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DEWEY
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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Slack-based Lower
for
Bound and
Heuristic
Permutation Flowshop Models
Anantaram Balakrishnan,
Srimathy Gopalakrishnan,
and Atsushi Kurebayashi
Sloan
WP# 3573-93
June, 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Slack-based Lower
for
Permutation
Bound and
Heuristic
Rowshop Models
Anantaram Balakrishnan,
Srimathy Gopalakrishnan,
and Atsushi Kurebayashi
Sloan
WP# 3573-93
June, 1993
MASSACHUSETTS INSTITUTE
OF TFrHWQi OGY
NOV
1
4 2000
LIBRARIES
Slack-based Lower Bound and Heuristic
for Permutation Flowshop Models^
Anantaram Balakrishnan
Sloan SchcxDl of Management
M.
I.
T.,
Cambridge,
MA 02139
Srimathy Gopalakrishnan
Operations Research Center
M.
I.
T..
Cambridge,
MA 02139
Atsushi Kurebayashi
Nippondenso Co.
Ltd.
Kariya-city, Japan
ABSTRACT
This paper addresses optimal job sequencing decisions for various classes of permutation
We first describe a framework to classify flowshop
flowshops.
the level of intermediate storage, job transfer
the interrelationships
bound
mechanism, and objective function.
heuristic to generate effective job sequences.
locally
a trend,
is
and the assignment-patching
effective
results
show
that,
compared
when job processing times
heuristic performs well for
all
Supported
in part
ai^e
within
10% from
by a research grant from
are
models.
improving the heuristic solution using a two-exchange procedure, we are able
generate solutions that
'
discuss
and describe an assignment-patching
Our computational
previous approaches, the slack-based lower bound
By
We
between various flowshop models, develop a new slack-based lower
for the total processing time of each machine,
random or have
scheduling problems based on
optimal for most scenarios.
MITs Leaders for Manufacturing program
to
to
Introduction
1.
Flowshop scheduling involves sequencing
that follow the
same processing
flowshop scheduling
a given set of jobs (or products, items, batches)
literature cites
many manufacturing and
service applications such as
sequencing programs on a computer system, managing operations
at
chemical and other
continuous processing plants, and assembhng consumer durable products on an assembly
Flowshops are inherently easier
The
route through successive stages (or operations, machines).
to
manage compared
line.
to job shops; they offer advantages of
lower inventories, quick response, and rapid feedback for process control and improvement
Consequendy, companies with complex manufacturing operations seek
to streamline their
product flows by creating "factories within the factory" (Skinner [1974]) using, say, group
technology methods;
this rationalization exercise often
network of embedded flowshop operations. The
transforms a huge job shop into a
accentuated this trend. For instance, several electronics companies
assembly lines
that
can each assemble
With more than two machines,
jobs in a permutation flowshop
Kan
is
the
many
machines has
availability of flexible
now
operate mixed-model
different circuit boards in small lot sizes.
problem of minimizing the makespan for
known
to be
NP -complete
(Lawler, Lenstra, and Rinnoo\
[1982]). Consequently, research has focussed on determining
developing exact (enumerative) and heuristic algorithms
(see, for
good lower bounds, and
example,
McMahon
Burton [1967], Campbell, Dudek and Smith [1970], Reddi and
Ramamoonhy
[1975], Dannenbring [1977], Lageweg, Lenstra, and Rinnooy
Kan
[1980], Szwarc [1983],
Nawaz, Enscore, and
Ham
and
[1972], Baker
[1978], King and Spachis
[1983], Park, Pegden and Enscore
Hundal and Rajagopal [1988], and Rajendran and Chaudhari
literature deals
a given set of
[1990]).
The
1
19841.
vast majority of this
with the problem of miniming makespan for a permutation flowshop with
infinite intermediate storage capacity
(hence, machines are never blocked).
and unrestricted or
free flow of jobs
We refer to this model
between machines
as the "standard"
flowshop
model. Other models covered
be processed without waiting
in the literature
include the no-wait case in which each job must
intermediate operations
at
(e.g.,
Reddi and
Ramamoonhy
Szwarc [1983]), and systems with limited or no intermediate storage between stages
Leisten [1990]).
environments
Some
(e.g.,
recent
McCormick
This paper develops and
We
scheduling models.
work has addressed optimization models
problems based on the
lower bounds and heuristics for several different flowshop
describe (in Section 2) a scheme for classifying flowshop
We
between these eight models and explore
programming formulations. Section
their interrelationships,
3 develops a
the slack or forced idle lime of
minimum makespan
mechanism,
consider four different intermediate storage-job transfer
combinations, and two objective function s-makespan and cycle time.
examines
(e.g.,
for cyclic scheduling
level of intermediate storage capacity, type of job transfer
and optimization objective.
1972|,
et al. [1989]).
tests
first
f
To clarify
we examine
the distinction
their integer
new slack-based lower bounding method
each machine
to
compute a lower bound on
that
the
or cycle time. Section 4 describes an assignment-patching heuristic
algorithm, motivated by principles underlying no- wait scheduling, that solves a traveling
salesman subproblem using estimates of the slack time between successive jobs.
we
In
Section
5.
apply the lower bounding and heuristic method to different flowshop models, and compare
their
performance with previous approaches
method
for the standard nxxiel).
