Discrete unreliable transfer lines with exogenous random unit demand

Pergamon Computers md Engng Vol 26, No 2, pp 307-320, 1994

Copyright © 1994 Elsevter Sctence Ltd 0360-8352(93)E005-S Pnnted m Great Bntmn All rights t~--scrved

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D I S C R E T E U N R E L I A B L E T R A N S F E R L I N E S W I T H

E X O G E N O U S R A N D O M U N I T D E M A N D

ABRAHAM MEHREZ and B. EDDY PATUWO*

Graduate School of Management, Kent State Umvers~ty, Kent, OH 44242-0001, U S A

(Recewed for pubhcatwn 23 November 1993)

Abstract--We evaluate the performance of unrehable production lines with intermediate buffers facmg a random umt demand which causes blocking and starvation to the production hne. It is assumed that the stations m the hne are respected periodically and ff a station fails to meet predetermined operational standards, it ts shut down for that entire period To increase output, fimte size buffers are placed m between stations and after the last station. A Markov cham ts used to model the system and an efficient algorithm to compute the marginal hmttmg distributions of the buffer contents is presented These hmJtmg probabdiues are then used to study the behavior of the system via experimental analys~s This paper also demonstrates the determination of the destgn variables (capactttes, station rehabihtses) and operational variables (work-m-process, capacity utthzatlon) that do not have a umque optimal pattern Th~s is done m contrast to opt~mlzauon via Markov decision processes

1 PROBLEM DESCRIPTION

In this paper, a model for transfer line with exogenous demand is considered. The motivation of the model formulated stemmed from an observation of the operation of a chemical industry in the Israel. The plant consists of five facilities/stations processing a chemical raw material in a fixed sequence. The raw material, which is potassium and sodium salt from the Dead Sea, is transformed at each station into an intermediate compound through a set of operations and into a final product (fertilizer) at the last station through a push mechanism, and then is shipped abroad. The processing stations are subject to breakdown due to mechanical failures. The push system is used instead of the pull system because of an agreement with the labor union to avoid worker strikes. The plant is located near the Dead Sea and the warehouse is located at the port, about 50 km away. It is assumed that the supply of raw material is infinite and hence there is always raw material available to be processed at the first station and the demand per period for the finished product is quite frequent. Actually, this demand rate is measured by the number of dedicated ships per period used to transport the finished unit abroad. At most, a single unit leaves the port per period (per day). Figure 1 illustrates this system.

There is extensive literature on the analysis and design of transfer lines. Most of the analytical studies focused on only two-station or three-station lines. A review and developments of analytical models of these sizes are provided by Buzacott and Hanifin [1], Gershwin and Berman [2], Gershwin and Schick [3], Jafari and Shanthikumar [4]. Recently, Hillier and So [5, 6] studied the effect of machine breakdowns and interstage buffers on the performance of production lines with more than three stations. They, like most other approaches, consider a continuous-time model and use a Coxian distribution to represent the processing time, which is equivalent to assuming that the service time, the up-time and the down-time of each station to be independent and exponentially distributed. With this model, the last station can be defined as an exogenous demand process; in this case, it is an interrupted Poisson process. One drawback is that the state space in this model grows exponentially as a function of the number of stations and the buffer sizes. This curse of dimensionality prevents them from computing exact analytical results for lines with large number of stations and large buffer sizes. This is the motivation for constructing a discrete-time Markov chain model is to alleviate this problem of state space explosion. For the discrete-time model

*Author for correspondence

307

308 ~BRAHAM MEHREZ and B EODY P~I-L wo

constructed in this paper, the memory requirement, the modeling effort and the computatIonal time requirement for finding stationary probabilities are minimal

At the design level, a decision may be made to determine buffer capaclues. Th~s problem has been studied by Altiok and Sudham [7] where the total profit is to be maximized. The determination of buffer capacities takes into account its relationship to various performance measures such as the production rate per period, the capao ty utlhzatlon, the average work-m-process inventory, and the excess or shortage of the production rate relative to the expected demand rate per period. The measures mentioned above have been considered by many authors in the production hne hterature (see e.g. Gershwln and Berman [2] which discuss these measures m reference to a two-machine. single-buffer transfer lines)

Tradluonal literature on production lines assumes that upon breakdown, a machine Is repaired. The model investigated here considers the case where an off-hne inspection of the machines Is accomplished at the beginmng of each production period and, depending upon the condition of the machine, a deoslon to operate or not to operate ~s made by the producuon line controller. In order for the ruth station to produce one umt t, the following three condmons must be met

(l) at least one unit of work-in-process inventory is available at the preceding buffer at the end of period t - 1,

(n) the ruth station is declared to be reliable to operate in period t; (m) the ruth buffer is capable of absorbing one unit produced at the ruth station at the end of

period t. This lmphes that the mth buffer is not full or, if it is full, the (m + 1)-st station ts not blocked (it can process one unit and transfer it down-stream or m the case of the last station (m = M), an exogenous demand of one unit occurs in period t)

Gwen that the above three condiuons are satisfied, we assume the ruth station operates w~thout interruption throughout period t

The discrete structure of the problem does not restrict the analysis since the length of the period can be selected to be arbitrarily small. A station is said to be starved if there ~s no unit available to be processed. So station 1 is never starved. Since the demand is exogenously determined, the blocking and the staring probabihties depend on the probabihty dls tnbuuon function of the demand. Clearly, both blocking and starving decrease the output of a production hne. For a prescribed buffer capacities and station reliabihtles, various performance measures are evaluated by first calculating the steady state probabilities of a Markov chain representing the states of the buffers. These probabilities are calculated under the assumption that in a given period, the failure probability of a station is independent of the states of the other stations. We also allow the failure probability of a station to depend on the state of that station m the previous period.

In the next section, the model is formally introduced. After deriving the t ransmon probablhues of the Markov chain, we present an algorithm to calculate the marginal steady state probabdmes. An experimental study and its mare results are reported next. This study considers a Bernoulh demand probabihty distribution on eight different structures of the producuon line, one of which contains an embedded parallel structure. Comparison among the structures is linked w~th ~ssues such as the effects of the number of stations, the station rehabihties, centralized versus semi- centralized storage policies, the sequencing of the stations and the design of the capacities based on various performance measures. Finally, three pohcy lmphcauons for the design of unrehable production lines are discussed.

2 MODEL FORMULATION

Alphabetical listing of all notations used in this section and their respective definitions are given in Appendix B. Consider an M-stat ion serial production system as depicted m Fig. I. There is an

f:lg 1 Serial produchon hne with exogenous demand

Transfer hnes with random umt demand 309

infinite supply o f raw mater ia l that feeds into the first station. Let c~, c2 . . . . , CM be the buffer capacit ies after s tat ions l, 2 . . . . . M, respectively. The processing t ime o f each s t auon is one unit per period. At the beginning o f per iod t and for m = l, 2 . . . . . M, let YT' = 1 if the m t h stat ion is declared to be reliable to opera te in per iod t, YT' = 0 if the ruth s ta t ion is down, P[YT' = l] = r m and PlY ? = 0] = 1 - r m . We will subsequent ly refer to rm as the reliability o f s tat ion m. We assume Y7 to be independent o f Y7 and o f the states o f the buffers if m 4= n. We can also assume { YT', t = 0. 1, 2, . . } to be a M a r k o v chain, i.e. YT' is dependent on Y",_~ only. In this case,

• m we interpret r , , = h m , ~ P [ Y , = 1], the steady state probabi l i ty of {Y'~,t >10}. Define D as the s ta t ionary demand per per iod with Bernoulli mass function, P[D = l ] = d and P[D = 0] = 1 - d . We can easily generalize D to take on integer values 0, 1 . . . . . n with minor modif ica t ion to the der ivat ion of the t ransi t ion probabil i t ies, but this will not be done in th~s paper .

