TWO-STAGE PRODUCTION PLANNING IN A DYNAMIC ENVIRONMENT
Stephen C. Graves, Harlan C. Meal
Sriram Dasu, Yuping Qui
August 1985
1698-85
2
INTRODUCTION AND OVERVIEW
In this document we report on research to develop and study
mathematical
models for production smoothing in a dynamic production
environment.
This effort was part of a larger study whose goal
to investigate
the production planning practices for an electronic
equipment manufacturing firm, and
mechanisms
was
for improvement.
in particular to explore
To motivate
research, we need to indicate
possible
the presentation of our
the nature of this production
environment.
The production process can be viewed as a multi-stage system.
In
its most aggregate form there are two
fabrication and equipment assembly.
uncertainty and variability over
time.
Nevertheless,
stages:
component
Customer demand has significant
the cumulative manufacturing
lead
customers demand a high level of service.
manufacturing process
is not totally reliable;
in particular.
is significant yield uncertainty in component fabrication.
The
there
Finally,
production capacity for both component fabrication and assembly
(including final test)
is very capital
intensive.
As a consequence,
it is expensive and/or difficult to adjust the production level.
As a consequence of these characteristics,
make-to-stock system that would
and
reliable delivery times
hierarchical
one to set
one to
lots of
_VI__
·
planning system
to
smooth production and provide
customers.
schedule the daily lot starts
...
1_----_1_111___-
).
Our
study then
short
We considered a
(Hax and Meal,
the aggregate production level
item k
one must consider a
1975) with two
levels:
(e.g 400 lots/month)
(e.g. one
focused on
lot of item j,
the design of
and
two
such
a
3
make-to-stock system.
Key questions in
items should
should
the design of a make-to-stock system are what
be stocked,
at what point(s)
stock be accumulated,
provide
and what
satisfactory service and
smoothing.
in the
production process
stock level
is
needed to
permit adequate production
Our research addresses these questions.
To answer these
questions, we must specify the rules or algorithms that will set the
aggregate output rate and determine
develop mathematical models,
termed production smoothing models,
set the aggregate
output rate.
smoothing models,
see Silver 1967.)
"disaggregating" the
individual
items.
the daily production starts.
(For a review of
into production starts
for
for
these models determines the
level(s) necessary for a desired service
level.
to
-
We then give a mechanism
aggregate output
The analysis of
production
We
stock
In addition, from
these models one can quantify the tradeoff between the smoothness of
aggregate production and the stock level.
In the next
section we describe the production smoothing models
and give their analysis.
production system.
treat
We first present a model for
We then give
two extensions
a two-stage production system.
evaluate
their performance,
computational
to this model
the
to
models and
we report in section 3 on a
study based on data gathered from the electronic
equipment manufacturer.
a wide range of parameter
generate
To compare
a one-stage
insight
In particular,
we simulate
the models
over
settings to assess their accuracy and to
into their
behavior.
4
DEVELOPMENT AND ANALYSIS OF PRODUCTION SMOOTHING MODELS
We develop here a set of production smoothing models that can
be
used to
look at the
tradeoffs between
inventory requirements.
at an aggregate
processing
We assume that production smoothing is done
level for families
requirements.
production smoothing and
of
products that have similar
For now, we will make
the following
simplifications:
we ignore
lot-sizing considerations;
we assume no uncertainty
in the build
time for the
products;
we assume no constraint on
production capacity;
other
be introduced as needed.
The models
simplifications will
should
considerations
permit focus on the production
in the
light of forecast
variability, and yield uncertainty.
examine
the
smoothing/inventory
uncertainty, demand
We will use a simulation to
consequences of the simplifying assumptions.
We model the production activity in a very gross way.
not concerned with detailed scheduling issues,
planning issues.
decomposed
inventory
Figure
stage:
We assume that the production
into a series
all
Then we
assume a known and
production started
stage at time t+1,
for
system can be
in finished goods
fixed lead
(see
time for each
at a stage at time t finishes
being the lead
concerned both with setting
stage,
but rather with the
of stages where we may maintain an
between successive stages and
1).
We are
time of the stage.
the aggregate production
and with deciding how to disaggregate it
level
that
We are
for each
into a production
5
level
much
Furthermore, we want to understand how
for individual items.
is needed as
inventory
We desire
to keep the
aggregate production
possible at each stage,
levels as
is
smooth as
the safety stocks as
and to keep
needed.
low as
providing satisfactory customer service.
while
possible,
it
safety stock, and where
developed a production smoothing model to
illuminate
We have
these
tradeoffs.
We then consider production
We use the production
stage.
we first describe
that drives the production smoothing
forecast process
the aggregate
models.
smoothing models,
the production
To present
smoothing for
one production
smoothing model for one stage as the
building block for developing two distinct models for smoothing two
production stages.
Forecast Process
input
A key
of
aggregate demand.
the process
define
so
for
this
it is
forecast
revised
is the
forecast
is by month
To model
monthly.
we define Ft(s) as
t of aggregate demand in month s (s>t).
the
for
forecast
Then, we
St(s) by
= Ftl(S) + St(s)
Ft(s)
that
that
forecast revision,
of
made at month
We assume
12 months, and that
the next
up to
(1)
to the production smoothing process
St(s)
time s.
denotes the change at
We assume
that
following characteristics:
E [St(
)] = O ,
St(s)
time t in the
aggregate forecast
is a random variable with
the
6
Var
and
t(s)
and
2
=
[(s)]
t(s)
(s-t)
,
independent for all t #
are
t'
We denote the aggregate demand in month t as Ft(t):
'forecast'
of demand
in t made at time t.
From
(1),
the
we can express
the aggregate demand as
k
(2)
Ft(t)
= Ft-k(t)
+
6
E
t-k+i(t)
i=1
where Ftk(t)
ago.
is the initial forecast for demand
in t made k months
The k-month forecast error is given by
k
Ft(t)
- Ft-k(t)
=
&t-k+i(t)
E
i=l
k
and has zero mean and variance equal
to
E
2
a (k-i)
i=1
Thus,
the forecast not only is
unbiased, but also
k-1
= E
o2 (j).
j=o
improves over
time.
We assume that the demand process, D(t)
with E
and
[D(t)]
Ft(t)
From
= D, and Var
[D(t)]
=
= Ft(t),
D for all t.
is stationary
(We will use D(t)
interchangeably to denote aggregate demand in month t.)
(2) we now can compute
the variance of
Ftk(t):
k-1
Var
[Ft-k(t)]
=
-D
E
a2(j)
j=o
Thus,
the variance of the initial forecast depends upon the forecast
horizon length k.
The
longer is the forecast horizon,
is the forecast variance;
there
more
presumably,
for a
the smaller
longer forecast horizon,
is less information available on aggregate demand and there
likelihood to use D as the forecast.
is
7
Smoo'thing for One Stage
Production
We now develop a production smoothing model for one production
stage.
We assume that the production stage
produces it
a finished goods
stock, i.e.,
smoothing model
production
aggregate
to
family and
We use a
inventory.
set an aggregate production
to
starts per month)
produces one
rate
for the family for each month.
To set
the actual
production starts we must disaggregate the aggregate
production
plan according to
the net requirements
(e.g.
for individual
items.
production, we must maintain an
To smooth
the aggregate production rate,
aggregate demand rate,
over short
inventory stock since
being a smoothed average of
will deviate from the aggregate demand rate
time intervals.
production,
In
inventory
To determine the aggregate
the production
requirements we need to analyze the behavior of
smoothing model.
particular, we expect that the more we smooth
the greater will be the stock requirements.
customer service
is
the
But since
determined by the stocks for individual
we also need to understand how the aggregate plan is to
items,
be
disaggregated.
