Scheduling in PSE:Before and After the State-Task Network

Pedro M. Castro

His most successful articles

July 18, 2019 2In Honor of Professor Roger Sargent

Number of citations per source

July 18, 2019 3In Honor of Professor Roger Sargent

Web of Science

49

Scopus

Google Scholar

#5 of all time CACE#3 excluding reviews

Number of citing articles from each country

July 18, 2019 4In Honor of Professor Roger Sargent

39 countries

4624 are mine#6 in author’s list

#1-Ignacio#2-Christos

Impact of articles citing seminal paper

July 18, 2019 5In Honor of Professor Roger Sargent

Total citations CACE articles in 2018: 13,224

Before the STN• Review article of Rippin (CACE ‘93)

– General considerations• “Customers’ requirements will be more specific and more demanding

in terms of specification, quality and delivery, requiring manufacturers to be much more flexible- a goal that… may be more readily achieved in batch rather than in continuous production.”

• “Additional requirements in batch operation… allocation of production tasks to equipment items and the sequencing and timingof the production of products through the plant.”

• “…the treatment of variable time and capacity requirement for tasks as a function of equipment or batch size is by now relatively routine.”

– Short-term scheduling as a type of batch processing problem• “It seems likely that future systems will provide a synthesis between

algorithmic and heuristic methods… The best balance… will depend upon the nature of the particular batch processing environment.”

• Trend : “comparison between exact algorithms and heuristics”

July 18, 2019 6In Honor of Professor Roger Sargent

Multiproduct plant design & scheduling

• Network flowshop configuration (Birewar & Grossmann ‘90)– Multiple products with same recipe– Task-unit allocation to be decided by

optimization (same for all products)• Possible to use equipment in parallel

– Merging of tasks is possible• Same for all products

– Continuous-time MINLP modelsolved with DICOPT

• Upgrade of single product approach (Yeh & Reklaitis ‘87)

– Batch size dependentprocessing times

– Heuristic solution procedure

July 18, 2019 7In Honor of Professor Roger Sargent

Mixing1

Reaction2

Crystallization3

Drying4

Multiproduct scheduling with storage

• Flowshop plant (Ku & Karimi ‘88)

– Single unit per stage– Same product sequence

in all units• Permutation schedules

– Continuous-time MILP model• Heuristic solution approach preferred

– Storage policy affects the makespan• Unlimited intermediate storage (UIS)• Finite intermediate storage (FIS)• No intermediate storage (NIS)

July 18, 2019 8In Honor of Professor Roger Sargent

Unit/stage1

Unit/stage2

Unit/stage4

Unit/stage3

Storage unit

107

UIS

111

NIS

FIS

107

Scheduling in a multipurpose plant• Precedence network

structure (Rich & Prokopakis ‘86)

– Product 𝑗𝑗 obtained from one or more products 𝑖𝑖• Stoichiometric factors

– Amount of 𝑖𝑖 required for one unit of 𝑗𝑗

– Product-unit assignment is known– Aggregate task per product 𝑖𝑖

• Variable quantities– Multiple runs 𝑘𝑘, 𝑁𝑁𝑖𝑖,𝑘𝑘 batches

with fixed size/time (𝑡𝑡𝑖𝑖)

– Multiple intermediate due dates– Continuous-time MILP with

general precedence variables• Big-M disjunctive constraints

July 18, 2019 9In Honor of Professor Roger Sargent

5

1

2

3 4

Unit #2

Unit #1

Unit

Unit

Unit

𝑆𝑆𝑗𝑗,𝑘𝑘𝑘 − 𝑆𝑆𝑖𝑖,𝑘𝑘 + 𝑀𝑀 1 − 𝑦𝑦𝑖𝑖,𝑘𝑘,𝑗𝑗,𝑘𝑘𝑘 ≥ 𝑁𝑁𝑖𝑖,𝑘𝑘𝑡𝑡𝑖𝑖𝑆𝑆𝑖𝑖,𝑘𝑘 − 𝑆𝑆𝑗𝑗,𝑘𝑘𝑘 + 𝑀𝑀 � 𝑦𝑦𝑖𝑖,𝑘𝑘,𝑗𝑗,𝑘𝑘𝑘 ≥ 𝑁𝑁𝑗𝑗,𝑘𝑘𝑘𝑡𝑡𝑗𝑗