Let X m. m = 1.2 . . . . . M, be the state of the m t h buffer (the content o f the ruth buffer) at the beginning of per iod t. I t is easy to see that {XI, YI, X, 2, Y ~ , - . . , X ~ , y M} is a mu lmd imens lona l discrete-t ime M a r k o v chain. Finding the steady state joint probabi l i tes o f this M a r k o v cham ~s not an easy task due to the explosion o f the state space. Readers interested m numerical solutions to large M a r k o v chains are referred to Schweitzer [8]. Fo r our exper imenta l study, it is sufficient if we can compu te the margina l s teady state probabil i tes ~ o f X , , X, . . . . . and XT. Knowledge of the more informat ive jo int s teady state probabi l i t ies are not required. Mos t o f the existing me thods are devised to find the joint probabi l i t ies f rom which the marg ina l probabdi t ies can be extracted. Below we give the transi t ion probabi l i t ies and present a simple and efficient procedure to compu te the marg ina l s teady state probabil i t ies.

Not ice that X,"+ ~ depends on XT'- ~, XT' . . . . . X~" (and o f course on D and YT'- ~ . . . . . yM). This allows us to write the margina l t ransi t ion probabil i t ies recursively backward f rom the last buffer to the first. To simplify notat ions, we define {B~" = 1} as the event that given {X," >/1] at the beginning of per iod t, one unit can be t ransferred out o f the ruth buffer to be processed at the m + 1st s ta t ion and stored at the m + 1st buffer at the end of per iod t (or in the case of the last buffer, the unit is consumed by the demand) . Hence, the complemen ta ry event {BT' = 0} ~s the event that the m t h buffer ~s blocked. The probabil i t ies o f these events are gwen recursively s tar t ing with the last stage:

P[B~ =0] = P[D =01 = l -d, (I)

P[B~ = 1] = P[D = 11 = d , (2)

and for rn = M - i , M - 2 . . . . . 2, 1,

P[B'~ = O] = P[Y7 +' = O] + P[Y7 +' = l ] . P[X7 +' = c,.+ ,]. P[B7 +' = O]

= ( l - r m + ~ ) + r m + , ' P [ X T + t = c , , + t ] ' P [ B T + ' = O ] , (3)

P[B~' = 1] = P[Y7 '+m = 1]. {P[X7 +' < c,,+,] + P[X7 +' = cm+,] " P[B7 +' = 1]}

= 1 - P[B'r = 0], (4)

We also define {A ~" = 1 } as the event tha t at the beginning of per iod t, the rn th stat ion is capable of p roduc ing one unit at the end o f per iod t, regardless o f the state of the rn th buffer, and cA-, , . . , = 0 1 is its c o m p l e m e n t a r y event. The probabil i t ies o f these events are: for m = M, M - 1 . . . . 2.

P [ A T = O ] = P [ Y T = O ] + P [ Y T = I ] . P [ X T - t = O ] = ( I - r m ) + r , , . P [ X T - ' = O ] , (5)

P[A'~ = 1] = PLY'7 = I ] ' P [ X , - t > 0 ] = 1 - P [ A 7 =0], (6)

and for m = 1

e[al = l] = P[Y~ = 11 = r,, (7)

e [ A ~ = 0] = e[Y~ = 0] = l = r , . (8 )

310 ABRAHAM MEHREZ and B EDDY PATUWO

Then, the marginal transit ion probabilities P[XT'+~=jIXT' = t] for m = 1.2. given by

IP[A7=O] if j = 0

otherwise.

For i = 1,2 . . . . . Cm-- 1,

. M. and t = 0 are

(9)

f P [ B ' = 1. P[A 7 = 0] l f j = l - - 1 ,

P[XT'+, = j l X 7 = t] = P[BT'=O]'P[AT'=O]+P[BT'= !]" P[A m= 1] i f j =l, (10)

P[B7 = 0]. P[A 7 = 1] i f / = i + i,

0 otherwise.

and for i = cm, (P[BT' = 1 ] ' p [ A m 7 0] l f J = c , . - 1 ,

P[X~'+I = j I X 7 ' = c~,] = {P[B'~ 0] + P[B'[ = I ] - P [ A m = 1] f f j = c.,, ( l l ) (o otherwise

Transi t ion probabilities (10) and (l 1) are close approximat ions to the actual t ransinon probabilities due to weak dependency between XT'-~ and X7 '+ ~ (they are separated by the mth buffer).

3 COMPUTATION OF THE MARGINAL STEADY STATE PROBABILITIES

Given r~ . . . . . rM and d, we notice that the probabilities given in (1)-(8) are functions o f P[X, = i], l = {0, c,, }, m = 1 . . . . . M and so are the one-step transition probabilities (9)-(11). Also, the probabilities in (1)-(4) can be computed recurslvely f rom the last stage to the first stage. This observat ion points us to a way o f comput ing the steady state probabilities iteratively for t = 0, 1, 2 . . . . . until convergence. We start with the following initial probabilities: P[X'~ = 0] = 1 and P[X~ = t] = 0, for i = 1 . . . . . Cm, m = 1, 2 . . . . . M. Next compute the probabdit ies m (1)-(8) for t = 0 and m = M, M - 1 . . . . I. The marginal probabilities for t = 0 and m = 1, 2 . . . . M are computed using

tm P[X.+, = d = ~ P[x/_ , = j I x T ' = ~] P [ X . = ~]. (12)

i=0

Repeat the above procedure for t = 1, 2, 3 . . . . until convergence, i.e.

m a x { I P [ X L , = j ] - e [ X , =J]l < c, Vm = 1 . . . . . M, and j = I . . . . . c,,]. (13)

There is no need to store the marginal transit ion probabdit les (9)-(11) m a square array for each m. They can be generated just before comput ing the marginal probabdlt ies in (12). Moreover , these transit ion probabilities are not constant with respect to t. The p rogram to compute the marginal steady state probabilities was written in For t ran-77 and the runs were done in single precision on an IBM 4381 using ¢ = 1.6 • 10 -4 as the s toppmg criterion for the iterations. In all o f our test cases, convergence is achieved within 40-500 iterations. As an example, one o f the test cases with M = 5, c~ = c2 = ' " = c5 = 5, converged after 165 iterations and the execution time was 0.45 sec.

