LIDS-R-1395
September, 1984
A HIERARCHICAL SCHEDULING POLICY
APPLIED TO
PRINTED CIRCUIT BOARD ASSEMBLY
by
Stanley B. Gershwin
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ACKiNOWLEDGEMENTS
This research was supporReted by the Manufacturing
search
Center of the Thomas J.
of
Laboratory
Watrson Research
the
International
Business
U.S.
by the
Co rpo ra t ion and
Army Human Engineering Laboraunder
contract
DAAK- 11tory
82-K-0018.
presented at the
Material Handling Research Forum
Material Handling Research Center
Georgia Institute of Technology
Atlanta, Georgia 30332
September 11, 1984
~_1.. .
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I;~. 'j~.
.--. '-,
-!.
.-
A HIERARCHICAL SCHEDULING POLICY APPLIED
TO PRINTED CIRCUIT BOARD ASSEMBLY
by
Stanley B. Gershwin
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
This paper discusses the performance
of a new hierarchical production scheduling policy for flexible manufacturing
systems.
A detailed simulation of an
automated printed circuit card assembly
line is used as an experimental test bed.
This simulation is a model of a line is
currently being installed at the International Business Machines Corporation
(IBM) plant at Tucson, Arizona.
The hierarchical strategy is effective in meeting production requirements
(both total volume and balance among part
types) while limiting average work-inprocess (WIP).
This is a consequence of
the nature of the policy whose key elements are a discipline that, at each
level of a hierarchy, keeps material
within
capacity,even in
the
p re s e n
feedback
e
of
.
un
e r tainty,
bdy
ubsi n g
1. Introduction
While the technology of manufacturing
-- including processes and computer hardware and software -- is improving rapidly,
a basic understanding of the systems
issues remains incomplete. These issues
include production planning, scheduling,
and control of work-in-process.
They are
complicated by randomness in the manufacturing environment, particularly due to
machine failures and uncertainty and variability in production requirements.
The
main thrust of the research in manufacturing systems theory now being performed at
the MIT Laboratory for Information and
Decision Systems Is aimed at the random
disturbances that rob plant managers of
sleep and factories of productivity.
(See
the references.)
It is our belief that such disturbances can have a major effect on the operation of a plant.
Scheduling and planning
must take these events into account, in
spite of the evident difficulty in doing
5so
Early research in manufacturing 5y5tems was directed at a limited range of
issues.
In particular, a great deal of
the work on production scheduling and
planning was concerned with the mathemati-
cal problem of fitting together the production requirements of a large number of
discrete, distinct parts.
Such combinatorlal optimization problems are very
difficult in the sense that they often
require an impractical amount of computer
time.
Furthermore, they are limited to
deterministic problems so that random
effects, including machine failures and
demand uncertainties, cannot be analyzed.
Some work that attempted to represent
the random nature of the production process was based on the Jackson network-ofqueues theory.
This had some success, but
modeling assumptions restrict its appllcability to a narrow range of problems that
do not include many important issues, for
example, machine failures.
A wide variety of methods are available to industry to deal with scheduling
and planning.
Such methods as MRP tend to
be highly computer-intensive, but based on
simplistic assumptions.
They are often
ua'nui'e'l'dy-V- makpoor use -bf---computer and
factory resources.
MRP systems can run
for hours or days on a mainframe computer,
but must be periodically rerun because the
conditions in the factory inevitably
conditions
in the
factory
inevitably
change.
Such
updates
are necessarily
infrequent because they take so long to
perform.
Simulation is also widely used in
industry to determine scheduling strategies, floor layout, and for other planning
problems.
It is expensive in both human
and computer time since simulations, to be
credible, tend be complex and require a
great deal of data.
Many simulation runs
are required to make a decision; the decision parameter must be 'tuned" until optimal, or at least satisfactory, behavior is
found.
To summarize:
researchers use sophisticated analysis methods to solve the
wrong problems, and managers run their
plants with clumsy tools.
This state of
affairs may be due to the fact that neither managers nor manufacturing software
developers have the background to evaluate
the kinds of problems that arise in this
context.
Both tend to look at scheduling
as a da-ta processing problem, rather than
a problem which can be treated by methods
of dynamic programming.