The production
(3)
P(t+l)
= n+
where t denotes
P(t+i)
[Ft(t+l,t+t+n)
the
current
-
is given by
P(t+l,t+t-1)
-
I(t)
+ SS]
time period, and
= the planned aggregate production
to
x
smoothing model
be completed by time t + i;
= the lead time for
production;
started at time t+i-I
9
rule in
its control
terms of
show that it
is both an
In this respect,
al
(1984).
it
is simple and does permit analysis of
In particular,
assumptions
Second,
we will
there
more general
is evidence that
be the
production capacity.
1974).
to express
all variables in
For instance,
in
common
to express planned
However, the actual inventory is likely
translate, using yield factors,
Thus, we need
this actual inventory into the
equivalent wafer starts for the items and
Similarly,
then aggregate over all
for the assembly of electronic
equipment, we might express
planned production
assembly hours;
fabrication facility, the units might be
machine
(4)
where
D(t)
I(t)
=
P(t)
is
for a parts
hours required
To analyze
·9
(3)
for the manufacture of
to be known in terms of chips for individual items.
family.
Third, while
the units need be aggregate measures of
production in wafer starts.
the
for
form.
integrated circuits it might be natural
in
form are optimal
form, we nevertheless can find the
parameter choice for this
In particular,
rule.
cost functions but also
1971,
et
its
implications fro this
(Schneeweiss
optimal
We need to be careful
items
consider.
suitable
linear rules of this
only quadratic
cost functions
the rule may not
units.
see that under
we obtain the disaggregation
or near optimal for not
optimal
interesting and reasonable rule to
it is very-similar to the study by Cruickshanks
First,
behavior.
and will try to
parameters, n and SS,
on the bottleneck
(3) we use the
I(t-1)
is the
+ P(t)
-
----_--1__
terms of
required
facility.
inventory balance
equation,
D(t)
actual family production
the family demand at
in
time t.
completed at time t and
We assume that
8
Ft(t+l,t+t+n)
P
= Ft(t+l)
+
...
(t+l,t+1-1) = P (t+l)
+
... + P
I(t)
+ Ft(t++n
)
;
(t+-1) ;
inventory on-hand at start of
= aggregate
time period t;
SS
= target safety stock level;
n
= window length.
Thus,
in period t we set P(t+l),
completed
periods
later.
the planned production to be
We assume that the
uncertainty in the
production yield
deterministic,
but that
so
production completed at time t+i will deviate
that actual
there is
lead time is
from
the term within the brackets represents the
P(t+t).
In
(3),
forecast
over the time
interval t+l,
... t+.+n minus the cumulative
by time t+X-1 and minus an
production planned for completion
inventory adjustment (the difference between the actual and target
inventory).
We term the quantity within the brackets to be the net
requirements over
set
the time
P(t+t) to be the
window t+t,
to the
interval t+t,
average of
length of the time window.
a decision variable that
on all
items
We note that
In this
respect
it
The value of n will
set to
time
decision variable equal
determine the
The safety stock target, SS,
this production
is
Thus, we
provide acceptable
is also
customer
in the family.
similar to
developed by Holt et al.
differs
is
... t+l+n.
the net requirements over this
... t+t+n where n is an integer
level of production smoothing.
service
t+1+1,
smoothing model
the
minimizing some cost function.
rule.
linear-decision-rule model
(1955,1956).
from that of Holt et al.
is a linear
However,
in that it
Rather,
our decision rule
is not a consequence of
we just pose this decision
10
(5)
= P(t) -
P(t)
where
(t)
(t)
We assume that c(t)
P(t+l)
-
and identically distributed over
(3) to consider the difference
Now we use
(6)
is independent
mean and has variance ap
has zero
time,
the yield uncertainty.
is a random variable that reflects
P(t+1-1)
= n+1
-
t(t+l,t+l+n-1)
We use
(4),
P(t+1-1)
+ I(t-1)}
I(t)
... +
t(t+t+n-1).
and the substitution Ft(t) = D(t) to simplify
(5)
P(t+l)
+
= 6t(t+l)
-
P(t+1-1):
t(t+l,t+l+n-1)
-
Ft-l(t)
+ P(t)
where
+
{Ft(t+l+n)
P(t+l) -
- P(t+1-1)
= n+1
{Ft(t+l+n) +
-
P(t+t-l)
(6)
to
t(t,t+±+n-1)
+
(t)}
from which we obtain
P(t+.) =
(7)
i
+
Thus,
to
we see
F
+
t(t+l+n)
+
--- (P(t+1-1)}
n+l
that the production smoothing model
which is a simple smoothing equation.
(7),
for time period t+t is a weighted average of
for the
t+t+n,
previous
modified
interval
time period t+1-1,
__ ___
is equivalent
Planned
production
the planned production
initial
forecast for
(t,t+l+n-1) and by the realized yield deviation for the
completed in t.
weights
that
so
1/(n+l)
significance of
The window length n determines
is the
(7)
is
1
the
smoothing parameter.
not
simple smoothing equation, but also
_
and the
(3)
by the change in the cumulative forecast over the
production
The
c(t,t+)+n-1)
(t)}
1_---_·^_-_·1111__11I-
only
its
interpretation as a
its usefulness for
11
long-term behavior of
characterizing the
model.
(7),
From
(8)
we can show that
E[P(t+t)]
this production smoothing
in steady-state
= D
and
(9)
= (D
Var[P(t+l)]
2
2
+ ap)
/ (2n
+ 1)
We defer the details of this derivation to the appendix.
also use
[P(t+l)
(7)
to characterize
- P(t+-1)], as well
demand process
is not
the
(8)
and
'step',
as study other cases
stationary
trend component or a seasonal
From
production
(e.g.,
in which the
the demand process has a
component).
(9) we see that the window length n does not
affect the expected aggregate production rate, but does
variability of
We can
aggregate production
the
rate.
control the
As we increase
n,
the
production variance decreases and the aggregate production becomes
smoother.
Thus,
the choice of n dictates the degree of production
smoothing given by
Aggregate
(3).
Inventory Behavior
In addition to
the aggregate production
level,
we also want
understand the behavior of the aggregate inventory level.
necessary tQ see the
service for
effect of
individual items.
planned aggregate
to be
forecast
through t+X:
we focus on
To do this,
at time
t+1.
inventory at time t,
completed over t+l,
is
the production smoothing on customer
inventory level
defined as the current
This
to
... t+X,
minus the
I(t+l):
At time t,
the
I(t+l)
is
plus planned production
cumulative
demand
12
(10)
I(t+l)
By using
(3)
(11)
From
(12)
= I(t)
+ P(t+l,t+l)
to substitute for
- Ft(t+l,t+l).
P(t+l,t+x-1)
(10),
- nP(t+)
I(t+l) = SS + Ft(t+l+l,t+l+n)
(11),
in
we obtain
.
we see that
E[I(t+.)] = SS
In addition, we can show its variance
(after substantial
algebra)
to be
(13)
Var[I(t+)]
=
n2
2n+
2
(
+
2
n-1
)
2
2
n+l
j=0
This characterization
be
useful
of the planned aggregate
level
for a
While the expectation of planned inventory level
equals the
target safety stock,
length n.
