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Heavy
Traffic
Policies:
A
Analysis
of
Dynamic Cyclic
Treatment of theSingle
Scheduling Problem
Unified
Machine
David M. Markowitz and Lawrence M. Wein
#3925-96-MSA
September,
1996
'1.
r
n,;y
4-,(4;p
HEAVY TRAFFIC ANALYSIS OF DYNAMIC
CYCLIC POLICIES: A UNIFIED TREATMENT OF
THE SINGLE MACHINE SCHEDULING PROBLEM
David M. Markowitz
Logistics
Management
Institute
McLean. \'A 22102
Lawrence M.
We in
Sloan School of Management.
ALLT.
Cambridge. ALA 02139
ABSTRACT
This paper examines how setups, due-dates and the mix of standardized and customized
products
affect the
environment.
We
scheduling of a single machine operating in a dynamic and stochastic
restrict ourseh'es to the class of
machine busy/idle policy and
d}'namic cyclic policies, where the
lot-sizing decisions are controlled in a
but different products must be produced
in
a fixed sequence.
dynamic
.As in earlier
fashion,
work, we
conjecture that an a\-eraging principle holds for this queueing system in the heavy
limit,
and optimize over the
class of
dynamic
cyclic policies.
The
traffic
results allow for a
detailed discussion of the interactions between the due-date, setup and product
facets of the problem.
September 1996
mix
This paper focuses on what we consider to be the three most important structural
machine scheduling problems. The
features of single
characteristic
first
the presence or
is
absence of setup costs and/or setup times when the machine switches from one type of
product to another. Setup penalties force the scheduler to exploit the economies of
which leads to dynamic
lot-sizing policies.
The second
factor
is
scale,
the presence or absence
of advanced information regarding future demands, which gives rise to problems with
and without due-dates,
nature of the object
function
function:
If
This aspect of the problem essentially dictates the
advanced information
based on due-date considerations
is
such information
measures
i\'e
respectix^ely.
a\'ailable
is
(e.g.. earliness
then the objecti\'e function
(e.g.. inA'entory costs,
is
is
pro\ided then the objecti\'e
and tardiness
and
if
no
expressed in terms of system
waiting time, throughput).
The
whether products are customized or standardized. This feature
the make-to-stock/make-to-order distinction:
costs)
third characteristic
is
is
intimatel\- related to
Customized products must be made-to-
order whereas standardized products can (but do not need to) be made-to-stock.
The aim
of this paper
is
to pro\'ide a unified (with respect to these three features)
treatment of single machine scheduling
in a
dynamic and stochastic environment. More
we consider a manufacturing system consisting
specificall}'.
multiple types of goods, which we refer to as products.
customized, which requires the request
standardized,
machine
is
in
for
is
machine that produces
Each product can be either
an order before production can begin, or
which case they can be pre-stocked
in a finished
goods inventory. The
limited in capacity and can only produce one unit at a time.
machine switches from producing one product
penalty
of one
to another, a setup cost
Whenever
the
and/or setup time
incurred. Orders arrive to the system, each requesting a unit of the product at
a specified due-date.
must be held and an
a completed unit
is
A
completed unit assigned
earliness cost
is
to
an order before the order's due-date
incurred. Similarly, a tardiness fee
delivered to an order after
its
is
levied
whenever
due-date. In addition, unassigned items
held in finished goods in\'entory also incur a holding cost equal to the earliness fee of that
(standardized) product. Interarrival times. ser\"ice times, due-date lead times (an order
due-date minus
by product.
its
time of
arri\'al)
and setup times are random
x'ariables that
s
can vary
Customized
Xo Setups
No Due-Dates
Standardized
Figure
In this setting, the
it
is
up
current!}' set
The
1:
\
subproblems under consideration.
eight
machine
at an}' point in
idle,
how much
time can
idle,
produce the product that
a setup for a different product. In other words, the
for. or initiate
scheduler makes three t}'pes of decisions
should be bus\' or
Due-Dates
/
to
in a
make
dynamic
fashion:
of each product
(i.e..
Whether the machine
lot-sizing)
and which
product to produce next. In this paper, we restrict ourselves to dynamic cyclic policies.
which allows
full
discretion o\'er the
fixed c\'clic sequence.
We
first
two decisions, but produces each product
in a
wish to optimize the queueing system with respect to long run
expected average costs due to earliness. tardiness, holding and setups.
The
b}'
our
\'enn diagram in Figure
anal}'sis.
Although
the body of the paper. §5
\"enn diagram.
X'enn diagram,
onl\'
is
1
delineates the eight subproblems that are incorporated
the general scheduling problem
is
considered throughout
de\'oted to a discussion of each of the eight regions in the
Since the existing literature has only examined specific regions of the
we delay our
literature re\'iew to this discussion.
This paper expands upon the methods of Markowitz. Reiman and Wcin (1995) (abbrex'iated hereafter
b}'
MRW'j. which analyzes the stochastic economic
lot
scheduling
problem (depicted
Reimans
is
MRW
4 in Figure 1).
fast
one where time
is
sped up by a factor of 0(n) and a slow one where time
increased by a factor of 0(s/n) (where n goes to
is
to analysis. .As in
and then
fluid limit
in
the heavy traffic limit).
to a fluid limit.
averaging principle couples these two processes and makes this
problem amenable
paper
infinit\- in
and the slow scaling leads
fast scaling leads 1o a diffusion limit
traffic
applies Coffman. Puhalskii and
(1995a. 1995b) heavy traffic averaging principle, which considers two sets of
A
scalings:
subproblem
as
MRW.
in the diffusion limit.
to determine
how due-dates
MRW
The primary
affect the
our previous applications of the heavy
1994. 1995.
we optimize the
difficult
control problem,
hea\'y
scheduling
the
first in
analytical contribution of this
system beha\'ior under the
traffic
The
The
fluid limit. .\s
averaging principle (Reiman and W'ein
and Reiman. Rubio and Wein 1995). we conjecture that
this principle
holds for the system under study, without prox'iding a rigorous proof of convergence:
see
Reiman and Wein
(1996) for a heuristic justification of this conjecture.
closed form solution has eluded us.
for soh'ing the control
we
resort to de\'eloping a
Because a
computational procedure
problem. Howe\'er. to gain further insight, we analyze the special
case where each product has a different deterministic (as opposed to random) due-date
Finally, a simulation study
lead time: in this case, the results simplify considerably.
performed to assess both the
the heavy
traffic
in §2.
our proposed policies and the accuracy of
approximation.
The scheduling problem
duced
effect i\-eness of
is
formulated
in §1
and heavy
traffic
preliminaries are intro-
W'e perform a hea\'y traffic analysis of the scheduling problem
work through the deterministic due-date lead time case
in §4.
Section 5
discussion of the subproblems outlined in the \'enn diagram in Figure
results of a
§7.
computational stud\- on due-date
Readers interested only
§2. §3
and
in
effects.
1
is
in §3
and
devoted to a
and §6 contains
Concluding remarks are offered
the qualitati\-e insights derived from this work
in
may omit
§4.
1.
A
is
single
PROBLEM FORMULATION
machine produces
assume that products
1.2
.V
different products.
V' are customized and
3
Without
.V"
-I-
1
loss of generality,
V'
-I-
A"'
=
we
A' arc
standardized. Each product
and
jj'l'^
has
;
coefficient of \ariatioii
demand
process.
c,j.
For each product
mean
tributed with
its
and
A,"^
own
Orders
i.
mean
generally distributed service time with
goods arrive from an exogenous renewal
for
the interarri\'al times of orders are generally dis-
coefficient of variation c,^.
The
and service time
arrival
processes for each product are assumed to be independent, although they need not be
Reiman 1984
(see
utilization of product
Orders
is
compound renewal
for the incorporation of correlated
arri\'e to
i
is
=
p,
X,l jJ,
and the system
utilization, or traffic intensity,
The due-date
the system with a specified due-date.
the time interval between an order's arrival and
variable with density f,(s). cumulati\-e distribution function F,{s)
independent of the
a
We
abo\-e the quantity denotes the diffusion scaling.)
time distribution has bounded support, and denote
6,.
further
assume that
a,
>
completed unit
)
of
tardy.
is
If
if
an order
6,
is
present.
Once
and
is
scalings.
minimum and maximum by
its
the unit
a,
demand, we
a customized order
is
onl>-
is
work
serviced, the
either held until the order's due-date or delivered immediately
if
the
held, the order incurs an earliness fee (holding cost) h, per
if
the order
is
past due.
it
incurs a tardiness fee (backorder
per unit time late. In anticipation of our subsequent analysis, we also express
these cost parameters in terms of units of work:
The
/,.
0.
unit time until the due-date:
cost
random
assume that the due-date lead
customized goods are immediatel}- queued. The machine can
for
on a customized product
is
a
and mean
respectively: because due-date information typically reflects future
Orders
order
is
denotes a lack of scaling, a "bar" denotes the fluid scaling, and no symbol
"tilde"
and
i
and service processes. (For quantities that undergo
arrix'al
is
lead time, which
due-date, for product
its
The
processes).
flow of physical product
and orders
is
h,
—
hifj,
and
more complex
b,
for
=
b,iJ,.
standardized products,
and we need to keep track of both orders and completed items. Completed items can be
pre-stocked
in
the finished goods inventory, where each unit accrues holding costs at a rate
of h, per unit time. Orders for standardized products are
filled b}'
queued upon
arri\al.
and can be
an item either from the finished goods in\-entory or directly from the output of
the machine.
Once an item
is
assigned to an order, the s\'stem incurs either an earliness
fee h, for
each unit of time until the order's due-date or a tardiness fee of
=
tardy. Again, these costs have workload equivalents h,
we have made the natural assumption
the finished goods inventory
item
is
and
h,fj,
6,
=
per unit time
Notice that
b,jj,.
that the holding cost for an unallocated item in
identical to the earliness cost incurred
is
b,
allocated to a standardized order before
its
due-date
(in
when
a completed
the latter case, one can
Hence, no
envision the item sitting on the shipping dock until the order's due-date).
benefit can be gained by assigning a completed item to a standardized order before
due-date: Delaying the allocation of completed items to orders results in
and a lower cost
Therefore, without loss of generality,
policy.
that incur no earliness fees for standardized products,
the system
when completed items
The scheduler can observe
notational burden,
will
be used
a
in this section for
set
in
up
those quantities that
Let the slack of an order be
traffic anal}'sis.
is
its
identical to its due-date lead
number
which we denote
(or being set up),
is
unit rate throughout the order's sojourn in the
at
system. The system state includes the
in\entor_\-.
exit
the system, but the slack, which can be positi\-e or negative,
dynamic quantity that decreases
product's queue, the
policies
To ease the
the system state at each point in time.
we only introduce notation
arri\'es to
it
we only consider
and so standardized orders
due-date minus the current time. Hence, an orders slack
time when
flexibility
are assigned to them.
our subsequent hea\y
in
more
its
arri\'al
time and the slack of each order
in
each
of items of each standardized product in finished goods
b\- I,(t) for
;
=
V. the product that
.V" 4-1
and the residual service and setup times
(if
currently
is
they are currently
progress).
The machine
cycle.
follows a
At any point
in
dynamic
c\xlic polic>'.
.\11
time the scheduler can deploy the machine
Produce the product currenth-
set
up
for
(this
The
policy
is
dynamic
in that
the
in
lot
set
up
sizes
in a fixed
one of three ways:
might not be possible
customized product and there are no orders present),
cycle, or idle.
products are serviced
for the next
if
set
up
product
and the busy/idle
for a
in
the
polic}-
can
be molded to address the changing needs of the system.
A
in
the
penalty
c}'cle.
is
incurred ever>- time the machine switches production to the next product
This penalty can be a cost, a period of downtime or both. The hea\y
traffic
performance of the system depends on these penalties
total switchover
and the a\erage
onl\-
through the average setup
time per cycle.
cost
only one form of setup
per cycle.
A',
penalty
present and setups \-ary by product, then the best cycle order can be found by
is
If
5.
sohing a traveling salesman problem, where the distance between two
to the setup penalty
the situation
between two products.
more complex.
is
We
also
the appro.x'imation scheme used here
is
If
cities
corresponds
both forms of penalties are present then
assume that the policy
preemptive-resume, but
is
too coarse to differentiate between a system with
a preemptive-resume polic\' \'ersus one without preemption.
We
wish to minimize the long run expected average cost of the system, and additional
notation
is
required to describe our objecti\-e function. Let T^, be the time that the n'
unit of product
i
we assume
it
this will
t.
The
at
is
minimize
define the
queue
that
assigned to an order.
is
t.
a completed unit
is
assigned to an order,
assigned to the order with the earliest due-date within
slack L,{t) to be the smallest slack
among
J(0 be the cumulati\-e number
Finall\-. let
long run ax'erage cost
its
product, as
and Teneketzis 1995 and Righter 1996). For
cost (sec Pandelis
minimum
time
When
all
t
>
0.
orders in product Ts
of cycle completions
by time
is
^^^
1
E
limsup-|X:
T y,^j
T->oo
T->c
+
Z
=.V'=
where x+(0
+
[l^h.l(t)dt+
'^°
x'{t)
product of cost and queue length
inventory, whereas
+
b,L:{Tr.,)]
(1)
Yl
b,L;{Tn,)]
+ I<J{T)
{n\T„,<T}
1
= max{j(O-0} and
[h.Lt{T^.)
{n|r„,<r}
we have chosen
is
=
ma.x{-x(t).0}. Notice that the multiplicative
integrated o\er time for the items in finished goods
to
sum
the product of cost and time
over orders. This "reversal of the order of integration"
is
(e.g. /i,I, (Tn.,))
a necessary step in analyzing
the effects of due-dates.
2.
HEAVY TRAFFIC PRELIMINARIES
This section prox'ides an over\'iew of the hea\-y
scribed in §1.
The approach outlined here
traffic anal\-sis of
relies heavily
upon
results in
the problem de-
Coffman. Puhal-
and Reiman and
skii
MRW.
and we
Workload Processes. We
heavy
traffic
analysis.
begin by defining the key stochastic processes for our
Recall that the system has queues of orders for
and a finished goods in\entory of items
products
;
=
A'^. let \Vi{t)
1
for all
be the total amount of time required
queue
I
=
+
A'"^
standardized orders of product
;
minus the
total
workload process
The
workload process.
product
for
total
t.
for the
machine
For standardized
machine time already invested
units in product Ts finished goods inventory at time
as the
time
products
be the total amount of time needed to process
A', let \\',(t)
1
at
all
For customized
standardized products.
to process all of the orders that are in product Ts
products
papers for more details.
refer readers to these
=
;
V.
1
workload at time
t
t.
and
We
let
=
the
W, = {ir,(0-' ^ 0}
refer to
it'
in
all
J2',=i ^^i
represents the total
be the total
amount
of work
currently being requested (in the form of orders) minus the total work stored in in\-entory
(this
work
will
This one-dimensional process
eventually be assigned to orders).
is
the
natural definition of workload for systems with customized and standardized products:
readers should note that, in contrast to
"make-to-order" point of view,
goods inventory
Heavy
traffic
in that
MRW.
work
in
the total workload
orders
is
positi\'e
is
defined from the
and work
in finished
negative.
is
Traffic
Averaging Principle.
Coffman. Puhalskii and Reiman"s hea\y
HTAP)
averaging principle (abbreviated by
has augmented the understanding
of hea\'ily loaded multiclass queueing systems that incur setup costs or setup times.
Although rigorously proved
for a two-class
queue
(in
the absence and presence of setup
times in 1995a and 1995b. respectively) that employs an exhaustive service discipline,
numerical results
in
Reiman and Wein
(1994. 1996).
support the conjecture that the HT.AP holds for a
these applications,
we conjecture
that the
MRW and
Reiman. Rubio and Wein
much wider
class of systems.
HT.AP holds without providing
.As in
a rigorous proof
of convergence.
The HT.AP
is
based on two
sets of limits,
synchronized by a scaling parameter
p)
approaches a constant
c.
n.
The
taken as the total utilization goes to one and
first
limit of the
HT.AP
states that as \/n{l
—
the normalized total workload process \\'(nt)/\/n weakh'
con\'erges to a diffusion process. 11(0. with parameters defined by the system data
and
policy.
It is
the diffusion limit and
calleil
The second
is tli(>
theorem.
result of a functional central limit
same
limit, called the 6iiid limit, states that for the
scaling parameter
and
ii
for
a given total workload, the individual workloads \\',(\/nt)/ \/n. converge almost surely
to
l'V'i(0'
process;
fluid
3,
this result
related to the functional strong law of large
is
numbers.
