W.I.T LJSH
DEWEY
HD28
.M414
The
Economic Lot Scheduling
Problem:
Heavy Traffic Analysis of
Dynamic Cyclic Policies
Stochastic
David M. Markowitz. Martin I. Reiman and
Lawrence M. Wein
#3863-95-MSA
October,
/
1995
MASSACHUSETTS INSTITUTE
JAN 23
1995
THE STOCHASTIC ECONOMIC LOT SCHEDULING PROBLEM:
HEAVY TRAFFIC ANALYSIS OF DYNAMIC CYCLIC POLICIES
David M. Markowitz
Operations Research Center, M.I.T.
Martin
AT&T
I.
Reiman
Bell Laboratories,
Murray
Hill
Lawrence M. Wein
Sloan School of Management, M.I.T.
We
nomic
consider two queueing control problems that are stochastic versions of the eco-
lot
scheduling problem: a single server processes A^ customer classes, and com-
pleted units enter a finished goods inventory that services exogenous customer demand.
Unsatisfied
demand
is
backordered, and each class has
demand
tribution, renewal
problem, a setup cost
is
its
own
general service time dis-
process, and holding and backordering cost rates. In the
incurred
when
the server switches class, and the objective
first
is
to
minimize the long run e.xpected average costs of holding and backordering inventory and
incurring setups.
the objective
is
to
The setup
cost
is
replaced by a setup time in the second problem, where
minimize average holding and backordering
restrict ourselves to a class of
dynamic
cyclic policies,
where
costs. In
both problems we
idle periods
and
lot sizes
are
state-dependent, but the
N
heavy
these scheduling problems are approximated by diffusion con-
trol
traffic conditions,
in
problems. The approximating setup cost ])roblem
dynamic
lot sizing
policy
procedure are derived
to
must be served
classes
is
found
in
for the setup
a fixed sequence. Under standard
is
solved exactly, and the optimal
closed form. Structural results and an algorithmic
time problem.
compare the proposed policy and several straw
A computational study
is
undertaken
policies to the numerically
optimal policy.
October 1995
computed
We
consider a queueing system scheduling problem that
commonly found
and multiple customer
product has
its
own
we assume
is
that an
Each product has
available.
the inventory, and unsatisfied
We
which
classes,
will
be referred to as products. Each
general service time distribution, and completed units are placed into
a finished goods inventory;
is
motivated by a situation
make-to-stock manufacturing. The system consists of a single server,
in
or machine,
inventory
is
demand
is
ample amount of
(costless)
own renewal demand
its
raw material
process that depletes
backordered.
analyze two variants of the scheduling problem. In the setup cost problem, a cost
incurred
when
machine switches production from one product
the
setup time problem, a random setup time
incurred
is
when
We restrict ourselves to the class of dynamic cyclic policies,
to another; in the
the server switches product.
where each product
is
serviced
once per cycle and the order of production does not change. Thus, the server has three
scheduling options at each point in time: Produce a unit of the product that
set up.
change over
to the next
problem), or remain
backordering a unit
idle.
product
in the
Each product has
in inventory.
The
These problems are prevalent
make-to-stock
mode
setup time problem
is
also less
is
its
own
in
costs per unit time for holding
objective in the setup time problem
to
currently
cycle (and initiate service in the setup cost
the long run expected average inventory costs (that
objective in the setup cost problem
is
is,
is
to
minimize
holding and backorder costs); the
minimize the average inventory and setup
many
costs.
industries because facilities that operate in a
typically produce standardized products that require setups.
is
amenable
more
realistic
to analysis.
than the setup cost problem
The setup
and
in
cost problem, however,
most
is
The
situations, l)ut
relevant for
manu-
facturing systems that have embraced just-in-time (JIT) manufacturing and internalized
their setup times; that
is,
they incur significant material, labor and/or capital costs to
achieve essentially instantaneous switchovers.
This dynamic scheduling, or
economic
lot
lot-sizing.,
problem
scheduling problem (ELSP), which
is
is
a stochastic version of the classic
NP-hard (Hsu 1983) and has not been
solved in general. Despite the vast literature devoted to the
by Elmaghraby 1978, and Zipkin 1991
inistic
for a list of
viewpoint has probably prevented
its
ELSP
more recent
(see the survey
references),
its
paper
determ-
widespread industrial use: The solution to
a deterministic problem in a make-to-stock setting will not hedge against uncertainty
in
future service times (e.g., machine failures) and
backorders (see the numerical results
Not
in
When
the state space
ELSP
many
costly
appears to be analytically
taken to be discrete, the stochastic
is
in
Federgruen and Katalan 1993).
surprisingly, the stochastic version of the
tractable.
demand, resulting
ELSP
(or
in-
SELSP)
can be viewed as a make-to-stock version of the dynamic scheduling problem for a polling
system, which
is
a traditional
(i.e.,
make-to-order) multiclass queue with setups. In fact,
our paper can be viewed as a companion to Reiman and Wein (1994).
namic scheduling problem
for a two-class polling system.
who analyze
The SELSP
is
the dy-
more challenging
than the polling scheduling problem, which also appears to defy exact analysis, because of
the nonlinear cost structure and the lack of an
its difficulty, this
imposed boundary
problem has been the subject of a recent
at the origin.
flurry of activity.
develops a Markov decision model for a one-product problem, and uses
heuristic for the
lego (1990)
ELSP
the
SELSP
in a
periodic review setting.
with stochastic demands that are rooted
first
Sharifnia,
it
to develop a
(1988), Gal-
lot-sizing algorithms for the
in the solution to the
of these papers considers a discrete time
Graves (1980)
Leachman and Gascon
and Bourland and Yano (1995) develop heuristic
Despite
deterministic
ELSP;
problem with nonstationary demand.
Caramanis and Gershwin (1991) employ a hierarchical approach
heuristic policies for a stochastic fluid version of the problem.
to develop
Federgruen and Katalan
(1993, 1994) develop accurate distributional approximations for polling systems, and use
these to analyze the performance of a class of periodic base stock policies for the
SELSP.
Anupindi and Tayur (1994) also consider a
and use
class of periodic base stock policies,
a simulation based approach (infinitesimal perturbation analysis and gradient search) to
obtain good base stock policies for a variety of performance measures. Sox and Muckstadt
(1995) formulate the
SELSP
tion algorithm to solve
it.
as a stochastic
Qiu and Loulou
decision process, and numerically
this
is
to the
As
(
compute
program and propose a heuristic decomposi-
1995) formulate the problem as a semi-Markov
the optimal solution in the two-product case;
the only paper to date to gain any insight into the nature of the optimal solution
SELSP.
in
Reiman and Wein, we employ heavy
traffic
approximations
in
an attempt to
make
further progress with this problem. This approach assumes that the server
busy the great majority of time
draw upon
order to meet average
in
We
the long term.
and Reiman (1995a, 1995b)
the results of Coffman, Puhalskii
the setup cost
demand over
must be
approximate
to
Their results
and setup time problems by diffusion control problems.
On
allow the analysis of the problems to separate onto two different time scales.
the
time scale over which the total workload content of the inventory varies, we solve a onedimensional diffusion control problem to determine the machine idling policy; this time
heavy
scale corresponds to the standard
compressed by the factors n and
expand time by a factor of
time
scale, the total
normalization, where time and space are
y/n, respectively. Starting with this normalization,
y/n, so that both time
and space are compressed by
workload content of the inventory
levels vary according to a fluid
the state-dependent lot sizes
in
traffic
model. Here, deterministic optimization
when
the server
is
busy.
lot sizes
is
we develop some
is
used to derive
In the setup time
derived.
derived under the fluid scaling impact the drift of the one-
dimensional diffusion process that arises under the standard heavy
In this case,
this
These two controls are decoupled
the setup cost problem, and a closed form solution
problem, however, the
on
and the individual inventory
fixed
is
\/n;
we
traffic
normalization.
structural results about the optimal solution
and provide an
algorithmic procedure to solve the one-dimensional diffusion control problem. Although
no weak convergence
no doubt that our
results are provided to rigorously justify our analysis,
we have
we conjecture
that our
results are essentially correct; consequently,
proposed policies are asymptotically
(in
the heavy traffic limit) optimal for the two-
product problems, where the optimal policies are dynamic cyclic policies.
In an attempt to both assess the effectiveness of
ize
some
of the existing literature on the
two straw
some simpler
policies
SELSP, we perform a heavy
policies that are closely related to those considered
(1993) and by Sharifnia, Caramanis and Gershwin.
and synthes-
traffic analysis of
by Federgruen and Katalan
Then a computational study
is
under-
taken that compares our proposed policies and these two straw policies to the numerically
derived optimal policy for a variety of two-product problems; several five-product prob-
lems are also examined.
The
explicitness of our results reveals a
number
of
new and unexpected
insights into
SELSP. Readers who are not curious about the
the nature of the optimal solution to the
SELSP may
mathematical details but who wish to obtain a deeper understanding of the
find
useful to bypass the heavy traffic analysis
it
and focus on
§1.8, §2.5
and
where
§3.4,
the key insights and observations are collected.
The remainder
of the paper
is
divided into four sections. §1 and §2 are devoted to
The computational
the analysis of the setup cost and setup time problems, respectively.
study
described in §3 and concluding remarks are
is
made
in §4.
THE SETUP COST PROBLEM
1.
A
Problem Description.
1.1.
products. Each product
z
=
1.2
.
.
.
,
N
single server, or machine, produces
has
its
own
general service time distribution with
service rate n, and coefficient of variation (standard deviation divided by the
The demand
for
easily generalize to correlated
pt
—
^i/Pi
Let
I,(t)
time
we say
t
at
time
depletes It(t) by one.
that this unit of
cj,,
processes; see
c^,.
respectively.
Reiman
increases
t
If
demand
a
is
is
p
Our
results
(1984) for details.
=
Yl',=i{^i/^'t) a'ld
i.
denote the number of units of product
i
and
long run average server utilization,
the utilization for product
completion of product
at
compound renewal
traffic intensity, or
is
mean)
each product follows a renewal process, where the mean and coefficient
of variation of the interdemand times are given by A~^
Hence, the
types of
A'^
[,{t)
demand
for
i
in inventory at
time
by one, and a unit of product
a unit of product
i
A
t.
i
service
demanded
occurs when
/,(/)
<
0.
backordered.
Since the scheduler follows a dynamic cyclic policy, only three options are available
at
each point
initiate
in time:
Produce the product that
production of the next product
is
currently set up, switch over and
in the cycle, or sit idle.
Because a setup
and instantaneous, the option of switching to another product and then idling
suboptimal and
will
not be considered.
The
server
resume manner, although the subsequent heavy
between
A
cost
this
cost h,
b, is
and the non-preemptive
is
is
traffic
assumed
analysis
to
is
work
in
costly
is
is
clearly
a preemptive-
too crude to distinguish
discipline.
incurred per unit time for holding a unit of product
incurred per unit time for backordering a unit of product
i.
i
in inventory,
and a
Let us also define the
cost indices
/?,
=
unit of expected
^,7^,
and
work
Without
=
which represent holding and backorder cost rates per
6,/^f,,
These indices play a key
in inventory.
the analog to the classic
1961).
b,
index
"c/^i"
in stochastic
loss of generality, the
role in
scheduling theory
our analysis, and are
(e.g.,
products are numbered so that
For notational convenience, we assume that
6/v
=
hj\!
Cox and Smith
=
we show
from that of the more general
that this restriction does not change the analysis
will often
h,.
mini<,<Arft!, so that the product with
the smallest holding cost index also has the smallest backorder cost index;
Consequently, product A^
mini<,<,v
later
case.
be referred to as the least cost product.
Because products are produced once per cycle
in
a fixed sequence, the performance
of the system depends on the setup costs only through the total setup cost per cycle,
a quantity we denote by
Of
A'.
course, the optimal sequence of products can be found
by solving a travelling salesman problem, where the "distance" between
is
given by the setup cost incurred
when switching from product
K/N
convenience, we simply assume that a setup cost
is
i
"cities"
i
and
j
For notational
to j.
imposed whenever the server
switches from one product to another.
The scheduling
policy determines the inventory process
/,
and the counting process
J,
where J{t) denotes the cumulative number of setups incurred by the server up
t.
