scheduling and the resource-task network -...

SCHEDULING AND THE RESOURCE-TASK

NETWORK

Pedro M. Castro ([email protected])

Invited Assistant Researcher

Department of Process Modeling and Simulation

Lisbon/Portugal

OUTLINE

Introduction Characterization of scheduling problems and solution strategy

Resource-Task Network process representation Fundamental concepts Instructions for generating the process network

Single time grid formulations Discrete-time Continuous-time Unit-specific approaches

Industrial case studies Optimizing the cooking process of a batch pulp mill Byproducts recycling on a tissue paper mill Equipment allocation on a fine chemicals plant

Conclusions

September 18, 2008 EWO Seminars: Scheduling & the RTN 2

3

INTRODUCTION

BASIC CONCEPTS

“Scheduling is concerned with allocation of resources over time so as to execute the processing tasks required to manufacture a given set of products.” (Pinedo, 2001)

A variety of methods can be used to solve a problem

Solution is represented in the form of a Gantt chart

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September 18, 2008 EWO Seminars: Scheduling & the RTN

U1

U2

time

U3

U4

U5

U6

U7U8

Equip

ment

units Stage 1

Stage 2

Stage 3

U1

U2 U5

U4

U3

U8

U7

U6

RM FP

SCHEDULING PROBLEMS VERY COMPLEX

Wide mix of features


multistage

multipurpose

flowshopjobshop

short-term

short-term

periodic

due dates

unlimited storage

finite storage

changeoversmanpower

utilities

batch

continuous

batch mixing/splitting

variable batch sizes

minimize makespanmaximize profit

just in time

VISION

Develop a general model that can cope with such a variety of features

Mostly done

Explore ways of improving efficiency for special types of problems

Ongoing

Research on decomposition techniques that can allow to solve large-scale problems, fast

Future work


STRATEGY TO FOLLOW

Separate problem description from mathematical formulation Use the Resource-Task Network

(RTN) to represent the process Collect information from flowsheet

and process recipe Convert real entities into virtual

entities (resources and tasks)

Use/develop RTN-based mathematical formulations Handling of time is a critical issue

Discrete-time Continuous-time


Process RTN

Process Information

+

RTN Model

TtRr

vNvN

RRRR

tRr

outtrTtRr

intr

Ii

tiir

Ii

tiirtiirTttiir

RRrtrRr

endtrtrtr

FPUT

t

UTCTCT

,

)(

1,||,

,,1,,,,||,,

)(1,1,1

0,

? ? ?

8

RESOURCE-TASK NETWORK

PROCESS REPRESENTATION

RTN (PANTELIDES, 1994)

Views all processes as bipartite graphs with two entities Resources (R)

Represented as a circle General concept that includes equipments, materials, utilities, cleaning

states, material location, etc.

Tasks (I) Represented as a rectangle Transforms one set of resources into another

Heat, React, Clean, Transfer, etc.

First step Identify resources and tasks

Second step Relate resources with tasks

Tasks consume and produce resources


STATE-TASK NETWORK (STN) VS. RTN

STN: Units implicit in model constraints

RTN: Equipment units treated explicitly Disaggregate tasks if multiple units are suitable


Make BA

E

B

Make DC D

Make E

0.4

0.6

Make BA

E

B

Make DC D

Make E_U10.4

0.6Make E_U2

U1

U2

CRITICAL POINTS

Distinguish between resource types Consumed temporarily (e.g. ) Consumed/produced permanently

(e.g. ) Have an availability profile ( )

Types of tasks

Useful for changeovers and storage

Instantaneous Useful for material transfer between

units or to meet demands

Some imagination may be required to find a proper set of tasks/resources


Continuous interaction

Discrete interaction

RM Make_P1_M1

RateP1,M1

P1M

M1 EL

Dispatch_P1

InstantaneousFP

HoldinStorage_P1

Duration=1 time int.

