Volume of each product •Pumping rates •Production and storage tasks timing MAXIMIZE PRODUCTION PROFIT REQUIRES SIMULTANEOUS SCHEDULING AND BLENDING • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 77. 77 Proposed optimization approach fi’’,t fi’,t fi,t FI i’’,p,t FI i’,p,t FI i,p,t FP p,t i’ i’’ B2 blenders product tanks component tanks p i i’ i’’ B3 B1 p’ p’ p’’ • Linear approximations for product properties 0 Due date 1 Due date 2 Due date 3 Due date NO ... Production Horizon Product Due Dates D1 D2 D3 D4 time DISCRETE TIME FORMULATION CONTINUOUS TIME FORMULATION • MILP multiperiod optimization method • Discrete or continuous time domain representation • Discrete decisions for resource allocation and operating rules • Iterative procedure for improving predictions • Integrated production logistics and quality specifications • Variable recipes and min/max component concentrations • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results Product Due Dates D1 D2 D3 D4 time T1 T2 T3 T4 T5 T6 T7 T8 T9 SLOTS Sub-interval
  • 78. 78 Product property prediction 86 87 88 89 90 91 92 0 0.2 0.4 0.6 0.8 1 Component 'A' volume fraction property Non-linear correlation linear volum. average linear volum. average + bias Property value Comp A: 92.1 Comp. B: 86.1 Correction factor ‘bias’ = 0.24 BLEND 40% COMPONENT ‘A’ 60% COMPONENT ‘B’ tkpvprpr I tpi i kitkp ,,,,,,,   tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFbiasFpr FprFbiasFpr P tptkp P tpkp I tpi i ki P tptkp P tpp,k ,,,,,, max , ,,,,,,, min    •Volumetric average tpiFFv I tpi P tp I tpi ,,,,,,,  •Non-linear flowrate-composition matching Linear approximation for non-linear product properties Volumetric product property correlation (Linear) • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 79. 79 Iterative procedure Generate initial guess recipes Compute non-linear properties for all products and time slots Compute correction factors 'bias' Solve MILP model using correction factors Compute non-linear properties for all products and time slots Fix product recipes for products on-spec All products on-spec or iteration limit reached Solution for the Scheduling and Blending problem component volume fractions in blends component volume fractions in blends NO YES • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 80. 80 Integrated scheduling and blending model Discrete time domain representation Product Due Dates D1 D2 D3 D4 time tnA B t p tp  , tpFF P tp i I tpi ,,,,  tpiFrcpFFrcp P tppi I tpi P tppi ,,, max ,,,, min ,  tpAserateFAserate tpttp P tptpttp ,)()( , max ,, min  tiFefinvV ttp I tpitii I ti , ', ',,,   tiVVV i I tii ,max , min  tpddFinvV td pd tt p tpp P tp , ' ',,    tpVVV p P tpp ,max , min  tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFprFbiasFprFpr P tpkp I tpipk I tpi i ki P tpp,k ,,, max ,,,,,,,, min   dpddF dd dp dt P tp ,' ' ,,             t p I tpi i i P tpp FcFpMax ,,,          i t I tii p t P tpp t p I tpi i i P tpp VspVspFcFpMax ,,,,, ALLOCATION COMPOSITION CONCENTRATION, MATERIAL BALANCE AND STORAGE CAPACITY VOLUMETRIC FLOW-RATE, MATERIAL BALANCE AND STORAGE CAPACITY SPECIFICATION PRODUCT DEMANDS OBJECTIVE FUNCTION MAXIMIZE PROFIT • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results C O M P O N E N T S P R O D U C T S
  • 81. 81 Integrated scheduling and blending model Continuous time domain representation Product Due DatesD1 D2 D3 D4 time T1 T2 T3 T4 T5 T6 T7 T8 T9 tpSErateFAhrateSErate ttp P tptppttp ,)()1()( max ,, minmin  tpAhrateF tpp P tp ,, max ,  tiFEfiniV ttp I tpitii I ti , ', ',,,   tiFSfinvV ttp I tpitii I ti ,' ', ',,,   tiVVV i I tii ,' max , min  p dd pd dt p tpp P tp dpddFinvV ,' ' ',,    tpVV P tpp ,' , min  tpAlSE tpptt ,, min  tAhSE p tptt   , tSE tt   1 dt TtdS  1 dt TtdE  d p tp B t p tp TttdAnA    )1,(,,1, FLOWRATES STORAGE CAPACITY MATERIAL BALANCE DURATION SEQUENCING SUB-INTERVALS BOUNDS ALLOCATION tnA B t p tp  , tpFF P tp i I tpi ,,,,  tpiFrcpFFrcp P tppi I tpi P tppi ,,, max ,,,, min ,  ALLOCATION COMPOSITION CONCENTRATION tiVVV i I tii ,max , min  tpddFinvV td pd tt p tpp P tp , ' ',,    tpVVV p P tpp ,max , min  tkpFprFprFpr P tpkp I tpi i ki P tpp,k ,,, max ,,,,, min   tkpFprFbiasFprFpr P tpkp I tpipk I tpi i ki P tpp,k ,,, max ,,,,,,,, min   STORAGE CAPACITY SPECIFICATION MATERIAL BALANCE OBJECTIVE FUNCTION MAXIMIZE PROFIT • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results C O M P O N E N T S P R O D U C T S S L O T S SLOTS
  • 82. 82 Treatment of infeasible solutions Penalty for preferred recipe deviations Penalty for min/max specification deviations Penalty for component shortages tpiFDFrcp tpi R tpi P tpip ,,,,,,,   tpiFDFrcp tpi R tpi P tpip ,,,,,,,       t p i R tpi R ip R tpi R ip DpltyDpltyPenalty ,,,, tkpFprDFprop I tpi i ki S tpk P tpkp ,,,,,,,, min ,    tkpFprDFprop I tpi i ki S tpk P tpkp ,,,,,,,, max ,        t p k S tpk S pk S tpk S pk DpltyDpltyPenalty ,,,,,, tiSFeprodiniV ti ttp I tpitii I ti ,, ', ',,,     t i ti SH i SpltyPenalty , Relax some hard constraints that can generate an infeasible solution • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 83. 83 Examples and numerical results 12 STORAGE TANKS 3 BLEND HEADERS 8-DAY TIME HORIZON 6 DUE DATES • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results i’ i’’ B2 blenders product tanks component tanks G2 C3 C4 C5 B3 B1 G1 G3 C2 C7 C6 C1 C8 C9 0 Day 1 8 days Day 3 Day 4 Day 5 Day 7 Day 8 0 Discrete time formulation 0 Continuous time formulation SLOTS