From
inventory level
length n,
inventory level will
the target safety stock SS and of the smoothing
specification of
(linear)
i=0
in determining the customer service
parameter n.
+j
(13)
its variance depends
on the window
we argue that the variance of the planned
essentially increases
since the second term
linearly with the window
in (13)
is dominated by the first
term.
Disaggregation and Customer Service
To determine the proper choice for the target safety stock SS,
we need to know how
individual
· _··
items
_________I___
the aggregate
or equivalently,
inventory is spread over
the
how the aggregate production
level
13
is disaggregated
into
item production.
We must set
starts for each item so that the planned production
family is achieved;
that
the production
level for
is,
E P(k,t+X) = P(t+l)
k
where P(t+l) is given by (3) and P(k,t+l)
is the production
for item k started at time t for completion by time t+1.
implies
that the planned inventory level for
call
I(k,t+l),
it
(14)
E
the
level
This
item k at time t+X,
must satisfy
I(k,t+l) = I(t+I)
k
where
I(t+1)
is given by
(10).
To characterize the service
level for
item k in period t+1 for
a particular disaggregation, we can express its planned inventory as
I(k,t+t)
where
I(k,t)
= I(k,t)
is current
+
- Ft(k,t+i)]
inventory and Ft(k,t+i)
demand forecast for month t+i
will
~ [P(k,t+i)
i=1
.
is
the current
The actual inventory at time t+t
be
I(k,t+t)
= I(k,t)
+
E [P(k,t+i)
- D(k,t+i)]
i=l
where for time t+i,
P(k,t+i)
demand, and where negative
that the
difference
is actual
production and D(k,t+i)
inventory denotes a backorder.
I(k,t+l)
- I(k,t+l), which reflects
uncertainty and cumulative forecast error
of
time periods,
zero
2
mean and a variance ck.
I(k,t+t)
/
k
to
is
the
We assume
yield
for k over the lead time
is a normally-distributed
random variable with
Consequently, we can use zk =
indicate the service
level
for k at time t+t from
14
a particular disaggregation
deviations of
(14).
zk denotes
safety stock protection provided by
Now suppose we set P(kit+1),
the
likelihood of stockout
I(k,t+l) = z
(15)
for all k.
Thus,
I(t+t)
we can use
specified by
/
(12)
standard
I(k,t+l).
I(k,t+2), so that
is the same for all
items k,lor equivalently
E
k
(15)
into
(14),
we find that
rk
and
(13)
to characterize the service
z) for a choice of SS and n.
level
The expected service level
the
greater will be the
month.
depends only on the safety stock target
But the amount of variability in
smoothing parameter n;
(as
We see that z is a random
variable, and thus the service level will vary from month to
SS.
items.
rk
By substituting
z =
or equivalently
in period t+1
we want zk = z for all
Thus,
the number of
the service
level depends
the more that we smooth production,
variability in service
from month
on
the
to month.
We have assumed here that it will always be possible to
disaggregate production
Indeed,
if current
to
satisfy (15).
inventories are
simultaneous satisfaction of
production
(P(k,t+l) < 0)
Obviously,
this
assumption of
(14)
and
(1981)
for items with large current inventories.
is not possible.
However,
that we produce to stock;
'balanced' inventories
ordering policies
we expect that the
be appropriate for the
we will
test this
We note that a similar
is made by Eppen and Schrage
and by Federgruen and Zipkin (1984)
centralized
the
(15) may imply negative
assumption with the simulation exercise.
assumption of
is not guaranteed.
'out-of-balance',
'balanced' inventories will
family of items
This
in their studies on
for multilocation distribution
15
systems.
Two-Stage Models
In this
section we extend the one-stage model
operation with
two
stages,
e.g. fabrication
and assembly.
an operation we may have not only a finished goods
also an intermediate
1).
inventory between the
The intermediate
partially,
inventory permits
the production of
smoothing model
stages.
disaggregates
we will
need to
stages.
The
first approach
without an intermediate
model;
include
it
is to
upstream stage smooths
stages.
Thus,
the intermediate
level for the
smoothing by the
The notation will be very
stage
the previous
but will
the various models.
The
intermediate inventory to permit
They differ
inventory,
upstream stage
for both
stages as one
and then to use
its production.
downstream stage.
the production.
plan.
in terms of how
In the
and on forecasts of
In the
third model,
information
the production
production
is based on the echelon
similar
to that
the
second model,
production smoothing by the upstream stage relies on
about
least
(parts fabrication)
this approach any further,
second and third approach use the
the
at
characterize how each stage
view the two
in the simulation study of
some decoupling of
(see Figure
for production smoothing for two
inventory,
we will not discuss
stages
production rates
the aggregate production
We mention three approaches
stage
(assembly).
needs to set aggregate
In addition,
two
For such
inventory, but
one to decouple,
the upstream
from that for the downstream stage
to a production
inventory.
for the one-stage
16
model.
We
let
1
downstream stage,
we want to
that
find
starts
respectively, and define 1t=
for the
and
completed by t+1 1,
inventory,
at
environment.
these
Then at time t
planned production
denote the planned
in t to
be completed
intermediate
to denote the planned finished-goods
the times
We present
1 +1 2 .
and for the downstream
planned production that starts
I(t+1 2 )
for the upstream and
upstream stage Q(t+t 1 ),
We use J(t+1l) to
inventory,
items
2 denote the lead times
in t to be
stage P(t+12),
by t+t 2.
and
t+1
and t+1 2,
1
models for the
respectively.
simplest two-stage production
We assume that there is a one-to-one mapping between
produced in the upstream stage and items produced by
downstream stage.
receives
That
is,
each
the
item from the upstream stage
further processing at the downstream stage and results
unique end
item.
are defined so
in a
Furthermore, we assume that the production units
that
one unit
of downstream production
starts
requires as
input one unit of
completed upstream production.
Although we
present the models for this setting, we can extend it to
more complex environments, such as when the downstream stage
performs
assembly of sets of
components produced by the upstream
stage.
To extend the one-stage model to a two-stage system requires
the specification and analysis of
a production
each stage.
in a way that permits one
We desire
to do
how the planning at one stage
intermediate
discuss
first
this
impacts
inventory can be used
the downstream stage,
smoothing rule for
the other stage,
and how the
to decouple the stages.
since
to see
We
it will be simpler.
17
Downstream Stage
for the downstream stage is
The production smoothing model
by
2
rule
{(Ft(t+l,t+1 2 +n)
= n+
P(t+t 2 )
(16)
where n is the window length and SS
Thus,
stock
target.
equal
the average of
...
we set the
level at time t for completion by time t+.
planned production
the
Namely,
that given for a one-stage system.
identical to
t+
2
as
(3) we set
in
from the previous section
analyses
given
directly to P(t+1 2 )
The extension of
(3)
+
I(t)
by
(16)
is the finished-goods safetyplanned production
the
of (16).
and
to
demand on
the
intermediate
I(t+t 2 )
the
implied by
is less clear.
is not
time t.
extend (3) we need an appropriate forecast of
,
inventory apply
as
the downstream stage:
inventory at
2
inventory is
As a consequence,
to the upstream stage
by the production by
to
the time window t+1
of the planned
is because the demand on the upstream stage
is set
,
2}
sufficient component
available to permit the execution
2 -1)
P(t+l,t+1
requirements over
the net
We assume that
+n.
2
-
independent,
P(t+1 2 )
(16).