4 MEASURES OF PERFORMANCE

After comput ing the steady state probabilities p[,ym = j ] = l i m , ~ P[X'[ = j ] , we calculate the following 13 performance measures o f the system:

(1) E[X u] = Y.~ ~j" P[X u = j ] , expected content o f the last buffer (warehouse); (2) P[X u >~ 1], probabil i ty o f meeting the demand, (3) ~' = y ~ u ~ c,./M, average design capacity; (4) ~" = y.M ~ E[X"]/M, average work-in-process and inventory; (5) ~ = {Y.g= j E[X"]/c,. }/M, average capaci ty utilization;

Transfer lines with random unit demand 311

(6) Y.,,~M ~ P [ X " = 0]/M, average s tarving probabi l i ty ; (7) Y'm=M I P [ X " = c,,]/M, average b locking probabi l i ty; (8) Sign{c~, c2 . . . . . CM }, sign funct ion o f the design capacities; (9) Sign{E[XJ], E [ X 2] . . . . . E[XM]}, sign funct ion o f the expected work- in-process ;

(10) Sign{E[X~]/ct, E [ X 2 ] / c 2 , . . . , E[XM]/cM]}, sign funct ion o f capaci ty utilizations; (11) X,,ffiM i Ic,, -- "~I/M, mean absolute deviat ion ( M A D ) o f the design capacities; (12) £m~M ~ IE[X"q - ~ / M , M A D of the work- in-process ; (13) M IE[Xm]/c,, Eq/M, M A D of the capaci ty utilizations.

These 13 pe r fo rmance measures are repor ted in Table 2 as co lumns 1-13. The sign function, Slgn{c~, c 2 . . . . , CM}, is calculated as follows: for i = 1 . . . . . M - 1

1 i f c , + ~ > c , ,

s ,= - - 1 i f c , + l < c , ,

0 i fc ,+l = c , ,

and Sign{c~, c2 . . . . . cM} = X~=q t s,. In pe r fo rmance measures 9 and 10 we use

1 i f l a , + l - a , l > 0 . 0 1 ,

s ,= - 1 i f l a , + t - a , l <O.O1,

0 i f l a , + l - a , I ~<0.01,

and Sign{a~, a2 . . . . . aM } = X,M=q t S,, where a, = E[X'] or a, = E[X']/c, . This is done because the a,s are not integers.

In all o f the test cases per formed, we choose d = P[D ---1] = 0.9 and P[D = 0] = 0.1, i.e. the expected d e m a n d rate is 0.9 units per period. Pe r fo rmance measure 2, P [ X M >t 1], is o f p r imary concern, since it measures the long-run ability o f the p roduc t ion line to meet the external demand. Since E[shortage] = P[D = 1]. P [ X M = 0], zero shor tage can only be accompl ished if P [ X u = 0] can be made to be a lmos t zero at the expense o f very large buffer sizes, which clearly does not m a k e good economic sense. In Section 5.2, we will consider only p roduc t ion line structures (configurations) with prescr ibed s tat ion reliabilities and capacit ies which results in P [ X M >I 1]/> 0.9. The cu t -of fprobabi l i ty o f 0.9 is chosen on the basis o f trying to match the long-run probabi l i ty o f having at least one unit to meet the d e m a n d to the probabi l i ty o f having a demand . Again, choosing P [ X u >i 1]/> P[D = 1] does not guaran tee that there will be no shortages. In fact, the choice o f P [ X M >1 1] i> 0.9 results in E[shortage] ~< 0.09.

4.1. A theoretical upper bound

Given the s ta t ion reliabilities rt . . . . . r M, and the expected demand rate, d, we can find the upper bound o f P [ X ~ >>, 1] by considering a line which consists o f one stat ion with reliability r = min {rt . . . . . r M }, an infinite supply tha t feeds the s ta t ion and an infinite buffer capaci ty after the station. I t can be easily verified tha t the buffer size {X, } forms a M a r k o v chain with the probabi l i ty t ransi t ion matr ix given by

l m r r 0 " ' '

d . ( l - r ) d ' r + ( 1 - r ) ( l - d ) ( 1 - d ) r . . .

0 d ( l - r ) d . r + ( l - r ) ( 1 - d ) ( 1 - d ) r • . . " . . " . .

p =

I f r < d, the s teady state probabil i t ies exist with P [ X = 0] = 1 - r/d, and P [ X >_. 1] = r/d. Thus the m a x i m u m p[xM>~ 1] = rid. For example , a three-s tat ion line with d = 0.9 and r~ = r3 = 0.9, r2 = 0.729, the m a x i m u m probabi l i ty o f meet ing the d e m a n d is P [ X u >i 1] = 0.729/0.9 = 0.81.

4.2. Structures o f the experimental analysis

We const ruc ted 8 different s t ructures (configurat ions) to be used in our exper imenta l study. They are:

• Structure 1: 5 -s tat ions ( M = 5) in series with s ta t ion reliabilities:

r 1 = 0.95, r 2 = r 3 = r 4 = 0.90, r s = 0.95,


Fig 2 Semi-parallel production hne with exogenous demand

• Structure 2: 5-stations ( M = 5) in series with s tat ion reliabllities:

r~ = 0.95, r2 = 0.85, r3 = 0.90, r4 = 0.9529, r5 = 0.95.

• Structure 3: 5-stations (M = 5) in series with s tat ion reliabllitles:

r~ = 0.95, r, = 0.9529, r 3 = 0.90, r4 = 0.85, r5 = 0.95.

• Structure 4: 5-stations (M = 5) in series with s tat ion reliabiliues:

rt = 0.90, r2 = r3 = r4 = 0.90, r5 = 0.90.

• Structure 5" 5-stations (M = 5) m series with s tauon reliabiliues:

r~ = 0.85, r 2 = r3 = r4 = 0.90, r5 = 0.85.

• Structure 6: 3-stations (M = 3) in series w~th s tat ion rellabilities:

rl = 0 . 9 5 , r, = 0.729, r 3 = 0.95.

• Structure 7' 3-stat ions (M = 3) m series with s tat ion reliabdities:

rl = 0.95, r , ----- 0.999, r 3 = 0.95.

• Structure 8: 5-semi-parallel s tat ions (M = 5, see Fig. 2) with s tat ion reliabihties.

rl = 0 95, r2 = r3 = r4 = 0 . 9 0 , r5 = 0.95.

The rehabdities of the middle s tat ions m structures 1, 2, and 3 are chosen such that r 2 . r 2 " r 3 = 0 . 9 3 = 0.729. In structure 2, the middle stat ions are ar ranged in ascending order of their s tat ion reliabilities, whereas in structure 3, they are ar ranged in descending order of their s tat ion reliabilities. In structures 4 and 5, the two end s tauons are less reliable than those of s tructure I. In addi t ion, s tructure 4 captures the case where all s tat ions are identical, and structure 5 has the two end stat ions as bottle-necks.