These practioners
are not solely to blame for this; many of
us in the research community are guilty of
talking more than we listen and of solving
the problems we know how to solve rather
than those that need solving.
We propose an alternate approach to
factory management that is based on the
following fundamental ideas:
DISCIPLINE Specified operating rules are
required for complex systems.
Manufacturing, communication, transportation and
other large systems degenerate into chaos
when these rules are disregarded or when
the rules are inadequate.
In the manufacturing context, all participants must be
bound by the operating discipline.
This
includes the shop floor workers, who must
perform tasks when required; and managers,
who must not demand more than the system
can produce.
CAPACITY An important element in the
discipline of a system is its capacity.
Demands must be within capacity or excessive queuing will occur, leading to excessive costs, and possiby to reduced effective capacity.
We have developed a concept of capacity which is specific to
manufacturing systems.
HIERARCHY There are many time scales over
which planning and scheduling decisions
must be made.
The longest term decisions
involve capital expenditure or redeployment.
The shortest involve the times to
load individual parts, or even robot arm
trajectories.
While these decisions are
made separately, they are related.
In
particular, each long term decision presents an assignment to the next shorter
term decision-maker.
The decision must ;.e
made in a way that takes the resource.-ie, the capacity--explicitly into account.
The definition of the capacity depends on
the time scale.
For example, short-timescale capacity is a function of the set of
machines operational at any instant.
Long-time-scale capacity is an average of
short-time-scale capacity, taking machine
availability into account.
UNCERTAINTY All real systems are subject
to random disturbances.
The precise time
or extent of such disturbances may not be
known, but some statistical measures are
often available.
For a system to function
properly, some means must be found to
desensitize it to these phenomena.
In a
manufacturing system, machine failures,
operator absences, material shortages, and
random demands are examples of such uncertainties.
Desensitization to disturbances
is one of the functions of the operating
discipline.
In particular, the system's
capacity must be computed while taking
such disturbances into account, and the
discipline must restrict requirements to
within that capacity.
The kinds of disturbances that must be treated differ at
different levels of the time scale hierarchy:
at the shortest time scale, a machine failure influences which part is
loaded next; at the longest scale, economic trends and technological changes influence marketing decisions and thus capital investments.
FEEDBACK In order to make good decisions
under uncertainty, it is necessary to know
the current state of the system.
At the
shortest time scale, this includes the
conditions of the machines and the amount
of material already processed as compared
with demand.
Loading decisions are made
on the basis of this information.
t is
essential, especially at the short time
scale, that these decisions are calculated
quickly.
2. Flexible Manufacturing Systems
A flexible manufacturing system (FMS)
consists of several machines and associated storage elements, connected by an
automated materials handling system.
It
is controlled by a computer or a network
of computers.
The purpose of the flexibility and versatility of the configuration
is to meet production targets for a variety of part types in the face of disruptions such as demand variations and machine failures.
In an FMS, individual part processing
is practical because of the automated
transportation system and because the
setup or changeover time, the time required to change a machine from doing one
operation to doing another, is small in
comparison with operation times.
The
combination of these features enables the
FMS to rapidly redistribute its capacity
among different parts.
Thus, a properly
scheduled FMS can cope effectively with a
variety of dynamically changing situations.
All production systems are subject to
disruptive events ranging from sudden
changes in demand to machine failures.
These disruptions are inevitable and affect the productivity of the system.
Their times of occurence cannot be predicted in advance: at best, only an approximate historical record can provide guideA
lines on when they can be expected.
scheduling policy must provide for these
factors.
The purpose of the hierarchical
policy described in this paper is to efficiently use the available information and
system flexibility to anticipate and react
to machine failures.
3.Capacity
For similar reasons, defining and
respecting capacity are important at
all levels
of the hierarchy.
No
system can produce outside its capacity
and it is futile at best and damaging at
worst to try.
It is essential, first, to
determ:,ine what the capacity is and then to
develop a discipline for staying within
it.
Consider a set of
cess
T]
J
part types.
states.)
From feasibility considerations, the
parts can be processed with minimal internal inventories if
,,DI(T) + ji2D2(T) + ... + TJDJ(T)
All operations at machines take a
finite amount of time.
This implies that
the rate at which parts can be introduced
Into the system is limited.
Otherwise,
parts would be introduced into the system
faster *than they could be processed.