Although the scheduling
policy,
and hence the stochastic processes
/,
to time
and J, can be
rigorously defined in terms of a sequence of starting and stopping times for each product,
we omit
this detailed specification
Our problem
because
to find a scheduling policy,
is
it
is
not required in our subsequent analysis.
which
is
nonanticipating with respect to
/;,
to
minimize
\imsup
^E\
"r J2ChJ+(t)
+ bj-it))dt
+ ^J{T)i
EC
'^
(1)
1=1
where
x'^
1.2.
described
in
heavy
=
max(.r,0) and
The Heavy
in §1.3 allows us to
traffic.
Since
and
it
=
min(.r,0).
Traffic Normalizations.
we
to precisely formulate the
traffic,
,r~
will
The heavy
compactly characterize the
traffic
class of
averaging principle
dynamic
cyclic policies
be optimizing over this class of policies, there
(more general) control problem that approximates
will suffice to describe the
heavy
traffic
is
no need
(1) in
heavy
conditions and normalizations.
The standard heavy
condition states that the server must be busy the majority
traffic
meet average demand. To prove a heavy
of time to
defines a sequence of control problems indexed by
??,
system approaches unity. Because a limit theorem
traffic limit
where the
theorem, one typically
traffic intensity of the n'
not be proved here,
will
we ease
notational burden by assuming the existence of a large integer n satisfying \/n\l
where
c is positive
and 0(1). Our recommended policy
of the heavy traffic scaling parameter
embedded
sum
the
in the finished
goods inventory
amount
/,.
If /,(/)
standard heavy
and
c,
turn out to be independent
W] denote
is
— p) =
the workload content
positive then W,{t) represents
if /,(/)
is
negative then W,(t)
of time that a continuously busy server requires to clear product
Vs backlog. The process W^
—
Let the process
of the service times for units in inventory,
represents the
I,(t)
/).
will
the
will
be referred to as the workload process
traffic scalings will
for
product
The
i.
be employed to define the normalized inventory process
I,{ni)/y/n and the normalized workload process W,{t)
=
W^int)/ \/ri (throughout
the remainder of the paper, a "tilde" denotes the unsealed version of the normalized
process).
Although the workload process
inventory
is
H',
known
is
employed
The
to hold for a
is
expressed
in
1^.
heavy
is
traffic control
in
heavy
problem
terms of the unsealed inventory process
(/i,
ial
magnitude
heavy
in the
traffic analysis,
we need
I,
=
for
i^iiW,,
traffic, will
be
into a scheduling
.
.
.
,
objective function (1) consists of inventory costs and setup costs.
a nontri\
when
more convenient
However, the linear identity
wide variety of queueing systems
for translating the solution of the
policy that
not observable by the scheduler
backordered, the normalized workload process W^
analysis than the normalized inventory process
which
is
these two costs to be of the
/,v)-
To perform
same order
of
approximating control problem. Since the inventory contents and time
are normalized in the heavy traffic control problem, the relative magnitude of the inventory
costs
and setup costs also need
to
The appropriate normalization
cost
A
is. if
traffic,
(see
in order to
achieve the aforementioned balance.
Reiman and Wein
for details) is to
reduce the setup
by a factor of n relative to the inventory costs, and consequently we leave the
inventory costs h, and
That
be scaled
6,
the setup cost
and
if
unsealed at 0(1), and define the normalized setup cost k
is
=
h'/n.
smaller than 0{n), then setup costs will be negligible in heavy
they are larger than 0(»), then they will dominate the inventory costs in
6
heavy
to be
Thus, heavy
traffic.
traffic
conditions for the setup cost problem require the server
busy most of the time, and require the setup cost to be very
A
1.3.
Preliminary Heavy Traffic Result. A
recent heavy traffic result obtained
by Coffman, Puhalskii and Reiman (1995a) provides the basis
result,
result
which
be referred to as the
will
=
only proved for the A^
is
CPR
2 case,
large.
our analysis.
for
This
result, considers a traditional multiclass (the
and
is
conjectured for
A'^
>
2), single
server
queueing system with renewal arrival processes, general service time distributions and
A
no setup times.
cyclic exhaustive polling
mechanism
each class to exhaustion and then switches to the next
notation
and
/u,
characterize the arrival process to this queueing system.
workload
Reiman
for class
/'
in the
traffic
and
A,
If
we
in the
Cd,
SELSP
to also
denote the normalized
let Vj
queue, then limit theorems in Iglehart and Whitt (1970) and
(1988) imply that the total workload process
under heavy
Let us reuse the service time
class.
and allow the demand parameters
Cj,,
employed: The server serves
is
V =
Ylv=i
K
is
well
approximated
conditions by a reflected Brownian motion on [0,oo) with drift
—
and variance
-^
See Harrison (1985)
The
process.
CPR
It
=
£^(4 + 4)-
for a definition of this process,
result allows us to obtain information
(2)
which we denote by
RBM
moves
infinitely quickly (in the
for the total
workload, the
linear path connecting points
case, the path
is
N— dimensional
On
the time scale giving
workload (V'i,...,Vyv) process
asymptotic limit) along a constant workload, piecewise-
where the server exhausts a product.
In the
two-product
the line segment from (O.V) to (V, 0); in the case of three identical
products (with the same service and
connecting the points
We make
{—c,a^).
about the multidimensional workload
provides a time scale decomposition for this system:
rise to
RBM
(0, V'/S,
demand
rates), the path consists of the line
2V/3), (V73,2V73,0) and (2V73,
0,
segments
V/S).
the crucial assumption that the time scale decomposition holds not just for
the cyclic exhaustive policy (which corresponds to a cyclic base stock policy in the
problem), but
for
more general
cyclic policies;
objections because the time scale decomposition
this
is
SELSP
assumption should not raise any
inherent in the heavy
traffic scaling.
The time
scale decomposition gives rise to an averaging principle that facilitates our
analysis in the following way:
expand the time scale
stays constant
(i.e.,
If
slow
we consider
and then
the heavy traffic normalization
down time) by
workload
a factor of x/n, then the total
and the A^-dimensional workload behaves
as a fluid
(i.e.,
moves
at a finite
rate in a deterministic fashion).
1.4.
An Overview
as consisting of
of the Analysis.
two interrelated decisions:
policy that specifies
The
CPR
useful to view the control policy
busy/idle policy and a dynamic lot-sizing
We
of the heavy traffic control problem
which measures the
total
begin by characterizing
and the well known relationship between queueing
result
systems and production/inventory systems
inventory.
is
what the server should do while working.
the busy/idle policy.
Xli=i 11'm
A
It
is
Morse 1958) imply
(e.g.,
that the system state
machine time embodied
in the
current finished goods
Furthermore, since setup times are zero, the total workload process
affected by the server's busy/idle policy, not by
how
often the server switches
products. Hence, the only reasonable form of the optimal busy/idle policy
to stay
u'o.
busy
if
< wq and
\V(t)
The quantity Wq
to idle
will often
if
W{t) >
uiq, for
The
CPR
result
Although the
only
among
for the server
the unspecified control parameter
and the one-to-one relationship between
queueing systems and production/inventory systems
W
is
is
be referred to as the idling threshold, and can be viewed
as an aggregate base stocA' level.
the total workload process
W=
the one-dimensional total workload process
is
a
RBM
(c, cr^)
(i.e.,
W{t) = wq —
on (—00. Wq] under
lot-sizing policy does not influence the total
the rate at which inventory costs and setup costs are incurred
V'(/))
imply that
this busy/idle policy.
workload
W
when W(t) =
.
it
does affect
w. Therefore,
a two-step procedure, with each step being performed at a different time scale, can be
used to find the optimal dynamic cyclic policy. In the
is
first
step,
CPR's averaging
principle
used to find the lot-sizing policy that minimizes the average (inventory plus setup)
cost incurred as the individual inventory levels oscillate deterministically while the total
workload remains constant
at w. let us call the resulting
minimum
cost c{w).
a family of deterministic optimization problems indexed by the total workload
able to construct a
dynamic
(i.e.,
workload-dependent) lot-sizing policy.
By
iv,
solving
we
are
In the second
step,
we
find the aggregate base stock level ivq that
c(ti.)
/- oo
where we have used the
minimizes the long run average cost
4e-''< ""'-"'>/''
f/u),
(3)
(7^
fact (e.g., Harrison) that
RBM
{—c,a^) on [0,oo) has an expo-
nential stationary distribution with parameter '2\c\/a^.
In §1.5,
when
§1.6.
we
dynamic
calculate the average cost incurred by a generic
The optimal heavy
the total workload equals w.
Finally, the
heavy
traffic
cyclic policy
traffic cyclic policy is
found
in
normalizations are reversed in §1.7 to obtain the proposed
scheduling policy as a function of the unsealed inventory levels and the server location.
1.5.
is
Construction of Dynamic Cyclic Policies.
to find the cost associated with
\'V(t)
=
traffic
normalization Wi{t)
w.
The
CPR
fluid scaling, W,{t)
is
=
=
N—
and the
A
is
Because idleness
stock level,
cyclic policy
we assume
is
=
\V(nt)/y/n
at a finite rate in
occur before the diffusion process
W
one cycle.
essentially characterized by the lot size for each product (or ecjui-
valently, since the analysis
cycle).
fluid cycles
suffices to analyze the deterministic behavior of
cyclic policy
factor of y/n to obtain a
dimensional workload W, moves
many
workload
Starting with the heavy
we slow down time by a
Wt{nt)/s/n,.
a deterministic manner. Because
it
this calculation tractable:
total
Wi{y/nt)/ ,/n. At this time scaling, the process W{t)
fixed at the value u\
changes value,
goal of this subsection
any dynamic cyclic policy when the
makes
result
The
is
deterministic, the length of time each product
only incurred
is
that no idleness
when
is
is
served in a
the total workload reaches a certain base
When W{t) =
incurred during a cycle.
il\
a
best viewed as a closed A^ — dimensional deterministic path in the constant
workload hyperplane J2i=i
'^'-'i
=
^'-''^
that
is,
the process traverses the
same path repeatedly,
once per cycle.
Although a cyclic policy can be specified
acterization that
is
convenient for analysis.
in
A
many
cyclic policy (or, equivalently, the closed
loop generated by the policy) will be defined by A^
and the cycle center x^
=
(
jj,
.
.
of the total workload w, but this
.
,
ways, we choose a particular char-
-|-
1
quantities:
The
cycle length t
x%). These control parameters are actually functions
dependence
will
be suppressed
for
improved
readability.
time
Figure
The
cycle length r
is
1:
Workload fluctuation over a
cycle.
the length of time recjuired to perform a cycle, and
is
.r^
product
Vs "center of fluctuation", or equivalently. the average amount of this product's inventory over the course of a cycle. Because the transient effects associated with initiating or
temporarily moving a cycle vanish
can be placed anywhere
We
levels
heavy
traffic
cyclic policy
when
\\'{i)
of each product produced per cycle
throughout this
=
w. For the system to remain balanced, the
must equal the amount demanded, and hence
of the time;
p,
fluid analysis, so that the server is
an arbitrary instantaneous
total
workload
w and
the work content in this product's inventory
at rate one,
and W, increases
the remaining
(1
—
at the fixed rate
p,)t time units in the cycle
the workload inventory
is
decreasing at rate
the
machine
depleted at rate
—
p, for
p,
i
is
level, as the reference point.
10
must be
servicing product
and
is
replenished
not being produced,
To uniquely determine
a cyclic policy, a reference starting point also needs to be specified.
average inventory
is
i
for
p,T units of time per cycle. For
when product
p,.
that p equals one
busy throughout the cycle. Thus,
when
is
1
we assume
cycle time r, each product
serviced for p,T units of time per cycle. Therefore,
?'s
.r'"
constant workload hyperplane.
in the
each product must be produced a fraction
i,
time scaling, the cycle center
begin by examining the deterministic behavior of the individual product workload
W, under a
amount
in the
We
the behavior of
use
.r^,
product
Readers are referred to Figure
1
for a
reinforcement of these notions.
The
center
cost of a cycHc policy can be expressed in terms of the cycle length r, the cycle
(an A^— dimensional vector) and the total workload w.
