Production
quality
performance
in
manufacturing
systems
processing
deteriorating
products
Marcello
Colledani
(2)
a,
*,
Andras
Horvath
b
,
Alessio
Angius
b
a
Politecnico
di
Milano,
Department
of
Mechanical
Engineering,
Via
la
Masa,
1,
20156,
Milan,
Italy
b
Universita
`di
Torino,
Department
of
Computer
Science,
Corso
Svizzera
185,
Torino,
Italy
1.
Introduction,
motivation
and
objectives
Production
quality
has
been
proposed
recently
as
an
emerging
paradigm
to
achieve
desired
service
levels
of
conforming
products
in
advanced
manufacturing
systems,
by
simultaneously
consider
ing
quality
and
productivity
requirements
[1].
With
respect
to
this
background,
the
importance
of
an
integrated
analysis
of
production
logistics,
product
quality
and
equipment
maintenance
to
achieve
balanced
manufacturing
system
solutions
has
been
pointed
out.
This
problem
is
particularly
relevant
in
manufacturing
systems
producing
deteriorating
products.
Product
quality
and
value
deterioration
due
to
excessive
residence
times
(or
lead
times)
during
production
is
a
signiﬁcant
phenomenon
in
several
technology
intensive
industries,
including automotive,
food
manufacturing,
semiconductor
and
electronics
manufacturing
and
in
polymer
forming.
For
example,
in
automotive
paint
shops
a
car
body
that
is
affected
by
prolonged
exposure
to
the
air
in
the
shop
ﬂoor
caused
by
excessive
lead
times
between
operations,
is
prone
to
particle
contamination,
leading
to
unacceptable
quality
of
the
output
of
the
painting
process.
Moreover,
food
production
is
pervaded
by
strict
requirements
on
hygiene
and
delivery
precision
requiring
a
maximum
allowed
storage
time
before
packaging.
If
the
production
lead
time
exceeds
this
limit,
the
product
has
to
be
considered
as
defective
and
cannot
be
delivered
to
the
customer.
In
these
systems,
higher
inventory
increases
the
system
throughput
but
also
increases
the
production
lead
times,
thus
increasing
the
probability
of
producing
defective
items.
Therefore,
a
relevant
tradeoff
is
generated
between
production
logistics
and
quality
performance
that
requires
advanced
engineering
methods
to
be
proﬁtably
addressed.
In
spite
of
the
relevance
of
this
phenomenon
in
industry,
the
analysis
of
production
quality
performance
under
product
deterioration
has
received
relatively
low
attention
in
the
literature.
The
manufacturing
system
is
considered
in
a
highly
aggregate
way
in
advanced
Economic
Production
Quantity
(EPQ)
models
considering
quality
deterioration
[2].
In
these
works,
the
quality
deterioration
due
to
the
parts
residence
time
along
the
stages
of
the
manufacturing
system
is
neglected.
Other
works
considered
supply
chain
coordination
mechanisms
in
presence
of
product
obsolescence
[3].
Furthermore,
production
control
policies
based
on
WIP
[4]
and
part
release
[5,6]
regulation
could
support
the
achievement
of
improved
production
quality
performance
under
product
deterioration,
although
they
do
not
provide
mechanisms
to
directly
control
production
lead
times.
The
ﬁrst
model
considering
this
interaction
is
proposed
in
[7]
that
analyzed
unbuffered
systems
where
the
material
under
processing
is
scrapped
after
long
machine
failures.
Moreover,
the
performance
of
serial
lines
with
product
deterioration
is
considered
under
Bernoulli
reliability
models
of
production
stages
in
[8],
and
in
small
two
machine
lines
in
[9].
While
all
these
works
are
important
to
shed
light
on
the
problem,
they
do
not
provide
methods
to
predict
and
control
production
lead
time
distributions
under
realistic
manufacturing
system
features.
