HD28
.M414
q3
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A Dynamic
E.
Stochastic Stock Cutting
Krichagina, R. Rubio,
Problem
M. Taksar
and
Lawrence M. Wein
WP#
3512-93-MSA
January, 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A Dynamic
E.
Stochastic Stock Cutting Problem
Krichagina, R. Rubio, M. Taksar
and
Lawrence M. Wein
WP#
3512-93-MSA
January, 1993
tiflj
^^FC
1993
A DYNAMIC STOCHASTIC STOCK CUTTING PROBLEM
Elena V. Krichagina,
Rodrigo Rubio, Operations Research
Michael
I.
Moscow
Institute of Control Sciences,
Center, M.I.T.
Taksar Department of Applied Mathematics,
SUNY Stony
Brook
and
Lawrence M. Wein,
Sloan School of Management, M.I.T.
Abstract
We consider
a stock cutting problem for a paper plant that produces sheets of various
sizes for a finished
when
to shut
goods inventory that services customer demand. The controller decides
down and
into sheets of paper.
restart the
The
paper machine and how to cut completed paper
objective
is
to minimize long run expected average costs re-
lated to paper waste (from inefficient cutting), shutdowns,
A
finished goods inventory.
Brownian control
mal. solution.
in the
The
linear
and backordering and holding
two-step procedure (linear programming in the
second step)
is
developed that leads to an
program greatly
restricts the
number
effective,
first
step and
but subopti-
of cutting configurations
that can be employed in the Brownian analysis,
and hence the proposed policy
implement and the resulting production process
is
tive
rolls
is
easy to
considerably simplified. In an illustra-
numerical example using representative data from an industrial
facility,
the proposed
policy outperforms several policies that use a larger ntunber of cutting configurations. Finally,
we
discuss
some
alternative production settings where this two-step procedure
be applicable.
December 1992
may
A DYNAMIC STOCHASTIC STOCK CUTTING PROBLEM
Elena V. Krichagina,
Institute of Control Sciences,
Rodrigo Rubio, Operations Research
Michael
I.
Moscow
Center, M.I.T.
Taksar Department of Applied Mathematics, SUNY Stony Brook
and
Lawrence M. Wein,
We
consider a problem that
is
Sloan School of Management, M.I.T.
commonly faced
in a
paper factory: how to cut com-
pleted rolls of paper into individual sheets for customers.
traditionally
modeled as an integer program: given a
as 8.5" X 11",
and an unlimited amount of paper on a
into the individual orders so as to minimize the
This stock cutting problem
set of orders of various sizes,
roll of specified
amount
such
width, out the
of paper waste.
A
is
roll
huge literature
has emerged on various generalizations of this problem; readers are referred to the recent
special issue of
European Journal of Operations Research (1990) and references
Our formulation
fact that
differs greatly
many paper factories
paper plant that motivates
sheet cutting
is
from the existing literature and
cut rolls in anticipation of customer
this
dri\-en
by the simple
demand. The particular
study guarantees same day delivery of orders, and because
a time-consuming operation, completed paper
into sheets of various sizes
is
therein.
and placed
in a finished
rolls are
immediately cut
goods inventory that services the actual
customer demand. This make-to-stock view of the problem leads to three complexities.
First, since
and
demand
is
uncertain, the problem
stochastic one, rather than as a static
is
more
realistically
formulated as a dynamic
and deterministic mathematical program.
Sec-
ond, the stock cutting decisions are intimately related to the amount of paper produced,
and consequently these two decisions need
faced by this factory
is
to
be considered
jointly.
Finally, the
problem
multi-criteria in nature. In addition to wasted paper that cannot
be cut into sheets, there are several other costs, which are described below, that are of
significant concern.
1
Since paper machines are extremely expensive, plant managers ideally like to keep
them rvmning
continually.
However, at
particular factory, plant capacity
tliis
higher than demand, and hence inventory would build up indefinitely
if
the machines were
never shut down. Since several hours are required to turn an activated machine
one working
least
needed to get an
shift is
of paper are wasted (although the paper
idle
off
and
at
machine into working order, many pounds
partially recycled)
is
slightly
is
and
significant labor costs
axe incurred during a shutdown.
Furthermore, as in
concern.
Managers
many
industries,
prompt and
Our
customers
at this facility believe that
facility to carry millions of
goal in this study
is
to find a
customer delivery
who do
is
not receive
of
utmost
same day
These large shutdown and backorder costs
delivery often pursue future orders elsewhere.
have led this
reliable
pounds
of
paper sheets
in finished
dynamic scheduling policy
goods inventory.
to minimize shutdown,
waste and inventory backorder and holding costs, where a scheduling policy simultaneously
decides
when
to turn the paper machine on
and
off
and how
to cut the completed paper
rolls.
The problem considered
ways.
here,
which
is
formulated in Section
Whereas most plants have many paper machines, we
paper machine. Also, a finished sheet of paper
and changing grades on
take several minutes.
idealized in several
will consider
characterized by
its
only a single
grade, color and size,
paper machine can take about an hour and changing colors can
a
We
is
1. is
will ignore these
two set-ups and assume that the paper machine
under consideration produces only a single color of a single grade. However,
plants, a popular grade, or even color-grade combination,
is
in
most paper
often processed on a single
dedicated paper machine. Finally, shutdown times will not be explicitly modeled.
Rather than attempting to find an optimal solution to
we look instead
procedure
is
for
an
effective solution that
taken: in the
activity rates to
first step,
which
is
is
this very difficult
relatively easy to
implement.
carried out in Section
2.
we
A
problem.
two-step
find the a\-erage
minimize the average paper waste subject to meeting average demand. In
this linear
(that
is,
program, a different activity
is
defined for each possible cutting configuration
combination of sheets that simultaneously
an activity rate
is
fit
on the width of the paper
roll),
simply the fraction of time that the paper machine employs this
In the second step, which
is
described in Section
3,
and
activity.
these activity rates are taken as given,
and we consider the problem of finding the dynamic scheduling policy that minimizes the
long run expected average shutdown and inventory backorder and holding costs.