These parts would then be stored in buffers (or worse, in the transportation
system) while waiting for the machines to
become available, resulting in undesirably
large work-in-process and reduced effective capacity.
The effect
is
that
throughput (parts actually produced) drops
with increasing loading rate, when loading
rate is beyond capacity.
Thus, defining
the capacity of the system carefully is a
very important first
step for on-ln
scheduling.
sing
machine
Let
uI
the
time
J.
ETjU
l
j
The
of the
(4)
Figure I shows how the instaneous
capacity set
[(a) varies as the machine
state a changes.
I
1u2
i be
of
3
()
dem.ndrooe
-
-
2
type J must be processed at machine i
during a period of T seconds.
That is, an
4
2,
2 3 4
2 3 4
oa(222)
of
o.1122)
)
,2
be
achieved
during
this
period.
2
c
4
Let T, be the the time available at
machine i during the total time period T.
(T i
is
less
less because
Denote
chine i at
than
of
or
the
equal
failures
to
T.
that
u
It
the operational state of matime t by a i (t).
That is,
i
UZ
a-(212)
212)
2
,
'
I'
I,
412
02'1
3
u2 0I (112)
2
4
mu,
2
at
m
I if machine I is up at time t.
(More
valued for
machines.
(1)
generally, a, is
integera pooled group of identical
a is a vector of individual
3
(1121)
uZ
4
3
1e
")
2
a..,
I
234
(a,(i1l)
PARTS PER
0 if machine i is down at time t,
234
a- (221)
I-I
2
is
occur.)
()
u4
T 14
2
must
hierarchical
u(t) e ll(a(t))
the j'th part type at machine
. Assume that
Dj (T)
parts
Dj(T)
(3)
i
policy is to impose the following dscipline:
at all times t, choose u[t) to satisfy
4
rate
V i- and uj 2 0
S a,
key element
pro-
production
instantaneous
It
is
feasible
-
to
average
the
(2)
only if it is a member of the instantaneous capacity constraint set
I machines proces-
Let
be
; T1
PRTS PER
234
O,(21)
MINUTE
NUTE
Figure 1. Instantaneous Capacity Sets
- 3 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-
Let the av
be given by
ej
e
era ge
4.Introduction to the
Hierarchical Scheduling Policy
availability
(5)
T,
order for the average demand to
In
feasible over a long period T, (2) must be
This is equivalent to
satisfied.
(6)
d e D(e).
The average
ted in Figure 2.
capacity set is represen-
Part 2
(parts/minute)
The policy described here incorporates both machine status and demand deThat is, scheduling
viation feedback.
d e c s ons are d e t e r m i n e d o n - i n e
based on the current status of each machine in the system and the current difference between production production and
demand.
Figure 3 outlines the hierarchical structure of the policy which reflects
the discipline that must be imposed in
Parts are loaded
scheduling the FMS.
into the system at rates that are constrained to be within the current capacity, which is determined by the current
This preset of operational machines.
vents congestion from occurring.
5
SYSTEM
REQUIREMENTS
MACHINE PARAMETERS
CONFIGURATION
(OPERAT!ON TIMES,
4
MTBF, MTTR)
IGENERATE
3
TOP
DECISION
PARAMETERS
[2_
1
d
*'CALCULATE
2
3
4
5
II11
RATES
~~~.
MIDDLE
REQUIREMENTS
LEVEL
6 Part I
(parts/minute)
-L
|SCHEDULE TIMES
Figure 2.
LEVELOFF-LINE
UPDATES
_-0
1
DEIS
_ __ _.
Average Capacity Set
DISPATCH PARTS
PART
ATIONLOWER
ON-LINE
Note the similarity between (4) and
(6).
These two statements are written in
this way to emphasize their relationship;
They are two sets of capacity constraints
that
function at two different
time
scales.
In order for the required production to be achieved, the loading process
must satisfy the short term (instantaand the
neous) capacity constraint (4)
average production rate must satisfy (6).
The first constraint is the responsibility
of the workers and foremen on the floor or
of the on-line scheduling system; the
second is the responsibility of the managers.
Respecting capacity, then, is a
universal discipline that must be applied
in different
ways at all levels.