This
but
is the
Consequently,
to
the planned production
for the downstream stage over the smoothing window for the upstream
stage.
We suggest
Upstream Stage:
below two approaches.
Production
The first approach is
an explicit
forecast
of
to
to
Intermediate
Inventory
smooth the upstream production using
the downstream production
over the
smoothing
18
window.
The production smoothing
(17)
Q(t+t
= m+1
1)
{Pt(t+X
-
where
for i=l,
...
t1 +m.
at time t, net of
J(t)
+l,t+X+m)-
Q(t+l,t+t 1 -1)
+ SS 1 }
J(t)
m is the window length,
Pt(t+g+m) and Pt(t+1 2+i)
2
rule is
Pt(t+t 2 +1,t+X+m)
= Pt(t+1 2 +1)
+
...
is a forecast made
at time t of P(t+1 2+i)
denotes
intermediate
the downstream
the actual
+
inventory
production started at time t;
that
is,
(18)
J(t)
and SS1
We note
is
J(t-1)
-
here that P(t+12+i) will
P(t+X2)
Hence,
for the
Pt(t+1 2 +1,t+l+m)
(18) we can rewrite
that is analogous to
_m_
upstream stage
is
-
denotes
interval
the demand forecast
t+1,
...
t+ll+m.
as a simple smoothing equation
+
[Pt(t+12,t+l+m-1)
Pt-l(t+1 2 ,t++m-l)] +
l(t)
+
{Q(t+1-1) )
Q(t)
.
Thus,
planned production for the
a weighted average
the previous period and
requirements
(17)
{Pt(t+l+m)
-
= Q(t)
inventory.
(7):
Q(t+ 1l) = m+l
where el(t)
intermediate
be the demand on the upstream stage
upstream stage over the time
Using
(19)
+ Q(t)
the target safety stock
at time t+i.
for the
=
the
of
the production
initial forecast
level
of cumulative
from
net
in time t+l+m.
To specify Pt(t+1 2 +i),
simple smoothing equation
we note
that
(16)
is
equivalent
to a
19
P(t+1 2 )
(20)
= n+1
{Ft(t+1
+
=
t(t,t+X
+
2 +n-1)
2(t)}
(P(t+12-1))
{
) is the forecast revision as defined before, and
where St(
C2(t)
+
2 +n)
-
P(t)
Now the forecast of
P(t).
P(t+1 2 +1)
made at time t
should be
since
n
(n+1)
+
2 +n+1)
Ft(t+1
i=2,
(22)
Similarly we can use
is zero.
...
forecast P(t+12+i)
to
R1+m:
=
Pt(t+1 +2)
2
=
Pt(t+l+m)
In spite
(21)
P(t+12 )
the expected-yield
the expected forecast revision is zero and
variation
for
1
= (---)
Pt(t+X 2 +t)
(21)
of the
(n;)
(-)
1
that
fact
Ft(t+t 2 +n+2)
Ft(t+l+m+n)
+
+
(
1)
n
( -)
Pt(t+12+1)
+m -
Pt(t+I
l)
the smoothing equations for the upstream
form as those for
a single-stage
[(17)
and
(19)]
have the same
system [(3)
and
(7)],
we have not analyzed fully the random variable
stage
for
planned production Q(t+1 1 ).
upstream production Q(t+l1 )
planned production,
demand.
a way
to
which
itself
We will need
this
smoothing model.
do expect that
to
'decode' this double smoothing in
the
or
any other measure of
it
is
easy to show that E[Q(t+
simulation
to
see
(However,
to use
D.)
the planned
is a smoothed average of customer
the variance of Q(t+t 1 ),
production smoothing.
is because
is a smoothed average of the downstream
We have not been able
find
This
Despite
behavior
this
of this
exactly the
behavior
1]
=
of
lack of an analytic result, we
smoothing model
for the
upstream
20
stage will
system.
closely parallel
that predicted for a single-stage
In particular, due
to the double smoothing, we expect that
the variability of the upstream production will depend upon the sum
of
the window lengths,
m+n.
J(t+
1)
= J(t)
is given
inventory at time t+. 1
The planned intermediate
+ Q(t+l,t+l
1)
-
Pt(t+1
2
by
+l,t+t),
which can be rewritten as
J(t+ll)
= SS 1
+ Pt(t+1+l,t+l+m)
by substituting from
clear that
E[J(t+t 1 )]
(17)
-
mQ(t+t
1)
Again, although it is
for Q(t+l,t+t1 -l).
= SS 1,
we have not been able to
characterize
analytically the variability in the planned intermediate
Nevertheless,
we need the planned intermediate inventory to specify
how the upstream aggregate planned production Q(t+
disaggregated
Q(k,t+t 1 )
for
inventory.
into planned
production for
1
)
individual
is
items,
e.g.
item k.
For this model we disaggregate the upstream production to
to equalize the service
across all
items.
levels
To do this,
from the intermediate
we set Q(k,t+t
1
)
try
inventories
such that
£ Q(k,t+tl) = Q(t+11), and such that the planned intermediate
k
inventory for item k, J(k,t+ll), provides the same level of
protection against
variance of
stockout for all k.
Define aklto be
the
the deviation between the planned intermediate
and the actual
intermediate
inventory at time t+1 1 :
inventory
21
~2
akl1= Var[J(k,t+l1)
= Var[
- J(k,t+l
1
{Q(k,t+i)
E
1 )]
- Q(k,t+i)
i=l
Pt(k,t+1
Then,
for
~2
given &ki
intermediate
where
+ P(k,t+1
+i)
we disaggregate Q(t+l 1 )
inventory for k
J(k,t+tl)
2
= z
2
+i)}]
so that the
planned
is given by
kl'
z is
z = J(t+t 1 )
/
E
kl'
k
Unfortunately, though, we are not able
^2
ak1'
to determine analytically
since we cannot determine the forecast error for downstream
production, Var[Pt(k,t+12+i)
need to
- P(k,t+1 2 +i)].
estimate it empirically or substitute
error for
demand, Var[Ft(k,t+1 2 +i)
error for
downstream production.
Upstream Stage:
Production to
Hence, we would either
the known forecast
- D(k,t+t 2 +i)],
Echelon
for the
forecast
Inventory
The second approach for smoothing the upstream production
based on the notion of echelon inventory.
a production
stage equals
all
inventory,
that is downstream of the stage.
inventory for the upstream stage
is
The echelon inventory for
including work-in-process,
In a two-stage system, the
is the intermediate
the work-in-process within the downstream stage,
echelon
inventory, plus
plus the finished-
22
goods
inventory.
Each period this echelon inventory is increased by
the amount of production
completed by the upstream stage, and
decreased by the amount of customer demand.
production
level for the
requirements
...
the
We set the planned
upstream stage to be the
echelon
inventory over
the
average of
the net
interval t+l,t+1+1,
t+l+m, where m is the window length:
(23)
Q(t+ 1 )
To explain
+
on
is
(J(t)
-
=
-
l {Ft(t+l,t+t+m)
-
Q(t+l,t+l1-1) -
-
J(t)
stocks), and
is the current
Q(t+l,t+1 1 -1)
+ SS 2
I(t)
-
(I(t)
echelon inventory
The sum of the current echelon
(t+l,t+1 1+1 2 -l)
since
2
should cover
is the
lead time for
Production at the upstream stage that is
started at t will be input into echelon inventory at time.t+
will not be available for
customer demand until
since Ft(t+l,t+l+m) = Ft(t+l,t+1-1)
bracketed term
echelon
in
2)
(exclusive of safety
inventory and the planned input over (t+l,t+1 1 -l)
the downstream stage.
