Stochastic modelling and simulation of production lines: a computer ...

Forsch Ingenieurwes (2007) 71: 205–213 DOI 10.1007/s10010-007-0059-3

ORIGINALARBEITEN · ORIGINALS

Stochastic modelling and simulation of production lines: a computer-based approach towards model validation Mahyar Mahinzaeim

Received: 18 September 2007 / Published online: 10 October 2007 © Springer-Verlag 2007

Abstract This study employs a simulation-based design methodology to investigate the performance of two models of manufacturing systems. In the first model, the dynamic behaviour of a single parallel-machine stage with unreliable work stations is modelled as a Markov process. A similar analytical method for evaluating the performance of a buffered production line is presented in the second model. A simple approach towards coding and simulating the models is presented, and numerical examples based on these simulation models indicate that the approach is viable. Stochastische Modellierung und Simulation von Produktionslinien: Ein computerbasiertes Verfahren zur Modellvalidierung Zusammenfassung Dieser Bericht stellt eine auf Simulation basierte Entwurfmethodik vor, mit deren Hilfe das Verhalten von zwei Fertigungssystemen zu untersuchen ist. Im ersten Modell ist das dynamische Verhalten von zwei parallel arbeitenden Produktionslinien mit unzuverlässigen Maschinen als Markov-Prozess entworfen. Eine a¨ hnliche, analytische Methode wird zur Bewertung der Verfügbarkeit eines aus zwei, durch einen Puffer getrennten, Maschinen bestehenden Fließbandes im zweiten Modell herangezogen. Eine einfache Methode zur Simulation der Modelle wird erläutert, und numerische Beispiele basierend auf dieM. Mahinzaeim (u) School of Mechanical and Systems Engineering, Stephenson Building, University of Newcastle upon Tyne, Claremont Road, Newcastle upon Tyne NE1 7RU, UK e-mail: [email protected]

sen Simulationenmodellen zeigen an, dass die Annäherung nützlich ist. List of symbols Standard Symbols [t, t + δt] Interval set {x | t ≤ x ≤ t + δt } A Matrices are represented in bold capital letters p Vectors are represented in bold lowercase letters Mathematical Expressions and Operations lim o(δt) δt

Limit Order relationship, h(δt) is o(δt) if h(δt)/δt → 0 as δt → 0 Short time interval

Notations from Queuing Theory B bm EB Em k Mm n¯ pk

Buffers are represented in capital letters Service rate of machine Mm Efficiency of buffer B Efficiency of machine Mm Number of states of the system Machines are represented in capital letters Buffer level Steady state probability or proportion of time in state k (analytical) pkS Steady state probability or proportion of time in state k (numerical) Pm Production rate of machine Mm ¯ TB Time in the buffer ε = 0, 1 States of the buffer B θm Repair rate of machine Mm λm Failure rate of machine Mm

13

206

1 Introduction In manufacturing systems that produce a few product types in large numbers the operation on a work piece is usually performed by a set of serially arranged machines that are subject to failure [1]. As a consequence, failures at any stage cause the failure of the entire manufacturing system, affecting the overall production rate. In order to regain a significant amount of the production rate, there are two common approaches: 1. The utilisation of parallel-machine stages; and 2. The provision of buffer storages between the workstations. Parallel machines are put into operation when one of the machines fails, whereas buffer storage between a pair of machines avoids that the production line breaks down due to the failure of one machine [2, 3]. There are a wide range of models that can be used to address manufacturing system design and operational problems. One modelling technique is queuing network modelling based on queuing theory that has its origins in the early 1900s for the design of telephone systems [4]. The queuing network models discussed in this study are used to analyse mass manufacturing systems that are characterised by very high production rates [5]. Mass manufacturing systems have a product-flow layout, as is illustrated in Figs. 1 and 2. Flow lines composed of parallel-machine stages (Fig. 1) are similar to the classical flow line (Fig. 2) except for the fact that a given stage may consist of parallel machines [6]. Within the scope of this report, the notions of queue and server in standard queuing theory are replaced by the notions of buffer and machine. Queuing network modelling of manufacturing systems has been addressed by a large number of researchers. Buzacott [7] noted that production lines were first studied analytically via a probabilistic approach by Vladzievskii [4]. He introduced the idea of the loss transfer coefficient and developed a simple model to determine it in the case of