The three-stat ion line in structure 6 corresponds to the five-station hne case where there ~s no buffer storage after staUons 2 and 3. In this case, we can combine the three middle stat ions into one s tat ion with reliability 0.729 (see e.g. [9]). Structure 8 is a semi-parallel structure w~th the three middle s tat ions ar ranged in parallel and a buffer storage after each stat ion (see Fig. 2). The margina l t ransi t ion probabil i t ies for s tructure 8 are given In the Appendix. Finally, s tructure 7 is like s tructure 8 with one c o m m o n buffer storage after stat ions 2, 3 and 4. In th~s case, the three middle s tat ions can be treated as one s tauon with rehabdlty of 1 - ( 1 - 0 . 9 ) 3 = 0 . 9 9 9 (see [9]). The results of our experimental study are reported in the remain ing sections of this paper.

Trans fe r hnes wi th r a n d o m u m t d e m a n d 313

5 R E S U L T S O F T H E E X P E R I M E N T A L A N A L Y S I S

In this section, a discussion of the main results from the experimental study is provided. First, we present the aggregate results based on all of the test cases. Second, we present the disaggregate results for those line configurations with probabilities of having at least one unit on-hand to meet the expected demand of 0.9 units per period (with the exception of structure 6). Here, we chose the best 10 cases for each structure, where "best" means that P[XU>~ 1] is as close as possible to 0.900 and, in the case of ties, we chose those cases with smaller design capacity. As noted at the end of Section 4, the choice of 0.900 probability does not imply that there will be no shortages. Last, we present some specific observations that were not previously captured.

5.1. Aggregate results

Table 1 reports on both the averages and the standard deviations of the probabilities that the last station has at least one unit available in a representative steady state period, P[X u >>, 1]. These probabilities also measure the long-run ability of the production line to meet the demand. The averages T'j, and the standard deviations 8j, where

N

Z e,t xM 1]

N

and

IP'tx"> 11- PJI2'N-

are calculated for each structure j, j = 1, 2 . . . . . 8. Structures 1-5 and 8 with 5 stations, has N = 5s= 3125 observations (patterns) corresponding to all permutations of buffer capacities (ct, c2 . . . . . Cs) where cm = 1, 2 . . . . . 5, and m = 1, 2 . . . . . 5. Structures 6 and 7 with 3 stations, has 73= 343 observations corresponding to all permutations of buffer capacities (c~, c2, c3), where C m = 1 , 2 . . . . . 7 , rn = 1 , 2 , 3 .

Without conducting staustical tests, here are the main Implications of Pj and 8j:

(i) Structure 1 has both higher Pj and higher standard deviation 6j than those of structures 2 and 3. Note that for these three structures, the joint reliability of the three middle stations is 0.729. This implies that the structure with equally reliable stations results in a higher efficiency and variablility of meeting the uncertain demand across the sample relative to the structures with unequally reliable stations. Also, the aggregate results for structures 2 and 3 demonstrate that interchanging positions of statmns 2 and 4 does not effect the considered measures, although differences regarding P[X u >1 1] may exist on the individual observation level.

(ii) The effect of reducing the reliabilities of the two end stations by 10% each, reduces the Pj by about 6% (see the results for structures 1 and 5).

(ii0 Decreasing the number of stations to 3 (see structures 6 and 7) or introducing a semi-parallel structure (structure 8) reduces significantly the average variability 6j of Pj when compared to the structures with five stations.

Table I Aggregate calculations P[X u >1 1]

Structure O) N ?/

I 3125 0 894 0 050 2 3125 0 880 0 039 3 3125 0 880 0 039 4 3125 0 869 0 045 5 3125 0 832 0 049 6 343 0 792 0 017 7 343 0 989 0 017 8 3125 0 986 0 016

314 ABRAHAM MEHREZ and B EDDY PA'I'UWO

Table 2 Selected observat ions with P[XM>~ I] ~ 0 900

Obs Capacit ies Col I 2 3 4 5 6 7 8 9 10 11 12 13

I 1 2 2 3 4 2 1 4 3 2 2 3 2 4 2 2 2 4 1 2 5 3 2 5 2 3 3 1 4 6 4 3 2 2 2 7 2 1 4 4 3 8 2 2 4 1 5 9 5 3 2 2 2 10 2 1 4 5 3

Structure I P [ D e m a n d = 1] = 0 9

N u m b e r of stat ions m series, M = 5 Rehablhty of the stations = 0 9500 0 9000 0 9000 0 9000 0 9500

538 0 9 0 0 2400 1 235 0599 0 0 9 2 0 3 5 9 3 0 - 2 0 8 8 0 0 157 0 176 227 0 9 0 0 2400 1 384 0633 0 0 7 7 0369 - 1 0 0 0 8 8 0 0348 0 129 227 0 900 2 400 I 548 0 656 0 065 0 381 0 0 0 0 640 0 384 0 086 227 0 9 0 0 2600 I 316 0601 0083 0 3 5 7 0 2 0 1 120 0226 0 170 541 0 9 0 0 2 6 0 0 1 547 0662 0 0 6 9 0437 I 0 - 2 0 8 8 0 0278 0 157 226 0 9 0 0 2 6 0 0 1 835 0676 0 0 6 4 0394 - 2 - 2 0 0 7 2 0 0731 0 0 8 2 421 0 9 0 0 2800 1 380 0 5 9 4 0087 0385 - 1 0 - 2 I 040 0273 0 2 4 6 608 0 9 0 0 2800 I 512 0648 0073 0445 1 0 - 2 I 360 0303 0 196 227 0 900 2 800 2 030 0 680 0 063 0 393 - 2 - 2 0 0 960 0 984 0 087 423 0 9 0 0 3000 1 382 0583 0087 0383 0 0 - 2 1 200 0271 0255

Structure 2 P [ D e m a n d = I] = 0 9

N u m b e r o f stat ions in series, M = 5 in series Reliability of the stations = 0 9500 0 8500 0 9000 0 9529 0.9500

I 2 2 2 2 4 1 541 0 9 0 0 2400 1 336 0591 0 0 9 7 0316 0 - 4 0640 0265 0 139 2 2 2 3 2 3 1419 0 9 0 0 2400 I 331 0581 0095 0305 - 2 - 2 0 4 8 0 0221 0147 3 1 3 3 2 5 1611 0 9 0 0 2800 1228 0529 0 1 0 7 0 2 6 9 0 - 2 1040 0 2 3 6 0172 4 I 3 3 4 3 1420 0 9 0 0 2800 1216 0513 0105 0256 2 - 2 0720 0 1 7 0 0178 5 I 4 2 2 5 1609 0 9 0 0 2800 1249 0543 0 1 0 6 0 2 8 2 2 - 2 1360 0288 0 1 6 6 6 I 5 2 2 4 1537 0 9 0 0 2800 1257 0545 0 1 0 4 0 2 8 2 2 - 2 1360 0 2 9 4 0165 7 1 5 2 3 3 1422 0 9 0 0 2800 1252 0 5 3 4 0103 0 2 6 9 2 0 1040 0251 0169 8 4 3 3 2 2 1229 0 9 0 0 2800 1757 0 6 0 2 0081 0308 - 2 - 2 0 0 6 4 0 0761 0 130 9 4 4 2 2 2 1 229 0 9 0 0 2800 1 789 0 6 1 8 0 0 8 0 0319 - 1 0 0 0 9 6 0 0778 0 116

