a comparative study of two deadlock avoidance for assembly process non-sequential hsieh

8/10/2019 A Comparative Study of Two Deadlock Avoidance for Assembly Process Non-sequential Hsieh

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A Comparative Study

of

Two Deadlock Avoidance

Controller Synthesis Methods for Assembly Processes

*

Fu-Shiung H sieh

Overseas Chinese Inst i tu te

of

Technology

Taiwan,

R.O.C.

[email protected]

analyze the control policies generated by applying the

two

sufficient conditions to the same class of assembly processes.

ur

analysis shows that the synthesis metho d

proposed

in

[4]

is

less conservative th n th t proposed

in

[8]-[9] for assembly

processes.

Organization of the remainder of this paper is as follows.

Section I1 reviews the CAPN model. Section

111

presents

the liveness condition for CAPN . Section IV compares with

an existing result. Section V concludes this paper.

Abstract -

Although there are a

few

results regarding

deadlock avoidance problem for non-sequential

producrion

processer in literature, there is a lack of

camparicon

f o r the

existing results. The goal

of

this paper

i s to

compare

hw

existing results concerning deadlock awdance in assembly

nianufaccnrring syst em , This pa pe r con siders a

subclass

of

the existing Controlled Assembly Petri Nets (CAPN) with

each operation requiring only one unit

of

arbitrary number of

Wpes

of resources called WW Analysis shows that the

. -

svnthesis method Drouosed

in

our

previous

work

is

less

.

conservative than that pro posed in an existing result for the

class

of

CAPNU.

2

CAPNModel

Let

J

denote the set of assem bly processes in the system.

types in the system. To capcure

the interactions among resources and jobs in assembly

Keywords:

Flexible manufacming system, deadlock,

Let

be the set of

control policy, assembly process.

1

Introduction

processes, an operation denoted as that merges two

Deadlocks cripple the progress

of

production activities.

Guarantee of deadlock-free operations is essential for

achieving high resource utilization in flexible

manufacturing systems. Although there are a few results for

non-sequential production processes in literature

([I)-[4], [SI-

[9]),

there is a lack of comparison for the existing results. The

goal of this paper

is

to compare the results of 181-[9] with those

appeared in [4].

In

[8]

and [9], the authors defined a Petri net

model for assembly/disassembly processes to be realizable if and

only if there exists a feasible execution sequence. The authors

proved that the computational complexity to determine the

realizability of an assembly process

(RAP)

is NP-complete. The

authors also proved that to find the a l l y permissive

deadlock avoidance control policy for assembly/disassembly

systems is NP-hard. A sufficient condition (Theorem 1 of

[SI)

was proposed to maintain the deadlock free

prom of

thisclass

of assemhlyidisassembly

systems.

The sUac ient condition

is

based on the zoned structure

of

the production processes and

states that if

the

resource capacity is no less than the number of

zones involved, there exists a tansformation to maintain the

deadlock k e ropem of the given system. In [4], the author

proposed a conlrolled Petri net model called Controlled

Assembly Petri Net (CAPW for a class of assembly processes

and a suboptimal polynomial complexity deadlock avoidance

algorithm based on a sufficient liveness condition for CAPNs.

However, no comparison has been made for

[8]-[9]

and

[4]. In

this paper, we will fist re-examine the sufficient conditions

proposed in [SI-[9] and [4] for the class of CAPNs.

Then we

PNs through common places, transitions, or arcs is defined

as follows. Given

two

Petri nets

GI

=

P , , ,

1 1 , 0 1 ,lo

)and

Gz

=

( P 2

T 2 , , , 4 20 ), where

we assume that m l o ( p ) mzo p ) V p

E 4

n 2 .