Under the heavy
traffic
conditions that the paper machine must be busy the great
majority of the time to satisfy customer
large relative to the inventory costs, the
demand and
the shutdown cost
dynamic scheduling problem
a dynamic control problem involving Brownian motion.
is
The Brownian
is
sufficiently
approximated by
control problem
is
equivalently reformulated in terms of workloads, and in the course of finding an optimal
solution to the workload formulation,
we provide the
first
complete solution to an impulse
control problem addressed in Bather's (1966) classic paper.
solution
down
is
easily interpreted in terms of the original
policy,
where the paper machine
is
The workload formulation
problem to propose an {s.S) shut-
shutdown when the weighted inventoiy process,
which dynamically measures the amount of machine time invested
tory, reaches the value S,
drops to the level
s.
and the machine
The determination
is
restarted
in the current inven-
when the weighted inventory process
of the values of
.s
and S
is
reduced to the solution
of two equations that are expressed solely in terms of the original problem parameters.
Since a one-to-one correspondence does not exist between cutting configurations and
sheet sizes, interpreting the solution of the workload formulation to obtain a cutting policy
is
not straightforward. Consequently, we propose a heuristic cutting poHcy in Section 4 that
incorporates two main concepts. First,
from the setting
second idea
is
of a multiclass
we adapt
Zipkin's (1990) myopic look ahead policy
make- to-stock queue to our more complicated
to find target inventory levels for each sheet size at the
to minimize the inventory costs incurred during the
attempts to attain the target inventory
levels
3
moment
setting.
of
The
shutdown
shutdown period. The proposed policy
when the machine
is clo.se
to shutting
down
(that
is,
when the weighted inventory process
approach S), and uses the myopic
starts to
poHcy otherwise.
The proposed poHcy
from the
facility that
is
model that uses representative data
tested on a simulation
motivated this study. The machine makes three different sheet
and our proposed policy uses only three
of the
many
possible cutting configurations.
simulation results show that the average cost imder the derived values of
6
sizes,
Our
and S are
very close to the corresponding cost under the best values of s and S, which were found
via a search procedure. Surprisingly, our proposed pohcy outperforms
heuristics that
employ more than three cutting configurations
manner; details are given
problems
in a
here can be applied to a broad range of multi-
where each
activity generates a certain
each product, in either a deterministic (as
example, when random yield
is
many
employed to produce
a
number
of
useful
completed units of
present) manner. Each different production process has an
is
associated with paper waste,
other settings the cost reflects the variable cost of employing the process. Average
production costs are minimized in the
and holding
costs (and
In Section 6,
we
shutdown
briefly describe
and blood separation
industries)
number
first
costs,
some
no more than
K
optimal
appropriate) are minimized in the second step.
alternative settings (from the steel, semiconductor,
LP
may be
applicable.
activities, say J, will typically
of different products, say A',
tive activity rates in the
if
step of the procedure and inventory backorder
where our procedure
Although the number of potential
the
a
is
in the stock-cutting case) or stochastic (for
associated variable cost; although in our setting this cost
in
when
The approach
make-to-stock setting.
large set of possible activities, or production processes, can be
set of products,
seemingly intelligent
in Section 5.
The two-step procedure proposed
criteria scheduling
in a
more complicated
no more than
solution in the
be much greater than
A' production processes
first
have posi-
step of the procedure, and hence
production processes are considered in the second step of the procedure.
In fact, only the processes selected in the
first
step are required for system stability: for
rigorous stability proofs along this line for
(1989) and Courcoubetis and
et al.
is
somewhat related problems,
Rothblum
(1991).
Therefore, the proposed solution
very easy to implement and will lead to a production system that
manage. Moreover, Johnson
axid
set-up costs are often incurred
costs,
which
until recently
will
when using an
be incurred
Kaplan (1987) and others have observed that
when an
additional production process
is
significant
employed. These
have been largely ignored by the cost accounting community,
no
or
little
in-
cost
is
additional activity in our stock cutting example, these set-up costs
in all the
chical approach,
relatively easy to
is
clude administrative, labor, purchasing and engineering costs. Although
incurred
see Courcoubetis
examples sited
Hence, the top level of our hierar-
in Section 6.
by greatly restricting the number of activities employed, aids
in
reducing
these often considerable costs and leads to a simpHBcation of the production process. Of
course, this restriction on the
may
number
of activities, or processes, that
can be employed
lead to a suboptimal solution to the original multi-criteria problem. However, in our
computational study
in Section 5,
employing more than A' of the J
1.
we were unable
to improve
upon the proposed
activities.
Problem Formulation
Consider a paper machine that produces A' different types of sheets.
sheet will be referred to as a product and
and
dfc2,
where dki <
dk2-
We
is
in
=
11.
will call d^i the
product width and
In keeping with industry conventions,
terms of pounds, not sheets.
merely assume that
the asymptotic
Dk
satisfies
demand
rate
square of the mean) of the
we
will
rfjt2
t.
No
specific distributional
of
f/^i
the length. These
(f^j
=
measure the quantity
Let Dk(t) denote the cumulative
product k demanded up to time
Each type
characterized by the sheet dimensions,
dimensions are measured in inches so that for standard writing paper,
dk2
policy by
number
assumption
is
of
8.5
of
and
paper
pounds
required:
of
we
afunctional central limit theorem, where Xk and vl denote
and squared
demand
coefficient of variation (variance divided
process. For simplicity,
we assume
that the
by the
demand
for individual products
independent, although this assumption can be easily relaxed.
is
number
Let S{t) be the cumulative
is
paper
of
continuously busy during the time interval
process with asymptotic service rate
fi
We
[0,t].
and squared
produced
rolls
assume that S{i)
coefficient of variation v^
policy dictates the cutting configuration for each completed
activity for each possible cutting configuration.
dimension
1,
...,
J
is
Jjti
or
characterized by
dimension, and nj
roll,
t2,
w
number
the
number
the
weigh p pounds and be
"j.fci,
a renewal
is
.