The time
two limits
scale decomposition inherent in these
approaches one, the amount of work
in
the system
is
intuitive: .As utilization
is
and the
large
workload cannot
total
change quickly; yet, as the machine switches between products, work can
among
The
the individual queues and inventories.
shift rapidly
shifting of individual workloads occurs
an order of magnitude. 0{\/ri). more quickly than the total workload, and so the
exam-
limit evolves for a period while the total workload remains relatively constant. For
workload might change on the order of weeks, while individual workloads
ple, the total
change
daily.
Dynamic
Cyclic Policies and the Fluid Limit. Under a dynamic
a service cycle corresponds to the setting
the machine
will
fluid
is
following a
dynamic
be completed before there
there are
many
.V products. If
many
that
dynamic
cyclic [)olicy can
be expressed
workloads;
level, not the individual
i.e.,
by an idling threshold
cycle center
.r'^{w)
process by
W
u'o
.r';(tr)
traffic
feasible total
of the cycle center
in
the fluid
liiuit.
is
is
workload by w).
the average
The
The
workload
total
fluid
process
amount
of
cycle length t{w)
is
work
the
in
product
amount
in ser\ice.
the process
U',
decreases at rate
8
i
is
i^^
over the
of lime required
and queues to remain balanced,
the machine must allocate p,T{ir) units of time per cycle producing product
is
The
dictated by the latter two functions.
to complete a cycle in the fluid limit. For the inventories
;
level.
can be completely characterized
and the cycle length t{w) (throughout, we denote the
and an arbitrary
course of a cycle
heavy
lot
and two workload-dependent functions, the .V— dimensional
unaffected by the idling threshold, and
product
the
each product and the busy/idle policy depend only on the total workload
policies in
cycles
Because
a significant change in total workload W{t).
workload
.VIRW show that dynamic cyclic
element
HT.\P implies
cyclic policy, then the
cycles for a given total workload, a
as a function of only the total
sizes for
is
up and processing of each of the
cyclic policy.
(1
—
p,):
/.
While
while the machine
is
set
up
for
other products.
=
iV'^+
the fluid limit:
The
inventory for
in
/
relatively infrec[uently.
increases at rate
II',
1,
.
.
N). Setu]) times are unsealed
,
.
large
amount
of
work
in
of .r';{w)
+ p,{ — p,)t(w)/2. The
must
ecjual the total
cycle
must be nonnegative
if
workload, J2,=\
for a
•''''iiw)
remain 0(1)) and vanish
level fluctuates
minimum
—
of x'i{w)
by p,[\
/y,(l
—
—
p,)t[w)
p,)t{w)/2 to a
cycle center and cycle length are parameters
1
that can be set by the scheduler, subject to
(i.e.,
the system causes setups to be performed
Thus. i)roduct Ts workload
over the course of a cycle, and ranges from a
maximum
(p(M-haps by depleting the finished goods
p,
some
=
restrictions:
of the cycle centers
minimum amount
the
u';
The sum
customized product,
.r"-{w)
—
^,(
I
work over the
of
— p,)t{w)/2 >
0;
and
there are setup penalties then the cycle length t{w) must be greater than zero.
Cost per Cycle.
CJiven the
dynamics described above, we can
a fluid cycle. c{ic). in terms of the parameters
composed
and
of individual product (earliness, tardiness
The
and t{w).
cost per cycle
and holding) costs and
is
.setup costs,
given by
is
N
A-
where
total
r'-'{w)
e.xpress the cost of
w)
c,(.r'\ r.
workload
and Wein 1991
is
«-.
product
and
A'
=
/"s
;^-
average cost rate for a cycle given a policy
l\ /
n
is
.r'
and r and
the normalized setup cost per cycle (see
for a justification of this scaling).
In
MRW,
we calculated
Reiman
c,(.r''. r, u')
by
integrating the cost over one production cycle and then dividing by the cycle length. In
the next section,
we perform the same type
of calculation for our due-date
"interchange the order of integration" as discussed before. That
over time in the cycle, we integrate over the orders
Dynamic
Reiman show
When
filled
is,
instead of integrating
during the cycle.
Cyclic Policies and the Diffusion Limit.
Coffman. Puhalskii and
that the drift of the diffusion process for total workload level
there are no setup times, the diffusion limit
is
problem but
w
is
.</T{ic)
a reflected Brownian motion
— c.
(RBM).
regardless of the policy in use: this result coincides with classic results (Iglehart and Whitt
1!)70).
Since the cycle length T(tr)
is
dictated by the
dynamic
cyclic policy, the drift of
the diflfusion process depends on both the policy and setup times
when setup times
are
The variance
present.
of the diffusion process
a^
is
=
J2',=i
^(^ai
+
f"si)' '^"^^
^^^''^ '^
"^"ly
dependent on system parameters.
Although the idhng threshoki
u'o
does not affect the fluid process, this control pa-
rameter does impact the one-dimensional diffusion process
barrier.
There are
system.
In this case,
restrictions on u\) only
of the cost on our policy,
we can
m sup
T->co
given
functions
.(''^(ir)
minimize
(2)
is
Our
in (2).
and
there are no standardized goods in the
Suppressing the notation illustrating the dependence
T'
1
is
"(((').
c{W{t))clt,
/
7;
J/
(3)
Jo
controls arc the idling threshold
.As in .\I1{\V.
with respect to
the optimization
is
ii-u
and the
;V
carried out as follows.
to find the cycle center in terms of r(u')
,r'
— dimensional
and
We
This
lcq.
a constrained nonlinear optimization problem. Then (3) becomes a diffusion control
problem with
a drift control (via the cycle length t{u-))
idling threshold
The
in several steps:
In
^(.].i
is
and
to optimize the cost in
§3.2.
we
('^).
policies
on
a fixed total
3.1.
is
performed
c,{x'^ .t.
w)
in (2) for
To ease readability
in
these two
we suppress the notation depicting the dependence
of the fluid processes
workload w. Then we perform the cost minimization
atul translate the solution into a
problem
This optimization
calculate the cost function
customized and standardized products, respectively.
and
(via the
OPTIMAL DYNAMIC CYCLIC POLICIES
goal of this section
subsections,
and a singular control
icq).
3.
is
reflecting
its
e.xpress the objective function in equation (1) as
11
c{ir)
by acting as
wq must be nonnegative.
The Optimization Problem.
where
if
11'
dynamic
cyclic policy for the original
in §3.3
queueing control
in §3.4.
Customized Products.
to determine
how due-dates
affect
The
i)rimary challenge in calculating c,(.r'\T.
this cost
by normalizing the due-date lead times.
typically 0(i/77) in heavy traffic mocU'ls.
function in the fluid limit.
We
ir)
begin
Because queue lengths and waiting times are
it
is
10
appropriate
for
due-date lead times to also
be 0{\/n). Therefore, we assume that the due-date lead time density f,(\/ns) converges
to the nontrivial density /,{<). witli cumulative distribution function F,{s)
and mean
=
/,
converges to a point mass at ^
appear
is
=
and
Hence, due-dates do not
zero elsewhere.
is
the diffusion process and are isolated
in
F,{^/us)
under a diffusion time scaling the density f,{ns)
In contrast,
l,/\/n-
—
in
the fluid limit. This state of affairs
consistent with the heavy traffic snapshot principle (see Reiman), which states that a
customer's sojourn
=
Let Li{t)
in
the system
is
L,[\/nt)l s/n denote the
define the process D,(s.t) to be the
time
t
that
has slack
is
due
at
time
.•<
=
Let D,{.sJ)
.s.
instantaneous under the diffusion scaling.
-|- f:
in
minimum
amount
slack process in the fluid system,
of product
other words. D,{s.
D,(s/ns.
^t)/
^
t) is
lenote
work present
/
in
and
the system at
the work present at time
t
that
Notice that D,{sJ)
its fluid limit.
provides a more detailed description of the orders than 11,(0-
Equation
(1)
shows that the key to calculating the
the behavior of the
minimum
function
is
c,(.r'^. r. »•)
slack
derixed
tion (11), respectively):
We
in
to
it.
function
is
to determine
throughout the course of the cycle. The cost
/!,(/)
three steps (in Proposition
2,
Proposition 3 and equa-
calculate the process D,{s.t) in terms of the original problem
parameters and the mininumi slack
process D,(.s.t) (in these
cost
first
at the start of the cycle, derive L,(t) in
two steps, we replace D,{s.t)
as explained after Proposition 1).
l>y
terms of the
a process closely related
and express the cost function
c,(.r'^.T. ic) in
terms
of L,(n.
Without
and service
loss of generality,
is
we assume
received from time
(1
—
p,)T to r. Hence, the
the system at the start of a cycle, which
The
initial
work
,r'
the system at time zero.
is
we denote by
nuist be stored in the
greater than or equal to
zero
that a cycle for protluct
Z.,(0).
The
system
/
amount
,rf, is
of product
equal to
.r^"
—
work
—
in
p, )/'-.
as orders with due-date lead times
between D,(s.t) and the workload
given by the following proposition, where f]'{s)
11
/
"^,(1
by definition the smallest slack among product
relation
time zero
starts at
is
defined as
1
—
/
orders in
n',(/) at
F,{.-<).
time
Proposition
1
time
.1/
t
=
0,
=
A(.s.O)
(4)
ifs<LdO)
and
r
1=
P.F[(s)ds.
(5)
JL.iO)
Proof: Because the |)roduct
under the
at rate pi
.s
workload arrival process
/'
any
fluid scaling, at
dt instant of
The work
units of time arrive in the fluid limit.
bounded by wie maximum amount
At time zero, the
work
maximum amount
work plus work that arrived
+
of
which
r)dr.
/
is
of
in
deterministic and flows in
is
time f,(^)p,dl units of work due
the system at time
could have arrived with a due-date of
th;.l
work present with slack
units in the past with a due-date of
equal to p,F^'{s). Thus, D,(.s,0)
<
.s
.s
^-f-^-
+ 7\
Xotationallw this
p,F[{s). This inequality
/o^/3,/,(-s
is
strict
is
being used and the machine processes work at rate one. which
orders due after time L,(0) were worked on. Since an earliest due-date policy
if
is
strictly greater
follows that D,(s.t) at the next instant either vanishes or
it
service and only affected by arrivals.
producing product
is
than
untouched by
time zero, the machine has just switched out of
.\t
implying that the work due after time L,{0) has not been touched
/,
and that there has been no opportunity
for orders to arri\'e
with due-date lead times
below L,(0~), the smallest slack the instant before the switchover. Thus, equation
.r^
=
/£^m) D,{s,0)ds.
Because the interaction of D,[s,
t)
and
holds.
By
is
the recently arrived
is
.s
is
PiF'^ls),
with slack
t
in
construction,
we have
Combining
this
with equation
(4)
(4)
yields equation (5). |
L,(t)
is
complex, we streamline our calculations
by creating a simplified variant of D,(s.t) that evolves through the course of one cycle.
Let Di''(s.t) be the
amount
work
from
for
Di^{sJ)
D;^(.s. /)
product
i
= D,[sJ)
is
useful for
directly from
it.
for
.s
of
work
=
t
>
time
at
nnf;7 r;
L,(f) but
two reasons:
The evolution
of
Its
is
/
with slack
D'^(sJ)
is
/);^(.'<./) is
12
is
if
the machine performs no
only defined for
not necessarily
behavior
.s
for
.s
<
easy to describe, and
t
G
L,(t).
I^,{t)
[O.r).
The
Thus.
process
can be derived
described by the following proposition.
Proposition 2 Fort G
vv
^"(.^.0
[O.r).
=
(6)
p,{Fns)-F[{s +
[
if,<L,{0)-t
t))
Proof: By construction, Di'{i>J) evolves according to
+
D;'(../
That
+
/
s
is,
+
amount
the
df
with slack
is
5
of i)rodurt
,//)
=
work
/'
D;^(5
+
+
f/^/)
amount
dt phis the
D-'isJ)
the solution to eciuation
+ Ml)
D,(s
in for
/
equal to the amount of work that was previously
+
of
in (8).
=
(7)
+
dt that
in
-
p,(F:{s)
F^{s
+
t))
t
+
=
0.
D,(.s
due
time
at
.s
readers can verify that
+
/,0)
(8)
follows by using equation (1) to substitute
|
tures regarding the evolution of D]^(s,t) in Figure
2.
of o,
= aj \JTi
D\ {s,t) as a function of
Although
and
6,
=
6,/x/n.
Figures 2b-2d display, for a fixed value of
and
is
perhaps more instructive to describe the dynamics of D'^ {s.t) as
t.
this figure represents
progresses through the cycle, the existing orders age and
metaphor
is
as waves,
and new orders
/
increases.
new orders
arrive.
t.
/.,(0).
snapshots of D^" {s.t) at a fixed time
a,
a, -f
minimum and
under three cases depending on the relative value of
s
fea-
For the sake of concreteness,
Figure 2a contains a uniform due-date lead time distribution, with fluid
maximum
t
that has just
an attempt to enhance the reader's intuition, we point out some noteworthy
In
t
is
the system at time
work with a due-date lead time of
The proposition
(7).
equation
p,/(.s).
the system at time
in
arrived. Using the fact that D'^is.t) equals D,{s.t) at
is
tlie differential
t,
it
As time
.A
useful
to imagine the area under the curves in Figures 2b-2d as water, the curves
work
travel to the left
grow
in
height as
as raindrops
(i.e.,
slack
new orders
time zero has slack
Z-,(0)
—
are
/
at
is
accumulating on the waves. Then the waves of
decreasing)
added
to
time
if
/
in
Figures 2b-2d as time progresses, and
them. Notice that work with slack
no work
13
is
/-i(O) at
performed dining the cycle.
By
equation
is
(6), for slacks
Z)f (s,<)
—
exceeding this
.s
—
is
invariant over time
/.
(.s,/)
Proposition
and
/,(0)
its
at
at
1,
maximum
time
a,
all
values of
>
,s
{sj)
is
of
L,(0)
— /,
—
are
t
propor-
=
=
t
value oi
p,
.^
(see Figures
t
[L,{0)
2b and
—
t.a], D'^'isJ)
is
2c); here, all of the
has arrived.
/
0,
has the e<:|uilil)rium value
Z)f'(.s,0)
depending on
greater than the initial
then no new orders
Also, for
6,.
p,F[{.->)
for
However, the evolution over the cycle of D^{s,t) has two
otherwise.
is
-|-
.s
time
different qualitative structures
lead time o,
arriving for
.s
Figures
The amount
eciuiUbrium:
in these eciuilibrium regions, D'^
/
drops to zero at the point 5
work that could be due
>
is
in
Figures 2b-2d to the right of the barrier Zi(0)
For a fixed time
constant and takes on
i:
in
that has slack
/
complementary cunudative distribution function of the due-date lead time,
tional to the
By
Z);^(.s,/) is in
aging ecjuals the amount of work that
and so the shape of the wa\es
and so /)f
the cjuantity
/,
the work at time
which corres[)oiKl to the regions
p,F^'{s). For these slack values,
2b-2d to the right of Z,(0)
work that
critical value,
minimum
have a slack
will
the earliest possible arriving due-date
if
less
slack Z,(0), or less than
If
it.
L,(0)
than those that had a slack of L,(0)
<
at
time zero. Therefore, orders arrive with just the correct due-date lead time distribution
to maintain the equilibrium behavior of /)-^(.s,0), as displayed throughout Figure 2b.
However,
the
minimum
a small
—
a,
then new orders can arrive with a due-date lead time smaller than
slack of the orders present at time zero. In this case, as time
amount
p,F^{L,{0)
of orders with slacks less than Z,(0)
which
/).
leadtime Z,(0)
barrier
>
if Z/,(0)
—
/
is
the (juantity of work
in
—
/
accumulates but never exceeds
the system at time
with due-date
t
(see Figure 2d). Thus, the critical slack value of Z,(0)
between the
ec|uilil)rium
work
progresses,
i
—
t
marks the
level p,F^{s) (to the right of the barrier)
accumulation of new orders with small due-date lead times (to the
left
and the
of the barrier).