As
a
matter
of
fact,
a
methodology
to
support
the
design
of
manufacturing
systems
under
lead
time
dependent
product
deterioration
that
integrates
quality
and
production
logistics
implications
has
never
been
proposed.
Important
questions
like
‘‘What
is
the
impact
of
buffers
on
the
production
rate
of
conforming
products
with
product
quality
deterioration?’’
remain
unsolved,
resulting
in
subperforming
system
conﬁgurations.
To
overcome
these
limitations,
in
this
paper
an
integrated
model
of
manufacturing
systems
affected
by
product
deterioration
and
a
new
method
for
the
prediction
of
the
production
lead
time
distribution
and
the
throughput
of
conforming
products
in
these
CIRP
Annals

Manufacturing
Technology
64
(2015)
431–434
A
R
T
I
C
L
E
I
N
F
O
Keywords:
Manufacturing
system
Quality
Deteriorating
product
A
B
S
T
R
A
C
T
In
several
manufacturing
contexts
including
food
industry,
semiconductor
manufacturing,
and
polymers
forming,
the
product
quality
deteriorates
during
production
by
prolonged
exposure
to
the
air
caused
by
excessive
lead
times.
Buffers
increase
the
system
throughput
while
also
increasing
the
production
lead
time,
consequently
affecting
the
product
quality.
This
paper
proposes
a
theory
and
methodology
to
predict
the
lead
time
distribution
in
multistage
manufacturing
systems
with
unreliable
machines.
The
method
allows
to
optimally
set
inventory
levels
to
achieve
target
production
quality
performance
in
these
systems.
The
industrial
beneﬁts
are
demonstrated
in
a
real
manufacturing
system
producing
microcatheters
for
medical
applications.
ß
2015
CIRP.
*
Corresponding
author.
Email
address:
marcello.colledani@polimi.it
(M.
Colledani).
Contents
lists
available
at
ScienceDirect
CIRP
Annals

Manufacturing
Technology
journal
homepage:
http://ees.elsevier.com/cirp/default.asp
http://dx.doi.org/10.1016/j.cirp.2015.04.122
00078506/ß
2015
CIRP.
systems
are
developed
for
the
ﬁrst
time.
This
approach
allows
setting
inventory
levels
to
achieve
desired
production
quality
performance
in
real
systems.
2.
System
description
The
considered
system
is
formed
by
K
manufacturing
stages
and
K
1
buffers
of
ﬁnite
capacity,
conﬁgured
in
serial
layout
(Fig.
1).
Stages
are
denoted
as
M
k
,
with
k
=
1,
.
.
.,
K,
and
buffers
are
denoted
as
B
k
,
with
k
=
1,
.
.
.,
K
1.
The
capacity
of
buffer
B
k
is
N
k
,
that
is
an
integer
number.
Finite
capacity
buffers
can
either
model
physical
conveyors
or
the
implementation
of
tokenbased
production
control
rules,
such
as
kanban,
regulating
the
material
ﬂow
release
at
each
stage
[10].
Single
stage
model.
The
dynamics
of
each
stage
is
modeled
by
a
discretetime
and
discretestate
Markov
chain
of
general
complexi
ty.
This
setup
allows
to
analyze
a
wide
set
of
different
stage
models
within
a
unique
framework.
For
example,
stages
with
unreliable
machines
characterized
by
generally
distributed
up
and
down
times
and
also
stages
with
nonidentical
processing
times
can
be
considered
within
the
same
framework,
thus
making
the
proposed
approach
applicable
to
a
wide
set
of
real
manufacturing
systems.
In
detail,
each
stage
M
k
is
represented
by
I
k
states,
and
thus
the
state
indicator
a
k
assumes
values
in
{1,
.
.
.,
I
k
}.
The
set
containing
all
the
states
of
M
k
is
called
S
k
.
The
dynamics
of
each
stage
in
visiting
its
states
is
captured
by
the
transition
probability
matrix
l
k
,
that
is
a
square
matrix
of
size
I
k
.