The
and we define a
Since each product can have
length dimension dk2 placed along the
its
the paper machine
if
cutting
different
width
its
=
width, activity j
roll's
of sheets of product k placed along its width
of sheets placed along its length.
inches wide. Since each activity must
fit
Let each paper
on the
roll,
roll
the waste
Wj associated with activity j must be nonnegative, where
K
= w
Wj
-'^(riju-idk] +nj_k2dk2)-
(1)
k=\
Define the A' x J matrix
F=
Fkj
so that Fkj
paper
roll
is
the
number
of
by
[Fi^j]
=
^'
•'•
(
pounds
of product k
(2)
)p.
w
on
a roll cut
according to activity j
cut according to activity j will often be referred to as a type
j
.
roll.
Recall that our scheduling policy dynamically decides whether the paper machine
busy or
is
idle
and how
to configure each completed
the bottleneck, and the
by a
it
In particular, the scheduling policy
vector of nondecreasing processes {Tj{t),t
t.
number
of
machine
they are produced by the
>
0}.
cumulative amount of time that the paper machine allocates to type j
Let Zkit) be the
is
convenient to assume that the production and cutting
undertaken simultaneously.
J— dimensional
hi practice, the paper
rolls are typically cut directly after
paper machine. However, we find
of a roll are
roll,
A
pounds
of product k in finished
where Tj(t)
rolls
during
goods inventory
where a negative quantity represents backordered demand; the vector
G
defined
is
Z =
(
Z^
is
[O.t].
at
)
the
time
will
be
referred to as the inventory' process.
If
we assume
that Z(0)
=
then
0,
J
Zk{t)
= Y,^kjS{T,{t))-Dk{t)
for
^•
=
1,...,A'
and
/
>
0.
(3)
Also, let the cumulative idleness process I(t) be the cumulative amovtnt of time that the
paper machine
is
idle in [0,
<],
where
K
I{t)=t-Y^Tk{t) for/>0.
(4)
Let J{t) denote the cumulative nvunber of times that the paper machine
during
[0,^];
the cumulative shutdown process J can be recovered from the cumulative
idleness process I.
the machine
down
shut
is
is
For concreteness, we assume that shutdown costs are incurred when
turned
As
off.
Harrison (1988). a scheduling policy
T
is
continuous, nondecreasing and T(0)
T
is
nonanticipating with respect to Z.
=
(6) implies that the scheduler
T must
satisfy
0.
(5)
(6)
J are nondecreasing with 7(0) = J(0)
/ and
where constraint
in
=
0.
(7)
cannot observe future demands or
ser^•ice
times.
Define the cost function c^ for k
,
,
=
1.....A'
/
-bkx
[hi^x
where
is
bk represents the
problem
cost
is
and
let
if
X
if
J
<
>
0,
,p,
0,
backorder cost per pound per unit time for product
the inventory holding cost per
shutdown
by
C^
pound per
unit time for product
k\
k.
and hk
Let Cs denote the
denote the cost per pound of wasted paper. Then the scheduling
to find a policy
T
mm limsup-E[ /
to
y
Ck{Zk[i))df
+ C^J(T) +
^^y
irjS(Tj{T))]
subject to constraints (3)-(7).
The
2.
First Step:
The number
Lineeir
Program
of possible cutting configurations, J,
step of our procedure
will actually
A
is
can be huge, and the goal of the
first
to select a small subset of these configurations, or activities, that
be employed
Brownian analysis
in the
in the
second step. These activities
will
be selected by solving a linear program that has J decision variables and A' constraints.
K activities will be used in the Brownian analysis, which greatly simplifies
Hence, at most
both the analysis and the resulting production process.
The
program are the
decision variables of the linear
activity rates Xj.
where Xj
fraction of total time (not just busy time) that the paper machine produces type j
the
is
rolls.
Let
Rkj=fiF^.j.
so that R^;j
rolls are
is
(10)
the rate, in pounds per unit time, that product k
produced. The linear prograni finds the
is
generated when type j
minimize the average
activitj' rates that
paper waste subject to meeting average demand:
J
min y
Xj
we assume
(11)
^Rkj^j =
subject to
Hereafter,
Wjx,
>
for
J
for
>^k
=
k
=
l
A',
(12)
(13)
1,..., J.
that a unique nondegenerate solution pi. ...,pj exists, although
the subsequent analysis can be employed with any optimal, possibly nondegenerate. solution.
pj
>
Without
for j
loss of generality,
—
1.
....
A'
and
pj
assume that the
=
for j
which we denote by the trafRc intensity
able to meet average customer
demand
=
A'
+
p. is less
original
1,
....
J.
activities are
We
also
indexed so that
assume that X^,Lj
Pj.
than one. so that the paper machine
over the long run.
8
J
is
The Second
3.
In this section,
Step:
A
Brownian Analysis
we use the LP
two ways: only these
(3)-(9) in
was minimized
in the long
K
solution p\,-.-,pK to simpHfy the scheduHng problem
activities
can be employed and the waste
run average sense in
(11), will
LP
policy proposed here will automatically enforce the
=
K
(1988),
Many
and Cu,
=
0.
which
be ignored. Since the scheduling
solution, our procedure minimizes
the average waste cost. Hence, the remaining control problem
J
cost,
is
identical to (3)-(9) with
Following the procedure taken in Sections 3 through 5 of Harrison
we approximate the scheduling problem
of the details of the formulation
as a control
and analysis
of the
be omitted since they are similar to those described
in
problem for Brownian motion.
Brownian control problem
will
Wein (1992) and Ou and Wein
(1992).
The Limiting Control Problem.