We define GlIIG2 =(P

,T ,

,

0

o ), w h e r e P = P, u p 2 ,

The operationI1 also appeare d in [3]-[7]. By extending

the

CPPN

model proposed in [SI, we propose a CAPN

model based on the Petri

net G = IIrEnGR,

IIjeJGJ,

constructed by merging the

resource subnets GR,

,

r E R , with job subnets GJ, ,

j

E

J

,

described as follows. The procedure

to

construct

CAPN also resembles that of RCN-merged nets ([7]).

Definition 2.1:

A

job subnet G J j = ( P , , T j , I j , O j , m j o )

is

an

acyclic marked graph ( M G ) nd is of tree structure as

each place has exactly one input and one

output

transition,

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where

m j o ( p )= 0

V p E

Pj

. That is, there is no job in

process under m o .

1;

;if;

P> p ,

t5

Figure 1 A job subnet

Assume that a unit of resource can only he involved in one

operation at a time. A type- r resource, r E

R

, may he

involved in

NR,.

activities, where each activity consists of a

sequence of ope rations using type- r resources sequentially.

Suppose there are n, distinct operations in the k

-th

activity

of

type- r

resources. The set T,

=

{ tf 1) , 2) ,...,

r, (n:) }of transitions are used to represent

the

operations

in the k -th activity of type- r resources. The states of the

resource before and after transition t , (n)are denoted by

places

p : ( n - I )

and

p : ( n ) ,

respectively. Letp,(O) be the

idle state place of type- r resources. The Petri net of the

k -

th activity can he represented by a type- r resource activity

circuit

p f ( n l

-l)t;(n,")p,(O)

.

To model synchronization of

operations, we assume two distinct resource activity

circuits C> and C? may have multiple common transitions

hut have one and only one common place p r ( 0 ) , he idle

state place. The type-

r

resource suhnet

GR,

is

GR, =

C: C: I . CrRr Remark that

GR,

allows modelling of

production activities that cannot be modeled with state

machines.

Definition 2.2: Let

GR,

=

( e ,

,,

,,

O r ,

m , , )

denote

the

type- r resource suhnet, where m,,(p,(O)) > 0

and

m,o p) =

0 Vp E P,

\ {p , (O)} .

W e will use

Po

=

{p , (O)lr

E

R}

to denote the set of resource idle state

places.

Given a

set

of job subnets G J j , j

E J

, and a

set

of

resource subnets

GR,

, r

E R

we construct a Petri net

model

G =

c, =

P,(O)t ; (I)P:(1) t ; (2)P:(2) . . .

GR,

j e J G J j

=

(P,,

,0

mo ) .

Definition

2.3:

A control place

p ,

is a control point to

enable or disable a controlled transition. We use a small

square box to represent a control place. There is a transition

input arc between pe and the corresponding controlled

hansition.

A

controlled transition is disabled if no token is

placed in the control place preceding it and may he fxed as

many times as the number of tokens in the control place.

1; *

Figure 2

Figure 3

Defmition

2 . 4

A

CAPN

is defined as an eight

tuple G,=

(P,Pc,T,,T,,I,O,mo,u)

abbreviated as

Gc(mo,u)

where mo is the initial marking of G , and mo

E

Mo(G,),

U,(C,) denotes the set

of

initial markings of G,, andu

is

a control policy defined based on control action ofG, as

follows.

Defmition 2.5: Mo G,) ~ { m m ( p ) = O

V

p

E P - P o and m(p)

0 V

p E P o } ,where Po denotes the

set

of all idle state places

of

all types of resources of

G, .

We w ill

abhreviateM,(G,) asMOwhenever it is clear from

the context.

U, G,)

denotes the set of initial markings

ofGCunderwhich all resources are in idle state.

Delinition 2.6 A conhul action a is a vector in Z' that

determines how many times each transition in T, may be fired

simdtaneously under a reachable markingm of a CAPN G, .

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The number of tokens in the control place p , of

transitionfunderaisdenoted asa(p , )ora( t ) . Atransitionthat

can be fued undera is called an admissible transition. A control

policy U is a mapping that generates a control action a

for G, based on its current marking. That

is,

U :R ( m o ) -f 2 ' .

