stochastic petri nets

MODELING AND SCHEDULING OF FLEXIBLE MANUFACTURING SYSTEMS USING TIMED/STOCH~STIC PETRI NETS

Hiroyuki Ta~ra and Itsuo Hatono Department of Precision Engineering Osaka University 2-1 Yamada-oka, Suita, Osaka 565, Japan Phone: +81-6-877-5111 Ext.4626, 4628 FAX: +81-6-878-3819 E-maiL:[email protected] [email protected]

Abstract. This paper deaLs with methodoLogies for modeLing and scheduLing of FLexibLe Manufacturing Systems (FMS) which are the typicaL discrete event dynamic systems. Since Petri nets provide a powerfuL tooL for modeLing dynamicaL behavior of discrete concurrent processes, we use them for modeLing FMS. For the purpose of off-Line scheduLing of FMS we use timed Petri nets, and for on-Line reaL-time scheduLing of FMS we use timed-stochastic Petri nets with hierarchicaL structure where we deaL with uncertain events in FMS such as variation of processing time, faiLure of machine tooLs, variation of repairing time, and so forth. We deaL with off-Line and on-Line ruLe-based scheduLing. The roLe of the ruLe base is to generate an appropriate priority ruLe for resoLving confLicts, that is for seLecting one of enabLed transitions to be fired in a confLict set of the Petri net. This corresponds to seLect a part type to be processed in FMS. Towards deveLoping more InteLLigent Manufacturing Systems (IMS) we propose a conceptuaL framework of a futuristic inteLLigent scheduLing system. Key words. Timed Petri net; stochastic Petri net; FMS (FLexibLe Manufacturing Systems); ruLe-based scheduLing; inteLLigent scheduLing; IMS (InteLLigent Manufacturing Systems).

1. Introduction

1982) are necessary for design, performance evaLuation,

HighLy computerized FLexibLe Manufacturing Systems (FMS)

have been

recent The

deveLoped to meet

high-variety/middLe-to-Low

voLume

~uch

Timed

ences

various and

and demands, and so

ture

forth.

manufacturing systems are

assessment such

even

for gLobaL

environment job

for

scheduLings are

tions

important tasks

1991b)

scheduLing discrete 1987).

and

their extensions

on

that is,

We construct

ruLe whenever we to seLect one

in a confLict set of

a ruLe

base to

need to resoLve

of enabLed transi-

the Petri net. In FMS this

corresponds to seLect a part type to be processed. Towards

This paper summarizes our recent works (Nakamura aL. 1988, Tamura et aL. 1988, Hatono et aL. 1989,

1991a,

a priority

confLicts,

In

for achieving various objectives of FMS. et

on-Line scheduLing.

generate

to consider technoLogy environment.

are used for the pur-

nets (Marsan et aL. 1984; Archetti and Sciomachen 1987)

varying prefer-

Very recent and fu-

naturaL

Petri nets (Sifakis 1977)

pose of off-Line scheduLing, and timed-stochastic Petri

as sociaL/economic environment

production, customers

pLanning, and production scheduLing of FMS.

We use Petri nets for modeLing and simuLating FMS.

production.

main needs for FMS are to adapt to various changes

of externaL factors of

p~oduction

the requirement of

deveLoping more InteLLigent Manufacturing

Systems (IMS) we describe a conceptuaL framework of a futuristic inteLLigent scheduLing system.

modeLing and

methodoLogies for FMS which are the typicaL event dynamic systems (Varaiya and Kurzhanski

The events in FMS are such as Loading, process-

2. Modeling of FMS Using Ti.ed/Stochastic Petri Nets

ing,

transporting, unLoading, and so forth.

The beha-

In this section we summarize Nakamura et aL.(1988)

vior

of FMS is so compLex that it is difficuLt to ana-

and Hatono et aL.(1989), where we deaL with an FMS with

Lyze it theoreticaLLy. FMS

(PanwaLker

Thus, simuLation techniques for

and Iskander

1977; BLackstone

et aL.

-96-

fixed routings so that sequences of machines to process are given for each part type.

