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Such as:
⎧
⎨
P
N
1
ξ
1
=1
−
P
K−
1
0
ξ
K
=1
−
(23)
⎩
P
j−
0
)
P
j
+1
N
j
+1
∀
j
=2
...K
−
1
,ξ
j
=(1
−
×
(1
−
)
Similarly to the two-machine-one-buffer model, the throughput of the production
line
ψ
is defined as the bottleneck between the effective production rates of all
machines:
ξ
i
μ
i
+
ξ
i
×
μ
i
×
ψ
=min
{
ω
i
×
λ
i
}
,i
=1
...K .
(24)
The equivalent machine method proposed in this paper to evaluate the system
throughput of a buffered serial production line is summarized by the non-linear
programming algorithm given below (see Algorithm 1). This algorithm can be
solved by LINGO software.
Algorithm 1.
Aggregated Equivalent Machine Method
K
Number of machines
K −
1 Number of buffers
M
i
Number of failure modes of machine
M
i
λ
im
Failure rate of machine
M
i
in mode
m
μ
im
Repair rate of machine
M
i
in mode
m
ω
i
Processing rate of machine
M
i
N
j
Capacity of buffer
B
j
for all
each machine
M
i
do
λ
i
←
M
(
i
)
m
=1
Require:
λ
im
μ
i
←
1
M
(
i
)
m
=1
λ
i
λ
im
×
1
μ
im
end for
for all
each buffer
B
j
do
for all
each machine
M
i
do
P
j
←
the steady probability that the buffer
j
is empty
P
N
j
←
the steady probability that the buffer
j
is full
α
j
←
the processing rate ratio related to the buffer
j
ρ
i
←
the equivalent throughput of the machine
i
end for
end for
return
ψ
=min
{ρ
i
;
i
=1
...K}
5 Numerical Results
The proposed method has been validated by a comparative study based on both a
simulation and an aggregation approach proposed by Belmansour and Nourelfath
