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...Papadopoulos (1993) Altiok (1997), among others). However, Markovian analysis is applicable for solving only small systems with up to six or seven stations in series. The reason for this limitation is that the resulting state space explodes exponentially as the number of stations and the capacities of the intermediate buffers increase. As a result it is very difficult to handle analytically large systems via Markovian analysis in order to compute the various performance measures of the production systems.
Approximate methods have been developed to overcome this major limitation of the Markovian analysis technique. One such method is the decomposition method, originally introduced by Gershwin (1987) and further developed by Dallery et al. (1988). The decomposition method decomposes a flow line into a set of virtual two or three-machine sub-lines, depending on whether or not note the flow of material is linear, as well as on whether or not the machines are serially arranged. Decomposition techniques have also been presented by Perros (1994), Altiok (1997), Helber (1999) and Tan and Yeralan (1997) among others.
An alternative approximation approach is the aggregation method (Lim et al., 1990). In this method, the first two machines of a production line are combined to form an aggregated machine. This new aggregated machine is then combined with the third machine of the production line to form a new aggregated machine and so on. This procedure is called forward aggregation and terminates when the last machine is reached. A similar aggregation going from the last machine to the first machine of the line (backward aggregation) may also be applied. The algorithm stops when the results from both the forward and backward aggregations are identical.
There is a vast literature on the analysis of flow lines where each workstation consists of a single machine and the flow of material is linear (see an excellent review paper by Dallery and Gershwin (1992) and the books of Papadopoulos et al. (1993) and Gershwin (1994). Some recent work on these systems includes Tolio and Matta (1998) who presented an elegant decomposition approach for the performance evaluation of automated flow lines with multiple failure modes. The decomposition block that was used in their analysis was solved exactly by a method that is independent of the buffer size. An extension of the decomposition approach for the performance evaluation of a flow line with linear flow of material and two part types was presented by Nemec (1999). A different efficient decomposition analysis for serial flow lines with two part types, deterministic identical processing times and multiple failure modes was proposed by Colledani et al. (2003). Flow lines with single machine workstations and non-linear flow of material were examined in Helber (1999) (where a detailed analysis of flow lines with split and merge operations is presented), Gershwin et al. (2001), Tan (2001), Helber and Mehrtens (2003) and Helber and Jusic (2004).
Literature on the analysis of flow lines with multiple identical parallel-machine workstations is relatively scarce. Friedman (1963) presented a reduction method that reduces a queuing system with parallel-machine workstations to corresponding problems for a system of fewer stages. It was also assumed that for any sequence of customer arrival times, the time spent in the system was independent of the order of stages. Forestier (1980) examined automated flow lines where each station consists of two parallel machines. Dubois and Forestier (1982) considered similar systems using Markovian analysis. Iyama and I to (1987) considered a flow line where some workstations have different numbers of parallel machines and unequal service rates. They presented the effects of server allocation on the maximum average production rate by using a Markovian model.
In Van Dijk and Van der Wal (1989) computationally attractive lower and upper bounds for finite multi-server exponential tandem queues were presented. A proof of the bounds and related monotonicity results were also presented, which were based on Markov reward theory. Gosavi and Smith (1995) developed computationally efficient bounds and approximations for the performance measures of series parallel queuing networks. They approximated analytically the throughput of a system with two tandem exponential queues and extended their analysis to elementary merge-and-split queuing networks.
Ancelin and Semery (1987) described a method that replaces each parallel-machine workstation by an equivalent single machine workstation. The processing rate of the equivalent workstation equals the sum of the processing rates of all parallel machines in the workstation. The failure rate and repair rate of the equivalent workstation are given by a formula which incorporates the failure and repair parameters of the parallel machines in the workstation. Burman (1995) applied a similar method that replaces each parallel server workstation by a single equivalent workstation for the case of continuous flow of material. The author assumed that the equivalent workstation has a maximum processing rate which equals the sum of the processing rates of the parallel machines. The failure and repair parameters of the equivalent workstation are calculated by using the assumption that all parallel machines at a specific workstation operate independently. Patchong and Willaeys (2001) presented a technique that replaces each parallel-machine workstation by an equivalent single machine workstation. The sets of equations necessary for this replacement were derived. The goal was to obtain a flow line with the classical structure with machines that are serially arranged.
A similar analysis for evaluating the throughput of Assembly/Disassembly (A/D) systems with parallel machines at each workstation and discrete flow of material was presented by Jeong and Kim (1999). Their method transforms each workstation consisting of multiple parallel machines into an equivalent single machine workstation and, therefore, the A/D system with the parallel machines is converted into an A/D system with single machines. After the transformation, a decomposition algorithm is used to calculate the performance measures of the transformed system.
Tempelmeier and Burger (2001) examined nonhomogeneous asynchronous flow production systems and presented an analytical approximation for the performance of such systems. They assumed generally distributed stochastic processing times as well as breakdowns and imperfect production. The proposed approximation was based on the decomposition of an M-station-line into M-1 two-station-lines that were analyzed using a GI/G/1/[N.sub.max] queuing model. They also presented numerical comparisons with exact and simulation results which indicated that the procedure provides accurate results. In Kuhn (2003) an analytical approach was given for performance evaluation of an automated flow line system which considers the dependency between the production and the repair system. The proposed model and solution approach may be used in the initial design phase as well as during a redesign process in order to evaluate various configurations of the production and repair systems.
Cheah and Smith (1994) showed how a M/G/C/C state-dependent queuing model is embodied into the modelling of large-scale facilities where the blocking phenomenon can either be or not be controlled. They also presented an approximation technique based on the expansion method to incorporate the M/G/C/C queuing models into series, merge and splitting topologies of production lines. Jain and Smith (1994) presented an analytical technique to calculate system performance measures of M/M/C/K queuing networks. They analyzed series, merge and splitting topologies and in addition they explored the optimal order of the M/M/C/K servers in such systems.
All the above papers employed approximate techniques for flow lines with multiple stations. Two studies that employ exact methods are Buzacott and Shanthikumar (1993) and Vidalis and Papadopoulos (2001). Finite buffer tandem queuing models with multiple stations per stage and exponential processing times were examined by Buzacott and Shanthikumar (1993) in their excellent book. They developed an algorithm for the performance evaluation of such systems. Vidalis and Papadopoulos (2001) developed a recursive algorithm for generating the transition matrices of multi-station multi-server exponential reliable queueing networks. Their algorithm can generate the transition matrix of an N-station network for any number of stations, N. This process can be used to calculate exactly the system throughput.
There have been a number of studies on flow lines with multiple stations. Hillier and So (1996) examined how the server and workload allocation affect the multi-servers throughput of serial production lines. They studied the simultaneous optimization of server and workload allocation to maximize the throughput of the production line, by applying a Markovian analysis to evaluate the throughput of the lines. Magazine and Stecke (1996) considered small flow lines with two and three workstations consisting of parallel machines. They examined how the throughput of such systems may be improved if specific parameters of the system such as the allocation of machines among the workstations, allocation of workload to the workstations and buffer allocation between workstations vary. The optimal allocation of parallel servers with different non-exponential service time distributions at each workstation was considered by Futamura (2000). The effect of the coefficient of variation (cv) of the service time distribution on the throughput of systems, where cv varies from one workstation to another, was examined. A generalised queuing network algorithm, the expansion method, developed by Kerbache and Smith (1987), was used as an evaluative procedure in conjunction with simulated annealing for optimizing large production lines configurations with parallel server facilities by Spinellis et al. (2000). The buffer allocation as well as the server allocation problem...
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