Finally, referring to the portion of the curve to the left of the barrier in Figures 2b-2d,
in
the region
.s
<
Z,(0)
—
D^(^.t) must equal zero
/.
all
for
.'^
arriving at time zero with the
.\ldous. Kelly
mance
of the
<
—
t:
minimum
and Lehoczky
of a single-class (il
a,
work
in
1,(0)
is
due
after
this critical value of s
due-date lead time of
time
.s
+
/,
and hence
corresponds to an order
a,.
(1!)95) use heav\- traffic theor\' to
analyze the perfor-
/G/[ fpieue with randcMU due-date lead times that operates
14
Due-date Lead Time DensitA'
/
(a)
bi
^(0) <
a,
N
D^^^t.)
(b)
1,(0)
-f
h
(I,
a,
<
<
L,(0)
a,
+t
(c)
A'(^-0
"^5
a,-/
L,(0)-t
a,
a
+t <
L,(0)
W
wv
Df(..^)
«,
Figure
2:
-
/
a,
Z,,(0)
The function D]
15
-
(.s./)
/
as a function of
s.
They
uiuler the first-come first-served discipline.
cal to
deriv(> a figure that
essentially identi-
is
Figure 2b. Because there are no setups and only one product, the non-equilibrium
region to the
of the due-date lead time barrier L,{0)
left
—
/
does not appear
in their
problem.
Our next
When
step
the machine
sponds either
/
(i.e.
is
minimum
terms of
slack process L,{t) in
[s.t).
D',
servicing other products, the order with the earliest due-date corre-
to the earliest
<
Z,(0)
is
to calculate the
due-date request just as the server switched out of product
or the request w^ith the due-date lead time of «, that just arrived after
a,)
the machine switched out
l,{t)
(i.e.,
L,{0)
>
«,).
Thus, we have
= m\n[a,-t,L.{0)-t]
for
f
-
e [0.t{1
(9)
p,)).
Recall that D['{s.t) was constructed under the assumption that the machine does not
work on [product
consumes the
left
/
throughout the cycle.
most part of the curves
in
When
product
t
—
t([
—
/>,
)
L,{t) has
Proposition 3
(5),
Ecpiations
ov^er
/.,(/)
/--^
(!))
(.5),
and (10)
(9)
and
1"^'
t
(^"(l
—
the server
/^,),r).
work corresponding to D; {sj)
D^'{sJ)cl..
be used to express the
r(
initial
minimum
minimum
for 5
1
(|ualitative behavior of the
— p, no
)
Since orders age and get closer to their due-date, the
16
minimum
one
cycle.
and the primitive
this expression here).
minimum
and only orders
services occur
(10)
slack process L,(t)
slack
problem (although we do not write out
we can summarize the
zero until time
/e(r(l-p,).r).
for
iiniqucly det(rnunt L,[t) outr tht course of
(10) can
probabilistic processes of the
From time
t
fhf ^inallrst qunntittj that satisfies
the course of one cycle solely in terms of the
In addition,
For
been completed. Thus we have the following proposition.
t-T(l-p,)=
Equations
being served, the machine
Since the machine works according
units of work.
to the earliest due-date rule within each product,
below
is
Figures 2b-2d (in the context of our metaphor,
the machine swallows the water at a constant rate).
would ha\e completed
/
slack process.
arri\'e to
the (pieue.
slack in product
/.
f.,{f).
is
monotonically decreasing at unit rate
—
r(l
until r,
)
/>,
Z,(/)
greater than Dt{s.t)
behavior
orders are being
more comple.x.
is
<
L,{0)
If
o,
product
/,
L,(t) increases linearly until the
machine
is
consuming the
of the cycle.
If
Z,(0)
the region to the
left
when the
is
barrier
>
flat
a,
step
tii.al
—/?,))'
seen in (9).
<^s
then,
when
end of the
part of the curve
and
of the barrier Z,(0)
—
in
t
to
always
is
cycle; referring to Figure 2b, the
L,(t) does not reach a, before the
is
moving quickly through
volatile,
Figures 2c and 2d, dramatically slowing
tail
determine product
.rj'
i's
(represented by
average inventory cost per cycle
and cycle length
r.
It
is
in
terms
the average of the
ecjual to
Since the machine
earliness or tartliness costs associated with orders as they are filled.
follows an earliest due-date policy, the earliness or tardiness of an order filled at time
/.,(/)
or the due-date lead time associated with a current arrival
a due-date lead time
than
L,(t),
than L,(t).
less
end
reached.
is
of Li{t) for a given cycle center
either
its
the machine begins working on
passed and then speeding up again as the right
is
From time
than they can age), although
filled faster
then the rate of increase
p,F[{s) in Figures 2c and 2d)
Our
G (0,r(l
t
monotonically increasing because the service rate
is
(i.e.,
for
then the machine spends
If
if
t
is
that arrival has
there are arrivals with due-date lead times ^ less
p,f,{-s) fraction
of effort on
Thus the average
fraction of effort on orders with slack Li{t).
them and
1
— p,F,(L,{t))
cost for a product
/
order
is
r,«,r,»0 = i
r
(b,L;(t)
T Jt(\-p,) \
+
h,
['^'^'\>J.{s)sds ]dt.
(l-p,F,(L,[t)))L:{t)+
'
[^
Ju
\)
(11)
Although
HTAP
in
this cost function
is
complex, the average cost per cycle
has dramatically simplified the problem. As noted
a Markov decision process framework
is
The
fluid limit
iD,(.s./).
point of view, the ideas are similar because D,[-J)
is
L^
.
The
is
a point
fluid limit
in tin-
the state of the system
in §1.
progression through this complex state space into
so
computable. The
unwieldy l)ecause the evolution of orders
with due-dates needs to be tracked over time.
domain and
is
has transformed order
From
a functional analysis
a bounded function on a compact
infinite-dimensional space of sc|uare integrable functions
thus approximates the e\-olution of orders
in
the system as a path
L^
in
.
parameterized by
the path in L'
index
tlie
t
in
made computationally
is
"reversing the order of integration."
we
Standardized Goods. The
more complex than
in
of product
/
embedded
work
machine time embedded
.An
abstract.
is
xMoreover, by
1-3.
are able to take advantage of this tractability
cycle.
cost calculations for standardized products are
the customized case because the ciueue of orders and the inventory
of completed items are
amount
relationship
tliis
tractable by Propositions
and translate D,{s.t) into an average cost per
3.2.
Althongh
D,(-J].
important aspect of
in
finished goods inventory at time
in
in /,(0)- a'ld let
U
/(/)
Let W/{t) represent the
the products' workload.
that
is
it
W/{t)
=
^y/{\/nt)/
must be nonnegativc
inventory; backorders are in the form ol unfilled orders
^
as
it
f
be
(i.e.,
the
its fluid
amount
of
counterpart.
represents actual goods
We
assume, as
in
the customized case, that at time zero the server has just switched out of product
/.
in
Because D]
(.^.t)
=
D,{.^.t) for
.s
>
in
D,(s,t).
workload process
L,{t). the
IV',(0 f^^n
'^f'
expressed
as
r
\V,{t)^
r)^'{s.t)ds-\V,'{t)
/G[0,r).
for
(12)
Jl,{i]
Recall that
we do not
assign completed units to standardized orders before their
due-date. This leads to the imj^ortant observation that L,{t)
for this
is
simple:
If
for
some unexplainable reason
event suddenly shifts the total workload le\el
to orders
and instead place them
in finished
\\',(t))
>
Z,(/)
<
for all
(for
/.
instance
The
rational
when
a rare
then we assign no completed units
goods in\entory. The
then decreases to zero and never again goes higher. This
is
minimum
slack
/.,(/)
a transitory effect that will
be washed away after the repetition of several cycles and so can be ignored. Hence, by
equation (6) we can conclude that
D;^'(.s,/)
=
/j,/'7(.s)
for
.s
>
and
/e[0,r).
Because units from in\entory are allocated to |)revent backorders,
only
when 117(0 =
it
follows thai
(13)
/.,(/)
<
0: i.e.,
\\\'{t)L,{t)
=
for
18
all
/.
(14)
So
<
Z,(^)
if
product
then the next unit of product
completed by the machine
/'
order with the earHest due-date, and a tardiness cost
/'
is
assigned to the
is
incurred.
Completed
units that are not assigned to ord<^rs are placed in the finished goods inventory represented
by \V/{t) and accumulate a holding cost. Therefore, the cost per cycle
product
1
r
W]
T.
b~i-{t)dt^-
r hxv![i)dt
Ecpiations (r2)-( 14) lead to the useful relation that li'/(/)
=
PiF[(s)ds, and so \\'/{t)
know the behavior
of 11,(0
(n'',(/)
—
pj,)
cost portion of the cost per cycle in (15)
equivalently, cycle center
(SELSP) holding
Now we
,r^')
by
p,l,
It
is
in (15).
r
.s
in
>
Z,(0)
<
—
(9)
/
Thus, the liolding
economic
lot
scheduling problem
Because
<
/.,(/)
when
tardiness costs
D^{sJ)ds+ rp,F:{,)ds
in
and the
fact that
the
integral,
first
in this integral,
:i6)
Jo
during these times. To determine D^^isJ)
-'*
MRW.
follows by (12)-(14) that
it
by
<
just a translation in terms of workload (or
is
JL,[t)
t
\V,{t)
cost.
U-(/)=
—
if
important to note that we already
of the stochastic
turn to the tardiness costs
are incurred,
.
positive only
is
standardized products from
fui"
(15)
Jo
Jt(\-p,)
Z,(0)
a standeirdized
is
C,(X%
Ji]^
for
Z,(0
is
in
the
first
integral,
nondecreasing
and so D^" (sj)
equation (16) becomes \\\{t)
=
=
for
p,F^'(s)
/
we observe that
€ (t(1 —
/>,).r).
by Proposition
jl^^,)p,ds
+
pJ,.
We
2.
L,(t)
>
Hence.
Because
conclude that
the backorder regions
p,L;(t)
By equations
(15)
and
(17). the
= {\\\{t)-pjY
(17)
.
time average tardiness cost
r
for a
given total workload
(\\){t)-p,tydt
is
(IS)
/'.
.Again, this
is
a translation of the
SELSP
cost per cycle.
Therefore, the cost per cycle for standardized goods with due-dates
same
as the
SELSP
cost
with a cycle center
19
shift
by
/>,/,.
which
is
is
just
exactly the
])ro(luct
/
s
utilization times its
products,
case,
is
it
follows
and the cost
t
is
mean due-date
For a system with only standardized
lead time.
hat the optimal switching curves are shifted by pj, from the
independent of the due-date lead times;
discussed further in §4.5, §5.6 and §5.8. Thus, as
into three regions based on
the cycle.
The
there
is
c,(.r"', r, tr)
is
broken down
only holding, only backordering or mi.xed costs over
cost per cycle can be expressed as
f'dpJ^
c,{x'-\T.ir)
if
important observation
this
.MRW,
in
SELSP
=
-
-1-1)
2
ifoe
[/>,/,
-,r:-±^i^4^]
<^
if /j,[,
-
.r;'
<
TpAi-P.)
(19)
3.3 Optimization. With an expression
in
for
average cost
for
each level of workload
hand, we can optimize over the dynamic cyclic policy as determined by
U'o-
The
generalit\- of the
due-date lead time distribution prevents an
numerical procedure, however,
is
both the
levels.
fluid
and the diffusion
.;"',
r
and
e.xact solution.
.A
Policy optimization must be performed on
possible.
I'nder the fluid scalings. the cycle center
,;•"
cycle length r and total workload level u\
can be optimized with respect to a given
This nonlinear program can be stated as
follows:
min
such that:
Z'XiC,(x'.T.
E;^=
.r?
The
(20)
(21)
i-K
> iMizM
f^.)r
;
highly noidinear aspects of L, and thus c,{x''.t.u')
V"
1,
make
(22)
this
problem complex.
Nonetheless, the problem can be solved numerically using standard descent methods,
and we denote the solution
In the dilfusion
Ci\en
In' .;"'( r, lu).
scheme, the long run average cost of the
that the optimal cycle center x''{t.
ir) is
20
entirt^
problem
is
minimized.
known, the minimization of equation
(3)
reduces to a diffusion control problem with a
(via Wo). Let
V (»')
drift control (via
(see
Mandl
1968),
<
^("'1
y^Cj(a-'' (r, w). r,
w)
unbounded, we assume that standard
is
and express the associated Hamiltou-Jacobi-Bellman
optimality equations, after using equation
min
)
be the potential (relative value) function and g be the gain (optimal
long run average cost). Although the drift of \V
arguments apply
t{w) and a singular control
(2), as
g
-\
~
c\''[u')
+
—
V'"(fr)
(
=
for tr
<
Wq
U=l
(23)
and
V"(('')
MRW,
.\s in
we numerically
=
for u'
(24)
u'o.
solve the diffusion control problem by approximating
the diffusion process by a Marko\' chain.
This Markov chain approximation algorithm
was developed by I\ushner (1977), and we
for details of the algorithm.
>
.MRW
refer readers to I\ushner
contains a
full
description of
its
and Dupuis (1992)
application to solving
the corresponding optimality equations for the SELSP. Because the optimality equations
in
MRW
and
refer readers to
When
and the
we omit a
are very similar to equations (23)-(24),
.Appendix
description of the algorithm,
1.
there are no setu]) times, however, the diffusion process
diffusion control
cycle center
.r"
must
still
problem
is
easy to solve.
be optimized
for
W
becomes a
.-Mthough the cycle length r and
each total workload
level,
they can be done
individually without the need to refer to or update the potential function
the steady state distribution of the total workload process
times are zero, the optimal idling threshold
k'q
=
argmm,^,,
r
3.4.
and
a""'
are the parameters for the
Proposed
Policy.
The
W
is
—
o-
f^BM
e
^^
'
for interpretation,
Since
exponential when setup
'aw,
U'.
solution outlined in §3.3 needs to be unsealed to
be implemented. .-Mthough the translation to an unnormalized
room
V''(a').
given by
c(a')
/
Jw'
where
is
REM
we propose an
intuitive policy.
polic\'
The heavy
has considerable
trafiic analysis
hnds
a
minimum
work experienced by an
every total workload
for
it.
lexel of
we propose
that the
minimum amount
of
level.
product over the course of a cycle
iiulividual
Because work
depleted while the machine
is
is
machine continue production on the current product
work
serving
until this
reached, and then begin setting up for the next product
is
the cycle. In addition, the machine idles
when the
workload
total
in
level has fallen to the
idling threshold.
We define the
0,{l),
which
which
is
I
number
number
.V.
\
identity O,
known
the
is
the
= iW +
policy in terms of
-
I,
=
to hold for a
of orders of product
For ease of exposition,
/;,ir,,
where 0,{t)
in
we
let
= 0,(nli/^
policy
Switch out of product
scheduling policy
.V
< A/',
/
queue
in
for
/
=
=
for
and
/,(/)
in
heavy
terms of
/
l....,N and
=
1, ...
=
[,{nt)/s/^
traffic, is
6',(/)
in practice:
/,(/),
goods inventory
in finished
/
7,(0
wide variety of queueing systems
traffic solution to a
o,{t)-i,{t)
/
of completed units of product
the heavy
is:
two quantities that are naturally observed
and
,
/V"-".
The
,
for
linear
which
is
used to translate
/,(/).
The proposed
when
..-w/>.,^
,-..^^
^.^
.e-,i:;li//;^(Q.(0-/.(0)
E; = ^- (O.(O-MO)
,
v/n
(26)
and
idle the
machine when
X:/'r'(^.(o- A(0) = v^u'o^
c-^")
1=1
where
.rf (tr).
in §3.3.
t'(w) and
((\"
are determined from the optimization procedure outlined
Because a computational i)rocedure
of the heavy traffic parameter
/;,
and we
is
let
»
being employed, we must choose a value
=
(1
—
/>)"'.
.A.s
in
MRW,
exploratory
computations (not shown here) reveal that the performance of the proposed policy
is
very insensitive to our choice of n.
.Although we do not do so here,
point
in
more complicated
time over the cycle, our analysis yields
policies can
tlie lluid
workload
be created.
level of
.At
any
each [jroduct.
This type of information can be used to determine when random events throw the cycle
0>
manner
off course in a
cyclic
not predicted by the
approach can thus provide "red
specif\-
when
warnings
flags" or
to skip jModucts in the cycle or
We
an unusually high number of orders.
when
fluid scaling.
i)olicy
to
The dynamic
unusual surges
for
A more complicated
or slow-downs in machine processing.
would
HTAP's 0(s/n)
in
demand
using this information
jump
to a product that has
leave the specification of such [)olicies for future
investigation.