Deterministic ModeLing Using Timed Petri Nets

start

and operationaL end,

Let P={ P1,P2, ••• ,Pn} and T={ t 1 ,t 2 , ••• ,t m} be finite sets of pLaces and transitions, respectiveLy. The

notes

an immediate

transition.

input

Loading

2.1

and output functions

pLaces. tion

reLate the transitions and

are

The input function is a mapping from a transi-

tion.

input pLaces I(t i ) of the transiThe output function 0 is a mapping from a tran-

sition

ti to a set of output pLaces O(t i ) of the tran-

that can

fire immediateLy after

and timed

transtions that

can fire

time

after they

become enabLe.

This time

and

represents the state

the state that

the machine is

P3' respectiveLy, represent

parts in each buffer.

combinations of this kind of

some fixed

Petri

net

is caLLed

state

of the FMS in each time step by tracing the dis-

representations,

and we

can

simuLate the

tribution of tokens in the timed Petri net. tokens that

the pLace

2.2

Pj can contain.

Stochastic ModeLing Using Timed-Stochastic Petri Nets In

defined as foLLows: Firing

timed-stochastic

Petri

nets

(Marsan

et aL.

1984; Archetti and Sciomachen 1987), the firing time of

Condition: A transition ti is enabLed when

~

a marking

and waiting pLaces, re-

in pLace P2

FMS is mode Led by the

the firing conditions of enabLed transitions are

Then,

buffers for

PLaces P2 and P4

be a marking and L(Pj) be the upper bound of

~

number of

the

A token

pLace P4 represents

An

firing time. Let

referred to as processing

in

they become en-

abLe,

P3 represent

waiting for a part to be processed. Tokens in pLaces P1

are cLassified into two types: immediate transi-

ions

P1 and

that a part is under processing by the machine. A token

sition. In timed Petri nets (Sifakis 1977), the transitions

PLaces

denotes a timed

and unLoading, respectiveLy.

spectiveLy.

ti to a set of

respectiveLy, where t1 de-

transition and t2

a

hoLds the foLLowing two conditions

timed transition is a random variabLe with an appro-

priate prObabiLity distribution. By using such stochastic Petri

we can modeL an FMS under uncertainty as

net~

a discrete event dynamic system whose eLements are such as

where Up.

)=

{

)

Firing

chine

buffer capacity, if Pj is a buffer pLace 1,

When

change the marking according

to the foLLowing evoLution ruLe. EvoLution RuLe: When a transition t . is enabLed at 1

time

with

marking

~'

a

marking~,

firing ti resuLts

~1(Pj)= ~ (pj)-1

for aLL Pj ' I
in a new

2.

~1(Pj)=~(pj)+1

for aLL Pj ' O(t i ).

An

FMS is

a discrete

modeL

tooLs,

and so

~Part

under uncertainty, the

simuLate by using one Petri

the increase

of the

number of

To cope with this difficuLty,

stochastic Petri nets are used for modeL-

into severaL submodeLs,

LeveL

such as a transporting

modeL, a processing LeveL mOdeL, a controL LeveL

modeL, and so forth.

event dynamic

The

transporting LeveL modeL

workpieces tooLs

exampLe of a

The

and

mutuaL

describes a fLow of

excLusion

or transports (automated

controL

of machine

guided vehicLes, AGV).

processing LeveL modeL describes the processing of

a workpiece with machine tooLs, repairing of a troubLed

represent events of operationaL

Part a-+j Machine

because of

and transitions.

hierarchicaL

net representation for an FMS eLement. In Fig. t1 and t2

with an FMS

net

whose eLements are such as Loading, processing,

transitions

we deaL

modeL is difficuLt to

tioned

unLoading, and so forth. shows a simpLest possibLe Figure Petri

of machine

of ma-

ing FMS under uncertainty, where an FMS modeL is parti-

1.

system

repairing

FMS

pLaces

as

regarded as

faiLures,

unLoading, occurrence

forth.

otherwise.

a transition wiLL

Loading, processing,

machine

(machine tooL or transport), and so forth. The

controL

LeveL modeL

computers

b

To

describes the

which controL machine

reduce the compLexity of

chicaL

systems,

the

data

pro~essing

on

tooLs and transports.

the modeL of such hierar-

hierarchicaL

structure

of

the

system must be described in stochastic Petri nets. Figure 2 shows an exampLe of hierarchicaL stochastic type

Petri net representation for an FMS where one part is processed with one

machine tooL. We introduce

the foLLowing conditions in the hierarchicaL stochastic Fig. 1.