  • 84. 84 Examples and numerical results COMPONENT C1 C2 C3 C4 C5 C6 C7 C8 C9 COST($/BBL) 24.00 20.00 26.00 23.00 24.00 50.00 50.00 50.00 50.00 PROD.RATE (MBBL/DAY) 15.00 33.00 20.00 14.00 18.00 10.00 0.00 0.00 0.00 INITIALSTOCK (MBBL) 48.00 20.00 75.00 22.00 30.00 54.00 12.00 20.00 15.00 MIN.STOCK (MBBL) 5.0 5.0 5.0 5.0 5.0 5.0 0.0 0.0 0.0 MAX.STOCK (MBBL) 100.00 250.00 250.00 100.00 100.00 100.00 100.00 100.00 100.00 PROPERTY P1 93.00 104.00 104.90 94.80 87.40 118.00 87.30 95.20 93.30 P2 92.10 91.90 91.90 81.50 86.10 100.00 79.50 85.80 81.90 P3 0.7069 0.8692 0.6167 0.6731 0.6540 0.7460 0.7460 0.8187 0.7339 P4 3.60 1.00 100.00 94.90 91.50 15.00 0.00 1.30 34.30 P5 16.30 4.50 100.00 97.10 95.50 100.00 0.00 6.00 57.10 P6 94.30 93.50 100.00 100.00 100.00 100.00 0.00 93.90 95.90 P7 35.00 22.70 351.10 117.10 93.00 31.30 63.30 16.00 52.40 P8 0.007 0.00 0.00 0.009 0.0002 0.05 0.0063 0.1805 0.057 P9 0.00 88.60 0.00 2.30 0.20 0.00 43.98 65.30 21.30 P10 0.00 0.1 61.30 48.90 36.00 0.00 1.04 0.60 33.30 P11 0.00 3.30 0.00 1.10 0.10 0.00 3.33 0.90 0.80 P12 0.00 0.00 0.00 0.00 0.00 15.40 0.00 0.00 0.00 9 Components 12 properties Min/Max stock Initial stock Flowrates Cost • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 85. 85 Examples and numerical results 8 Linear and 4 non- linear properties Min/Max component concentrations Min/Max storage capacities Multiple product demands PRODUCT G1 G2 G3 PRICE ($/BBL) 31.00 31.00 31.00 REQUIREMENT (MBBL) MIN MAX LIFT MIN MAX LIFT MIN MAX LIFT DAY 1 (MBBL) 5.00 45.00 10.00 5.00 50.00 12.00 5.00 50.00 10.00 DAY 3 (MBBL) 5.00 50.00 25.00 DAY 4 (MBBL) 5.00 45.00 25.00 5.00 50.00 23.00 DAY 5 (MBBL) DAY 7 (MBBL) 5.00 45.00 30.00 DAY 8 (MBBL) 5.00 45.00 10.00 5.00 50.00 22.00 INVENTORY (MBBL) 5.00 150.00 5.00 150.00 5.00 150.00 RATE (MBBL/DAY) 5.00 45.00 5.00 50.00 5.00 50.00 RECIPE (%) MIN MAX MIN MAX MIN MAX C1 0.00 22.00 0.00 25.00 0.00 25.00 C2 0.00 20.00 0.00 24.00 0.00 24.00 C3 2.00 10.00 0.00 10.00 0.00 10.00 C4 0.00 6.00 0.00 23.00 0.00 23.00 C5C 0.00 25.00 0.00 25.00 0.00 25.00 C6 0.00 10.00 0.00 10.00 0.00 10.00 C7 0.00 100.00 0.00 0.00 0.00 0.00 C8 0.00 100.00 0.00 0.00 0.00 0.00 C9 0.00 100.00 0.00 0.00 0.00 0.00 SPECIFICATIONS MIN MAX MIN MAX MIN MAX P1 95.00 98.00 98.00 P2 85.00 88.00 88.00 P3 0.72 0.775 0.72 0.775 0.72 0.775 P4 20.00 50.00 20.00 48.00 22.00 50.00 P5 46.00 71.00 46.00 71.00 46.00 71.00 P6 85.00 85.00 85.00 P7 45.00 60.00 45.00 60.00 60.00 90.00 P8 0.015 0.015 0.008 P9 42.00 42.00 42.00 P10 18.00 18.00 18.00 P11 1.00 1.00 1.00 P12 2.70 2.70 2.70 3 Products Price • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 86. 86 Ex. 1: Blending problem (fixed scheduling) INITIAL RECIPE C2 20 C3 3.598 C4 6 C6 7.246 C7 C8 16.156 C1 22 C5 25 ITERATION 1 C1 22 C2 20 C3 2 C4 4.938 C5 25 C6 10 C7 5.181 C8 1.114 C9 9.767 ITERATION 2 C1 22 C2 20 C3 2 C4 4.847 C5 25 C6 10 C7 5.198 C8 0.958 C9 9.997 C1 C2 C3 C4 C5 C6 C7 C8 C9 Convergence to the preferred recipe for product G1 MIN. SPEC. INITIAL RECIPE ITERATION 1 ITERATION 2 MAX SPEC. BLEND COST ($/BBL) 29.30 29.97 29.99 QUALITY Value Value approx. Value Approx. P1 95.00 97.891 97.898 97.7737 97.893 97.8928 P2 85.00 88.417 88.470 88.0493 88.438 88.4335 P3 0.72 0.7418 0.7325 0.7324 0.775 P4 20.00 34.455 35.418 35.409 50.00 P5 46.00 46.00 50.80 50.833 71.00 P6 85.00 96.460 91.797 91.780 P7 45.00 60.00 60.00 60.00 60.00 P8 0.0378 0.0152 0.0150 0.0150 0.0150 0.015 P9 28.458 22.974 22.923 42.00 P10 14.256 15.974 16.005 18.00 P11 0.8964 1.00 1.00 1.00 P12 1.1223 1.5684 1.5488 1.5687 1.5684 2.70 Objective: Find product recipes that satisfy all specifications at minimum cost • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results
  • 87. 87 Ex. 2: Blending and scheduling (limited production) G1 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime Objective: Make scheduling and blending decisions that maximize profit Min/max production for each time interval Product specifications Operating conditions Demands at specific due dates Discrete time formulation Continuous time formulation Profit: $1,611.21 Operates blenders at full capacity for 2.67 days less than discrete time G2 G3 G1 G2 G3 Mbbl Mbbl EVOLUTION OF COMPONENT STOCKS 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime
  • 88. 88 Ex. 3: Blending and scheduling (flexible production) 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime 0 40 80 120 160 200 240 0 1 2 3 4 5 6 7 8T ime Objective: Maximize profit of the refinery Product specifications Operating conditions Demands at specific due dates Profit: $2,448.05 (increased by 52%) Operates blenders at full capacity for 2.6 days less than discrete time Increase production by 36% Discrete time formulation Continuous time formulation G1 G2 G3 G1 G2 G3 Mbbl Mbbl EVOLUTION OF COMPONENT STOCKS
  • 89. 89 Computational results Modest number of 0-1 variables Very low CPU time requirements EXAMPLE DISCRETIME CONTINUOUSTIME Binary, Continuous, Constraints Objective CPU Binary, Continuous, Constraints Objective CPU 2 9,757,679 1,611.21 0.26 9,841,832 1,611.21 0.26 3 18,757,679 2,448.05 0.23 18,841,832 2,448.05 0.26 4 9,757,679 1,234.49 0.23 9,841,832 1,234.49 0.26 • Motivation • Modeling issues • Problem statement • Proposed optimization approach • Product property prediction • Integrated model Discrete MILP formulation Continuous MILP formulation • Treatment of infeasible solutions • Numerical results Seconds on Pentium IV (2.4 GHz) with GAMS 21.2 / CPLEX 8.1
  • 90. Multiperiod Refinery Planning Model • LP planning models (PIMS) – Fixed yield model – Swing cuts model • NLP planning models – Aggregate distillation model – Fractionation index (FI) model 90 Alattas, Palou-Rivera, Grossmann (2012) Goal: Develop nonlinear multiperiod planning
  • 91. FI Model (Fractionating Index) – 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 – Generates cut point temperature settings for the CDU – Adds few additional equations to the planning model 91
  • 92. 92 Multiperiod Refinery Planning Problem • Time horizon with N time periods • Inventories and changeovers of M crudes • Given: refinery configuration 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?
