strengths & drawbacks of milp, cp and discrete-event...

Strengths & Drawbacks of MILP, CP and

Discrete-Event Simulation based

Approaches for Large-Scale Scheduling

Pedro M. Castro

Assistant Researcher

Laboratório Nacional de Energia e Geologia

Lisboa, Portugal

Outline

• Basic concepts on scheduling – Types of scheduling problems

– Classification of scheduling models

• Sequential facilities

• Network plants

• Approaches other than mathematical programming – Constraint Programming

– Discrete-Event Simulation

• Full-space models & decomposition algorithms – Hybrid models and solution approaches

• Different concepts or methods are effectively & efficiently combined

• Extensive testing through a case-study – Automated Wet-etch Stations

November 3, 2011 Pedro Castro, EWO Seminar, Carnegie Mellon University 2

Introduction

• Scheduling plays an important role in most manufacturing and service industries – Pulp & Paper, Oil & Gas, Food & Beverages, Pharmaceuticals

• Type of decisions involved – Define production tasks from customer orders

– Assign production tasks to resources (not only equipments)

– Sequence tasks (on a given resource)

– Determine starting and ending times of tasks


Demand (orders)

A

B

C

D

E

A1 A2 A3

B1 B2

C1

D1 D2

Batching

How many batches? What size?

Batch-unit Assignment

Where each batch is processed?

A1A2

A3

B1

B2

C1D1

D2E1 E1

Sequencing & Timing

In what sequence are batches processed?

U1

U2

A1 A2 A3 C1

D1 D2 B1 B2 E1

Batches

Maravelias et al. (2011)

Classification of scheduling problems (I)

• Structure of production facility

– Sequential

• Lot identity is kept throughout processing stages

– Network

• Mixing and splitting of materials is allowed


M1

M2

(a) Single-stage (b) Multi-stage

M11

M12

M22

M21MK1

MK2

M23

…M2

M1 M4

M5M3

(c) Multi-purpose

M6

M7

A A

B

B

Make B A

E

B

Make D C D

Make E

0.4

0.6

Maravelias et al. (2011)

Classification of scheduling problems (II)

• Production mode – Batch, continuous or hybrid

• Operation mode – Short-term for highly variable demand

– Periodic (cyclic) for stable demand


Batch TaskCharacterized by:

duration (h)BA

Continuous TaskCharacterized by:

processing rate (kg/h)BA

Time (h)

Inve

nto

ry (k

g)

Start of task End of task

Fill Draw

Time (h)

Inve

nto

ry (k

g)

Start of task End of task

Fill & Draw

Classification of scheduling problems (III)

• Type of operations – Production but also material transfer (e.g. pipelines)

• Other aspects – Storage policies

• Fixed capacity (shared or not), unlimited or no storage

– Changeovers

• Sequence-dependent (e.g. paints) or not


Classification of scheduling models

• Time representation – 4 major concepts

– In generality: Single grid > discrete > multiple grids > precedence

– Solution quality function of # slots for time grid based models

– In # slots: Discrete > single grid > multiple grids


1 1332 54 76 98 1110 12

T1 T3 T4U1

U2 T5 T6

Discrete-time grid

T2

1 52 3 4

T1 T3 T4U1

U2 T5 T6

Continuous-time with single grid

T2

12 3 4

T1 T3 T4U1

U2 T5 T6

Continuous-time with multiple grids

T2

12

T1 T3 T4U1

U2 T5 T6

Precedence (through sequencing variables)

T2

Immediate General

Models for sequential facilities

• Precedence concept Méndez et al. 2001; Harjunkoski & Grossmann 2002; Gupta & Karimi 2003

– Provide high quality solutions with limited computational resources

– Favored when preordering can be performed a priori (e.g. due dates)

– Set of binary variables for processing tasks can also be used for other discrete resources (e.g. transportation devices)

– Difficult to prove optimality

• Multiple time grids Pinto & Grossmann 1995; Castro & Grossmann 2005; Liu & Karimi 2007; Castro & Novais 2008

– A few options available

– Tighter and computationally superior

– More difficult to understand


M0 M3 M4 M5 M7 M9 M35 M37 OUTM10 M11 M20M15

P3

P4

P5

P2

Models for network facilities (I)

• Most complex arrangement – May involve resource constraints

other than equipment

• Linked to systematic methods for process representation – State-Task Network (Kondili et al. 1993)

– Resource-Task Network (Pantelides, 1994)

• Bear in mind – OPL Studio (Constraint Programming)

similar to RTN • Activities (tasks), resources (materials),

unary resources (units)

– RTN process model feeds a timed automata model (Subbiah et al. 2011)


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,

? ? ?

