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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.
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,
-7803-7952-7/03/ 17.00 0 2003 IEEE.
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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 )
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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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