_ M_4
MACHINES AND
TRANSPORT
SYSTEM
STATUS
SYSTEM
Figure 3. Hierarchical Scheduling Policy
The middle level is the heart of the
scheduler.
It determines the short term
production rates, taking the capacity conof the system into account.
straints
Based on these rates the lower level determines the actual times at which parts
are loaded into the system.
The middle
level uses machine status information and
demand deviation for its computations.
It
also needs certain longer term information.
This is supplied by the higher
level, which computes it from machine data
such as failure and repair rate information, and part data such as operation
times and demand.
S.The IBM Automated
Card Assembly Line
At IBM's General Products Division at
Tucson, an automated card assembly line is
being built up in stages, through a series
of "minilines." The portion of the system
of interest to us is the stage consisting
of insertion machines.
Printed circuit
cards from a storage area upstream arrive
at the loading area of the insertion
stage.
Each card is placed in a workholder and it is then introduced into the
system.
These workholders move through
the
system
from
machine
to
machine
along
A SPI
SIP2
'Q i
-
..
(
I...
E E,[-
Th'.,
.
-1)[1
,
8
.
-
[D8
F
.
l.
[fig()'"-
UL~W'
()-"
,7S
/
[
IAC.IN.E
transportation elements which are controlled by a hierarchy of computers and micro-
VCD
RECTILINEARCONVEYOR
ROTARYI
CONVEYOR
TRANSPORTATON
ELEMENT
TYPE-.-- EFERENCE
ORIENTATON
OFELEMENTS
WBUFFER
.
processors.
At each of these machines electronic
components are inserted into the card.
Each type of card goes to a specific set
of machines.
The processing time of each
card at any machine depends on the number
and type of components that are inserted.
If a machine is busy or otherwise unavailable, the workholders are stored in a
buffer near the machine.
Finally the
workholders exit the system and go to the
downstream stages, which consists of testing and soldering machines.
There are several types of insertion
machines, each of which inserts one mechanically distinct type of component.
The
common ones are SIPs (Single In-line Package Inserter), DIPs (Dual In-line Package
Inserter), MODIs (Multiform Modular Inserter) and VCDs (Variable Center Distance
Inserter).
By loading diff erent components, the line can be used to assemble a
variety of cards.
In order to concentrate on the operational issues of the FMS, we assume that
component loading has already been determined.
The changeover time is small amongT
the family of parts producible with a
given component loading.
We also restrict
our attention to the miniline whose schematic is shown in Figure 4.
This consists
of a DIP, a VCD and two SIPs.
Each of the
machines also has an associated buffer,
which can hold 30 parts.
Figure 4. Card Assembly Line
6.Scheduling Objectives
An FMS is normally only one stage of
a production process, with other stages
preceding and following.
This necessitates co-ordinated production scheduling.
The schedule must determine the part types
and the number of each type to be produced
by the FMS over a period of several days.
The objective of the short term schedule
is to track demand over the course of each
day so as to meet the production targets
set by the long term schedule.
The production target is specified
for each j as Dj(T) parts of type j to
be made by time T, the production period.
The cumulative production WJ(t) is the
total amount of material of type j actually produced by time t.
The cumulative
production must equal the total demand at
to
ensure
that
W (T)
is
is
equaI
equal
to
to
DJ ( T )
The production
PJ '
is
of
percentage,
defined as
WE(T)
D (T) x 100%, for all j
primary
_~~Tis
importance.
is
_h
(7)
This
is
the
production of type j parts expressed as a
percentage of total demand for type j.
The closer this measure is to 100I7, the
better the algorithm is judged to be.
H e d g i n g
Also of interest is the average workin-process, i.e., the average number
of parts of each type present in the sysThe smaller the WIP, the better thetem.
algorithm.
total number of parts of type J produced
and the total number of parts required:
Finally, to compare various control
policies, it is necessary to aggregate the
performance measures by part type, into
They are
total performance measures.
total production percentage
iW
p-_
(8)
x 1QOo
xj(t)
At time t, the production surplus
between the
is
the difference
(
Dt
Figure 5 illustrates the cumulative
demand Dj[t) being tracked by the cumuOur objective
lative production W(t).
is to meet production targets as closely
as possible at the end of time period T,
or, equivalently, to keep xj(T) close to
We also assume that it is desirable
zero.
to keep xl(t) small, for t < T.