+ P(t+l,t+1
SS 2 )
is the planned input to the echelon
inventory over (t+l,t+1 1 -1).
customer demand over
2)
+ SS 1 }
the bracketed term, note that
SS 1)
P(t+l,t+t
(23) corresponds
inventory over
set planned production
t+1 1+1
+ Ft(t+l,t+l+m),
to
(t+l,t+l+m).
1
. but
= t+R.
2
Now,
we see that the
the net requirements
on the
As we have done previously,
to be the average of
the net
we
requirements on
the appropriate inventory over the chosen smoothing window.
The analysis of
stage
__
is
identical to
system where we replace n by m,
(SS1+SS2 )
_
(23)
and
I(t)
by
((t)
that
for
(3)
for the
P(t+l) by Q(t+t 1 ),
+ P(t+l,t+1
2)
+ J(t)).
one-
SS by
In particular,
23
we can rewrite
(23) as
Q(t+£ 1)
1
= m+l
+
{Ft(t+t+m)
t(t,t+l+m-1)
m
+ el(t)
from which
+
+
2(t)}
(mQ(t+l1 -1))
we can obtain
E[Q(t+l1)
D
]
and
Var[Q(t+l 1 )]
2
p
where
.
Q(t)]
= P(t)
= Var[e2 (t)
as expected,
Thus,
+
-
to the window
echelon
planned
In
parallels
that for
(10)
-
(13).
define the
(24)
fact,
analysis
we use
I(t+
)
+
the planned intermediate
this,
To see
+ P(t+Z1 +l,t+t)
I(t+l 1 )
inventory
echelon
for the planned
this, we write the counterpart
(23) to
P(t +
P(t+l+l,t+)
to
(10)
to
inventory:
+ P(t+ll+l,t+l) + J(t+t 1 )
we find
for the
it is
model,
for this
= I(t)
substitute for
+ J(t+)
Q(t+l,t+
=SS
1
From
is inversely
its variance
while
+ P(t+l,t+z 2 )
+ Q(t+l,t+l
Then,
rate
the planned inventory for the one-stage system.
planned echelon
I(t+l 1 )
-
length for smoothing.
is not immediate
the
= Q(t)
the aggregate upstream production
inventory at time t+R 1 ,
J(t+t 1 ).
(2m+1)
2
and aQ = Var[e l (t)
P(t)]
While the characterization of
inventory J(t+l 1 )
/
)
the demand rate,
equals, on average,
proportional
+
= ( D
that the mean
echelon
1)
in
1-1)
+ SS
2
(24)
+ J(t)
Ft(t+lt.11)
to
obtain
+ Ft(t+tl+l,t+l-m)
m Q(t+ )
1
inventory equals
SS 1
+ SS 2
24
+ 12D, while
its
variance is given by
m2
2m+l
2
2
+
(
+
2
Q)
-
I +j
m
2(m_
12+m-
E
[1
j=om+1
+
To disaggregate Q(t+1 1)
each item k, Q(k,t+l 1 )
E Q(k,t+l
We can do this
Namely,
for item k,
I1D(k),
equals Zkl
l
(i)]
1)
1)
manner to that for the one-stage system.
so that the
planned echelon safety stock
+ P(k,t+
I(k,t+t 1 )
1l+l,t+l )
+ J(k,t+1
1
)
-
where z is given by
E 0
kl equals
v 2
i=O
I(t
P(t+
1) E
1 +l,t+)
and
It1+j
E-
such that
in a similar
defined as
2
[aD
]
we need to find the planned production for
) = Q(t+
we set Q(k,t+
j+l
)
+ J(t+t
D
kl
the variance of
1
[Q(k,t+i)
-
+ [Ft(k,t+i)
Q(k,t+i)]
-
D(k,t+i)]
i=l
In words,
echelon
time
kl
is the
variance of
inventory for time t+t
t+t1
for item k.
planned echelon
Thus,
1
the deviation between the planned
and the actual
safety stocks
for all
yield uncertainty and forecast
for
items,
safety stock
(i.e.
the
where we have
the cumulative upstream
error over the
this disaggregation
planned echelon
inventory for
this disaggregation equalizes
normalized by the standard deviation of
rationale
echelon
scheme
either
is
lead
time t 1.
to try to
in the
The
spread the
intermediate or
25
finished-goods
intend for
level
inventory),
each item's
items.
By
'evenly!, we
echelon safety stock to provide the same
and from yield uncertainty in
lead time
As
before, the measure of
+SS 2 )
the upstream stage over
its
1.
disaggregation,
1
the
of protection against uncertainty both from the forecast
errors
(SS
'evenly' over
lerel
z, is a random variable.
akl,
a
/
the
while
its variance
is
of
protection from the
Its mean is given
by
directly proportional to
(25).
to term the first
It will be convenient
(produce to
intermediate inventory) approach as the decoupled model and to term
the
second
the first
(produce to echelon
case,
the
intermediate
The upstream stage produces
downstream
stages
forecast.
But the stages
produce
to the
are
the
inventory based on
stage draws
stages.
a forecast
of
its raw materials
demand
namely the
same forecast,
'nested'
In
both the upstream and
stage
in that the upstream
inventory which contains
produces to the echelon
stage and its
to this
In the second case,
inventory.
the nested model.
inventory decouples
the downstream
downstream usage, while
from this
inventory) as
the downstream
inventory.
COMPUTATIONAL STUDY
To examine
the proposed production smoothing models.
conducted a computational study based
items produced
computational
on data gathered on the set of
in one manufacturing facility.
study
is
threefold:
we
The purpose
of
the
to understand how much inventory
26
is needed
in a make-to-stock system and where that inventory should
be positioned;
to compare the effectiveness of the different
production smoothing models for a two-stage system;
determine,
via a comparison to a simulation,
approximate analyses of these
the test data,
and finally
Test
the
computational
the accuracy of the
smoothing models.
then the simulation
and to.
We first describe
program that we have developed,
results.
Scenario
To test
the production smoothing models we abstracted a test
scenario based on data gathered
the results given here,
from a manufacturing operation.
we have disguised this data to protect the
identity of the manufacturing firm.
are representative of
In Table
the results
I we give
family, we consider only
system.
We exclude all
200 units
these
per month,
However,
the reported results
obtained using the actual data.
the mean demand rate
its standard deviation for the
this
For
family of
(units
items.
Of
25 as candidates for
per month)
the
38
and
items
in
the make-to-stock
items that have a start rate of less than
since
it would
be
too risky
to plan to
stock
low-demand items.
We assume that there are two production stages and that each
item requires processing from both stages.
first stage
stage
(
2
is three months
=1).
·1_
find it
1 =3)
and is one month for the second
The first stage processes each
exactly 500 units.
We will
(
The lead time for the
The second stage can process
convenient to express
__1___1__·_1_11_1_____-___-·----
item in a lot of
the
lots of any
simulation results
size.
in
27
terms
of
the lots
For the
per month
for
the upstream stage
sample of
25
items,
28.5 lots per
month
For the test scenario
and
revision step
(
81
lots
= 811).
we assume that the forecast process
that for eac h item k,
the
variance of
is
the forecast
is given by
a2(o)
=
2(1)
2
.1
QD
2
a2(2)
.13
D
a2(3)
.2
oD
cr2(4)
=
2(5)
=
.23 aD
acr2 ( j )
Thus,
the mean monthly demand is
(500 units/lot) and the standard deviation of aggregate
demand aD is
unbiased,
(500 units/lot).