DETERMINISTIC DUE-DATE LEAD TIMES
4.
In this section
we assume
that due-date lead times for each product are deterministic,
thereby allowing us to write a closed form expression for the cost per cycle
In a(^hlition,
in §1.1.
optimal cycle center
is
found
terms of
in
to
some
The
and
.r"^'
\
method
are able to state a quick
'(»•)
and the
still
jn'oposed solution
is
total
equation (2)
§4.2 for determining the
in
provitle a formulaic solution.
problem must
diffusion control
algorithm.
we
in
With
this,
cycle length r
However, the solution to the
workload w.
be computed by the Markov chain ai)proximation
stated in §4.3. .Subsections 4.4 and 4.5 are devoted
structural properties of the optimal solution and the value of due-date lead times,
respectively.
4.1
/,,
Cost per Cycle.
which
is
/,
=
Let each product
For a given cycle center
or 1,(0)
=
products
.r*
/,
is
.
'
\
The cumulative
<
if-s
1
—
is
^
cycle length r and total workload level w. the cost per cycle
is
given by (19). with
and D[^(s.t) given
.r'/p,.
distribution function
/,
/,
in
place of
cost per cycle for a customized prodvict,
of Dt{.s.l). L,{t)
slack
.r;
standardized product
To Hnd the
have a deterministic due-date lead time of
f,/\/n under the fluid scaling.
'^
for a
/
.r"^
and
r.
By
we must determine the behavior
Proposition
1.
Therefor(>, Z,(0) has a range of [— dc.
greater than or equal to zero. This
must be no larger than
/,
makes
because no orders can
23
/,.
\\v
/,],
have
=
//'(o)
Z-*'^''*-
since for customized
intuitive sense:
arrive-
.r'
The mininuim
with larger due-date lead
times.
From Propositions
1
ami 2 respectively, we have
D.U.O) =
«(/-''/"/)
if.'
{''„
(.29)
otherwise
1^
and
if
f
=
D'^'i-^.t)
I
if
P,
if
[
We
can solve
Because
—
(I
bounded above by
the
initial
We
r(l
—
-
Pi))/p,
Xote that
/',.
inventory
L- -rl/p,
=
—
p,){t
/
by using Proposition
for L,{t)
^.(0
< /, - .r;V^>, / - -rj/p, -t<s<f,
/ < ^
.^
(1
—
-
(1
P.}T
—
For
3:
+
p,)T
G
/
-
(1
-
p.){t
—
/9,).r),
r(l
- /0)M.
(31)
than or equal to zero. L,(t)
less
is
(7"(1
(30)
.
is
depends on the the decision variable
this function
x']
also
via
.r^.
can now substitute (31
into equation (11) to calculate the average cost per cycle.
)
Since no orders arrive with due-date lead times L,{t)
than
less
/,.
equation (11) can be
rewritten as
c,(.r^r.».)
This structure
cycle
is
is
= -
r
(b.l;(t)
SELSP
similar to that in the
of the cycle
(i.e.. if />,/,
-
,r^"
>
rp,(l
If
—
!,(/)
is
h,l + (f))dl.
case discussed in
broken down into three cases depending on
both during the course of the cycle.
+
if
(32)
MRW'. The
cost per
orders are only tardy, only early or
always greater than zero over the course
pt)/2) then orders are always filled early and so
equation (32) implies
c,(.r^r, w)
=
-h,
which simplifies to
MRW: The
f
[./
c,(.r^\-, lu)
cost per cycle
is
- xUp, -
=
h,p,{f,
(
—
1
-
/>,
+
)r
(
1
This
.i"'Jp,).
ecpial to that in the
-
is
SELSP
p,){t
-
"(1
-
p,))/p,] dt
.
(33)
very similar to the results from
with axes shifted by
p,f,.
The
similarity can be viewed as an application of Little's law. In our notation, the utilization
p,
corresponds to the arrival rate of work and
24
/,
-
-i^l/p, is
the average waiting time, and
their product
is
Similarly,
if
cost
the nuniher
is
tardy then L,{t)
t)rders are alvvays
is c,(.v'-'.T,w)
then £,(/)
queue.
in
=
b,pt[(.r'i/pt)
—
If
/,].
is
less
than
for all
G
/
and the
[0,r],
orders are both early and tardy over the cycle
both positive and negative over the cycle and the average cost per cycle
c,(j-^r,
«•)
=
b,p
is
^-i(-/^-?) + i-(l-^^'
341
^''
'^'
^
I
+lhP,
2t(1-p,)
r
if
2^-l'
Again, this reduces to a Little's law version of the
For ease of reference,
we say
tiiat
e [pj,
-
product
.(;
±
i
r/j,(l
summary, the average
•)
+
SELSP
results.
the parameters are such that
if
condition
is
in
-
p,)/2]
and
Similarly,
I.
condition 3
in
we
5^(1-/3,;
/>,/,
-
customized products
cost per cycle for
x'^)
c,{x'-\t.w]
>\
I
+ h,
I
^
result:
/
Tp,(l
in
condition 2
for
condition
Equations (19) and
that for customized products the cycle center
.rj"
cannot be
4.2.
us to
Optimization. The
make
if
In
for
condition 2
1
35)
(3-"))
condition 3
imply that
less
than
for the de-
and standardized
/j,(
is
I
the restriction
—
p,)r/2. Due-
dates have transformed customized products into (juasi-standardized ones.
and
p,)/2.
,.c\2
f
ones are exactly the same. The only difference between the two types
in ^j5.7
p,)/'2,
is
terministic due-date case the cost structure for customized products
the interpretation of this resvdt
—
>
< -Tp,{[ -
.v";
for
This leads to a remarkable
x'^^
product
call
if /;>,/,
—
We
discuss
§5.9.
explicit derivation of average cost per cycle in §4.1 allows
further progress in optimizing this system. First,
we derive the optimal
cycle
center and cycle length, and then construct an algorithm for use in the dilfusion control
problem.
Cycle Center. By our
analysis
in §4.1.
the only ilifference between the optimization
of the cycle center in (20)-("22)
and
MRW
in
is
the inchision of the equation (22) con-
customized products.
straints representing the non-negativity restrictions on
MRW,
In
the solution to the program without inequalities (22) was found exactly. For later refer-
we
ence,
by
the "unconstrained" version of (20)-(22) and denote
call (20)-(21)
.As discussetl in
.r"'*".
MRW,
the form of the cycle center
three regions depending on whether the objective function
cheapest product at the point
it
used
is
in
of
and
.r^^'
We
broken down into
linear or quadratic in the
because
.r"
r.
solving (20)-(22)
the boundary conditions in equa-
is
exploit the structure of the objective function to determine the structure
to find
which
objective function
minimum
in
is
wish to find a similar expression for
determining the optimal cycle length
The major complication
tion (22).
We
.r*-'".
is
.r*^*"
solution
its
exists
it
is
piecewise quadratic with linear edges.
will
be a global minimum.
.As
convex and so
It is
,V
A.;0
=
c{r^,r.
.r)
+
A
(
^
w
is
(V^
\
,<
a local
if
per Bertsekas (199-5), the Lagrangian
associated with (20)-(22) with fixed cycle length r and total workload level
I(.r'-.
The
are binding with respect to the inequality constraints.
.r^~*"s
- u^ +
E/',(^^^\^
"
-rj)
(36)
•
:1
The Karesh-Kuhn-Tucker necessary
exist
Lagrangian multipliers A' and
where 0*
pj)l'l.
is
We
for j
=
•V''
I
there
=
0.
(37)
^i'
>
0.
(38)
^-
=
the set of non-binding cycle centers;
Vj€0',
i.e..
we
(39)
products j such that
suppress the dependence of 0" on r and
set of
.r"
such that
lv
for
x'j
binding products
categorize the binding products
in
The products with binding
>
Tpj{l
increased readability,
l)y
their condition.
0'' be the set of products with binding cycle centers and in condition
be the
3).
//'
minimum
V,ci(.r'--.A'./r)
additional ease of reference,
let
conditions state that for local
J
—
for
We
and 0"'
condition 2 (no binding product can be in condition
cycle centers are pushed to their limit,
26
in
the sense
more work can be performed on these products.
that no
+
^c,(.r''*, r, (r)
iV"
+
Thus,
.'V.
1
—
A
=
//,
for
for
=
i
e 0' and
/
A'",
I
+
and -^c,{x''\t,w)
l,...,yV'-"
=
i
Eciuation (37) imj^hes that
it
=
A
follows that T^c,(.r'',
r,
for
=
/
+A =
w)
0.
t
same
Therefore, the Karesh-Kuhn-Tucker conditions are the
the non-binding cycle
for
centers as for those in problem (20)-(22). This fact implies that the cycle center optimality
arguments
all
.MRW' hold
in
(minimum
but the cheapest earliness
condition
2.
non-binding 0' products.
for the
The condition
I,
where the cheapest product
is
product
is
condition
in
or tardiness
of the cheapest product
workload into 3 regions: Region
II.
/?,)
There
(minimum
condition
condition
in
is
and region
2;
3.
product.
For ease of notation,
let
0'
equal $1
let
if
is
= w—
J2jti.(-)'
'''Pji^
(
V.
=
—
l>j)l~
^pAi-r.)
n f
Pifi
then the non-binding product
w >
-
TO,
•
-;,
h,-h,
\
-
(E,e0- P,L
|(E,ee- P'L
rp,{i-l',)
Pif>
h,
\b,-k
(E,ee-
P>J>
and the cheapest product cycle center
=
(....(6,
the vector o,
-
is
-
(.
.
.
.0,^.
.
.
.)^ is
d^)
'^')
.r^*.
-//,)/2-(6,. ~hg.)/2
=
1.
\
I^;^,
/,
MRW.
and
6l
we
let
If
0'] cycle center
is
»')f>i
•
7-2
-
<l <
E,e{e-\^-}
(40)
^^""2"''"'
-bf>
+ h,
if
Oi
-
<
if
condition
1
/,
,
G {0"
/
in
0^ be the cheapest
otherwise; the inde.x 0' corresi)onds to the "X'^ product" referred to in
ir
region
1:
Here we use the same region terminology to denote when
S.
Let 0^ be the cheapest tardiness cost product in 0' and
earliness cost
in
where the cheapest
III.
the cycle center and cycle length are such that the cheapest product
2 or
product are
h,)
used to categorize the total feasible
is
where the cheapest product
in
was shown that
it
-
= w —
)^, 72
Ylie&'
<
-i'",
E,e{(-)-\(9-}
-''i
= (....(V
'
Tpg.{l-pg.
where, for
i
G {0'
\ ^"}-
+he')/{pe-{\ -pe')]...Y.
defined by
p.(i-p,) P)(i-pj)
6,+/t,
bj+k,
E '€«"
Pti^-Pi)
o,
for
b,
+ ki
;
^
J.
ana
(41-
/I
=
a„
+
^'
and
finally
=
have xf
It
Pi
is
is
where
rp,(l
.r';
-
=
/>,/,
—
ra,
\
•
(4-
L/G0- -fT^iir
"'
—
71
-
Z-'^/,
«')o,
•
Fo'"
-;2-
'
G {0"' U 0"^}, we
p,)/'2.
zero, the inclusion or absence of
/
in
;
is
binding yet
0' does not
Lagrangian multiplier
its
affect the cycle centers of the other
This can be seen by setting the borderline product's cycle center as derived
from equation (40) equal to
r/;,(l
—
p,)/2 and by subsequent algebraic manipulations. In
addition, on the border between regions
II
and
I
or
III
the cycle center does not change
using any of the two corresponding e.xpressions in equation (40).
conclude that the cycle center
\\ ith
in
(X]^ge-
important to note that when a product
products.
if
Pi(l-Pi)
V^
is
a continuous function of the cycle length
these expressions for the cycle center,
we can
terms of the cycle length r and the total workload
level a\
For
/
G 0"
pji^-p,)
we
define a f3*7l
r.
\ ^".
it
is
I
II
(43)
((•)
//.
If
can therefore
restate the average cost per c\cle
'(?•
c,(t-,
We
=
_u
/,
\LPAlzJhl
^P'Ji-p,)
I
-T..ie{(^'\<}-}<^'li
^'i<^l
\
h,-h,
'-^O'l^
28
=
•
I
1
h,-h,
\
I
-'L,e{(r)'\d-}<^''n2.
then the cost per
cycle for the cheapest product
r
is
+ rE,^(-).^=^V^
/,,.(„,,.
p,(l-p,)
-fid'
C0.{t.
E,ee-\0'
'
^+
h,-h
6.+/,
/'^
w)
(44)
+
+
{rao-ji
:^
(a'0.
+7-2^
jQe'ri]
III
The products with binding
for
cycle centers can be in condition
G {©'' U 0""} the cost per cycle
/
Ti;(l-P,)'
(
for the
/\'
/
for
,
€ 0"'
+
.s\
'(u').
e 0'-
we can
optimal cycle length r by differentiating equation (23) with
The optimal
cycle length
of basic system parameters, the set of binding products
=
for
of the cost per cycle in terms of r and u\
respect to t and solving against zero.
.S'
Thus
(45)
an expression
cycle
2.
r, «•
Cycle Length. Given the form
find
or in condition
is
ll:(p:.ft
c,
I
Thus, 0", 0"' and 0'" are
all
and the
is
expressed
in
terms
effective setup cost per
functions of
5 and
w.
If
we define
the constants
^--
(/),
+
/?,)/>,(!-/->,)
i->,(^
"^1
2{b,
ie&'\6'
-
p,) f
+
h,
b,
-
h,
(46)
(h0'-h,)+ 2^
+ Z^
(he.+b,).
iee'2
{pJ.)Hb,
+
l>.
(17)
S2
+
S3
8
x
—
r(«r;i)
2/>,(l -/-*,)
29
+
— "m
;^
(48)
~
^''
"'
/
,
„
0'
a
"
i€0*'
+
6,
/;
y-
ef
(6,
-Pj)
.^
r(«t7j
+
Pj(l
ie0*2
,
2^ iT^l
Y^
-
h,)p,{l
and
,
p,)
8
!e0*\e*
(49)
p,(l
-
2(6,
+
p,) /
6,
-
/;,
^
(50)
/;,
j60'
lee*
then the optimal cycle length can be stated as
s+a
I
+^f(E,p,,.^'/--"'^+<>
ol
II
III
c"
In order to use the
to be able to find r'
fintl
Markov chain approximation algorithm discussed
and
.!•"*
w and
given
for a
for varying V'(>r).
Thus,
the set 0* of non-binding products as a function of the setup penalty
and workload w.
In the
.\ppendix
we construct an algorithm
2.
in §3.3.
it
5'
is
we need
necessary to
= A
+-sV
'(ir)
that generates the set of
binding customized products as a function of effective setup cost for each total workload
level.
This algorithm allows the cycle length, cycle center and cycle cost to be calculated.
This cost can then be fed into the .Markov chain api>ro\imation algorithm so that
and hence the proposed
4.3.
Proposed
policy,
Policy.
0,{t)
—
in §4.1
/,(/)
machine based on
~
ft,\Vi{f)-
III
'(ic).
can be compuletl.
The mapping from
problem to the proposed policy
rule for the
\
the
is
\\\.
same
the solution of the diffusion control
as in §3.4.
which implements a switching
the current workload i)resent
in
individual orders, via
the presence of deterministic due-dates, however, the analysis
allows us to create an alternative switching rule based on the mininnim slack
of each product, via \\\{t)
^
p,
/',
(
—
L,{t)).
30
I'his alternative^
approach might be easier
to
implement
in
cases where due-date data are
more
order queues and inventories. Markovvitz (lODti) uses
readily available than the levels of
Monte Carlo simulation
to
compare
the two alternatives, and finds that the approach given in §3.1 clearly outperforms the
minimum
slack
method, although the discrepancy between the two
Markowitz
as the length of the due-date lead time grows; see
Structural Properties. Unfortunately,
4.4.
V''((r)
is
appears possible, even
0'
if
is
known
for
w —>
assume that
as
that for
(Ci.iCi
all
some
>
«•'.
there are only a finite
we have
in
0'((/^i)
=
w
five .sef
(this
MRW.
it
about the derivative of the potential
in
i.e.,
the .\ppendi.\
respect to
of possibilities for 0'.
We
there exists a
»''
such
.S
2.
which demonstrates
and w. and the
fact that
These structural properties are
Please see Markowitz for details of their proofs.