. Dj
J
work-in-process.
average
total
and
To measure the distribution of production between the various part types, we
as
balance
define
mn
CUMULATIVE PRODUCTION
DEMAND
-CUMULATIVE
ZC3
Pj
(9)]
x 100
B 5
---
<
2
Z
wO
max PJ
.
This is the ratio of the worst production
percentage to the best percentage.
0
Let T 1(used) be the time that machine i processes parts, during the time
Machine
is operational.
Tt that it
S[Z
is then given by
utilization
1 0
(
/Ai
LL>
//
-a
D
m Z
Tl(used) x
1001
)=T 14
If
is
an
with
this
ratio
efficient
very
little
is
use
idle
(10)
close
of
to
100X,
there
TIME
system resources,
time.
7.The Hierarchical Scheduling Policy
The objective of the hierarchical
scheduler is to meet production targets as
This is
to be
closely as possible.
achieved in the presence of machine faiFor efficient production, congeslures.
tion in the transportation system and in
The
internal buffers must be minimized.
hierarchical policy ensures this by respecting the system capacity constraints.
The loss of production due to machine
failures is compensated for by hedging,
We
that is, by building up safety stock.
discuss these important concepts in detail
below.
Figure 5. Production Tracking Demand
Keeping the production surplus xj
small is an effective way of tracking
However, failures result in a
demand.
One
shortfall in production capacity.
compensates by building up safety stocks
by overproducing
point.
when possible,
up to a
Thus, rather than maintaining x(jt)
at a value near zero for all t, it is
reasonable to maintain it near a level
(
O.
This is only possible if
Hick)
the
machine
state
a
is
feasible,
i.e.
L
if
d e i(a)
We
call
If
nents
Hi(a)
hedging
point.
(12) is not satisfied, some compoof x(t) must decline over time.
The scheduler is divided into three
levels, as shown in Figure 3.
The top
level generates the decision parameters ofe
the policy.
These include the hedging
The
points Hj(a) and other quantities.
repair and failure time data (i.e.the
MTBF and MTTR) of the machines and the
demand rate and processing times for each
part type are required for this calculation.
This top level is intended for offline computation.
It is designed to be
called just once, at the start of a pro-'
duction run.
However, if failure or repair rates change, it can be called to
update the decision parameters.
When there is a change in machine
state, i.e., when either a machine fails
or is repaired, the middle level is called
to compute the new values of the short
term production rates.
It takes the capacity constraints of the system into account and uses the off-line parameters
supplied by the higher level.
The resulting production surplus or buffer state
trajectory is also computed.
At the lowest level, parts are loaded into the
system so as to follow the buffer state
trajectory computed at the middle level as
faithfully as possible.
M id d l e
L e v el
At this level, the current production
rate uj(t) of each part type is determined for current machine state a(t) and
current production surplus x(t).
The
objective is to compute the production
rates such that x approaches and then
remains equal to H(a) whenever enough
capacity is present.
At that time, the
production rate uj is set equal to the
demand rate dj.
If too many machines
are unavailable for that, the scheduler
choses from among the available production
rates a set of rates to control the manner
in which the production surplus declines
and becomes a backlog.
P r or g
minimize c1ul + c2 u2 +
(12)
the
n e a r
subject to
u e
... +
am
(13)
CjUJ
l(a)
The cost coefficients of the linear
program, which are functions of production
surplus x, are given by
c (x
- Aj(a) (xj - Hl(a)])
(14)
at the higher level.
A1 (a)
is a positive quantity that reflects the relative
value and vulnerability of each part type.
Consequently, coefficients cj are negative when type j is behind and are more
negative for more valuable or vulnerable
parts.
Production rates generated according
to this program automatically satisfy the
instantaneous capacity constraints.
This
linear program is not hard to solve online since the number of constraints and
unknowns is not large.
by
The
(11).
x(t)
production surplus x(t) is
It is approximately
J
given
t
[uft) - d(t)l dt
(15)
since the function of the lower level is
to keep the actual production rate close
to the value calculated here.
As x(t) changes, the coefficients of
linear program change.
In principle, it
is necessary to solve the linear program
at every time instant.
This leads to
undesirable "chattering" behavior and unnecessary computation.