Moreover,
a
quantity
reward
vector
m
k
is
considered,
with
I
k
binary
entries.
While
in
the
generic
state
i,
M
k
produces
m
k,i
parts
per
time
unit.
Therefore,
state
i
with
m
k,i
=
1
can
be
considered
as
an
operational
state
for
stage
M
k
,
while
state
i
with
m
k,i
=
0
is
a
down
state
for
stage
M
k
.
Material
ﬂow
dynamics.
A
discrete
ﬂow
of
parts
is
considered
in
the
system.
Stage
M
k
is
blocked
if
the
buffer
B
k
is
full.
Stage
M
k
is
starved
if
the
buffer
B
k1
is
empty.
Stage
M
K
is
never
blocked,
i.e.,
inﬁnite
amount
of
space
is
available
to
store
ﬁnished
products.
Stage
M
1
is
never
starved,
i.e.,
unlimited
supply
of
raw
parts
is
assumed.
Operational
Dependent
Transitions
are
considered,
i.e.,
a
machine
cannot
make
transitions
to
other
states
if
it
is
starved
or
blocked.
State
transitions
take
place
at
the
beginning
of
the
time
unit
and
buffer
levels
are
updated
at
the
end
of
the
time
unit.
Part
quality
deterioration.
The
quality
of
parts
deteriorates
with
the
time
parts
spend
in
a
critical
portion
of
system,
denoted
by
two
integers,
e
and
q
with
1
e
<
q
K,
and
composed
of
those
buffers
that
are
between
stages
M
e
and
M
q
.
The
lead
time
of
a
part
is
the
time
spent
in
buffers
B
e
,
B
e+1
,
.
.
.,
B
q1
.
For
example,
for
the
system
in
Fig.
1,
e
=
2
and
q
=
4.
If
e
=
1
and
q
=
K,
then
the
time
spent
in
the
whole
system
is
considered.
The
probability
that
a
part
is
defective
at
the
end
of
the
line
is
a
nondecreasing
function
of
its
lead
time.
The
function
g(h)
indicates
the
probability
that
a
part
released
by
the
system
is
defective
given
that
it
spent
h
time
units
in
the
critical
portion
of
system.
Defective
parts
are
scrapped
at
the
end
of
the
line.
Performance
measures.
The
main
performance
measures
of
interest
are:
Average
total
production
rate
of
the
system,
denoted
by
E
Tot
,
Probability
that
the
lead
time,
LT,
is
equal
to
a
given
number
of
time
units,
h,
i.e.,
P(LT
=
h),
Average
effective
production
rate
of
conforming
parts,
E
Eff
,
which
is
given
by:
E
E
f
f
¼
E
Tot
X
1
h¼1
PðLT
¼
hÞ½1
gðhÞ
(1)
System
yield,
Y
system
,
i.e.
fraction
of
conforming
parts:
(E
Eff
/E
Tot
).
Total
average
inventory
of
the
system,
WIP.
3.
Lead
time
distribution
evaluation
In
this
section,
an
efﬁcient
and
exact
analytical
method
to
compute
the
distribution
of
the
lead
time
in
the
critical
portion
of
the
system
is
described.
The
rationale
of
the
approach
is
explained
in
the
following.
Firstly,
the
probability
that,
at
the
moment
when
a
randomly
selected
part
enters
buffer
B
e
,
the
system
is
in
a
given
state
is
derived.
Secondly,
the
probability
that,
given
that
state
of
the
system
as
initial
condition,
the
last
part
in
buffer
B
e
crosses
the
critical
portion
of
the
system
in
h
time
units
is
analyzed.
By
taking
the
product
of
these
two
quantities
and
summing
for
all
possible
states,
the
distribution
of
the
lead
time
in
the
critical
portion
of
the
system
can
be
reconstructed.
In
the
following,
the
steps
of
this
procedure
are
brieﬂy
described.