(n-)
=
—
u'
If region II
+
o[ic).
containing the standardized products and the cheapest hackorder
product could he customized or standardized) and
9
--
If region III
However,
-^ oc, then
\
product
levels.
Let the average setup time per cycle s he greater than zero.
conditions hold as
is
workload
G'iwo). This assumption follows naturally
observations 1-S
number
similar to the ones derived in
where 0'
no closed form solution to
qualitative insight about the proposed policy.
how the products smoothly become binding with
1.
0,
for details.
co the set of binding variables 0' stabilizes;
from the t>ehavior evident
Property
>
for all total
|)ossible to deri\e several structural properties
function ^'((r), which yields
.s
policies disappears
conditions liold as
U'
—>
V
2^jee'
;o3)
Pji^-Pj)
h;+h;
3c. then
V'(»')
= -
—w +
c
31
o{w).
(54)
Property
the setup cost per cycle
If
2.
then the policy at the idling threshold
/),
=
hj for all
///;, 7^
I) J
T-{Wo)
=
for
i
and
some
i
If this
J.
and
I\
=
ami
icq satisfies
all
of the products are staudardized.
the region
II
and only
ci)nditions if
il
condition holds then
j then the idling threshold satisfies the region
I
conditions and
0.
Property
3.
the limit as
tc
average setup time s
If
—
>•
oo
if
and only
if
greater than zero, region
II
conditions hold in
the tardiness costs of all the standardized products
and the cheapest tardiness
are ecpial
is
cost
among
the customized products
is
or greater than the standardized tardiness cost (or if the tardiness cost of
equal to
all
of the
customized products are equal and there are no standardized products).
Like their counterparts
in
MRW.
ture of the optimal cycle length.
customized i>roduct and of
tion
(.')1)
load w.
antl
If
Property
1
all
By
these three properties provide insights into the na-
Proposition
.'5.
if
the standardized products are ec[ual, then from ec[ua-
the oi^timal cycle length r" grows linearly for large total work-
the tardiness costs are not ec[ual then from Property
grows as the scpiare root of total workload. Property
near the idling threshold
Beyond
MRW,
these three properties
2,
all
describes the optimal cycle length
w —>
cc. the
the set of non-
optimal cycle length
idling threshold,
If
region
II
if
/\'
—
and region
conditions hold or
I\
>
0.
I
conditions
then 0"
The Value
of
Due-date Lead Times. Based on
is
a
we can comment on how due-date
the analysis of the oi)tinial
lead times impact the long run average cost.
For systems with standardized goods, due-dates do not influence the optimal costs.
a
is
difficult to describe.
4.5.
policy,
(51). r'
but potentially one customized product to become
around the
hold, then no products are binding.
prion
and equation
make statements about 0'.
In the limiting case of
growing and eventually causes
By Property
"2
1
(t'o.
binding product classes.
binding.
the tardiness costs of the cheapest
quick inspection of the cost
|>er
cycle in (Mpiation
32
(1!)).
we
By
see that modifications in
due-date lead times are
is
by a translation of the cycle center. Thus, the
offset
not alteretl by changes
the due-date lead times:
in
we describe the
oi)tinial cost
intuition behind
this result in §5.6.
The
situation For customized products
more complex.
is
centers influences the costs associated with binding products.
lengthen, the feasible regions for the cycle center
.r'
The
translation of cycle
.\s
due-date lead times
and cycle length r grow, thereby
allowing for a lower long run average cost. However, for the zero setup time case,
show that when due-date lead times become
and hence the long run average cost
by solving the control problem
this
(i.e.,
in
region
II
/,
the original problem
=
for the s
case, both ignoring binding constraints
O
V}
r{l
.V})
{1
u-^
of this no due-date
problem and
;i
i
+
P.{^
-
(56)
»o«.72.
be binding. Since the derivative of
will ever
non-binding
Similarly,
if
in
w
(Cq.
is
positive for the
at the smallest
.s
=
case,
.r^~
—
this.
region
I
policy,
)/'2
that
which
Therefore, for due-date lead
conditions hold at the idling threshold w^ and
'>
\ ^J'
/>,
for
the same
the standardized case.
P,(l
>
—
we need only check
workload experienced by the
Equation (56) guarantees
r/j,( 1
if
the due-date
leadtimes satisfv
l>J.
if
l>,)
times satisfying (56). the optimal long run average cost remains constant
reason as stated
and
S3
with respect to total workload
the idling threshold
=
Let Wq be the optimal idling threshold.
e(|ual to zero.
>
then none of the products
is
do
of the due-date lead times satisfy
all
\
a product class
We
independent of the due-date lead times.
conditions hold at the idling threshold
/'-/-
is
sufficiently large, no products are binding
treating each customized good as a sla idardized one so that 0"
setting the due-date lead times
If
is
we can
V
^^.
-
+
/>.)
'b,
-
h
+
/Mi
{1
^'
33
V}
+
/><(!
-
P:)^
for
/
^
()'
and
p0-fe'
-
p e-i^
>
pe-)
y-
{1
\f!
then no product
same reason
i^S'
will
-
Pi(l
V
6,
+
k -
pi
/^
+
h.
(58)
V
1
h,
be bintling (one only needs to check at the idling threshold for the
as in the region
II
case).
For positive setup times, Property 3 states that as long as there
is
product which does not have the cheapest tardiness cost, that product
be binding
for high total
workload
levels.
The
is
high for the workload levels
\.l;ere
We
date lead times increase, eventually the reduction
in
all
we
binding, the density
long run average cost asymptotically
Longer due-date lead times achieve diminishing returns
In this section,
is
can infer from this that as the due-
of the products are treated as
5.
eventually
Although the average
the product class
associated with these regions diminishes rapidly.
approaches the case where
will
longer the due-date lead times the larger
the total workload level before the product becomes binding.
cycle cost
a customized
if
they were standardized.
for large total
workloads.
DISCUSSION
step through each region of the
Venn diagram and
discuss the
relevant literature along with the insights derived from our analysis of the deterministic
due-date lead time case:
same way
for ease of reference, the subsections
as the regions in Figure
versions of these problems,
static stochastic scheduling.
we do
1.
not
Because our focus
is
below are numbered
in
the
on the dynamic stochastic
include the vast literature on deterministic or
For each subproblem.
we compute and
display a typical
set of switching curves
on the workload plane that characterize the proposed policy
the two-product case.
The
cost
parameters satisfy
//[
=
'llii
and
bx
=
in
262 in each of
these examples. .Mthougli our computations are restricted to the deterministic due-date
lead time case,
Markowitz undertakes
a partial
examination of the cost per cycle
for a
customized product with uniform due-date lead times, and the numerical results suggest
that the f|ualitati\e observations describe<l in this section carry over to the
date lead time case.
34
random due-
U',
switcli
from 2 to
switch from
1
to 2
1
idle
Wi
Figure
Customized products, no setups, no due-dates.
Customized Products, No Setups, No Due-dates. The
5.1.
Figure
3:
corresponds to systems
1
vvitli
outer area of
only customized goods, no setup penalties and no
Ihese traditional multiclass queues are the simplest of the systems displayed
due-dates.
in
the diagram and liaxe l)een extensively studied.
to
show the optimality
tardiness cost rate
cost from the
b).
system
of the
"T//
Cox and Smith (1961) were the
rule" (in our notation, the
r//
first
index corresponds to the
which gives priority to the product that, while serviced, removes
at
the highest rate;
.see
Bertsimas and .\iho-.\Iora (1996)
for
an
up-to-date set of references for this scheduling problem.
Our proposed
mined from
to be zero.
region
is
III
policy
The lack
§4.2.
old
is
u'o is
for a
ol
conditions can apply.
set to
zero.
a-.
/
policy parameters can be easily deterto vanish
and the cycle length r
for all jjroducts.
implying that only the
setups forces region
Thus
In this special case,
The implied dynamic
6i
/,
for
not ecpial to O^.
two-product example wliere
to exhaustion
The
similar in spirit.
.\o diu'-dates indicates that
set to zero for all
product
is
=
any total workload
and the cycle center
one can
h^):
u-
>
for
is
I
his policy
the cheapest backorder
Service
then simple
all l)ut
in
form (see Figure 3
the least expensive product
and switch out of producing the cheapest product
cost products present.
the cycle center .rf
0.
trivially calculate that the idling thresh-
cyclic policy
>
II
if
tliere are
can be interpreted as a two-level priority
rule: .\11 but
the least expensive product have high priority and are served to exhaustion
manner; the
least
expensive ])roduct has low priority.
3.0
any higher
in
a cyclic
worth noting that
It is
in
the heavy
traffic limit tlie (|nrue
products vanish and oidy the lowest priority product
length of the high priority
present (see W'hitt 1971).
is
follows that the heavy trafHc limiting behavior of the proposed policy
of a strict "b" |)riority policy: In both cases, the workload process
the heavy
Thus,
in
policy
and the
c/-t
traffic setting
nothing
lost
is
is
lies
It
identical to that
along the 0^ axis.
between our proposed dynamic
cyclic
rule.
5.2 Standardized Products,
No
Setups,
No
Due-dates. Multiclass
systems with standardized products are generally considered to be more
cjueueing
difficult to an-
alyze than systems with customized products because of the nonlinear cost structure
introduced by having both holding and backorder costs. .*\dditionally. when constructing
policies for these production/inventory systems, there are
aries as with
exhausting a cjueue
in
no natural switching bound-
customized systems. These obstacles have hindered
the analysis of even the simplest case:
Standardized products with no'setups and no
due-dates. Zheng and Zipkin (1990) analyze the intuitively appealing longest queue (or
smallest finished goods inventory) policy for a two-product system, which
is
optimal when
both products have identical cost parameters and operate under independent base stock
policies.
Ha
(1993) partially characterizes (in terms of switching curves) the optimal
Wein (1992) uses heavy
policy for the Markovian two-product case.
traffic
theory to de-
velop a dynamic priority policy and an aggregate base stock policy for the multiproduct
problem, and \eatch and Wein (1993) expand upon this by examining index policies
Markovian setting and
analyzing the
l)y
Finally.
lost sales case.
propose a policy that combines the best aspects of the policies
in a
Pena and Zipkin (1993)
in
Zheng and Zipkin, Ha,
and Wein.
Our proposed
more in-depth
this system.
is
policy
is
similar to that of Wein. and readers are referred there for a
discussion of the basic insights gained Irom the heavy traffic analysis of
Fri^m the calculations involving setup penalties and cycle length
again zero for this case and only regions
but the cheapest product
w <
total
0.
is
set to zero
and
1
and HI are
x^. equals
possible.
ir
for
ir
>
The
in §4.2. r
cycle center of
and
.rj,.
equals
w
all
for
Because there are no setup times, the total workload process, which measures the
work
in
backorders minus the total work
36
in linisheil
goods inventorv.
is
a
RHM
and
w,
switch from 2 to
1
\v
switch from
1
to 2
idle
Figure
the
iflling
threshold
the total work
of Wein).
icq is
in finished
The dynamic
unfilled orders with the
inventory.
Staiidardizecl products, no setups, no due-dates.
4:
It lias
available: 2)
all
negative
—Wq
(i.e..
represents an aggregate base stock level for
goods inventory) and has an explicit form (see eciuation (76)
system
cyclic policy can be stated as a two-level priority
added complication of a mechanism
the following rules:
1)
service
all
to build
for
up a finished goods
orders with finished goods inventory
if
cpieued orders for products other than product 01 have priority and are
serviced in an exhaustive cyclic maimer: 3) orders for the cheapest backorder product 01
have the lowest
priority: 4)
idling threshold, then
no orders are present and the total workload
if
produce product
typical two-product policy (where
6)
0^.
>
which
62, hi
is
>
is
above the
the cheapest to hold in inventory.
.\
pictured in Figure 4 and. as
in
hi)
is
the customized case, the strict priority rule between the products causes the switching
curves to nearly overlap on product 2's axis.
in
the heavy
traffic limit, identical to
Comparing
The performance
of the resulting policy
the one suggested by Wein.
the customized product and standardized i)roduct cases,
proposed policy a distinct role
for inventory:
It
in
we
see from the
hedges against the risk of backordering.
.\ccording to the IILAP. uncertainty in production and d(>maiid have
pact on the total workload
is.
tlieir
greatest im-
the system, and the time scale decomi)osition allows the
3<
machine
to flexiljly address requests for high cost pro(hicts without the need for storing
the jjrodurts themselves. Thus, inventory acts as a reservoir of reserve capacity, able to
absorb random tluctuations
capacity
in tlie
cost product.
policy
and demand, and
service
most economical manner
Similarly,
when
possible:
It
is
tlu" |)ro|)osed
placed
the finished goods inventory
is
in
policy stores this
the cheapest holding
exhausted, the proposed
able to flexibly service requests for high cost products by neglecting the
still
is
in
cheapest backorder product, effectively storing deficit inventory
in its
cheapest form.
Customized Products, Setups, No Due-dates. The models
5.3.
by region
the
in
.'5
Venn diagram
known
are
computer communication networks.
systems
as polling
in
represented
the literature on
Although considerable research has appeared on
the performance analysis of these systems
(e.g.,
l3oxma and Takagi, 1992), the dynamic
scheduling of these nudticlass (pieues with setups have not yielded to an exact analysis.
Hofri
and Koss (1987),
results,
Xain and Towsley (1992) and Koole (1994) derive structural
Liu.
Reiman and Wein
(199-!)
use heavy traffic analysis to develop policies for the
two-product problem. Boxma, Levy and Westrate (1994) and Bertsimas and
compute
static policies (i.e., polling tables),
dynamic index policy
Browne and
Xu
(1993)
Vechiali (1989) derive a quasi-
to choose sequences of products to service at the start of each cycle,
and Duenyas and Van Oyen (1995, 1996) develop scheduling heuristics based on myopic
reward rates
for
systems with
the literature review of
Our proposed
.*\s
in
setu]) times
Reiman and Wein
policy
is
and
costs, respectively: readers are referred to
(1994) for
more
a multiproduct version of
Reiman and Wein
equals zero for
for all
in
it
(1994) policy.
s
the previous customized case, the lack of due-dates and of standard goods limits
the cycle center and cycle length equations to the region
/,
details.
.^'
and
a cyclic
all
all
w.
manner.
\\
— T{w)pe'{ — P9'
1
but the cheapest product,
hen
set
)/-.
set
up
up
for Ol
where
for all
is
ir is
binding and so 0"
C
{01}
Service each product
it
until its
normalized workload reaches
the current normalized work
is
Ciiven that
but the cheapest backorder product, service
work on
there are only two products, the pokuy
The presence
B^.
formulations.
Thus, the policy can be stated as follows:
When
to exhaustion.
.r'^.(«')
all
/,
III
identical to
the system. \\ hen
Reiman and Wein's
of setui)s eliminat(^s the two-lexel priority
38
in
scheme seen
(1994).
in
the previous
\v,
switch from 2 to
switch from
1
1
to 2
idle
Figure
no setup
o:
Customized products, setups, uo due-dates.
The proposed
cases, because a strict priority rule leads to excessive setups.
policy avoids this by keeping to the cycle, yet minimizes cost by offering ciuick delivery
to high cost
products
at
the possible neglect of the 9l product.
previous two subproblems. the heavy
priority product in a similar fashion,
systems,
it is
exhaustive
more
beneficial to focus
in
the
traffic analysis essentially treats all
but the lowest
riiese results suggest that for heavily
loaded polling
on noncyclic exhaustive policies than dynamic non-
])olicies.
typical two-product j^olicy
shown
Figure
5.
has added breadth to the switching curves of Figure
3.
.A
However, as
is
in
The presence
The
only balancing setups and ([ueueing costs, as directly seen
in
of setup penalties
policy can be viewed as not
the formulation of r and j^.
but also as controlling the randomness of the system by isolating the effects of total work
fluctuation in the least cost product. This
is
seen
in
Figure 5 by the huge range of
low-
cost product cjueue length values (along the vertical axis) caused by the fluctuation of
total
workload
in
the system versus the relatively confined range of the
more expensive
product (along the horizontal axis).
5.4.
Standardized Products, Setups,
No
Due-dates.
Standardized goods
with setups and without due-dates has long been considered the prototype
for
modeling
make-to-stock maiuifact uring systems. The deterministic version of this problem, which
is
called the
Economic
1915 (see Elmaghraby
Lot Scheduling
1!)7S).