Make_P1_M2

Duration=P1,M2+

P1,M2×SizeP1,M2

M2

BRINGING RTN DIAGRAM INTO THE MODEL

Structural parameters generation Tasks will be characterized by two sets of variables

Ni,t (Ni,t,t’)- start of i at event point t (ending at t’)

i,t (i,t,t’)- amount handled by task

Five sets of structural parametersGive total resource consumption/production or proportion

relatively to amount handled by task r,i (r,i)- discrete interaction at start (end), linked to Ni,t

r,i (r,i)- discrete interaction at start (end), linked to i,t

r,i- continuous interaction during task, linked to i,t

Large majority=0, others mostly 1 or -1

Easily generated after some practice


EXAMPLE

Batch reaction 10 ton A produces 8 ton B and 2 ton C 5 ton/h cooling water needed

Option 1 Recipe in absolute terms Binary variables Ni,t suffice

Option 2 More general Recipe in relative terms Need also i,t

i,t=10 ton leads option 1

There may be more than one possible set of values


Option 1

AReaction (i)

Fixed duration

B

R

C

CW

A,i=-10

B,i=8

C,i=2

CW,i=-5

R,i=1R,i=-1

Option 2

AReaction (i)

Variable duration

B

R

C

CW

A,i=-1

B,i=0.8

C,i=0.2

CW,i=-0.5

R,i=1R,i=-1

=/10 =/10

14

MODELING TIME

IN RTN SCHEDULING FORMULATIONS

MODELS TYPICALLY DISTINGUISHED BASED ON

TIME REPRESENTATION FOLLOWED

Use of explicit time grid(s) Single time grid (a.k.a. global time intervals)

Discrete-time (Kondili et al., 1993; Pantelides, 1994)

Continuous-time (Castro et al., 2001; Maravelias & Grossmann, 2003; Sundaramoorthy & Karimi, 2005)

Multiple time grids (a.k.a. unit specific) One time grid per unit (Floudas & co-workers, 1998-2008; Giannelos &

Georgiadis, 2002, Castro & co-workers, 2005-2008)

Use of sequencing variables Immediate precedence (Gupta & Karimi, 2003)

General precedence(Méndez et al., 2001; Harjunkoski & Grossmann, 2002)


COMMON TO ALL RTN MODELS

Excess resource variables Rr,t

Keep track of resource availability over time

Excess amount immediately before end of interval Rr,tend

May be required when in presence of continuous tasks

Equipment units treated individually (Rr,tmax=1)

Initial resource availability Rr0

Often know for all resources (model variable otherwise)

Discrete inputs r,tin and/or outputs r,t

out can be handled

Heuristic: Fix Rr,t=0 for as many resources as possible


VITAL TO UNDERSTAND RESOURCE BALANCES

Structural parameters come into action Multiperiod material balance expressions

Depend on the type of resource/task involved

Illustration for equipment resources


TtRrv

NvNRRRR

tRr

outtrTtRr

intr

Ii

tiir

Ii

tiirtiirTttiirRRrtrRr

endtrtrtr

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,

)(

1,||,,,

1,,,,||,,)(1,1,1

0,

||,,)( *,,,,,,,, TtTtRrvRR CTtiir

Ii

tiir

Ii

tiirtrend

trsc

Task 1_M1 Task 2_M1

t= 1 2 3 4 5 6 7

RM1,t

+

1

1

0

-

1

+

1

-

1

+

1

DISCRETE-TIME MODEL

Most powerful approach overall Can handle problems of industrial relevance Simple, elegant and very tight MILP formulation Few sets of constraints

Besides excess balances

Critical modeling issue Uniform interval length δ may be difficult to select

Trade-off: data accuracy vs. problem tractability


TtRrRRR r,tr,tr,t , maxmin

TtIiVNVN r

Rr

rititir

Rr

ritiEQEQ

, max,,,

min,,

IiNTt

ti

1,

δ Binary

variables

Total

variables

Constraints Cost

[k$]

CPUs

10 4005 11242 7277 91 5.36

5 8077 22514 14477 90 150

2 20137 56174 36077 89 40

1 40341 112378 72077 89 1429Accurate data

Rounded-up data

Not true optimum

CONTINUOUS-TIME MODEL

More general approach Can handle a wider variety of features rigorously Significantly more complex overall Additional set timing constraints & variables