  • 93. 93 MINLP Model 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
  • 94. 94 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)
  • 95. 95 Simultaneous Cyclic Scheduling and Control of a Multiproduct CSTR Reactor Given is a CSTR reactor N products Lower bounds demand rates Dynamic model for reactions Determine cyclic schedule Cycle time Sequence Amounts to produce Lenghts transitions and their dynamic profile Objective: Maximize total profit Terrazas, Flores , Grossmann (2008) MIDO Optimization model
  • 98. 98 Simultaneous Approach for solving Dynamic Optimization Problems Problem • Few discrete decisions • Can exhibit instability due to: • Open loop models • Choice of control laws • Can add stability constraints • Requires simultaneous approach • Large, nonlinear NLP sub-problem • NLP expense >> MILP expense
  • 99. 99   K 1k iki0 z(t)zz(t) k x x x x element i k = 1 k = 2 Differential variables Continuous ´ ´ xxxx   K 1k ik(t)uu(t) k Algebraic and Manipulated variables Discontinuous ´ ´ Simultaneous Formulation Collocation on Finite Elements to tfCollocation pts · · ·· · · · · · · · · Polynomials · Mesh points hi ´ ´ ´ ´´ ´´ ´´ ´ ´ ´
  • 101. 101 Optimal sequence ADECB Cycle time = 124.8 h Profit = $7889/h Results
  • 102. 102 Technology 1 Technology I Technology 1 Technology I DMK lpt PUjpt Q PL jkpt Q WH klpt Suppliers Plants j=1,…,J Warehouses k=1,…,K Markets l=1,…,L Wijpt INVkpt CPL ijt CWH kt Model = Plant location problem (Current et al.,1990) plus Long range planning of chemical processes (Sahinidis et al., 1989) • Three-echelon supply chain • Different technologies available at plants • Multi-period model Multiperiod MILP formulation supply chain Guillen, Grossmann (2008)
  • 104. 104 1. Mass balances Plants Warehouses Markets 2. Capacity Expansion Plants Plants Binary variable (1 if technology i is expanded in plant j in period t) Multiperiod MILP formulation (I)
  • 105. 105 Transport links Multiperiod MILP formulation (II) Binary variable (1 if warehouse k is expanded in period t) Warehouses 3. Capacity Expansion Warehouses Binary variable (1 if there is a transport link between plant j and warehouse k in period t) Binary variable (1 if there is a transport link between warehouse k and market l in period t) 4. Transportation links
  • 106. 106 5. Objective function Summation of discounted cash flows Net Earnings Fixed cost Multiperiod MILP formulation (III)
  • 107. 107 Case study (I) Problem : • Redesign a petrochemical SC to fulfill future forecasted demand Case study Existing plant Potential plant
  • 108. 108 One step oxidation of ethylene Cyanation / oxidation of ehtylene Acetaldehyde Acrylonitrile Ethylene Ammoxidation of propylene Propylene Phenol HCN HCl H2SO4 O2 NH4 Hydration of propylene Reaction of benzene and propylene Isopropanol Oxidation of cumene Benzene Cumene Acetone 0.67 0.38 1.35 1 0.61 By-product 0.05 0.83 1.20 0.76 H2SO4 NaOH Others 0.010.010.01 1 Active carbon 0.01 1 0.43 0.15 1 1 0.900.17 0.83 0.83 0.40 1 O2 Technologies in each Plant Site
  • 109. 109 Multiperiod MILP Models: • Number of 0-1 variables: 450 • Number of continuous variables: 4801 • Number of equations: 4682 • CPU* time: 0.33 seconds *Solved with GAMS 21.4 / CPLEX 9.0 (Pentium 1.66GHz) Potential Supply Chain Horizon: 3 yrs
  • 110. 110 NPV = $132 million Optimal Solution
  • 111. 111 Multiperiod Production Planning LP Model i jt jt jt jt j J t T j J t T it ijt jt jt jt jt i I j JM t T j J t T j J t T Max PROFIT S P W V SF                             TtJMjJjIiWW iitijijijt  ,',,' TtJMjIiQW iitijt  ,, TtJj dSd aPa U jtjt L jt U jtjt L jt        , , 1 , j j j t ijt jt jt ijt jt i O i I V W P V W S j J t T           TtJjSdSF jt U jt jt  , TtJjSFSF U jtjt  ,0 TtJjVV U jtjt  , , , , 0jt jt it jtS P W V  Mass balance process Capacity Mass balance chemicals Shortfalls Purchases Sales Limit inventory Fixed price purchases What if prices not fixed but given by contracts?