Problem (ELSP). was
originally fornuilated in
and only recently has the stochastic version of
39
this
problem
- -
switch from
'2
switch from
1
to
1
to 2
Kile
Figure
(the
SELSP)
6:
Stanclarchzed prochicts, setups, no due-dates.
received attention.
decision model,
Graves (1980) develops a heuristic using a Markov
Leachman and Gascon
(19SS), Gallego (1990) and Bourland and
Yano
(1995) develop heuristic lot-sizing algorithms that are rooted in the deterministic EL.SP,
Sharifnia.
Caramanis and Gershwin (1991) use
anal\ze a stochastic
a hierarchical
tluid \ersion of the i)roblem.
decomposition approach to
Federgruen and Katalan (1995b. 1996)
use polling theory to analyze the performance of (periodic and cyclic, respectively) base
stock policies for the
SELSP. Qiu and Loulou (1995) numerically compute the optimal
solution to the two-product problem by modeling
.\nupindi and Tayur
base stock policies,
(
it
as a
semi-Markov decision process,
1994) develop a simulation based approach to
MRW analyze the problem using the
compute cost-effective
HTAP. and Sox and Muckstadt
(1995) formulate the problem as a stochastic program, and propose a decomposition
procedure to solve
Our proposed
that of
MRW.
it.
policy restricted to standardized goods and no due?-dates reduces to
Since
all
of the products are standardized, there are no orthant constraints
and hence no binding [)roduc1s;
i.e..
0" =
{1
\}. .Ml three regions are possible and
the cycle length, cycle center and idling threshold are nontrivial.
40
.\
sample policy
is
pictured
A
Figure
in
(i.
detailed description of the proposed [jolicy for the
key insights are given
larger problem.
As
in
in
MRW.
Here we
SELSP and
a
summary
of tlie
briefly cover the liighHglits as they pertain to the
the standardized goods, no setup case, the policy treats the total
workload inventory as a reservoir of stored capacity, used to hedge against demand and
The switching curves
service rate uncertainty.
is
stored in the cheapest holding cost product
the cheapest backorder product 0^.
are constructed so that e.xcess inventory
Oj^.
and
inventory
deficit
is
moved
into
Positive inventory also serves a secondary role:
A
small cache of inventory impedes backordering on an individual product level while the
machine
is
producing other products during a cycle. The amount of inventory required
dependent on the length of the cycle, which
as
shown
in
as seen in
.Also,
comparing Figures
4
an increasing function of the setup penalty
setups introduce breadth between the switching curves,
and
6.
Customized Products, No Setups, Due-dates. The systems
5.5
Figure
equation (51).
is
1
reflect
manufacturing
facilities that service
some future time. The
in region 5 of
customer requests on a make-to-order
basis with the additional feature that customers do not
at
is
want the goods immediately but
inclusion of due-dates causes an explosion in the dimension of
the state space, which makes this problem difficult. Pandelis and Teneketzis consider earliness
and tardiness penalties and examine
pro[)erties of an
optimal policy. Righter uses
Van Mieghem
stochastic ordering to further characterize aspects of an optimal policy.
(1995) studies a multiclass queueing system with costs based upon a convex nondecreasing function of each orders delay in the system.
that a generalized
ci^i
rule
is
Using heavy
asymptotically optimal, where c
traffic analysis,
is
a
he shows
dynamic function
that
represents each jobs marginal cost of delay.
.As in
both no setup, no due-date cases, the lack of setup penalty causes the cycle
length T and region
II
to vanish.
moves the switching curves
If
of (deterministic) due-dates, however,
into the orthant so that they
by equation (40) the cycle centers
described as follows:
The presence
.r^"
the total work
are shifted by
in
lie
/>,/,.
unfinished orders
1
on a new
he proposed policy can be
is
less
than
orders that are almost at their due-date have priority and are serviced
41
set of axes; i.e..
in
Yl\ = \ Piji-
then
a cyclic manner.
Wo
switch from 2 to
switch from
1
to 2
1
idle
due-date axis
IF,
Figure
and
if
7:
no orders near their due-date and the total workload
tlieie are
idling threshold, then the
earliness cost.
Custoniized products, no setups, due-dates.
If
machine works on orders
the total workload
is
for
level
is
above the
the product with the smallest
greater than J2[=i PiJi then
all
products other
than the cheapest tardiness product 0^ have priority and their orders are serviced
a cyclic
manner
product
OJl
just as their due-dates are reached or passed;
has lowest priorit\' and
its
orders are serviced only
in
the cheapest tardiness
when the due-dates
of the
higher priority goods are distant.
.A
two-product e.xample
is
given in Figure
7.
.\s in
the no due-date, no setup examples,
the switching curves for the two products nearly overlap. However, the switching curve
shift
onto the new due-date axes
The
shift of the
due-date axes
is
in
readily transparent by
comparing Figures
4
and
the workload plane represents the tolerance of the
policy toward aging orders not due in the near future.
The
region corresponding to the
"northeast" portion of the plane corresponds to the workload states where there
much work and
the plane
The
is
orders are completed past their due-date.
an area where orders are few and
intersection of the
new axes
7.
corresponds to the state where the wait
in
worked on
if
(the vertical one
The
is
is
too
"southwest"' portion of
will
be completed
early.
liidden t^ehind the switching curve)
(lueue for an order exactly equals
its
due-date
lead-time.
.•\s
in
the previous subproblems. the i)roposed policy minimizes the costs of the higher
cost products at the
expense of the cheapest product. The policy attempts to service the
42
high cost products
As
Ol.
in
the no due-date case, excess orders are shifted to the cheapest tartliness product
The presence
of due-dates allows the scheduler to avoid tardiness costs by staying
"ahead of schedule"
slack
such a manner that they are completed exactly when they are due.
in
when there
-
that
is,
by completing some of the orders early to allow
an unexpected surge
is
demand
in
against this uncertainty in the most economical
manner
when
how much
a product
boundary
in
is
"deficit"
possible:
workload they can hold: The policy
exhausted of orders, and, as
It
in the
the proposed switching curves in Figure
is
hedges
It
only completes early
7.
is
forced to switch setup
"kink" on the horizontal orthant
further hedging against tardiness
must be achieved by completing early the more expensive product.
that the low total workload
more
However, customized products have a
those products with the cheapest earliness cost.
limit on
or difficulty in production.
for
costly, the idling threshold
may
In addition, given
be nonzero so as to avoid
states with higli earliness costs.
It is
interesting to com])are these results to those of \ an
Mieghem. His
cost structure
can acconmuKlate a multiclass queueing system with customized products, no setups and
deterministic due-dates, where the earliness costs
this case.
Van Mieghems generalized
r//
orders (and early ones can be processed
group
is
determined,
in
/?,
are set to zero for
all
products.
In
rule gives priority to tardy orders over early
any manner), and priority within the tardy
in
out notation, by the highest "b" rule.
.A.lso.
the server works as
long as there are orders waiting.
Our
results provide the
dynamic
cyclic version of this policy just as
was seen
standardized and customized cases with no due-dates and no setups. There
priority
scheme
for
tardy orders, where
all
all
services the orders to exhaustion
is
set at zero
there
is
policy has the
Ol orders are serviced
do.
our policy and Van
when
only
when
there are no tardy orders.
policy:
I
The
as the gcMunali/.ed
Mieghems
Oj,
there are
idling threshold
he machine works as long as
.Again, as in the previous cases without setups, the
same beluuior
a two-level
of the earliness costs are eciual, the proposed policy
and so does not contribute to the
work to
the
but the cheapest tardiness cost product
have priority and are serviced cyclically: tardy
no other orders past due. Since
is
in
are similar
cfi
rule in the
when they are
43
heavy
dynamic
traffic limit.
Ijoth restricted to the
cyclic
I
hus,
one case
w.
switch from 2 to
switch from
\v
1
1
to 2
i.lle
due-datp axis
Figure
where
tlie
Standardizetl products, no setujjs. due-dates.
8:
models overlap.
No
Standardized Products,
5.6.
Setups, Due-dates. This
been explicitly studied to our knowledge,
the orthant boundaries.
shifted onto the
The
is
which has not
similar to the customized case, but without
cycle length and region
new due-date axes
case,
vanish.
II
The
cycle centers are
The presence of a finished goods inventory,
/;,/,.
however, modifies the proposed policy from the previous customized one. The policy can
be interpreted as the following
rules:
1)
only
fill
orders
when they
the finished goods inventory or directly from the machine output: 2)
present in orders minus that in finished goods inventory
two-level priority scheme:
Orders
for
is
are due. either from
if
above J2iPifi then follow
high tardiness cost goods
(i.e..
are closer to their due-date have priority and are serviced in a cyclic
Ol orders
priority
have lower
priority: 3)
and are serviced
if
cyclically,
on the lowest holding cost [Moduct
until the
in
Figure
This
workload idling threshold
total
and
O^
is
work
if
is
the total workload
all
a
but O^) that
manner and tardy
Ijelow Y^tPifi then tardy orders
have
there are no tardy orders the machine works
and stores them
reached.
A
in
the finished goods inventory,
typical two-product policy
is
depicted
8.
i>olicy
is
exactly the
same
as a shifted standardized,
44
no setup, no due-date
Moreover,
policy.
average
cost.
in
the heavy
traffic limit
both systems incur exactly the same long run
This perhaps surprising result
The due-dates we have considered have
makes
upon
intuitive sense
a special structure:
They
closer inspection.
are 0{y/n) and so only
influence the fluid limit. This implies that in our policy, orders will arrive,
and he serviced before the
total
late
workload has an opportunity to significantly change.
Moreover, the orders are continuously arriving and
not able to either get ahead or
become
fall
in
behind on orders.
that are due today and arrived 10 days ago, then
time frame the machine
this
Thus,
if
tomorrow we
we
is
are servicing orders
be servicing orders
will
due tomorrow that arrived nine days ago. Due-date lead times have not provided any
additional flexibility to the system; orders are not serviced earlier or later than usual,
only the absolute time of service has been shilted.
The
goods
ditTerence between this case and the customized one
for
customized case (see Figure
of future orders
product when
means
it
7).
the ability to pre-make
With standardized goods, the
a finished goods inventory.
allowed to hedge against backordering by investing
In the
is
in
stored work in
policy
its
always
is
cheapest form.
the inability to manufacture goods in anticipation
that the policy might be forced to store
runs out of orders for
0'^
work
in a
more
e.xpensive
products and so must finish early the next
cheapest holding cost product.
In a
system not
in
heavy
traffic,
however, the long run average cost
is
affected by the
due-date lead time. Buzacott and Shanthikmar (199:1 §4.5) find the long run average
cost for the single-product
Markovian version of
idling threshold (Buzacott
and Shanthikumar's target
cost
level),
If
one optimizes over the
then the long run average
is
ln/>
\b
+ hj
as long as the due-date lead time /
is
\
less
\np
J
than or equal to
condition guarantees that the idling threshold
is
this jjroblem.
(Cq is
.^'_
nonpositive).
.
In
The
(-^j —
first
»'o
term
('^i'^
in (59)
equal to the long run average cost for a system without due-dates and the second
the due-date contribution. Since i/|—^
-1-
A
is
is
alwavs negative, due-date lead times rcnluce
the cost of the svstem.
45
The
reason for
t
lie
difrerence between tlie heavy traffic system
and the nnscaled one
can be most easily nnd(>rstood by examining the nnscaled system with no orders and
fnll
inventory.
When
an order arrives, the total order workload increases above the idling
threshold and the server initiates prodnction.
is
Goods can be
dne, inflating the inventory level beyond the heavy
finished before the order
traffic prediction.
Holding costs are greater and the chance of baclvordering
effects:
a
This has two
The due-
decreased.
is
date leadtinie has allowed the system to hedge against bacfcordering, something that the
heavy
traffic
approximation has not allowed
to one, the long run average cost in (59)
for.
Nonetheless, as utilization grows close
dominated by
is
^
In
^,
which
is
independent
of the due-date lead time.
ft
is
also interesting to note the difference
model of Hariharan and
for-one replenishment
There
level.
is
Zi[)kin (lf)95).
scheme
between our system and the inventory
They consider
for a single
They
facility that
emj)loys a one-
product, and optimize the on-site inventory
a deterministic due-date lead time.
the facility's orders, L^.
a
L,i. for
find that increasing the
requests and a lead time for
due-date lead time decreases the
average cost of the system up until the due-date lead time equals the re-order lead time;
when Ld >
is further increases in the due-date lead time have no value.
be tempted to conclude that the two results are consistent, because Ld
system, ffowever.
also
our problem, the sojourn time
0{\/n) by the heavy
it
is
in
our system.
result
in
and ours
We
is
urgency
demand
5.7.
Ls does not always hold
uncertainty but docs not act as stored workload, freeing
for higher cost products. W'ith longer
be replenished quickly and so
hedge against backordering.
is
>
the role of Ls- and
that Hariharan and Zipkin consider an uncapacitated system, where
for orders to
stored work
conditions: hence, Ld
L^ for our
believe the reason for the difference between Hariharan and Zipkin's
inventory hedges against
machine resources
traffic
for orders plays
>
One might
In contrast, in
due-date lead times there
less
inventory
is
less
is
needed on-site
to
our capacitated model, the same amount of
needed indejK'ndent of due-dat(> lead time, as discussed above.
Customized Products, Setups, Due-dates. We know
analytically treating this problem.
1
his case contains all of the
of no previous
complexity of
work
§1.2.
.Ml
three regions can be present, cycle length, cycle center and idling threshold are nontrixial
46
W:
switch from 2 to
switch from
due-date axis
Customized products, setups, long due-dates.
Figvire 9:
and products can be binding,
it
is
interesting to note that
if
the due-date lead times are
the proposed policy looks like a shifted set of
9).
to 2
idle
I
long enough (Figure
1
1
SELSP
switching
curves (systems with setup times, however, will always be slightly different because the
expanding cycle length
by Property
1
in §4.4).
into the orthant
and
r will eventually hit
If
theorthant l)oundary
the due-dates are short (Figure
flatten out.
10).
for large total
workload,
the switching curves
bump
Readers should note that Figures 9 and 10 are based
on cases with setup costs but no setup times, whereas the other cases with setups
this section contain setup times but
a shifted
SELSP
policy,
it
no setup
costs.
in
In addition to viewing the case as
can also be thought of as the customized product, no setups,
due-dates case (Figure 7) with breadth added to the switching curves as was seen
in
the
transformation between the no due-date cases without setups to the case with setups
(Figures 3 and
5.8.
o).
Standardized Products, Setups, Due-dates. This subproblem corresponds
to the center of Figure
ized products
1.
and has received very
little
attention.
The switch
to standard-
from the customized case eliminates the orthant boundaries, thereby sim-
plifying the nature of the optimal
dynamic
cyclic policy.
47
I
he policy
is
merely the
SELSP
w.
switch from 2 to
switch from
1
to 2
1
idle
due-date axis
H.
Figure
policy
oil
10:
Customized products, setups, short due-dates.
and compare
it
the cost under the optimal dynamic cyclic policy
is
the shifted set of due-date a.\es (see the example
to Figure 6); moreover, as in
§").6.
insensitive to due-date lead times.
iu
Figure
11
.\Iarkowitz performs simulation runs for the p
=
0.9
case that support this iusensitivity conjecture.
It
is
worth noting that as due-date lead times increase, the entire
set of
switching
curves passes into the positive orthant. Hence, longer due-date lead times can transform
a make-to-stock
method.
When
method
of servicing
demand
for
due-date lead times reach this
standardized goods into a make-to-order
critical level,
it
is
optimal to
fill
orders
early and incur holding costs: these early orders jn'ovide a buffer against backordering as
would a finished goods inventory.
5.9
sider
Systems with Customized and Standardized Products.
Lastly,
we con-
mixed systems with both customized and standardized products, which would
respond to the border of the "Standardized"
customized and standardiz(xl
is
based
entir(>ly
circle in Figure
I.
Our
of
distinction between
on product design, and the derived switch-
ing curves dictate which standardized products are made-to-order.
work has treated the partition
cor-
In contrast, other
customized and standardized goods as a decision to be
4S
—
•-
•
•
Figure
11:
where
tlie
(MTO) goods
MTO/MTS
without setups or due-dates, and with
demand items
intrinsic priority rule allows for a
lost sales
instead of backordering. In a subsequent
for the
MTO
and
MTS
switching from
MTO
and
MTS
goods to
MTO. They
priority rules
also propose a heuristic for partitioning
items.
