Approximate models of assembly systems with finite inventory banks

European Journal of Operational Research 45 (1990) 143-154 North-Holland

143

Approximate models of assembly systems with finite inventory banks X i a o - G a o L I U a n d J o h n A. B U Z A C O T T

Department of Management Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

This paper describes the development of an analytical approach by which methods for the evaluation of the peformance of transfer lines with finite inventory banks can be adapted and applied to determine the performance of assembly systems such as are used in the automobile industry. The approach enables such real features as multi-station stages, unequal stage cycle times, finite bank transit times and non-serial work flow to be included. The approach is demonstrated by comparison with a simulation model of an actual automobile assembly system. Abstract:

Keywords:

1.

Transfer lines, quenes, manufacturing industries, approximations

Introduction

The research described in this paper was motivated by a project involving the development of complex simulation models of a new automobile assembly plant (see Kostelski et al., 1987). Because of the complexity of the simulations it was felt desirable to try and validate the simulation models by comparison with analytical models. Also, if appropriate analytical models were available, it was anticipated that these models could be used to give more rapid and unambiguous results showing the influence of some key design parameters: of particular concern was the influence of the size of the banks for in process inventory. The simplest of the simulation models consists of five stages where a stage is a sequence of work stations with no intermediate inventory. Between adjacent stages inventory banks are provided. Figure 1 shows the system configuration and it will be noted that stage 5 (including stations 51, 52, 53 and 54) is fed by both stages 3 and 4. Station 51 is an assembly station at which one part coming from stage 3 and one part coming from stage 4 are assembled together to create the combined part that then leaves the system. Stations in each of the

stages can break down and then must be repaired. Because there is no bank within the stage all stations in the stage stop if any station breaks down. The system described by Figure 1 and modelled by our simulation has a number of features that distinguish it from traditional transfer line models such as described in Buzacott and Hanifin (1978) or Gershwin and Schick (1980). In particular: - The cycle times of the stages are not identical. - Parts take a finite time to travel through the inventory banks. - There are multiple stations in each stage. - There is an assembly stage fed by two stages. The purpose of this paper is to describe the development of approximate analytical models

O $uttioa

Bank withtma-zei~¢almcity

flowd~etion

Received August 1989

Figure 1. A car assembly system

0377-2217/90/$3.50 © 1990 - ElsevierScience Publishers B.V. (North-Holland)

Bank with zero capacity

144

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

that enabled us to deal with each of these aspects of the real system and thus provide us with a useful tool for exploring key issues in the system design.

2. T r a d i t i o n a l

transfer line models

The starting point of our model development is the traditional multiple stage transer line model. This describes a system consisting of a number of single station stages connected in series. Between each stage there is a finite capacity inventory bank. The model assumes that - all stages have the same constant cycle time, - any stage can break down with the number of parts processed between breakdowns having a geometric distribution, - a stage which is unable to process a part, either because the next bank is full or the previous bank is empty, cannot break down (operation dependent failures), - the time to repair a broken down stage has a geometric distribution, - the time for a part to transit through an inventory bank is zero. Given these assumptions the first step in the transfer line modelling is the formulation and solution of a Markov chain model of the two stage line with inventory bank capacity z. The two stage line has closed form solutions for the state probabilities and also for the various performance measures of interest, in particular the system throughput or efficiency and what are known as the loss transfer coefficients. There are two loss transfer coefficients in a two stage line ~21(z), the ratio of the time stage 1 is forced down because the bank is full and stage 2 is broken down to the time stage 2 is broken down, and 812(z) the ratio of the time stage 2 is forced down because the buffer is empty and stage 1 is broken down to the time stage 1 is broken down. These results are given in Appendix A. For lines consisting of more than two stages no closed form solution to the Markov chain model exists. Because the state transition matrix is sparse and with a well defined structure, there has been some success recently in using special sparse matrix techniques to solve for the state probabilities directly (Maione, 1987); however, an approximate method, whose basic concept was proposed i n i -

tially by Zimmern (1956) and Sevastyanov (1962) and since extended by a variety of authors (Gershwin, 1987; Dallery, 1988) has proved to be able to give accurate results for the line performance measures. The method can deal with lines of arbitrary length and arbitrary buffer capacity. The essence of the approximate method is that to an observer placed at an inventory bank on the line the upstream station, the bank, and the downstream station appear to form a two stage line. However, as seen by the observer the failure and repair probabilities of, for example, the upstream station are not just these quantities for the actual upstream station next to the bank but also reflect the way in which stations further up the line result i n stoppages of the upstream station next to the bank. Thus the key to the approximate method is the way in which these apparent failure and repair probabilities are determined. Since the line can be divided at each inventory bank into apparently two stage lines, it follows that a further constraint is that the throughput of each of these two stage lines must be the same. The application of these concepts usually results in a set of equations which are solved iteratively. Since we intend to indicate how to modify the procedure to deal with more general systems in a subsequent section, the mathematical details will not given at this point.