When
(e.g.,
Lageweg
et al.'s
the job processing times are
increasing or decreasing trend, our slack-based lower bounding
models; the bound
is
random or have an
method performs well
for
all
not as effective for problem instances with correlated processing times
The assignment-patching
heuristic
[1970] heuristic algorithm for
local
[1978] lower bounding
improvement procedure,
all
compares favorably with Campbell, Dudek and Smith's
models and processing time
this heuristic
distributions.
Augmented with
appears to be a versatile and effective method to
generate near-optimal flowshop schedules for different contexts.
a
)
2.
Permutation flowshop scheduling: Alternative models and
their interrelationships
Given n available jobs and
sequencing the n jobs
to
m machines, the flowshop scheduling problem
optimize some criterion of
interest.
All jobs have the
sequence but different processing times, machines can process only one job
cannot undergo simultaneous processing
every machine processes the jobs
in a certain
in the
sequence, they follow the
that the schedule
at
order. Thus, after
we
common
a classification
i.e.,
machines are not
is
empty
commencing operations on
assume
idle unless
(i.e..
machines
the next job.
the optimization objective might vary.
flowshop scheduling models discussed
scheme
We
context, the scheduling problem might include certain constraints
on material storage and movement, and
between the
and jobs
release jobs into the system
they are "blocked" by downstream machines, or the upstream queue
Depending on the application
at a time,
discipline at all intermediate queues.
does not contain any "unforced" idleness,
are "starved"), or timing constraints prevent
same processing
multiple machines. In permutation flowshops.
same
FCFS
involves
that is
analagous to
Graham
et al.'s
in the literature,
To
we
distinguish
flrsi
descnbe
(1979) framework to classify
general machine scheduling problems based on the machine environment, job characteristics.
and optimality
criterion.
We
illustrate the differences
between the various flowshop models
in
terms of their integer programming formulations, and relate their respective upper and lower
bounds based on these formulation
differences.
We also introduce
a
new
objective function,
cycle time minimization, that applies to flowshops with cyclic scheduling policies.
2.1
Flowshop model
classification
We classify flowshop scheduhng models based on three problem charactenstics:
(
1
intermediate storage capacity, (2) job transfer mechanism, and (3) optimization objective.
Intermediate storage capacity
2.1.1
Intermediate storage capacity refers to the
maximum number of jobs
that
an intermediate
queue between two successive stages can accommodate. The standard flowshop scheduling
model (addressed by Johnson [1954] and
assuming
others) ignores such
that intermediate storage capacity
is
infinite (or
>
queue
limits, effectively
We denote
(n-1)).
problems with
unlimited intermediate storage capacity as IIS (infinite intermediate storage) problems. In
applications, upstream operations can be blocked
capacity, and so
we must
due
to inadequate
downstream storage
explicitly account for this capacity constraint in the scheduling model.
For instance, the length of the conveyor segment connecting adjacent of)erations
electronics assembly line limits the
number of boards waiting
We refer to tightly coupled flowshops with limited
FIS
(finite
intermediate storage) systems.
storage capacities are zero as the
that
we
some
consider in this paper,
NIS
we can
NIS problem instance by introducing
to be
processed
at
an
in
each
station.
(< (n-1)) intermediate storage capacity as
We distinguish
the special case
when
all
intermediate
(no intermediate storage) case. For the flowshop models
transform any FIS problem instance into an equivalent
dummy
intermediate operations with zero processing
times, one corresponding to each intermediate storage location. Therefore, in our subsequent
discussions,
we
consider only two options for intermediate storage: IIS and NIS.
Job transfer mechanism
2.1.2
The second flowshop
characteristic, yo^ transfer
mechanism,
associated with nx)ving jobs to ot from each machine.
refers to the timing constraints
We consider three types of job transfer
mechanisms: /ree/7ow (FR), no wait (NW), and synchronized (SYN). The
FR
case
corresponds to unrestricted transfer of jobs from machines except due to downstream storage
capacity constraints.
this
job
idle),
if
and
machine
the
When
a
machine finishes processing a job, we can immediately unload
downstream queue
start
is
not full (or, in the
processing the next available job
until the
in
downstream machine completes
4-
NIS
case,
if
the
downstream machine
queue; otherwise, the job waits
its
processing.
The FR
in the
discipline often
is
applies to flowshops with asynchronous materials handling systems that use pallets to transfer
products between machines. The standard flowshop scheduling model assumes an IIS system
with
FR job
The
transfers.
NW case represents the most restrictive job transfer mechanism.
once a job
is
staned
(i.e.,
any intermediate waiting
released for the
(at
first
processing step)
it
Under
scheme.
this
must be completed without
intermediate machines or in storage); hence,
NW systems do not
have intermediate storage capacity. Hot roUing operations provide one example of NW'
scheduling. In this context, the flowshop
is
a series of hot rolling mills connected by
conveyors, and each incoming job or workpiece
is
a sohd, rectangular metal ingot.