10 1 3 3 5 3 1422 0 9 0 0 3000 1218 0 5 0 2 0105 11254 I 2 - 2 0800 0169 0182

I 1 1 4 3 4 2 2 1 3 4 3 3 2 2 5 2 2 4 2 3 3 3 2 5 1 4 2 2 5 6 1 4 2 4 3 7 1 5 2 2 4 8 1 5 2 3 3 9 3 1 5 3 2

10 3 2 3 4 2

Structure 3 P [ D e m a n d = 1] = 0 9

N u m b e r of stat ions in series, M = 5 Reliability of the stations = 0 9500 0 9529 0 9000 0 8500 0 9500

536 0 900 2 600 I 357 0 639 0 083 0 436 422 0 900 2 600 1 450 0 648 0 068 0 438 228 0 900 2 600 1 831 0 702 0 053 0 438 227 0 900 2 600 I 723 0 670 0 053 0 407 610 0 9 0 0 2800 I 555 0 6 4 4 0 0 7 0 0 4 0 7 421 0 900 2 800 I 543 0 628 0 069 0 393 538 0 900 2 800 I 663 0 655 0 069 0 406 423 0 900 2 800 1 659 0 645 0 067 0 393

1 0 --2 1 280 0 4 2 9 0 2 3 9 0 0 0 0 880 0.323 0 212 0 0 --2 0 960 0 687 0 096 0 0 --2 0 4 8 0 0438 0 131 I 0 -- 2 1 360 0 460 0 178 0 0 0 I 040 0 443 0 196 1 0 - 2 I 360 0 634 0 164 l 0 0 I 040 0 636 0 174

226 0 9 0 0 2800 I 830 0 6 9 4 0058 0 4 5 7 - 2 0 0 1040 0 8 4 5 0 176 227 0900 2800 1 762 0663 0053 0417 0 - 2 - 2 0 6 4 0 0 4 6 6 0 160

I 2 2 3 3 3 2 2 2 2 4 4 3 2 4 4 3 2 4 3 2 5 2 3 5 5 2 2 3 3 6 1 3 3 4 5 7 1 3 3 5 4 8 1 4 4 4 3 9 3 1 4 4 4

l0 3 4 2 5 2

Structure 4 P [ D e m a n d = I] = 0 9

N u m b e r o f stat ions in series, M = 5 Reliability o f the stations = 0 9000 0 9000 0 9000 0 9000 0 9000

421 0 9 0 0 2 6 0 0 1 472 0 5 8 6 0 0 7 6 0 2 9 0 I - 2 - 4 0 4 8 0 0075 0 105 537 0 9 0 0 2 8 0 0 1 457 0571 0 0 8 0 0292 I - 2 - 4 0960 0 128 0 141 227 0 9 0 0 3 0 0 0 1 651 0 5 7 4 0065 0260 - I - 2 0 0.800 0253 0 0 7 9 424 0 9 0 0 3000 I 681 0 593 0065 0295 0 0 - 2 0 8 0 0 0 3 8 3 0 116 423 0 9 0 0 3000 I 887 0 6 2 0 0065 0 3 1 0 0 - 2 - 4 0 8 0 0 0 7 5 2 0 104 608 0900 3 200 1 410 0 522 0 0 9 6 0259 3 0 - 4 1 040 0 2 0 0 0 155 540 0 9 0 0 3 200 I 417 0 523 0 0 9 6 0 2 5 9 I 0 - 2 I 040 0203 0 155 423 0 9 0 0 3200 1465 0 5 2 6 0091 0 2 5 4 0 --2 - 2 0 9 6 0 0239 0153 536 0 9 0 0 3200 1615 0579 0081 0 3 3 4 0 0 - 2 0 9 6 0 0323 0220 227 0 9 0 0 3200 1757 0 5 8 2 0061 0271 0 0 0 1040 0383 0088

Structure 5 P[Demar td = 1] = 0 9

N u m b e r of stat ions in series, M = 5 Reliability o f the stations = 0 8500 0 9000 0 9000 0 9000 0 8500

1 3 2 4 5 5 1606 0 9 0 0 3800 1804 0 5 0 4 0 0 7 3 0181 I 0 0 1040 0332 0106 2 3 4 3 5 4 1 539 0.900 3.800 I 845 0 488 0 072 0 139 0 0 0 0 640 0 329 0 054 3 4 2 3 5 5 1607 0 9 0 0 3800 1872 0 5 1 7 0 0 6 9 0188 I 0 0 1040 0463 0082 4 5 2 3 5 4 I 538 0 900 3 800 1 963 0 530 0 065 0 191 0 0 0 I 040 0 585 0 064 5 3 3 5 4 5 1612 0 9 0 0 4 0 0 0 1828 0 4 7 7 0071 0143 l 0 - 2 0 8 0 0 0 2 3 0 0 1 0 0 6 3 4 4 4 5 1.611 0 9 0 0 4 0 0 0 I 834 0 4 7 0 0071 0 130 2 0 - 2 0 4 0 0 0 195 0 0 7 6 7 3 4 5 4 4 1 536 0.900 4 0 0 0 1 856 0 4 7 2 0 0 7 0 0 129 I 2 - 2 0 4 0 0 0231 0077 8 4 2 5 4 5 I 607 0 9 0 0 4.000 l 883 0 5 0 2 0067 0 179 0 0 0 0 8 0 0 0335 0 121 9 4 4 3 4 5 1 604 0 9 0 0 4 0 0 0 I 869 0478 0068 0 134 1 - 2 0 0 4 0 0 0219 0065

10 5 3 3 4 5 1611 0 9 0 0 4 0 0 0 1916 0 4 9 4 0065 0 151 1 - 2 0 0 8 0 0 0348 0075

Transfer lines with random unit demand

Table 2--Continued

315

Obs Capactttes Col 1 2 3 4 5 6 7 8 9 10 I1 12 13

I 4 4 2 2 4 4 3 3 4 5 2 4 5 4 2 5 4 4 4 6 4 5 3 7 4 6 2 8 5 4 3 9 5 5 2

10 6 4 2

I 2 3 4 5 6 7 8 9

10

1 I 2 1 1 2 1 3 3 1 4 1 2 2 1 4 1 5 5 1

I I 11 2 2 1 1 3 I I 1 4 1 1 2 5 1 2 1 6 3 1 1 7 4 1 1 8 5 1 1 9 1 1 3

10 1 3 1

Struetm~ 6 P[Demand - I] = 0 9

Number of stauons m senes, M = 3 Rehabdlty of the statmns ~ 0 9500 0 7290 0 9500

007 0 810 3 333 1.937 0 568 0 136 0 353 - 1 0 0 0 889 093 0 810 3 667 1 947 0 517 0.140 0 305 - I 0 0 0 444 007 0810 3667 1939 0552 0136 0352 0 0 0 I I l l 007 0810 3667 2271 0.571 0136 0353 - 2 0 0 I I l l 130 0810 4000 I 954 0489 0 141 0292 0 0 0 0000 094 0 810 4 000 I 948 0 502 0 140 0 304 0 0 0 0 667 007 0 810 4 000 I 939 0 542 0 136 0 352 0 0 0 1 333 094 0810 4000 2280 0520 0 140 0305 - 2 0 0 0667 007 0810 4000 2272 0555 0 136 0352 - I 0 0 1 333 007 0810 4000 2604 0573 0 136 0.353 - 2 0 0 I 333