.r*^
inventory cost for product
i
per unit time, c,(r,
The average
over the cycle and divide by the cycle length.
regions depending on whether the product
and backordered or
is
is
we
if),
j-j',
=
,r^, ty)
find the average
integrate the inventory cost
cost breaks
down
into three
entirely held throughout the cycle,
entirely backordered. Figure
is
held
allows us to deduce that
1
h,x'^
c,(r,
To
if .r^
iMiz^il
>
<
-M^
if
J^< - '^'^'-'^'^
(4)
The
total
1.6.
controls:
average cost, c{t,x'',w), which includes inventory and setup costs,
The Optimal Dynamic Cyclic
The aggregate base
arbitrary
(iii)
w
is
derived
and
r, (ii)
in
cyclic policy consists of three
stock level wo and, for each total workload level
length T and the jV — dimensional cycle center
cyclic policy
Our
Policy.
three stages:
(i)
In this subsection, the
x"'.
in
u',
the cycle
optimal dynamic
find the optimal cycle center
optimize over the cycle time r
is
x"^
in
terms of
terms of an arbitrary
u',
and
substitute the derived cost function c{ui) into equation (3) and find the optimal idling
threshold wq.
The Optimal Cycle Center. We
cycle center
.r^
tion c(t,x^,w)
for
is
begin by showing the existence of a cost-minimizing
a given total workload
w and
differentiable with respect to
x"^
is
x"^
and
its
derivative
is
continuous.
If
one
=
iv,
for fixed r the cost function in
piecewise-quadratic with linear edges;
its
second derivative
ignores the constant workload constraint X]!I,
terms of
cycle length r. Note that the cost func-
ive step function.
Thus c{t,x'^,w)
is
^'i
is
a nonnegat-
convex and the restriction of the cost function to
the constant workload hyperplane determined by
w
is
also convex. This fact implies the
existence of a solution to the constrained minimization problem: Choose
c(t,x'^,w) subject to Yli=i
Now we
—
^"l
x'''
minimize
to
w-
use the cost function
in
The
to find the optimal cycle center.
(4)
total
workload constraint can be used to eliminate one variable and express the cost function
as a piecewise
polynomial function of
N—
N—
Any
variables.
1
1
components of x'^ can be
used. Over the constant workload hyperplane, the polynomial order of the
between one and two depending on whether
fluctuates
respectively.
For the gradient to be equal to zero, each of the iV
quadratic. Consequently, at the optimal
.r*-",
at least
.'V
—
1
two, with the remaining component possibly being linear.
of Ci(r,
u')'s
.rj',
-Pi(i-p,)
— or
>
|x^'|
—
1
of the Ci{t,
To
N —I
<
x*:'
variables
.rj",
u')'s
variables
Tp,{i-p,
must be
are of order
see this, suppose that
are not of order two, and let j denote the index of such a term.
some
If
we
eliminate Xj, the gradient equation can then be written as
N
=
1
1
The
have a solution only
1
ff there
are only
function at the optitiial x"
Proof: This
Cj
if
is
h,(x1)^
/
the remaining A^
linear
fact
is
and
+ b,{xX +
—
\
theu^ the linear
,
most
c,\ is
ikx
N
+
easily seen by
By
quadratic.
quadratic Cj(r,
term must
x'i
—
1
c,{t^x'j, u')'s are
(4), the
- Pn)
rp,v(l
h Nl
tradeoff between x^ and
+ xl; =
w' (with
iv'
.r^v
.r^,u')
quadratic.
+
sum
bN
,
ter'ms in the total cost
be C!\]{t,x']^,w).
examining the function
8
The
.
1
following proposition greatly simplifies our analysis.
Proposition
where
=
V
<*1
will
Y.x%w
W
T,
1
This equation
\
(
V,c
Ct{T,
+ hN
2tpn(1 -
u''
^2,^'V-6N„.
+
'N-
H
1
^;^
2
Tp^{l
-
12
p?
.ry^,
w]
— x^
for
rX\,
'N
if
(6)
pj^
.r^'
N
X^=^r
sum
along the line
into equation (6)
the derivative with respect to x%- yields
-h,
a')+c.v(T,
is
can be examined by looking at this
arbitrary). Substituting
.r,",
> w\
and taking
Since the quadratic region of
j^
that the c}uantity in (7)
than or equal to
than or equal to
for
>
x'i^j
w' or
6,
—
is
Hence, neither
bj^.
equals
b,
less
.r^
+ x% ^
of these solutions
lie in
when
solutions occurs
<
6/v for j/y
then the optimal .r^ satisfies \x%\
along the line
restricted to the region
is
=
>
If
I
'^^'^^"^^'
it
.r*^,
be
will
c/v(t,
t*-^, u»).
product
product
<
;
A^
A^'s cycle
that product A^.
from
equals
hj^j
if
>
lM|z£il.
\x1\
and
A^ are linear,
Although many
one of the optimal
components are quadratic
there
is
(or
on the
a linear cost component at the
center, or average
—
amount
of inventory per cycle,
p,)/2,Tp,(l
—
p,)/'2].
whereas
which
is
the least cost product by our indexing convention,
amounts
of
work when the
is
w
workload
total
the product
fluctuates
zero.
We now
erality, the
.T^'
greater
center can be arbitrarily far from zero. Intuitively, this fact suggests
that will hold the excess or deficit
far
is
multiple optimal solutions
in
and
restricted to the region [— rp,(l
is
/?,
follows
|
As a consequence, the optimal cycle
for
i
of the cost
border between quadratic and linear). Hence,
optimal
in (8)
jj
the holding or backorder costs are equal,
the region where products
—
LpkL^ipnI^
and the quantity
LPml^Lmi^ resulting
w' with |x^|
at least A^
/?,,
<
nor (8) can equal zero unless
(7)
w'.
—
/?Ar
|.r5v|
= w —
use Proposition
1
to find the optimal cycle center
workload constraint
Y^tJl^
x'i.
To
is
.r"-".
Without
loss of gen-
used to eliminate .r^ from the cost function, so that
find the optimal center,
we take
the gradient of (5) and set
it
equal to zero:
N-l
N-l
V,c[X;Q(r,x^u;) + c^(r,a>- ;^
At
this point,
the (A^
—
c,(r,
u;)'s
J-"',
we do not know whether
7
.r^ u.)
+
-]
=
0.
c/v(t, .r^, ly) is linear or quadratic.
l)-dimensional vector that solves (9) under the assumption that
x^ satisfies
are quadratic in
{w and r multiply
jj^.
Taking the
their vectors
-^^"^
T
-
(9)
71
(A^
—
x"^
be
A^ of the
l)-dimensional gradient, we find that
component-wise
- -72 =
T
all
Let
0,
in the analysis
below)
(10)
where
b,+h,
A
+
/?A'(1
7i
;i2)
2
2
^N+^N
=
72
(11)
- Pn)
131
pn{^-pn)
Th us.
= tA
.r^
w here
the matrix elements of
Pi(l-P.
h,
w—
Z!!^i
^
-J"?!
the optimal center:
we must
With
^
x'i
=
)
Pj(l-Pj)
hj+h,
for
«
7!^
^^^''
x'j
Y^iV
h.
^1=1
^^^ then C/vIt,
for
;
<
A'
and
x^, w)
x'^
=
Pi(i-Pi)
hi+hi
is
indeed quadratic and
»'— Xl,-/
If
-'"i-
|ft'~I2,=/
solve the multivariate gradient equation with the linear
this substitution,
2
h,
2
'15)
and
J,
;i6)
+
6,
I
+ h,
Pi(i-P()
Y^.v
^^'=1
b,+h,
Oin
If
14)
72,
are
.4
On
+ wA
7i
.rj'
-^'i^l
form of
determines
>
~rpN(l-PN)
c.\'(r, ,r^
N
lU]
equation (9) decomposes into univariate expressions of the form
- p,)
6, + h^
+
Tp,{\ - p,
(17)
T/9,(l
6,
,
x:
+ bN
if
w-
E
1
'
!
^
2
(18)
Putting the results from (14) and (17)-(18) together, we obtain a complete expression for
14
the optimal cycle center. For
i
N
<
Tp,(l-p.)
h,+h,
(
we have
+ N
if »'
ll
-
E!IT'
^^
''^^2"'''^
>
(19)
rp.jl-p,)
b,
where a,
is
The
the ith row of .4"'.
The Optimal Cycle Time.
calculus.
if "'
}N
+ h,
The optimal
Substituting the optimal cycle center
it;.
'"^^2"''''
equal to
10
—
J2i=i^
7i
+
shorthand for the region
I^^d^fMl, and case
cost function
w-
III is
E,=7'
wa,
cycle time r can be derived by simple
into (4) yields the average inventory
.r^
For
N
<
i
this cost
is
given by
(20)
72]
^P<i^-P')
2{b.+h,)
Ih J- fit)
h \L£A1:zpA -t
_u
\iJ,-t
g
is
•^?-
11
^!h_Li^T(y,
I
and
is
< -
=
Ct{T,w)
where case
a^?
component, i^,
last
cost for each product as a function of r
- e!It'
4^-6 N th^-b N
— E!Ii'
»'
^1
>
""^^^2
>
< - ''^^[-"^l
4
''^^
^^^^
III
^^ '^
l^i'~E,Ii^
Similarly, product
N\
-^i^l
^
inventory
is
f
cn{t,w)
=
Mu'-rEf=T'13^[^ + /^N
PN{i-PN
<
II
(21)
.
2/
2TPfv(l-Pw)
i'
1
(1
hfj-b^
_
+ ^li^(u,(l
•r^N-l „
- E,=l «
E;17' a,
.
72)
•72
-
J
N-l
r E:IT' «.
71)
-M--rEr=7'^fe^N"-6
'/v
The optimal value
of r
is
determined by taking the derivative of the
III
total cost function
Y^iLi Cifr,
w)
+ k/r and
solving against zero.
2[b,
we
If
(22)
+ h,)\
b,
+
2
h,
•lp.[\-p.
+2/)Af(l
y^' (k
6
-
+
TI4
9
8
N-l
bi
+
hi
^,2p,{l-Pr)
=
1
—
Y^^-i^ o,
•
72
2
>'
and
72)'
•
=
7^
7-y
O'
(23)
li
(24)
_ p^il-p^) f b, 2
2{b, + h,)\
h,)p,(l-p,)
+
74,
?;
pat)
—-6,
—
/?,
2
,
o, -71)
e2
where 73
define the constants
,
+
— +-hN
^'V
Ylf=i'
<^i
•
_ ^^
(25)
2
T
2p^(l
h,
(26)
r73,
pn)
then the optimal cycle length
7i,
is
given
by
I
T
=
<
7«ligd±
11
III
Considerable insight can be gleaned from (27). However, a discussion of the insights from
our analysis
is
deferred until §1.8.
The Optimal
Idling Threshold. With an expression
ecjuation (3), which determines the idling threshold Wq.
lengths at the borders of region
expressed
levels
w >
in
II
and
its
terms of the total workload
u'l.
region
II
by
u'2
<
ic
<
By
for r in
hand, we can refine
using (27) to ec|uate the cycle
two adjacent regions, the three regions can be
level.
Region
Wi and region
III
I
by
is
characterized by workload
w <
W2, where the workload
level cutoffs are
Wi
6-ei
(28)
A-,
Ui
6Us
/
102
16
e.-
k.
(29)
By
and
(3)
(5), the total
for r in
is
given by
Y^c,(t.x%w) + -] -e-^^'^'°-U:w.
^/ '^'
/:
Having solved
average cost
(30)
u=l
terms of the system parameters and w, we can substitute (27) into
the inventory cost (20)-(21) and into the setup cost term k/r in (30) to express the total
average cost solely
in
terms of the total workload w; we
expression because
it
adds
Depending on the
terms based on the
little to
split of the total
closed form solution. Therefore,
we
not explicitly write out this
our understanding of the problem.
can be broken into three or fewer
idling threshold wq. the integral
Because of the irregular form of r
will
workload into three regions as defined
for
w G
[»'2,u'i],
in (28)-(29).
the integral over region
resort to numerical
methods
to
has no
II
determine the optimal
value of Wo-
Our
derivation of the optimal
Recall that
when
dynamic
cyclic policy in
the problem was initially defined,
the smallest holding and backorder cost indices.
heavy
we assumed
The
traffic is
now complete.
that product
—
1) to
A"
—
1
in
heavy
This
in
place of product
traffic policy
is
(for
example,
designate the product with the lowest backorder cost index, and use product
The Proposed
1.7.
had both
analysis would remain unchanged,
however, without this restriction. One only needs to introduce new notation
A'
N
done
in
two
N
when
Policy.
derived
stages:
the total workload level
The
in §1.6 to
We
final
is
less
step in our analysis
is
than zero.
to
employ the optimal
develop a proposed policy for the original SELSP.
reverse the heavy traffic scalings to express the solution
terms of the original problem parameters, and then interpret the resulting solution.