G , ( m o , u )

evolves by firing the admissible transitions under

no

= u ( m o ) . A marking m, is reached and a, = u ( m l ) s

applied nexf etc. Hence a sequen ce { ai of control actions will

be generated for G based on current marking under control

policyu

.

Let

R ( m o , u )

denote the set

of

all reachable markings

of

G,(mo ,u )

fromm, E

Mo(Gc

.

Let R(mo = U R ( m o , u )

where U is the set of all control policies ofG

.

A necessary condition for the existence of a control policy

to keep a given CAP N G, live is to have sufficient resources

available to fire each transition in G C .We defineMo(G,) as

the set

of

initial markings o f

G,

with sufficient resources to

keep G,live as follows, where

M , ( G , ) MO .

Definition 2.9: M o ( G , ) = ( m m E Mo(G ,)and there exists

a control policy U

under which

G , ( m , u )

is live.}. We will

abbreviate M o ( G , ) as

M O

whenever it is clear from the

context.

Given a C A F " G, with marking

m E R ( m o ) ,

wberem0

E

M o ( G , ) , determine a least restrictive allowed

control action

a

such that

G

can be kept live under the

marking reached after executing

a

under

m

. The problem

was proven to be N P-hard in [SI and

[9].

herefore, we will

develop a suboptimal algorithm with polynomial

complex ity to maintain the liveness of

G ,

.

U E U

3

A Liveness Condition for CAPN

Definition 3.1:Let VI E Z N and V2 E Z N . We denote

VI 2

V , ifV,(i)

2

V 2 ( i ) or eachi

E

{1,2,3 .._..

)

.We denote

V,

> V2 if

VI

2 V2 and there exists at least

oneiE{1,2,3 , _ _ _ _ _}such tha tV,( i )>

V 2 ( i ) .

Definition 3.2:Given a CA" G,, M , ' ( G , ) denotes the subset

of initial markings of G, with minimal resources for the

existence of a control policy

to

keep G, Live. Obviously,

M i ( G , ) M o ( G , ) .

M,'(G,)is abbreviated

as

M i w h e n it

is clear from the context. The set

of

resources in idle state

under m E M , ' ( G , ) can be represented

by

a vector in ZIRl

called a

MRR

of

G ,

.

For each m E M i ( G , ) , there exists a control policy

U

under

which G , ( m , u ) is live. For any

marldng

m' , f m < m for

somem

E

M i

( G , )

, here does not exist any control policy

U

'

under which G,(m',u') is live.

Property 3.1: G iven a CAPN

G ,

with marking

m E N ( m o ) ,

there exists a control policy

U

such that G,(m,u)s live if

and only if there exists a m* EM, and a sequence of

control actions that bring m to a marking m E M O

with

m' 2 m* .

A s a C A F " G, consists of a set J of processes, an upper

bound of

MRRs

of G, can be calculated based on a h4RR of

each process in

J

.

To

find a MRR of type-

j

process, every

transition in

T j

need to be fired. Figure 9 illustrates an

assembly transitiont of type- jprocess.

To

fire an assembly

transition requires all its immediate subassembly transitions

to be fired. We present a heuristic algorithm to compute an

upper bound of

MRR

for an assembly transition based on

the resource requirement to fire each immediate

subassembly transition o f t . We use + t

to

denote the set of

immediate subassembly transitions precedent to t . A formal

defin ition of't is as follow s.

Definition 3.3: The set of input places of f , not including

its

control place, is denoted as 't. The set of output places

o f t is denoted as f . The set of immediate subassembly

transitions precedent to t is denoted

as

+t = { t / t *s f } .or

the type-1 job subnet shown in Figure

I ,

+ f 2

= ( f l )

,

f 4 = ( f 3 )

and

+ t S = { t 2 , f 4 } .

t

Defmition

3.4:

Transitions in the subassembly processes

of

a tmnsitiont

The set

rj

t)

of all transitions in the subassembly processes

oft is the subset of transitions of Tj with each transition

in

T j t )

onnected to t with at least one directed path

in GJ,

.