Fig. 1 Series parallel flow line made of m work centers

Fig. 2 Serial production line composed of m workstations and (m − 1) buffers

13

Forsch Ingenieurwes (2007) 71: 205–213

two equal stations. The earliest English paper on production line modelling was by Königsberg [8] who described Vladzievskii’s results. Avi-Itzhak and Naor [9] and AviItzhak and Yadin [10] studied the production rate of production lines in the absence of buffers and in the presence of buffers of infinite capacity. Apart from the work by Buzacott [7, 11] there were then few publications on this topic until the late 1980s when a variety of improvements or alternative approximate algorithms were developed by Gershwin and Schick [12] and Choong and Gershwin [13]. Dallery et al. [14] proposed an algorithm for the analysis of discrete part production lines where machines had identical processing times. Alves [15] addressed the issue of performance evaluation of series-parallel systems using continuous-time Markov processes. Gopalan and Kannan [16, 17] and Gopalan and Kumar [18] studied the influence of defective parts on the performance of twostation systems in detail. Tempelmeier and Bürger [19] presented an analytical approximation for the performance of asynchronous flow production systems with finite buffers. Their research considered distributed stochastic processing times as well as breakdowns and imperfect production. AlHassan et al. [20] proposed a simple Markov model aiming at identifying the prime costs involved in production line downtimes. They investigated the influence of the performance of a single machine on a line and presented an approximate analysis of the benefits of a machine building up a buffer stock. Sörensen and Janssens [3] studied a production line with several machines in series. They developed a general Petri net model for an m-machine, (m − 1)-buffer system. In order to understand and better control production line systems the main contribution of this paper is the numerical approximation of the early analytical results outlined before through simulation studies. The objectives of this study are: 1. Modelling and analysis of an unreliable flow line composed of a single parallel-machine stage; 2. Reliability analysis of a two-stage production line with infinite buffer storage; 3. Simulation of the models utilising a simple, discrete r ; and Monte Carlo method in MATLAB 4. Performance evaluation of the two-stage production line with the help of the numerical results from the simulation. The remainder of this report is organised as follows. The second section deals with a detailed presentation of the proposed research methodology. In section three a single parallel-machine stage model is presented. The fourth section is dedicated to the analysis of a two-stage production line with infinite buffer storage. Finally, conclusions drawn on the outcomes of this study as well as recommendations for further research are presented in section five.


2 Description of the proposed method The methodology employed in addressing the following problems emphasises mathematical modelling and uses common mathematical methods as well as simulation techniques as an attempt to understand the dynamic behaviour of production lines. The models considered in this study are: 1. An unreliable production line composed of a single parallel-machine stage; and 2. A two-stage production line with infinite buffer storage. These are widely used in the literature and predominantly modelled via the use of Markov processes assuming that the times between successive arrivals and/or service times obey the exponential distribution or, equivalently, that the arrival rates and service rates follow a Poisson distribution. Using the state model method, which aims at obtaining exact analytical results, these can then be solved. The idea behind the state model method is straightforward. First, all the feasible states of the Markov process describing the model are identified. In the second step, the transition probability matrix is generated from analysing the states of the model via differential-difference equations. Once the transition probability matrix is obtained, the stationary equations together with the boundary conditions can be used to solve for the stationary distribution. As stated before, the main contribution of this paper is the numerical approximation of the analytical results through simulation studies. Hence, a simple simulation algorithm is r , which is to determine iteratively coded using MATLAB the states of the system by taking small time steps and generating the proportion of time the system spends in a particular state. In summary, the solution procedure considered in this paper consists of the following five steps: 1. 2. 3. 4. 5.