If
we
replace the normalized quantities
w,k and
r
spectively (r undergoes this normalization because time
by w/\/n,h'/n and i/y/n,
is
compressed by
\/n. in
re-
the fluid
model), then the optimal cycle length formula
=
r
<
becomes
in (27)
JiAEI+K
31)
ii
III
=
Motivated by (14), we define x
W(t) =
Then
lb.
fp,(i-p.)
TO,
.
ijy
c,{T,y, w).
is
7i
"^m^
equal to
is
^')i
+ibA
the unsealed cycle center
(
and
f.4
tt'
equal to
—
b,-h,
+
^72 for fixed total
for
x'^ is,
unnormalized workload
< N,
t
+ hN
U'Q,
[^ -
^n]
if
- - Ef=T^
-n
< -
2
Hence, the unnormalized average inventory cost,
Z]'=i'.r^.
r,(f,.r'', lu).
(32)
72
•
Upon
u'l
=
U'2
=
reversal of scaling, ecjuations (28)-(29)
\l-^—.
— A,
become
(33)
U^
/6-6,^.
(34)
^46
Therefore,
if
wq minimizes equation
(30), then
wq
=
\/nwo
minimize the unnormalized
will
long run average cost expression
A'\ 2(1
-
Notice that the heavy
§3,
we use Maple
V
traffic
p)
-e
U'
_m^,a.o-w
(u.o-u-)^/
"'
parameter n does not appear
W.
in (31)-(35).
(35)
To compute Wq
in
to numerically solve the first-order optimality conditions associated
with (35).
Our proposed dynamic
original
hit)
{N +
cyclic policy for the
SELSP must
l)-dimensional system state, which
/a'(0 and the server location. There are
18
is
be expressed
in
terms of the
given by the current inventory levels
many ways
in
which the unnormalized
policy in (31)-(32) and (35) can be interpreted for purposes of implementation. Perhaps
the most natural
maximum
By Figure
workload
way
to express a
inventory (or "produce-up-to"
when
1,
level for
lot-sizing policy
level for the
)
product
traffic identity
i
is
when
=
/,
+
.i'j
p,{l
—
IZ,=iMr'/i(0 ^
II is
W{t) equals w. the maximum
i
is
[^^'-i.ii'i]
in (33)- (34):
and region
product with the smallest holding cost index
for
product
i
The server should
b^.
<
N
f
^/^
achieved at the
Making use of the
finished.
Region
III is J2,=i 1^-7^
proposed policy can be described as follows: IfY2t=i l^~^h{t) >
the smallest value of
is
us define the three unnormalized workload regions in
terms of the inventory process and the thresholds
Wi, region
to specify a state-dependent
p,)t/2; of course, this level
the production of product
/ijM^j, let
is
product currently being produced.
the unnormalized total workload level
instant within a cycle
heavy
dynamic
h^;
otherwise, let
idle ifJ2^=i lJ-T^h[t)
h{i)
then
N
>
ZlUi /'r'^i(0 >
I is
<
let
W2-
N
Then our
refer to the
denote the product with
Wo\ otherwise, if set up
then produce this product as long as
\a: 6,+/i y
.
/'r'A(o<
I
+
{o,-'n
\IpAI
-
K
p,
«3
^^^) +
h
{a,-j2)Zl,^^7'kt)
b,-b
^-''^'
b,+h,
III
(36)
Once
p^ '/,(0
reaches or exceeds this
product N, then produce
p,^^lN{t)
<
this
switch to the next product.
up
If set
for
product while
S^N
,,-lf//\
2^,=i
A',
''I'^
I
+
$2
<
level,
ff^i P;v('-P.v)
V^i^
>>
2
Y^N-1
^t=i
1^1=1
2
p,(1-p,)
b,+h,
"i
\
b,-h,
[~2
,
i
^ "'^
I
II
III
+(E!Ii/ir'^.(0)(i-E;=7'".-72)
n^l PnI^-Pn)
E^t=i Piii~^fli\-L
^'y-i'^y^^y
2
v^iV-1 P,(l-PpA
2^1=1
b,+h,
b.~h,
'iV
)
III
(37)
and then switch to the next product when
1.8.
Insights.
optimal policy
in
Our
heavy
p]^^ij^(t) reaches or
analysis reveals
traffic.
numerous
exceeds this
level.
insights into the behavior of the
Before reading this subsection, readers
may want
to
digest the graphs
marked "proposed"
policies in both the
in
Figures 3 and 4 in §3.2, which depict the proposed
symmetric (each product has the same parameters) and asymmetric
two-product settings.
An
Three Workload Regions.
essential feature of the
heavy
traffic
policy
characterization via three workload regions, as described in (28)-(29). There
workload
in region
I,
significant backorders in region III,
intermediate case where the total workload
State- Dependent Lot Sizes.
cycle
in
is
an interval containing zero.
SELSP).
lot sizes for the
when
that the optimal lot sizes are state-dependent
contrast, the lot sizes are constant in regions
I
and
thereby generating constant
remains constant
lot
/
within a
I
and
points in the intermediate area of region
the deterministic
ELSP
literature
can observe from (27)
III; in
is in
region
II.
In
these regions, surplus or deficit
lot sizes
and setup costs
stabilize,
This observation and (19) imply that the cycle
sizes.
in regions
We
the total workload
unavoidable, and the trade-off between
is
.r'^
sufficient
represents the
Because the time spent producing product
(and determines the optimal expected
center
II
its
the optimal cycle length r determines the optimal lot sizes in heavy traffic
is /9,r,
inventory
and region
is
is
III,
It is
II.
(Dobson 1987
and gradually
shifts
between these two
worth pointing out that
is
in nearly all of
a notable exception), the analysis
is
restricted to policies with constant lot sizes.
Relationship to the
the lot size
region
As
in the
in
is
Focused
in the
Least Cost Products.
the product with the smallest
h,,
hold (small h,) and lengthy to process (small
inventory
6,;
In
this
economic order quantity (EOQ) model,
I
and
III.
In
the setup cost again appears in the numerator of the square root term.
II,
up
Model.
proportional to the scjuare root of the setup cost in regions
is
Inventory
built
EOQ
(i.e.,
backorders)
product
is
is
which
f/,).
In region
is
excess inventory
is
inexpensive to
III,
excess negative
a product that
Similarly, in Region
is
held in the product with the smallest backorder cost index
inexpensive to backorder and has a long expected processing time.
both regions, inventory
is
held in the least cost product so as to reduce the absolute
value of the inventory of the higher (holding in region
I
and backorder
products. In this regard, the dynamic lot-sizing policy derived here
traffic policy
I,
is
derived for the corresponding problem without setups
20
in region III) cost
similar to the heavy
in
Wein
(1992).
In
that paper, instantaneous switching causes the inventories of
heavy
to vanish in the
added
is
When
normalization.
traffic
normalized cycle length and,
to the
setup costs are introduced, breadth
r
=
=
then region
0,
and
.r^
Lot Sizes
optimal
=
In fact,
w >
(region III) corresponds to
I
workload, a "corridor" of
for a fixed total
possible inventory states replaces the least cost axes.
case A'
the higher cost products
all
if
we consider
0(w <
in
0);
the special
both regions,
w, and the solution reverts to that of Wein.
Grow
lot size is
By
with Absolute Value of Total Workload.
smallest
when
absolute value of the workload.
(27),
we
see that the
the total workload equals zero, and grows with the
When
the total workload
near zero, costly backorders
is
can be avoided by switching frequently between products. In contrast, when the absolute
value of the workload
large,
is
it
is
possible to
employ large
without adversely
lot sizes
affecting the inventory costs (because inventory tends to be held in the
product
regions
in
produce products
and
I
III); in this case,
it
i
product
for
< N, product
product
/'
and
i
is
i
advantageous to avoid setup costs and
is
In
heavy
under our proposed policy
inventory
positive
cost
batches.
in large
Inventory Levels at Switching Epochs.
workload
minimum
negative
is
when
(i.e.,
is .r^
the
traffic,
maximum
+ /9i(l — p,)t/'2.
It
normalized
follows that for
backordered) when the server switches into
the server switches out of product
i.
For product
N
the
sign of the inventory level during the switching epochs depends on the region: In region
II
product
positive
N
when
inventory
is
backordered when the server switches into product
N.
the server switches out of product
always positive
in region
I
Cost-Symmetric Case.
=
and
is
Our
=
always negative
and
is
In contrast, product A^ inventory
is
in region III.
results simplify considerably
=
A'^
under the cost-symmetric
example,
case,
where
if all
products are relatively indistinguishable, except for their color. Then each product
h,
h and
would be expected
demand
h^
h for all
to have the
rates, since
some
same
colors
;'
1,...,A'^.
This case
service rate and inventory cost rates, but different
may
be more popular than others.
and, by (28)-(29), the workload always resides in region
workload resides
we
in
only one region (region
define the constant p
=
Xlfli Pi(^
will arise, for
~
Pt)-
II) if
then
II.
and only
when
In fact,
if
Then
^i
=
ifs
=
one can show that the
the costs are symmetric.
the server
is
busy and
is
set
up
If
for
product
I,
this
product
is
produced as long as
(E^, ^-'i,(t)y ^
It is
T:l,^i7'h{t)
(38)
evident from this expression that the cost structure effects the
K/{b +
the ratio
h)\
not surprisingly, the
lot size is
lot sizes
only through
an increasing function of
this quantity.
THE SETUP TIME PROBLEM
2.
Problem Description.
2.1.
(rather than a setup cost)
is
In the setup
when
incurred
another.
In all other respects, the setup cost
identical,
and
we
^
f>{b+K
P"
\
-IK
all
time problem, a random setup time
the server switches
dynamic
As
cyclic policies. Let 5 denote the
Coffman, Puhalskii and Reiman (1995b) show that the heavy
system depends upon the setup time distributions only via
parameters are unaffected by a permutation
the desired permutation
given by the
is
mean setup
deterministic service times,
Under general
Fuhrmann
mean setup time
traffic
per cycle:
performance of
this quantity,
Thus
this
and other key
in
heavy
traffic
traffic conditions,
Poisson arrivals and
(1992) and Cooper, Niu, and Srinivasan (1992)
time over the cycle. Also, numerical calculations
performance of a cyclic policy
setup cost problem,
where the intercity distances are
show that the mean waiting time depends on setup times
that the
in the
in the cycle order.
the travelling salesman tour,
times.
to
problem and the setup time problem are
relevant notation from §1 will be retained.
shall only consider
from one product
is
in
through the
onl\'
total setup
Federgruen and Katalan (1993) show-
quite insensitive to the chosen permutation.
The
server has three scheduling options at each point in time: Produce a unit of the product
that
is
currently set up, initiate a setup for the next product in the cycle or
control problem
is
to find a nonanticipating scheduling policy to
\\msupl;E\rY.ih.!^{t) +
T_>co
Once
again,
i
'Jo
^
^
we avoid unnecessary notation by omitting
inventory vector
/ in
terms of an arbitrary cyclic policy.
99
The
minimize
b,i-{t))dt].
^
sit idle.
(.39)
J
the equations that express the
The
2.2.
same
as in §2,
The form
Diffusion Control Problem.
and the setup times are not rescaled
of the proposed policy
and works otherwise, where
the server
busy and the
is
The heavy
is
also the
same
in
heavy
traffic
traffic; that is,
The
as before.
server idles
workload equals w, the cycle time
of average inventory workloads
is
they are 0(1).
if
is
t{iu)
x'^(w).
and Coffman, Puhalskii and Reiman (1995b) show
is
\'V{t)
equals
traffic limit,
that the time scale decomposition
described in §1.3 also carries over to the setup time problem.
normalized workload
When
and the vector
Since the setup times are 0(1), they occur instantaneously in the heavy
total
W{t) > wq
an unknown normalized threshold parameter.
ioq is
total
normalizations are the
More
specifically,
the ./V-dimensional fluid process {Wi,
u',
when
.
,
.
.
the
Wj^j)
identical to the A'^-dimensional fluid process in the setup cost problem; hence, the
optimal placement of cycle center
c,(.r",r, u') is
We now
x'^ is
given by equation (19) and the inventory cost rate
defined in equations (20)-(21).
characterize the normalized total workload process
problem under the
class of
dynamic
workload process follows from existing heavy
and Reiman (1995b)
position holds for
all
after
Our
cyclic policies.
for the setup
time
characterization of the total
theorems
traffic limit
W
making the innocuous assumption
in
Coffman, Puhalskii
that the time scale
decom-
multi product cyclic policies, not just for two-product cyclic base
stock policies. For make-to-order systems, readers are referred to Coffman, Puhalskii and
Reiman 1995b)
(
and
case,
to
for a derivation of this characterization in the
Reiman and Wein
the approximation.
process.