3. Adaptation

of the traditional transfer line model

3.1. Determining parameters of a stage from station parameters As indicated above a stage consists of a set of stations in series with no intermediate inventory banks. We are given the parameters (failure and repair probabilities) of each station, but in order to use the transfer line models we require to find the parameters of a stage since the model require each segment between inventory banks to be considered as equivalent to a single station. Our stage is equivalent to a multiple station transfer line with zero banks between stations. It can be shown that with operation dependent failures the up times of the stage will have a geometric distribution with parameter a given by a=l-

fi(1-ag), j=l

(1)

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

where aj is the failure probability of station j. However, because of the possibility of more than one station failing in the same cycle the down times of the stage are not geometric. Thus one approach to determining the equivalent stage parameters is to use the a given by equation (1) above and for the repair time parameter (b) use the reciprocal of the mean down time of the stage. This can be obtained by solving the Markov chain model of the transfer line with no inventory banks to determine the line efficiency A and using the relationship that in a single station system, A = 1/(1 + a/b), thus b = aA/(1 - A ) . Another approach is to look at the distribution of the time between units leaving the line. If T, is a random variable denoting the time required to process a unit at station i plus the time required to repair any breakdowns occuring during the processing of the unit, then the distribution of the time T between successive units leaving the line will be given by Pr{ T~< n } = Pr{max(T1, T2 . . . . . T3) ~< n }.

145

For a single station line with parameters a and b, this distribution is given by

Pr{T=I} = l - a , Pr{T---n} = a b ( 1 - b )

"-2,

with

E ( T ) = 1 + a/b, and

E ( T ( T - 1)) = 2a/b z. In Appendix B the corresponding results are given for multiple station lines. The approach is thus to determine the equivalent a and b by fitting the first two moments for the single station line to the first two moments of the multiple station line. It can be seen that both approaches will give the same first moment for the interdeparture time distribution but the second moments will differ. In some cases, where a stage has a very large first moment the second aproach can fail in the sense that it is not possible to find an a, b

Table 1 Comparing methods for fitting equivalent stage Case a 1

a

b

0.010 0.020 0.020 0.04920 0.04938 0.04938

0.110 0.110 0.110 0.10917 0.10956 0.10956

0.010 0.020 0.020 0.04920 0.03987 0.03989

0.160 0.060 0.160 0.09511 0.07708 0.07710

0.001 0.002 0.002 0.00497 0.00403 0.00403

0.160 0.060 0.160 0.09591 0.07739 0.07739

1.05205

2.39722

60.9978

1.05205 1.05205 1.05205

2.13738 2.39722 2.39722

35.0001 53.2004 53.1998

0.001 0.002 0.00300 0.00263 0.00263

0.160 0.060 0.07576 0.06656 0.06656

1.03957 1.03957 1.03957 1.03957

2.22872 2.08433 2.22872 2.22872

58.0587 42.4121 54.6382 54.6383

(3.1) (3.2) (3.3)

M1 M2 M3 2

(3.1) (3.2) (3.3)

M1 M2 M3 3

(3.1) (3.2) (3.3)

M1 M2 M3 4 M1 M2 M3

(3.1) (3.2)

a Mi: Method i in Section 3.1.

1st moment

2nd moment

3rd m o m e n t

1.45071

9.67825

226.312

1.45071 1.45071 1.45071

9.70778 9.67825 9.67825

228.356 226.736 226.737

1.51731

14.9405

600.585

1.51731 1.51731 1.51745

12.3950 14.9405 14.9407

344.612 523.973 523.843

146

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

pair with both a and b in the range (0, 1). However, this seems not to happen with real data. A third method can be obtained by adapting the second method in a way which is easier for computation because it uses a recursive scheme. That is, replace stations 1 and 2 by an equivalent station then add station 3 and replace the resulting two station system by an equivalent single station and so on. In Table 1 we present results comparing the three methods. For small breakdown probabilities and large repair probabilities all methods work well and give good third moment approximations. However, the methods 2 and 3 work better as the stations become more unreliable. In most cases methods 2 and 3 given the same results, thus for our system model we have used method 3.