Starting
from
the
parameters
of
the
stages
and
according
to
the
modeling
assumptions,
a
discrete
time
Markov
chain,
denoted
by
D
1
,
can
be
determined
that
characterizes
the
overall
behavior
of
the
system.
A
state
of
D
1
is
described
by
a
vector
s
1
=
(n
1
,
n
2
,
.
.
.,
n
K1
,
a
1
,
a
2
,
.
.
.,
a
K
)
where
n
k
is
the
number
of
parts
in
buffer
B
k
and
a
k
is
the
state
of
machine
;
k
.
The
set
of
all
the
system
states
is
denoted
by
V
1
and
the
transition
probability
matrix
by
P
1
.
The
entries
of
P
1
can
be
obtained
as
shown
in
[11].
The
row
vector
of
the
steady
state
probabilities,
denoted
as
p
1
,
and
the
total
production
rate,
E
Tot
,
can
be
calculated
as
in
[11,12].
The
subsystem
including
only
that
portion
of
line
that
is
downstream
machine
M
e
is
considered
next.
This
subsystem
comprises
buffers
B
e
,
B
e+1
,
.
.
.,
B
K1
and
machines
M
e+1
,
M
e+2
,
.
.
.,
M
K
.
Machines
;
1
,
.
.
.,
;
e
are
not
considered
because,
once
a
part
is
put
in
buffer
B
e
,
these
machines
have
no
impact
on
its
lead
time.
For
this
subsystem,
a
Markov
chain,
denoted
by
D
2
,
with
initial
probability
vector
p
2
and
transition
probability
matrix
P
2
is
determined.
D
2
can
be
considerably
reduced
with
respect
to
D
1
.
A
state
of
this
model
is
given
by
a
vector
s
2
=
(n
e
,
n
e+1
,
.
.
.,
n
K1
,
a
e+1
,
a
e+2
,
.
.
.,
a
K
).
The
set
of
states
of
D
2
will
be
denoted
by
V
2
.
Note
that,
since
no
machine
feeds
parts
in
the
ﬁrst
buffer
of
this
model,
all
parts
will
eventually
leave
and
the
system
will
get
empty.
In
other
words,
D
2
is
an
absorbing
Markov
chain
[12].
The
subset
of
states
in
S
2
in
which
there
are
no
parts
in
the
buffers
of
the
critical
portion
B
e
,
B
e+1
,
.
.
.,
B
q1
will
be
denoted
by
V
0
.
The
lead
time
in
the
critical
portion
of
system
can
be
expressed
as
follows:
PðLT
hÞ
¼X
s
2
2
V
2
p
2;s
2
f
ðs
2
;
hÞ
(2)
where
p
2;s
2
is
the
probability
that
a
random
part
that
enters
in
buffer
B
e
ﬁnds
the
system
in
state
s
2
and
f(s
2
,h)
is
the
probability
that
a
part
that
enters
the
critical
portion
in
state
s
2
leaves
machine
M
q
in
at
most
h
time
units.
The
ﬁrst
term
in
Eq.
(2)
is
obtained
by
properly
mapping
the
states
s
1
in
D
1
into
the
states
s
2
of
the
sub
system
in
D
2
.
For
this
reason
the
matrix
F
of
binary
entries
is
deﬁned
that
links
the
states
in
s
1
and
s
2
if
the
states
of
all
the
stages
M
j+1
and
buffers
B
j
,
with
j
>
e,
are
the
same.
In
other
words,
the
initial
probability
of
the
state
s
2
in
D
2
,
p
2;s
2
,
has
to
be
equal
to
the
sum
of
the
probabilities
that
a
part
is
put
in
the
buffer
B
e
in
the
connected
set
of
states
s
1
in
D
1
.
Therefore:
p
2
¼p
1
Q
1
E
Tot
F
(3)
where
Q
1
is
identical
to
P
1
,
except
that
those
transitions
that
do
not
cause
the
release
of
a
part
in
buffer
B
e
are
set
to
zero.