The
Its
drift of
W
is
for a discussion of the controllable case; here
The normalized
variance a^
is
the
same
no longer v^(l
workload process
total
—
W
is
the total workload equals w.
is
assumed
as in the setup cost problem,
but x/'^(/(«')
p).
—
p),
of time during a cycle that the server serves customers, as
when
two-product uncontrollable
When
and
state
to be a diffusion
is
where f{u')
opposed
we only
defined in (2).
is
the fraction
to incurring setups,
the normalized workload equals
lo,
this fraction
given by
'nri w)
where t{w)
is
the cycle length, which
is
+
s
both deterministic and controllable.
Hence,
\/n{f{iv)
—
p) -^ c
— s/t{w)
as n -^ oo,
W has the state-dependent
and we assume that
drift
;^(uO
= c--i-.
(41)
t(w)
Notice that this drift approaches the drift in the setup cost problem as the setup times
vanish or the cycle length becomes arbitrarily large.
=
Define the cost function c(t, w)
are defined in (20)-(21).
J2',=i c,(r.
The approximating
>
state-dependent cycle length t{w)
w), where the individual cost components
problem
diffusion control
and the threshold wq
is
to choose the
minimize
to
1
Imis up ^E[j\{Ti\V(t)),Wit))dt],
T
W
where
a (i^i{w),a^) reflected diffusion process on
is
(
(42)
— ocwq]-
Hence, the controllable
cycle length t(w) affects both the drift i.i(w) and the cost c{t, w) in a nonlinear fashion.
The Optimality Conditions.
2.3.
and a singular control via the
as
t{w] -^
0,
we proceed
as
if
Problem
reflecting barrier wq.
(42) involves a drift control t{w)
Although the
drift is
unbounded
standard arguments apply (Mandl 1968) and state the
Hamilton-Jacobi-Bellman optimality conditions:
min
{
'(u')>o
c{t(w). w)
-g+
[
[c-
-—
\
t{w)J
<r\..,
\"(w)
+ —V"(w)
2
}
=0
for
w < wo
(43)
J
and
V'{w)
where g
the gain and
is
V'(.r)
is
{g.V{w),T{w). Wq) can be found
to (42),
is
g
is
=
for
that V
(44)
to (43)-(44).
then (r(«'),a'o)
is
If
a solution
the optimal solution
the optimal long run average cost (independent of initial state), and
when
the initial state
G C^, and define p{w)
jjrinciple of
Wo,
the potential (relative value) function.
the cost incurred under the optimal policy
incurred
w >
smooth
fit
when
the initial state
which we take
is
some reference
=
V'{w); this assumption, which
state,
is
(Benes, Shepp and Witsenhausen 1980),
solving diffusion control problems.
24
is
to
w
is
minus the cost
be Wq.
known
often
V{w)
We assume
as the heuristic
imposed when
Using the workload cutoff
unknown
{wi and W2 are
and
levels ivi
u'2
to distinguish
among
regions
I,
II
and
III
and are not given by (28)-(29)), we substitute the
at this point
three forms of cost function c{t,iu) into (43) to obtain
min
<
+ h^w -
<fir(u')
r{w)>0
min
<^
+
g
^-Tiw)
ic
\
^2Tiiv)
+ ^sw +
w
^4—~ -9+
t{w)
t(i())>0
+
p{iv)
[c
——
\
t(w)J
-—p'{iu)
2
p{w)
-
\
=
for
lui
+ —-p (w)) =0
2
< w <
wq,
for
< w <
it;2
(45)
uh,
J
(46)
min
r(w)>0
(3t{w)
<
-
bjwiv
-
g
+
^
c
(
where the new constant
N-l
b,
+
-7,
p{iu)
)
+
Tiw) J
\^
[
-—p'(iv) >
2
J
=
for
w <
(47
1V2,
^5 is
h,
-
h,
z{Olz
7l
•
".
)
72
+ h^
:7473 +
p.v(l - Pn)
b^
b,
+
,
CVi
72
•
/?/v
-
^Af
::;
73-
(48)
Solving the first-order optimality conditions for t{w) yields
sp(w}
for 10
=
r [w]
1
for u;2
<
< w
<
It'
<
(49)
ifi
«2
sp{v
for
w < W2
43
we obtain
Substituting t'{iv) into (45)-(47),
the nonlinear ordinary differential equations
(ODE's)
\/~^isp(w)
'2y/^2U^^
-
+
^2Sp(w)
hiMiv
—
+ ^5W +
g
+
cp{io)
Cp[w)
+
+
—p'(io)
—p'{lo)
-
^
=
for
Wi
< w <
Wq,
(50)
=
for
W2
< w <
Wi,
(51)
2
-y-^sspiw) - bNW - g + cp{w) + —p'{iv) =
2.4 Structural Properties.
for iv
<
W2.
(52
Before pursuing a solution to the diffusion control
problem, we derive several useful structural properties.
Property
— —00
If 102
1.
=
p(w)
if u'2
/ — oc
then
w+
o(W)
w—
as
>
—00;
(5o)
f/jen
p(it')
=
w+
o(w)
as
IV
-^
— cxo.
(54)
c
The
derivation of this property, which assumes an asymptotic monotonicity property,
nearly identical to that in the Appendix of
is
Reiman and Wein
(1994); see
Markowitz
(1995) for details.
Property
if
and only
if
wq
>
u'l
2.
The idhng threshold wq
if h,
=
hj for all
i
and
At the
j.
and t'(wo) = y'6/G»'o
>
it'i
and
r*(ft'o)
=
\/^4/^2Wo
=
'Tft'o
and equality holds
=
"'i-
and equation (49) imply that t'(wo)
<
if u'o
it'i,
idling threshold, the cycle length r"(u'o)
—
Proof: The boundary condition p(wo)
Wq
greater than or equal to
is
In order for iuq
u'l-
<
u'l,
must
r'(»'o)
=
if
satisfy
the cycle placement conditions originally specified in the conditioning of equation (19).
We
can rewrite the centering condition as
\wo
- l^T
!
Rearranging terms,
=
(u'o)a,
•
-/I
+
tt'oQ,
•
72I
<
^
this inequality
.
is
pn{1--pn)
[ii,
73<\/— (^4 +
,
i^f..
)
(5")
7,
^
V V2
Algebraic manipulations (see Markowitz) show that this inecjuality
holding costs are not identical and holds at equality only
cost-symmetric case, the placement conditions
by the idling threshold
that
is,
»'o
Property
=
3.
U'l.
u'2
(55)
•
1
u'o,
and hence wq
is
in
if h,
if
and only
if b,
—
false
hj for all
i
the boundary between regions
bj
26
if
and
inventory
j.
In the
equation (55) are satisfied at equality
I
= — oc
—
is
for all
i
and
j.
(I)
and
(II);
Derivation: The constant
imply that
T'(tu)
Hence,
= —00
as
u'2
w — —00
(II),
unbounded
for
in the
w <
u'2.
cost-symmetric case.
1,
This fact and (49)
which would violate the placement conditions.
cost-symmetric case.
in the
by Property
>•
region
is
equals
^3
-J 7^
and t''{w)/w ->
= — cxd
W2
If
w —> -00 by
as
**
then p(w)/w -^
(49).
^^.
—
To remain
-
in
42
the placement condition
AT-l
IV
-
V
T'{iu)a,
7i
-I-
a»Q,
•
72I
t'{w)Pn{1
<
- pn
r
(57)
1=1
must be
For
satisfied.
approaching —00, we can substitute the limit of t'[w) and by
10
algebraic manipulation reduce equation (57) to
73
1^,Pn{1 - Pn)
^
< \/^(
T
V
As
in
Property
equality
=
if 6,
2, this
inequality
bj for all
The nonlinear ODE's
in
V2
false
j (see
in (50)-(52)
if
The approximate
satisfies the
(5^)
backorder costs are not identical and holds at
Markowitz
for details). |
do not appear
to
Reiman and Wein, we pursue both an approximate
solution.
is
and
;
is
,^^.
.
74)-
"
analytical solution aims to
optimality conditions (44) and (50)-(52).
admit a closed form
1
compute a function
The
this analysis did not
Hence,
here,
this
basic idea behind our
problem,
terms
in (50)-(52),
nearly
method
and then
fit
holds.
However, the scheduling policy arising
test cases
considered
in §3.
approximate analysis and the corresponding numerical results are not given
The next
2.5.
smooth
perform consistently well over the
and the interested reader
procedure
p[.r) that
through 3 to paste together the solutions of the resulting ODE's so as
to ensure that the principle of
from
As
analytical solution and a numerical
to use Taylor series expansions to linearize the square root
to use Properties
solution.
is
referred to
for details.
subsection contains insights from our structural results, and an algorithmic
for solving the diffusion control
Insights.
it is
Markowitz
more
problem
is
Because a closed form solution
difficult to
provided
is
in §2.6.
not obtained for the setup time
develop insights into the behavior of the optimal solution.
Nevertheless, several noteworthy comparisons can be made.
Cost-Symmetry
The
Cost-Asymmetry.
vs.
solution for these two cases are surpris-
ingly different; see the graphs entitled "proposed"
cost-symmetric case the workload stays
threshold,
in
Figures 5 and 6 in §3.3.
and the cycle length
in region II,
at the idling
proportional to wq. In the cost-asymmetric case, the policy
r(ii'o), is
In the
char-
is
acterized by three regions, and the lot size approaches zero as the workload approaches
the idling threshold; these results follow from Properties 2
used
ling threshold, small lot sizes are
whereas
in the
this option is not available in the
and 3 and equation
grows with \/^w
when backorders
symmetric
case,
asymmetric case
workload
asymmetric case and with
are large, there
and larger
is
less
3.
Evidently, near the id-
to reduce inventory costs,
cost-symmetric case. Finally, by Properties
(49), as the current total
in the
and
w
—w
tends to
in the
— oo,
the cycle length
symmetric
case.
opportunity to reduce inventory costs
lot sizes prevail in
order to reduce the
amount
1
Hence.
in the cost-
of time devoted
to setups.
Setup Costs
differences
Setup Times.
vs.
is
interesting to note both the similarities
problems can be distinctly broken down into three workload regions
(one region, respectively)
lot sizes are
when
asymmetric (symmetric, respectively). For both
costs are
state-dependent and inventory
is
focused
in the least cost products;
moreover, the description of the inventory levels at the switching epochs
to the setup time problem.
The proposed
similar in the region around
the
w =
two extreme regions (regions
(i.e.,
I
and
policies for the
region
III). In
II),
the
time problem, the cycle time contracts to zero as
grows as \/— ic when
As noted
tions because
in
w
tends to
effects
but have different characteristics in
asymmetric setup cost problem, the
There
In the
asymmetric setup
approaches the idling threshold and
the
two problems lead
to qualitatively different solu-
cause setup times to consume available capacity
nonlinear manner. Therefore, the effective cost of a setup time
the setup time problem:
is
that the total workload will
over
— oo.
Reiman and Wein,
queueing
iv
in §1.8 carries
two problems are qualitatively
cycle time r remains constant throughout these two regions.
ility
and
between the setup cost and setup time problems: The behavior of the proposed
policies for both
problems,
It
no direct penalty
fall.
As the
total
28
for
is
in
a highly
woMoad-dependent
in
a set up, only an increased probab-
workload approaches
— oo, many
items
are backordered and the effective cost of a setup
to use the capacity efficiently
workload
is
by running large
level. In contrast, as the total
effective cost of a setup time decreases,
very high; thus, the scheduler attempts
lot sizes, so as to
workload approaches the idling threshold
ioq,
the
and the scheduler can afford to employ small
lot
reduce inventory costs. As a consequence of our analysis,
sizes to
costs should not be used as a surrogate for setup times in the
practice
is
common
quite
Setup Times
policy
vs.
in the
No Setup
deterministic
Times.
ELSP
it
SELSP;
is
clear that setup
unfortunately, this
literature.