  • 112. 112 . Discount after d jt amount. TtJRjPPCOST d jt d jt d jt d jt d jt  ,2211  TtJRj P P y P P y d jt d jt d jt d jt d jt d jt d jt d jt                              , 00 0 2 1 2 2 1 1  TtJRjPPP d jt d jt d jt  ,21 Disjunctive model TtJRjPPCOST d jt d jt d jt d jt d jt  ,2211  TtJRjPPP d jt d jt d jt  ,21 TtJRjPPP d jt d jt d jt  ,12111 TtJRjyP d jt d jt d jt  ,0 111  TtJRjyP d jt d jt d jt  ,212  TtJRjUyP d jt d jt d jt  ,0 22 TtJRjyyy d jt d jt d jt  ,21 }1,0{, 21 d jt d jt yy MILP model (Convex hull) Price drops after amount 
  • 113. 113 Bulk discount TtJRj P PCOST y P PCOST y b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt b jt                              , 0 2 2 1 1     Disjunctive Model TtJRjPPCOST b jt b jt b jt b jt b jt  ,2211  TtJRjPPP b jt b jt b jt  ,21 TtJRjyP b jt b jt b jt  ,0 11  TtJRjyUPy b jt b jt b jt b jt b jt  ,222  TtJRjyyy b jt b jt b jt  ,21 }1,0{, 21 b jt b jt yy MILP Model (Convex Hull) Price for total amount drops after amount 
  • 114. 114 Fixed-duration contracts TtJRj P P P PCOST PCOST PCOST y P P PCOST PCOST y P PCOST y l jt l tj l jt l tj l jt l jt l tj l jt l tj l tj l jt l tj l jt l jt l jt l jt l jt l tj l jt l jt l tj l jt l tj l jt l jt l jt l jt l jt l jt l jt l jt l jt l jt                                                                              , 3 2, 3 1, 3 2, 3 2, 1, 3 1, 3 3 2 1, 2 1, 2 1, 2 2 1 1 1             Disjunctive Model TtJRjPCOST LCp T lp tj lp j l jt p t     ,   TtJRjPP LCp T lp tj l jt p t     ,   LCpTTTTtJRjyUPy p t plp j l j lp tj lp j lp j  ,,,   TJRjyy l j LCp lp j   , }1,0{lp jy  , }3,2,1{LC MILP Model Price gradually drops if amount  purchased fixed number of months
  • 115. 115 Case 0-1 variables Cont. variables Constraints CPU time [s] Time periods Solution [105 $] 1 0 12,606 13,416 0.18 10 18,085.95 2 6,160 40,606 46,002 0.95 10 22,073.06 Computational Results 0 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 9 10 Time periods Qauntities[kton] fixed Ethy fixed Acet fixed Naph discount Ethy discount Acet discount Naph bulk Ethy bulk Acet bulk Naph length Ethy length Acet length Naph Purchases raw materials 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 PROFIT REVENUES PURCHASES OPERATION STORAGE*100 SHORTFALLS 1E5$ Without Contracts With Contracts Comparison without/with contracts
  • 116. 116 Computational Results ILOG CPLEX Dec 1, 2008 22.9.2 LNX 7311.8080 LX3 x86/Linux Cplex 11.2.0, GAMS Link 34 MIP Presolve eliminated 44425 rows and 38571 columns. Reduced MIP has 909 rows, 1367 columns, and 3267 nonzeros. Reduced MIP has 270 binaries, 0 generals, 0 SOSs, and 0 indicators. Implied bound cuts applied: 3 Flow cuts applied: 40 Gomory fractional cuts applied: 15 MIP Solution: 22073.060039 (959 iterations, 21 nodes) MODEL STATISTICS BLOCKS OF EQUATIONS 47 SINGLE EQUATIONS 46,002 BLOCKS OF VARIABLES 30 SINGLE VARIABLES 40,606 NON ZERO ELEMENTS 85,033 DISCRETE VARIABLES 6,160 S O L V E S U M M A R Y MODEL MULT OBJECTIVE NPV TYPE MIP DIRECTION MAXIMIZE SOLVER CPLEX FROM LINE 878 **** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 22073.0600 RESOURCE USAGE, LIMIT 0.470 10000.000 ITERATION COUNT, LIMIT 1437 1000000
  • 117. 117 Decomposable MILP Problems A D1 D3 D2 Complicating Constraints max 1,.. { , 1,.. , 0} T i i i i i c x st Ax b D x d i n x X x x i n x        x1 x2 x3 Lagrangean decomposition complicating constraints D1 D3 D2 Complicating Variables A x1 x2 x3y 1,.. max 1,.. 0, 0, 1,.. T T i i i n i i i i a y c x st Ay D x d i n y x i n          Benders decomposition complicating variables Note: can reformulate by defining 1i iy y  and apply Lagrangean decomposition Complicating constraints
  • 118. 118  MILP optimization problems can often be modeled as problems with complicating constraints.  The complicating constraints are added to the objective function (i.e. dualized) with a penalty term (Lagrangean multiplier) proportional to the amount of violation of the dualized constraints.  The Lagrangean problem is easier to solve (eg. can be decomposed) than the original problem and provides an upper bound to a maximization problem. Lagrangean Relaxation (Fisher, 1985)
  • 119. 119 n Zx eDx bAx cxZ     max bAx Assume that is complicating constraint n LR Zx eDx AxbucxuZ    )(max)( 0where u Lagrange multipliers (IP) Assume integers only Easily extended cont. vars. Lagrangean Relaxation (Fisher, 1985)
  • 120. 120 0uwhere ( )LRZ u Z n Zx eDx bAx cxZ     max n LR Zx eDx AxbucxuZ    )(max)( bAx  ZuZLR )( 0)(  Axb 0u This is a relaxation of original problem because: i) removing the constraint relaxes the original feasible space, ii) always holds as in the original space since and Lagrange multiplier is always . Lagrangean Relaxation Yields Upper Bound Complicating Constraint  Lagrangean Relaxation
  • 121. 121 n LR Zx eDx AxbucxuZ    )(max)(Relaxed problem: min ( ) 0 D LRZ Z u u   Lagrangean dual: )( 1uZLR )( 2uZLR )( 3uZLR Z n Zx eDx bAx cxZ     maxOriginal problem: DZdual gap Lagrangean Relaxation
  • 122. 122   n xu D Zx eDx AxbucxZ      )(maxmin 00 min ( ) 0 D LRZ Z u u   n LR Zx eDx AxbucxuZ    )(max)(Relaxed problem: Lagrangean dual: Combined Relaxed and Lagrangean Dual Problems: Graphical Interpretation
  • 123. 123   n xu D Zx eDx AxbucxZ      )(maxmin 00  n ZxeDxx  ,  n ZxbxAx  , Optimization of Lagrange multipliers (dual) can be interpreted as optimizing the primal objective function on the intersection of the convex hull of non- complicating constraints set and the LP relaxation of the relaxed constraints set . 0 ),( max       x ZxeDxConvx bAx cxZ n D Nice Proof Frangioni (2005) Graphical Interpretation
  • 124. 124  eDxx  Conv n ZxeDxx  ,  bAxx  cx ZLP ZD Z 0 ),( max       x ZxeDxConvx bAx cxZ n D dual gap Graphical Interpretation
  • 125. 125 Lagrangean relaxation yields a bound at least as tight as LP relaxation  eDxx  Conv  n ZxeDxx  ,  bAxx  cx ZLP ZD Z ( ) ( )D LR LPZ P Z Z u Z   Theorem
  • 126. 126  Lagrangean Decomposition is a special case of Lagrangean Relaxation.  Define variables for each set of constraints, add constraints equating different variables (new complicating constraints) to the objective function with some penalty terms. n Zx eDx bAx cxZ     max n n Zy Zx yx eDy bAx cxZ         max New complicating constraints n n LD Zy Zx eDy bAx xyvcxvZ        )(max)( Dualize x = y Lagrangean Decomposition (Guignard & Kim, 1987)
  • 127. 127 n n LD Zy Zx eDy bAx xyvcxvZ        )(max)( n LD Zx bAx xvcvZ    )(max)(1 n LD Zy eDy vyvZ    max)(2 Subproblem 1 Subproblem 2  )()(min 21 0 vZvZZ LDLD v LD   Lagrangean dual Lagrangean Decomposition (Guignard & Kim, 1987)