3.1.
problem, we need to assume that the server
shutdown
heavy
cost
traffic
intensive
is
is
To obtain a
limiting
Brownian control
busy the great majority of the time and the
sufficiently large relative to the inventory costs.
We
suspect that these
conditions typically hold in practice, since paper machines are very capital
and shutdown
indexed by n where the
such that y^(l
costs are substantial. Formally, one defines a sequence of s}'stems
traffic intensity
— p") and C" /n'^'^
p" and shutdown cost
C"
of the
77"^
system are
converge to positive constants, while the inventory costs
remain unsealed. The original system under study
is
identified as a particular
system
in this
sequence. Since we will not be attempting to prove convergence of our optimal controlled
processes to limiting controlled diffusions, the limiting control problem will be informally
stated without introducing the substantial
to define a sequence of systems.
a proof of convergence for
amount
of additional notation that
Readers are referred to Krichagina
et al.
Brownian control problems that approximate
(
is
required
1992a, b) for
single prf)duct
examples.
Define
o^.
=
pk/p to be the proportion of the paper machine's busy time that
9
is
)
devoted to producing type k
=
by k
1,...,A',
rolls.
Although
no special relationship
exists
Y
the scaled centered allocation process
activities
and products are now both indexed
between activity k and product
amount
Zk{t)
=
^^^
/(/)
=
^^
V"
for
J{t)
=
^^
for
is
for
process, respectively. Let A'
=
V^(l^j=i RkjCtj —
=
fc
t
and /
=
The
with
left
{Ikj)
is
t>0,
(A'l,
(15)
>0, and
(16)
^>0.
and J
(17)
?7
—
>
oc.
as the inventory, idleness,
and we
will
and shutdown
...,
A'jt)
be a A'— dimensional Brownian motion process
=
1
K
^i;)^k
K
^kj
l,...,A'and
obtained by letting the parameter
refer to the limiting scaled processes Z, /
drift
sides of these equations
of notation required)
The Brownian approximation
with
Define
by
and defined the scaled processes (the same symbols are used on both
to reduce the
k.
= ^kvlhj +
^i^^
and covariance matrix E. where
K
Yl X^
niin(p/.
Fji^km
(IS)
/9„,
the A' x A' identity matrix.
limiting control problem
hmits) process
mm
V
is
to choose a
A'— dimensional
RCLL
(right
continuous
to
limsup
r_oc
^E[f
i
Jo
yck{Zkit))dt + CsJiT)]
(19)
jfr;
k=\
K
su'bject
to
J
"..[',
Zjt(/)
=
A';^-(/)
".. V /
-
Y^ RkjYjit)
for k
=
1
A
and
t
>0.
(20)
K
Iit)
= J2^'k{t)iort>0,
(21]
k=\
I and J
l'(0)
=
are nondecreasing with 7(0)
and
Y
is
=
J(0)
=
0.
and
nonanticipating with respect to A.
10
(22)
(23)
The Workload Formulation.
3.2.
in
terms of workloads.
by restricting attention
step
,
to reformulate (19)-(23)
is
(Rkj) denote the invertible A' x A' matrix obtained
—
in definition (10) to activities j
elements of the matrix R~^
for
R =
Let
The next
and define the expected
1,...,A'.
Let i?r^ denote the
consumption
effective resource
rrif;
product k by
mfc
K
= ^i?-^ iovk =
Define the one-dimensional Brownian motion
B
(24)
hy
K
= ^mkXk{t)
B{i)
l,...,K.
for
>
f
(25)
0,
k=l
B
SO that
has
drift 6
=
y/n{l
—
p)
>
A
A
(7^
=
^ Xk^iWk +
it
R
=
//^'^
A
k=\
/=1
,
or
A'
A'
FjiFkm min(p;. pm
XI "'>} X! "^'^ X] X]
1-1
l
= m'^m^
and variance a^
1
of
Wein (1992)
formulation of the limiting control problem (19)-(23)
is
to choose the
+
C,J{T)\
Since
process
is
Z and
invertible,
it
follows
from Proposition
limsupi^r/
T-oc 1
subject to
Jo
y
Ck{Zk{t))dt
that the workload
A— dimensional
'^nnZk{i) = B{t)-
Z and
I(i)
{or
t
>0,
(28)
=
J(0)
/ are nonanticipating with respect to
The two problem formulations
(19)-(23)
=
0.
and
A'.
is, all
(29)
(30)
and (27)-(30) are equivalent
objective functions are identical and every feasible (that
for the limiting control
(27)
j^i
I and J are nondecreasing with 7(0)
y
(26)
the one-dimensional process / to
mm
policy
).
m=l
in that their
the constraints are satisfied)
problem yields a corresponding feasible policy {Z.I)
for
the workload formulation, and every feasible policy {Z,I) for the workload formulation
yields a corresponding feasible policy
Y
for the limiting control problem.
11
The cumulative
,
shutdown process J
is
not an explicit control in (27)-(30), since
cumulative idleness process
The Solution
3.3.
is
it
dictated by the
/.
Workload Formulation.
to the
Not only
is
the workload
formulation easier to solve than the limiting control problem, but the workload formulation
solution,
/, is easier to interpret in
process
Z and
which consists of the scaled inventory process
The
}'.
control process
cumulative idleness process
terms of the original problem than the scaled centered allocation
solution to the workload formulation requires two steps.
Z
is
derived in terms of the control process / in the
optimal control process /
carried out in Section 4 of
derived in the second step.
is
Wein
(1992), only the solution
one-dimensional weighfed mvenior}- process
Since the
Z*
will
The optimal
first step,
first
and the
step has been
be displayed. Define the
W by
A
W{i) = ^nnZk(t)
This quantity
the weight
is
is
the weighted
the expected
sum
eff"ective
resource consumption.
hi;
it is
possible for j
=
/.
Then
Zm =
f
=
—
mj
=
—
mi
nik
mm
i<k<K
for
and
loss of generality, define the indices j
mm
i<<r<A'
where
t>0.