Like the setup cost policy, the proposed setup time
a generalization of Wein. As setup times vanish,
is
recover from the low
all
of the inventory
becomes
stored only in product N. Using the symmetric cost results, the optimal idling threshold
Wo goes
2.6
—
to
An
ln(
^)o-^/('2c), which
is
Algorithmic Solution.
we pursue a numerical
the
same
as that derived
Since problem (42) cannot be solved analytically,
solution. In particular, the
developed by Kushner (1977)
is
by Wein.
Markov chain approximation technique
employed. This method systematically discretizes both
time and the state space, and approximates a diffusion control problem by a control prob-
lem
for a finite state
Kushner and
Markov
chain.
Weak convergence methods have been developed by
his colleagues to verify that the controlled
Markov chain (and
its
corres-
ponding optimal cost) approximates arbitrarily closely the controlled diffusion process
(and
its
corresponding optimal cost); we refer readers to Kushner and Dupuis (1992) for
an up-to-date account of
reference.
approacli
It is
is
this research area,
and retain most of their notation
for
ease of
worth emphasizing that the computational complexity of the algorithmic
independent of the number of products; this important fact
state space collapse inherent in the
diffusion process,
CPR
result,
is
due
to the
which leaves us with a one-dimensional
and our optimization of the cycle center
in (19).
Before describing the method, we introduce a slight modification to the heavy
traffic
analysis to account for the fact that setup times do not vanish in the original problem.
The
cycle length t{w) consists of the time devoted to processing and the time allocated
to setups. In the fluid scaling, sj ^Jn units of time are spent setting
a cycle; although this quantity vanishes
in the limit,
we include
intended refinement. More specifically, we replace t[w) by t{w)
it
-\-
up over the course of
in
our analysis as an
s/s/n.
Let h denote the finite difference interval, which dictates
One can
space and time are discretized.
indexed by the interval
h,
finely
both the state
consider a sequence of controlled
Markov chains
and as the value of h becomes smaller the resulting discrete
Markov chain described below becomes a
time, finite state
how
better approximation of the
controlled diffusion process.
To numerically
W to a bounded
Markov chain
of h. For
now
such that Wq
be
us
let
{
fix
M and
resides in { — A/,
we need
to confine the one-dimensional diffusion process
the state space of the controlled
W resides on a
M — M}, where M a positive integer multiple
— M, —M + h,
region.
will
<
solve (42),
Since
halfline,
h,
...,
is
the idling threshold wq (this parameter will be optimized later on)
ioq is
an integer multiple of
—M + h,...,Wo — h,iUo}.
Hence, the Markov chain actually
h.
The approximating Markov chain has nonzero
transition probabilities
+ 2h(c-
a'
2(j2
+
2h c
+
2h\c
.f
,
t(iu)
,
^Y
+ s/^
and
P''iw.w-h]
2a^
on the interior of the state space, and the time
t{w) + s/^
intervals, or interpolation intervals^ are of
length
At
h
(61)
a'
Two
chain:
h
T{w) + s/s/n
issues need to be addressed to obtain our approximating controlled
for
(i)
of the state
w and
control t(w) (see
Markov chain
issue,
we
define
is
h,
Markov
an ergodic cost problem, the interpolation interval At'' must be independent
of the
of t(io)
+
we
at the
boundary states w
Q^ = a^ + max^,
let Q''
=
Kushner and Dupuis, page 209), and
a^
+
^(^,1
\ch
—
h\c
s\
,
,
=
^
f
—N
;
/-
and
•
w =
Wq.
To
the behavior
deal with the
first
Since the smallest nonzero value
and define the new nonzero
30
(ii)
interior transition
probabilities
+ 2h(c-
(T^
P\iv.w-h) =
'
,
,^ )^
r(,.)^siv^)
V
^g3^
2(5
and
=
P'^iiiuw) =
and the new interpolation
+
[a^
—
-
1
^'
'
threshold.
A
consider the boundary states.
;("-)+-/^
^
(64)
,
^
(65)
.
reflecting
boundary
is
employed
at the idling
However, the Markov chain appro.ximation method assumes that
for a reflecting
diff'erent
"
interval
A<' =
Now we
h
boundary
than (65) at
tt'o,
Because the interpolation interval
state.
this
boundary
state
must be eliminated.
A/'' takes
We
A^''
=
on a value
define the transition
probability (see page 212 of Kushner and Dupuis)
P^iwo -
We
also
impose a
reflecting
P^i-M +
h.
Wo
if
h)
boundary
h.
=
- P\wo -
1
at state
—M,
h,
wq - 2h)
=
1
—M
is
artificial in the
at
-
P(-M +
/?,,
-M + 2h)
visited sufficiently infrequently.
if
the value of
M
is
(67)
.
sense that P{
the boundary was chosen to be larger than
approximation should be negligible
(66)
.
and define the transition probability
-M + /O
Although the reflecting barrier
would be positive
-
M,
suflficiently large,
In our implementation, the size of the
— M,—M +
h)
the effect of this
and consequently
Markov chain
is
M does not change the optimal solution {t{w),wo). In
—M +
summary, our approximating Markov chain has state space — M +
wq —
chosen so that a further increase
in
{
2/i, u'o
—
/?},
/?,
2/?,
...,
interpolation interval defined by (65), and nonzero transition probabilities
= P^{w,y)
P''{w,y) defined by (66)-(67) and P^(w,y)
otherwise, where P''{w,y) are
defined in equations (6'2)-(64).
The dynamic programming optimality equation
Markov chain
for the controlled
is
given by (see equation 5.3 on page 204 of Kushner and Dupuis)
Ey PHw, y)V{y) +
Viw) =
+ s/^) +
(6(r(u')
Ey P\io, y)V(y) + iUriw) +
h^io
s/,/1^.)
-
g)Ai'
for
+ 6u.
u'l
for i«2
<
<
lUo
^ w <
it'i
lu
I
Ey PHw.y)V(y) + iUriw) + s/^) -
bj^ic
- g)At'
—N<w<
for
ii'2
(681
and the Markov chain control problem can be solved using the following policy improve-
ment algorithm.
First,
we choose
the initial policy:
T{iii)
=
ioq
— w
for
w < wq and
arbitrary Wq. In the evaluation step of the algorithm, a generic policy {t{ui), Wq)
ated (that
V(w) and g
is,
are found) recursively. Since the
process, the stationary probability distribution
7r„
rTti'o
Y^'('o~^
iro
set
w G
{
—M +
—
P
/i
(k,k — h) \
is
for
w > Wq —
h
for
w <
Wo
—
h
=
Wq
—
h
for IV
a birth-death
M — h}
h
evalu-
is
(69)
is
g^
We
for
P''{k-h.,fc)
_[_
and the gain
tt,^,
Markov chain
is
V'(A/)
=
0,
so that
equation (68) implies that
V(w-h:
In the policy
-h
Y^
w=-\ + h
it„,c(t(w)
V{w) =
V{w —
g-c{T(w),w) +
for
+
w >
s/^/n,w).
u'o,
by equation
,70)
For
(44).
w <
wq,
h) can be calculated recursively by
{l
-
P''(w,iv)]V(w]
- P''{w,w +
h)V{iu
+
h]
(7i;
PHw.wimprovement step we
idling threshold wq.
The
cycle length
first
is
h)
solve for the cycle length t{w)
and then
for the
determined by minimizing the right side of (68)
32
w ith
>
respect to t{iv)
0.
If
the drift c
—
+
s/{t'(id)
slV(w+h)-V{w)]
=
il,n2
_
s\V{w+h)-V(u,)]
for IV2
slV{w+h)-V(w)]
and
if
the drift
<
u'l
<
u'
<
iv
(72)
,
102
negative then
is
for U'l
U
r'iw)
s[V{w)-V{w-h)]
2
Notice that (72)-(73) converges to (49) as
/?
—
)•
0, as
< w <
Wq
<
lUi
for
W2
iov
—N
s[V{w)-V{w-h)]
<
it-1
<
(73)
< W2
u'
expected. Keeping t{w) constant,
we determine an improved idHng threshold wq by evaluating
all
u'o
— A'^ <
for
s\V(w)-V(w-h)]
g for
< w <
Wi
for
t'{w)
positive then
s/y/ri.) is
(via equation (70)) the gain
values of wq, and searching for the gain-minimizing value.
The evaluation and improvement
steps of the policy
peated until the new and old gains are sufficiently close
remains constant.
In the
implementation of
improvement algorithm are
in value
algorithm
this
and the
in §3,
re-
idling threshold
we chose
M
=
1000
and terminated the algorithm when the difference between the new and old gains was
less
than 0.05 and the idling thresholds changed less than
difference interval h
is
w =
{
—M +
h,
.
.
.
,
Wq —
/?}.
u'l
and
In the next subsection,
our choice for the
The output
discussed in the next paragraph.
procedure includes the workload cutoffs wq,
0.1;
of our algorithmic
as well as t(w)
102,
we show how
finite
and V(w)
for
to translate this output
into a proposed solution.
2.7.
The Proposed
proposed policy
is
less
an analytical solution
way
to develop a
parameter
7j:
Policy.
The mapping from a
straightforward
is
derived.
More
when a numerical
specifically, with
proposed scheduling policy that
The
drift of the
is
diffusion control solution to a
solution
is
obtained than when
a numerical solution there
independent of the heavy
underlying diffusion process
is
\/n.{l
is
no
traffic scaling
— p) — s/{T{w) + s/\/n),
and a value of n must be chosen
in
order to compute a numerical solution to the
chain control problem. This quandary
ecjual to
one and
let
n
~
{1
—
dealt with in the
is
p)~^. Moreover,
We
set c
set the finite difference interval h equal
Markov chain approximation procedure
to l/\/n, so that the discretization in the
ponds to individual units of inventory
we
most natural way:
Markov
corres-
problem. Exploratory computations
in the original
revealed that the performance of the proposed policy was very insensitive to our choice
of
77.
Recall that
vve
replaced the cycle lengih t{w) by t{io)
-\-sI y/n in
the
Markov chain
al-
gorithm. Therefore, in creating our proposed policy, we employ the cycle center Xc[t{w)-\s/y/n^iu).
The proposed
policy for the setup cost case produced product
ventory reached x1{t{iv),w)
time
in
-\-
f(iy)/9,(l
In order for the
p,)/'2.
a cycle to be equal to r{ih), the s/\/n. term
the two terms in this expression, and
.rUr(w)
—
+ s(l-
/>), Cu)
The proposed
+ f(w)pAi -
policy
is
is
p)J2'i=iP7^f>{0
greater than
Then our proposed
policy
we produce product
is:
II
I,
u'l,
If (I
up
for
product
i
<
—
and
III
not added in the second of
i
until its inventory reaches
p) X]i=i p7^
idle if{\
[77'2,7i'i]
MO
—
>
let
or less than
f^^" ^^^
N
^
tt'2,
(1
—
respectively.
refer to the product
denote the product with the
p) Yl,^=\ P7^ U[^)
then produce this product as long as
34
Define the
according to whether the quantity
in the interval
The server should
N
expected total busy
the setup cost problem.
in
with the smallest holding cost index h,: otherwise,
smallest value of b,.
until its in-
p,)/2.
constructed just as
unnormalized workload regions
is
7
>
Wq; otherwise, if set
+
^
""
{i-p){b,+h,)
•((1-p)E7=i^
+ hN
''^C')"'**""'*
2{l-p)
II
1-p
A'r'/.(0<
(74)
<
2(1-P)
fe.-A,
{\-p)(b,+h,)
-6»
III
_
2{l-p]
Once
jj.^
product
'/,(0 reaches or exceeds this level, switch to the next product.
i'V,
up
for
then produce this product while
•((i-p)Er=.^"'^i('))p^<i- PN)
^N
I
^i=i
-
<
E^,
[2
ri-oU6.+/i.t
(l-p){b,+h,)
^A'
/i^'/A'(0
If set
N
'"
(^((i-p)Er=,M7'^.(o)+-(i-p))"-
.^_
Tl
II
;';
75)
'
,-
^
^((i-p)Er-i''7''fj(o)pA'(i-PN)
,
2(1-P)
(1-pn)
E
j
= lMj
^jlU
+
III
2(l-p)
m7'/,(0)+»(i-p))p.(i-p,)
^,v-i ^(<'-^'Er=,
and then switch
The
to the next product
solution proposed above
and the constants
ure.
is
Since t(w)
u'o, u'l, u'2
is
is
when
i^ijj^
I[^{t)
-6 N
reaches or exceeds this level.
specified in terms of the original
problem parameters,
and the function t(w) generated by the algorithmic proced-
only defined on a discrete state space, the argument of this function
rounded to the closest discrete value
approximated by r(y*), where y'
in the
satisfies |y*
algorithmic discretization; that
— u;| <
|y
— lol
for all y
G
[—N + h,
is,
.