Models for network facilities (II)

• Discrete-time – Handled problems of industrial relevance

(Glismann & Gruhn 2001; Castro et al. 2008-09; Wassick 2009)

– Simple, elegant and very tight MILP

– Easy integration with higher level planning

– Major drawback related to accuracy

• Continuous-time with single grid (Maravelias & Grossmann 2003,Castro et al. 2004; Sundaramoorthy & Karimi 2005)

– Most general

– High sensitivity to data makes it more appropriate for integration with lower level control layer

– Computationally inefficient

• Continuous-time with multiple grids (Ierapetritou & Floudas 1998; Susarla et al. 2010; Seid & Majozi 2011)

– Fewest # slots & better performance

– Issues have been raised related to generality


1 1332 54 76 98 1110 12

T1 T3 T4U1

U2 T5 T6

Discrete-time grid

T2

1 52 3 4

T1 T3 T4U1

U2 T5 T6

Continuous-time with single grid

T2

12 3 4

T1 T3 T4U1

U2 T5 T6

Continuous-time with multiple grids

T2

12

Other solution approaches (I)

• Constraint Programming (CP)

– Not as broadly applied as mathematical programming

– Has specific scheduling constructs for easy model building and

problem solving with constraint propagation (OPL Studio 3.7)

•

•

•

– Easy to develop specific search strategy for an efficient

integrated approach (Zeballos & Méndez, 2010; Zeballos et al. 2011)

– Can be classified as precedence based, discrete-time

– Excels at makespan minimization

• Single variable in objective function

– No optimality gap being computed


Other solution approaches (II)

• Discrete-event simulation

– Heuristic, rule based approach

– Problem represented as a set

of interlinked modules featuring

algorithms for decision making

– Extremely useful for visualizing

system behavior

• Generate feasible solutions for

complex problems

– Cannot guarantee optimality


Problem definition


W CC

ZW ZW

NIS NIS

j=0 j=1 j=2 j=3 j=M j=M+1…

… W

Input

buffer

Output

buffer

i

Jobs

W CC

ZW ZW

NIS NIS

j=0 j=1 j=2 j=3 j=M j=M+1…

… W

Input

buffer

Output

buffer

i

Jobs

CC

ZW ZW

NIS NIS

j=0 j=1 j=2 j=3 j=M j=M+1…

… W

Input

buffer

Output

buffer

i

Jobs

WC

ZW ZW

NIS NIS

j=0 j=1 j=2 j=3 j=M j=M+1…

…C W

Input

buffer

Output

buffer

i

Jobs

W CC

ZW ZW

NIS NIS

j=0 j=1 j=2 j=3 j=M j=M+1…

… W

Input

buffer

Output

buffer

i

Jobs

W

i1

i1

i3i1

i1

i1

i3i1

i1

i1

i3

i3

Un

its

Processing Time Transfer Time

Timej1

j2

j3

Holding Time

Job Sequence i1-i3-i2

Robot

i1

i1

i3i1

i1

i1

i3

i3

i2

i2i3

i2

MK

... ... ...

...

...

...

robot schedule

bath schedule

ZW “Zero Wait ”

MIS “Mixed-intermediate Storage”

Buffer

Bath j C = “Chemical Bath”

j=1,3,5...M-1

W = “Water Bath”

j=2,4,6...M

Input buffer j=0

Output buffer j=M+1

NIS “Non-intermediate Storage”

m

m

m=1 m=2 m=3 m=|M|-1 m=|M |

m1

m2

m3

• Automated Wet-Etch Station (AWS)

Objective function: minimize makespan

Best MILP model (Castro et al. 2011)

• Hybrid in terms of time representation concept (Bhushan & Karimi, 2003)

– Multiple time grids for processing tasks

• Why is it a good approach?

– Single unit per stage

» No uncertainty in # time slots to specify

» Global optimality ensured with # slots= # wafer lots

(no iterative search procedure)

– Lot sequence unchanged throughout stages due to storage policies

– General precedence for robot transfer tasks

• Why?