Petri net modeL of an FMS eLement.

Petri net in Fig. 2.

-97-

wa-iting to process

l}arta~Partb

o~ ~ 000 Po.

tal

laJ

Pal

PaJ

buffer for loading

loa-ding starts

Pa3

processing

unloa-ding buffer for ends unloading Pl (a ) A

(a) An example of transporting level model (Discrete-time stochastic Petri nets).

eo,

operational

'0'

.:~.~"

. . ,"

o~ ~o· p,

/)

h

JI ~,

a fadure

Fig_ 3_ mode Led,

A hierarchicaL stochastic Petri net modeL.

time of each

timed transition in

2(b) is obtained using a random number associated

with the foLLowing probabiLity distribution. tb2: a normaL distribution

with mean of the processing

time determined in the process pLanning ance

0

2

Transporting LeveL modeLs of four basic FMS eLements.

we need onLy one processing LeveL modeL. This

wiLL simpLify the overaLL modeL enormousLy. Figure

The firing

( d ) A trall sfe r Jiltf' ..... i t h two pil rt ty pt ·s

( c ) A dis a....;sc mhly lin e.

(b) An example of processing level model (Continuous-time Stochastic Petri nets).

Fig.

'~'

Po>

pro cessIng

O .P . r-"-------------------/

Fig. 2.

/h ) ,\n il ..sst'rnhl y lin e

lin e.

I'a.nb Ma.c hlne " a.rt c ~

6

r:::rr'" repairing

Pa. rt a.

operational

tra.n ~ fe r

and vari-

LeveL

typicaL exampLes

contains

a hierarchicaL

a

white box,

thick

timed transition

as shown by

hence contains

a processing

and

as shown

of transporting

FMS eLements. Each eLeme nt

LeveL

modeL

These

four basic stochastic

are

•

tb4: an exponentiaL distribution with mean 1/ y, where y

3 shows

modeLs of four basic

in Fig.

2(b) as

its submodeL_

Petri net representations

combined to modeL a transporting LeveL modeL of an

FMS.

denotes faiLure rate. tb6: an exponentiaL distribution

with

mean r, where r

2.3

ConfLicting Transitions

If

transition t1 fires then

denotes mean repair time. If

In

the negative firing time is Obtained, it is assumed

abLed

to be zero_ 2)

In Fig. 2(a), we

hierarchicaL ta2

caLL the transition ta2 the

timed transition

is obtained as

and the

the time spent by

from pLace Pb1 to Pb4 in Fig. 2(b). A token in pLaces Pa1' Pa2'

Pb1' Pb2' Pb3' Pb4 in Fig. 2(b) represents the state of a workpiece under operation. Therefore, the processing LeveL

modeL shown in Fig. 2(b) pLays a submodeL of the

transporting

LeveL modeL shown in Fig. 2(a). Moreover,

aLL

the processing LeveL modeL

can

be represente.d

and vise versa_ We identify a set of tra ns iti ons ConfLicting

tion

of any process in FMS,

by the stochastic

Petri nets that

Transitions: A set of enab Led tra ns i-

is confLicted

transitions Pa3 in Fig. 2(a) and

and t3 are enabLed.

tran s iti on t3 bec omes un-

in such situation as foLLows:

firing time of a token to move

Fig. 3(d), transitions t1

at time

ti and tj

if, for

in the set,

when ti is fired at time In

T,

T,

every pair of

tj become s una bL ed

and vi ce versa.

the timed/stochastic Petri net modeL of an FMS

with fixed routings, confLicted sets of enabLed tr an sitions

are disjoint and uniqueLy

operationaL-start pLaces.

identified

transitions with

a~

sets of

the com mon wai t in g

To resoLve confLicts, we need t o seLect one of

enabLed transitions to be fired in each confLicted s et_

has the same connective reLations of pLaces and transi-

Since

tions

to each part type, we can resoLve confLicts by appL ying

when

a confLicted set correspo nds

in Fig. 2(b)

with different parameter

of mean

and variance

of probabiLity distribu-

priority

ruLes to

associated with firing times. ConsequentLy, even

priority

ruLe for resoLving a confLict may change over

vaLues tions

each transition in

as shown

an

FMS with

many

part types

and

processes is

-98-

time

a~d

seLect a part

type. An ap pr opria t e

may depend on the different confLicted set.