The
successive rolUng mills progressively reduce the thickness and elongate the workpiece: the
finished products are plates or sheets with varying thicknesses and lengths.
time
at
The processing
each rolUng mill varies by job, depending on the starting and finished dimensions of the
workpiece. Each ingot
elevated temperature
to the next without
staggered ensure
The
is
preheated before
is critical
it
is
released for processing; since maintaining the
for hot rolling, the
workpiece must proceed from one rolling
any interruptions. Consequently, the release of ingots
to the line
mill
must be
NW processing.
third job transfer
mechanism, synchronized flow,
is
common
in
assembly systems
that
use a single conveyor belt with equally pitched fixtures to transpon workpieces from one station
to the next. In the
SYN
system, the process
coordinated across machines,
at the
same
instant of time.
until all other
machines
i.e., all
When
a
start
times (and job unloading times) are
machines stan processing the next available job
machine completes
its
operations on a job,
finish processing their current jobs; the
unload the current job, and load the job
Since having infinite storage capacity
at
(if
it
queue
holds this job
conveyor then moves foi^vard
available) in the next position
on
any stage effectively decouples the
to
the conveyor.
line at that stage,
consider only the NIS case (and hence the FIS case using our transformanon) with
-5
in
SYN
we
tlov.
2.1.3
Optimization objective
the job sequence that minimizes
The standard flowshop scheduling model seeks
i.e.,
the last machine.
on
the completion time of the last job in the sequence
Some
makespan.
researchers
have considered alternative performance measures such as minimizing the weighted sum of job
completion times, or reducing lateness or tardiness (with respect to given job due dates).
For periodic or cyclic scheduling environments
intervals, the interval time for
start
interval time
machine
is
of the
the
first
last
interval time over all machines.
the length of the scheduling cycle
job on that machine. Thus,
(i.e.,
To
The
cycle time for a group of n jobs
proponional
all
of
its
we
focus on the
M and C, respectively.
is
relevant
interval time, for
other machines after a machine completes
discussions,
the
the time between the release of successive batches) must
machines. This objective
to the length
is
achieve regular (repeating) processing schedules.
be greater than or equal to the cycle time. Another objective function of interest
interval time of
time
processing times on the machine plus any forced idle time due to
starvation, blocking, or timing constraints.
maximum
at regular
We define the interval time of a machine as the elapsed
job and the completion of the
sum of job
same schedule
each machine and the cycle time for each batch of n jobs are
important pjerformance metrics.
between the
that repeat the
all its
when
example,
if
processing
minimum makespan and
Our observations concerning
is
the total
the cost of operating a
machine
is
operators can be reassigned to
in a cycle.
In
our subsequent
cycle time objectives which
we denote
as
cycle time also apply to the total interval
time objective.
We can
use these three problem characteristics-intermediate storage capacity, job transfer
mechanism, and optimization objective-to distinguish between various flowshop scheduling
contexts. Thus,
NIS/FR/M
no intermediate storage but
refers to the
free
problem of minimizing makespan
flow job transfers. As
we have
in a
indicated, not
flowshop with
all
combinations
of options are compatible or
practical; for instance,
infinite intermediate storage is
ingore the
nS/SYN
IIS/NW/M
is
not meaningful since having
unnecessary when jobs are not permined to wail. Similarly, we
combination.
We will consider four combinations of intermediate
storage
and job transfer characterisiics-IIS/FR. NIS/FR, NIS/NW, and NTS/S YN-with each of
M and C.
objective functions
cycle,
i.e.,
we
For the NIS/SYN/C nxxiel,
many
consider only a single scheduling
of the flowshop scheduling models discussed
not necessarily exhaustive. For instance,
literarare, but is
different batch sizes for processing
and intermachine
we have
In this section,
not considered systems with
transfers, or batch processing stations
we
and bounds
first
for different
mixed
present a
models
integer formulation for the standard model, and
discuss enhancements needed to incorporate intermediate storage capacity limitations
NW and SYN flow, and cycle time minimization. We are given n jobs,
require processing at machines k
Pjj^ at
in the
heat treatment ovens) that can simultaneously process multiple jobs.
22. Formulations
NIS),
machine
for the standard
k, for all
j
=
=
1, ..., n,
m in
1
and k =
j
=
1
(i.e..
.
.
n. that
sequence. Job j requires a processing time of
1, ...,m.
The
integer
programming formulation
flowshop scheduling problem uses the following three
sets
of decision
variables:
if we assign job
otherwise;
|0
=1^
X
'J
2.2.1
two
ignore the synchronization requirements to dovetail adjacent cycles.
This classification scheme covers
(e.g.,
we
the
Tj^
=
starting time of the
S^^
=
processing time of the
i
j
to the
job
i'*^
i'^
in the
job
position in the sequence, and
sequence on machine
in the
k;
and,
sequence on machine
k.
Minimum makespan models
We first formulate the standard model, and then indicate how
makespan models
differ
from the standard formulation.
-7-
the other three
minimum
The Standard model: IIS/FR/M
minimize
subject
^nm
"*
(2.1)
^nm
to:
Sequencing constraint:
for all
j
=
1, ...,
n.
[2.2)
for all
i
=
1, .... n,
(2.3)
for
k =
1=1
n
Processing rime equations:
all
1, ...,
m,
1,
m-1,
i
=
1,
....
(2.4)
n.
J=l
Job
starting time constraints:
^i(k+l)
Machine starting time
"^(i+Dk
^
"^
"^ik
for all k
^ik
=
...,
i
=
n.