(31)
of the finished goods inventory for each product, where
W{t) = B{t)-I{t)
Without
ioT
rrtk
/
t
>
By
also have
by
hj
and
(33)
(34)
>
is
if
A-=j and
ii
k^j
if
A-
=
/
and W{t) <
0.
if
A-
7^
/
and H'(0 <
0.
ir(/)
0.
'"'
(O
we
(32)
0.
the optimal solution Z*{t)
inn
(28).
(35)
and W{t)>0.
and
W(t)
Zt{i)=
I
""'
(30
12
Notice that the optimal control process Z*
is
expressed in terms of the control process I
via (32).
To describe the remaining
bi/mi, where the indices j and
control problem for J, let us define h
are defined in (33)-(34),
/
(
hr
1^
—
/(^)=
The second
if
..
,
ox
II
X
>
X
^n0.
<
step of the workload formulation solution
ticipating (with respect to
X)
and
=
hj/nij and
b
=
let
0,
is
(37)
to find a nondecreasing, nonan-
control process I to
T
min limsup^£[/
W{t)
subject to
:^
B{t)-
+ CsJ{T)]
f{W{t))di
for
I{t)
/
>
0.
(39)
Recall that the shutdown process J measures the cumulative
control process /
lem
for
is
exerted.
Problem
(38)- (39)
is
a
exerted at certain points in time.
A
form
Bather's problem
S whenever
is
an
the level
.s
is
the determination of
s
of instantaneous
is
reached, where
level s
S >
s;
of
.s
and notes that equation
The
solution to
1(f)
and
this suggests that
an optimal pohcy to
whenever the
level
S
is
reached, where
S >
s.
A
does not appear possible for this problem. Bather reduces
and S (and the optimal long run average
to the solution of
+
displaced to the level
Puterman (1975) employs a regenerative argument
and S
B(t)
is
equations (6.1), (6.2) and (6.4) of his paper, and asserts that
assertion,
=
negative, was solved by Bather (1966).
is
5
jumps, or impulses.
changed to \V{i)
S) policy: the controlled process W{t)
closed form solution to 5 and
due to the fixed
fact that,
(s,
our problem displaces W(t) to the
of times that the
nearly identical problem (essentially a mirror image
of our problem), the only differences being that (39)
the drift of the Brownian motion
number
one-dimensional impulse control prob-
Brownian motion. The word impulse stems from the
cost Ca, the optimal control process / takes the
(38)
cost) to the solution of
5 >
>
Bather appears to contain
13
a
this
to reduce the determination
two equations that are displayed on page 156
(6.4) of
Under
.s.
of his paper.
minor misprint.
The same procedures can be used
to derive analogous equations for our problem.
>
>
Using Putermcin's approach and Bather's assertion that S
5,
we
find that
5
is
the
solution to
2^ = (^)(1
and
5
_ ^5 -
e-^^)^
_
2(1
- ^5 +
^
- e--%
(40)
can be determined from
S-s^i^){S-<f>-\l-e-^')),
where
it
=
4)
26/a^.
we
If
let
(41)
g denote the long run average optimal cost of (3S)-(39). then
satisfies
g
=
b{<f>-'-s).
(42)
However, upon numerically solving various problem instances of (40)-(41
tered several cases where
S >
>
.<:
0.
S >
our problem
the mirror image of his problem, (40)-(41)
The reason
(39).
the drift of
s.
is
Hence,
B
is
if b is
much
that
possible for 5
5
< 5 <
it is
0,
sign
is
< S <
can be optimal for
in
.s
>
0.
incomplete: since
explained as follows. Although
still
An
is
attain values below
not very large,
and
analogous argument shows
Bather's problem; for the inventory problem motivating
highly unlikely for the parameter values to satisfy the conditions that
and
To complete the
it
is
only a partial solution to (3S)-
greater than h in (37). the shutdown cost
not far from zero,
Bather's study,
same
of the
is
positive in (38)-(39). the controlled process can
is
imply
does not always hold, and hence his solution
and 5 can be
that 5
the drift
it is
s
we encoun-
Closer examination reveals that Bathers original
assertion that
>
).
so
it
is
not surprising that he did not consider this case.
solution to (38)-(39),
under the assumption that
S>
s
>
0.
we simply use Puterman's regenerative approach
In this case, the parameter
A = S—s
can be found
by solving
.--)--•
^^A^(
2J
VI -e--^-^
h
'
(?
14
(43)
and then
5=
--In
4>
general theory of variational inequalitiies, which
The
now
is
well developed, shows
that the Hamilton-Jacobi-Bellman equation of the impulse control problem has a unique
solution
and
a feasible
this solution yields
(i.e.,
an
(-5,5)
optimal policy. The latter implies the existence of
5 is
the sign constraint on s and
satisfied) solution to either (40)-(41) or (43)-
(44). This, however, does not exclude the possibiHty that
both (40)-(41) and (43)-(44),
to feasible solutions to
tangential issue for this problem,
It
exist.
must be employed. Whether or not exactly one
solution
local
all
minima, corresponding
In this case, the minimal cost
local
minimum
and therefore we have made no
should be mentioned, however, that
(40)-(41)
two
exists
effort to
is
prove this fact.
of our numerical computations with equations
and (43)-(44) have yielded one solution that
violates the sign constraint
solution that satisfies the sign constraint. Moreover, the two solutions coincide
optimal value of
S
a rather
and one
when
the
equals zero.
For completeness, the solution to Bather's problem when
5
< 5 <
is
given by (using
Bather's notation)
S-s
^
p
'
Vl
-
e-A(S-^)
;45)
2/
A
and
\(S-
1
s)
^^="^"[(^)(^^^^^)JFinally, note that
to (43)
is
S =
5 (i.e.,
if
the shutdown cost Cj
A =
0).