Enter R• J

Rk

Fig. 4.

HierarchicaL s"ructure of a ruLe base.

3. Rule-based F"S Scheduling

This Hatono

section summarizes Nakamura et aL.(1988) and

et aL.(1991a),

where we deaL

FMS model ing sys tem

with an off-Line

and on-Line ruLe-based scheduLing system for generating

FMS

Rul e base

simulator

appropriate priority ruLes to seLect a transition to be fired to

from a set

of confLicting transitions according

the states of the

timed/stochastic Petri net modeL

of an FMS. 3.1

Fig. 5.

Constructing a RuLe Base For

generating an

construct

appropriate priority

a hierarchicaLLy structured

ruLe, we

ruLe base. Each

responding priority ruLe is seLected for resoLving con-

ruLe in the ruLe base has the foLLowing form:

fLicts of enabLed transitions.

ifPithenQi·

Job scheduLing is carried out under various objec-

A condition Pi is given by a predicate which describes the states of the Petri net such as throughput, remaining

number of processes, remaining

date,

remaining processing

each

buffer, and so

either

A ruLe-based scheduLing system.

time up to the due

time, occupation

such as

minimizing compLetion

pending

upon the

statement Qi is group number of the ruLes

exampLe

of the

to be checked next.

time, fLow time,

number of tardy jobs, maximizing production rate, and so forth. We need to find an appropriate ruLe base de-

ratio in

forth. An action

a priority ruLe or a

tives

the

different scheduLing ruLe base for

objectives. An

on-Line scheduLing with

major objective JIT (Just-in-Time) can be found in

Hatono et aL.(1991a).

A hierarchicaL structure of the ruLe base is shown in

Fig.

4 (Nakamura

et aL.

1988). The

procedure to

3.2

FMS ScheduLing System

check

whether each predicate Pij in group is true or faLse is as foLLows: In the first stage, the first pre-

duLing

dicate

the data

P11 in group is checked. If it is true then the predicate P21 is examined next as shown by a directed arc T; otherwise P12 in group 1 is examined as

shown true ing ruLe the

by a

directed arc F,

where arc T

and F denote

and faLse, respectiveLy. In generaL, after checka predicate

P ,..J in

base, if it group

of

group

in this hierarchicaL

is true then

the first predicate in

Lower LeveL

foLLowed

by an

arc

A ruLe-based scheduLing system for off-Line sche-

ruLe

in the

a group is faLse, then

a predicate in the group of

upper LeveL foLLowed by an

Petri net

modeL of

an FMS

with fixed routings

taking into account the machining informations. The FMS simuLator conducts simuLation run for each time step as foLLows: Step 1. Search enabLed transition at time t. Step 2. If there exist confLicting enabLed transitions,

then caLL the

ruLe interpreter; otherwi,e fire

aLL the enabLed transitions.

arc F is checked. If a

predicate in the group of LeveL 0 is true then the cor-

the ruLe base and

interpreter. The FMS modeLing system constructs a

timed

T is

by an arc F is checked. If the Last predicate

Fig. 5. Besides

base, the system consists of three subsystems

such as an FMS modeLing system, an rMS simuLator, and a

checked; otherwise the next predicate in the same group foLLowed

is shown in

~.

for

-99-

If the

present time

simuLation, then stop; otherwise

is the finaL time set

t+1~t

and go

to St ep 1. The

modify the meta-ruLe for generating an appropriate ruLe ruLe

interpreter generates

a

priority ruLe

base.