The
second
term
in
(2)
can
be
easily
computed
as
the
distribution
of
the
ﬁrst
passage
time
to
V
0
in
the
absorbing
Markov
chain
D
2
[12].
The
probability
that
the
lead
time
is
exactly
h
is
given
by
P(LT
=
h)
=
P(LT
n)
P(LT
n
1).
The
other
performance
measures
deﬁned
in
Section
2
can
be
derived
based
on
the
lead
time
distribution
and
the
total
throughput.
Fig.
1.
Representation
of
the
analyzed
manufacturing
system.
M.
Colledani
et
al.
/
CIRP
Annals

Manufacturing
Technology
64
(2015)
431–434
432
4.
Numerical
results
and
system
behavior
In
the
ﬁrst
experiment,
the
distribution
of
the
lead
time
in
multistage
manufacturing
systems
is
investigated.
A
production
line
with
ﬁve
stages
is
considered
with
parameters
reported
in
Table
1.
The
stages
correspond
to
machines
with
one
operational
state
and
one
failure
state,
with
failure
probability
p
=
1/MTTF
and
repair
probability
r
=
1/MTTR
(MTTF
is
Mean
Time
to
Failure
and
MTTR
is
Mean
Time
to
Repair).
The
machines
produce
one
part
per
time
unit
if
operational.
The
results
are
reported
in
Fig.
2.
The
following
considerations
hold:
The
distribution
of
the
lead
time
is
multimodal
with
a
number
of
peaks
equal
to
the
number
of
production
stages,
K.
The
vth
peak,
with
v
¼
1;
.
.
.
;
K,
is
located
in
correspondence
to
the
following
number
of
time
units,
h
v
:
h
v
¼
K
v
þX
v1
i¼1
ðN
i
1Þ
(4)
The
peaks
appear
for
the
following
reasons.
The
ﬁrst
peak
corresponds
to
the
situation
where
the
system
is
empty
and
the
parts,
after
being
processed
by
stage
M
1
,
cross
the
remaining
K
1
stages,
each
one
spending
one
time
unit
for
processing.
The
vth
peak,
with
v
¼
2;
.
.
.
;
K,
corresponds
to
the
situation
where
stage
M
v
failed
for
sufﬁciently
long
time
to
ﬁll
up
all
the
upstream
buffers
and
empty
all
the
downstream
buffers.
After
the
stage
is
recovered
from
the
failure,
an
incoming
part
has
to
wait
N
i
1
time
units
in
the
upstream
buffers,
with
i
<
v,
and
1
time
unit
in
each
of
the
K–v
downstream
stages,
if
no
other
failure
occurs.
The
knowledge
of
this
behavior
can
be
exploited
to
correctly
operate
and
design
the
production
line.
In
the
second
experiment,
the
impact
of
the
buffer
size
on
the
effective
throughput,
E
Eff
,
is
investigated
for
a
simple
system
with
two
machines
and
one
buffer,
with
parameters
reported
in
Table
1.
The
quality
deterioration
function
g(h)
assumes
value
zero
for
h
=
1,
.
.
.,
h*
and
value
1
for
h
>
h*,
with
h*
=
40.
In
other
words,
if
the
parts
spend
more
than
40
time
units
in
the
system,
they
turn
into
defective.
In
the
experiment,
the
buffer
capacity
N
is
varied
from
5
to
50.
Results
are
reported
in
Fig.
3.
The
following
considerations
hold:
The
effective
throughput
curve
is
a
concave
function
presenting
a
maximum
corresponding
to
a
buffer
capacity
equal
to
25.
This
behavior
is
explained
in
the
following.
It
is
well
known
that
the
total
throughput
of
the
system
is
monotonically
increasing
with
the
buffer
size,
with
a
marginal
gain
that
reduces
while
increasing
the
buffer
size.
