Modeling and Analysis of Manufacturing Systems with Limited Buffers

219

MODELING AND ANALYSIS OF MANUFACTURING SYSTEMS WITH LIMITED BUFFERS Ke

LlU

University of Science I Technology Beijing 100083 - Beijing, PRC

Mei 'e

SHEN

Beijlng Information I Technology Institute 100101 - Beijing, PRC

Keywords. ordered event model, manufacturing systems, limited buffer size, period Abstract, Based on max -algebra. state equations are established for serial productlon lines with limited buffer size. This model Is expIict and can be constructed directly. It has the same state dimensions and equation order as that with infinite buffers, thus has the same computationa] burden. Some examples are given to verify this model. Extensions of this model and open problems are debated in this paper.

INTRODUCTION By means of max -algebra Cohen et a1. (1985) established a kind of linear models for deterministic and decision - free Discrete Event Dynamic Systems
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parts can be obtained for the K -th time. The ordered event model be set up as bellow,

can

X(K+1)=AX (K+1HBBU (Kt1) EBTX(K) By means of which and the period of determined easily.

(1)

we can learn the change law of all state variables system output. Thereby the control law can be

Example 1. A serial production line with two machines and two parts, shown in Fig.1. Procelsing time matrix t21

Fig. 1. A Serial Line

J

t 12 t 22

Tt=[tll

where t IJ Is the time used by machine MJ while precesBlng part PJ • The ordered event graph of this serial production line can be obtained immediately as Fig. 2. E IJ refers to the even t M I processing P J • The sequence of events is E 11 , En , EZl Fig. 2, Ordered Event Graph and E22 . There is no direct relationship between E 12 and E 21, so they can be ordered arbitrarUy and are always ordered according to machine number. The earliest time of EIj possibly starting for the K -th tlme is denoted as state variable xq(K), where q=(i-l>mtj. Actions from E 11 to Ell and En to E2l are denoted a8 feedbacks. The ordered event model is ,

~~~:f;].. [~'u: 11' [xaCKtl) t

x,

•

EB [~11 b=J [u 1CK+l>] EB[: t12 , . J[:~~:~:J x 3CKtl) .• Ua (Kt1) . :

·' .] [Xl(K)l • xz
:

t12 t a '

x,(Ktl)

here, ..... represents the null element

.. 8

'

of

e

operation.

•

·

'.

8

t:za

.

X3

(K>J

x,(K)

.-00 .

Changed some elements of matrices A and T, recurrence formula (1) can be used to describe the system with llmited buffer size efficiently. This method of modeling is of traits as follow, CD the form and most elements of formula (1) remain the same. Only some 8 '8 in matrices A and T need to be changed as e ' s ( unit element in EB operation ). The computational burden remains the same. ~ the changed elements correspond directly to the added relationships caused by the existence of limited buffers. Every changed element has clear physical meaning. State variables need not any transformation. @ the modifying of elements follows a simple regulation and can be carried out directly on basis of formula (1). MODELING AND ANALYSIS In this paper the performance measures considered are system throughput and utilization ratio of machine. For serial lines with n machines and m kin-ds of parts, shown in Fig.3 , Liu et al. (1991) gave some resultsl with sufficient buffers and pallets the output can reach the period of bottleneck machine (s) after several batches when the parts are input

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intensely. If we control the input period of parts in accordance with the period of bottleneck machine t system can perform under the largest throughput with the need for every buffer size no larger than m -1. So we only consider the case hI , size of the i-th buffer, within (0, m-I). When a buffer is bigger than m -1 units, it will has no restrictiono n system while approaching its largest throughput, i. e .. , approaching the period of bottleneck machine. hI MI h2 M2 hD MD Pm·····PI ~~~ ••• -~~ Fig.3.

A Serial Line With

n

Machines

And m Parts

The existence of buffer h lu between MJ and Mltl has direct influences only on events both of Ml and Mltl • So buffer hlH can be analysed separately and so do the other buffers. The partial ordered event graph corresponding to MI and Mitt with infinite intermediate buffer is shown in Fig. 4. Whether an event can start is only determined by if the corresponding machine and part are all ready for use. While the finl te buffer exists, the starting time of MI processing PJ is determined not only by machine and part as above, but also by whether the h It1 +1st part P k ahead P J begins to be processed on Ml+l . P k and P j may not belong to the same batch of parts, thus the Fig. 4~ Ordered Event corresponding activity events may not Graph (partial) happen for the same times. The relationship between them is possibly a common action or a partial feedback· Let hl+I .. 1 , the times when Ml starts processing Pl , can not earlier than the times when Mitt starts processing P 1I1-I(K ) ••• PIl1(K) respectively, corresponding to states variables Xlmttn- I(K)··· Xlmtm(K) , these restrictions can be viewed as feed backs and are denoted in matrix Tb' [he times when Ml starts processing P:t% (K+l)· .. Pm, corresponding to state variables x (\-1)mtlt2
-~

(2)

Compared

with

matrices A and

T

in (1),

the

ID

-dimensional

square

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sub-matrices A I. I +l and TI • I+1 ( in Ala and TIa respectively ) are no longer compoaed of 8 'a. Some 8'a become e'a regularly. AI.I+1 and TI • ltl form a combinatorial matrix H lt1 - [AT 1.I+1J used to describe the added I.IH