from

the ruLe base when the FMS simuLator detects con-

fLict

among enabLed transitions. According to the pri-

centraL

ority

ruLe generated, one enabLed transition is chosen

system

to be fired and the confLict is resoLved. For

ReaLization

this

simuLating an on-Line FMS scheduLing, the FMS

of an effective "Learning box" is the

probLem

to obtain

of an FMS. As direction,

an

inteLLigent scheduLing

an introductory research towards

a seLf-tuning

mechanism of

ruLes in

ruLe-based scheduLing of FMS is being deveLoped (Hatono

modeL ing system in Fig. 5 constru.cts a timed-stochastic

et

Petri net modeL of an FMS as described in 2.2.

parameters in the ruLe base automaticaLLy anaLyzing the

aL. 1990). This mechanism can adjust some heuristic

scheduLing resuLts obtained previousLy. 4. Towards InteLLigent ScheduLing

Another

A conceptuaL bLock diagram of a futuristic inteLLigent

scheduLing system is shown in Fig. 6 (Tamura et

aL.1989). This system consists of four bLocks as 2) KnowLedge base box

duLing

3) FMS simuLator

duLe, is to

a meta-ruLe

time-varying

(Tamura

utiLity

muLtiattribute

function

can

box".

The muLtiattribute

preference

socio-economic

may vary and

in

time depending

poLicy determining moduLe, each of which

ing, preference of the customers, the demand for

vari-

upon the change

fuzzy ruLes. De-

of scheduLing objectives user

easiLy correct the knowLedge

base by revising the

evaLuation moduLe.

s.

ConcLuding Re.arks In

on are

the typicaL discrete event

dynamic systems. It is

fuL tooL for modeLing such systems.

production scheduLes as discussed in the previous where the ruLe base generated by the "Learn-

ing box" is impLemented in the "knowLedge base box".

In duction in the

"FMS simuLator" is evaLuated whether the scheduLe meets

possibLe.

the decision

maker or

ControL) are quite im-

which the in-process inventory and the inventory of

tion scheduLe generated by the "knowLedge base box" and

information is fed back

flexibLe manufacturing JIT (Just-in-Time) proand TQC (TotaL QuaLity

portant (Monden 1983; Ebrahimpour and Schonberger 1984)

In the "muLtiobjective evaLuation box" the produc-

objective of

this paper we have summarized our recent works

modeLing and scheduLing methodoLogies for FMS which

shown that timed/stochastic Petri nets provide a power-

The "knowLedge base box" and "FMS simuLator" gene-

this

and dispatching mo-

is described by

membership functions of the fuzzy ruLes in the scheduLe

upon the

naturaL environment of manufactur-

ous products, and so forth.

the

devide the knowLedge

utiLity function

identified represents the preference of the decision maker who manages the manufacturing system concerned.

sections,

this appraach we

et aL. 1988) identified in the "muLtiobjective

evaLuation

rate

inteLLigent scheduLing

into 3 moduLes; scheduLe evaLuation moduLe, sche-

pending

behave as

which generates an appropriate ruLe base depending upon

The

1991b). In

base

"Learning box"

towards

we propose an appLication of fuzzy inference (Hatono et aL.

4) MuLtiobjective evaLuation box.

approach

reaLizing scheduLing objectives with high variety,

1) Learning box

The the

for

not. If not,

to the "Learning box" to

finaL products

sources

At the

are tried

to decrease

same time, saving

and energy as much

as possibLe and production

of high quaLity pruducts and services with high variety

:' ... i~t'~ Ll i gent' '5', hed'u'L i ~g' ·s;' s't'e~"

... .............. ..... .... .

:' "R~'l~::B·~~~d' 'S~'h~d'~ l·i·~!i' Sy'~'te~ So c i 0Economic and NaturaL Envi ronment

~

h/

;

KnowLedge Base

Learning

~( ; e.g. RuLe Base (Meta-ruLe) : Frame base)

Information

Fig. 6.

I

as much as

the necessary re-

r-

r-- SimuLator

Production

MuLtiobjective

F MS

~ :

,

.

''

.'