Critical modeling issues Batch tasks characterized by indices t (start) and t’ (end) Global optimal solutions only for |T|→∞


|T| Binary

variables

Total

variables

Constraints Cost [k$] CPUs

8 462 957 495 Infeasible 0.57

9 528 1089 562 27.222 7.18

10 594 1221 629 27.008 369

11 660 1353 696 26.911 4131Not optimum

1 o

rde

r

ma

gn

itud

e

UNIT-SPECIFIC MODELS

Multiple time grids better in special types of problems Sequential processes (multistage)

Competitive with sequence-based models

Multipurpose plants without shared resources

Critical modeling issue All tasks last one time interval (one index)

Fewer event points to find global optimal solutions


Model Time

grid

|T| Binary

variables

Total

variables

Constraints Cost

[k$]

CPUs

Discrete-time Single 1501 27383 60406 33054 793 165

Continuous-time Single 7 453 615 348 793 6214

Multiple 4 165 198 295 793 0.82

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CASE STUDY 1. OPTIMIZING THE COOKING

PROCESS OF A SULPHITE PULP MILL

PROBLEM CHARACTERISTICS

System of 4 parallel batch digesters for pulp production

Heating stage was bottleneck 2 digesters sharing steam

simultaneously

The digester sequence affects the cycle time

Different digester capacities


Tinitial T(H0)

H0

h1 h2 h3

H1

90ºC @TTx15 Tcook

Steam for the cooking section

Di

Dj

MODELING OF THE HEATING STAGE

Duration of heating tasks from dynamic simulation

RTN superstructure Tasks

H0- heat till 90 oC

H1- final heating

ResourcesS3- initial temperature

S4- 90 oC

S8- cooking temperature

S5-S7- Temperature after H0


OPTIMAL PERIODIC SCHEDULE

Production rate 1% higher, cycle time H= 564 min

Optimal sequence: D3-D6-D5-D4

From discrete-time formulation in 393 CPUs (δ=1 min)

Continuous-time approach only finds H=584 min in 41 h of CPU

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K2 H0 H1

K11

K11

K11

K12

K12

K12

K13

K13

K13

K13

K14

K14

K14

K14

K1

K1

K1

K2

K2

K2

Steam Sharing

H0 H1

K1

H0 H1 K11 K12

Steam Sharing

H0 H1 K11

0 50 100 150 200 250 300 350 400 450 500 550

D3

D4

D5

D6

Time (min)


25

CASE STUDY 2. OPTIMIZING BYPRODUCTS

RECYCLING ON A TISSUE PAPER MILL


Scandinavian tissue paper mill (continuous plant) 5 products (P50 darkest quality to P85, brightest quality)

Part of the fiber lost as broke in converting lines

Current broke recycling policy Mix broke with old newspaper (ONP), for low quality products (P50, P60)

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De-inking line 1

De-inking line 2

Intermediate Storage 1

Tissue Machine 1

Tissue Machine 2

Broke

Storage

Ash

Raw material ONP

Raw material MOW

Raw material VF

P50

P60

P75

P80

P85

P50

P60

P75

P80

P85

Intermediate Storage 2

Sludge & reject


MODELING FOR A NEW RECYCLING POLICY

Do not mix broke from different qualities

BR, BR80 and BR85 recycled with ONP, MOW and VF

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SP2_ONP67

SP1

SP2

VF

MOW

ONP

ONP

50

ONP

60

ONP

67

VF

85

MOW

80

S67

S80

GTS_67

RateMax=48 t/day

GTS_80

RateMax=48 t/day

DW

GFS_80

L80

L67

GFS_67

P80

P75

P85

P50

P60BR

TM2_P85

TM1_P85

TM1_P80

TM2_P80

TM1_P75

TM2_P75

TM2_P60

TM1_P60

TM1_P50

TM2_P50

TM1

TM2

0.1

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

SP2_ONP60

SP2_ONP50

SP2_VF85

BR

85

BR

80SP1_MOW80

0.224

0.224

0.896

0.896


EVALUATION OF RECYCLING POLICIES

Profit 1.5% higher

Double benefit of lower raw-material costs and no disposal costs

With a MINLP continuous-time formulation

Better solution in significantly less time than discrete-time formulation

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91.95

93.51

94.87

86

88

90

92

94

96

98

100

No recycling Current strategy Future strategy

Pro

fit

(r.m

.u./

yr)