K we
let
A
=
->
in
^''^
problem (3S)-(39). then the solution
in (44),
then L'Hopital's rule yields the
limiting solution
S = ^ln(l +
When
Cs
=
0,
(47)
-^).
problem (3S)-(39) becomes a singular control problem, and the solution
(see Section 6 in
Wein)
is
I(t)= sup [5(..)--ln(l-f-)l^.
o<s<r
<P
15
/'
i>0.
(4S)
so that the controlled process
is
a reflected Brownian motion in the interval ( -oo,
{b/h)). Hence, as expected, the
4.
two solutions coincide as the shutdown cost
C'a
<?
-^
'
ln(
1
-|-
0.
The Proposed Scheduling Policy
we propose a scheduling
In this section,
policy that
is
partially based
on the preceding
Recall that a scheduling policy dictates the shutdown/startup times for the
analysis.
machine, and dynamically chooses a cutting configuration for each
roll
of paper that
is
produced.
The shutdown
policy
is
easily interpreted in
terms of the optimal control process J*,
which represents the scaled cumulative idleness process. By (31
)
and
(39). the [s.S) policy
derived in (40)-(41) and (43)-(44) suggests that the machine should be shut
down when
the scaled weighted inventory process W{t) reaches the level 5, and should be reactivated
when W{t) drops
for the original
specifically,
v/n(l
-
if
to
.s.
Moreover, by reversing the heavy
traffic scalings.
the (^.S) policy
system can be expressed solely in terms of the problem parameters. More
we substitute s/y/n
p) for 6 in (40)-(41)
S/y/n for 5, A/v^" for A. CJn^l'^
for s,
for
C. and
(43)-(44), then the scahng parameter n vanishes,
and
and
these four equations become, respectively,
a\h +
b)
~^
h
\
_ 2fl _
(1
- P)C,
)
G^
^[LZ^ + 2{^-^—^f
2f
\
-
e-2(i-p)57'^'y
(49)
g^A
1\
(5i;
and
^"
2(1
-p)
\
L
L'/?
+
,
I-H_2/,2(l-o)A/(T=
6'V2(e2<i-p)A/'^^
IG
-1 J
._
1
[52)
The proposed shutdown/startup
employs the resulting
{s,
policy calculates s
and S from these equations, and
S) policy with respect to the original (that
unsealed) weighted
is,
L'
inventory process Xlit=i ""^k^kit)-
Brownian approximations to queueing system scheduling problems
In
ple,
is
exam-
Harrison 19SS and Wein 1992), the policy for "who to serve next," which corresponds
to the cutting policy here,
the
(see, for
LP
that
is
embedded
given by (35)-(36).
there
is
is
typically interpreted in terms of the pathwise solution to
workload formulation. In our case, the pathwise solution
in the
However, unlike the previous problems that have been analyzed,
no one-to-one correspondence between the A' cutting configurations and the A'
products.
Consequently, we have been unable to interpret (35)-(36) to obtain a reliable
cutting policy.
Instead,
first
idea,
we propose
which
service time look
is
is
due
a heuristic
to Zipkin.
ahead policy
to use a
is
here.
myopic look ahead
for a single server multiclass
essentially a special case (where A'
problem considered
pohcy that incorporates two
=
J,Cs
—
essential ideas.
policy.
The
He proposes
a
make-to-stock queue, which
R
and the matrix
is
diagonal) of the
This policy dynamically serves the class that minimizes the
expected reduction in cost rate per unit of time after one service completion. The policy
has
its
roots in Miller "s (1974) transportation look ahead policy for the decision of which
base to send an item repaired at a central depot, where transportation time plays the role
of service time.
Numerical results
in
Veatch and Wein (1992) show that Zipkin"s policy
performs very well for multiclass make-to-stock queues.
We
propose a simple extension to Zipkin"s policy that makes
cutting problem considered here. Since a paper
it
is
roll's service
time
eventually cut, the policy chooses the cutting configuration
configurations identified by the
LP
in Section 2 will
is
j
it
adaptable to the
independent of how
=
1
K
(only the
be considered) that minimizes the
expected inventory cost rate incurred after one service time: that
is.
at
time
/.
we
cut a
type j*
roll,
where
r
and the expectation
Although
= axgminE
is
+
Y^Ck(^Zk{t)
Fk,
- Dk{S})
taken over both the service time
this policy vi'orks well
when no shutdown
(53)
,
S and
the
demand
costs are incurred,
processes
an
effective
cutting policy for this problem needs to look beyond one service time and, in particular,
must prevent the vector inventory process from entering a vulnerable position
incurring
we
many
costly backorders) during the potentially long
derive a target inventor}' level tk for each product k at the
when W{t) = S)
e^.,
we assume
example,
shutdown period. Hence,
moment
to minimize inventory costs incurred during the
order to obtain a closed form expression for
(for
that
of
shutdown
(i.e.,
shutdown period. In
demand
is
deterministic
during the shutdown period. Under this assumption, the length of the shutdown period
is
S-s
(54)
and the target inventory
levels are
found by solving
mm
in
^
I. _ 1
=\
k
/
Q(ei.
-
\ki)dt
(55]
''0
such that
y
rrik^ii
=
S,
(56)
k=l
where S
is
the unsealed
solution to (55)-(56)
When
maximum
weighted inventory
level derived in (49)-(52).
The
is
the weighted inventory process gets sufficiently close to S,
we use the cutting
con-
figuration that minimizes the Euclidean distance between the expected resulting inventory
levels
and the target
levels; that is, cut
arg
mm
a type j*
roll,
where
A,
J](Z,(0 +
F,,
--^-6,)-.
^'
\k=\
18
(5S)
.
Our proposed cutting
policy uses the myopic policy (53)
K
Y^mkZk{t)<s +
I
k=l
when
'r{S-s),
(59)
K
J2mkZk{t)>s + ^{S-s),
(60)
and uses the target inventory policy
(58)
when
ife=i
where the parameter
-)
€
[0,1] quantifies the
proximity to machine shutdown.
simulation study in the the next section, we set the parameter )
policy's
5.
and found that the
->
An Example
we compare our proposed scheduling policy
on a simulation model of
with a manager of the
facility
for the simulation
under study.
produces three different products. Table
cost rates for each product.