Most general algorithm before the STN

• SRSP program (Egli & Rippin ‘86)– Products with alternative routes

• Intermediates & final products– Units shared by different products– Time-dependent resource

consumption• Specified for a time period 𝜃𝜃 relative

to the start of the task• E.g. electricity & steam demand

– Sequence-dependent times/costs– Preemption over weekends– Multiple intermediate due dates– Production scheduled on a hourly

basis over 20 days• Algorithm, not a mathematical model

– Ready for re-scheduling

July 18, 2019 10In Honor of Professor Roger Sargent

Seminal article of the State-Task Network

July 18, 2019 11In Honor of Professor Roger Sargent

AbstractA general framework for handling a wide range of scheduling problems arising in multiproduct/multipurpose batch chemical plants is presented.Batch processes involving a variety of complexities are represented using a state-task network. The novel feature of this representation is that both the individual batch operations (“tasks”) and the feedstocks, intermediate and final products (“states”) are included explicitly as network nodes. Processes involving sharing of raw materials and intermediates, batch splitting and mixing and recycles of material, can be represented unambiguously as such networks.The short-term scheduling problem is formulated as a mixed integer linear program (MILP) based on a discrete time representation. Flexible equipment allocation, variable batch sizes and mixed intermediate storage policies involving both dedicated and multipurpose storage vessels are taken into account. Limited availability of raw materials, both at the start and during the time horizon of interest, is accommodated. Product deliveries may take place at any time during the horizon, and the amounts involved may be either fixed or variable. The use of utilities by the various tasks may vary over the task processing time, and may be constant or proportional to the batch size. The availability and/or cost of utilities may vary over the time horizon of interest.The objective function is the maximization of a profit function involving the value of the products, and the cost of raw materials, utilities and material storage.The formulation may result in MILPs involving large numbers of binary variables. Issues pertaining to the efficient solution of these problems are discussed in Part II of this paper.

State-Task Network (STN) (Kondili, Pantelides & Sargent ‘93)

• New representation model– Superstructure featuring all

alternatives (Sargent & Gaminibandara ’76; Grossmann & Sargent ‘78)• Removes ambiguities of recipe

networks (Reklaitis ‘91)

– Allows for complex recipes, multiple processing routes, shared intermediates, recycles• Material states as circles• Tasks as rectangles

– Process units and task-unit suitability not shown explicitly• Other resources also not shown

– STN not necessarily connected graphs (disjoint sub-graphs)

July 18, 2019 12In Honor of Professor Roger Sargent

State-Task Network (STN) (Kondili, Pantelides & Sargent ‘93)

• New discrete-time MILP formulation– A first in PSE literature– Easy to handle time-dependent profiles/costs– Drawback

• “Even the solution of a small example using a state-of-the-art generic MILP solver was found to require substantial amounts of computation”

July 18, 2019 13In Honor of Professor Roger Sargent

12 |T|-1

t=|T|3 4 |T|-2|T|-3

time pointsft1 ft2 ft3 ft4 ... ft|T|ft|T|-1ft|T|-2

time of each time point is known a priori

δ

...

uniform slot size (time units)

𝑆𝑆𝑠𝑠,𝑡𝑡 = 𝑆𝑆𝑠𝑠,𝑡𝑡−1 + �𝑖𝑖

�𝑗𝑗

�𝜃𝜃=0

𝜏𝜏𝑖𝑖

(�̅�𝜌𝑖𝑖,𝑠𝑠,𝜃𝜃𝐵𝐵𝑖𝑖,𝑗𝑗,𝑡𝑡−𝜃𝜃 − 𝜌𝜌𝑖𝑖,𝑠𝑠,𝜃𝜃𝐵𝐵𝑖𝑖,𝑗𝑗,𝑡𝑡−𝜃𝜃) + 𝑅𝑅𝑠𝑠,𝑡𝑡 − 𝐷𝐷𝑠𝑠,𝑡𝑡 ∀𝑠𝑠, 𝑡𝑡Material balances (multiperiod)