Definition of the states of the model; Analytic solution of all the transition equations; Computation of the steady state probabilities as t → ∞; r ; and Simulation of the model using MATLAB Determining the exact solution of the relevant performance measures by means of simulation, thus providing a means to compare the accuracy of the two approximations.

207

3 Modelling and analysis of an unreliable flow line composed of a single parallel-machine stage 3.1 System description Adopting the approach proposed by Papadopoulos and Heavey [17], two lines with different failure rates λ1 , λ2 and repair rates θ1 , θ2 are considered. Here, the following events may take place: 1. A machine breaks down; and 2. A machine is repaired. The machines can be in three states: operational, under repair, or broken down. The probability that a failure occurs during an interval [t, t + δt] while a machine is operating is λ1 δt for machine 1 (M1 ), and λ2 δt for machine 2 (M2 ). The probability that a repair is completed during an interval [t, t + δt] while a machine is broken is θ1 δt for M1 , and θ2 δt for M2 . A schematic diagram of the flow line considered is given in Fig. 3. This process requires the solution of a system of k linear differential-difference equations in k unknowns, where k is the number of states of the system (Table 1). It is now necessary to introduce k = 5 states. 3.2 Analytical model The system stays in state 1 if both machines stay up. The probability of this is (1 − λ1 δt) (1 − λ2 δt). It can move from state 2 to state 1 if the first machine is repaired and the second stays up. The probability of this transition is θ1 δt (1 − λ2 δt). The probability of the transition from state 3 to state 1 is (1 − λ1 δt)θ2 δt since the first machine stays up and the second is repaired. The system cannot, however, move from state 4 to state 1 since the probability of this transition is θ1 δtθ2 δt = θ1 θ2 δt 2 = o(δt). Accordingly, the system cannot get from state 5 to

Fig. 3 A flow line composed of a single parallel-machine stage

Some common characteristic features of the two models under investigation are summarised in the following: 1. In all models of this study failures are assumed to be operation dependent; 2. The probability of two events occurring in the same time interval equals o(δt), which becomes negligible when compared to δt as δt → 0; and 3. In all models it is assumed that there is only one repair crew.

Table 1 Feasible system states k 1 2 3 4 5

M1 operational under repair operational broken down under repair

M2 operational operational under repair under repair broken down

13

208


state 1. Consequently, the first equation is (for λ1 = λ2 and θ1 = θ2 ): p1 (t + δt) = p1 (t) (1 − λδt)2 + p2(t) θδt (1 − λδt) + p3(t) (1 − λδt) θδt + o(δt) . The other equations are similar: p2 (t + δt) = p2 (t) (1 − θδt) (1 − λδt) + p1(t)λδt (1 − λδt) + p4(t) θδt + o(δt) , p3 (t + δt) = p3 (t) (1 − λδt) (1 − θδt) + p1(t) (1 − λδt) λδt + p5(t)θδt , p4 (t + δt) = p4 (t) (1 − θδt) + p3(t)λδt (1 − θδt) + o(δt) , p5 (t + δt) = p5 (t) (1 − θδt) + p2(t) (1 − θδt) λδt + o(δt) . Rearranging these equations and letting δt tend to zero, leads to the differential equations, p˙1 (t) = p1(t) (−2λ) + p2(t) (θ) + p3(t) (θ) , p˙2 (t) = p2(t) (−λ − θ) + p1(t) (λ) + p4(t) (θ) ,

subject to the boundary condition, 5

pk = 1 .