Structm'e 7 P[Demand = I] = 0 9

Rehabtht 0 921 0 921 1 000 0 946 0 946 1 333 0 947 0 947 1 333 0 952 0 952 I 667 0 952 0 952 1 667 0 953 0 953 2 000 0 954 0 954 1 667 0 954 0 954 2 000 0 954 0 954 2 333 0 954 0 954 2 333

Number of staUons m series, M = 3 of the stauons = 0.9500 0 9990

1.265 259 264 709 254 258 264 702 708

2 153

0.260 0.294 0 271 0 264 0313 0 304 0 278 0 298 0 274 0 266

0 9500 946 0946 0054 0946 0 - 2 - 2 0000 0017 0017 223 0 934 0 025 0 894 - I - 2 0 0 444 0.341 0 045 219 0 926 0 036 0 889 0 0 0 0 444 0 357 0 033 519 0 930 0 032 0 875 0 0 0 0 889 0 753 0 031 522 0940 0018 0882 - 1 - 2 0 0889 0731 0045 838 0947 0017 0879 - 1 - 2 0 I 333 I 150 0038 533 0 925 0 019 0 870 - I 0 0 0 444 0 386 0 044 834 0936 0031 0870 0 0 0 I 333 I 172 0025 152 0 939 0 030 0 866 0 0 0 1 778 I 596 0 020 163 0953 0016 0877 - 1 - 2 0 I 778 1 583 0031

Stnlcture 8 P[Demand = 1] = 0 9

0 9500 000 0 850 0 850 0 150 0 850 0 0 0 0 000 0 092 0 092 200 1 039 0 867 0 107 0 842 - 1 0 2 0 320 0 274 0 053 200 1030 0861 0119 0841 0 - 2 0 0320 0264 0080 200 I 015 0874 0096 0844 0 0 0 0320 0 160 0067 200 0 985 0 845 0 108 0 798 0 2 2 0 320 0 166 0 101 400 1 226 0877 0098 0842 - I 0 2 0640 0555 0051 600 1 418 0 883 0 095 0 842 - 1 0 2 0 960 0 858 0 049 800 1 614 0888 0094 0842 - 1 0 0 I 280 I 171 0050 400 1 188 0891 0071 0845 0 0 0 0640 0414 0066 400 1 104 0850 0087 0786 0 2 2 0640 0322 0 113

Number of stations m semi-parallel, M = 5 Rehabtht, of the stations = 0 9500 0 9000 0 9000 0 9000

0 948 0 948 0 952 0 952 0 953 0 953 0 953 0 953 0 953 0 953 0 953 0 953 0 953 0 953 0.953 0 953 0 954 0 954 0 954 0 954

Column I E[X M] = expected content of the last buffer (warehouse) Column 2 P[X ~ >i I] = probabdlty of having one umt m the last buffer to meet the demand

Column 3 i~" = ~ cm/M = average design capacity

Column 4 ~ '= ~. E[XM]/M = average work-m-process and inventory m - I

Column 5 ~={~, E[XM]/c.}/M =average capac,ty utthzat,on

M

Column 6 ~" P[X ~ = O]/M = average starving probabdlty m.i

Column 7 ~ P[X~= ct]/M = average blocking probabdlty

Column 8 Stgn{cl, c:, . e~} =Stgn function of the design capacity Column 9 Sign{E[Xl], E[X2], , E[X~]} = Stgn functton of the work-m-pr~ess and inventory Column 10 Stgn{E[Xtl/tl, E[X2]/c2, , E[XM]/c~¢]} =Stgn funcUon of the capacity utdtzatton

Column I I ~ Ic,~ - C"I/M = mean absolute deviation of the design capaoty

Column 12" ~ IE[X ~] - YO/M = mean absolute devtauon of work-m-process ~,.~

Column 13 ~ IE[X"]/cm - UI/M = mean absolute capactty ut th~uon

(iv) Based on the ~j and Oj measures, structures 1-5 are clearly inferior to structures 7 and 8 which have higher station reliabilities. As noted in (ii), the ordering of the ]~j with respect to station reliabilities is expected. However, the results regarding the standard deviations, Oj, are not easy to explain and predict.


5.2. Dtsaggregate results whwh ,~attsfy demand

For each structure, ten "'best" observations are selected and reported in Table 2. Through exhaustive search, we chose these ten observations whose P [ X ~ >~ 1] values (column 2) are greater than and as close as possible to 0.900. In the case of ties, we chose the observations with smaller average design capacity (column 3) Exception is made for structure 6 since the maximum observed values for P[X~>~ 1] Is 0.810. This is the theoretical upper bound achievable by one-station production line with station reliabxlity of 0 729 and an infinite buffer capacity (see Section 4.1). These ten "best" observations are chosen to ensure that the probabilities of these lines having at least one unit to meet the demand in the long run match the expected demand of 0 9 the observations chosen does not guarantee that there will be no shortages (see the discussion at the end of Section 4) The choice of P [ X ~ >>, l] ~> 0.900 is arbitrary in some sense since choosing P [ X ~ >1 1] I> 0.950 results m lower expected shortages. However, larger design capacities will be needed to produce lines with P [ X ~ >1 1] ~> 0.950 than those with P [ X ~ >~ I]/> 0.900. We chose 0.900 probabd~t~ as a compromise to trade-off between required design capaclues and low shortage.

Here are the results from Table 2

(v) For structures 1-5, 7, and 8. various combinations of capacities meet the expected demand of 0 9 units per period. This lmphes that economic cost analysis of the cost of the capacities may improve the design of a production line. Furthermore, the reqmred capacities (column 3) for structures 1-4 to meet the expected demand are lower than those for structure 5, which has lower rehabilmes at the first and the last stations.

The three-station structure 6 achieves the theoretical upper bounds of 0810 for P [ X ~ >1 i] with relatively small design capacity (c~ = 4, c_~ = 4, c3 = 2). Larger capacity will generally increase the work-m-process inventories (columns 1 and 4). For structure 7. the smallest P[X">>. 1] IS 0.921 which is obtained using the minimal structure with c~ = c , = c~= I For the same mmlmal structure, c~ = - = c 5 = 1, the semi-parallel structure obtains higher P [ X ~ >>, 1] = 0.948 than that of structure 7. For larger capacities. there is only a margmal improvement

(Vl) Results for structure 2 and structure 3 allow for comparison of station arrangements Note that P [ X '~ >1 I] = 0 9 for these two structures. However, puttmg the more reliable station at the second position (structure 3) results in lower average starvmg probablhties (column 6), higher average blocking probabilities (column 7) and higher capacity utihzatlon (column 5), than putting it in the fourth posmon. This is an indication that stations should be arranged in the descending order of their station rehabilitles.

