Buffer store of line-type modelling

Franciszek MARECKI

D epartm ent o f Com puter Science Systems Technical University o f Łódź

Bielsko Biała Branch

BUFFER STORE OF LINE - TYPE MODELLING

S um m ary: The paper presents the mathematical model o f a buffer store. The store consist o f autom atic lines o f FIFO type, l oading and unloading contr ol consist in the choice o f the line.

1. Introduction

Buffer stores o f various types occur in the com puter integrated production [3], [4], The process o f storing is m odelled by means o f arithm etical-logical equations o f state [1], These equations allow to construct com puter sim ulator program for control purposes [2], The paper presents m athem atical model o f buffer store. The store consist o f type FIFO lines and o f control points: loading and unloading ones.

2. Problem formulating

Let's consider the system com posed o f two m achines with buffer stores betw een them.

The m achines serve object o f various types. During the change o f the type o f the served object the m achine expects the change o f the tool. The expectation tim e o f the machine depends on the type o f the two objects follow ing each other.

From the point o f view o f m inim alization o f the total expectation tim e o f the machine - the optimum sequence o f serving the objects may be marked. The optim um sequences o f serving objects for every m achine are different in the general case. For this reason com prom ising solutions are marked.

The buffer store betw een m achines plays im portant part in the optim ization o f the production schedule. This store allows to minimize tim es o f expectation o f the second machine. The object that exist in the store and m inimizes the expectation time, is subjected to the service in the second m achine.

Buffer stores have lim ited capasity. Moreover, various constructions o f these stores are used, e.g. linear (FIFO), pile (LIFO), round robin etc.

A model o f the m ultilinear buffer store will be presented in the paper. The store consist o f many independent lines with the com mon loading point and with the com m on unloading point.

Let's assume that the objects, that have been served with the optim um sequence for the first machine, are delivered to the store. These objects will be placed in the chosen store line.

The objects are m oved to the furthest free position in the buffer store lines. The object from the last position o f the line may be delivered to the unloading point - and

further on, to the service in the second machine.

M om ents and types o f the objects occuring in the loading point are random . The stream o f object in the loading point determines extortion o f the system . The m om ents of taking object from the unloading point to the second machine may be treated analogically.

The objects may be introduced into the store on the first position o f the chosen line.

The objects m ay be taken out from the last position o f the store o f the chosen line in the analogical way.

We assum e that all the times o f changes o f tools in the second m achine are known before starting the service o f the next object. The problem o f control a buffer store consist in defining:

- the line w here the next object should be unloaded - the line w here the next object should be loaded.

The m inim ization o f the global shutdow n time o f the second m achine is accepted as the criterion o f control.

3. M athem atical model

Let's assume that the buffer store consist o f M lines, and eveiy line contains N positions.

We accept the follow ing designations:

m - num ber o f lines, ( m = l,...,M ) n - num ber o f positions, (n = l,...,N )

Let's assume that there are I object types in the served system. The m atrix o f shutdown tim es for the change o f the tool is given:

where: ti.j - the time o f the machine shutdow n after the service o f the i-th object and before the service o f the j-th object.

The cycle "c", after which the object moves one position further w ithin the buffer store line, is given. W e accept that during the "c" cycle the object may be taken out:

I

J (

1

58

- from the loading point to the first position o f the chosen line, - from the last position o f the chosen line to the unloading point,

A buffer store w orks in cycles. Let's denote by: k - the cycle number, (k -l,...,K ).

We accept the follow ing priorities o f cycle k:

I). The object is unloaded if the unloading point in k - 1 cycle was em pty and if the object was on the last position o f the chosen line.

II).The object is m oved to the following position if this position has been free in the cycle k-1 III).The object is loaded if it was in the loading point in the cycle k - I, and w hen the first

position o f the chosen line was free.