(2.5)
n -1,
(2.6)
1
constraints:
^
"^
^ik
^ik
for
all
k =
for
all
i,
j
i
=
m,
1
i
=
1
Non- negativity and integrality constraints:
Xij
€
{0,1}
for all
The objective function
n'*^
job
in
every job
sequence on the
is
(2.1)
last
=
1, ..., n,
1, ...,n,
minimizes the makespan, which
and
k =
is
1
we
j
in the
to the i'^ position in the
equal to the processing time of job
downstream machine (k+1) can
j
start
i'^
that
sequence, and each position contains only one job.
sequence, then
on machine
k.
processing the
job, while constraint (2.6) ensures that
finishes the
(2.7h)
machine. The sequencing constraints (2.2) and (2.3) ensure
assigned to one position
assign job
m.
the completion time for the
Equations (2.4) determine the appropriate processing time for each job
If
(2.7a)
X- =
1
-8
chosen sequence.
and equation
(2.4) sets
S|^,
Constraint (2.5) specifies that the
i'*'
job only after machine k completes
machine k stans processing the
job.
in the
(i+1)^'
this
job only after
it
The NIS models:
With
soon as
infinite intermediate storage
its
op>erations at that
machine k immediately
However,
the
i'*'
for the
must wait
and k can then
NIS
in
machine
after the
i'^
job
is
unloaded from machine k as
are completed. Therefore, the (i+1
job
i'*'
free flow, the
is
)^'
job can
completed; constraint (2.6) expresses
start
on
this condition.
case, since the system does not have storage capacity between machines.
machine k
start
and
machine (k+1) completes the (i-1)^ job; machines (k+1
until
processing the
i'*^
and
(i+1)^' jobs.
)
The following machine blocking
constraint imposes this requirement.
Blocking constraints:
^
Vl)k
Adding
this set
forallk=l
T,(^^l)
of constraints to the
nS/FR/M
i=l
m-1,
n-1.
(2.8)
formulation gives the formulation for the
NIS/FR/M model. The NIS/NW/M and NIS/SYN/M models
that
we
discuss next also
contain these constraints.
No-wait and Synchronized flow models with NIS:
In addition to the blocking constraints (2.8), the no-wait
starting time for the
machine,
i"^
job on machine (k+1) to be equal to
its
model
NIS/NW/M
requires the
completion rime on the previous
i.e..
No-wait constraints:
%+!)
For the
=
NIS/SYN/M
Ty^+^ik
forallk=l
m-1,
i=l
n-1
(2.9i
model, the job staning times must satisfy both the blocking
constraints (2.8) and the following synchronization constraints:
Synchronization constraints:
Vl)k
=
%.l)
forallk=l
-9-
m-1,
i
=
l
n-1. (2.10)
processing consecutive jobs (since the
Consffaints (2.10) require adjacent machines to
start
system has no intermediate storage)
time.
To summarize,
the standard
at the
same
IIS/FRAI model
the core formulation to
is
which we add
various constraints on the job starting times in order to nxxlel the NIS/FR/M,
NIS/SYN/M flowshop
constraints (2.8).
while the
scheduling problems. These
In addition, the
NlS/SYN/M model
However,
we
as
discuss
NlS/NW/M model
models
all
require the blocking
requires the no-wait constraints (2.9i.
contains the synchronization constraints (2.10). Note that the
minimum makespan models do
idleness" condition since the
latter three
NIS/NW/M. and
not require an explicit constraint to enforce the "no unforced
problem always has an optimal solution satisfying
later, certain
minimum
this condition.
cycle time models require a separate constraint
for this purpose.
Lower bounds:
IIS/FR/M
is
a relaxation of the remaining three
additional blocking constraints (2.8)), and
NIS/SYN/M.
NIS models
NlS/FR/M
is
(all
the NflS
a relaxation of
models contain
NIS/NW/M
the
and
Therefore, for a given set of problem parameters (number of machines and jobs.
processing time on each machine),
optimal value (makespan) of the IIS/FR/M model or any lower bound on
(a) the
is
a valid lower bound on the optimal values for
the optimal values of the
In Section 3,
observation
NIS/FR/M model
optimal value of the
(b) the
we
(a),
NIS/NW/M
and
or
all
its
three
is
a valid lower
bound on
models.
describe a slack-based lower bound for the IIS/FR/M model which, b>
also underestimates the optimal
subsequently improve these
constraints in the
latter
makespan
for the
NIS models.
We
lower bounds by accounting for the additional timing
NIS models.
10
value
NIS models; and.
lower bound
NIS/SYN/M
this
Upper bounds:
Given any job sequence
O,
we can choose
timing constraints for each of the four models.
optimal for nxxlel P
constraints,
if
it
appropriate job starting times that satisfy the
We
the objective value (makespan),
NIS/FR/M, NIS/NW/M, or NIS/SYN/M. What
and
for the different
SYN models can
model
to the
between the a-optimal
the relationship
likewise,
schedules for the
\W
same
any a-optimal NIS/FR schedule
relationship that
upper bounds. In particular,
if
we observed
Mp denotes
^IS/FR/M
-
is
for the lower
the a-optimal
"^^ '^slIS/NW/M' ^NIS/SYN/M
"
For the special case of two-machine flowshops, we can show
and NlS/SYN/M models
Minimum
The
one of IIS/FR'M.
makespan
P, then:
^IS/FR/M
2.2.2
is
be adjusted to produce a feasible (but possibly not a-optimal) for the
feasible for the IIS/FR case. Therefore, the
for
is
where P
We note that the a-optimal
models?