3.2. Allowing for different cycle times m different stages In real systems the cycle times of the stages are often different. Unfortunately with the discrete time Markov chain approach, the model with different cycle times is usually intractable. Hence approximation is again needed for solving the problem. A natural extension of the approach of the previous section is to use the idea of equivalent stages. There are several possible ways to obtain equivalent stages to use in the transfer line model. Suppose we have a stand-alone stage with cycle time c o and let the failure and repair probabilities in this cycle time be a 0 and b0. The question then is: if instead of observing the stage every co time units we observe the stage every c (c < Co) time units, what would be the equivalent stage in the new observation interval. "' Equivalent" here means that at least the production rate of the stage per unit of time should be the same. However, there are several additional ways in which equivalence can be defined. (1) We can require the equivalent stage to have the same second moment of interdeparture times. Denoting the moments (measured in units of the new cycle time) by m i ( c o / c ), we have

m , ( c o / C ) = Co 1 + C ( a-~0°) m2(c°/c)=(~-)

ao Oo)

1 + - ~ o + b---~ '

and thus the new failure and repair probabilities over time c will be given by

b=

2(ml(Co/C ) -- 1) m 2 (Co/C) -- m, ( Co/C)'

a = b(m,(Co/C ) - 1). (2) We can require the equivalent stage to have the same mean down time. Let

1

1 c0

l+a

b=b o c'

( b=

ao)c o --c-"

1+-~o

Then

a=ao+bo(1-~o

)'

b = CO C---bO"

(3) Similarly, we can require the equivalent stage to have the same mean up time. We obtain c

a = --ao, Co

b=

aobo¢o/C

( a o + bo)co/C - bo "

(4) If Co/C is an integer then we can observe the system at intervals c apart. Following an instant at which a unit is produced we know that at the next instant no unit will be produced, thus we take a = 1. In order to find b we can require that

m'(c°/c)=

Co( a°) v l +-~o '

and we have b=

b0 (Co/C) ( ao + bo ) - bo "

In Table 2 we show a comparison of simulations of two stage lines with unequal cycle times with the predicted performance using the above methods and the two stage line formula. We found that in most cases the equivalent stage from method 1 works best. We note that in cases 5-8, method 4 fails to find correct equivalent repair rates (~< 1), although it seems to provide good estimates of throughput.

3.3. Allowing for travel time through the bank Existing transfer line models assume that the time for a unit to travel through the inventory bank is zero. This is valid for shunt type banks or magazines where units only enter or leave the

X.-G. Liu, J.A. Bueacott / Approximate models of assembly systems with inventory banks

147

Table 2 Results for two-stage lines with different cycle times Case

a1

b1

a2

b2

z

cl : c2 a

Efficiency

Ave. bank lev.

1

0.030

0.090

0.010

0.100

5

1 :2

Sim. b M1 c M2 M3 M4

0.42286 ___0.01109 0.42764 0.41727 9,38838 0.43135

4.34 _+0.09 4.27 3.84 2.61 4.38

2

0.030

0.090

0.010

0.100

30

1 :2

Sim. M1 M2 M3 M4

0.44799 + 0.00448 0.45353 0.44945 0.41653 0.45405

27.88 5:0.62 27.84 25.91 18.12 28.30

3

0.050

0.060

0.005

0.100

5

1:3

Sim. M1 M2 M3 M4

0.28297 + 0.01335 0.28636 0.27134 0.25206 0.28790

3.98 + 0.17 4.02 3.58 2.75 4.05

4

0.050

0.060

0.005

0.100

30

1:3

Sim. M1 M2 M3 M4

0.31485 + 0.00625 0.31576 0.30670 0.25650 0.31612

26.63 + 0.77 26.89 24.14 16.84 27.10

5

0.200

0.100

0.050

0.100

5

9 : i0

Sim. M1 M2 M3 M4

0.30785 + 0.00723 0.30809 0.30574 0.30158 0.32785

1.05 + 0.03 1.01 1.00 0.99 1.03

6

0.200

0.100

0.050

0.100

30

9 : 10

Sim. M1 M2 M3 M4

0.32217 + 0.01395 0.33257 0.33214 0.33092 0.33333

2.10 + 0.92 2.61 2.82 3.21 1.28

7

0.005

0.100

0.005

0.100

5

6 :7

Sim. M1 M2 M3 M4

0.79523 + 0.02933 0.79794 0.79588 0.79126 0.79915

4.63 + 0.14 4.49 4.06 2.86 4.68

8

0.005

0.100

0.005

0.100

30

6 :7

Sim. M1 M2 M3 M4

0.81541 + 0.02552 0.81465 0.81345 0.80804 0.81513

27.40 + 1.47 27.66 26.30 21.25 28.24

a ci: Cycle time of stage i; c o ~ m a x ( q , c2) , c = min(cl, c2). b Sire.: Simulation with mean and 95% confidence interval. c Mi: Method i in Section 3.2.

bank if either the upstream down.