Remark that

+ f r T j ( t ) .

Let

s Tj t))

enote the

projection of firing sequence

s

on the set T j ( f ) f

transitions.

Example 1:

For

t h e C AF " in Fig. 3

TI =

{f[,tg

,f1,t2,f3~t4.fs.f(l

TI (ts 1=

{t;,t; ,r,,t2,t3,t,,t ,f

.

Suppose

s = t ; t ; t l t 2 t 3 t 4 t s t [

a n d t = t S .T h e n s ( q ( f S ) ) =

f ; t ; t l f 2 f j t 4 f s .

Let N , be a vector in ZIRl that denotes the resource

requirement

to

fire a sequence s of transitions, with N, ( r ) ,

r

E

R ,

as the number of type- r resources required. We

will let R, , vector in 21Rl, denote the resources involved in

firingjwtt,withR, r)asthenumber

oftype-r resources

required,where

r

E R .

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Definition 3.5: A firing seqwnces is said to be a m i n i i

firing sequence

to

lire t

E T, if I ) S s an enumeration of the

set T, t) of transitions,

(2)

t is the

last

transition ins , and (3)

there does not exist any enumeration

S'

of the set

T,

( t ) of

transitions withNs,N , . A minimal firing sequence to f r e t

is

denoted

ass; .

Remark that

s:

contains

t .

Defmition 3 .6 Let R( S) denote the set of resources

involved with the set S of transitions. That is,

R (S )= { r l R , ( r ) 0 , wheret

E S

n d r E R } ,

The following heuristic Pair-wise Interchange algorithm

finds an upper bound of

MRR

for GJ with polynomial

End For

End For

s,

J

A U

{t)

End For

U p d a t e 5

=

{tit E

T,

\Aand't E A }

If

E

CJ Then

Go to Step 1

Else

End If

Exit

To

calculate an upper bound of MRR for G , with a set J of

processes, we define the following operator.

Definition 3.7: Operator@ takes the larger of two vectors

element by element.

Let

s j

denote the firing sequence constructed by applying

the Pair-wise Interchange Algorithm to fire the final

transition f f we- Process. A n

bound

Of Mm

complexity. Let s -

t)

denote the prefvt of s preceeding

t E a n d

s r

-) denote the suffut of s a f ie rt . Lets, denote

the firing sequence constructed by the algorithm

to

fire

t

.

Let A be the set of transitions whose firing sequence have

~

been constructed by the algorithm in some iteration.

for G, can

be calculated

as

follows.

Let B be the set of transitions whose firing seq uence

is

to be

Interchange A lgorithm is very intuitive. Consider a Exam ple l(Continued): Consider the CAPN GI shown in

transition

t E T , .

Let

t = { f i 2 f 2 , h , . . . J ~ } .

Figure 3. The algorithm initializes

A

with

{ : , t i } .

fuins

f

after

t i

is fired may require more resources as the

same h e

f

resources are required byf, andr' .

In

this case,

it is reasonable

to

fue t before ti is fired. Although the

algorithm does not guarantee generation of minimal f i n g

sequence, an upper bound of MRR can be obtained wing

the firing se quence generated by it.

Pair-wise Interchange Algorithm

N = N ,

@

N ,

@

N ,

e...@ N

.

constructed in some iteration. The idea of the Pair-wise

IJI

I f R ( ( t i l ) n R ({ r ' ) ) fC J fo r

Some

t ' E T j ( t k ) - { t k l

9

Fort , ,s , ,

=t;ti,andfortj,s,,=f;t).

For t , , s ,~= t ; r , t , , and

for t 4 ,

sq

=

t ; t3 t4

. For fs , the above algorithm

initializes s , ~ with

s , ~ =

sl, s,, =

t ; t l t 2 t ; t 3 t 4 .