NIS/FR model with equal or lower makespan;
bounds also applies
o-
is
processes jobs in the given sequence o, satisfies the model's timing
and minimizes
makespan values
say that a schedule of starting times
integer
all
have a
common
that the
'
'-^''
NIS/FR/M, NIS/NW/M.
optimal schedule.
cycle time models
programming formulations
for the
minimum
cycle time models contain the
previous sequencing and timing constraints (2.2) to (2.7), and the special constraints (2.8) to
(2.10) for the
NIS
versions. In addition, the cycle time formulations require a
new
set
of
decision variables and constraints to model the cycle time objective, and two of the four models
require special constraints to prevent unforced idleness.
To
express cycle time in terms of the sequencing and timing decision variables,
interval time variables
l^,
for all k
=
1, ...,
the schedule. Since the interval time for
m, and a variable
machine k
-
11
is
C
we
define
representing the cycle time of
the difference
between the completion
time for the
last (i.e., n'*')
job and the
start
time of the
first
job on that machine,
we
introduce
the following defining equations:
Interval time equations:
Since cycle time
is
T„,^S^-T„
=
k
maximum
the
interval time l^
forallk=l
over
all
the
m.
machines
k,
(2.12)
we add
the cycle time
constraints:
C
and the optimization objective
>
for all k
I,^
=
1,
m,
...,
(2.13)
is
minimize C.
(2 14)
For the NIS/SYN/C model, the synchronization constraints (2.10) ensure
not have unforced idleness. For instance, the
machines 2 and
idle
only
when
1,
respectively, at the
the input
queue
is
same
first
and second jobs
start
any schedule
that
machines
are
empty. However, the DS/FR/C and NIS/FR/C models
queue (otherwise, we can reduce the cycle time by postponing the
that
processing on
time. Similarly, in the no-wait case,
require additional constraints to prevent machines being ready but idle
Note
machines do
that
when jobs
first
are waiting in
and subsequent
does not have unforced idleness must process the
first
job
jobs).
in the
sequence without intermediate waiting. Therefore, for the IIS/FR/C and NTS/FR/C models we
add the following
set
of constraints to enforce the no- wait requirement for the
fu-st
job.
No-wait processing for first job:
Tlk
These constraints are
Adding
makespan
forallk = 2
"^Lk-l+Si^
sufficient to ensure feasibility of the schedule,
that processes the first
remaining jobs
=
job without intermediate waiting,
to satisfy the
we can
m.
i.e.,
(2.15)
given any schedule
adjust the starting times for the
no unforced idleness condition without increasing
the cycle time.
constraints (2.12), (2.13) and (2.15) to formulation (2.2) to (2.7) of the standard
tTKxlel
IIS/FR/M, and replacing the objective function (2.1) with (2.14) gives the
12
integer
programming formulation
for the
IIS/FR/C model. The NIS/FR/C model has the
additional blocking constraints (2.8); replacing constraints (2.15) with the no- wait constraints
(2.9) or the synchronization constraints (2.10) in the
NIS/NW/C
or
NIS/SYN/C
NIS/FR/C formulation gives
the
formulations, respectively.
Lower bounds:
IIS/FR/C
is
a relaxation of the
NIS/FR/C model, which
in turn is a relaxation
of
MS/NW/C
(since constraints (2.15) are a subset of the no-wait constraints (2.9)). Therefore, the optimal
value or a lower bound for IIS/FR/C
However, unlike
the
to the other.
maximum
value, over
We can
set
bound
is
for
NIS/FR/C and NIS/NW/C.
"unrelated" to
NIS/FR/C
since these
of constraints each (constraints (2.10) and (2.15)) that do not
A naive lower bound on the optimal cycle time of NlS/SYN/C
all
also underestimates the
models).
a valid lower
makespan models, NIS/SYN/C
two formulations have one
belong
is
machines
minimum
k,
of the
total
processing time TPj. on machine k
cycle times for the IIS/FR/C, NIS/FR/C, and
is
the
(this
value
NIS/NW/C
possibly improve this bound by adding a lower bound on the machine idle
times due to synchronization effects.
Upper bounds:
To
relate the
compare
upper bounds for various models,
we
consider a specific job sequence a. and
the cycle time of the o-optimal schedule (with respect to the cycle time criterion
)
for
each model. For any given job sequence a, the job starting times for the NIS/N'W/C model's
O-optimal schedule
feasible for
is
feasible for
nS/FR/C. Therefore,
^IIS/FR/C
-
The o-oprimal schedule
NIS/FR/C; likewise, the o-optimal schedule for NIS/FR/C
if
C^ denotes
SlIS/FRyC
for
-
the o-optimal cycle time for
S(IS/NW/C-
NIS/SYN/C might entail some
model
is
P. then:
'" '^'
intermediate waiting time for the
first
job (due to synchronization constraints) and, therefore, does not necessarily satisfy the no
unforced idleness assumption for the NIS/FR/C model. However, we could develop an upper
13-
bound
for
NIS/FR/C by adding
that the first
In
job must wait
summary,
at
for
NIS/S YN/C cycle time an estimate of the
any machine
the different
and solution principles
to the
to ensure
to the other
related,
models as
well.
develops a new slack-based lower bound for the IIS/FTl model, and extends
and the NIS/FR cases. Section 4 proposes a
time
synchronous operation.
flowshop scheduling models are closely
one model might apply
maximum
common
heuristic
it
and the bounds
The next
section
NIS/NW
to the
method.