Recent research
has found a simple technique for eliminating much of the computation and all of
the chattering.
The system operates on a random
cycle:
when the machine state a is feasible (i.e.
when (12) is
satisfied),
the production surplus x approaches H(a)
and then stays there.
The production rate
then equals d.
When a machine fails so
that the machine state is not feasible, x
moves away from H and eventually some or
all componennts may become negative.
Hi g h e r
These desirable characteristics are
the result of choosing the production
rates as the solution to the following
linear programming problem.
Level
The purpose
-7-
~~~~~'-----~~~~~~~~~~~---~~~~~~~-I-c~~~~~~~~
-I-----~~~~~~~~~~~~~~~~~~~~~~~--~'-"I
of
the top level of
the
parameters to the middle level.
From an
analysis of the random cycles described
above the value of Hi can be chosen,
based on MTBF, MTTR, the penalties for
being ahead or behind, and the operation
times,to keep xl small.
The coefficients
Aj a)
can be
computed from the number of machines that
type j parts visit and the relative value
of
The part
more . machinto
of part J.
The
more
mvulnerable that part
type visits, the more vulnerable that part
type is to failures.
Also, the smaller
the more
failures, the
between failures,
time between
the mean time
the vulnerability.
To simplify our analysis, we assumed that the mean times between failures of all the machines are the
same and the values of all the parts are
all the same.
In our experiments, we
chose Ai(a) to be number of machines
that type j parts visited and then varied
it .
vaIThe present methodjs ifor selecting
values for A and H are simple and incomplete, but, as the simulations show, they
work very well.
Lo w e r
L e v e I
The lower level has the function of
dispatching parts into the system in a way
that agrees with flow rates calculated at
the middle level.
The middle level of the
scheduler calculates the projected trajectory, xP(t), the best possible future
behavior of x(t) if no repairs or failures
would occur for a long time.
The lower level treats the projected
trajectory xP(t) as the value that the
actual production surplus xA(t) (11)
should be close to.
A part of type j is
loaded into the system whenever the actual
production surplus xjA(t) is less than
its
projected
value
XJP (t)t
Wh e n
there is a machine state change, a new
projected trajectory is calculated starting at the time of the change, and the
same loading process continues with the
new .trajectory.
8.Alternative Policies
In this section we discuss three
simpler policies.
All of them limit the
number of parts in the system.
The differences lie in the amount of information
they use about system status and how they
use this information.
The important differences between the
hierarchical policy and those described in
this section are:
1.
These policies are not explicitly
based on satisfying the capacity constraints although they limit the number
Consequently,
of parts in the system.
there is congestion in the system.
2.
They require
tuning
to perform
well.
3.
3. The
The policies
policies are
are not
not hierarchical
hierarchical and
and
do not separate the scheduling problem
a set of time
problems
characteristic
scales. with
As different
a consedifficult to analyze
quence, they are difficult to analyze
difficult to preand their performance
simulation.
This policy loads a part whose type
is furthest behind or least ahead of cumulative demand.
A limit N is set on the total number
of parts in the system in order to avoid
filling up the buffers and transportation
system.
work verybuffers
Also,
well.
upstream and downstream
of the FMS may be have limited capacities,
or the cost of extra inventory may be
high.
Thus even if production is ahead of
on excess
is
set
demand, a limit
production.
P o I i c y
Y
Policy Y is the same as
except that there is a separate
Nj for each part type.
P o I 1c y
policy X
threshold
Z
This differs from Policy Y in that
when machine fails, the flow rate of parts
going to it should be set to zero.
Equivalently, the limit NJ is set to zero
when a machinen that type j goes to fails.
9.Simulation Results
To test the hierarchical policy and
to compare it with Policies X, Y, and Z,
simulation experiments were performed in
which the system was heavily loaded.
That
is, machines had to be used for a large
percentage of the time they were operational to satisfy demand.
This is the only
situation in which it is meaningful to
Under lighter loading
compare policies.
conditions, any strategy may be effective.