However,

text that motivated

or downstream

in the automobile

with

con-

this study, banks were arranged

i n a p a r a l l e l s e r i e s f o r m , i.e., a n u m b e r parallel

s t a g e is

assembly

a finite

number

of lanes in

of places

in each

lane. Units would take a minimum of t cycle times to pass through the bank, where t < n, the capacity of the bank. reduces

Effectively,

the capacity

be captured

the finite travel time

of the bank.

by assuming

The impact

can

t h a t t r a v e l t i m e is e q u i v -

148

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

alent to processing at a number of workstations in series where the number of workstations is equal to t. However, these workstations cannot fail although they may be forced down if, for example, the upstream stage is down or if the bank is full and the downstream stage is down. That is, the actual bank capacity of n is divided into two parts, the t stations representing travel time and the remaining n - t spaces in the bank representing the actual usable capacity and thus the effective bank capacity to use in modelling performance. Note that if the inventory bank consisted of a single lane with t spaces, then the effective bank capacity would be zero. We used this bank capacity reduction approach in modelling the automobile assembly plant and found good agreement with simulation. However, further work is needed in developing models to validate this approach. We also note another possible approach to this travel time issue. We can model travel time by stations which are not necessarily forced down by breakdown of the upstream stage; that is, if those stations hold parts, they would keep on passing these parts independently of the state of the upstream stage unless they are blocked. But this approach is not easy to use with the transfer line model. In writing this paper, it has come to our attention that a similar approach has been suggested by Commault and Semery (1987). In contrast to our discrete part result, these authors used a continuous flow model of transfer lines. 3.4. Modelling an assembly stage f e d by two (or more) input stages

As was pointed out by Buzacott (1968) an assembly system with converging (or diverging) structure and no inventory banks is equivalent to a system with all the stages of the assembly system arranged in series. Now, consider a system consisting of three stages with stages 1 and 2 both feeding stage 3. Then if there is no bank between stages 2 and 3 but a bank between stages 1 and 3, the system is equivalent to a two stage system with the upstream stage having the parameters of stage 1 and the downstream stage having the parameters of stages 2 and 3 combined as if they were two stations in series in a stage with no inventory bank between them, i.e., the parameters can be determined using the methods described in Section 3.1.

When we introduce a bank between stages 2 and 3 as well as between stages 1 and 3, then an observer stationed at the bank between stages 1 and 3 would observe that units cannot leave the bank when either stage 3 is down or stage 3 is up and stage 2 down and the bank between stages 2 and 3 is empty. However, in a system with stages 1, 3 and 2 arranged in series, the observer stationed at the bank between stages 1 and 3 would observe that units cannot leave the bank when either stage 3 is down or stage 3 is up and stage 2 down and the bank between stages 3 and 2 is full. It can in fact be shown that, in the system with stages 1 and 2 feeding stage 3 with banks of capacity z I and z 2 between stages 1 and 3 and stages 2 and 3, respectively, the probability of a state (Y1, Y2, I/3, x y ) where Y~ is the state of stage i (Yi~ (W, R ) ) a n d x and y are the inventory levels in the banks between 1 and 3 and 2 and 3, respectively, is identical to the probability of the state (Y~, Y2, Y3, x, z z - y ) in a serial system consisting of the stages arranged in sequence 1, 3, 2 and banks of capacity z 1 and z 2 between stages 1 and 3 and stages 3 and 2, respectively (Figure 2). It follows that the two systems will have the same throughput or efficiency and thus the converging system can be analysed by analysing the equivalent series system. However, if in the system with stages 1 and 2 feeding stage 3 there was a stage 4 following stage 3 (see Figure 3), then the system would no longer be equivalent to a series system if there are nonzero banks between 1 and 3, 2 and 3, and 3 and 4. Now if either 4 is down and the bank between 3 and 4 is full or 2 is down and the bank between 2 and 3 is empty, then stage 3 can be forced down and prevent units leaving the bank between stages 1 and 3. That is, our system of two stages feeding stage 3 which then feeds stage 4 is equivalent to a system in which stage 1 feeds stage 3 which in turn feeds stages 2 and 4 in the sense that one unit entering stage 3 becomes two units leaving the stage, with one of the units feeding stage 4 and the other unit feeding stage 2 (Figure 4). Unless there