– Provides very good solutions in early nodes of the search

» Often difficult to prove optimality (high integrality gap at termination)

– Alternative of a robot grid with too many time slots (|I|×|M|)

» Resulted in a much worse computational performance


No big-M constraints for processing tasks


• Slot duration greater lot’s processing time

• Difference in time in consecutive units equal to processing + transfer

• Ending time greater starting time + processing

• Starting time in next unit equal to ending time + transfer

• Exactly one lot per time slot

• Time of last slot in last unit lower than makespan

Un

it m

ZW

LS

Time

i=11

T1,1

2

T1,2

1

T2,1

i=2

i=1 2

T2,2

1

T3,1

i=3

Te1,2

p1,2

2

Te2,2

p2,2

2

T3,2

i=3

Te3,2

p1,2

Robot r

1 1 12 2 2

can hold lot past processing time

do not hold lot past processing time

Robot assignment & sequencing constraints


• Binary variables – Wt,m,r – assigns robot r to the transfer to unit m of the lot in slot t

• 4 sets of big-M constraints –

– • If same robot, lot i’ to m after transfer i to m+1

–

– • No overlap between transfer of any two lots to different units

m

Tt,m

(i,t)

m+1

m

Tt+1,m

(i',t+1)

(i,t)

m

m+1

2 transfers between processing of

consecutive lots

One Robot Models

• Three alternative formulations – ORM (current work)

• Hybrid time slots/general precedence model

– BK (Bhushan & Karimi, 2003) • Hybrid model with slightly different sequencing variables

– AM (Aguirre & Méndez, 2010) • Pure general precedence model

• New approach clearly better – Only 6 problems can be solved to optimality

– BK better in smaller problems (P2-P4), in P4 by one order of magnitude (as tight as ORM)

– AM finds good feasible solutions in 4 cases where BK fails (P7, P9-P11)


Motivation

• Industry requires decision-making tools that generate good solutions with low computational effort – Guaranteeing optimality looses importance

• Only a subset of the production goals are taken into account

• Implementing the solution as such often limited by dynamic nature of industrial environments

• Real life applications should take advantage of state-of-the art, full-space models – Ability to handle almost all the features that may be

encountered at a process plant

• Need for efficient decomposition approaches that keep number of decisions at a reasonable level – Tunable parameters

• Specific AWS problem – Full-space models only useful up to 12 lots in 12 units


New scheduling algorithm

• Main components – Heuristic approach

• Does not guarantee optimality

– Solves constrained versions of full-space models R-ORM & ORM

• Rescheduling through neighborhood search to approach optimality

– Schedule of transportation tasks first determined by Discrete-Event Simulation

• Ensures feasibility

– Tradeoff computational effort vs. solution quality achieved with tunable parameter NOS

• Number of lots per iteration


Iterations |J|Lots/iteration

NOS

Best Sequence Processing Tasks(Neglecting Robot Availability)

posi

Neighborhood SearchR-ORMM

ILP

GA

MS

Discrete Event Simulation(Considers Robot Availability) A

ren

a

Feasible Solution(One Robot Problem)

Sequence of Transfer Tasksslott,m

Neighborhood SearchORMM

ILP

GA

MS

Best Solution(One Robot Problem)

Full schedule

Neighborhood Search

• Systematic decomposition strategy – Solves highly constrained versions of

full-space model

– Keeps number of decisions at a reasonable level

– Also being called • Solution polishing

• Local branching

• How does it work? – Starts from a feasible solution

• Most binary variables are fixed

– Deciding which variables to free is the challenging part

• Knowledge about problem structure

• Example for R-ORM – Acting solely on processing sequence


t=1 t=2 t=3 t=4 t=5 t=6

I1 I2 I3 I4 I5 I6j=0

Ij=1={I2,I3,I5}

t=1 t=2 t=3 t=4 t=5 t=6

I1 I3 I5 I4 I2 I6j=1

Ij=2={I1,I3,I6}

t=1 t=2 t=3 t=4 t=5 t=6

I6 I3 I5 I4 I2 I1j=2

.

.

