Computing the productivity of multistation serial manufacturing systems: A heuristics approach

COMPUTING THE PRODUCTIVITY OF MULTISTATION SERIAL MANUFACTURING SYSTEMS: A HEURISTICS APPROACH

S. Yeralan

*

and S. Khajenoori

University of Missouri, Industrial Engineering

** University of Central Florida, Industrial and Computer Engineering

ABSTRACT

Analytical models to evaluate the performance of multistation serial production systems are difficult to build and solve.

Based on the insights obtained from a two-station continuous

materials flow production line, a heuristic is developed to approximate the production rate of longer lines.

KEYWORDS

Serial manufacturing systems; reliability; buffer storage; stochastic processes.

INTRODUCTION

Production lines are serial arrangements of workstations.

Materials pass through successive

stations as a specific operation is performed at each station. ciency is station breakdown (Koeningsberg, 1959).

A major cause of line ineffi-

When a station breaks down the following

stations may be forced down, or starved, since the broken down station is unable to feed the downstream stations.

Similarly preceding stations may be forced down,

or blocked,

since the

broken down station is unable to remove the semiprocessed materials from its upstream stations. Interstation buffer storages are often used to stations.

However,

finite

capacity

buffers

reduce

cannot

this

potential

entirely

interference between

decouple

adjacent

stations.

Blocking and starving can still occur when the buffers become either empty or full.

Most of the past research on production lines assumes discrete materials flow which leads to a Markov chain models. probabilities.

The production rate o f such models is computed from the steady-state

The production rate is defined as the amount of materials produced in unit time

in the long run.

Since the number of states are directly proportional to the product of the

capacities of the buffers, the computational effort involved in evaluating the production rate is nontrivial.

A general state nonhomogeneous discrete flow model is given by Yeralan and Muth

(1984). There are few models of production lines with continuous materials flow.

Wijngaard (1979)

found the production rate of a production line with continuous materials flow.

Fox and Zerbe

(1974) developed a production line model based on rather restrictive assumptions that buffers are empty while all stations are operating and that only one station can be repaired at any given instant.

Murphy (1975) developed a recursive equation to estimate the expected improve-

ment in the production rate due to added buffers.

Gershwin and Schick (1980) computed perform-

ance measures, including production rate, the average level of the buffer, ities of starving and blocking.

and the probabil-

Recently Yeralan, Franck, and Quasem (1984) presented a two

station continuous materials flow production line.

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Their model is relatively general since no

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PROCEEDINGS OF THE 8TH ANNUAL CONFERENCE ON COMPUTERS AND INDUSTRIAL ENGINEERING

restrictive assumptions are made about the distributions of the station breakdown and repair times when stations are blocked or starved.

The production rate and the expected level of the

buffer are given in closed-form.

An exact mathematical formulation for production lines with more than two stations is extremely difficult to obtain (Hunt 1956, and Gershwin and Schick 1983).

In this study, based on the

two-station model given by Yeralan, Franck, and Quasem, a heuristic approach to evaluating the production rate of lines with more than two stations is presented.

MODEL DESCRIPTION

First consider the case of two unreliable workstations connected by an

intermediate buffer.

Let the time to breakdown for station i be exponentially distributed with rate A i for i=I,2. Similarly, When

let the time to repair be exponentially distributed with the rate ~i for

station

i is run

Ai=ui/(Ai+Pi ),

i=i,2.

loss of generality,

independently of the

line,

i=I,2.

the fraction of time it is productive is

Here, A i is known as the stand-alone availability of station i.

Without

the capacity of the intermediate buffer storage is assigned to be I.

We

assume that the rate at which the first station feeds the buffer is equal to the rate at which the

second

station

depletes

the

buffer

provided

that

both

without loss of generality, this rate is taken to be I.

stations

are operating.

Again,

This can be achieved by letting the

time unit be equal to the time for the first station to fill the buffer when the second station is kept down.

With this arrangement, one can measure the buffer level in terms of time, that

is the time it would take the first station alone to fill the buffer to that level. At any given time, the buffer can be in three possible states, full, empty, or partially full. While the buffer remains full, the first station may be blocked. remains

empty,

the second station may

be

starved.

Let

Similarly, while the buffer

T e and Tf

be

two

random variables

denoting the time the buffer remains empty and the time the buffer remains full, respectively. It is assumed that station breakdowns are operation dependent, stations may not break down. Tf.

While

states.

the

buffer

and thus, blocked or starved

This assumption uniquely specifies the distribution of T

is partially

full

there

are

four

possible

combinations

of

and e station

Each combination is assigned an index as shown below.

Table 1

Station States when the

State

Buffer is

Station 1

Partially

Full

Station 2

1

Up and operating

Up and operating

2

Up and operating

Down and under repair

3

Down and under repair

Up and operating

4

Down and under repair

Down and under repair

There are two other combinations of the system states.