Finally, there
and due-dates.
MTS
products
base stock levels. Federgruen and Katalan (1995a)
examine mixed systems with setups and no due-dates, and compare several
for
that
a heavy traffic analysis of mi.xed systems
paper (1995b), she examines different priority rules
for setting
system with no setups
partition, but does not involve optimal schedul-
Nguyen (1995a) performs
and suggests an algorithm
to 2
due-date axis
represent low
have priority over the make-to-stock (.MIS) ones. This
ing of the products.
1
idle
(1!)93) consider a cpieueing
rt a/.
make-to-order
performance analysis of the
switch from
1
Standardized products, setups, due-dates.
optimized. For example. Carr
or due-dates,
switch from 2 to
is
.\11
the
full
i)roblem:
Customized and standardized products, setups
of the previous cases are subsets of this general model.
It
is
the
proper setting to ask questions of balancing inventory costs and setup penalties, of setting
base stock inventory levels and avoiding backordering. of determining due-date lead time
effects
and natural due-date-based
]jartitions of
49
.MTS/MTO
goods.
\'et. this
generality
Wo
switch from 2 to
-
-
switch from
1
1
to 2
idle
due-elate axis
Figure
12:
Mixtxl products. setui)s. due-dates.
does not complicate the system beyond the previous cases. The product mix
for in
accounted
our calculations by the presence of orthant constraints on some products and not
on others (see P'igure 12
for
an example of a mixed system with one customized product,
one standardized product, setup times and due-dates).
earlier
is
Most of the insights described
about the interactions of setups, due-dates and orthant constraints carry over
directlv to the
combined svstem.
6.
Computational
COMPUTATIONAL STUDY
results in
and robustness of the HT.AP
Reiman and Wein
for
(1994) and
MRW
confirm the accuracy
make-to-order and make-to-stock systems, respectively,
with either setup costs or setup times, but no due-dates.
Here we report on a com-
putational study that tests our methods with respect to mixed systems and due-dates.
Unfortunately, the presence of due-dates prevents an exact derivation (via dynamic
gramming)
of the optimal policy
consideration. Therefore,
and an
we employ
(^xact e\aluatiou of the various policies
discrete event simvdation for these tasks.
50
i)ro-
under
For sim-
plicitv,
we only examine two-product systems with deterministic due-date
and setup times are independent and exponentially
Interarrivai times, service times
tributed.
The
service
preemp'ive-resume and there are no setup
is
we
scenarios in this section,
lead times.
For
costs.
dis-
of the
all
systems void of both orders and inventory and
start with
then perform ten independent runs. Each run contains
a 100,000
time unit initialization
period and statistics are recorded for the next 10,000.000 time units.
We
wish to examine three issues
mixed systems: The accuracy
for
of
the heavy
traffic
approximation, the effectiveness of our proposed policy, and the value of increased duedate lead times. The
the
same parameters
tardiness costs to
as a case in
=
b,
product
first
customized and the second
MRVV, we
oh,, serxice rates
a\erage setup time per cycle
for values ecpial to 0.
is
—
.s
20.
We
arrival rates Aj
1.
the due-date lead times
.set
The
(27). to three straw policies.
date considerations; that
set to zero.
produced
level.
uj) to
The base
due-date case
(product
is
The second straw
t')/[/;i(
a stationary
)/cr'
(see
1
is
j)olic\- is
in (26)-(27)
policy
a
is
A,)
=
/2
/i
=
0.3
and
=
1,
and
test /,
in §3.1 of
is
MRW.
pi)!-
+
I'-
as follows.
—
If
Pi)/2.
we
The
The
calculated with the due-date lead
hybrid base stock/exhaustive policy
is
equivalent to a
—r
order workload
cycle center for the customized product
cycle center for the standardized product
define v
— pi) + /'2(l — Pi)\- Inder
gamma
the proposed policy without due-
calculated in a fashion analogous to the standardized, no
set to r((r)/n(f
—
is
defined in ecjuations (26)-
is
serviced to exhaustion and the standardized product
a base stock level //"'r. which
stock level
set to t(u-)p2{\-
2(tr—
/;,
is
1)
straw
first
the polic\-
is.
where the customized product
is
0.6,
2,/?2
20 and 100.
an optimal policy, we compare the [proposed policy, which
/,
=
=
/?i
Since we do not have a convenient point of reference provided by
.Sfraw Policies.
time
standardized. Using
set the earliness costs to
=
//,
is
=
vj \/n. then the cycle length
this policy, the total
density with parameters o
CofTman. Puhalskii and Heiman
=
'lsjn[\
l!)95b).
51
—
is
t[w)
—
normalized workload U' has
p)lo- and 3
The average
=
-sll'-i /',(1
cost for this policy
—
is
given by
.
+c,(r
-
+
Pi(i-p.)ELi ""-'''"
—
^^i.^^—
—
-.2=-^^^^^
"EL,
—
r
set
/?
equal to
( 1
which minimizes
The
—
p)~'^
results of
is
^*''+'*
/
and use a steepest descent
is
e-"^"-^\liw.
parameter
algoritliin to find the
again a hybrid base stock/exhaustive policy where the base
determined by an exhaustive search using nuiltiple simulation runs (the
which are not shown here). These two hybrid
as the bybri'f
"'p ':::•'
,r)
(60).
third straw policy
stock level
(60)
3
-.
^('-''-''EL,"'(i-''-)"""EL,^'"-''''"
We
w
''-(i-''-)
HTAP
policies,
which
be referred to
will
policy and the hybrid ^'^nrch policy, together offer the opportunity
to determine the accuracy of the HT.\P. Numerical results are contained in Table
and switching curves
for the
Figures 13 and 14 for the
/,
hybrid search policy and the proposed policy are given in
=
and
=
/,
100 cases, respectively.
Due- Date
Lead
Cost of
Hybrid
Cost of
Hybrid
Time
HTAP
Search
290.3 (± 3.2)
201.8 (± 2.7)
122.8 (±1.4)
290.0 (± 2.7)
20
100
Table
Observations:
Straw
I:
I
Cost of
Proposed
vv/o D-date
199.5 (± 3.2)
121.3 (±L. 3)
Simulation results
Policies.
The
274.6 (± 1.9)
186.3 (± 2.1)
158.3 (± 0.9)
for the
Cost of
Proposed
Policy
274.6 (±
1.9)
184.1 (± 2.7)
109.7 (± 1.1)
mixed system.
long run average cost for both hybrid policies
decreases with longer due-date lead times.
With
zero due-date lead times, orders for the
high cost customized product are immediately backordered. driving up costs.
date lead times increase, orders
for
.As
the due-
the customizerl jiroduct are less tardy and costs
However, one would expect that eventually they would
start to increase again
fall.
(under the
hybrid policies) as the diu^-date lead times become inordinately long and holding costs
become
excessive.
The two
hyl)rid policies incm- nearly identical costs.
search policy are
4').
37 and 13 for the
/,
=
corresponding HT.-\P levels are 30. 28 and
52
0.
•").
The base stock
levels for the
20 and 100 cases, respectively, and the
.Mthough the base stock
levels for the
MTS
Proposed Policv
iVITS
Ilvhiid Policy
W2
I
40-
MTO
MTO
Wi
40
Base Stock Level
switch from 2 to
switch from
1
1
1
to 2
due-date axis
Figure
Switching curves
13:
for a
mixed system with zero due-date lead times.
search policy are consistently higher than those of the
in
long run average cost
is
HTAP
])olicy,
the actual difference
very small, for the hybrid exhaustive/base stock policy, the
long run average cost as a function of base stock level appears to have a shallow slope
about the optimal solution, and the heavy
level that
performs
Finally,
it
is
traffic analysis
is
able to identify a base stock
(piite well.
not surprising that the performance of the proposed policy that assumes
zero due-date lead times deteriorates as due-date lead times increase. This deterioration
suggests that
it
is
Observations:
not advisable to ignore due-dates
Proposed Policy.
when due-date
lead times are large.
For the reasons cited above, the proposed policy
also has decreasing long run average costs as due-date lead times increase.
policN'
14:
has an important advantage over the hybrid policies that can be seen in Figures 13-
It is
able to avoid large buildu|)s of product
be shipped. The cost reduction achieved
hybrid policies
the
/,
The proposed
=
is
in
in
fable
I
1
orders, both in (lueue
l)y
and waiting to
the pro[)Osed policy r(4ative to the
the o-lO'/c range, and increases with the due-date lead time
/,.
For
case, the proposed policy avoids se/ere backordering by switching to product
53
MTS
MTS
Hybrid Policv
W2
Proposed Policv
W2
':
40
MTO
-40-
11
Base Stock Level
:
switch from 2 to
switch from
to 2
1
due-date axis
Figure
i
if
there
Switching cur\cs
14:
is
for a
mixed system
an excessive luunlx'r of orders
setup penalty, the cycle length r
is
in (jueue.
witli
due-date lead times of 100.
However, due to the severity of the
long and so the product
buildup must be large.
1
Thus, the marginal benefit of the proposed policy over the hybrid policy
in this case.
greater opportunity to avoid excessive product
is
.As
seen in Figure 14. the
near the due-date axis pi
The hybrid
The
in
1
costs.
Tlu^ cycle length r
is still
large
able to attain a balance l)etween earliness and tardiness costs by cycle
center placement.
order to wait
not great
the due-date lead time increases, however, the proposed policy has a
.As
but the policy
is
queue
/i
for
.
which
is
amount
of product
the level of product
1
workload
is
maintained
workload necessary
1
an amount of time exactly equal to
its
for a new-
due-date lead time.
policies are not able to [jerform this type of cost minimization.
\'aluc of
Due-date Lead Times. As due-date lead times increase, our heavy
traffic
analysis (see §4.6) predicts that the long run av(M'ag(> costs umlfn' the
mixed system should
beyond
this threshold value,
decrease until the due-date lead tinu^s
hit a critical value;
costs should be very insensitive to due-date lead times and
almost ecjual to the cost under the corresponding SELSP.
54
t
he total system cost should be
.\Ioreo\'er.
because the hybrid
optimize the staiulardizfHl and rustoniized products independently, the inventory
policies
costs for the standardized protluct should be independent of the due-date lead time under
these policies.
For the hybrid
product
Table
in
I:
Its
respectively, for
11.
there
polici(\s.
change
little
is
the costs due to the standardized
in
average holding and backorder costs remain approximately 23 and
all
values of the due-date lead time, and almost
of the savings
all
are due to the customized product. In contrast, the inventory costs for the standardized
product
the proposed policy change with the due-date lead time (in the
in
earliness cost
is
46.7 and the tardiness cost
16.2, respectively).
Unlike the hyl
This difference
rid policies, the
due
is
proposed
is
the
12.7: in
to the
/,
=
/,
=
20 case the
100 case, they are 34.7 and
dynamic nature of the proposed
polii.}' is
policy.
able to shift inventory costs between
the two products by changing the cycl<^ center and cycle length for each workload level.
To analyze the
system to
case.
cost of
101).
dynamic
cost under the
7 incurred in the
MRWs
than
/,
=
due-date lead time
-An alternative
critical
I nfortunatelx'.
100.
optimal
for the
cyclic policy
mixed system
discrepancy suggests that the
less
is
asymmetric, setup time,
.According to the d\namic programming results
SELSP
the
corresponding SELSP. which
its
it
is
in
row 13 of Table
106.5.
in Taljle
I
present in the hybrid
was found to
t'l
is
\'ery difficidt to calculate
polic\'
when setup times
in ^^3.1 of
MRW.
Ije
i)olicy.
By our
it
49.0.
making
SELSP
cost.
MRW.
for this
problem
the critical threshold
SELSP under
is
to contrast
the generalized
is
similar
includes a nontrivial idling threshold not
analysis in §4.1. the threshold due-date lead time
SELSP
tlu^ critical du(^-dat(^
case. For this
lead time.
/,
/i.
—
the SF.LSP cost under the generalized l)ase stock polic\-
is
1
17.0.
SELSP
scenario,
equal to SI. 6.
is
t'l
Hence,
100 shoidd be rougldy
Referring again to row 13 of Table X'lII of
00
is
are present.
heavy trafhc theory predicts that the mixed system cost of
ec[ual to the
10
This small cost
100.
This generalized base stock policy
the base stock level from the
is
=
=
fairly similar to the
is
due-date lead time threshold
form to the hybrid HT.AP policy, except
where
/,
mi.xed
6i
\'III of
comparison where we can calculate the threshold value
base stock policy described
''i/pi-
which
when
the mixed system under the hybrid search policy with the
in
we compare the
costs incurred by the customized products,
.MRW. we
.\s
lind that
we predicted,
this
value
is
reasonably close to the mixed system cost of 122.8
part of this discrepancy
is
in
Table
particularly since
I,
prol)ably due to the increased sophistication of the generalized
base stock policy relative to the hybrid
HTAP
policy.
CONCLUDING REMARKS
7.
Theories for scheduling a manufacturing
facility
have examined the issues of customized-
standardized product mix. setup penalties and due-dates, hut have typically focused on
each of them separately.
averaging principle
is
paper, Coffman, Puhalskii and
In this
Reimans heavy
used to investigate the composite problem. With
computational method to optimize within the class of dynamic cyclic
case where each product has
describe the policy.
affect the
its
own
behavior of the
liuid
we
particularly the determination of
system,
may be
we outline a
policies.
deterministic due-date lead time,
Our methodology,
it,
traffic
For the
qualitatively
how due-dates
useful for studying other systems with
delay constraints, such as due-date scheduling problems arising
inventory-routing problems with time windows, and inventory
in
computer applications,
management
of perishable
products.
Our heavy
traffic analysis
dom demand and
suggests that the risks inherent
service processes cannot be removed,
in
the uncertainty of ran-
^et. by proper scheduling, the
impact of the variability can be reduced by channeling the fluctuations
in
order queues
and finished goods inxentories into low cost regions of the state space. The presence of
setups, due-dates and product
Our
mix each dictate how
results yield a simple interpretation of
how
this
dampening
of cost
is
performed.
these facets affect the switching curves
that characterize the proposed policy:
due-dates
setups
customized/standardized goods
The
shift
it
is
simplicity of the
first
=
=
=
shifts
breadth
presence/absence of orthant boundaries.
of these three observations
the optimal switching curves
-
is
to analyze due-date scheduling
-
that deterministic due-dates merely
particularlx' striking, given
problems
in
a
how notoriously
dynamic stochastic
three-point guide allows us to ciualitatively understand the nature of
56
difficult
setting.
This
dynamic stochastic
scheduling problems with
deteriiiiiiistic
due-date lead times without explicitly calculating^
the solution.
Our
analysis also sheds light on the value of foreknowledge of customer
the form of due-dates)
affects the
when the system
We
optimal policy.
find that the costs incurred
independent of the due-date lead times
increase
is
beyond a certain
no change
the
same
in cost;
Although costs
when the
level,
orders are
role as a finished
for
how
this
(in
information
by standardized products are
When
these products.
due-date lead times
the standardized products are made-to-order, but there
filled
before their due-date, and these early orders play
goods inventory: They provide a buffer against backordering.
customized products
for
heavily loaded, and
is
demand
initially
decrease as due-date lead times increase,
(deterministic) due-date lead tin es reach a critical value the costs level off and
the customized product incurs nearly the
same
cost as standardized products. In essence,
large due-date lead times blur the make-to-order/make-to-stock distinction:
They cause
standardized products to be made-to-order and provide customized products with the
flexibility to
he produced
early, thereby imitating a
make-to-stock
mode
of production.
ACKNOWLEDGMENT
This research was supported by
at
MIT and
a grant
We
N.SF grant DDM-00572*)?.
and thank Frank Kelly
for
from the Leaders
for
.Manufacturing Program
thank Marty Reiman
for helpful
comments
sharing with us an early draft of .Aldous. Kelly and Lehozsky
(1995), which deepened our understanding of the problem.
APPENDIX
1:
THE MARKOV CHAIN APPROXIMATION ALGORITHM
This supplement describes the Markov chain approximation algorithm that solves
equations
total
('23)-(24).
The Markov chain
is
workload state space into intervals of
define Q''
=
rr-
+
\ch
—
s\
created by discretizing the one dimensional
size
/;
and time into blocks of
then the transition probabilities of
P"(a..u.-/0
^
=
01
th<'
^,
size
A/
.