.

t=1 t=2 t=3 t=4 t=5 t=6

I2 I3 I6 I1 I5 I4j=|J|

Free assignments

Position has changed

Random selection of variables NOS=3

Neighborhood Search for ORM

• Two sets of interconnected binary variables – Chemical and water baths processing sequence

– Robot transportation sequence

• Knowledge about problem structure needed to – Free binaries of transportation tasks involving the lots being freed

– Allow transportation tasks of fixed lots to change position • If one of the lots to be rescheduled is immediately before or after in current

processing sequence


NOS=2

Robot grid

1 2 3 4 5 6 7 8 9

I1,M1j=0 I1,M2 I2,M1 I1,M3 I2,M2 I2,M3 I3,M1 I3,M2 I3,M3

Ij=1={I1,I2}


Ij=2={I2,I3}


Just one transportation task remains fixed

I3 remains the last lot to be processed but transfer of I3 to M1 may change position

Discrete-Event Simulation

• Very attractive and powerful tool to model, analyze and evaluate the impact of different decisions

• Major advantages – Representation of complex manufacturing processes

– Visualization of the dynamic behavior of its elements

• Arena Simulation Model of entire AWS process – Set of operative rules and strategic decisions on each sub-model

• Internal robot logic to coordinate and effectively synchronize the transportation of jobs between consecutive baths (ensure feasibility)


Neighborhood search using R-ORM

• Starts with lexicographic sequence (LP) – Major improvements when compared to initial schedule in <60 CPUs

• NOS=7 lots/iteration, 100 iterations

• Similar performance to full-space model up to 60 CPUs

• Best found solution => Arena Simulation Model


Discrete-Event Simulation model (Arena)

• Outcome from R-ORM is a lower bound – Schedule may feature transfers occurring simultaneously

• Increase in makespan

• Solution quality rapidly degrades with # baths

• Advantage: Very low computational effort


Indication of good the approach is!

Neighborhood search using ORM

• Major improvements in solution quality with respect to initial schedule from Arena

• All problems solved in less than 30 min (NOS=2)

• NOS => solution quality & CPUs

• 10 different runs for each NOS value

• Significantly better solutions than CPLEX solution polishing after 60 CPUs1h – With increase in problem size


Bet

ter

than

so

luti

on

po

lish

ing!

Constraint Programming Approach

• Integrated approach with CP model & efficient domain-specific search strategy

• Competitive full-space approach – Good quality solutions in 1-h CPU

• Less likely for solution to keep improving given additional computational time when compared to neighborhood search


Best found solution for largest problem


Search for the optimal solution


• Most improvements in first 20% of CPUs – Reaching a plateau towards the end

Conclusions

• Wide variety of approaches for scheduling problems – Mathematical programming, Constraint Programming,

Discrete-Event Simulation, Heuristics, etc.

• A few alternative efficient models – Good for academic research, bad for industrial problems

• Effective decomposition methods much needed – Good quality solutions with few computational resources

• Tunable parameters for best tradeoff

– Critical to incorporate knowledge about problem structure

• Major improvements are possible


Method Heuristic algorithm (A2’)

Bhushan & Karimi (2004)

DES CP NS

NOS=2 NOS=3

Better NS

(submitted)

Makespan 478.6 529.9 443.4 428.2 410.7 396.8

Improvement (%) 0.0% -10.7% 7.4% 10.5% 14.2% 17.1%

Acknowledgments

• Carlos Méndez, Luis Zeballos, Adrián Aguirre – Results & animations shown on this talk

• Sponsors – Fundação para a Ciência e Tecnologia & Ministerio de Ciencia,Tecnología e Innovacion

Productiva • Bilateral cooperation agreement Argentina/Portugal (2010-2011)

– Luso-American & National Science Foundations • 2011 Portugal – U.S. Research Networks Program

• References – Scope for Industrial Applications of Production Scheduling Models and Solution Methods.

Review paper on scheduling. Multiple authors. To be submitted to CACE.

– Pedro M. Castro, Luis J. Zeballos and Carlos A. Méndez. Hybrid Time Slots Sequencing Model for a Class of Scheduling Problems. AIChE J. doi:10.1002/aic.12609.

– Adrián M. Aguirre, Carlos A. Méndez and Pedro M. Castro (2011). A Novel Optimization Method to Automated Wet-Etch Station Scheduling in Semiconductor Manufacturing Systems. Comp. Chem. Eng. 35, 2960-2972.

– Pedro M. Castro, Adrián M. Aguirre, Luis J. Zeballos and Carlos A. Méndez. (2011). Hybrid Mathematical Programming Discrete-Event Simulation Approach for Large-Scale Scheduling Problems. Ind. Eng. Chem. Res. 50, 10665-10680.

– Luis J. Zeballos, Pedro M. Castro and Carlos A. Méndez. (2011). An Integrated Constraint Programming Scheduling Approach for Automated Wet-Etch Stations in Semiconductor Manufacturing. Ind. Eng. Chem. Res. 50, 1705-1715.


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