.
.
t{w)
is
,Wo — h].
COMPUTATIONAL STUDY
3.
we evaluate
In this section
the effectiveness of our proposed policies by conducting
a series of two-product and five-product experiments for both the setup cost and setup
time problems. For the two-product cases, we compare the performance of our proposed
policy and two "straw' policies against a numerically derived optimal policy.
programming value
performance of
iteration algorithm
four policies.
all
is
^From
A dynamic
used to find the optimal policy and evaluate the
this
data we compute the suboptimality for the
proposed and two straw policies by
.
policy
s
suboptmiaiitv
policy's cost
=
—
optimal cost
x lOO/o
;
:
.
optimal cost
In
implementing the value iteration algorithm (readers are referred to Markowitz
for
a detailed specification of the algorithm), the inventory state space was truncated to
[-150.150] by [-150,150] for the setup cost cases and [-250,250] by [-250,250] for the
setup time cases. To achieve three-digit accuracy of the suboptimalities, 7,000 iterations of
the algorithm were required for the setup cost
lem.
Due
to the large
number
of inventory states, a
not feasible for the five-product cases,
crete event singulation
To evaluate a policy
time units
for the
first
for the
is
for the
setup time prob-
dynamic programming algorithm
and thus no optimal policy
is
for a particular scenario,
we perform
5
is
derived. Instead, dis-
used to evaluate the proposed policy and the two straw
policies.
independent runs of 6,000,000
setup cost problems and 10 independent runs of 6,000,000 time units
setup time problems; each run starts with an empty system and statistics from the
10,000 time units are discarded. For
demand
service
problem and 14.000
interarrival times, service times
is
preemptive
in the
all
scenarios in this section,
we assume
that the
and setup times are exponentially distributed:
two-product cases and non-preemptive
in the five-product
cases.
For systems with two products,
we consider
20 setup cost cases and 14 setup time
cases; all but
two cases
parameters,
.'\lthough nearly all of our cases are
for
Reiman and Wein suggest
each type of problem assume that the products have identical
symmetric, the numerical results
that the heavy traffic analysis
36
is
ecjually accurate for
in
symmetric
and asymmetric problems. For systems with
We
and four setup time cases.
five products,
we consider
in this setting,
which allows us to assess
the suboptimality of our proposed policies; since the optimal policy
we allow
options that
optimal
in
in the
heavy
at
(i.e.,
each point
traffic limit.
two dimensions (see Figures
in time),
we conjecture
that our proposed policies are
Also, the graphical depictions of the various policies
through
2
Our key observations
6) help us to
understand the subtleties of the
and setup time problems are given
3.1
Straw
are
summarized
To help
Policies.
a dynamic cyclic
is
the optimal policy chooses one of the three scheduling
behavior of this system. The two straw policies are described
results for the setup cost
The
focus on the two-product setting for several reasons.
optimal solution can be numerically computed
policy in the two-product case
six setup cost cases
in §3.1,
in §.3.2
and
and the numerical
§3.3, respectively.
in §3.4.
assess the effectiveness of the proposed policy,
we
consider two simpler classes of cyclic policies, and use heavy traffic analysis to optimize
within these classes.
One
is
a generalized base stock policy and the other
is
a
fi.xed size
corridor policy similar to one considered by Sharifnia, Caramanis and Gershwin. Neither
straw policy employs the s/^/ri refinement that was introduced in §2.7; we discuss this
issue in §3.4.
Generalized Base Stock Policy. The generalized base stock
as follows.
^Vi{t)
>
workload
If
f'i,
the server
then idle
level
Wj(t)
if
>
is
set
up
product
Vj
—
yj\
for
product
i,
policy can be stated
then serve this product
if
W,(t)
this policy
is
optimal
in
the generalized base stock policy contains
parameters, the heavy
Reiman and Wein
the iV
heavy
-|-
1
y.
normalized parameters
traffic, this
policy
is
for details)
depends on the
Hence, we set each
(ci,
.
,
.
.
vj^.y),
of as a refined
(
1993), in the
a state-independent manner. Although
sense that their policy can only insert idleness
2A'^
two-product
in a
version of the cyclic base stock policy considered by Federgruen and Katalan
us denote this quantity by
If
otherwise, switch to product j at this point. Hofri and
symmetric polling system. The generalized base stock policy can be thought
let
v,.
next product to be produced in the cycle, has a
j, the
Ross (1987) prove that the make-to-order version of
this policy (see
<
y,
where y
yj's
equal to
=
traffic
behavior of
only via maxi<;<;v yn
y,
and optimize over
y/\/n and
v,
=
vj\/n. In
equivalent to one that completes production of product
/'
when
its
inventory level reaches
Figure
and employs the workload idling threshold
Vi,
X],=i
~
*•'!
y\
see
2.
To
we move from
calculate the cost associated with this policy,
meters to those used
and
in §2
CPR
I'nder the
§3.
these natural para-
a given total workload
results, for
w
a generalized base stock policy has cycle center
j,(u')
=
V,
-
—^
-
E;=i^;(l
-(2^
Vj
-
w)
(/6)
Pi) j=i
and cycle length
t{iu)
Product Vs average inventory cost
tion (4);
I.e.,
c,(2
V"'
CPR
Using the
v,
results,
is
^.
=1
^'='"
^'
-'t
=
2
(77)
obtained by substituting these parameters into equa-
^n
:'
we can derive
,
u'
,
the total average cost for the generalized base
stock policy for both the setup cost and time problems. In the setup cost case, total aver-
age cost
is
calculated by integrating the average inventory costs plus average setup costs
over the stationary distribution of the normalized total workload.
total
workload
W
approximated by a
is
/£- "-'
where a
=
2>/n(l
—
RBM,
the total average cost
e;I, c,(2 ^^.;;-
p)/cr^.
'-"
,,)
For the setup time problem, the average inventory cost
W
Coffman, Puhalskii and Reiman 1995b). The
/S"'
is
- ^^^-:;!^^Uy\
.,
similar, although the stationary distribution of
(see
Since the normalized
(el .(2*^^.
E, =
,''-<1-''')
V
=v
is
no longer exponential, but
total
-
average cost
-'^"^p-r'
E;=,^;<>-Pj)
is
gamma
is
.
...))
J
(79)
"(°(E.l,".-))%,-a(E;:.-'.-'^-)^^,
where q
is
as
above and
f3
=
sJ2^=\Pi{^
—
Pi)/^^-
^Ve set n equal to
use a steepest descent algorithm to find the parameters
38
(I'l,
.
.
.
,
I'/v,
y) that
(1
—
p)~^ and
minimize (78)
Wo
V2
Vl
Wi
Generalized Base Stock Policy
Figure
and
(79).
2:
The two straw
The Corridor
with
we
For both the setup cost and setup time cases,
scaling to obtain the proposed parameter values
in the
policies.
Vt
=
This policy can be stated
Policy.
—
vj(l
reverse the heavy traffic
p)
and y
=
i//(l
—
p).
terms of switching hyperplanes
in
product workload space. The hyperplanes are created to form a fixed width corridor
its
long axis orthogonal to the constant workload plane (see Figure
represents a natural
embodiment
stochastic framework, and
The
of the "constant lot size" philosophy within a
^+2
defined by
is
2).
parameters:
width), the idling threshold Wq and the parameters (yi,
We
intercept of the corridor's axis.
The
.
.
.
policy
dynamic
cycle length r (or corridor
,yAf),
which determine the
can use these variables and the notation of the
previous two sections to formulate the average inventory cost of the policy. For a given
workload
i«,
the cycle center
Ts average inventory cost
.r^
equal to wj
workload
for
Because the cycle length
is
is
w
is
N+
y,
and the cycle length
then c,(t,w/N
independent of workload,
time cases the diffusion process
W
is
in
approximated by a
an exponential steady state distribution. The
drifts of the
+
is r.
Product
yi,w).
both the setup cost and setup
RBM
RBM
on
(
— cxd,u'o], which
are c and c
—
sjr
has
for the
setup cost and time problems, respectively. Thus, the total average cost
/
for the
Ci(^. "IT
+
yn"')
+ -
e
-.
is
'dw
-=
setup cost problem, and
'"'
X-
/
for the
«'
-p)-
2(v/^(l
,^
I
s/t)
_ AV^[^-P)-'lr)
^^,^_^,^
w=i
The cost-minimizing parameters
setup time problem.
for (80)
and (81) are de-
termined by a steepest descent algorithm. .Although we were able to use
in
our computations,
traffic scaling factor
in the
77
77
— p)
so that \/r?(l
is
y,
=
—
y,/(l
/>)
=
and f
r/(l
//i
=
^2
also set
/?2
=
p).
=
1
1
standardize the two-product scenarios, we set the service
and control the
and each case
and, by modifying
is
demand
utilization rates p, by varying the
/i), b\
and
62, select
in the IS synj/iietric cases,
characterized by three parameters: Backorder cost,
setup cost per cycle.
We
examine
all
permutations of values shown
The second asymmetric
doubled to
Table
6,
II
=
20 and
case
62
=
Ai
=
0.6, A2
is
the
same
=
0.3,
/7i
=
2, 61
=
and
traffic intensity
in
Table
I;
notice that
The parameters
of these scenarios grossly violate the heavy traffic conditions.
asymmetric case are
We
rates A,.
product 2 as the least cost product.
Inventory costs and arrival rates are identical across products
first
—
Finally, the
The Setup Cost Problem.
Two-Product Cases. To
the
p)~-^
simply the stability condition that the fraction of time
proposed parameter values are given by
some
—
(1
greater than sjr, thereby guaranteeing a well
is
the server spends processing units and setting up must be less than one.
3.2.
=
setup time problem one must be careful to choose the heavy
defined integral; this inequality
rates
(bO)
oo
10, 62
=
=
5 and A'
200.
as the first, except that the backorder cost
is
10.
displays the results for the 20 two-product cases and Tables III to
V
show
the averages (for the 18 symmetric cases) over individual parameters for each policy.
switching curves for the optimal and proposed policies for the (6
two-product symmetric case
for
is
depicted
in
Figure
40
.3,
=
5, A'
=
200, p
and corresponding curves
=
The
0.9)
for the
Backorder Cost
Setup Cost
Traffic Intensity
b
A^
p
Low
20
Medium
High
10
Backorder
5.6%
Optimal
Proposed
W2
Wo
10
101
/
' -
J\
f
10
10
idle
if
setup for
idle
if
setup for 2
switch from
1
in the
asymmetric
In the setup
The
Switching curves for the
4:
to 2
=
10
1
asymmetric setup cost
time cases, the average suboptimality over the 12 symmetric cases
policy performs very well (1.8% average suboptimality)
asymmetric cases
somewhat
(1.5%)
Switching curves:
remarkably similar
stock policies.
in
In the
same general shape
and
in the lighter traffic cases.
3. 3%-
Figure
It
7.2%.
the traffic intensity
is
also performs well in the
of the proposed and optimal policies are
Figures 3 to 6 and are unlike either the corridor or generalized base
two symmetric problems (Figures 3 and
as predicted
a
by our heavy
wide cycle time
and a small cycle time about the zero
in
when
is
suboptimalities).
The switching curves
workload idling threshold,
problem
case.
cases.
high, but degrades
is
61
~Wi
1
switch from 2 to
Figure
-^-±
Wi
4, the three- region
total
A
traffic analysis:
for large positive
workload
these curves have the
5),
distinctive constant-
and negative inventories
level. In the
asymmetric setup cost
categorization predicted by the heavy traffic theory
easily recognizable in the optimal policy. Figure 6 confirms that lot sizes shrink as the
44
Back-
Backorder
Cost
_b
5
Setup
Traffic
Optimal
Time
Intensity
Policy
Proposed
Suboptimality of
Corridor
Gen. Base Stock
^
()
Gain
Policy
Policy
Policy
Setup Time
Backorder Cost
Traffic Intensity
b
^
2
Low
Medium
12.8%
8.4%
High
12.1%
16.5%
18.7%
12.1%
6.7%
Overall Average .Suboptimality
=
12.5%
Table X: Average suboptimality of the corridor policy: Setup time problem.