Slack-based lower bound for Makespan and Cycle Time
3.
Since the
minimum makespan problem
for flowshops with
more than
3
machines
is
NP-
complete, several authors have focussed on developing efficient procedures to estimate the
smallest possible
in an
makespan
for a given
problem instance.
enumeration procedure such as branch-and-bound
eliminate suboptimal job sequences or stop the search
optimal. Lageweg, Lenstra, and Rinnooy
Kan [1978]
We can then
(e.g.,
when
McMahon
the
generalizes and outperforms previous lower bounds on the
bound
3.1
for both the
close to
will
denote as the
to generate the
bound;
minimum makespan
that
LLR
it
for the
provides a lower
makespan and cycle time minimization models.
The slack-based bound belongs
for the IIS/FR/M
to the class of
model
machine-based bounds
processing time of a particular machine. Machine-based bounds
McMahon
we
new slack-based procedure
Machine-based lower bounds
[1965],
is
to
describe an effective lower bounding
method, solves several two-machine flowshop subproblems
section proposes a
and Burton [1967])
incumbent solution
procedure for the standard flowshop model. This method, which
IIS/fWM model. This
use these lower bounds
that focus
(e.g., Ignall
on the
total
and Schrage
[1969]) underestimate the makespan by adding the following two terms to
the total processing time TPj^
on a machine
k:
14
(i)
pre-k processing time: the
first
(ii)
minimum
(k-1) machines and reach machine
post-k processing time: the
the remaining
minimum
(m-k) machines
after
The maximum value of this sum over
makespan,
the
time for the
i.e.,
we compute
the
ail
k,
first
job
to
complete processing on
the
and
time required to complete processing the
machine
last
job on
k.
machines k gives a valid lower bound on the minimum
machine-based lower bound
M on the minimum makespan for
IIS/FR/M model as foUows:
M
= ^^'^^^^^ [M(k)],
M(k)=
where
TPu +
,•"'"
(3.1)
{Qik + R,k}
for all k
i^k
for
all
=
k =
1, ...,
m,
(3.2)
1,
m,
(3.3)
...,
(3.4)
(3.5)
M(k)
is
is
the
makespan lower bound obtained by using machine k
the total processing time
processing time for job
first
and
j
on machine
and Qm^ and
Rj-j^
correspond respectively
and the post-k processing time for job j'. Jobs
jobs in the sequence. The
last
k,
as the reference machine.
minimum
value of [Q-^ +
The
LLR method
generalizes this machine-based
in
and
over
R..|^}
gives an underestimate of the total pre-k and post-k processing time
j
j'
all
TP|,
to the pre-k
are candidate
job pairs
j
and
any feasible schedule
bound by considering
a block of
intermediate machines, say, from machine kl to machine k2, instead of a single reference
machine
k.
makespan
Suppose we can evaluate a lower bound
to
complete
all
(or the optimal value)
jobs on machines kl to k2, assuming
available (at time 0) at machine kl. Then, adding this
jobs become simultaneously
makespan or
smallest possible pre-k j and post-k2 processing times over
15
all
all
on the minimum
its
lower bound
job pairs
j
and
j'
to the
gives a lower
j'
)
bound on
the
minimum makespan
for the original
m-machine problem.
lower bound (or optimal value) of the makespan for machines kl
computes the overall lower bound
M
denotes the
LLR method
to k2, the
for as follows:
maximum
=
If L(k:l ,k2)
(3.6)
M.(kl,k2),
kl,k2=I....m, k2>kl
where
M(kl,k2) = L(kl,k2) +
how
Let us discuss
kl = k2 =
(i) if
LLR bound
k,
is
compute
the
same
(iii) if
^^^ ^'^ ^^^^^
(Qjkl+RjTcl)
the kl-to-k2
=
^^
^
^'^
bound
the
LLR method computes
(kl + 1) to (k2-l), to transfer each job firom
L(kl,k2) by treating
all
(i.e.,
intermediate
these machines
machine kl
to
machine
k2).
modifying the job processing times, Johnson's algorithm determines the
machines
By appropnately
minimum makespan
two-machine flowshop relaxation with intermediate job transfer delays.
The LLR bound remains
all
the
L(kl Jc2) for these two machines, and
effectively introduce a delay, equal to the total intermediate processing time on
of
and
(3.2);
machines except machines kl and k2 as non-bottleneck machines
this
k,
that:
kl and k2 are adjacent machines, then Johnson's algorithm computes
i.e.,
k2 > (kl+1), then
sequence for
- ^1.(3.7)
makespan lower bound L(kl,k2). Note
as the previous machine-based
minimum makespan
the
^
then L(kl,k2) equals the total processing time TP,^ on machine
k2 = (kl + 1),
(ii) if
to
..^j'"
.
valid even
m(m-l)/2 possible machine
if
we
pairs.
consider only a subset of machine pairs k
Since
instead of a single reference machine, this
it
bound
1
.
k2 instead
considers a block of consecutive machines
is
superior to the machine-based bound
(3.