relationships brought by hl t l . It is of regularity to construct HItl. CD if h lt1 =m -1, then AI.I+l = Q)m ( m -dimensional square null matrix T I,ltl IS Em ( m -dimensional unit matrix ). (2) with the size of h lt1 decreased one unit, the diagonal line composed 6 8 o 8, of me's in H Itl moves vertically o "8 up one unit, as shown in Fig.5. A1.I+lj a:: Ht+l = •••••••• ··t~·· h HI 0 [ T1.I+l cl) when h It 1 =0,

A 1.1+1 =

J 8 :~~8 [ a

T I• lt1

..to ° O""t

[8-8 10] =

6

8

Fig. 5

I

0

= 1 hi+! =m-l

Change Regularity of H 1+1

Becauee of the existence of finite buffera, A h ia no longer a strictly lower triangular matrix. There are non - 8 elements in upper triangular region of A la' The directed graph corresponding to Ah does not Include circuits. For example, a serial Hne with two machines and m parts, ha =0, the event relationships described by Ah are shown in Fig. 6. Although Ell events numbered larger act on events numbered smaller, such as E 21 -- E 12 , En all theae actions correspond to arrows directed obliquely down. No matter how Eza large buffers are, there are not any arrows directed up or obliquely up. It is impossible for circuits to exist. The same conclusion can be analogized to 8erial lines with multi -machine. We have A ~ D =Q) and can remove the term Fig.61 Event Relationships X (K +1> in the right hand of (2) by Described by Ab recurrence. Theoreml For serial lines of n machines and m parts with limited buffers, hI
X CK+1>=AiiTh X

If the parts are input inteneively,

(3)

<3} can be eimpHfled as

XCK+l>=AiiTb X(K) The periodicity of system is directly of A ii T h • Because the order of state state vector do not increase with the burden remains the same as that with

(4)

concerned with the characteristics equations and the dimensions of relationships added, computational infinite buffers.

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EXAMPLES Example 21 A serial production line of 2-machine and 3-part, its ordered event graph is shown in Fig. 7. The processing time matrix

~

Tt=[!

---Y2

3 10J·

---- Y3

The

period of bottleneck machine The upper limit of buffer h2 is 2 units.

Fig.7 I Ordered Event Graph of System in Example 2

M2 is ). =6+2+10=18.

variables and events I X 1'+- -+ Ell. x3----E13' X4+---+ E21 , x5+---+E22 , Xs+---+ E2:J. When h2 =0 we have I

The

relationships

X
A~Th=

between state

···· 5 ·· ·5 9 · °· 0 · · ·· 9· 3· 6 2· ·· j · 3 8

· · · · ·

• 19

• 20 •

• 22

·

.-

:l m+1lJ ~ ~rX+1>m

· · 100 · •• 1019

·7 ·8 · 17··

Yl

-t--~

0

I •

I l.-

,

let

··3 · · · · · · · · · ·

n X'l ~

l~l

J

XZ .....- ... En:,

· 0'1 •

·

I

I

· 10' \ X (K) ·· :J (22'-

132 X (2)= 3~ l4<

, then X (1). 19 10

" -".

41 4(1

19

22

For K >2, there exists X (K +1)=22X (K). Now that the practical period is 22, the bottleneck machine M2 can not be utilized sufficiently. Example 31 System is the same as that in example 2. ordered event model is changed as

( 5. · · · · X(K+1)=

· ·

9

5

9

·

· ::lI ·

··

0

.

r~ . ~tK+l>m .I

X(K+l)$ : ·6 · · 3 · 2 :J l:

0

:j

( I

Let h 2 =1,

the

· 0 o·'1 · · · · · · · · 10"I X (Xl · ·, · . ··· .) ·· · '

·

3

I

Let X (0) = (0 0 0 0 0 0) r, we have X =18X (K) for K ~ 4, Now J.. =18, which is just the inherent period of bottleneck machine M2 . In thi8 case M 2 is utilized sufficiently after four batches of parts processed. The utilization ratio of M 2 approaches to 1.

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By comparing the results in example 1 and size of buffer h 2 is one unit.

2

we find that the optimal

CONCLUSION This method of modeling can be analogized to assembly lines. disassembly lines and some other complicated manufacturing systems. In this paper we only study serial production lines with the same processing sequence, i. e., a batch of parts pass every machine in the same order. It can be proved that the change regularity shown in Fig. 5 holds for serial production lines with arbitrary processing sequences except that the lower limit of h I is possibly not zero. If some buffers are too small, deadlocks may occur during the performance. For serlal production lines with identical processing sequence, slzes of all buffers fall on the interval (0, m -1) . Most of the buffers needn't reach the upper limit m -1. How to minimize the size of a buffer without lowering the performance efficiency of system is an important problem in the optimization of system. The number hi does not appear in the model as a variable, so it's difficult to determine the optimal size of hi by analytical method. We still have some work to do on setting up direct relationship between A. and hi' REFERENCES Cohen, G., Dubois, D., Quardrat, LP., Viot, M., 1985, - A Linear -System -Theoretic View of Discrete -Event Processes and Its Use for Performance Evaluation in Manufacturing -, IEEE Trans. on AC , Vol. 30, No. 3. pp. 210-220. Liu, K., Zheng, Y.P., 1990, - Periodicities and Control Laws of Basic Production Lines with Multi-Part - , Proc. of Annul Meeting of Control Theory and Its Applications (Chinese), Hangzhou, China. Liu, K., Shen, M.E., Zheng, Y.P., 1991, - An Ordered Event Prco. of IFAC Workshop Model of DEDS and Its Applications -, Applications in Manufacturing and Social on DES Theory and Phenomena, Sbenyang, China. Liu, K. , Shen, M. E. , Zheng, Y. P. I 1991, Modeling and Analysie of Manufacturing Systems with Finite Buffers - I Journal of Decision ! Control_ (Chinese), Vol. 6, No. 4, pp. 241-246.