k=1

This gives p1 = θ 3 + 2λθ 2/∆ , p2 = λθ 2 + 2λ2θ/∆ , p3 = p2 , p4 = λ2 θ + 2λ3/∆ , p5 = p4 , where ∆ = θ 3 + 4λθ 2 + 6λ2 θ + 4λ3 . In order to verify these results a numerical code of this r according to system has been implemented in MATLAB the algorithm shown in Fig. 4. Figure 5 is a graph of pkS (k = 1, 2, . . ., 5) as a function of the ratio λ/θ for various values of the failure rate λ. It compares these with corresponding values of pk (k = 1, 2, . . ., 5). As can be seen from these results, the simulation model ( pkS ) provides a remarkably good approximation to the analytical

p˙3 (t) = p3(t) (−θ − λ) + p1(t) (λ) + p5(t) (θ) , p˙4 (t) = p4(t) (−θ) + p3(t) (λ) , p˙5 (t) = p5(t) (−θ) + p2(t) (λ) . As t → ∞, the following stationary equations are obtained: 0 = p1 (−2λ) + p2 (θ) + p3 (θ) , 0 = p2 (−λ − θ) + p1 (λ) + p4 (θ) , 0 = p3 (−θ − λ) + p1 (λ) + p5 (θ) , 0 = p4 (−θ) + p3 (λ) , 0 = p5 (−θ) + p2 (λ) . The preceding equations represent a five-state Markov process having a transition probability matrix, ⎛

⎞ −2λ θ θ 0 0 ⎜ λ − (λ + θ) 0 θ 0 ⎟ ⎜ ⎟ ⎜ A=⎜ λ 0 − (λ + θ) 0 θ ⎟ ⎟. ⎝ 0 0 λ −θ 0 ⎠ 0 λ 0 0 −θ The steady state probability or proportion of time pk in state k is then given by the solution of Ap = 0, ⎛

⎞⎛ ⎞ ⎛ ⎞ −2λ θ θ 0 0 p1 0 ⎜ λ − (λ + θ) ⎟ ⎜ p2 ⎟ ⎜0⎟ 0 θ 0 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ λ ⎜ ⎟ ⎜ ⎟ 0 − (λ + θ) 0 θ ⎟ ⎜ ⎟ ⎜ p3 ⎟ = ⎜0⎟ , ⎝ 0 ⎠ 0 λ −θ 0 ⎝ p4 ⎠ ⎝0⎠ 0 λ 0 0 −θ p5 0

13

Fig. 4 Simulation algorithm


209 Table 2 Feasible system states k 1 2 3 4 5 6 7 8 9 10

Fig. 5 Comparison of — p1 , – – p2 , - - - p3 , − · − p4 , — p5 with × p1S , + p2S , p3S , ∗ p4S , p5S respectively

results ( pk ). Furthermore, Fig. 5 shows that the overall line efficiency decreases as the failure rate of both the first and second machine increases, which makes sense.

4 Reliability analysis of a two-stage production line with buffer storage 4.1 System description Consider now a flow line manufacturing system organised as a two-stage production line separated by an infinite buffer. A schematic diagram of such a system appears in Fig. 6. A machine can either be operational, under repair, or broken down. Additionally, M2 is said to be starved when the buffer is empty. The following assumptions are central in the development of the model: 1. Both machines operate at equal rates (b1 = b2 ); 2. A starved machine cannot break down; and 3. The behaviour of machine Mm (m = 1, 2) is characterised by three exponentially distributed random variables: the time to fail (with mean 1/λm ), the time to repair (with mean 1/θm ), and the buffer service time (with mean 1/bm). In other words, when a machine is under repair, for example, it remains in that state for a period of time which is exponentially distributed with mean 1/θm . When a machine is

M1 operational operational operational operational under repair under repair under repair broken down broken down under repair