Consumption Batch size Raw-material supply & product demand

Material state availability Production

�𝑖𝑖𝑘

�𝑡𝑡𝑘=𝑡𝑡

𝑡𝑡+𝜏𝜏𝑖𝑖−1

𝑊𝑊𝑖𝑖𝑘,𝑗𝑗,𝑡𝑡𝑘 − 1 ≤ 𝑀𝑀 1 −𝑊𝑊𝑖𝑖,𝑗𝑗,𝑡𝑡 ∀𝑖𝑖, 𝑗𝑗, 𝑡𝑡Equipment allocation constraints (Big-M)Assigns start of task 𝑖𝑖 to unit 𝑗𝑗 time 𝑡𝑡Processing time

�𝑖𝑖

�𝑡𝑡𝑘=𝑡𝑡

𝑡𝑡−𝜏𝜏𝑖𝑖+1

𝑊𝑊𝑖𝑖,𝑗𝑗,𝑡𝑡𝑘 ≤ 1 ∀𝑗𝑗, 𝑡𝑡Fewer & tighter constraints (Shah, Pantelides & Sargent ‘93)

After the STN: representation improvements

• mSTN (Barbosa-Póvoa & Macchietto ‘94)

– Design and scheduling• Explicit location of material states to

identify connection between units• Unambiguous representation of

recipe/flowsheet/transfer information

• Resource-Task Network (Pantelides ‘94)

– Unified treatment of production resources (states, units, etc.)

– Tasks pre-assigned to units• Multiple tasks for alternative units

– Structural parameters linktasks & resources• Excess resource balances

July 18, 2019 14In Honor of Professor Roger Sargent

Hh_C1Cast_Gg_CC1

Duration=154 min

Hh

PW ENCC1

Hh´_C1 Hh´

After the STN: handling of time• Periodic scheduling

(Shah, Pantelides & Sargent ‘93)– Wrap-around operator

• Continuous-time, single grid (Zhang & Sargent ‘94)– Mockus & Reklaitis (‘95)– Schilling & Pantelides (‘97)

• Also handles continuous tasks– Castro et al. (’01, ‘04)– Maravelias & Grossmann (’03)

• Continuous-time, multiple time grids(Ierapetritou & Floudas ‘98)– Giannelos & Georgiadis (‘02)

July 18, 2019 15In Honor of Professor Roger Sargent

12 |T|-2 |T|-1

slot 1

3

time slot 2 slot |T|-2 slot |T|-1

event points t=|T|

T1 T2 T3 T|T|-2 T|T|-1 T|T|

timing variables to be determined by optimization

Sucessful applications in industry• Discrete-time RTN model

– Dow Chemical(Wassick ‘09, Wassick & Ferrio ‘11)• Drumming facility for 15

businesses• Liquid-waste treatment network

– ABB (Castro et al. ‘09)• Handling time-dependent

electricity costs in a cement plant

– Johnson Controls(Rawlings et al. ’18)• Heat recovery system providing

heating and cooling to Stanford University

July 18, 2019 16In Honor of Professor Roger Sargent

individual silo

multiple silos

individual silo

grinding mill

grinding mill

grinding mill

Final stage of cement manufacturing

cement clinker

cement grade

cement grade

cement grade

Storage

truck

boat

train

Highlights of contributions in scheduling• State-Task Network process representation

– Engineers can solve a scheduling problem without knowing about mixed-integer linear programming

• Same concept of process simulators

July 18, 2019 17In Honor of Professor Roger Sargent

• STN-based discrete-time formulation– Covers a wide variety of scheduling problems– Recommended approach when facing a new problem

• Essentially the same formulation

• STN-based continuous-time formulation– Far less applicable than its discrete-time counterpart

• Significant developments have occurred since seminal paper

• Still, each real-life scheduling problem has its own specific constraints, so we are not done just yet– Generalized Disjunctive Programming

helps to derive computationally efficient formulations