D efinition 1:

The state o f the buffer store after cycle k is the follow ing matrix

Z m =

1

n = 1 N ( 2)

where

*• m .n*k

I i if there is the object o f type i in k cycle on n position j o f m line

[O if the n position o f the m line is em pty in the k cycle

(2.a)

We also accept:

- the state o f loading point

ii if the object o f type i exists in the loading point { in this cycle

lo

( 3 )

- the state o f unloading point

i i if the object o f type i exists in the { unloading point in cycle k

[O if the unloading point was empty in this cycle

( 4 )

The initial state o f the system is defined by:

The num ber o f cycles results from the tim e o f the work o f the system.

Definition 2:

The follow ing vector is the control in cycle k

U‘ = K J 1= u (5)

w here

and

Cm if the object in cycle k should

uki = 1 be loaded into line m (5.a)

lO in the oposite case

im if the object in cycle k should

uk2 = ( be unloaded from line m (5 b)

lo in the oposite case

The adm issible control m ust fulfil the following condition:

(« * = /» ) =>[(-¿f: =

0

0

6

and

(,/‘ = m) => [ ( i ‘ . > 0) a ( / ' 1 = 0)] (6.b)

M oreover we assume that

[(* * -'> 0 )a( « * > 0)] => (X* = 0 ) (7.a)

and

[ ( / - ' = 0 )a(i/‘ - 0 ) ] => ( / > 0 ) (7,b)

The record o f the state is kept at the end o f the cycle.

4. Equations of state

The buffer store state changes after every cycle. These changes depend on the accepted control.

Equations o f state have the general form:

60

x* Z " ' r t , d x) (8) (9)

. * = / I ( x* -|y - |, z * - , ,i#1‘ ,«/i) (io)

where: dv, dx- exterior interference

The state o f the unloading point is m arked in the following way:

if ( / * ' > 0) a

(dr

/ = ( * , i f ( / " = 0 ) a ( « , ? = » » ) ( 1 1 )

\o, if ( / - ' > 0 ) A

(dr

The state o f the store is m arked in the follow ing way:

if ( C = o)

„ V ( i - V . v = ( < v , if ( * V v > 0 ) a ( / / ( = 0 ) ( 1 2 )

\o, if (< -;. >o)a(,,‘ = W)

- the positions from 2 to N - 1 :

if ( C = o )

= if ( - L ' , > 0 )a( C > ° ) (13)

\0. if ( C > 0 ) A = 0)

- the first position

* . if ( < 7 = 0 ).a ( « ,* * » )

¿V . if ( < V > 0 ) (14)

\o, if = 0) A ( « * = 0) V ( < - ' > 0) A (? *;' = 0)

The state o f the loading point is m arked in the following way:

I d , , if ( x * - ' = 0 ) A•i' = / . V k- ', if (X1- ' > 0 )A (» ,‘ = 0 )

\ o , if (i/* >

0

( 15)

Equations o f state allow to mark the state o f the store in the follow ing cycles from k=l to k=K

5. Final remarks

Equations o f state allow to create a com puter sim ulator program o f control a buffer store. In the simulator, control: Ujk and u2k should be defined for every cycle k. The total tim e o f the change o f tools in the machine is accepted as the criterion o f control. Therefore the follow ing may be added up:

<7

kr.

where: qk - tim e o f the machine expectation in the cycle k C om ponents qk are marked in the following way:

j t i r if ( x 1' = j ) a ( x ' = /) and for 1 < r < k

q k = l

\ 0, in oposite case

H euristic contr ol rules may be applied for minimizing index (16).

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" I n ż y n i e r i a W i e d z y i S y s t e m y E k s p e r t o w e " , I S i T S P o l i t e c h n i k i W r o c ł a w s k i e j . W r o c ł a w 1 9 9 3 , t o m II st r . 4 9 3 - 5 0 1 .

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19 9 3 , t o m II, s t r . 3 7 6 - 3 8 1.

R e v i s e d b y : J a n K a ł u s k i