  • 128. 128  Lagrangean decomposition is different from other possible relaxations because every constraint in the original problem appears in one of the subproblems. Subproblem 1 Subproblem 2 Graphically: The optimization of Lagrangean multipliers can be interpreted as optimizing the primal objective function on the intersection of the convex hulls of constraint sets. Notes
  • 129. 129 Z Graphical Interpretation Subproblem 1  eDxx   bAxx  cx ZLP ZLR Conv n ZxeDxx  , Conv n ZxbxAx  , ZLD Subproblem 2 Note: ZLR, ZLD refer to dual solutions
  • 130. 130  The bound predicted by “Lagrangean decomposition” is at least as tight as the one provided by “Lagrangean relaxation” (Guignard and Kim, 1987)  For a maximization problem LPLRLD ZZZPZ )( Solution of Dual Problem ZLR or ZLD u or  minimum Piecewise linear => Non-differentiable Theorem
  • 131. 131 1,... max{ ( ) , } max { ( )}k k x k K cx u b Ax Dx d x X cx u b Ax         Assuming Dx  d is a bounded polyhedron (polytope) with extreme points 1,2...k x k K , then How to iterate on multipliers u? 0 01,.. min max{ ( )} min{ ( ), 1,.. }k k k k u uk K cx u b Ax cx u b Ax k K           => Dual problem Cutting plane approach 1 min . . ( ), 1,.. 0, k k ns t cx u b Ax k K u R          Kn = no. extreme points iteration n subgradient Note: xk generated from max{cx + uk(b-Ax) subproblems
  • 132. 132 Update formula for multipliers (Fisher, 1985) 21 ( )( ) / [0,2] k k LB k k k k LD k u u Z Z b Ax b Ax where          Subgradient ( )k k s b Ax  Steepest descent search 1k k k u u s   Subgradient Optimization Approach Note: Can also use bundle methods for nondifferentiable optimization Lemarechal, Nemirovski, Nesterov (1995)
  • 133. 133 Solution of Langrangean Decomposition Select MaxI, ε, ak Set UB = +, LB= -  Solve (RP’) to find v0 | ZLD - ZLB |<ε? or k=MaxI? Solve (P) with fixed binaries or use heuristics: Obtain ZLB Solve (P1) and (P2): Obtain ZLD k = k+1 Update uk For k = 1..K Return ZLB & Current Solution YES NO Iterative search in multilpliers of dual Notes: Heuristic due to dual gap Obtaining Lower Bound might be tricky Remarks 1. Methods can be extended to NLP, MINLP 2. Size of dual gap depends greatly on how problems are decomposed 3. From experience gap often decreases with problem size. Upper Bound Lower Bound
  • 134. 134 Multisite planning with sequence dependent changeovers is an integrated problem Month 1 Month 2 Month n -1 Month n Multi-site Network Market 3 Market 2 Market 1 A D B C Continuous Processes Production Cycle Given: • Network of production sites and markets • Capacities in each plant for producing different products (e.g. polymers) • Forecasted demands • All costs and product prices Determine: • Product assignment to sites • Production and shipment volumes • Inventory levels • Sequence of production in each site Objective: Maximize profit Profit = Sales - Operating costs - Inventory costs - Distribution costs- Changeover costs Production Site 1 Production Site 2 Production Site n-1 Production Site n
  • 135. 135 The solution coordinates sites and markets across time periods A B A BSite 1 Site 2 A 20 20 100 100 50 Site 1 Site 2 Market 1 Market 2 Site 1 Site 2 Market 1 Market 2 10 10 20 100 20 60 40 40 Month 1 Month 2 Changeover time Production time Inventory Distribution Sales Month 1 Month 2 Changeovers Inventories Decrease Capacity Increase Cost
  • 136. 136 Mathematical Model  Objective: Maximize Profit subject to  Market constraints • Balance of sales vs. shipments to markets  Production constraints • Capacity constraints: Limited capacity at each production sites • Inventory constraints: Penalties for inventory over or under target  Links across periods: a) Carry over inventories from last month b) Changeover to first product in next month • Time Balances: Task should not take longer than available time • Sequencing Constraints: Traveling Salesman Problem Constraints Source of complexity of the model: TSP constraints
  • 137. 137 Mathematical Model (MILP) sk t i ski t m smi t si t si t si t xfzzz finvinvp , 1 ,, ,,,, 1 ,          m s t tsmtsmtmtmtsts fgxcxcprofit ,,,,, 2 , 2 , 1 , 1 m,tfx s tsmtm   0,,, 2 Links across time periods Objective Function            i s t si t si t i k s t ski t ski t ski t ski t m s smi t smi t t i m s si t si t si t mi t mi t tpCOP zzzzzzCTR f penpsl profit ,, ,,,,,,,, ,,,, ,,,,, ] )([   Market Constraints  s smi t mi t fsl ,,, tsfxJxJxJ m tsmtstststststs ,0,,1, 1 , 3 , 1 , 2 1, 1 , 1   Sites Variables Market Variables Shipments Shipments Market Variables Site Variables Site Variables Site Variables Shipments
  • 138. 138 Mathematical Model (MILP) si t si t si t si t si t si t yCAPp tpCAPp ,,, ,,,   sisi t si t si t sisi t QUOTAinvpen invQUOTApen ,,, ,,,     t s t i si t i ski t ski t ski t k kis t HTRTtp zzzzzztrt     , ,,,,,,,  Capacity Constraints Inventory Time Balance Sequencing             ik k sk t si t sii t sk t sii t sii t si t ski t ski t i k ski t i ski t sk t k ski t si t yyz yz zy zzz zz zy zy ,,,, ,,, ,,, ,,,, ,, ,,, ,,, 1 1 si t k ski t i si t i si t k ski t si t i ski t sk t xlzzz xl xf zzxl zzxf ,,, , , ,,, ,,, 1 1           tsyBxA tstststs ,e ts,,,, 1 ,  Market Variables Product assignment Variables
  • 139. 139 Decomposition strategies are required Multisite production planning with sequence dependent changeovers integrates functions but involves large-scale optimization problem Computational expense Performance (Economic Profit) Problem size More sites, markets, products and time periods Decomposition strategies are required Options: Benders, Lagrangean, Two - Level. Problem can become intractable 3 markets 3 sites 3 time periods 3 products 6 markets 3 sites 6 products 6 markets 6 sites 6 products 6 time periods 6 time periods Solved in 1 s Solved in 780 s Not solved to optimality in 10,000 s 982 cont. vars. 36 discrete vars. 1,042 constraints 5,437 cont. vars. 126 discrete vars. 4,984 constraints 10,621 cont. vars. 252 discrete vars. 9463 constraints
  • 140. 140 • Temporal and Spatial Lagrangean decompositions for multi-scale problem Two types of Lagrangean decomposition Simultaneous solution of sites, markets across time periods Production Sites Markets Time period 1 Time period 2 Production Sites Markets Time period 1 Time period 2 Time periods Solved Indepen- dently Time periods Solved Indepen- dently Sites solved independentlyProduction Sites Markets Time period 1 Time period 2 Markets solved independently Spatial Decomposition Temporal Decomposition Full space solution Which is better ?