At
the
same
time,
if
the
ﬁrst
machine
is
more
reliable
than
the
second,
while
increasing
the
buffer
size
also
the
average
lead
time
increases,
as
the
average
inventory
increases
almost
linearly.
Therefore,
the
scrap
rate
will
increase
as
a
function
of
the
buffer
size
since
the
probability
that
the
lead
time
exceeds
the
threshold
increases.
The
result
of
this
combined
effect
is
the
following.
When
the
ﬁrst
phenomenon
is
more
relevant
(small
buffers),
the
effective
throughput
curve
increases
with
the
buffer
size.
When
the
second
phenomenon
is
more
relevant
(large
buffers),
the
effective
throughput
curve
starts
decreasing,
as
the
system
produces
more
parts
but
also
a
higher
fraction
of
defective
parts.
Therefore,
a
tradeoff
is
generated
that
is
optimized
for
a
speciﬁc
buffer
size.
This
optimal
buffer
size
could
not
be
found
without
the
proposed
approach.
It
is
worth
to
note
that
for
N
>
h*
+1
the
curve
drops
abruptly
in
a
discontinuous
way
and
shows
a
different
slope.
This
phenome
non
is
due
to
the
fact
that,
according
to
equation
(4),
the
lead
time
distribution
curve
in
a
twomachine
line
shows
a
probability
peak
at
level
N1.
Therefore,
operating
a
system
with
N1
>
h*
does
not
make
sense
in
practical
settings.
The
third
experiment
shows
the
impact
of
the
repair
probability
of
the
ﬁrst
machine
on
the
effective
throughput,
for
different
buffer
sizes
in
a
line
with
two
machines
and
one
buffer.
Results
are
reported
in
Fig.
4.
The
following
considerations
hold:
For
all
the
cases,
it
is
possible
to
notice
that
faster
repair
operations
are
not
always
beneﬁcial
in
terms
of
effective
throughput
of
the
system.
Indeed,
a
faster
repair
of
the
ﬁrst
machine
increases
the
total
throughput
of
the
system
but
also
increases
the
amount
of
inventory
stored
in
the
buffer,
as
the
buffer
will
be
more
frequently
full.
This
phenomenon
increases
the
lead
time
and,
consequently,
the
scrap
rate.
This
behavior
is
more
prominent
for
large
buffers.
Indeed,
in
these
cases,
the
positive
effect
of
shorter
repair
time
on
the
total
throughput
is
less
signiﬁcant
than
the
negative
effect
on
the
lead
time
increase.
This
analysis
suggests
that
improvement
actions
that
are
beneﬁcial
on
the
total
throughput
of
the
system
(faster
repair
operations)
can
be
detrimental
on
the
effective
throughput,
if
the
products
are
perishable.
Thus,
neglecting
this
aspect
can
lead
to
subperforming
system
conﬁgurations.
5.
Real
case
study
The
proposed
approach
has
been
applied
to
the
production
of
microcatheters
as
high
value
medical
products
for
the
aging
society
in
the
medical
technology
sector
at
ENKI
s.r.l.
[13].
The
manufacturing
process
is
composed
of
three
main
phases:
(i)
Table
1
Data
of
the
systems
analyzed
in
the
experiments.
Exp.
no.
System
data
1Balanced
line:
l
k
¼1
p
k
p
k
r
k
1
r
k
;
m
k
¼
½1;
0
with
p
k
=
0.01
and
r
k
=
0.1,
N
1
=
N
2
=
N
3
=
N
4
=
10.
e
=
1,
q
=
K
=
5.
2
Same
machine
model
as
exp.
1
p
1
=
0.01;
r
1
=
0.05;
p
2
=
0.02;
r
2
=
0.2.
N
1
=
[5:50].
e
=
1,
q
=
K
=
2.
3
Same
machine
model
as
exp.
1
p
1
=
0.01;
r
1
=
0.25;
p
2
=
0.02;
r
2
=
[0.02:0.5].
e
=
1,
q
=
K
=
2.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
Pr(LT=h)
h
Fig.
