Download - Academy Publisher

(2) T ' = T ∪{
~
t } − Tt
;
~ • ~
•
(3) F ' = F ∪{(
p,
t ) | p ∈ ti}
∪{(
t , p)
| p ∈to
}
•
o
} − Ft
( , M
0
•
−{( p,
ti)
| p∈
ti}
−{(
to,
p)
| p ∈t
.
Definition 4.3 ( N ', M
0'
) obtained from N ) by
t-subnet reduction operation comprises net N'
and
marking M
0'
, where M 0' = M
( P \ Pt
) 0
(where M
( P\
Pt
)
is obtained from M by deleted the vector corresponding
to P
t
).
Definition 4.4 A net ( N
t
, M t 0)
is said to be a t-closed
net if we add a transition t
t
and arcs
•
•
{( p , t t
) | p ∈t
o
} ∪{( tt
, p)
| p ∈ ti}
to t-subnet
( N
t
, M t 0)
, and the marking of ( N
t
, M t 0)
is preserved.
Note that in this section, let
( N ', M
0
') : the original net; N
t
= ( Pt
, Tt
; Ft
, Wt
) : the
p-subnet ; ( N
t
, M t 0)
: the p-subnet system; ( N
t
, M t 0)
:
the closed t-subnet system; ( N , M
0
) : the reduced net.
Theorem 4.1 Suppose that ( N , M
0)
is obtained from
(
0
N ', M ')
by t-subnet reduction operation. Then
( N ', M
0'
) is bounded iff ( N,
M
0)
and ( N
t
, M t 0)
are
bounded.
Proof. (1) Since N , M ) is bounded, then ∀ p ∈ P ,
(
0
∃k > 0 1
such that ∀ M ∈ R ( M
0)
, M ( p)
≤ k1
.
Since ( N
t
, M t 0)
is bounded, then ∀ p ∈ Pt
, ∃k > 2
0
such that ∀ M
t
∈ R( M t 0)
, M t
( p)
≤ k2
.
Let k = k 1
+ k2
, ∀ p ∈ P'
, ∀ M ' ∈ R(
M
0'
) such that
M '(
p)
≤ k . So, ( N ', M
0'
) is bounded.
(2) Suppose that N , M ) is unbounded, then
(
0
∃ p ∈ P , ∀k > 0 , M ∈ R M )
∃ such that
(
0
' ( M
0
M ( p)
> k . That is ∀k > 0 , M ∈ R ')
that M '(
p)
> k
N ', M ') is bounded.
(
0
∃ such
. This contradicts with the fact that
Theorem 4.2 Suppose that N , M ) is obtained from
(
0
(
0
N ', M ')
by t-subnet reduction operation. If
•
( N ', M
0'
) is live and t i
⊆{ p|
p∈P'
∧(
M'(
p)
> 0)}
,
then ( N , M
0)
and ( Nt , M t 0)
are live.
Theorem 4.3 Suppose that N , M ) is obtained from
(
0
(
0
N ', M ')
by t-subnet reduction operation. If
( N , M
0)
and ( Nt , M t 0)
are live, then ( N ', M
0'
) is
live.
Ⅴ. APPLICATIONS
In this section we apply results of Section Ⅲ and
Section Ⅳ to reduce a flexible manufacturing system.
The manufacturing system consists of one workstation
WS for assembly work and two machining centers for
machining. Machining center_1 and WS share robot R
1
.
Machining center_2 and WS share robot R
2
. The
system runs as follows:
In the machining center_1, the intermediate parts are
machined by machine M
1
. Each part is fixtured to a
pallet and loaded into the machine M
1
by robot R
1
.
After processing, robot R 1
unloads the final product,
defixtures it and returns the fixture to M
1
.
In the machining center_2, parts are machined first by
machine M
3
and then by machine M
4
. Each part is
automatically fixtured to a pallet and loaded into the
machine. After processing, robot R
2
unloads the
intermediate part from M
3
into buffer B. At machining
station M
4
, intermediate parts are automatically loaded
into M
4
and processed. When M
4
finishes processing a
part, the robot R
2
unloads the final product, defixtures it
and returns the fixture to M
3
.
When workstation WS is ready to execute the
assembly task, it requests robot R 1
, robot R 2
and
machine M
2
and acquires them if they are available.
When the workstation starts an assembly task, it cannot
be interrupted until it is completed. When WS completes,
it releases the robot R1
and robot R
2
.
Firstly, we give the Petri-net based model of the
manufacturing system. Secondly, a reduced net system is
obtained by p-subnet reduction method and t-subnet
reduction method. Thirdly, we will analysis property
preservation of the reduced net system.
The Petri-net based model ( N , M
0)
of the original
manufacturing system is illustrated in Fig.5.1.
p 11
t 11
p 12
p 13
t 12
p 14
p 15
t 13
p 16
p 17
t 14
t 31
p r11
t r11 t r12 p 33
t
p 32
r13
p 31
p 34 p32
p r12
t 33
t r13
t r14
p r14
t 34
p 35
p 21
t 21
p
p 23
22
p r21
t 22
t r21 p 24
p r22
p r23 t 23
p 25
p 26
t 24
t r22
p 27
t 25
p r24
t 26
Fig. 5.1 The original net system
p 28
p 29
252