  • 141. 141 Example 1: 3 sites, 3 markets, 3 months & 3 products Price [$/ton] Operating Cost [$/ton] Inventory Cost [$/ton] Distrib. Cost [$/ton] Product A 1.20 0.010 0.010x10-2 0.2 Product B 1.00 0.008 0.008x10-2 0.2 Product C 1.00 0.015 0.015x10-2 0.2 A B C A - 100 | 0.2 200 | 0.4 B 100 | 0.2 - 100 | 0.2 C 200 | 0.4 100 | 0.4 - Changeover Costs | Time [$] | month fractionsPrices and Costs Month 1 Month 3 Market 3 Market 2 Market 1 Production Site 1 Production Site 2 Production Site 3 Month 2 Forecast for Market 3 0 50 100 150 200 250 300 350 400 450 500 Month 1 Month 2 Month 3 Tons Product A Product B Product C
  • 142. 142 B Example 1 - Output B C BSite 1 Site 2 0.2 Site 1 Site 2 Market 1 Market 2 Month 1 Month 2 Month 2 Site 3 Market3 Site 3 Month 3 0.780.02 0.59 0.390.02 A 0.51 0.470.02 A A 0.182 0.7980.02 0.133 A 0.8470.02 Profit: $ 2.576 million
  • 143. 143 10 20 30 40 50 60 70 80 90 100 2 2.5 3 3.5 x 10 4 Iteration ObjectiveFunction[$] Temporal vs. Spatial Decomposition Temporal Lagrangean Subproblem Feasible Solution (Temporal Decomposition) Spatial Lagrangean Subproblem Feasible Solution (Spatial Decomposition) Example 2 Full space Temporal Spatial Upper Bound 28,820 27,970 31,026 Best Feasible Solution 27,373 26,838 27,322 Gap [%] 5 4 13 CPU time [seconds] 10,000 + 804 5489 Temporal Spatial Sum of LP multipliers 171 1176 Size: 6 Sites, 6 Market, 6 Months, 6 Products (10,621 cont. vars., 252 discrete vars., 9463 constraints) Objectives: Compare computational performances of full space model, temporal and spatial decomposition Upper Bounds Lower Bounds Temporal is superior Selection criterion predicts better performance of the temporal decomposition Numerical results confirm expected results Temporal decomposition obtains better result than full space solution Observations: STvTw  
  • 144. 144 Time Safety Stock Reorder Point Order placed Lead Time InventoryLevel Replenishment • Inventory System under Demand Uncertainty  Total Inventory = Working Inventory (WI) + Safety Stock (SS)  Estimate WI with Economic Order Quantity (EOQ) model Stochastic Inventory System
  • 145. 145 Time InventoryLevel Average Inventory (Q/2) Order quantity (Q)  F = Fixed ordering cost for each replenishment  h = Unit inventory holding cost Replenishment Constant Demand Rate = D Economic Order Quantity Model
  • 146. 146 Total Cost 2 * 2     Q h Q D F Q FD h Order Cost Curve Order quantity Q Annual Cost Optimal Order Quantity (Q*) Minimum Total Cost Economic Order Quantity (EOQ) Economic Order Quantity Model
  • 147. 147 Time Inventory Level Lead Time Order Quantity (Q) Reorder Point (r)  When inventory level falls to r, order a quantity of Q  Reorder Point (r) = Demand over Lead Time Order placed Replenishment (Q,r) Inventory Policy
  • 148. 148 Reorder Point (r) Time InventoryLevel Lead Time Place order Receive order Safety Stock Reorder Point = Expected Demand over Lead Time + Safety Stock Stochastic Inventory = Working Inventory (EOQ) + Safety Stock Stochastic Inventory Model
  • 149. 149 Safety Stock (Service Level) Lead time = L Safety Stock Level
  • 150. 150* GD Eppen, “Effect of centralization in a multi-location newsboy problem”, Management Science, 1979, 25(5), 498 • Single retailer: Retailer Retailer Retailer Retailer Retailer • Centralized system:  All retailers share common inventory  Integrated demand • Decentralized system:  Each retailer maintains its own inventory  Demand at each retailer is Risk-Pooling Effect*
  • 151. 151 • Given: A potential supply chain  Including fixed suppliers, retailers and potential DC locations  Each retailer has uncertain demand, using (Q, r) policy  Assume all DCs have identical lead time L (lumped to one supplier) Suppliers RetailersDistribution Centers Problem Statement
  • 152. 152 • Objective: (Minimize Cost)  Total cost = DC installation cost + transportation cost + fixed order cost + working inventory cost + safety stock cost • Major Decisions (Network + Inventory)  Network: number of DCs and their locations, assignments between retailers and DCs (single sourcing), shipping amounts  Inventory: number of replenishment, reorder point, order quantity, neglect inventories in retailers retailer supplier DC Supplier RetailersDistribution Centers Problem Statement
  • 153. 153 Annual EOQ cost at a DC: ordering cost transportation cost Working inventory cost v(x)= g + ax EOQ cost
  • 154. 154 The optimal number of orders is: The optimal annual EOQ cost: Annual working inventory cost at a DC: ordering cost transportation cost inventory cost Convex Function of n Working Inventory cost
  • 155. 155 • Demand at retailer i ~ N(i,  i) • Centralized system (risk-pooling) • Expected annual cost of safety stock at a DC is: where za is the standard normal deviate for which Reorder Point (ROP) Time InventoryLevel Lead Time Safety Stock Cost for DCs
  • 157. 157 retailer supplier DC DC – retailer transportation Safety Stock EOQ DC installation cost Supplier RetailersDistribution Centers Assignments Nonconvex INLP INLP Model Formulation