0

40

80

120

160

0 2 4 6 8

Am

ou

nt (t

)

Time (days)

BR

0 2 4 6 8

SP1

SP2

TM1

TM2

DW

Time (days)

MOW8020% BR80

MOW80 MOW80 MOW80

VF9.2% BR85

ONP507.6% BR

VF9.2% BR85

ONP67 ONP6020% BR

P85P50 P85 P75 P60

P80P80 P75 P75 P80

GTS80 GTS67

MOW80

0

40

80

120

160

0 2 4 6 8

Am

ou

nt (t

)

Time (days)

BR

0

40

80

120

160

0 2 4 6 8

Am

ou

nt (t

)Time (days)

BR

Excess

49 t

0 2 4 6 8

SP1

SP2

TM1

TM2

DW

Time (days)

MOW80MOW80 MOW80 MOW80

VFONP5020% BR

VF ONP67 ONP6020% BR

P85P50 P85 P75 P60

P80P80 P75 P75 P80

GTS80 GTS67

MOW80

0 2 4 6 8

SP1

SP2

TM1

TM2

DW

Time (days)

GTS80 GTS67

MOW80 MOW80 MOW80 MOW80 MOW80

ONP50 VF VF ONP67 ONP60

P50 P85 P85 P75 P60

P80 P80 P75 P75 P80


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CASE STUDY 3. OPTIMAL EQUIPMENT

ALLOCATION IN A FINE CHEMICALS PLANT


Portuguese fine chemicals batch plant How many units to allocate to production of API?

The more units the lower the makespan But fewer units for other APIs (flexibility decreased)

Virtual equipment units in the RTN Model will make correspondence to real plant units


OPTIMAL COST AS FUNCTION OF CYCLE TIME

Three solutions deserve further analysis From discrete-time MILP formulation (total CPU=1078 s)

31September 18, 2008 EWO Seminars: Scheduling & the RTN

Un

its n

ot

allo

ca

ted

to p

rod

uctio

n A

PI

ANALYSIS OF PROMISING SOLUTIONS

Total production time for 20 batches of the API (E)


78 days + 2 shifts

91 days + 1 shift

116 days + 2 shifts

33

CONCLUSIONS & REFERENCES

CONCLUSIONS

There is a wide variety of complex scheduling problems out there

The underlying process/production recipe can be described as a Resource-Task Network

This procedure is mostly independent on the mathematical formulation used to solve the problem

Three conceptually different models can be used

Guidelines given to select most appropriate

Industrial problems have been tackled

There is still a lot of research to be done…


IMPORTANT RTN REFERENCES

Pantelides, C.C. Unified Frameworks for the Optimal Process Planning and Scheduling. In Proc. 2nd FOCAPO; Cache Publications: New York, 1994; pp 253.

Castro, P. et al. Simple Continuous-time Formulation for Short-Term Scheduling of Batch and Continuous Processes. Ind. Eng. Chem. Res. 2004, 43, 105.

Castro, P. et al. Simultaneous Design and Scheduling of Multipurpose Plants Using Resource Task Network Based Continuous-Time Formulations. Ind. Eng. Chem. Res. 2005, 44, 343.

Castro, P.M.; Grossmann, I.E. New Continuous-Time MILP Model for the Short-Term Scheduling of Multistage Batch Plants. Ind. Eng. Chem. Res. 2005, 44, 9175.

Méndez, C.A. et al. State-of-the-art Review of Optimization Methods for Short-Term Scheduling of Batch Processes. Comput. Chem. Eng. 2006, 30, 913.

Shaik, M.; Floudas, C.A. Unit-specific event-based continuous-time approach for short-term scheduling of batch plants using RTN framework. Comput. Chem. Eng. 2008, 32, 260.


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