B 0 1 0 1 1 0 1 0 1 0

M2 operational operational under repair under repair operational broken down broken down under repair under repair starved

operational, it operates on a work piece if it is not starved. It continues operating until either it completes the piece or a failure occurs, whichever happens first. Either event can happen during the time interval [t, t + δt] with probability bm δt or λm δt respectively, for small δt. The state of the buffer B is denoted by the integer ε and does not depend on the capacity of the buffer. If ε = 1, the buffer is not empty; if ε = 0, it is empty. Note that machine M2 is starved at the instant when it has completed a piece and there is no work piece in B, see Table 2. It is now necessary to introduce k = 10 states. 4.2 Analytical model The system stays in state 1 if both machines stay up and the buffer stays empty. The probability of this is (1 − λ1δt)(1 − λ2 δt)(1 − b1δt). It can move from state 2 to state 1 if both machines stay up and the buffer empties. The probability of this transition is (1 − λ1 δt)(1 − λ2 δt) b2 δt. The probability of the transition from state 3 to state 1 is (1 − λ1 δt)θ2 δt(1 − b1 δt) since the first machine stays up, the second is repaired, and the buffer stays empty. Furthermore, the system can move from state 10 to state 1 if the first machine is repaired, the second stays up, and the buffer stays empty. The probability of this is θ1 δt(1 − λ2 δt)(1 − b1δt). No other transitions are possible. The equilibrium equations for the steady state condition for this model are (for λ1 = λ2 , θ1 = θ2 and b1 = b2 ): 0 = p1 (−2λ − b) + p2 (b) + p3 (θ) + p10 (θ) , 0 = p2 (−2λ − b) + p1 (b) + p4 (θ) + p5 (θ) , 0 = p3 (−λ − θ − b) + p1 (λ) + p6 (θ) , 0 = p4 (−λ − θ) + p2 (λ) + p3 (b) + p7 (θ) , 0 = p5 (−λ − θ − b) + p2 (λ) + p9 (θ) , 0 = p6 (−θ) + p10 (λ) , 0 = p7 (−θ) + p5 (λ) , 0 = p8 (−θ) + p3 (λ) , 0 = p9 (−θ) + p4 (λ) ,

Fig. 6 A two-stage production line with infinite buffer

0 = p10 (−λ − θ) + p1 (λ) + p5 (b) + p8 (θ) .

13

210


The preceding represents a ten-state Markov process having a 10×10 transition probability matrix, ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−2λ − b b θ 0 0 b −2λ − b 0 θ θ λ 0 −λ − θ − b 0 0 0 λ b −λ − θ 0 0 λ 0 0 −λ − θ − b 0 0 0 0 0 0 0 0 0 λ 0 0 λ 0 0 0 0 0 λ 0 λ 0 0 0 b ⎞ 0 0 0 0 θ 0 0 0 0 0 ⎟ ⎟ θ 0 0 0 0 ⎟ ⎟ 0 θ 0 0 0 ⎟ ⎟ 0 0 0 θ 0 ⎟ ⎟. −θ 0 0 0 λ ⎟ ⎟ 0 −θ 0 0 0 ⎟ ⎟ 0 0 −θ 0 0 ⎟ ⎟ 0 0 0 −θ 0 ⎠ 0

0

θ

0 −λ − θ

In order to determine the proportion of time pk in state k = 1, . . ., 10, analytical solutions were obtained. Additionally, a simple numerical simulation of this system based on the algorithm in section three was also performed to study the behaviour of the model developed above; several examples were performed, and a set of representative results is shown in Fig. 7. From the graphs one concludes that the simulation ( pkS ) provides a useful, yet less reliable approximation to this analytical model ( pk ), when compared to the single parallelmachine stage in section three. As pointed out before, the capacity of the buffer, which is denoted as the time in the buffer, i.e. the time that the current level of inventory could sustain operation of M2 , taking work pieces from the buffer at some rate, was not considered in the model. In order to overcome this lack of information and give useful results in the following case study example, the simulation code accounted for the time in the buffer by iteratively determining the buffer state in δt. 4.3 Performance measures The analysis of the performance of production line systems which are subject to breakdowns has a long and interesting history. Many previous papers on reliability analysis of production lines concentrate on non-financial performance measures. The major non-financial performance measures in flow line analysis are [21]:

13

1. 2. 3. 4.