One of these combinations, denoted by

station state index 5, corresponds to the station states while the buffer remains empty. other combination,

denoted

buffer remains full.

by station state 6,

corresponds

The system state at time t, Xt, is expressed by the 2-vector (i,x).

second variable x denotes the amount in the buffer. 6 denotes the state of the stations.

The time dependent system state probabilities while the

Fi(t,x)=P[(Xt=(i,s), where 0
(i)

The time dependent system state density functions are also defined as: i=1,2,3,4.

(2)

For the states where the buffer is either empty or full we define, P5(t) = P[Xt=(5,0)]

The

The first variable i, for i=1,2,3,4,5, and

buffer is partially full are:

fi(t,x)=SFi(t,x)/Sx

The

to the station states while the

(3)

YERALAN & KHAJENOORI: P r o d u c t i v i t y o f M u l t i s t a t i o n

S e r i a l M a n u f a c t u r i n g Systems

P6(t) = P[Xt=(6,1)]

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(4)

Since the long-run characteristics of the system are of interest, the steady-state density functions are defined. fi(x)=lim fi(t,x) t-,~

i=1,2,3,4

(5)

The steady-state probability mass functions corresponding to the state where the buffer remains empty or full are respectively defined. P5 = lim P5(t) t-~

and

P6 = lim P6(t) t-~

We assume that the process

(6)

is ergodic and,

thus, the steady-state density functions exit.

Although it is possible to prove the ergodicity for this class of models, a rigorous analysis will not be attempted in this report.

Let the conditional steady-state density functions gi(x)

be defined as: gi(x) = fi(xl0
for i=1,2,3,4

(7)

The conditional steady-state density functions for the extreme values of the buffer level are defined as, gi(0) = lim gi(x) x~0

and

gi(1) = lim gi(x) x~l

for i=1,2,3,4

(8)

FORMULATION AND SOLUTION OF THE STEADY STATE EQUATIONS

It can be shown from following a renewal argument that in the long-run the fraction of time that the buffer remains empty and the fraction of time the buffer remains full are expressed as: P5 = [g3(0)E[Te ]]/[I + g3(0)E[Te]+g2(1)E[Tf ]]

(9)

P6 = [g2(1)E[rf ]]/[I + g3(0)E[Te]+g2(1)E[rf ]] (i0) For the case Te=Tf=0, the steady-state density functions fi(x) become identical to the steadystate conditional density functions gi(x).

Conditioning the probability density of the event

{Xt+h = (l,x)} on the state of the system at time t gives rise to the defining equations for the conditional density functions gi(x).

gl(X)=[U2/(XI+X2)]g2(x) + [Ul/(XI+X2)]g3(x)

(ii)

8g2(x)/Bx= -(Xl+U2)g2(x)+X2gl(X)+Ulg4(x)

(12)

8g3(x)/gx = (X2+~l)g3(x) - Xlgl(X) - ~2g4(x)

(13)

g4(x) = [XI/(UI+U2)]g2(x) + [X2/(~l+U2)]g3(x)

(14)

Equations (11-14) are a set of simultaneous equations whose solution yields the steady-state density functions gi(x).

Since we have two first order differential equations, we also require

two boundary conditions.

The

first boundary conditions

follows

from the

conservation

of

materials:

~

I [gl(x)+g2(x)]dx

0

= / 0

[gl(x)+g3(x)]dx

(15)

The second boundary condition is the normalizing equation. 4

1

~.

f

i=l

0

Let O=X2HI-XI~ 2.

Notice that 8=0 when the line is balanced, i.e., AI=A 2.

conditional density functions (ii-16).

(16)

gi(x)dx = i

The steady-state

are evaluated in closed form by solving the defining equations

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gI(X):CI[iUI+U2)/iAI+,\2 )j

exp(~®x)

g2(x)=C1 exp(~ex) g3(x)=Cl exp(~@x) 17)

g4(x)=Cl[(il+%2)/(u1+u2 )] exp(~®x) where

C1=~8/[exp(~e)-l]

When

and @=i/(~i+~2+ui+~2).

then

line

is balanced,

i.e.,

8=0

the

solution becomes: gl(X)=C2u2/i 2 g2(x)=C2

g3(x)=C2 g4(x)=C212/U2

(18)

where C2=~i~i/(~i+~i )2 for either i=l or 2.

The line is productive whenever the second station is up and operating.

Let Pe' Pf' and Po be

the fraction of time the line is productive while the buffer remains empty, full, or partially full, respectively.