0
for j
> 5.
any forecast for 6 or more months
monthly demand,
and is not revised
ahead
equals the mean
until there
is 5 or less months
to go.
We assume that
in each stage.
data
there is no uncertainty in
There are two reasons
from which to estimate
for this.
the parameters of
Second, we had observed in an earlier
the production yield
First,
we had no
the yield uncertainty.
study of a comparable
production operation that the uncertainty in the forecast errors
dominated
that
in
the yield realization.
including only the uncertainty in
capture
given
this
the primary
behavior of
Hence, we hope
our study.
by
the forecast errors we will
the smoothing models.
information on yield uncertainty, we could
into
that
Nevertheless,
easily incorporate
28
We can
for
find ok
item k over a given lead time 1,
the variance
of
2
ak
, the variance of
D
for
of the
item k
2
2
a (0) + ( 2 (0) +
2
cumulative forecast
from the above
forecast revision
(Table
the
specification of
steps and from the knowledge
1).
For instance,
(1))
+
2
error
if
t=3 then
2
2
a ()
r(2))=
.63a
D
For later reference, we note that the sum over the 25 make-to-stock
items of the standard
E ak' is
25 lots
when
deviations of
t=1 month,
the cumulative
63 lots when
forecast errors,
=3 months,
and 86
k
lots when
=4 months.
Simulation Program
We have written in PL-1 a program to simulate the production
planning process
for a two-stage production system.
is a discrete-time simulation where the time unit
Each month
the
simulation generates
The simulation
is one month.
customer demand and a demand
forecast for the next twelve months for each item.
inventories
for the items
are increased by completed downstream
production and are depleted by the demand amount.
created when insufficient
inventories exist.
disaggregation to determine
The intermediate
the production
inventories
for the
starts by the downstream stage.
(3)]
and
its
items are increased by
by usage from the
Note that the
intermediate
inventory cannot have backorders;
insufficient
inventory exists,
___1__11__^____11_l_l_-__--
[e.g.
starts for the downstream
completed upstream production and are depleted
production
Backorders are
The simulation then
applies a specified production smoothing model
stage.
The end
rather,
if
then the downstream production
starts
29
the simulation then applies a
Finally,
need to be reduced.
specified production smoothing model
(17)] and
[e.g.
disaggregation to determine-the production starts
stage.
the
Note
simulation.
of the
for the upstream
then repeated each month for the run length
process is
This
its
that,
in
effect, the simulation acts as
if
events, customer demand, production starts, and completed
production,
occur at
the start
(or end)
For our tests the simulation is
of
every month.
run for
1000 months where
the
first 40 months are for initialization and various statistics are
collected over the remaining 960 months.
forecast time series
common demand and
comparability of
Thus,
simulation runs
in order
for different
to
increase the
smoothing models.
in performance between two simulation runs
any differences
will be
the smoothing models and not due to any differences
due to
To generate
in the realization of the demand or forecast process.
forecast time series,
the demand and
obtains
Ft(s)
as a lognormal
with variance
In the
relies on a
The simulation
2
[see
(s-t)
previous
random variable with mean Ftl(s) and
(1)].
section we develop the disaggregation procedure
with no restrictions on the sign of
this procedure may
the forecast revision process
suggest the impossible,
namely negative
The
production for an item with an excessively high inventory.
simulation does not permit negative production.
disaggregation results
the
in this outcome
production starts for that
disaggregation procedure is
Similarly,
Indeed,
the production outcome.
Rather,
for a particular
item are set
repeated for the
to zero and
remaining
the disaggregation procedure assumes
if the
item, then
the
items.
that
sufficient
30
raw material
is available to make the desired production starts.
For the upstream stage,
However,
the simulation retains this assumption.
for the downstream stage,
more production of a particular
item than is available in its
intermediate
inventory
intermediate
inventory).
item exceeds
the available raw material, we set
to
the raw material
procedure
for the
(i.e.
the simulation will not start
Rather,
level and
if the desired production for
then repeat
remaining items
aggregate production
impose a lot
the upstream stage.
its production
Thus,
the planned
for the excluded
size of
500 units
500 units.
is to compute
units to start, divide by 500 to
the nearest
integer.
item).
for all
To accomplish
this we have to modify the disaggregation procedure to
The modification
equal
for each item the monthly
production starts must be a multiple of
restriction.
an
the disaggregation
(after we reduce
by the amount preset
For the simulation we
items for
we do not permit backordering on the
reflect this
the desired number of
convert to lots, and
then round to
We assume no such restriction for the
downstream stage, although we could
impose a fixed
lot size,
if
appropriate.
The simulation also has the capability to limit the aggregate
production for each stage.
not be capable
case,
production starts equal
production smoothing model
to the
and the capacity
computational work
but assume
the upstream stage might
of starting more than 90 lots per month.
we would modify the
from the model
For instance,
minimum of
limit,
to
In
set
the desired start
say 90 lots.
this
rate
However.
in the
that we report, we do not use this capability,
that there are no
limits
to
the production at each stage.
31
The disaggregation of upstream production for the decoupled
(produce to intermediate
forecast
item-level version of
replace
-
(21)
by the amount that would be started
the intermediate inventory.
P(k,
starts
In this
is necessary because
This modification
level;
from their desired
hence,
compare
the
the actual_
of future
Results
section we present
and discuss our
First, we
in two parts.
one-stage model
Second, we
consider the application of a
and study its behavior on the test
them against
work.
computational
scenario.
and
consider the two versions of the two-stage model
each other and versus
the one-stage model for
test scenario.
the one-stage model
To apply
to the test
make-to-stock system with a finished-goods
intermediate
inventory.
scenario, we assume a
inventory but with no
months.
rate,
in the
In effect, we combine the two stages
test scenario into one stage with a production lead time
via
(21)
by the downstream stage.
Computational
We do this
in
t+1t2 ),
if there were no shortages in
production starts may not be an accurate reflection
production
We
intermediate inventory perturb the actual
stockouts in the
production
for item k,
starts
actual production
the
with one modification.
(22),
via an
this forecast
We generate
downstream production.
of
requires an item
inventory) approach
Each month we use
(3)
to set
=
4
the aggregate production start
which is disaggregated based on the finished-good inventories
(14)-(15).
32
We used the simulation to compare the "actual"
found
from the
simulation)
of the one-stage model
behavior from our analysis of
We report our results
window
lengths n=0,1,2, ...
120 and
150
lots
(recall
show the consequences
standard deviations
aggregate
the fill
e.g.
(9),
(13).
We have run the simulation for
6 and for safety stock targets of SS = 80,
ak = 86
that
from production
lots
= 4 months).
for
smoothing, we
To
report the
of aggregate production P(t) and of the actual
inventory position
rate,
(as
with the predicted
the one-stage model,
in Table 2.
behavior
I(t).
As a measure of service we
report
which equals the fraction of demand that is satisfied
by inventory without any delay.
From
the simulation
two decision parameters,
results in
this table we can see how the
the window length and the safety stock
target, affect the reported performance measures.
Namely,
the
window length essentially determines the measures of production
smoothing
[the standard deviations of P(t)
and
safety stock target determines the fill rate.