2.
Lead
time
distribution
for
a
ﬁve
machine
line.
0.7
0.72
0.74
0.76
0.78
0.8
0.82
5 10 15 20 25 30 35 40 45 50
Effective Throughput, E
Eff
Buffer size, N
Fig.
3.
Effect
of
the
buffer
size
on
the
effective
throughput.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Effective Throughput, E
Eff
r
1
N=20
N=30
N=40
N=50
Fig.
4.
Effect
of
the
repair
probability
on
the
effective
throughput.
M.
Colledani
et
al.
/
CIRP
Annals

Manufacturing
Technology
64
(2015)
431–434
433
material
compound
preparation
and
control,
the
(ii)
micro
extrusion
of
the
microtubes
and
(iii)
ﬁnal
microcatheter
assembly.
Defects
are
mainly
geometrical,
generated
within
the
microextrusion
process.
The
above
defects
lead
to
extremely
high
defect
rates
(up
to
70%
in
standard
production).
During
the
phase
of
preparation
of
the
raw
material,
in
forms
of
granules,
the
moisture
level
is
measured
and
if
the
value
exceeds
a
ﬁxed
limit,
the
granules
are
dried
to
reduce
the
moisture
level
before
the
downstream
microextrusion
process.
The
lead
time
between
the
drying
and
extrusion
processes
should
not
exceed
a
certain
limit
to
avoid
increase
of
the
moisture
level
by
exposure
to
the
air.
A
maximal
moisture
level
of
300
ppm
is
usually
the
accepted
limit.
Excessive
moisture
level
affects
the
material
viscosity.
This
results
in
ﬂuctuations
of
the
material
shear
rate
and
instability
in
the
extrusion
process,
thus
affecting
the
quality
of
the
tube
section.
The
line
under
analysis
is
composed
of
three
main
stages.
Stage
M
1
is
the
compounding
stage
provides
additives
to
the
material
and
makes
the
size
of
the
pellets
homogeneous.
Stage
M
2
is
the
drying
stage,
which
reduces
the
moisture
level
of
the
material.
The
residence
time
of
the
material
in
the
dryer
depends
on
the
material
type
and
properties.
For
example,
for
PEBA
pellets
the
drying
time
is
usually
4
h.
After
the
drying
process,
the
material
is
loaded
in
the
input
hopper
of
the
extruder
at
stage
M
3
.
The
extrusion
speed
selection
is
one
of
the
key
competences
of
the
company,
directly
affecting
the
output
quality
of
the
tube.
It
is
strongly
dependent
on
the
material
type
and
tube
section
complexity,
usually
ranging
between
3
m/s
and
20
m/s.
After
the
extrusion,
each
tube
undergoes
a
set
of
very
severe
EOL
(End
of
Line)
inspections
and
nonconforming
tubes
are
scrapped.
However,
the
company’s
strategic
goal
is
to
avoid
the
generation
of
defects
by
improving
the
control
of
the
material
properties
before
the
extrusion
process.
For
this
reason,
the
lead
time
between
the
end
of
the
drying
process
and
the
feeding
of
the
granules
in
the
extruder
hoppers
must
be
limited
to
avoid
gradual
increase
in
the
humidity
on
the
granules.
After
a
deep
study
of
the
material
behavior
in
extrusion,
the
company
assessed
that
the
maximal
residence
time
of
the
material
between
these
stages
is
45
s.
Therefore,
the
main
objective
of
the
analysis
is
to
properly
design
the
inprocess
buffer
included
between
the
drying
stage
and
the
extrusion
stage
in
order
to
decrease
the
fraction
of
defects
due
to
material
deterioration
in
extrusion,
thus
increasing
the
effective
production
rate
levels.
The
microtube
forming
line
under
the
current
conﬁguration
has
been
modeled.
Two
types
of
tubes
have
been
considered
with
parttype
dependent
process
parameters.