Figure 2. Equivalentthree-stage systems

X.-G. Liu, J.A. Buzaeott / Approximatemodels of assemblysystemswith inventorybanks

149

divided by the probability that the process was not stopped at the end of the previous cycle, i.e., Figure 3

z

z

E e(RW, x) + E V(WW, x). x=l

x=O

That is, is space for both units in the respective banks, then stage 3 will be forced down. Thus, while it so happens that the system of Figure 1 is equivalent to a series system, this would not generally be true in modelling assembly systems so an approach is required which will enable us to analyse general configurations of assembly systems. For a more general discussion of the equivalent class of manufacturing systems, see Ammar (1980), and Ammar and Gershwin (1981).

3.4.1. Zero bank equivalent systems. A key element of the approach is an observation about the two stage system with finite inventory bank. In such a system an observer located at stage 2 will observe the stage stopping, (1) because of breakdowns of itself, (2) because of units can enter the stage from the bank (because stage 1 is failed and the bank is empty). That is, the observer would perceive the system as equivalent to two stations in series with no intermediate bank. Obviously the parameters of the second station in the equivalent system are the same as the parameters of the second stage in the original system. However, the parameters of the first stage in the equivalent system are determined by the output process from the inventory bank in the original system. The distribution of down times will appear as if it is geometric with parameter b 1. To find the equivalent failure rate it can be observed that the probability that the output process will stop during a cycle will be given by (1 - b , ) P ( R W , 1) + a , P ( W W , O)

(1 - b l ) P ( R W ,

1 aeq ~

z

1 ) + a l P ( W W , O) z

E ,,(Rw, x)+ E p(ww, x) x=l = al~12 (z)

x=0 .

It can be seen that these values are consistent with the efficiency formula for the two stage line. Using a similar argument, to an observer stationed at stage 1 in a two stage line with a finite inventory bank the line appears to be equivalent to a two stage line with no inventory bank and with parameters of stage 1, a I and b 1, while the parameters of the equivalent stage 2 are a2321(z ) and b z. Thus in two stage systems it is possible to replace a bank and the upstream stage or a bank and the downstream stage by a single equivalent stage with no inventory bank. The idea of replacing a bank and associated stage by an equivalent stage with no inventory bank is the basis for the analysis of assembly (or disassembly) systems such as Figure 3 (or equivalently Figure 4). An observer located at stage 3 will see stage 3 stopping through its own failure and also because there is no output from bank 13 or from bank 23 and because no input is possible into bank 34. So each of these banks and the adjacent stage is replaced by an equivalent stage with no bank, e.g., stage 1' replaces stage 1 and bank 13, stage 2' replaces stage 2 and bank 23, stage 4' replaces stage 4 and bank 34. The resulting system (Figure 5) has no inventory banks and hence is equivalent to the system with all four stage 1', 2', 3 and 4' arranged in series. The above replacement requires the observer to know the behavior of stage 3 in the system with interaction between stages. In order to obtain this

(3--<3----O--@ ¢o) Figure 4

Figure 5

150

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

(a)

Co) Figure 6

(a)

Co) Figure 8

information, we can use the two stage system model in the following way. We can replace banks 23 and bank 34, giving the two stage system of Figure 6(a) where, because there are no internal banks in the segment downstream of bank 13, the segment can be replaced by an equivalent segment with all stages in the segment in series (Figure 6(b)). The series stages can then be replaced by an equivalent stage using the methods described in Section 3.1 above. From this two stage system we could then determine 813. Similarly by replacing banks 13 and 34 we could determine 823 (Figure 7) and by replacing banks 13 and 23 we could determine 843 (Figure 8). Since in determining 813 the downstream stage consists of stage 3 with parameters a 3 and b 3, stage 2' with parameters a2823 and b 2 and stage 4' with parameters a4843 and b4 in series, it can be seen that 813 will be determined as a function of 823 and 843. Similarly in determining 823 it will be determined as a function of 813 and 843, while the determination of 843 will involve expressing 843 as a function of 813 and 823. Thus in fact we have three (non-linear) equations:

813 = f ( Z l 3 , al, b 1, a2, b2, a3, b3, aa, b4,

823,843), 823 = f ( z 2 3 , al, bl, a2, b2, a 3, b3, a4, b4,

8., 843), ~43 = f ( z 3 4 , al, bl, a2, b2, a3, b3, a4, ha,

813,823). These can be solved quite readily using an iterative technique, e.g., set B23 and 843 equal to zero, determine a value of 813, then use this value to

determine a value for 823 , then use these values of 813 and 823 to determine 843 and so on. This approach can be extended to other configurations of assembly or disassembly such as one stage fed by a number of stages (Figure 9), one stage feeding a number of stages (Figure 10) or both (Figure 11). In each case analysis requires determining all the two stage equivalent systems in order to find the associated 8 's, e.g., the system of Figure 9 with four inventory banks adjacent to the assembly station can be considered as a two stage system in the four ways shown by Figure 12.

Remarks. (1) The approach just described is a heuristic. It can be shown that replacing a bank and associated stage by an equivalent stage with no inventory bank is correct (in the sense of having an equivalent Markov chain) when there are no interdependencies between certain stages in the system. For example, for the system shown in Figure 3, we need to assume mutual independence between stages 1, 2 and 4. It seems that for many real systems this independence assumption is acceptable as long as stages in a system have similar reliability parameters. (2) The correctness of the iteration equations must be demonstrated by satisfying conservation of flow. It can be shown that this would not be true if we directly use the equivalent stage methods of Section 3.1. However, the error of using these methods is of second order. Hence, it can be ignored in many real cases, especially for small failure probabilities.

Co)

(a)

Figure 7

Figure 9

151

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

> Figure 10 Figure 12

3.4.2. Analysis of general systems containing assembly-disassembly stages. A general system would consist of a number of segments of stages in series separated by inventory banks with the segments meeting at assembly or disassembly stations. Analysis of such a system requires writing down equations for each bank /j describing how 8,j and ~j, are determined by the apparent failure and repair rates a" and b" of the stage upstream of bank ig, and the corresponding P! IP quantities aj and bj for the stage downstream of the bank. To determine the apparent parameters of either an upstream or a downstream stage it is next necessary to check whether the stage concerned is an assembly or disassembly stage. If it is not then determination of a, and b,' if stage k is the stage upstream of i will be found from the parameters of the series arrangement of stage i and a stage equivalent to the bank ki and the stage upstream of this bank, i.e., a stage with parameters a~ and b~ in series with a stage having parameters a~, 6k~ and b~ are the apparent failure and repair rates of the stage upstream of bank ki. If stage i is an assembly-disassembly stage, then its apparent parameters will be determined by considering a stage with parameters a, and b~ in series with a set of stages corresponding to each of the stages adjacent to i (except of course for j). If k, l, m are these stages then it will be necessary to determine the parameters of a system with no inventory banks and consisting of stages with I ! I t. ! P parameters a~, b~; akSk,, bk; ate1,, bt, am~rnn, b~ in series.

Figure 11

Thus associated with every b a n k / j there will be two equations for 8ij and 6,j and the four equations determining a', .b" and a jP,! b]'. The equations are non-linear but can be solved using an iterative technique.

4. Numerical results

The above approaches were combined and applied to the car assembly system and compared with simulation. Table 3 shows the results. It can be seen that the agreement is good, quite sufficient for design purposes in investigating the effect of bank capacity and station parameters. Tables 4 and 5 show the results of two four stage systems, in which system 1 is a system with three stages (1, 2 and 3) feeding one assembly stage (4) and system 2 is similar to system 1 except that the flow of materials between stages 3 and 4 is reversed. From cases 4-6 and 4r-6r, we can see that the reversi-

Table 3 Simulation and approximate results for the car assembly system Simulation

Analytical

Mean

95% conf. inter,

approximation

Travel time = 0 Ave. Bank Lev. 1 Ave. Bank Lev. 2 Ave. Bank Lev. 3 Ave. Bank Lev. 4 Throughput (jph)

6.84 13,56 16.92 57.64 79.23

(6.66, 7.02) (12.79,14.34) (15.47, 18.36) (56.23, 59.05) (78.95, 79.51)

6.91 13.64 15.73 59.02 79.17

Travel time > 0 Ave. Bank Lev. 1 Ave. Bank Lev. 2 Ave. Bank Lev. 3 Ave. Bank Lev. 4 Throughput (jph)

11.14 12.80 16.81 62.77 76.93

(11.07,11.21) (11.60,13.98) (15.18, 18.43) (61.16,64.38) (76.30, 77.55)

12.00 13.75 17.61 62.97 77.54

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

152

Table 4 Results of four-stage system 1 Case

(al, bl) (a2, b2)

z12

z24

z34

(a3' b3) (a4' b4)

Efficiency Sim. a

Ave. bank lev. App. b

Bank 14 Sim.