However, light loading is not generally
realistic; the cost of capital equipment
is such that managers will need to get the
most
these
'ving
effectiveness
ture.
from an FMS.
they can
vs
Hierarchical
Policy
X
Our runs correspond to an 8-hour
examine the
We first
production shift.
policy
performance of the hierarchical
during a given run, with different values
This is
of the hedging and A parameters.
compared with the performance of Policy X
of the threshold
values
for different
The highlimit N on parts in the system.
lights of the performance are summarized
in Figures 6 and 7.
a plot of total producFigure 6 is
tion percentage versus in-process-inventory, for different parameter values of the
The reference values of
two strategies.
Hj were
points
Aj's and hedging
the
They
chosen as described in Section 7.
were then varied.
objectives
of the
demonstrates the
hierarchical struc-
The points corresponding to different
parameters are clustered close together.
This shows robustness to parameter perturThis is noteworthy because the
bations.
parameters are computed from demand, machine, and part type data, which are not
Any strategy not
always known accurately.
is
preferred.
to these
unduly sensitive
This is a very important characteristic.
policy's
the simpler
contrast,
In
results are more scattered and correspond
to a combination of higher WIP and lower
production percentage.
The hierarchical policy and Policy X
are compared with respect to balance and
The
production percentage in Figure 7.
total production percentage of the hierarchical policy is uniformly high and it is
robust with respect to variations in the
, hedging point and Al parameters.
100 HIERARCHICAL
O 0
x POLICY x
U
x
0
0
0M0
cra:
X
HIERARCHICAL
lOG
~
z z
xPOLICYx
<tLX
L)i
·* I
z
90 _
z
u
x
a.80
"o
u "U
80
1j
_j
Figure 6.
I
I , I
I
22
20
18
16
14
IN-PROCESS INVENTORY
I
,
010
12
80
24
7
100
TOTAL PERCENT PRODUCED
Production Percentage and WIP
All the points corresponding to the
s chedule r lie In t h e u ppe r
hierarchical
graph in Figure 6.
of the
region
left
productioncy
a high total
This indicates
percentage, and a low WIP (work-in-process). Both high production percentage and
low WIP are highly desirable, as we IndiSimultaneously achlecated in Section 6.
Figure 7.
Production Percentage and Balance
X has lower balance, producpercentage, and machine utilization,
percentage, and machine utilization,
as well as greater sensitivity
parameters (N) than the hierarchical
ting
policy.
tion
-7-.
Comparison
With
Different
Seeds
The same type of comparison is conducted between the hierarchical policy and
Policy X, but for a set of different seeds
Each seed
of the random number generator.
corresponds to a sequence of machine fallures
and repairs.
That is,
each seed
The same value
represents a unique day.
of N (16)
Is used with each seed.
The
hierarchical policy is
run with the same
set of seeds.
The results,
shown in Figures 8 and 9 and are essentially similar
to those seen In the previous sub-section.
The hierarchical policy achieves higher
production percentages with lower WIP and
better
balance.
.
BALANCE
D
MIN PRODUCED PERCENT
MAX PRODUCED PERCENT
90
80
70
60
. HIERARCHICAL
x POLICY X
50-
D SEED 123457
a SEED 987654
o SEED 320957
40
0
The
sA
70 65
O 10
1
1f
,1
,I
12
i
I
13
l00
performance
of
Y
the
A
II
14
15
and
,
Z
hierarchical
values of the
compared with
Figures 10 and 11 show the
tive performances
of all four
The hierarchical
strategy
has
performance.
It
is better than
which is better than Y, which, in
better than X.
A
,,
With. Policies,
policy with the reference
hedging parameters is also
. that of Policies Y and Z.
I-
I-
I ,,
I X*, II , ,
1
70
80
90
TOTAL PERCENT PRODUCED
Comparison
o SEED 320957
o SE 3
.80BO.J
0
,
Figure 9. Production Percentage and Balance
-- Different
Seeds
x POLICY X
c SEED 123457
SEED 123987654
:a100
u:
a.
90
so
Z
i. I
65
comparapolicies.
the best
Policy Z,
turn, is
16
IN-PROCESS INVENTORY
Figure 8. Production Percentage and WIP
Seeds
-- Different
There is a particularly great
difference between the performances of the
hierarchical and Policy X on certain days.
The performance of the simpler policy is
more variable.
I.e., less
predictable,
from day to day.
Thus, even if a policy is tuned carefully for a given run, Its performance is
not guaranteed to be good in runs with
other seeds.
This shows the impracticality of parameter tuning.