The efficiency of machine Mm in the system; The production rate of machine Mm in the system; The efficiency of buffer B in the system; and The work-in-process level of buffer B.

The efficiency of machine Mm in the system (sometimes called availability) is defined as the probability of Mm working in the system. Here, it corresponds to the proportion of time during which Mm is neither in standby nor down (m = 1, 2): p + p2 + p3 + p4 Em = 1 . p1 + p2 + p5 The production rate of machine Mm in the system is the average number of parts on which Mm finishes operations in the system per unit of time (m = 1, 2): Pm = bm E m . The efficiency of buffer B in the system is defined as the probability of B sustaining operation of the following machine. It corresponds to the proportion of time during which B is not empty, hence, E B = p2 + p4 + p5 + p7 + p9 . Another important performance measure is the average buffer level, also called work-in-process level of buffer B: n¯ = T¯B P2 . T¯B is the time in the buffer. It is important to note that the system properties and performance measures studied in the following section are all concerned with steady state behaviours of the system. 4.4 Case study application A production line survey has been conducted in a largesized enterprise in Germany, employing nearly 7000 people worldwide. It is one of the biggest suppliers of suspension joints, stabiliser links, front and rear axle systems, and gearshift systems for manual and automatic transmissions to the world’s major automakers. The company considers buffer stocks and is interested in keeping its stock level in the stores fixed. Therefore, a feature of the production lines is that the machines can work at different rates. Within the scope of this case study, however, the machines worked at a nearly constant rate of 48 parts per unit of time. Observing the operating line for ten minutes per day, empiric data about the failure, repair and service rates were collected and the first three rows in Table 3 summarise these estimates from a two-stage production line with buffer storage, as shown in Fig. 8.


211

Fig. 7 Comparison of (a) — p1 , – – p2 with + p1S , p2S respectively; (b) — p4 , – – p10 with + p4S , p10S respectively; (c) — p6 , – – p9 with + p6S , p9S respectively; (d) — p3 , – – p5 with + p3S , p5S respectively; (e) — p7 , – – p8 with + p7S , p8S respectively

Running the simulation code based on these empirically determined data for δt → 0 and using the performance equations outlined before, the production rate of the real system is calculated and P1 = 0.0208 parts per unit of time, while

P2 = 0.0174 parts per unit of time. The difference is 16.35% and the first machine appears to be more reliable than the second. As a logical consequence, one may assume that the buffer is likely to accumulate stock due to this effect. How-

13

212


Table 3 Parameter values for a two-stage production line with buffer storage Mean time between failures Mean time between repairs Service time Efficiency Production rate Time is the buffer Buffer level

M1 4320 15 48 0.9986 0.0208 – –

M2 1200 300 48 0.8369 0.0174 – –

B – – – 0.5379 – 0.0055 0.0001

the exponential distribution is a statement of ignorance. In other words, the fact that a system has gone so long without a transition gives us no information about whether a transition is any more likely. Therefore, a natural extension of the research presented here would be to try to relax this assumption concerning the times between successive events. The new approach would have to take into account the history of a production run. Furthermore, there is the issue of a buffer rate regarding the capacity of the buffer. This was derived from an earlier model, in which the buffered two-stage line could not be treated as a Markov process due to interdependencies between the states of the machines and the capacity of the buffer. Therefore, further research could focus on the use of other queuing network analysis approaches or queuing formulas in the buffer calculation procedure, especially procedures that more accurately estimate the amount of time in the buffer. References

Fig. 8 A two-stage production line with buffer storage (case study)

ever, it is interesting to note from Table 3 that the buffer level is almost equal to zero. This is indeed what has been observed in terms of the real production line in the company, and it may be a result of the assumptions considered when modelling and simulating this particular production line configuration. In other words, since the states of the buffer are considered only, regardless of the number of parts in the buffer, the performance measures regarding the buffer are not sufficiently accurate.