Backorder Cost
Setup Time
Traffic Intensity
6
s
()
Low
Medium
9.5%
High
11.8%
14.0%
9.8%
8.2%
9.8%
'
11.6%
Overall Average Suboptimality
=
10.7%
Table XI: Average suboptimality of the generalized base stock policy: Setup time problem.
will increase
while producing product
workload
exceed the reduction
will
in
2;
that
is,
the increase in product 2's inventory
product Ts inventory workload. The optimal policy
takes advantage of this imbalance by allowing product 2's inventory to grow beyond the
product
1
idling threshold; this extra product 2 inventory allows the server to idle while
setup for product
1.
Performance of the corridor policy.
The
The
corridor policy exhibits erratic behavior.
policy performs very well in the symmetric setup cost cases
proposed policy
in six of the 18 scenarios, all of
but degrades slightly at high utilization.
A
(it
outperforms the
which have low or medium
comparison of Figures
2
and
utilizations),
3 leads us to
believe that the parameters of the policy are being set correctly at high utilizations, and
the performance degradation
is
due
to the corridor policy's inability to
employ mean
lot
sizes that are state-dependent.
The
is
corridor policy does not perform as well in the symmetric setup time cases;
it
not able to increase the cycle length r for small total workloads and so has difficulty
recovering from this high backorder region. In contrast to the symmetric setup cost cases,
the corridor policy's performance diminishes in light traffic;
we have not determined how
Optimal
Proposed
W2
W,
10
TT
-to
ly,
idle
if
setup for
idle
if
setup for 2
switch from
1
1
to 2
switch from 2 to
Figure
much
5:
Switching curves
of this degradation
low utilizations, and how
The
ality
is
is
a symmetric setup time case.
for
is
in the
approximation
at
when asymmetry
is
present:
Its
suboptim-
setup cost cases and increases to 30.6% and 60.9%i in
Comparing Figures
the setup time cases.
traffic
intrinsic to the policy.
corridor policy performs nuich worse
13.1% and 14.9%
1
due to the inaccuracy of the heavy
much
H/',
2, 4
and
6, it
would appear that the corridor
policy would never be very close to optimal for an asymmetric problem. In fact, Figure
6 suggests that a hyperplane corridor policy (see Figure 7 of Sharifnia,
Gershwin) might perform reasonably well
in
the
asymmetric setup time problem;
two-product case, the two lines forming the corridor
in
Caramanis and
in
in the
Figure 2 would not be parallel
the hyperplane corridor policy, but would intersect at an idling point in the upper
right portion of the
graph and generate a cone-shaped corridor emanating out
in the
southwesterly direction.
Performance of the generalized base stock policy.
performs better
in the
setup time cases than
in the
48
The
generalized base stock policy
setup cost cases:
Its
average suboptim-
Optimal
Proposed
Wo
4o;
Wi
40 f'
VV'i
r
idle
if
setup for
idle
if
setup for 2
switch from
1
1
to 2
switch from 2 to
Figure
ality is
23.6%
6:
for the 18
setup time cases.
use of large
Switching curves
for the 6i
=
10
1
asymmetric setup time
symmetric setup cost scenarios and 10.7%
case.
symmetric
for the 12
In contrast to the corridor policy, the generalized base stock policy's
lot sizes
when
the total workload
is
negative
is
a key reason for
its
ability to
avoid poor performance in the symmetric setup time cases; however, these large
lot sizes
lead to considerable backordering in the setup cost scenarios. Like the corridor policy, the
generalized base stock policy's performance deteriorates at high utilizations in the setup
cost cases and at low utilizations in the setup time cases.
The
generalized base stock policy's suboptimality
metric setup cost cases and 11.5% and 15.8%i
in the
is
.35.8% and
45.8%
in the
asymmetric setup time
asym-
case.
It
is
interesting to note that the generalized base stock policy handles expensive inventory in a
manner opposite
good
is
to that of the
proposed
set larger than those of less
policy.
The order-up-to
level of the
most costly
expensive products to reduce the risk of expensive
backordering; in contrast, the proposed policy minimizes the excess or deficit amounts
of expensive inventory.
It is
clear that the generalized base stock policy
is
incapable of
closely approximating the optimal setup cost solution in Figure 4.
Five-product examples: In the six setup cost cases in Table VI, the proposed and
corridor policies are roughly comparable in the two A'
corridor policy
8% more
about 10% more costly
is
expensive
in the
in the
two A'
=
=
500 symmetric cases and the
50 symmetric cases and about
two asymmetric cases. The generalized base stock policy does
not fare as well in the four symmetric setup cost cases, incurring an 18.7% cost increase
relative to the
proposed policy, on average. Once again, the generalized base stock policy
performs very po'^-ly
in the
asymmetric setup cost
cases.
In contrast, the generalized base stock policy performs slightly better than the pro-
posed policy
in the
two symmetric setup time cases
more
12%)
To compare
we can
cases,
in
=
setup time cases.
5.
identify the six
A'
=
is
8% more
about
The
corridor policy
is
asymmetric cases, but performs
II
symmetric cases
and VIII;
20.
p
=
in
for
the two-product cases and the five-product
in
example, the
0.9 case in Table
II.
Table VI with their two-product counfirst
scenario in Table VI corresponds
For the four setup cost cases, the cost
increases of the straw policies relative to the proposed policy are
somewhat
larger for
the two-product cases: the generalized base stock policy's average cost increase
for the
about
one of the two symmetric cases.
the relative cost differences
terparts in Tables
to the 6
in the four
costly than the proposed policy in the two
extremely poorly
Table VI, and
two asymmetric cases. Both of these policies out-
costly than the proposed policy in the
perform the corridor policy
in
two-product cases versus 4.7%i
for the five-product cases,
is
5.8%
and the corresponding
quantities for the generalized base stock policy are 26.3% and 18.7%, respectively. For the
two setup time cases, the average cost increase of the generalized base stock policy
for the
two-product scenarios and -l.l%i
performance of the corridor policy
in
for the five-product cases.
is
2.7%
Disregarding the poor
one of the five-product symmetric setup cost scen-
appears that the relative cost advantage of the proposed policy degrades slightly
arios,
it
when
the
number
of products increases
from two
to five;
however, further experiments
are required to fully investigate this issue.
LacA' of Robustness.
Simulation results not reported here show that the performance
of the three policies are rather sensitive to the policy parameters, particularly in the setup
50
time problem; this
(e.g.,
EOQ
the
Because
policy
is
it
somewhat
is
surprising, given the robustness of
model) that capture the tradeoff between inventory costs and setups.
unable to increase
its lot
sizes as the total inventory decreases, the corridor
clearly the least robust of the three policies:
is
some simpler models
narrow
(as apparently
can set
in (notice the
happened
if
the corridor width
is
set too
eighth row of Table VI) then stability problems
in the
confidence intervals for this case).
The s/y/n Refinement. Recall
that the s/ s/n refinement described in §2.7
ated into the proposed policy, but not the two straw policies.
We
tested
all
is
incorpor-
three policies
with and without the refinement, and summarize our findings here. The refinement had
a minor effect on the performance of the proposed policy in the p
by decreasing the cycle length,
ation cases.
significantly
it
The refinement had
=
improved performance
0.9 cases; however,
in the
lower
utiliz-
a mixed influence on the generalized base stock policy,
sometimes improving and sometimes degrading performance;
overall,
it
slightly
impaired
performance. The refinement had a negative effect on the corridor policy, and
let to
a
severe stability problem in the eighth row of Table VI.
Summary.
Although additional asymmetric cases need to be investigated before
drawing definitive conclusions
summarized
for the
two-product problems, our observations can be
as follows.
The proposed policy performs very
confirm that
ality
is
it
captures nearly
all
well in the 34 two-product cases:
of the complexities of the optimal policy,
6.0% over the 34 cases (and 2.1% over the
the heavy traffic conditions), and
it is
tions, especially considering that
most potential
in settings
with high
Figures 3 to 6
12 cases that
suboptim-
its
do not obviously violate
quite robust with respect to the heavy traffic condi-
traffic intensities.
industrial applications for the
SELSP
are
However, the relative superiority of the proposed
policy appears to degrade slightly as the
number
of products increases,
and
this issue
requires further investigation.
The two straw
the optimal policy.
policies are not flexible
The
enough
to consistently capture the subtleties of
corridor policy outperforms the generalized base stock policy in
24 of the 34 two-product examples, and
its
average suboptimality
to 19.5%) for the generalized base stock policy.
is
11.2%i as
compared
Nonetheless, in the setup time cases the
corridor policy
fails to
use large
when
lot sizes
the total workload
negative and can
is
perform erratically (see Table VIII). The generalized base stock policy
optimal and performs poorly
4.
in the
asymmetric setup cost
is
never close to
cases.
CONCLUDING REMARKS
From
a practical standpoint, the stochastic economic
lot
scheduling problem
is
argu-
ably the most important scheduling problem in operations management. Unfortunately,
there has been no success in obtaining an optimal solution to this problem using standard
techniques
(e.g.,
semi-Markov decision process theory).
to the class of
dynamic
in time:
produce the product that
product
Idle,
in the fixed
is
in
restrict ourselves
at
sequence. Using heavy
traffic analysis, in particular the
closed form, and the idleness policy
is
traffic lot-sizing policy is
reduced to the numerical calculation of
some key
qualitative characteristics
number
of
the problem to a one-dimensional diffusion control problem that
is
of the optimal heavy traffic policy are derived; moreover, regardless of the
we reduce
averaging
Reiman, we make considerable progress on
a single threshold value. For the setup time problem,
products,
each point
currently set up, or switch over to the next
problem: For the setup cost problem, the optimal heavy
derived
we
where the server has three options
cyclic policies,
principles derived by Coffman, Puhalskii and
this
In this paper
solved numerically.
The
explicitness of our results, coupled with the surprisingly intricate behavior of
the optimal policy, leads to a
SELSP. These
insights are
lot-sizing policy
cost structure
embodied
is
number
of
summarized
depends upon whether
new
insights into the optimal solution to the
in §1.8
(i)
and
§2.5,
and describe how the dynamic
setup costs or setup times are incurred,
symmetric or asymmetric across products and
in the current finished
goods inventory
is
much
less
(iii)
(ii)
the
the total workload
than zero,
in a
neighborhood
containing zero, or larger than zero and near the optimal idling threshold.
We
also
perform a heavy
traffic analysis of
related to ones analyzed by Federgruen
two classes of
policies that are closely
and Katalan (1993) and Sharifnia, Caramanis
and Gershwin: the unified treatment of the optimal policy and the two straw
(see. in particular,
policies
Figures 2 through 6) makes transparent the relative strengths and
52
weaknesses of the straw
policies.
A
computational study
is
undertaken to compare the
proposed policy and these two straw policies to the numerically computed optimal policy
in 34
two-product examples. The computational study (see §3.4 for a description of the
key observations) confirms that the insights summarized
in the
in §1.8
and §2.5 do indeed occur
optimal policy. Moreover, numerical results for the two-product examples show that
the proposed policy
traffic conditions,
is
reasonably close to optimal,
and outperforms the two straw
is
robust with respect to the heavy
policies;
two straw
in contrast, the
policies lack the sophistication required to imitate the subtleties of the optimal policy,
their behavior
is
somewhat
erratic.
The computational study
also includes 10 five-product examples,
and the
results sug-
gest that the relative superiority of the proposed policy degrades slightly as the
of products increases.
robustness of heavy
ments
(e.g.,
In the last decade, researchers
traflfic
have improved the accuracy and
approximations through the use of various heuristic
refine-
§2.7),
we
believe that other refinements to the proposed policy
lead to significant improvements and should be investigated; such refinements
be necessary
moderate
in
order to develop effective policies for problems with
once per cycle
in a fixed
(or polling tables),
products and
method
for
policies,
where each
class
is
served
sequence. More generally, the cost of dynamic periodic policies
where each product can be produced more than once
be evaluated using the methods developed here.
efficient
many
may
traffic intensities.
This study has only considered dynamic cyclic
of
number
Harrison 1988, Nguyen 1995, Lennon 1995). Although we have investigated
one such refinement (see
may
and
in a cycle,
can
However, we have not yet found an
optimizing within this class of policies, and a thorough investigation
dynamic periodic and nonperiodic
policies
is left
for future research.
ACKNOWLEDGMENT
This research was supported by a grant from the Leaders
at
for
Manufacturing Program
MIT, NSF grant DDM-9057297 and EPSRC grant GR/J71786. The
the Statistical Laboratory at the University of
of this
work was carried out.
Cambridge
last
author thanks
for its hospitality
while some
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