(wi) Results for structures I, 4 and 5 allow us to see what happens when the reliabdmes of two end stations are reduced (0.95, 0 90 and 0.85) We see that larger design capacity (column 3) ~s needed to compensate for the reduction of two end-station rehablhtles and to achieve P [ X ~ >~ I] = 0 900. As a consequence, the average work-in-process (column 4) increases However, the average capacity utlhzat~on (column 5) decreases. Th~s reflects the Inefficiency of the line with reduced rehabditles. There is no slgmficant decrease in the starving probabilities (column 6), but the blocking probabdmes does Increase

(VUl) To meet the expected demand of 0.9 umts per period, a relatwely high level of average work-in-process inventory (column 4) and warehouse inventory (column 1) are reqmred. Thus, as expected, given an unrehable production line. there is a trade-off to be considered between the cost of holding inventories and the designed capacmes on one hand and the desire to meet the expected demand on the other. Also, as illustrated by structures 7 and 8, this trade-off depends on both the networks configuration of the production system and the number of statxons in the system These two structures are more rehable and their P[X~f>~ l]s are higher than the other structures, yet their inventories are lower (column 4 and column 1) This is because the last station ~s the bottle neck.

0x) To capture the patterns and the varmblhty of various measures such as the sizes of the capacity (column 3), the average work-In-process inventory (column 4) and the average capacity utilization (column 5), the following two measures are calculated for each observation, the mean absolute deviation (MAD) (columns I l, 12 and 13), and the sign functions (columns 8, 9 and 10) of the design capacity, the expected work-in-process, and

T r a n s f e r l ines w i t h r a n d o m u n i t d e m a n d 317

Table 3 CorrelaUon coeflictents between columns 8, 9, and 10 o f Table 2

Type of correlation Columns Columns Columns Structure co¢t~oent 8 and 9 8 and 10 9 and 10

I Pearson 0 576 - 0 266 - 0 034? Spearman 0 562 - 0 262 - 0 038t

2 Pearson 0 405 - 0 336 0 067 Spearman 0 391 - 0 332 0 077

3 Pearson 0 580 - 0 161 0 153 Spearman 0 569 - 0 164 0 147

4 Pearson 0 766 - 0 555 - 0 115 Spearman 0 764 - 0 546 - 0 I 12

5 Pearson 0 529 - 0 638 - 0 145 Spearman 0 520 - 0 625 - 0 140




?Not slgmficant at 95% level

(x)

the capacity utilization, respectively. The MADs and the sign functions demonstrate the non-constant patterns. It is interesting to observe that the sign functions of the average capacity utilization (column 10) is nonpositive for all observations (with the exception of structure 8). This means that in a serial production system, to meet the expected demand, the capacity utilization is a nonincreasing function of the buffer position, or the utilization decreases more frequently than it increases as a function of the buffer position. This phenomenon indicates that it is harder to utilize the downstream buffer capacity than to utilize the upstream ones.

Results from columns 8 and 9 show inconclusive behavior of the sign functions of the designed capacities and of the work-in-process inventories. The mean absolute deviation measures the presence of location shift and scale effect. The results from columns 11-13 indicate the instability of these measures across the various structures and the sampled obserations. In Table 3, we report the correlation coefficients (Pearson's product moment correlations and Spearman's rank correlations) between the sign functions given in columns 8, 9 and 10. These correlations are computed using SAS across all of the observations (see Table 1 for the number of observations used for each structure). The sign functions of the work-in-process and the capacity utilization (columns 9 and 10) are statistically correlated at a 95% significant level (except for structure 1) but the correlation coefficients are rather small. Moreover, the sign of the correlations varies with structures and thus no conclusive statement can be made on the consistency of the pattern of behavior between the sign functions related to the two measures.

On the other hand, we see consistent positive correlauons between the sign functions of the pattern of capacity (column 8) and the sign functions of the expected work-in- process (column 9). Also we see consistent negauve correlations between the sign functions of the capacity pattern and the capacity utihzauon (column 8 and column 10).

5.3. Other specific observations

The exhaustive search conducted for various combinattons of capacity (see Table !) reveals various interesting phenomena. Several of these are reported here without providing specific examples except for item (xiv).

(xi) In structures i-7, the ordermgs of the observations with respect to the magnitude of P[X M >1 1] and E[X M] are not identical within a structure, ~.e. given any two observations h and i2, P, I [X M >t 1] ~< P,2[)( M >t 1] does not necessarily imply that E,~[X M] <~ E , 2 [ ) ( M ] .

(xii) E[X M] and P[X ~ >>. 1] are not necessarily monotonic nondecreasing functions of the sum of the available capacity, M Y~,. = t era.

318 ABRAHAM MEHREZ a n d B EDDY PATUWO

(xm) E[X M] and E[XM]/c,,, are not necessarily monotonic nondecreasing functions of the station posmon m. Also E,,M= ~ E[X M] and EmM= ~ E[XM]/c,, are not necessardy monotomc non-decreasing funcuons of M ~',n = 1 Cm.

(XlV) In Table 4, we pick two specific examples to compare structure 1 and structure 5 (structure 1 has more rehable two end stations than structure 5). The more reliable structure I is able to achieve the reqmred P[X 5 >1 1]/> 0.900 with a minimum total capacity of Z~= ~ c,, = 12, whereas structure 5 requires a minimum total capacity of 19. Also, the corresponding E[X 5] is smaller for structure 1 than that of structure 5. This means that the more reliable production hne ~s able to meet the expected demand with less total capacity and less end-product inventory when compared to the less reliable production line. For the same capacity configuration, the P[X 5 >t 1] of structure 1 is larger than that of structure 5. Th~s is no surprise since the more reliable production line should be able to meet the demand better than the less reliable one.

6 D I S C U S S I O N

The experimental analysis presented in this paper provides some policy implications for the design of production lines w~th unreliable stations. First, the derivation of an economic model to evaluate an optimal design of buffer storages through experimental analysis is useful if the various capacity patterns result in identical or similar stationary probabilistic behavior gwen a fixed stationary demand probability distribution and reliability parameters. A model formulated for the case of the unit demand may have the following form:

M ~/

Minimize: z = ~ .{P[D = 1]. P[X M : 0]} + ~ tim" E[ XM] + ]"" ~ TC(c~, c 2 , . , cM) m = ] r t r = l

s.t. O<~cm<~6,,,rn= 1,2 . . . . . M,

where ~ is the penalty (shortage) cost per period, tim is the inventory holding cost coefficients of work-in-process in the mth buffer per unit per period, y is the capital recovery factor per period, and TC is the investment cost function evaluated for buffer sizes c~, c2 . . . . . c , . The long-run expected cost function per period, z, may be evaluated under various designed structures such as serial and semi-parallel, under technical constraints and storage constraints. 6m, of the capacities. The optimization of the above model can be done through exhaustive search using the procedure given in this paper provided that the size of the problem is not prohibitively large.

The second policy imphcation is that increasing the capacities may ~mprove significantly the ability of an unreliable production line to meet the expected demand. Employing the theoretical upper bound of P[X">>. 1] using the unlimited capacity case (see Section 4.1) can indicate that for some structures, the expected demand, d, cannot be met for given station reliabdities of the production line. Here, max. P[X M >1 1] = r/d, were r = min{r~ . . . . . rm} and r < d. This imphes that in order to meet the expected demand d, d 2 ~< r < d. For d = 0.9, 0.81 ~< r < 0.9 is required. For structure 6, r = 0.729 is not sufficient to meet the expected demand of 0.9.