I(t)],
This is
with the analysis of the production smoothing model.
the analytic results
of
the actual
(9),
(13)
while the
consistent
Furthermore.
are reasonably accurate predictions
(simulated) values. 1
approximations made by the analysis
This
indicates that the
(i.e.
ignoring lot-sizing. and
assuming that inventories remain balanced so that the proposed
disaggregation
this test
is always
feasible)
do not give
significant
errors
scenario.
Finally, from Table 2 we see the implications
stock system for
starts needed
for
this
family of
the sample of
items.
items
of
a make-to-
The average production
in the test scenario is 81
in
33
lots
of
per month.
80
90%
lots
fill
We see that maintaining a finished-goods inventory
(about one month of demand),
improves service to a 95% fill
length
for
rate.
To get to a 98%
investment in another 30 lots.
the
production smoothing mdel
(n=0),
production start rate,
while equal
lots
standard deviation of
nearly 30
to exceed 120 lots
below 50 lots
16% of
16%
lots.
Thus,
of the time,
possible by increasing the window length,
returns.
For instance, a window
using a six-month cumulative
deviation of production
the monthly
on average,
has a
we expect production
and similarly to fall
smoothing is
but with decreasing
length of n=2,
forecast
rate
extent of
Substantial production
the time.
lots
fill
dictates the
With no smoothing
to 81
in a
The choice of window
production smoothing.
starts
will result
Increasing this safety stock by 50% to 120
rate.
requires an
on average,
(6=n+l),
which corresponds
reduces,
the standard
starts by almost 60% over no smoothing.
The
cost from increased production smoothing is a slight degradation
fill
rate,
and an
increased variability in
the actual
to
in
inventory
position.
We
examined the two-stage models to
possible by inserting an intermediate
see what improvement
inventory between the
To examine the smoothing behavior, we first
smoothing models for
lengths.
effectively independent
levels.
approach
(produce to
results
for
stages.
simulated the two
a fixed safety stock but with varying window
As with the one-stage system,
stocking
is
of the
the
safety stock targets
Table 3 gives the results
intermediate
the nested
approach
smoothing behavior
inventory)
is
for reasonable
for the decoupled
while Table 4 gives
(produce-to echelon
inventory).
the
In
34
both cases
Table
the
safety stock
3 shows
= 100
targets are SS
and SS
2
= 120.
that for the decoupled approach the production
smoothing behavior for the downstream stage is as predicted and
independent of the upstream stage.
upstream
is
The production behavior for the
stage, for which we do not have an analytic
reflects the effect of double smoothing:
prediction,
the upstream production
is
set by smoothing the forecast of downstream production, which itself
is a smoothed average
of
the demand forecast.
Consequently, for a
fixed window length for the upstream stage there is
smoothing as the downstream window length grows.
the one-stage system,
In contrast with
the fill rate provided by the safety stocks is.
very dependent on the level
the fill
_
greater
of production smoothing.
In particular,
rate declines dramatically with longer window lengths
smoothing) for the downstream stage.
the item forecasts
This
is due to
(more
the fact that
of downstream production used by the upstream
stage become less accurate with longer window
downstream stage. 2
As a consequence,
frequent stockouts,
which ultimately results
the
lengths for the
intermediate inventory has
in poor customer
service.
In Table 4 we see
the comparable
for the nested approach.
Here,
production smoothing behavior
the analytic predictions of the
production smoothing for both stages are reasonably accurate.
Furthermore,
the level of smoothing for the upstream stage
totally independent of the downstream stage.
virtually independent
And
the fill
is
rate
of the choice of smoothing parameters,
is
as we
saw for the one-stage system.
To explore the impact of the safety stock
levels, we contrast
35
in
Table 5 the fill
rates
safety stock choices.
from the two approaches
Due to the fact
decoupled approach is very
that the fill
sensitive to the
smoothing, we simulated a set of
positioning of
these
stocks
stock target to be set
hope of
= 1).
to provide insight
production
(m=l,
smoothing
n=O).
below 40 lots,
results
We
into the proper
and to allow comparison with the
providing reasonable service
Based on the
smoothing
We did not permit the
(Table 2).
stage model
level of
rate for the
cases with substantial
(m=3, n=2) and another set with limited
chose the safety stock levels
for a series of
one-
downstream safety
since below that we have no
(recall that
in Table 5,
we make
ak = 25
lots when
the following
observations:
a)
(produce to echelon inventory),
For the nested approach
fixed
total
safety stock
insensitive with slight
(SS 1 +SS 2 ),
the fill
for a
rate is relatively
improvement as more safety stock is
placed downstream.
b)
the fill
rate
intermediate inventory)
(produce to
For the coupled approach
is very sensitive to both the amount of smoothing
and the
positioning of
service
again deteriorates with increased smoothing.
fixed
placed
total
the safety stock.
safety stock,
in the intermediate
(SS 2 =40)
As seen in Table 3,
For a
fill rate improves as more stock
inventory as long as a minimal
is kept downstream;
beyond this minimum,
is
level
service will
be degraded.
c)
see that the nested
In comparing the two approaches,
we
approach dominates the decoupled
approach for
substantial
smoothing.
When there
is
the case with
limited smoothing,
36
however, the
decoupled approach
is slightly better,
provided
the appropriate inventory positioning.
In comparing either of-the approaches for the two-stage model
with the one-stage model, we need more inventory with a two-stage
model
than with
a one-stage model fora given
instance, a safety stock of 120
a 95-96% fill rate.
provides a 90%-92%
lots with
the
fill rate for the same
However,
Thus, a two-stage model
total safety stock;
the cost
to hold
finished-goods
stock in
inventory.
can be prefereable to the one-stage model
for
the finished-goods
For
We can achieve this
with the one-stage model with a finished-goods safety stock of
lots for window lengths n=2 and n=3.
For comparable smoothing with
n=2 and with intermediate safety stock of 80
lots and a finished-goods
safety stock of
stage model would be preferable if
inventory is
goods inventory.
80
we would need the nested approach with
smoothing windows m=3,
intermediate
if
low enough
inventory.
a 90% fill rate is desired.
the two-stage model,
the
rate but with limited
the holding cost for the intermediate inventory is
instance, suppose
For
the one-stage model, gives
it typically will cost less
intermediate inventory than in the
relative to
rate.
The nested approach for the two-stage model
decoupled approach can provide a 94% fill
smoothing.
fill
40
lots.
Thus, the
two-
the holding cost for 80 lots of
less than that for 40 lots of
finished-
ll
37
FOOTNOTES
Note that we report the standard deviation of actual inventory,
Since the
rather than that of planned inventory as given by (13).
actual inventory is given by
I(t+Z)
= I(t+1)
2
+
{P(t+i) - P(t+i)}
i=1
+
{(F
i=l
we can express
by (13).
its
variance
(t+i)
- D(t+i))
t
in terms of
the variance
of
I(t+i)
given
2
To disaggregate upstream production for the decoupled approach, we
We obtained this
need an item forecast of downstream production.
By using an
- (22).
forecast via an item-level version of (21)
alternate forecasting method, namely proportioning the aggregate
forecast by the expected demand level, we could avoid the
However, this alternate
degradation in fill rate seen in Table 3.
forecast method resulted in system performance that was strictly
dominated by the nested model (Table 4).