The
number
of
lumens
ranges
from
4
to
26
and
the
external
diameters
ranges
from
2
mm
to
5
mm.
Timeconsuming
setup
operations
are
required
while
switching
between
part
types;
therefore
different
production
settings
have
been
considered,
one
for
each
tube
type.
The
operational
and
down
times
were
provided
by
the
company.
The
performance
of
the
system
was
evaluated
and
the
results
were
validated
through
comparison
with
historical
production
data,
showing
deviations
always
below
5%.
The
approach
proposed
in
this
paper
has
been
applied
to
design
a
different
production
control
policy
to
manage
the
available
hopper
capacity
before
the
extrusion
machine.
In
detail,
for
each
tube
type,
the
hopper
capacity
that
maximizes
the
effective
throughput
of
the
system,
under
a
constraint
on
the
maximum
hopper
capacity
currently
available
in
the
line
(1500
g),
has
been
investigated.
Results
are
reported
in
Table
2.
Similarly
to
what
was
shown
for
experiment
2
in
Section
4,
for
each
case,
the
effective
throughput
is
maximized
for
a
hopper
capacity
that
is
smaller
than
the
currently
adopted
capacity.
With
the
proposed
optimal
policy,
in
all
the
cases
only
1%
of
the
tubes
is
defective
due
to
excessive
lead
time,
against
the
6%
and
17%,
respectively,
for
each
tube
type,
in
the
current
conﬁguration
(a
reduction
of
more
then
80%
for
both
types).
Since
defective
tubes
cannot
be
repaired
and
are
sent
to
recycling,
this
solution
entails
a
large
environmental
beneﬁt.
Moreover,
beneﬁts
are
achieved
in
terms
of
effective
throughput
(3.47%
and
16.8%),
while
also
reducing
the
work
in
progress
in
the
system
(50%
as
a
minimum).
This
directly
translates
into
a
reduction
of
production
costs,
thus
positively
impacting
on
the
company’s
competitiveness.
6.
Conclusions
This
paper
proposes
a
modeling
framework
and
a
methodology
to
predict
the
lead
time
distribution
in
multistage
manufacturing
systems.
The
lead
time
distribution
is
used
to
compute
the
system
production
quality
performance,
when
products
deteriorate.
The
theoretical
work
developed
in
this
paper
opens
the
way
for
applications
in
several
different
ﬁelds.
For
example,
supply
chain
management
approaches
can
beneﬁt
from
the
availability
of
a
solution
to
the
problem
of
deriving
the
lead
time
distribution
in
unreliable
manufacturing
systems.
Moreover,
production
planning
approaches
that
take
into
account
the
exact
distribution
of
the
residence
time
of
parts
in
the
system
can
be
developed.
Finally,
new
production
control
policies
to
provide
a
direct
control
on
the
lead
time
can
be
elaborated
from
these
results.
Acknowledgements
The
authors
would
like
to
thank
Eng.
Moreno
Camanzi
and
Mr.
Mario
Di
Cecio
from
ENKI
s.r.l.
for
the
support
in
this
research.
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Microtube
production
line
performance
under
optimized
buffer
size
(the
time
unit
is
omitted
for
conﬁdentiality
reasons).
Tube
type
Buffer
size
(g)
E
Tot
Y
system
E
Eff
WIP
(g)
Scrap
rate
1
N
Current
=
1500
0.862
0.94
0.812
884
0.06
N
opt
=
783
0.849
0.991
0.84
448
0.009
Difference
D
%
1.5%
+5%
+3.47%
49.3%
82%
2
N
Current
=
1500
0.868
0.829
0.719
901.7
0.17
N
opt
=
591
0.85
0.991
0.842
338
0.009
Difference
D
%
2%
+19.3%
+16.8%
62.5%
93.9%
M.
Colledani
et
al.
/
CIRP
Annals

Manufacturing
Technology
64
(2015)
431–434
434