5 Concluding remarks and further developments This study has reviewed a set of analytical models of serial production lines. A common and easily-applicable method to model and analyse a single parallel-machine stage, and a line consisting of two machines arranged in series and accumulating buffer has been presented. A simple, discrete Monte Carlo-based simulation algorithm has been coded r , in order to determine the proportion of using MATLAB time the system spends in a particular state. The accuracy of this method has been assessed by reference to comparative studies, and found to give very good estimates across the entire range of parameter values. Certainly, however, not everything is exponentially distributed, and in many cases the memory-less property of

13

1. Helber S (2001) Cash-flow-oriented buffer allocation in stochastic flow lines. Int J Prod Res 39(14):3061–3083 2. Baral SC (1993) Probabilistic modelling of the states of a buffer in a production flow system. IEEE Trans Eng Manage 40(4): 381–389 3. Sörensen K, Janssens GK (2004) A Petri net model of a continuous flow transfer line with unreliable machines. Eur J Oper Res 152(1):248–262 4. Vladzievskii AP (1952) The probability law of operation of automatic lines and internal storage in them. Automat Rem Control 13(3):227–281 5. Papadopoulos HT, Heavey C (1996) Queueing theory in manufacturing systems analysis and design: a classification of models for production lines and transfer lines. Eur J Oper Res 92(1): 1–27 6. Patchong A, Willaeys D (2001) Modeling and analysis of an unreliable flow line composed of parallel-machine stages. IIE Trans 33(7):559–568 7. Buzacott JA (1967) Automatic transfer lines with buffer stocks. Int J Prod Res 5(3):183–200 8. Königsberg E (1959) Production lines and internal storage – a review. Manage Sci 5(4):410–433 9. Avi-Itzhak B, Naor P (1963) Some queuing problems with the service stations subject to breakdowns. Oper Res 11(3): 303–319 10. Avi-Itzhak B, Yadin M (1965) A sequence of two servers with no intermediate queue. Manage Sci 11(5):554–564 11. Buzacott JA (1968) Prediction of the efficiency of production systems without internal storage. Int J Prod Res 6(3):173–188 12. Gershwin SB, Schick IC (1983) Modeling and analysis of threestage transfer lines with unreliable machines and finite buffers. Oper Res 31(2):354–380 13. Choong YF, Gershwin SB (1987) A decomposition method for the approximate evaluation of capacitated transfer lines with unreliable machines and random processing times. IIE Trans 19(2):150–159 14. Dallery Y, David R, Xie XL (1988) An efficient algorithm for analysis of transfer lines with unreliable machines and finite buffers. IIE Trans 20(3):280–283 15. Alves R (1990) Performance evaluation of series-parallel systems. J Manuf Oper Manage 3:224–250

Forsch Ingenieurwes (2007) 71: 205–213 16. Gopalan MN, Kannan S (1994) Expected duration analysis of a two-stage transfer-line production system subject to inspection and rework. J Oper Res Soc 45(7):797–805 17. Gopalan MN, Kannan S (1994) Expected number analysis of a two-server queuing network subject to inter-stage inspection and rework. Comput Oper Res 22(9):935–946 18. Gopalan MN, Kumar UD (1995) On the utilization of a split production system with inter-stage inspection. Comput Oper Res 38(2/3):107–116

213 19. Tempelmeier H, Bürger M (2001) Performance evaluation of unbalanced flow lines with general distributed processing times, failures and imperfect production. IIE Trans 33(4):293–302 20. Al-Hassan K, Swailes DC, Chan JFL, Metcalfe AV (2002) Supporting maintenance strategies using Markov models. IMA J Manage Mat 13(1):17–27 21. Papadopoulos HT, Heavey C, Browne J (1993) Queuing Theory in Manufacturing Systems Analysis and Design. Chapman and Hall, London

13