1
but requires additional computational effon (O(m^) applications of Johnson's algorithm).
3.2
The slack-based bound
for IIS/FR
The LLR method has proven very
models
effective in accelerating branch-and-bound solution
procedures for the IIS/FR/M models (Lageweg
et al. [1978]).
However, because
it
uses
Johnson's algorithm to solve the two-machine flowshop relaxation problems, the bound applies
-
16-
HS/FR/M model. The
only to the
difficulty, but the single
slack-based bound that
machine version
we
that
we
describe next overcomes this
discuss might produce infenor bounds for the
IIS/FR/M model.
The key observation
that motivates the slack-based
bound remains valid even
if
we
(i)
the machine-based
is that
replace the total processing time TPj^ in (3.2) with any valid
lower bound on the interval time of machine
components:
approach
The
k.
interval time
the actual processing time TPj^ for the n jobs
I,^
consists of
on machine
k,
and
two
(ii)
the slack or
"forced" idle time on machine k due to starvation, blocking, or job transfer timing constraints.
If
we can
then
use the processing time information to determine a lower bound on the slack time.
we can
strengthen the machine-based bound (3.2). Next,
lower bound for the slack time
in
an
nS/FR
system. Since
intermediate storage and job transfer system
single machine, this lower
bound appUes
other model types as well (except the
To
times,
1)
,
we
nS/FR
discuss
is
compute
interval time for a
makespan and cycle time versions of
NIS/SYN/C model
as
we
a
discussed
in
the
Section 2.2.2).
gain insights about the relationship between machine slack time and job processing
let
us examine the Gantt chart
and machine
machine
k.
Recall that
shown
S^ denotes
Figure 1(a) shows the case
k.
in
Figure
1
for
two adjacent machines, machine (k-
the processing time of the
when
completes on machine (k-1
Figure
after
machine k
busy with prior jobs when machine (k-1) completes
is
it
);
in
1
(b), the h"^
We let 5^ denote the cumulative slack time
between the
=
1
n,
and
all
k = 2,
...,
m. As Figure
job
shows,
following inequality:
-
17-
first
this
at
in
sequence on
on machine
k
job must wait since
this job.
IIS/FR system with no unforced idleness, the fu^t job does not wait
...,
i'*^
the h'^ job starts processing
immediately
2,
to
the least restnctive
and since we focus on the
to both the
how
Notice that
in the
any intermediate
and h^ job on machine
stage.
k. for all h
cumulative slack must satisfy the
2
1
^^
max
>
5jj
t^-'
k-1^
-I
0,
1=2
max
X
0,
(3.8)
s.k
i=l
^h
=
1
* ^
^i(k-l) ~ ^l(k-l)
Ah
A
k-l
1=1
i=l
=
,
I
.
where A,
Z^i(k-l)~Z
i=l
The parameter
(k-l) and the
and j' as the
first
difference in total processing times for the
last
=
aJoJ')
we
to calculate the parameter
A
last
on machines
6j >
[o,
k—
(k-2)
in
and
bound on machine
k,
(k-l),
Since the
2
n.
k's slack
and the slack time
k—
5.
time
k—
A^j,'^
+
max
+
5.
on machine (k-2) and
max
{ 0,
{
..
k'*'
machine takes
in
in
order
terms of the job
on machine
5.
and so on. Making these substitutions
Thus, the calculation of slack time on the
all
j
(3.11)
jobs (and not the entire sequence of jobs)
terms of the slack time
A^ + max{o,
necessary for
choose two jobs. say.
n, if
(j,j').
processing times on machines (k-l) and
5.
Notice that for h =
(h-I) jobs on machine
(TPk-i-TPk)-(Pj.k-l-Pjk),
Inequality (3.9) provides a lower
express
k.
last
jobs in the sequence, then (3.10) reduces to
only need to choose the fu^t and
we can
(3.10)
-(^Kk-D'^hk)-
^ik
(h-1) jobs on machine
and
(3.9)
m
i=l
Aj^ is the
first
n,
h
/ h
i.e.,
for all h = 2
and all k = 2
A +
[o.Ai^s-]
max
aJ +
(k-
In turn.
1 ).
the processing times
in inequality (3.9)
6|,}
}
}]
we
get
(3 12
into account the smallest slack times
previous machines.
first
machine does not have any slack time
And, for a given choice of the
first
and
18
last
in
jobs
j
an IIS/FR system. 5, =
and
j'
in the
sequence,
for
all
we can
h =
compute
1[
Aj^Od') using equation (3.1 1) for
machines
all
Substituting these values
k.
right-hand side of equation (3.12) gives a lower bound, say, a^(jj') on
time on machine k
of a^(j J') over
all
if jobs
and j'
j
the job pairs
j
and
j'
any feasible nS/FR schedule. Adding
lower bound, say,
Ij^
and
are the first
on machine
last
jobs
in the
in the
5^(j,j'), the total
slack
minimum
value
sequence. The
gives a lower bound on total slack time on machine k
lower bound
this
k's interval
to the total processing time
time in any feasible schedule,
TP^
in
gives a
i.e..
i*y
For the makespan minimizing model IIS/FR/M, we can funher add
the pre-k
and post-k
processing times as in (3.2) to get the slack-based lower bound on the optimal makespan.
summarize
this
We
slack-based bounding procedure below.