  • 158. 158 • Small Scale Example  A supply chain includes 3 potential DCs and 6 retailers (pervious slide)  Different weights for transportation (β) and inventory (θ) β = 0.01, θ = 0.01 β = 0.1, θ = 0.01 β = 0.01, θ = 0.1 • Model Size for Large Scale Problem  INLP model for 150 potential DCs and 150 retailers has 22,650 binary variables and 22,650 constraints – need effective algorithm to solve it … Illustrative Example
  • 159. 159 Non-convex MINLP Avoid unbounded gradient • Variables Yij can be relaxed as continuous variables (MINLP)  Local or global optimal solution always have all Yij at integer  If h=0, it reduces to an “uncapacitated facility location” problem  NLP relaxation is very effective (usually return integer solutions) Z1j Z2j Model Properties
  • 160. 160 • MINLP Heuristic Method  Solve the convex relaxation (MILP), using secant for convex envelope  Use optimal value of X and Y variables as initial point, solve the reformulated problem with an MINLP solver (BARON, Dicopt, etc.) Convex Relaxation Algorithm 1 – MINLP Heuristic
  • 161. 161 Supplier RetailersDistribution Centers • Lagrangean Relaxation (LR) and Decomposition  LR: dualizing the single sourcing constraint:  Spatial Decomposition: decompose the problem for each potential DC j  Implicit constraint: at least one DC should be installed,  Use a special case of LR subproblem that Xj=1 decompose by DC j Algorithm 2 - Lagrangean Relaxation
  • 162. 162 Solve the reduced NLP to get UB No Initialization Convergence or iteration limit Yes Stop Solve the | j | LR subproblem for Xj=1, set Vj as the optimal obj. fun. value Update subgradient Update LB, adjust step length parameter, fix all the X variables NoYes LR Algorithm Flowchart
  • 163. 163 • Case 1: 33 retailers and DCs  Each instance has the same number of potential DCs as the retailers 300 600 900 1200 1500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 θ ObjectiveFunctionValue DICOPT, SBB, BARON, Lagrangean 0.01 0.1 1 10 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 θ CPUTime(s) BARON (global) DICOPT (Algorithm 1) SBB (Algorithm 1) Lagrangean (Algorithm 2) 300 600 900 1200 1500 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 β ObjectiveFunctionValue DICOPT, SBB BARON, Lagrangean 0.1 1 10 100 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 β CPUTime(s) BARON (global) DICOPT (Algorithm 1) SBB (Algorithm 1) Lagrangean (Algorithm 2) Computational Results
  • 164. 164 • Case 2: 88 ~150 retailers  Each instance has the same number of potential DCs as the retailers No. Retailers β θ Algorithm 1 Algorithm 2 DICOPT (CONOPT) SBB (CONOPT) BARON (global optimum) BARON+CONOPT Lagrangean Obj. fun. Time (s) Obj. fun. Time (s) Obj. fun. Time (s) Obj. fun. Time (s) 33 0.001 0.1 398.64 0.22 398.64 0.20 398.64 53.31 398.64 15.8 33 0.001 0.5 580.46 0.23 580.46 0.22 580.46 74.27 580.46 17.9 33 0.001 1.0 728.21 0.17 728.21 0.17 728.21 39.14 728.21 15.85 33 0.001 5.0 1460.40 0.18 1460.40 0.20 1460.40 93.22 1460.40 37.28 33 0.003 0.1 770.26 0.23 770.26 0.27 734.60 75.67 734.60 42.79 88 0.001 0.1 935.02 20.89 935.02 20.94 867.55* > 10 hr 867.55 356.1 88 0.001 0.5 1386.28 42.16 1386.28 38.50 1295.02* > 10 hr 1230.99 322.54 88 0.005 0.1 2297.74 42.11 2297.74 42.16 2297.80* > 10 hr 2284.06 840.28 88 0.005 0.5 3082.19 42.35 3082.19 42.38 3022.67* > 10 hr 2918.3 934.85 150 0.001 0.5 2205.37 229.57 2205.37 228.31 1847.93* > 10 hr 1847.93 659.1 150 0.005 0.1 3689.71 397.88 3689.71 398.34 3689.71* > 10 hr 3689.71 3061.2 * Suboptimal solution obtained with BARON for 10 hour limit. Computational Results
  • 165. 165 No. Retailers β θ Lagrangean Relaxation (Algorithm 2) BARON (global optimum) Upper Bound Lower Bound Gap Iter. Time (s) Upper Bound Lower Bound Gap 88 0.001 0.1 867.55 867.54 0.001 % 21 356.1 867.55* 837.68 3.566 % 88 0.001 0.5 1230.99 1223.46 0.615 % 24 322.54 1295.02* 1165.15 11.146 % 88 0.005 0.1 2284.06 2280.74 0.146 % 55 840.28 2297.80* 2075.51 10.710 % 88 0.005 0.5 2918.3 2903.38 0.514 % 51 934.85 3022.67* 2417.06 25.056 % 150 0.001 0.5 1847.93 1847.25 0.037 % 13 659.1 1847.93* 1674.08 10.385 % 150 0.005 0.1 3689.71 3648.4 1.132 % 53 3061.2 3689.71* 3290.18 12.143 % • Case 2: 88 ~150 retailers  Each instance has the same number of potential DCs as the retailers * Suboptimal solution obtained with BARON for 10 hour limit. Computational Results
  • 166. Carnegie Mellon 166 Van Hentenryck (1988), Puget (1994), Hooker (2000) Variables: Continuous, integer, boolean Constraints: Algebraic h(x) ≤ 0 Disjunctions [ A1x ≤ b1 ]  A2x ≤ b2 ] Conditional If g(x) ≤ 0 then r(x) ≤ 0 Unusual Operators: All different all different(x1, x2, ...xn) Constraint (Logic) Programming 
  • 167. Carnegie Mellon 167 Constraint Programming 1. Declarative language with high level operators (OPL-ILOG) 2. Tree search: implicit enumeration Depth first search Lower bound: partial solution Upper bound: feasible solution 3.Constraint propagation: Domain Reduction Reduction of bounds/discrete values: a) Tighten bounds linear/monotonic functions b)"edge-finding" for jobshop scheduling Job i Job j Earliest Latest Earliest Latest Earliest Latest Earliest Latest {Job i before Job j} OR {Job j before Job i} {Job j before Job i}
  • 168. Carnegie Mellon 168 Constraint Programming Example Problem Find the order at which jobs A, B and C are sequenced on machine M Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7