!
This order is a direct result
of the
more effective use of information.
Policy
between part
X does not differentiate
types and does not make use of machine
repair
state
information.
It
performs
poorly in terms of all measures.
Policy Y
does much better in terms of average WIP
and total production percentage by differentlatlng
among part types.
Policy Z
also makes use of machine state and so has
The implilower WIP and higher balance.
cation is that effective feedback based on
more information results in better performance.
The series of policies culminates
in the hierarchical policy, whose sophisticated information usage helps it achieve
superior performance.
10.Conclusions
From the simulation results, we conclude that a hierarchically structured
policy described here and elsewhere (Kimemia, 1982: Kimemia and Gershwin, 1983;
Gershwin, Akella, and Choong. 1984; Akella, Choong, and Gershwin, 1984) can be
very effective in scheduling a FMS.
It
can achieve high output with low WIP in
the presence of machine failures.
Further
research is required to incorporate other
kinds of uncertainties and disturbances In
the hierarchical structure.
D HIERARCHICAL
x POLICY X
I-.
0 POLICY X
o POLICY Y
a POLICY
POLCY ZZ
100
The success of the policy is a result
of using feedback and adhering to the
discipline of respecting system capacity
constraints.
Capacity limits are not just
observed in the long run; they are considered as each part is considered for loading into the system.
All relevant machine and system status information is
fully utilized.
cum:
uJ
Hi
@
85o
t-: 90-I
D
x
X
85
0 12
85
1I.]}
Is
i
I5.
13
14
15
IN-PROCESS INVENTORY
I
16
=
This approach
is
robust
so
that
for a
wide range of policy parameters it works
very well.
This obviates the need for
precise machine and part data which may
It also eliminot always be available.
nates the need to use time consuming (and
thus infeasible) trial runs.
Further
research is needed in choosing hedging and
Aj parameters for larger systems.
The
grouping of parts into families when there
are a large number of part types Is another research issue.
Figure 10. Production Percentage and WIP
-- All Strategies
A variety of new problems arise when
M
BALANCENT'~l...L
PR
we
O
MIN PERCENT PRODUCTION
MAX PERCENT PRODUCTION
0 HIERARCHICAL
a SOPHISTICATED
100
0 LESS SOPHISTICATED
-xCOMMON
SENSE
El
95A
X
explicitly
consider
the
scheduling
of
an FMS in the context of a factory.
The
FMS is then one of the stages of the
automated production system.
It is supplied with raw material by an upstream
s100
tage.
It must supply the stage which is
downstream from it.
Co-ordinated production between the different stages becomes
a necessity.
90
85
80
0' 85
90
95
100
PRODUCTION PERCENT
Figure 11. Production Percentage and Balance
Acknowledgments
PRODUCTION
PERCENT
This research was supported by the
Manufacturing Research Center of the Thomas J.
Watson Research Laboratory of the
International Busfness Corporation and by
the U.S. Army Human Engineering Laboratory
under contract DAAK-11-82-K-0018.
References
R.
Akella, J.
P.
Bevans, and Y. Choong
(1984), "Simulation of a Flexible Manufacturing System," Massachusetts Institute of
Technology Laboratory for Information and
Decision Systems Report. To appear.
R.
Akella, Y. Choong, and S. B. Gershwin
(1984), "Performance of Hierarchical Production Scheduling Policy," Massachusetts
Institute of Technology Laboratory for
Information and Decision Systems Report
LIDS-FR-1357.
S.
B.
Gershwin, R. Akella, and Y. Choong
(1984), "Short Term Scheduling of an Automated Manufacturing Facility," Massachusetts Institute of Technology Laboratory
for Information and Decision Systems Report LIDS-FR-1356.
J.
G.
Kimemia (1982), "Hierarchical Control of Production In Flexible Manufacturing Systems," Ph.D. Thesis, Massachusetts
Institute of Technology Laboratory for
Information and Decision Systems Report
LIDS-TH-1215.
J.
G.
Kimemia and S. B. Gershwin (1983),
"An Algorithm for the Computer Control of
Production in Flexible Manufacturing
Systems, IIE Transactions,
Volume
15, No.4, December, 1983, pp.353-362.
-.,~~~~~~~~~~~~~~~~~
7--