Thirdly, different structures may demonstrate desirable and undesirable properties. Thus, realizing the difficulty of obtaining economic data to solve a cost model, a multiple-objective approach can be suggested to support managerial decisions regarding the design of the production systems. In this paper, the three basic objectives are studied: the probability of meeting the expected demand, the expected inventory of work-in-process, and the sizes of the capacities.

Finally, we note that the experimental study in this paper may be extended to cover structures other than structures 1-8 and the case where the demand per period is more than one unit. The probabilistic analysis done here is limited to the stationary problem. In the case of nonstationary

Table 4 A specific example Capacity configuration Total Structure I Structure 5

t'l c2 ~3 c4 t5 capacity P[X u>>. 1] E[X u] P[A"M ~> I] E[X u]

I 2 2 3 4 12 0 900 I 538 0 833 I 209 3 2 4 5 5 19 0 942 2 065 0 900 1 606

Transfer lines with random unit demand 319

demand with seasonal and trend patterns or time-dependent parameters which allow capacity expansion analysis, simulation studies can be used. The discussion of simulation models is not the purpose of this paper. We believe that the model used here, although restricted in its form (discrete time model assuming only single period inspection and repair policy, and a push rather than a pull production policy), demonstrates a guide for managerial decisions via experimental analysis.

Future research will consider the effect of reducing inventory holding costs by considering operational policies other than the push system (for classifications of the push and pull systems, see Pyke and Cohen [10]). It is worth noting that production managers sometimes prefer the push pohcy to increase the expected output although this policy may be relatively inefficient. However, incorporating policies that control the production rate by defining state dependent actions can be quite efficient for a production line with many stations. Here, Markov decision models may be used as the analytical framework to investigate control decisions.

We recognize that our study differs significantly from typical stochastic analysis of production lines. However, the models considered in this paper exist in our industrial society. Hence, there is justification to use them to shed light on the behavior of these systems. For example, the nature of a chemical process observed in the Israeh industry by one of the authors of this paper is characterized by relatively low holding costs of work-in-process and relatively high capital investment costs for the capacity. Under these conditions and given the managerial preference for push policy, the framework of the model suggested in this paper is appropriate.

Acknowledgements--The authors are indebted to the reviewers for their cnUcal and constructwe comments which help clarify the presentation in this article The authors also would like to thank Paula Spelrs for compdmg and orgamzmg the results and to Professor Michael Hu for insightful discussions

REFERENCES

1 J A Buzacott and L E Hanlfin Models of automatic transfer hnes with inventory banks--a review and comparison AIlE Trans 10, 197-207 (1978)

2 S B Gershwm and O Berman Analysis of transfer lines conslstmg of two unreliable machines with random processing times finite storage buffers. AIIE Trans 13, 2-11 (1981)

3 S. B Gershwm and I. C Schlck Modehng and analysm of three-stage transfer hnes with unrehable machines and finite buffers Opns Res 31, 354-380 (1983)

4. M A Jafan and J G Shanthlkumar Exact and approximate solutions to two-stage transfer hnes with general uptlme and downtime dlstnbutmns l iE Trans 19, 412-420 (1987)

5 F S Hdller and K C So The effect of the coett~oent of varmtlon of operation times on the allocation of storage space m production line systems l iE Trans 23, 198-206 (1991)

6 F .S . Hilher and K C So The effect of machine breakdowns and mterstage storage on the performance of producuon line systems Int. J Prodn Res 29, 2043-2055 (1991)

7 T Altlok and S. Stldham Jr The allocation of lnterstage buffer capacities m production lines HE Trans 15, 292-299 (1983)

8 P A. Schweltzer A survey of aggregatlon-dlsaggregation m large Markov chains In Numerical Solutwn of Markov Chains (Edited by Stewart W J ) Dekker, New York (1991)

9. K C Kapur and L R Lamberson Rehabdlty in Engmeertng Design Wiley, New York (1977) 10 D F Pyke and M A Cohen Push and pull m manufacturing and distribution systems J Opns Mgmt 9, 24-43 (1990)

APPENDIX A

For the semi-parallel structure 8 (see Fig 2), the marginal translUon probabilities P[X?+ I =/IX? = t] m = I, 2, ,5. are also by (9), (10) and (11) However, the probabdltles of events {B ? = 1} and {A? = 1} and their complementary events {B? =0} and {A? =0} are different and they are given below by (AI)-(AI0) If there are two or more of the 2nd, 3rd and 4th buffers nonempty, station 5 will draw one umt from the preceding buffer m the following order' the 2nd buffer first, the 3rd buffer next, and the 4th buffer last For example, if the 2nd and the 3rd buffers are nonempty, station 5 wdl draw one umt from the 2nd buffer

P[B~= 1]= P[D = I] = d , (AI)

PtB~=I]=P[Y~= 1] { P t X ~ < c ~ I + P [ X ~ = c s ] PtB~= I]}, (A2)

PtB~ = 11 = PtX~ = 01 PtB~ = 11, (A3)

P [ B ; = II = PtX~ = 0l eIB~ = II, ( A 4 )

P[B I = l I = 1 -- P[BJ = 0], where

PtB: = 01 = { t , lY~ = 0l + Pit ~, = i i P [ X ~ = c,.l PIB~ = 01} { t , l r ~ = 0] + PIY~ = !] P [ x ~ = c,] els~ = 01}

{e[Y;=Ol+ety4,=l] e[x4,=c,] PIB4,=OI} (AS)

CAIE 26/2--H


P[A~ = II = PtY~ = 11 {l - P[X~ =01 PIX] = 0 l PtX; =01},

P[A~= I ] = P [ Y ~ = 1] P[X] >OI.

P[A~= II= P[Y~= II P [ X ) > 0 ] [P[Y~=OI+ PIY2, = II P[X~=c,] P[B~=0]},

p[A4,=II=p[y4=II P [ X ~ > 0 ] {P[Y~=OI+P[Y~=I] PtX~=c2I P[B,2=0I}

{P[Y~=OI+ P[Y~= I] P[X]=cs]

P[A~ = 11 = e[rl = l]

(A6)

(A7)

~A8)

P[B~ = 0]}, (A9)

(Al0)

A P P E N D I X B

Notations {AT' = 1} is the event that the ruth station is capable of producing one umt at the end of period t, regardless of the state of the ruth buffer {AT' = 0} is the complementary event of {A, ~ = 1} {B 7 = 1} is the event that one umt can be transferred out o f the mth s tauon, gwen that {X" >1 1} {B. = 0} is the event that the ruth buffer is blocked cm, m = 1, 2, , M, is the buffer capacity for the ruth station D zs the demand per period for the fimshed product M Is the number of s tauons In the transfer hne r., = P[YT' = 1] Is the rehabdlty o f the ruth station XT' is the state (the content) of the ruth buffer at the beginning of period t X" Is the state (the content) of the m t h buffer at steady state YT' = 1(0) indicated that the ruth statxon is up (down)

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Discrete unreliable transfer lines with exogenous random unit demand