Bank 24 App.

Sim.

Bank 34 App.

Sim.

App.

1

(0.010, 0.10) (0.010, 0.10) (0.010, 0.10) (0.010, 0.10)

10

10

10

0.7996 + 0.0318

0.8005

6.81 _+1.06

6.84

6.76 + 0.48

6.84

6.78 + 0.49

6.84

2

(0.050, 0.20) (0.010, 0.10) (0.020, 0.15) (0.010, 0.10)

10

10

10

0.7368 + 0.0136

0.7497

3.46 _+0.67

3.52

8.05 + 0.63

8.03

6.96 _+0.53

7.10

3

(0.050, 0.20) (0.010, 0.10) (0.020, 0.15) (0.040, 0.09)

10

20

10

0.6252 + 0.0244

0.6365

6.12 _ 0.56

6.42

18.18 + 0.85

18.10

8.20 + 0.42

8.22

4

(0.050, 0.10) (0.060, 0.09) (0.120, 0.25) (0.004, 0.50)

30

30

30

0.5723 + 0.0261

0.5970

15.90 5:3.05

16.16

4.55 + 2.03

3.95

19.68 _+2.07

17.86

5

(0.150, 0.10) (0.160, 0.09) (0.120, 0.05) (0.004, 0.50)

10

10

10

0.2522 +0.0193

0.2620

6.67 ::t:0.53

6.44

5.35 +0.32

5.45

3.19 +0.40

3.13

6

(0.050, 0.10) (0.060, 0.09) (0.120, 0.25) (0.034, 0.10)

2

2

2

0.3890 + 0.0073

0.3978

1.45 5:0.03

1.44

1.24 + 0.06

1.27

1.36 + 0.04

1.34

a Sim.: Simulation with mean and 95% confidence interval. b App.: Analytical approximation.

bility is satisfied by the iterative technique (see Ammar, 1980). The convergence of all these resuits was very fast.

5. Conclusions

The techniques developed in this paper provide an effective means of modelling and analysing assembly systems, taking into account the characteristics that distinguish real assembly systems from the conventional transfer line model.

In another paper we intend to present results on the organization and convergence of the iterative scheme for solving the set of non-linear equations that describe assembly-disassembly systems and lines. In this paper we have only considered assembly and disassembly systems. However, it would be also desirable to extend these models to allow for nodes at which different jobs can take different paths, for example, an inspection station at which some jobs are removed from the line go to a repair station and then rejoin the line. This is a direction for further research.

Table 5 Results of four-stage system 2 Case

(al, bl) (a2, b2)

z12

z24

z43

(a3' b3) (a4' b4)

4r 5r 6r

(0.050, (0.120, (0.150, (0.120, (0.050, (0.120,

0.10) 0.25) 0.10) 0.05) 0.10) 0.25)

(0.060, (0.004, (0.160, (0.004, (0.060, (0.034,

0.09) 0.50) 0.09) 0.50) 0.09) 0.10)

30

30

30

10

10

10

2

2

2

Efficiency

Ave. bank lev.

Sim. a

App. b

0.5742 + 0.0260 0.2543 + 0.0139 0.4134 + 0.0119

0.5970

a Sim.: Simulation with mean and 95% confidence interval. b App.: Analytical approximation.

0.2620 0.3978

Bank 14

Bank 24

Bank 43

Sim.

App.

Sim.

App.

Sim.

App.

16.21 + 2.71 6.88 + 0.50 1.45 + 0.030

16.16

4.50 -/- 2.59 5.66 5:0.44 1.25 + 0.045

3.95

10.30 + 3.04 7.24 + 0.40 0.74 + 0.028

12.14

6.44 1.44

5.45 1.27

6.87 0.66

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

153

Appendix A Formulae f o r the two stage transfer line model

T h e formulae given below are due to Buzacott and Hanifin (1978) but written in a slightly different form. Only the results for the operation dependent failure case are provided. Symbols

the failure probability of stage i, i = 1, 2,

a i

b i = the repair probability of stage i, i = 1, 2, z the b a n k capacity, an integer, A 1 = a 1 +a 2-ala 2-b12 , A2= a 1 +a 2-ala 2-alb2, B 1 = b I + b 2 - bib 2 - alb2, b 1 + b 2 - bib 2 - bla2, r*

bla2B1 a~b2B 2 '