If
we
Markov chain are
(61)
P'{<r.,r
+
^
^
=
h)
(62)
and
P'iu'.w)
=
1
{-'
^
-
+
'^y-
""'
- V^^\)
',
(63)
,
Q
and
time
tlie
iiiter\al itself
additionallv must be set to
=
A/'
There
is
2
^
one exception: The reflection boundary
probability in
tlie feasible
region before
P'iwo +
All the other transitions
We
h
h.
Wo
+
//)
(64)
.
is
never reached and so the transition
u\) is
= !- P'(wo -
h.
,/•„
-
2//)
(65)
.
have zero probabilities.
can now write the dynamic programming optimality equations as
\-{u-)
= Y,P\>r.u)\'{<j)+{c{.v^\T.ic)-g)At''
Because the Markov chain
is
a birth-death process, for a policy r
(66)
.
and Wq the steady-state
distribution and gain can be easily calculated from the transition probabilities. Similarly,
the potential function can be recursively calculated by
With
V
,^
q-c{.v^^'.T{a-).w)
h
=-
(((•)
and
\{w +
+ {l~ P''{w.w))\-(w)-
cj
embedded
h)
.
in ecjuation (23)
threshold that minimizes the gain
APPENDIX
g.
2:
ecjual to zero
and an improvement on
The algorithm terminates when
((w)
tliat
and then tracks how the
by finding the
and
converge.
constructs 0'(.S') by finding an
initial set
set evolves as
58
u'o
»'o
AN ALGORITHM FOR
This ap])endix describes an algorithm
5
+ h)V{w +
determined, an improvement iteration on r can be achieved by perform-
ing the optimization
for
P'{>r.,r
J—
.S'
r
0"(.S)
increases; for clarity
we
re-
.
introduce the notation specifying the dependence of the
on
Our
5'.
jump
task
and there
in region,
leaves 0'(.S')
complicated by two
is
is
.S'
of non-binding products
increases the optimal solution can
nc guarantee that when a product becomes binding and
does not re-enter for larger
it
As
facts:
0"
set
We
5'.
simplify our calculations of 0'(5') by
including the type of region the cycle center and cycle length imply in our accounting.
0' (5) denote Q'{S) wdien the cycle center and cycle length
Let
0'"{S)
conditions,
algorithm
for the region
II
satisfy the region
conditions and (:)''"(S) for the third region.
based on the following eight observations (proofs are provided
is
I
The
in .Appendi.x
3):
The cycle center
1.
and cycle length
x'^'
t are continuous functions of the efi'ective
setup cost per cycle S.
monotonically increasing with respect to S
2.
The optimal cycle length
3.
If
product
t'
calculated with 0' satisfy .rf
length
and cycle centers
is
not in 0". then for 0'
and S" are such
If S'
4.
is
I
r
I
r'p,(l
-
{/} the cycle center
.r"'
and cycle
/>,)/2.
< S" and both
that S'
satisfy region
<
= 0' U
(III) conditions,
their respective optimal cycle lengths
C
then Q''(S")
C
Q''{S') (Q'"'(S")
0-'''(.s");.
IfS
5.
II to
such that the optimal cycle length and cycle centers imply a shift from region
is
region
(III),
I
then 0-'( liin,_u
.S'
+
= 0'"{\\m._^uS -
t)
(0-^'^(lim._o
>'
+
f)
=
0-"(lim._o5-f);.
6.
If S'
and S" are such
and cycle centers
7.
8.
region
When
If
I
S
their respective optimal cycle lengths
satisfy region II conditions, then Q''"{S")
a condition
is
< 5" ami both
that S'
1
binding i)roduct
/
C
0'"(.^'').
changes to condition
2. it
remains
such that the optimal cycle length and cycle centers imply a
or III to region
II.
tlien
0*"(lim,_yo
.S
+
e)
can be calculated
i)y
Ijinding.
shift
from
an iterative
algorithm.
With these
eighl observations, the algorithm
0"'(O). 0'"(O) and 0''''(O).
events can change
t/inding
and
we track how each evolves
0"(.s'): .\ shift in region, a
a condition
1
we suggest
l)in(ling
as
5
is
is
simple.
From an
initial
increased. Three types of
non-binding customized ])roduct can l)ecome
product can become condition
59
2.
Given equations
(10)
)
and
we can
(51).
The
ralculatr the range of
initial set
w
less
is
6-2,
We
find 0''(O) by
in
the
e,
Y2i=iPifi
reached
.
,
list,
<
shift
~
Jl^LiPifi
in the e, list or
Given an
.A
II
/',/,,
The
"'•
The
vanishes.
If
the total workload
of the cheaper customized products.
is
the next product
is
product
.A
t j
/?e,
is
insufficient for storing the
when
process stops
then 0-'(O)
For an effective setup
there iwe few orders remaining in the system and
removing products one-by-one.
set to
h,.^
<
removed
•
•
if it
•
Let
<
^f,v-
is
next
remaining work.
i.e.
either a standardized product
>
implies J2i=\ Pifi
€j
<
Yl',=i Pift
~
"'
it
is
'f "'
'^
V}. Both 0-''(O) and 0-''^(O) are
'I
V}.
set to {1
changes.
as region
make binding some
to
customizable and
is
greater than i:i=i
always
t'l*"'!
chosen as follows.
is
e\} be an earliness ordering of the products such that
{ti,
.
becomes zero
than Z!ili/^./c
may be advantageous
.
before any one of these events occur.
of non-biiuling products
cost of zero, cycle length r
level
.S'
initial
set
0"(U) we can find the range of
changes when the
minimum
.S'
set
such that one of the three events occur:
the binding of a customized product or the change in condition of a
in region,
binding product. For a gi\en S' with non-binding
c^-y'(S') /^E,e-:»-^-')'^'-^"'-"'
SI
Q'(S') a region change occurs when
set
S2
region
c
setup costs before the
eifectix'e
I
to region
'->a'l~'|-2(Z^,g,.,.//|g/)
'
II shift
PJ - "•
.
S3
I'u'l ,-'l('-ffl
&'"{S')
-^4
T.,e<.-i'"(S')Pifi
-
"
'^')
region
II
0'"(.s")
v2
to region
I
shift
(68)
":( s')'^2(E,ee-'/(5')'''^'""'
S3
.,5,,(i-Pe.,,-/)
,
(V^
pA}j1£i1\
region
w'"(.s') /^E, €«•//;,, /i/^./.
II
'!)'•
'
"6'
to region III shift
p(-)-"'(.S')
region
60
III to
region
II
shift
where
-
6,
E
/^
+
Peus')(^
-
Pe-js'))
(69)
>e{e-'{S')\e'^{s')}
E
iiQ''(S')
6,
-
h.
Pei(S'){l
—2
-
te{e-'i'{S')\ei{S')}
'
^
p,{^
A customized
'
-
"
I-
Pi
product becomes binding when
region
©*'(>") ( ^•e{-''{f')\n'^ts'))
I.
"'
binding
S2
region
f0-"(.S') /
S3
=
y^ 0'^
i
c'ry'(s')
f
SI
.s
pe-AS')]
(70)
boi(s')
^'/-(Z;6(e--f/(5')
I.
binding
B'^
Pj/;-'^'>'>'^'2
S2
<
(Ljee-"(5')Pj/j
-s^
-
i'K
tt')
region
&-"'{s')
^5
I
p,f,{h,+h.)
V
liinding
/
<,"'"'<''")
Pili-PiXfji-tg.)
region
c(-)-'"(>")
II.
III,
i
^
6^
binding
f T..et.^>-"Us'}\e-fs')}"'^--'"
S2
region
61
III.
0',
binding
Lastly, product
=
^
<
changes from condition
;
T^-ii
^3
to condition 2
when
-^2
iEjeG'"{S')Pjfj-^'^')
region
Thus the current
from 5' to the
and Q'^{S")
set of 0'(.S"). 0*'(.S")
minimum S
II
regi""
TT^-^2
^5
K
I
is
(-
.
III
valid for effective setup cost
greater than S' in eciuations (68), (71), (72). At that point,
Q''(S), Q'''(S) and Q'"^{S) are updated according to observations 5 through 8 and the
process
or
all
is
repeated.
The algorithm ends when
either
all
customized productes are binding
but one of the customizetl products are binding and the cheapest
3 product in region
III
with
-^(.r;'
—
"^'
|,~^
<
)
(in
is
the condition
the other regions, increasing cycle
length impies that either the customized product would
become binding
or that the
region would eventually shift).
APPENDIX
In this
PROOFS OF EIGHT OBSERVATIONS
3:
supplement we prove the eight observations stated
Observation
1.
For a given
tr
and
\
'('(')•
the (luantity
in
equation (23)
with continuous derivatives with respect to the cycle length t (t
effective setup cost per cycle
.s".
.r''".
r and
.'^'.
and
.S'.
for
a given
.S
and
ir.
This
is
is
continuous
cycle center
.r'^
and
also continuous
The only way the optimal
t' could be discontinuous with respect to an increase in
optimal solutions
0).
2.
Thus, the Karesh-Kuhn-Tucker
necessary oi)timality conditions change continuous!}' with
x'^
>
boundary conditions are
In addition, the
with continuous derivatives with respect to
.Appendi.x
in
5' is if
there were multiple
not the case since the objective function
is
convex with respect to cycle center and cycle length and has the unicpie optimal solution
presented
in
equations (40) and (51).
Obser\'afjon
2.
This
is
easily seen
from ecjuation
(-51)
and the
plying the continuity of r" during changes of binding products
Obser\'atjon
3.
Observation
I.
(-)"
lirst
observation im-
and changes
in region.
This follows from the construction of O".
We
show
this b\- inducting
62
on the cheapest products. Consider the
region
I
become
case: let 01 he the Hist cheapest [)ro(hict to
>''
setup cost
be binding
with
c\-ch'
condition
in
length r'. All other
2.
The
at effective
I
with smaller earliness costs must
i>r()(lucts
instant product
biiiiHiig in region
becomes binding, the cycle center has
Oj^
reached the posit ivity boundary and so
_i
P:(\-
-p.
z
pj\
-
T'p,(l-p,)b,-h.+2h
^'/>fl;(l
"
+
b,
-Pel)
—=
;
h,
(73)
minimum work
In order for the
the boundary as r
-
P,{\
increased,
is
P,)
.\s .S increases, r also increases,
only re-enter
in
implies that e([uation
The
p,{\- p,)b,-
(7:5)
b,
-
+ h,
will
+ 2/;^i
+ h.
h,
<
1.
region
in
equation (74)
remain negative
will
p,)l'2. to
reach
is.
1
-
Psj,
<0.
(74;
Additional products might
1.
l)ecome more negative. This
for larger r.
induction on the cheapest product results
/''^
/>()(
—
and hence the cheaper binding condition 2 products can
—j—r—
'•,
r/_),(l
must be negative: that
condition 2 and will not re-enter
leave 0"'(.s'') but since
(-)''.
derivative
its
E
+
—
experienced over the cycle, xl
in
and so
61
cannot re-enter
an argument identical to the
previous one where equations (73) and (74) are modified by re])lacing $1 with
0'^
and
.S'
with 5'.
The other more expensive products
re-enter Q''
Since hg'^s")
.
p,{l
-
p,)b,
-
>
^'8'(S')-
b,
+
b.
-l
+
+
h.
that
become binding
in
region
I
also cannot
from the above induction we have
he-^^s^
h,
,^
+
b,
2
+
+
h,
h -?:(>")
(75)
Therefore, the cycle center of the more expensive non-binding products are lower for
larger
"/),(!
.S'.
—
Since cycle length t increases with
p,)/2.
and hence binding,
The same argmncnt
Observation
ef[\iivalent to
5.
it
will
holds fur region
>'.
if
the cycle center becomes less than
remain lunding.
III.
Since cycle centers and
cN'cle
length are continuous in
the statement that no binding |)roducts
63
in
rt^gion
11
.^'.
this
is
become non-binding
after a transition to region
in
region
Binding condition
I.
and rcMuain binding by observation
I
4 (in region III, a condition
becoming non-binding would violate the region
region
I
have a cycle center of the form
will
the derivative of
,r','
r/>,(
Observation
We
6.
minimum amount
derivative
0'^'.
1
— p, )/2
Thus when
a
will
of
is
it
71
— {J^j^e-
Pjfj
~
"'')f*i
and so
72
"
always negative. This implies that
—
x'l
work over the course of a
that
•
in
continue to push against the orthant boundary.
prove this by examining
we show
negative,
is
— ro,
product
1
Condition 2 products
III definition).
with respect to r
2 binding cycle center
a condition
the
—
p, f,
products must have become binding
I
—
Tp,{l
the derivativ(> of
/>,)/2,
Once the
cycle, with respect to r.
remains so as further products are removed from
product becomes binding at
.rj'
—
Tp,{l
—
p,)/2
=
0. as
expands
r
it
cannot re-enter: The negative derivative would continue to push the cycle center against
the orthant boundary. Since no binding products in 0'"(.S') can
the other regions by obser\'ation
The
(d'''(S)
5.
minimum workload
derivative of
is
a nonincreasing
region
in
II
for a
become non-binding
set witii S.
non-binding product
P,(l-P.)
d_
.(•
-
-or;l
in
-
pM
Y.
-
is
Pj)
J^0-"(i-)
E
Ljew"(>)
V
where H
=
T.ie(r'>-"(S)
(2i>.
E
+ h.,-i>,)^^4^ri^+
b,
+
hj
pA^-pj:
(76)
ji<r)"'(S)
Pji^ —Pi)/i^'j +'' J- Therefore the derivative
is
negative
if
and only
if
E
Thus,
at efi"ective
i-^l>'
+
l'.<-'>
setup cost
expensive tardiness product
•>;
je0*"(5)u{/}
.^''.
E
/',(l-/>.)>0.
the derivative of product
;
is
negative and
(77)
a
more
leaves to form ©'"(>''). then equation (77) implies
/
,
if
/f~'''^ +
;
;
J'A^ "
'>j
+
Pj^
<>j
64
,
^
j^e'"(5)u{/}
p,(l-p,)>0,
(78)
which
is
E
Since
b,
+
{^i>.
—
bi
<
l'.-'>/-T^+
0, \vc
Q'"{S), the derivative of
E
left
side of (81)
is
+
less
than
because their ditference
i>,-!>
If
a cheaper tardiness prodnct
yf~,'''^
E
+
[-K^h.-b/f-P^K
/
leaves forming
E
pAI-Pj)>0.
(SI)
pM-p.)
(s-2)
is
Therefore the derivative of the
/
(79)
cheaper product implies
tlie
{^'"
E
negative after
PAl-p,) + 2pd}-Pi)^j^>0.
have
and so the derivative remains negative.
The
E
minimum workload
over a cycle lor product
i
remains
then product
;
remains
becomes binding.
OhservHt'ion
7.
binding because
in
If
this
change occurs
in
regions
these regions the (luantity -^(x",
condition 2 product.
If
the change occurs
in
region
-
II.
I
or
III,
^'''"^'''^
)
is
a condition
always negative
1
product
for a
in region II
implies that
^
^:7i^i''.-''.+-^''.i-'-^^f^<o.
jee'"(.S)\{,}-^'-'j + "j!
(SD
or
/Ml -/>-)+
E
je(-)-"(.s)\{<}
"'j^~'''\ h,-b,-2h,]<0.
"-'^"j
65
(85)
Subtracting equation (77) from equation (84) we get
V
which
is
positive.
Observation
As stated
S.
A
m+h yf~l'^\
observation
in
6, this
(86)
inipHes that T^(.r^"—
II.
.-Xt
these transitions, binding products
of Q'"{S), however,
region
I
is
not difficult.
or III to region
be re-included
in
)
conclusion from the previous seven observations
again enter
S
the effective setup cost
binding products not before
II. all
0'''{S).
.At
may
in
is
0'(.S').
is
negative.
that 0'(.S')
is
and
to
from regions
relatively predictable with the notable e.xception of transitions
region
'^^'''^~^''
I
111
Re-calculation
point of transition from
Q'"(S')
for S'
< S should
Fhe cycle center can then be re-calculated. Those products
such that either their cycle centers are infeasible or were previously binding and currently
have a negative cycle center deri\'ative (as determined by equation (77)) can be removed
from
0"''(,'^').
Q'''(S)
is
By observation
found such that
6.
all
cycle derivatives are removed.
they do no re-enter. This process can be repeated until
binding products with negative
The resulting
©"'"(.S')
is
minimum workload
equal to 0'
per
(.S'"*").
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69
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