As R((t2 1 nR(Ti(t4)) t.

J ,

the firing sequence found to

fire ts is

s , ~

= t ; t3r4 t ; t l t2 t5 The resource requirement to

f i re t ; r3 t4 r~t l t2 t s i s [2

I] .

Step 0:

A

=

LT,

4. Comparison

with An

Existing Result

Update5 = {tit E

Tj

\ A and't

c

A }

[SI and [9] deal with deadlock avoidance problem in

assembly systems. The pa ps s focus on the problems of process

realizability and of the least restrictive deadlock avoidance

policy. In 181 and 191, the anthors defined a Petri net model for

assembly

processes

o

be

realizable

if

and

only

ifthere

exists an

execution sequence U =m Otl m,t 2m zt3. .fnm n such that

{ f i l i ~ l , Z , , 3

...,

n}=T,whereTdenotesthesetoftransitionsin

the assembly process. The authors also deiined a system is

reversible if and only if for eachm E R(mo) , mo s reachable

fromm

.

n

181

and [9], the authors stated that

to

distinguish the

least restrictive model

to

which the deadlock avoidance problem

of removing firing can

be

addressed and to develop the

least

restrictive deadlock

avoidance policy, it's necessary to find algorith ms to so lve the

following

two

problems:

(1)RedimbiIity of Assembly Rocess (RAP): Given a

systemPT , s it &ble?

Step 1: For each t E E

Let't

= { t i , t z , t 3...,

N )

s:

=st,

st

st,

...St,.) SI SI,., . S I , sr, st, ,

'

S I N

Fori

E

{l,2,..,,

N)

Fork E ( i + l,i+ 2,,.., A }

For PE

T , ( t k )

\

{fk}

IfR({fi})nR({t')) CJ and t '@

sl - t i )

Let

Si

t

sequenceb from it.

St

(T,(f ' ) )

I Where a

b is the

s 2 = s i ( - t i ) s , ( T , ( t ' ) )

t i

$ , ( t i -)

st + 5 2

End If

End For

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(2)Reachability of ma (RIM): Given a realizable system PT ,

suppose

m E R(m,)

and f is enabled under marking m

.

Let

m ' be

the marking reached after firing f under marking m .

Is

mo

reachable h m m '

7

The authors proved that Problem

RAP

is NP-complete

(Theorem

4

f [9]) whereas Problem RIM is NP-hard (Theorem

2 of [9]). Based on the above results, the " a l l y permissive

deadlock avoidance control policy is NP-hard To reduce the

computational complexity, [8] and 191 proposed a sufficient

condition (Theorem

1

of [8] nd Theorem 3 of [9]) for a Petri

net model of assembly processes to

be

real ihle. The sufficient

condition proposed in [8] and [9] is as follows.

Theorem 3 of [9]: If c(b )

L

v(b) for each b

E

B then PT

is

realizable, where B denotes the set of buffers, c(b ) denotes the

capacity of buffer b

E B

and v(b) is a parameter that denotes the

number

of

zones in the job model involving buffer b

E

B .

The authors also proposed a sufficient condition (Theorem 2 of

[SI and Theorem 4

f

[9]) to guarantee deadlock free property

of the system by making the assembly processes r e a li bl e and

reversible. Application of the sufficient condition

requires c(b) 2 v(b) for eac hb

E

B

.

Theorem 4 of [9]: For each system AP such

that c(b) L v(b)

V b E

B , he system under the control of

Definition 5.4 in [9] is realizable and reversible.

We compare the result of

this

paper with the existing result of [8]

and [9]. We show that our result is less conservative than the

sufficient condition proposed in [8] and [9] for the class of

CAPNs

as the resources required by DA C algorithm is no more

than those required hy the sufficient condition of [8] and [9] .

We

i rst

illwkate

this

result by

an

example.