~ EvaLuation ;\

Know:~~.~fl.~~.. , ...................... .. ..... ~,~ t.~ rnat i ves

System

Decision Mak.ing

A conceptuaL framework of an intelLigent scheduLing system. -100-

are expected. For take

production of the of

the concerning

production in

the industriaL and factory

into

account. This consideration is We need

assessment before other gentLe Lize

~e

to take into

~astes

the gLobaL

have been taken

not enough in fu-

account the environmentaL

of consumption and househoLd

~astes

as

~eLL

start to produce the concerning products. In

~ords ~e

need to

deveLop

for our gLobe, that is,

ne~

~e

"sustainabLe deveLopment". For

teLLigent Ligent

shouLd

For the environmentaL assessment of production,

onLy ture.

~e

into account the technoLogy and environmentaL as-

sessment sense.

next generation,

products

~hich

are

are obLiged to reathis purpose, In-

Manufacturing Systems
job scheduLings are

Hatono, I., Tachibana, K., Yamagata, K., and Tamura, H. (1990): RuLe-based ScheduLing of FlexibLe Manufacturing Systems ~ith SeLf-tuning Mechanism, Proc. 34th Conf. of ISCIE, pp. 493-494, Kyoto, May 16-18. (in Japanese) Hatono, I., Yamagata, K., and Tamura, H. (1991a): ModeLing and On-Line ScheduLing of FLexibLe Manufacturing Systems Using Stochastic Petri Nets, IEEE Trans. on Soft~are Eng., VoL. 17, No. 2. Hatono, I., Suzuka, T., Yamagata, K., and Tamura, H. (1991b): ScheduLing of
highLy expected to contriMonden, Y. (1983): Toyota Production System, lIE.

bute.

Nakamura, Y., Hatono, I., Kohara, Y. , Yamagata, K. and Tamura, H. (1988): FMS ScheduLing Using Timed Petri Net and RuLe Base, Proc. 2nd USA-Japan Symp. on FLexibLe Automation, MinneapoLis, JuLy 18-20.

References Archetti, F., and Sciomachen, A. (1987): Representation, AnaLysis and SimuLation of Manufacturing Systems by Petri Net Based ModeLs, P. Varaiya and A.B. Kurzhanski eds.: Proc. IIASA Conf. on Discrete Event Systems; ModeLs and AppLications, Springer. BLackstone Jr., J.H., PhiLLips, D.T., and Hogg, G.L. (1982): A State-of-the-Art Survey of Dispatching RuLes for Manufacturing Job Shop Operations, Int. J. Production Research, VoL. 20, No. 1, pp.27-45. Canada, J.R., and SuLLivan, W.G. (1989): Economic and MuLtiattribute EvaLuation of Advanced Manufacturing Systems, Prentice HaLL, EngLe~ood CLiffs, N.J. Ebrahimpour, M., and Schonberger, R.J. (1984): The Japanese Just-in-Time/TotaL QuaLity ControL Production System, Int. J. Production Research, VoL. 22, No. 3, pp. 421-430. Hatono, I., Katoh, N., Yamagata, K., and Tamura, H. (1989): ModeLing of FMS under Uncertainty Using Stochastic Petri Nets, Proc. Int. Workshop on the Petri Nets and Performance ModeLs, pp. 122-129, Kyoto, Japan, Dec. 11-13.

Pan~aLker, S.S., and Iskander, W. (1977): A Survey of ScheduLing RuLes, Operations Research, VoL. 25, No. 1, pp. 45-61.

Sifakis, J. (1977): Use of Petri Nets for Performance EvaLuation, In H. BeiLner et aL. eds.: Proc. 3rd Int. Symp. on ModeLling and P~rformance EvaLuation of Computer Systems, North-HoLLand, Amsterdam. Tamura, H., Matsubayashi, K., Nakamura, Y. (1988): ModeLing MuLtiattribute Preferences for EvaLuating Production ScheduLes in FLexibLe Manufacturing Systems, Proc. 2nd USA-Japan Symp. on FLexibLe Automation, MinneapoLis, JuLy 18-20. Tamura, H., Yamagata, K., and Hatono, I. (1989): Decision Making for FLexibLe Manufacturing - OR and/or AI Approaches in ScheduLing , Syst. AnaL. ModeL. SimuL., VoL. 6, No. 5, pp. 363-371. Varaiya, P., and Kurzhanski, A.B. (1987): Discrete Event Systems ModeLs and AppLications -, Lecture Notes in ControL and Information Sciences 103, Springer, BerL in.

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