38
APPENDIX
Derivation of _81_and_91
By recursive substitution for P(t)
P(t+t)
=)(n+
i=o n+
({F
t
n+1
6t i(t-i,
in
(7) we obtain
(t-l+.+n)
t-i+I+n-1)
+
+
(t-i)}
where we assume that an infinite time history exists.
we have that
By assumption
i
for all
E[Fti (t-i+t+n) + 6ti(t-i, t-i+t+n-1)
+ E(t-i)] = D
and
VarF ti(t-i+t+n)
t-i
Furthermore,
+
ti
t-i (t-i , t-i+.+n-1)
+ C(t-i)]
=
we have assumed that these bracketed terms are
independent across time.
E[P(t+)
=
nDi
i =0
Thus,
we
obtain
n+1
DVar[P(t+
I
Var[P(t+l)
=
i=0
=(D
(n1)2 (n)2i
n l
n+i
+
P) /
(2n
+1)
(o 2
D +
2
O'p)
2
D
+
2
P
I1
39
References
Cruickshanks, A. B., R. D. Drescher and S. C. Graves, "A Study of
Production Smoothing in a Job Shop Environment," Management Science
30, 3 (March 1984, 168-380.
Eppen, G. and L. Schrage, "Centralized Ordering Policies in a
Multiwarehouse System with Lead Times and Random Demand," in
Schwarz, L. (ed.), Multi-Level Production/Inventory Control Systems:
Theory and Practice, North-Holland Amsterdam, 1981.
Federgruen A. and P. Zipkin, "Approximations of Dynamic,
Multilocation Production and Inventory Problems," Management
Science, 30, 1, (January 1984), 69-84.
Hax, A. C., and H. C. Meal, "Hierarchical Integration of Production
Planning and Scheduling," In Studies in Management Siet
Sciences, Vol.
I_ Logistics, M. A. Geisler (ed.). North Holland-American Elsevier,
New York, 1975.
C. Holt, F. Modigliani, and H. Simon, "A Linear Decision Rule for
Production and Employment Scheduling," Management Science 2, 1-30
(1955).
Iand J. Muth, "Derivation of a Linear Decision Rule
- -----for Production and Employment," Management Science 2, 159-177
(1956).
Schneeweiss, C.A., "Smoothing Production by Inventory Application of the Wiener Filtering Theory," Management
7 (March 1971), 472-483.
An
Science
17,
_____-"Optimal
…
Production Smoothing and Safety Inventory,"
Management Science, 20, 7 (March 1974), 1122-1130.
Silver, E. A., "A Tutorial on Production Smoothing and Work Force
Balancing," Oerations Research, 15, 6 (November-December 1967),
985-1010.
Figure 1: Two-Stage System
--4
I
II
#upstream'
II
I
·
II
'downstream'
customer
demand
TABLE 1: DEMAND FOR TEST SAMPLE.
500 demand units equals one lot.
Starred items(*) are excluded from Make-to-Stock System.
DEMAND
ITEM CODE
MEAN
STD. DEV
*1
2
2
*2
1
1
*3
18
21
*4
8
16
5
651
961
6
214
362
*7
137
298
*8
11
15
9
325
238
10
237
293
11
483
569
*12
2
6
13
1315
1121
14
1953
1685
15
6729
4793
*16
1
3
17
554
722
*18
48
49
*19
20
30
20
1887
1951
TABLE 1 CONTINUED ON NEXT PAGE.
TABLE 1 (continued)
DEMAND
ITEM CODE
MEAN
--
--------
STD. DEV
21
2193
1908
22
587
604
23
615
728
24
833
665
*25
47
151
*26
27
8
1206
11
1603
28
15691
12278
29
1142
2286
30
336
677
*31
81
242
32
246
660
33
749
890
34
817
1199
35
232
636
36
360
822
37
279
634
38
819
1336
Safety Stock
Window
Length
SS= 80
SS = 120
SS = 150
Predicted
Std.
Deviations
n = 0
29.8, 30.1*
.92
29.8, 30.2
.96
29.1, 30.1,
.98
28.5, 30.8
n= 1
17.4, 34.0
.91
17.3, 34.3
.96
17.2, 34.1,
.98
16.4, 34.9
n=2
13.1, 37.5
.90
13.1, 37.9
.96
13.0, 37.6,
.97
12.8, 39.4
n =3
10.8, 40.8
.90
10.8, 41.3
.95
10.8, 41.0,
.97
10.8, 43.6
n =4
9.3,44.0
.89
9.4, 44.4
.95
9.3,44.0,
.97
9.5, 47.4
n =5
8.3, 46.8
.89
8.4, 46.9
.95
8.2, 46.7,
.97
8.6, 51.0
n =6
7.6,49.0
7.6, 49.4
7.4, 49.3,
7.9, 54.4
.89
.94
.97
.96
.98
Predicted fill
rate.90
Table 2: Results from One-Stage Model
A
*
x,y
z
: x = standard deviation of Pt; y = standard deviation of it; z = fill rate.
downstream
upstream
window
n=O
length
upstream
up
n
n= 1
n=2
n=3
l
stream
stage
prediction
window length\
29.2, 27.4,* 24.1, 15.9,.
.99
98
m=O
22.0, 12.4,
.93
19.9, 10.6,
.88
m= 1
17.2, 27.4,
.99
15.9, 15.9,
.98
14.7, 12.4,
.93
13.6, 10.5,
.88
m=2
13.0, 27.4,
.99
12.6, 15.9,
.97
11.9,12.4,
.92
11.2, 10.5,
.87
m=3
10.7, 27.2,
.99
10.7, 15.9,
.97
10.2, 12.4,
.92
9.8, 10.5,
.87
downstream
stage
prediction
28.5
16.5
12.8
10.8
**
Table 3: Production Standard Deviations for TwoStage Model: Production to Intermediate
Inventory (SS1 = 100, SS2 = 120)
: x = standard deviation of Qt; y =
*LZ
standard deviation of Pt; z =
zt
*xy
customer fill rate.
**
No analytic prediction iscurrently available.
II
downstream
indow
window
engthn=
gup
upstream
window length
n= 1
n=2
n=3
upstream
stage
prediction
nstream
m =0
29.2, 27.4*
29.2, 15.9
29.2, 12.4
29.2, 10.6
28.5
m=
17.2, 27.4
17.2, 15.9
17.2, 12.4
17.2, 10.6
16.5
m=2
13.1, 27.3
13.1, 15.9
13.1, 12.4
13.1, 10.6
12.8
m=3
10.9, 27.2
10.9,15.9
10.9, 12.4
10.9,10.6
10.8
downstream
stage
28.5
16.5
12.8
10.8
prediction
Table 4: Production Standard Deviations for Two-Stage Model:
Production to Echelon Inventory (SS1 = 100, SS2 = 120)
*xy
x = standard deviation of Qt; y = standard
deviation of Pt.
Note, the fill rate is .98 for all instances.
SS
1
SS 2
FILL RATE
FILL RATE
(m=3, n=2)
(m= 1, n =0)
PRODUCE TO
PRODUCE TO
ECHELON
INTERMEDIATE
INVENTORY
INVENTORY
PRODUCE TO
PRODUCE TO
ECHELON
INTERMEDIATE
INVENTORY
INVENTORY
40
40
.83
.69
.84
.81
0
80
.83
.66
.85
.39
80
40
.90
.78
.91
.94
40
80
.91
.77
.92
.88
0
120
.91
.74
.92
.43
110
40
.93
.84
.94
.97
70
80
.94
.84
.95
.96
30
120
.94
.82
.95
.87
140
40
.95
.88
.96
.98
100
80
.96
.88
.97
.98
60
120
.96
.87
.97
.97
-
_
I
Table 5: Comparison of Fill Rates for Two-Stage Models