The demand
Poisson process; the average
against several other heuris-
a hypothetical system.
The data
The Problem Data.
all
0.9,
performance was quite robust with respect to the specification of
In this section,
tics
—
In the
demand
I
We
model are based on con^'ersations
consider a single paper machine that
provides the
rate for each product
rate of orders
is
demand
size,
is
assumed
given in Table
and inventory
rate
I,
to
a
compound
all
orders for
be
and
products are independent Poisson random variables with mean 200 pounds. Hence,
we denote the average demand
rate of product k orders by
equals 40, 200Ai.
19
A^..
then the term
A^^
if
r^ in (IS)
PRODUCT
ACTIVITY
1
8.5"
11"
17"
22"
28"
35"
WASTE
pj
The optimal
target inventory levels are given by
ei
=
68, 441 e2
,
= HI- 280 and
63
=47, 609
pounds.
Straw
Since our problem formulation
Policies.
policies to test against the
III.
We
proposed policy, which
consider three other policies;
all
is
is
new, there are no obvious straw
denoted by
PROPOSED
in
Table
three policies use a shutdown/startup policy
characterized by an (^,5) policy on the total unweighted inventory process, where the
values of s
most
and S are found via a search on the computer simulation model. The
simplistic policy
is
called the
LP
the solution (p\,P2-,p3) given in Table
repeats a cycle often
tenth
roll of
The second
where the
rolls,
each cycle
II.
first
More
III,
and
nine rolls are activities (2,1.2,1,2.1.2,1.2). and the
MYOPIC,
uses the myopic policy defined in (53). but considers
Wj < w. The cost minimizing value of the parameter
the
third policy, called
MYOPIC
MYOPIC+TARGET,
w
is
all
configurations j with waste
found using
a search procedure.
uses our proposed cutting policy, but as in
policy, considers all configurations j
Simulation Results.
deterministically enforces
specificallj^ the cutting policy continually
not the three cutting configurations specified by the LP, but
The
it
and
randomized according to the probabilities (0.193.0.572.0.235).
is
policy, called
policy in Table
first
with waste Wj
<
iv.
For three of the four scheduling policies, we ran ten independent
runs, each consisting of six years.
To obtain
made
LP
50 independent runs for the
policy.
sufficiently small confidence intervals,
we
Although we do not expect the demand
facing this facility to remain stable for six years, a sufficiently long period was required to
eliminate round
off"
effects
the machine activated.
policies (the
due to shutdowns. All runs began with the system empty and
The
parameter 7 for the
three policies,
minimizing values of the parameters for the scheduling
cost
PROPOSED
and the parameter
three runs of six years.
w
for the
poHcy, the parameters
two myopic
policies)
.<i
and S
for the other
were found
b}-
making
POLICY
PROPOSED
MYOPIC+TARGET
LP
WASTE
INVENTORY
SHUTDOWN
TOTAL
COST
COST
COST
COST
99,331(±195)
34,640(±575)
18,967(±361)
152,938(±663)
121,039(±383)
21,180(±380)
24,865(±589)
167,084(±535)
67,840(±8431)
20,434(±254)
1S7,541(±8484)
34,970(±703)
26,618(±551)
192.526(±850)
99,267(±80)
MYOPIC
130,938(±370)
TABLE
The simulation
nual cost, and
costs;
95%
its
III.
Simulation Results: Average Annual Costs.
results are reported in Table III,
breakdown
into inventory (holding
confidence intervals are reported for
all
which contains the average
and backorder), waste and shutdown
cost quantities. Notice that paper waste
accounts for roughly two-thirds of the total cost. For ease of comparison. Table
average costs as a percentage of the costs of the
POLICY
total an-
PROPOSED
policy.
l\
reports
the accuracy of our derived values for
which
is less
2%
than
minimizing values,
=
values, s
The
minimizing values.
for the cost
s
5
search yielded an average annual cost of $149,932,
lower than the cost under our derived values.
=
=
—17.5 and S
—39.74 and S
in lower inventory costs
=
175.0, are significantly different than the derived
and higher shutdown costs
cost minimizing value of
We
when 7 =
also tested the
and S resulted
7
0.95
in (59)-(60)
was
0.9.
The
PROPOSED
.s
and
less
policy's per-
The average
this parameter.
and increased by
PROPOSED
relative to the derived values of
IV are 100%, 75%, 131% and 98%.
formance was remarkably robust with respect to
increased by 0.3%
However, the cost
211.17. Consequently, the cost minimizing s
5; its four normalized costs for Table
The
and S, a two-dimensional search was undertaken
than 0.2%i when
=
-;
total cost
0.7.
policy under one slight variant: the (s,S) policy was
based on the total unweighted inventory ^j^-i Zk{t) rather than the weighted inventory
^j(._j ini.Zk{t).
A
search for the best {s,S) policy gave essentially the same cost ($150,037)
as the search under the weighted inventory policy ($149,932).
This close agreement
hardly surprising since the three mjt values are very similar.
Hence, any discrepancy
among
bases
the various policies cannot be attributed to the fact that the
its
PROPOSED
is
policy
(5,5) policy on the weighted inventory, and the other policies base their (s.S)
policy on the unweighted inventory.
As expected, the LP
However, since the
LP
same waste
policy achieves the
policy
is
an open loop policy
(i.e.,
cost as the
the cutting policy
The
of the state of the system), very high inventory costs are incurred.
values of
s
and S are
5
in imits of hours for the
= —15,000 and S =
235,000. Recall that
PROPOSED
and
policy,
PROPOSED
in units of
i;
pounds
is
policy.
independent
cost minimizing
and S are expressed
of paper for the other
three policies.