Example

1 (Continued): Consider the example

in

Figure

2,

which is similar to one of the assembly process in the example

that appeared in [SI (Figure 1) and [9]. For

thi s

example, there

are three types of resources and the set Po

=

(

ps,p,.ps)

According

to

the delinition of a zone in [8] and [9], there

are

three zones in thi s example: zi =

p I p 2 z

=p,p4.2

= p s .

As there are

two

zones ( z i = p 1 p 2 n d z 2 =

p 3 p 4 )

equires

bufferbi , v(bl) =

1.

Similarly, v(b2)= 2, and v(b3) = 1 (See

Example

6.2

on page 419 of [SI). The sufficient condition of [9]

requires the capacities of buffersb, ,b and b3 to be c(b,) ,

c ( b 2 )

2 2

, andc(b,)

2

1 , respectively. That is, the resource

requirement is [2 2 11. By applying the Pair-wise Interchange

Algorithm,

an

upper bound of M R R for thi s example is

[2

1 I].

As [2

1 I] 5 [2 2 11.The Pair-wise Interchange Algorithm yields

a less conservative result

th n

Theorem

3

of

[9].

To

formally

compare our results with Theorem 3 of [9], we consider the

following subclass of C A P " .

D e f ~ t i o n .1: The subclass

of

CAF'Ns with each operation

requiring only one unit of arbitrary number o f

types

of resources

isdenoted asCAPNU.

For the class of CAPNU, although the Pair-wise Interchange

Algorithm does not.guarantee

MRR

can be found, it always

3

yields less conservative results than Theorem

3

of 191. That is,

we have the following result.

Theorem 4.1: For CAPNU, the Pair-wise Interchange Algorithm

always h d s a firing sequences whose MRR

is

no greater th n

the resource required by Theorem

1

of

[SI.

That is,

N,

r )


6/6

w i t h l t l 2 2 , t ~ T ~ a n d =k+l. L e t t = (t ,, f ,, t , ,..., N } ,

where

N

2 2

.

Lets, denote the-firing sequence conshucted by

the Pair-wise Interchange Algorithm. Lets,- be the sequence that

sequentially fires the transitions in GJj starting with f and

-

ending with

t{

.

Let

s =

s, s,.

-

st,

sr,

st,

...

,,-, s,, s

,,+, .. s,,.]

s , ~

s

,,+,

.. sly s,-

.

AU

we need to

prove is that

N , ( r ) i v ( r )

V r c

R

. As 1 =k for each

f E { t , , t 2 , f 3...,

t N } according to the inductive assumption,

NS,

r )

5 v 5

r )

V r E R . Firing s , ~ requires at most

N,,,

( r )

units of type- r resources. If n o type

r

resource is held

afier firings,, , irings,, afters,, requires at mo stN,,> (r)unitsof

t ype r resources. Otherwise, firing s,, after s requires at

most

I Y ~ , ~r ) + N ( r )units of type- I esources.

Based on

similar reasoning, firings, +,fters,, s s

..

s,-,s , ~ equires

no greater than

Ns, ,

r )

N ,

( r )

+...+

r )

unts

of

*-

r

resources. Firing s,- er s, requires

at

most

N ( r )+

N

( r ) ...

N,, , r )unts

of type r resources as all

with the exception of at most one

unit

(being held) of

type-

r

resources will be released after S J is fired. ?herefore, the

number of type-

r

r e s o m s requmd

to

fire s s no greater

1

so

542

than I f s , , @ )

+

N , , > ( r )

+...+

N s , N ( r )

. As

N , , ( r ) +

N

N s r 2 ( r ) + , . . + N s , J r ) 5v , ( r ) , i t h o l d s f or l , = k + l .

QE.D.

4

5.

Conclusion

In this paper, we compare two existing liveness

conditions for the subclass of CAPNs with each operation

requiring only one unit of arbitrary numher of

types

of resources

called CAPNU. Analysis shows that our condition is less

conservative than the existing result

for

CAFNU.

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