The production rate o is then expressed as:

p = [I-P5-P6]p ° + P5Pe + P6p f

(19)

Clearly, while the buffer is partially full, 1 Po = ~ [ g l ( x ) + g 3 (x)]dx"

(20)

It can further be verified that, in this model: E[Te] = (XI+X2+~I)/(~2~ I) E[Tf] = (XI+~2+U2)/(XI~2) Pe = ~I/(~I+X2+~I) pf = u2/(XI+X2+~ 2) This concludes the solution of the steady-state probabilities and density functions for the two station case.

AN APPROXIMATION FOR LONGER LINES

The principal

idea is to replace a section of the production

If the entire production equivalent

station,

line with an equivalent

station.

line is divided into two sections and each section is replaced by an

then one could use the solution

cases to compute the production rate.

procedure

developed

for the two-station

and e e cannot reproduce all of the characteristics of a section of the production line containing more than one station.

Therefore,

Clearly, an equivalent station with parameters %

the method is only an approximation.

A section of N stations has

2N parameters compared to the 2 parameters of the equivalent station. Therefore, degrees of freedom in selecting criteria

in choosing

these

the parameters of the equivalent station.

two parameters.

Consider

the

section

of N

there are two

Now we discuss the stations

to

operate

independent of the rest of the production line. i.

Set the stand-alone availability

of the equivalent

station equal to the production

rate of the section of N stations. 2.

Set w e = 7 where 7 is the rate at which the line becomes productive after an unproductive period.

The stand-alone availability of the equivalent station is simply Ae=~e/(~e+%e ) Finding the production rate and y require the solution of the steady-state probabilities of the N

station

production

approximation

line.

procedures

can

Using be

the notion

designed.

of

Here

an equivalent we

present

station,

a simple

several

algorithm

to

recursive find

an

approximate production rate of an L-station production line. I.

Solve for the steady-state probabilities first two stations

of the L-station

of the production line that consists of the

production

line and the buffer

which

connects

YERALAN & KHAJENOORI: P r o d u c t i v i t y of M u l t l s t a t l o n S e r l a l Manufacturing Systems

them.

99

From the steady-state probabilities, compute the production rate p and X.

It

can be shown that the latter is evaluated in the following way. = [I-P5]u 2 + PbUl Note that the line becomes

productive

as

soon

as

station

2 becomes

productive,

provided that the buffer is not empty; in which case the line becomes productive as soon as station I becomes productive. 2.

Compute the parameters

of the equivalent

station.

The criteria

in selecting

the

parameters of the equivalent station is expressed as p=Ue/(Ue+%e ) and Ue=~. Therefore, select ~e=y and le=(7/p)(l-p). 3.

Combine the equivalent station with the next station and the buffer between them. Solve for the steady-state probabilities of this new arrangement, solution procedure for the two-station case.

~.

using the simple

Compute ~ and p.

Continue by going back to step 2 with the new equivalent station.

When the production rate of the system consisting of an LI station section and station L are computed,

the procedure terminates.

The production

rate

computed

in the

last step

is the

approximate production rate for the L-station production line.

CONCLUSION

The production rate predicted by the approximation were compared against lengthy simulation runs.

It is discovered that for relatively short lines (L~I0) the approximation gives good

results.

For longer lines, the approximation underestimates the production rate obtained from

simulation runs.

The solution for the two-station case is in closed-form.

solution procedure described above for longer lines is also expedient. tions are convenient for numerical work. an IBM/PC.

The approximate

All of the computa-

The approximate solution procedure is implemented on

Computation times are about three seconds for a i0 station line.

REFERENCES

Fox, R. J. and Zerbe, R.D. (1974). Some Practical System Availability Calculations.

AIIE

Trans., 6, 228. Gershwin, S.B. and Schick, I.C. (1980). A Continuous Model of an Unreliable Two-Stage Material Flow System with Finite Interstorage Buffers. MIT Laboratory for Information and Decision Systems Report LIDS-R-979. Gershwin, S.B. and Schick, I.C. (1983). Modeling and Analysis of Three-Stage Transfer Lines with Unreliable Machines and Finite Buffers. Operations Research, 31, 2, 354-380. Hunt, G.C. (1956). Sequential Arrays of Waiting Lines. Operations Research, 4, 674-684. Koeningsberg, E. (1959). Production Lines and Internal Storage-A Review. Management Science, 5, 410-433. Murphy, R.A. (1975). The Effect of Surge on System Availability. AIIE Transactions, 7, 439-443. Wijngaard, J. (1979). The Effect of Interstage Buffer on the Output of Two Unreliable Production Units in Series with Different Production Rates. AIIE Trans., ii, 42-47, Yeralan, S. and Muth, E.J. (1984). A General Model of a Production Line with Intermediate Buffer and Station Breakdown. Working Paper 841071, Industrial Engineering, University of Missouri. Yeralan, S., Franck, W.E., and Quasem, M.A. (1984). A Continuous Materials Flow Production Line with Station Breakdown. Working Paper 841073, Industrial Engineering, University of Missouri.