C = B1A2 ATB 2 •

Loss transfer coefficients 1 -r* 1-_r~Z

~12(Z)

if alb 2 4= bla2, bl + b 2 - b,b 2

=

~ - ~ ~2 -- gb72 ~ z ~ - +

a l / b l ) blb2

l-r* C~ 1 - r*C ~

~21(Z)

if alb 2 = b l a z ,

if alb 2 =I: baaz,

bl + b 2 - bib 2

=

b 1 + b 2 - bib 2 + z(1 + a l / b l ) b l b 2

if alb 2 = bia 2.

Line efficiency or throughput A = a2 ( al 1 + ~ + bl

ala2 )S12(z ) bl + b z _ bab2

1 al ( a2 1 + b-~l + b 2

a,a2

b I + b 2 - bib 2

82,(z)

Appendix B Results f o r the transfer line with zero banks

T h e results here are derived under the operation dependent failure assumption. Symbols

s ai

= the n u m b e r of stages in the line, = the failure probability of stage i, i = 1 . . . . . s,

154

X.-G. Liu, J.A. Buzacott / Approximate models of assembly systems with inventory banks

the repair probability of stage i, i = 1..... s,

bi

=

aij bij

= a i d j, = b i + bj - b ibj,

aij k =

aiajak,

biik = b i + bj + b k - bibj - bib k - bjb k + bibjbk '

f°(n)=

1-ao

ifn=l,

a o b o ( 1 - b o ) "-2

ifn>l.

The interdeparture time distribution s

g,(n) = E f , ( n ) - E L i ( n ) = i= 1

i
E

Lj (n) . . . .

, where i, j, k ~
i
Moments of interdeparture times E(N),--I+

-~i - i"~: -~ij + i=1

E(N(N-

"

1)), = ~

2ai

E

aiJk . . . .

i < j < k bijk

2aij

,=, b-Tf - ,
E(N(N-

1 ) ( N - 2)), = ~ i= 1

6ai(1-- b,) 63

'

2ai2~ 9


~_, 6 a i j ( 1 - b i J ) i
b3j

6aijk(1--bijk)

+ E

i
b3ijk

where i, j, k < s.

Acknowledgement This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-5176.

References Ammar, M.H. (1980), "Modelling and analysis of unreliable manufacturing assembly networks with finite storage", MIT Laboratory for Information and Decision Systems Report, LIDS-TH-1004. Ammar, M.H., and Gershwin, S.B. (1981), "Equivalence relations in queueing models of manufacturing systems", MIT Laboratory for Information and Decision Systems Report, LIDS-P-1027 (revised), OSP no. 87049. Buzacott, J.A. (1968), "Prediction of the efficiency of production systems without internal storage", Internal Journal of Productions Research 6, 173-188. Buzacott, J.A., and Hanifin, L.E. (1978), "Models of automatic transfer lines with inventory banks--A review and comparison", A I I E Transactions 10, 197-207.

Commault, C., and Semery, A. (1987), "Taking into account delays in buffers for analytical performance evaluation of transfer lines", to appear in IIE Transactions. Dallery, Y., David, R., and Xie, X.E. (1988), "An efficient algorithm for analysis of transfer lines with unreliable machines and finite buffers", l i E Transactions 20, 280-283. Gershwin, S.B. (1987), "An efficient decomposition method for the approximate evaluation of tandem queues", Operations Research 35, 291-305. Gershwin, S.B., and Schick, I.C. (1980), "Modelling and analysis of two- and three-stage transfer lines with unreliable machines and finite buffers", Working paper, MIT. Kosteiski, D., Buzacott, J.A., McKay, K.N, and Liu, X-G. (1987), "Development and validation of a system macro model using isolated micro models", Proceedings of Winter Simulation Conference "87, Atlanta, 669-676. Maione, B., Fanti, M.P., and Turchiano, B. (1987), "Large scale Markov chain modelling of transfer line", Working paper, Dipartmento di Elettrotecnica ed Elettronica, Universita di Bari. Sevastyanov, B.A. (1962), "Influence of storage bin capacity on the average standstill time of a production line", English translation: Theory of Probability and lts Applications 7, 429-438. Zimmern, B. (1956), "Etudes de la propagation des arrets aleatoires darts les chaines de production", Revue de Statistique Appliquee 4, 85-104.