The
MYOPIC
MYOPIC
and
policy achieves the worst performance of the fovir policies.
PROPOSED
policies
have similar inventory
ventory cost reductions gained by the
MYOPIC
24
costs,
policy while the
it
Since the
appears that the
in-
machine was operating
were
during the shutdown periods.
offset
activities that
is
do not minimize average waste
forces this policy to
The
costs,
poHcy employs
a significant increase in waste cost
shutdown periods
employ more frequent shutdowns and hence incur higher shutdown
cost minimizing values of s
For both myopic
and S are
roll,
=
—50, 000, 5
=
100, 000.
which
u),
is
the
This parameter value allowed ten cutting
3.
When w
was raised to
included and the overall cost increased by 6.3% for the
MYOPIC+TARGET policy. When w
3.0",
two more
MYOPIC
was raised to the next
activities
pohcy and
were
9.49c for the
level of 4.75". eight additional
were included for a total of 20, and the cost roughly doubled for both
The use
maximum
was found to be 2.75", which was the lowest value that
included any activities producing product
configurations to be employed.
s
minimizing value of
policies, the cost
allowable waste on a paper
activities
MYOPIC
Finally, the high backorder cost incurred during the long
incurred.
costs.
Moreover, since the
policies.
of the target inventory levels clearly enhances performance, since the
MYOPIC
MY-
OPIC+TARGET
policy achieves a
OPIC+TARGET
policy achieves significantly lower inventory costs than the other three
policies.
However, relative to the
by increased waste and setup
POSED
policy,
much
costs.
We
were
initially
somewhat surprised
poHcy, which uses only three activities, outperformed the
stricting the
number
by comparing the
in
the
costs, the
MYOPIC+TARGET
waste costs than the
between the two
policy to the
PROPOSED
policy,
PROPOSED
policies
is
PRO-
reason for
re-
performance between these two pohcies can best be seen
MYOPIC+TARGET
shutdown
that the
was to obtain a mathematically tractable
PROPOSED
minimizing values (as opposed to the derived values) of
identical
The MY-
MYOPIC+TARGET
activities; after all, the primary-
of cutting configurations
The discrepancy
policy.
PROPOSED policy, this cost reduction is more than offset
which can employ any number of
problem.
lower cost than the
and the
policy.
due to the
policy has
5
and
S:
policy under the cost
the two policies have
21.8% higher inventory costs than
MYOPIC+TARGET
policy has 21.9% higher
Hence, the 11% difference in average total cost
fact that
25
waste costs comprise roughly two-thirds of
the total cost. Therefore, since waste costs are minimized in the
the
PROPOSED
poHcy may not work as
tory and shutdown costs. However,
this simulation
It is
we
well
when waste
reiterate that the
three policies in the simulation study, but
policies. First, the
PROPOSED
poHcy
is
it
problem parameters employed
PROPOSED
is
step of our procedure,
costs are small relative to inven-
study are based upon our best estimates from
worth noting that not only does the
first
actual
aji
facility.
policy outperform the other
also simpler to
employ than the other
based on derived values of
s
and S, whereas the
Furthermore,
other policies require a two-dimensional search using a simulation model.
the
PROPOSED
which
specifies the proximity to
policies
roll.
policy's performance
is
very robust with respect to the parameter 7,
shutdown. In contrast, the performance of the two myopic
was very sensitive to the parameter w, which
Hence, the parameter 7
is
in
much
is
the
maximum
easier to specify than the
allowable waste per
parameter w.
Other Applications
6.
In this section,
may be
we
briefly describe several other settings
where the two-step procedure
applicable.
Semiconductor Manufacturing.
In semiconductor wafer fabrication, a batch of wafers
produced according to a particular process randomly yields chips of many
different prod-
uct types, which are partially ordered with respect to quality.
rejjresents the
Here,
Fi,j
expected number of type k chips in a batch of wafers produced according to process
These
facilities
usually have
many more
Each production process has
yield higher quality chips.
its
own
The LP
possible production processes than product types.
variable cost,
mode and
photoHthography workstation, and wafers
processing.
and
costlier processes will typically
in the first step of the
variable processing cost subject to meeting average
often operate in a make-to-stock
j.
procedure minimizes average
demand. Wafer fabrication
facilities
often have only one bottleneck station, the
visit
this station ten to twent}- times during
Hence, by focusing on the bottleneck workstation, we obtain
2G
a
scheduling
problem
for a production/inventory
The Browniaji
in
system with customer feedback and random
yield.
control problem in the second step of the procedure was expUcitly solved
Ou and Wein
(1992) and readers are referred there for details.
Similar problems also
occur in fiber optics, ingot and crystal cutting, and blending operations in the petroleimi
industry.
A
Blood Separation.
into its various
variety of separation processes can be used to separate blood
components (plasma, red blood
finished goods inventory servicing hospitals
separation process has
its
own
which are maintained
cells, etc.),
and other health care
facilities.
is
Here, each
variable cost, which are minimized in the first step LP,
produces specific amounts of each component. One key aspect that
model
in a
is
and
not captured in our
the perishability of the blood; typically, blood components (except for plasma)
have to be discarded after sitting in inventory for a certain number of days.
made from
Steel Industry. In a tube mill, each product type can be
several different
blooms, or starting stocks, and a particular starting stock can be used to produce several
different products. If
we
define an activity for each feasible combination of starting stock
and product type, then the
activity cost includes
both raw material costs for the starting
stock and processing costs, which differ by activity.
dynamic production policy
The second
for the bottleneck operation of the
step analysis develops a
tube
mill.
Acknowledgments
We are deeply
indebted to
Tim
for generously sharing his time
Loucks, plant manager at Strathmore Paper Company,
and expertise.
engaging in helpful conversations about the
We
also
thank Anant Balakrishnan
steel industry.
This research was partially
supported by a grant from the Leaders for Manufacturing Program at
Science Foundation Grant
Award No. DDM-9057297.
for
MIT and by
National
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