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of parts produced by the last machine of a manufacturing system over a given
period of time [1]. Throughput can be defined also in the steady state of the
system as the average number of parts produced per time unit.
Due to this crucial importance, analytical methods to investigate the per-
formance of production lines have been widely studied in the literature. An
interesting literature review which summarizes the recent studies devoted to
developing these analytical methods has been parented by Li et al. [10]. An im-
portant conclusion of this study is that most of these analytical methods are
based on aggregation or decomposition techniques. The first technique consists
of replacing a two-machine line by a single equivalent machine that has the same
throughput in isolation. Then, the obtained machine is combined with the next
machine to obtain a new aggregated machine. This technique can be done in
a forward as well as in a backward direction. One of its earliest applications
is that of Buzacott [4], who applied this method to a fixed-cycle three-stage
production line. De Koster [7] tested the aggregation approach to estimate the
eciency of an unreliable line with exponential failure and repair times. The
second technique consists of decomposing the longer production line into a set
of two-machine lines when the analysis of these subsystems is available [10]. The
set of two-machine lines is assumed to have equivalent behavior to the initial
production line. This method was proposed for the analysis of discrete mod-
els of long homogeneous lines by Gershwin [8]. The computational eciency
of this method was later improved by the introduction of the so-called DDX
algorithm [5,6].
Recently, Ouazene et al. [12] have proposed an equivalent machine method
based on the analysis of a serial production line using a combination of birth-
death Markov process to analyze the behavior of the different buffers in theirs
steady states. This approach allows the analysis of the serial production line
without decomposing or aggregating this line into smaller sub-lines.
Most of these methods deal with unreliable production lines considering single
failure-machines. Or, in many practical cases, machines can be subject to differ-
ent kinds of failures which occur with different frequencies and require different
amounts of time to be repaired [3]. Levantesi [9] proposed an analytical method
able to deal both with non-homogeneous lines and multiple failure modes for
each machine considering a discrete flow of parts. Their method is based on the
decomposition method introduced by Gershwin [8] which is an approximate eval-
uation of
K
-machine line based on the evaluation of
K
1 buffered two-machine
sub-lines. This method is also an extension of the decomposition technique based
on a multiple failure approach previously proposed by Tolio et al. [15] for the
analysis of exponential lines. Another method that considers performance eval-
uation of a tandem homogeneous production line with machines having multiple
failure modes has been studied by Belmansour and Nourelfath [2,3]. This ap-
proach consists of an aggregation technique. The aggregation algorithm is a
recursive approach that replaces each dipole (two-machine on buffer model) by
a single machine. Based on a comparative study with the simple aggregation
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