  • 169. Carnegie Mellon 169 A1 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A2 A3 B1 B2 B3 B1 B2 B3 B1 B2 B3 C1 C2 C3 C 3 ABC C2 ACB
  • 170. Carnegie Mellon 170 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A1 A2 A3 B1 B2 B3 B1 B2 B3 B1 B2 B3 C1C2C3 C3 C2 C1 ABC ACB BAC CAB BCA CBA
  • 171. Carnegie Mellon 171 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st A1 A2 A3 B2 B3 B1 B3 B1 B2 C1C2C3 C3 C2 C1 ABC ACB BAC CAB BCA CBA
  • 172. Carnegie Mellon 172 A{1,2,3} B{1,2,3} C{1,2,3} Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st C must be 2nd B2 B3 C2C3 A1 ABC ACB
  • 173. Carnegie Mellon 173 Job Processing Time Release Due A 2 0 5 B 4 1 9 C 3 2 7 A must be 1st C must be 2nd A{1,2,3} B{1,2,3} C{1,2,3} B3 C2 A1 ACB
  • 174. Carnegie Mellon 174 Constraint vs. Mathematical Programming Constraint Programming  Fast algorithms for special problems Computationally effective for highly constrained, feasibility and machine sequencing problems Not effective for optimization problems with complex structure and many feasible solutions Mathematical Programming  Intelligent search strategy but computationally expensive for large problems  Does not exploit special structure Computationally effective for optimization problems with many feasible solutions Not effective for feasibility problems and machine sequencing problems MAIN IDEA Decompose problem into two parts  Use MP for high-level optimization decisions  Use CP for low-level sequencing decisions
  • 175. Carnegie Mellon 175 Hybrid MILP/CP Models Two classes of scheduling problems: Multistage batch plants Continuous time State Task Network •Combine Constraint Programming (CP) with Mixed Integer Linear Programming (MILP) to overcome combinatorial explosion •Two subproblems: assignment (MILP) and sequencing (CP)
  • 176. Carnegie Mellon 176 Dvyx vyxGts xfM   ,, 0),,(.. )(min:)2( pmn T Rvyx avCyBxA aCvByAxts xcM    ,}1,0{,}1,0{ .. min:)1( MILP/CP Hybrid Models MILP: CP: Complicating rows Complicating variables (not in objective function) Hybrid: MILP (optimality) CP (feasibility) Jain, Grossmann (2001) The picture can't be displayed. Dvyx Rvyx vyxG xx aCvByAxts xcM pmn T      ,, ,}1,0{,}1,0{ 0),,( .. min:)3(
  • 177. Carnegie Mellon 177 Theorem: If the MILP/CP decomposition method is applied to solve problem (M3) with the above cuts, the method converges to the optimal solution or proves infeasibility in a finite number of iterations. Decomposition Hybrid Model MILP CP No-good Cuts (weak) Balas & Jeroslow (1972)
  • 178. Carnegie Mellon 178 Job 1 Job 2 Job n Time Unit 1 Unit m Time Release date Due date Processing time Scheduling of parallel units (Single Stage) (Jain and Grossmann, 2001) Given: n jobs/orders (release dates, processing times, due dates) m units (cost different for each unit) Find schedule that minimizes cost and meets all due dates
  • 179. Carnegie Mellon 179 Decomposition Strategy Job 1 Job 2 Job 3 Machine 1 Machine 2 Machine 3 MILP CP Job 4 Job 5 Job 6 Assignment Sequencing
  • 180. Carnegie Mellon 180 Hybrid Optimization Model Mmrdpx Iix Iixpdts rtsts xC iiii Ii imim Mm im Mm imimii ii Ii Mm imim             }{min}{max 1 .. min Assignment tasks to units: Mixed-integer linear programming     otherwise munittoitaskif xim 0 1
  • 181. Carnegie Mellon 181 MmIimzthenxif iim  ,)()1( IiDdurationiDstartiIiMz MmIitsx Iitrequiresi pdurationi Iipdstarti Iirstarti i iim z z zi i i i i       .,.; ,0},1,0{ . . . Sequencing of tasks in each unit Global constraint (implicit) Constraint Programming
  • 182. Carnegie Mellon 182 Decomposition Strategy Separate problem into assignment and sequencing problems Assign jobs Sequence jobs Feasible? Add cuts Stop YesNo Fix assignments Analyze solution Find the minimum assignment (production) cost MILP Find a feasible sequence for the chosen assignment CP
  • 183. Carnegie Mellon 183 Cuts No sequence can be found in machine 2 in the assignment of is infeasible and can be cut out The cuts also exclude large number of supersets CP-1 CP-2 CP-3 due date Cut out this assignment OK OK Inf. A B C ED F 3 2 1 (yD2 + yE2 ≤ 1)   k m m k k im k m m k Ii im IBxiI MmBx k m    ,1 1Cuts for infeasible assignments:
  • 184. Carnegie Mellon 184 Results Test Problems MILP: CPLEX 6.5 CP: ILOG Scheduler 4.4 SUN Ultra 60 Tasks Units MILP(s) CP(s) Hybrid (s) 3 2 0.04 0.01 7 3 0.31 0.04 12 3 926.3 3.84 15 5 1784.7 553.5 20 5 18,142.7 * 68,853.5 * 0.02 0.6 4.5 2.6 14.6 * Suboptimal Order magnitude reductions!
  • 185. Carnegie Mellon 185 Effect of new data Release dates, due dates and durations with arbitrary rational numbers (Harjunkoski et al., 2002) 2 machines 3 machines 3 machines 5 machines 5 machines 3 jobs 7 jobs 12 jobs 15 jobs 20 jobs Problem Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 MILP 0.04 0.04 0.31 0.27 926.3 199.9 1784.7 73.3 18142.7 102672.3 2M, 3J 3M, 7J 3M, 12J 5M, 15J 5M, 20J Problem Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 Hybrid 0.00 0.01 0.51 0.02 5.36 0.03 0.64 0.92 36.63 4.79 CPU times from Jain and Grossmann (2001) (integer numbers) CLP 0 0.02 0.04 0.14 3.84 0.38 553.5 9.28 68853.5 2673.9 Hybrid 0.02 0.01 0.52 0.02 4.18 0.02 2.25 0.04 14.13 0.41