Slack-based bounding method
For every machine k =
For every job pair
compute
A^(j,j')
compute
otj^CJo')
k
j,
1,
j'
....
=
1,
for
IIS/FR/M
m,
...,
n,
j
*
j',
using equation (3.1
1);
k
by substituting \ij,i') values and
Compute slack-based lower bound
M^'^'^(k) on the
c
1
5^^
=
in
RHS
of (3.12):
minimum makespan
using machine k
as the reference machine:
}^^Hk) = TP,+ jj,^;"„
{aJ(j,j')
+ Qjk +
(3.14)
Rj.k}.
j*j'
The slack-based bound on
the optimal value of the
IIS/FR/M model
is:
(3.15)
Since the slack-based bounding method adds the non-negative term
produces a lower bound that
is at least
as tight as the
machine-based lower bound
Observe from (3.12) and the definition of aJ (equation
bound M^'^'Hk) depends on
O-^ij.]) to TP,^.
1
).
(3.10)) that the slack-based lower
the differences in processing times at successive
19-
(3.
it
machines pnor
to
machine
We,
k.
therefore, expect the slack-based
bound when job processing times have
downstream). Our computational
Finally, as
minimum
the
a negative trend
indicated previously, the slack-based
-•-
any IIS/FR schedule
maximum
interval time
over
that
all
TP,.
^
n
in
to
outperform the machine-based
processing times decrease
(i.e.,
results, reported in Section 5,
cycle time as well. Since {a„(jj')
machine k
is
we
bound
pnxesses job j
this behavior.
method provides bounds on
the
an underestimate of the interval time on
is
}
confirm
first
and job j'
last,
and since cycle time
machines, the following expression provides a slack-
based lower bound on the minimum cycle rime:
max
_
-
(-slack
^
!
m
k=l
F^p
min
n
i,,=I
„k,.
,
L^^k
Improved Slack-based lower bound
3.3
..."I
(3.16)
+ ^n^'J^J
for
NIS models
Although the slack-based bounds of Section 3.2 also apply
the special structure of the
IIS/FR case
we
NIS system
to further strengthen these
intermediate storage, however, jobs must wait in the
machine becomes
determine a lower bound on
(3. 12),
thus increasing
discuss a
how
way
machine
set the total slack time 5. for the first
release) until the second
to
a
this
(jj')
value then
available,
method
first
it
to
machine
and so
we can add
6.
NIS models, we can
to the
NIS/FR
for
exploit
bounds. Recall that for the
for
(or
all
h
= 2
we must
might be
n.
With no
delay their
positive.
If
we can
to the right-hand side of inequality
and the overall makespan or cycle time lower bound.
compute a lower bound on 5
to adapt this
to
NIS/NW
We
first
systems, and subsequently show
tiKxlel.
Reddi and Ramamoorthy (1972) have shown
that the
NIS/NW/M problem
is
equivalent to
an asymmetric traveling salesman problem (TSP) defined over a network whose nodes
correspond
to jobs.
The
distance from node
i
to
node
20
j
in this
network represents the necessary
o
delay Ljj
in releasing
job j
immediately follows job
I
ij
max
_
~ l<g<m-l
(to
ensure that
in the
i
does not wait
it
We compute
sequence.
X ^^i(k+l.i)-Pjk)'0
at
L,.
for
all
intermediate stages)
when
it
as follows:
i,j
=
n,
1
i
*
(3.17)
j.
k=l
To develop
bound on
a lower
sequencing the jobs,
we
the first machine's total slack time for
and
first
last
jobs
and
j
smallest delay that any successor job will experience,
\
For the
first
job j,
if jobs j
of
total slack
minimum
/=i
n.
<
For each job
j'.
^ j. j',
let X,
be the
i.e.,
n
foralli=l
,
/^ij^i/
i
•
/-,
•,
i
n,
i^j,j.
,
(3.18a)
we defme
minimum
_
^
Then,
=
actually
use the following principle.
Consider a given choice of
1
NIS/NVV without
,
and j' are scheduled fu^t and
last in the
sequence, the
first
machine must have
a
at least
n,i*j'
IX,
=
SiCJO")
(3.19)
i=l
to ensure
NW job transfers.
side of (3.12) to
compute
improved value of
a
Therefore,
the lower
we
bound
values of the no-wait models
NIS/NW/M
proposed an alternate lower bound for the
NIS/NW/M
algorithm), analagous to the
cCj^OJ
)
o" ^^ slack time
in
machine
k;
using this
(3.14) and (3.16) gives tighter lower bounds on the optimal
(jj') in
of a series of rwo-machine
substitute 5j^(J0') i" place of 5^ in the nght-hand
and NIS/NW/C, respectively. Szwarc (1983) has
NIS/NW/M
problem based upon the optimal values
subproblems (solved using Gilmore-Gomory's
LLR bound for the
performance of the slack-based bound for
IIS/FR/M model.
NIS/NW/M
problems.
21
In Section 5,
[
1964]
we compare
with the Szwarc bound for our
test
the
.
For the NIS/FR case, jobs
will
However, because
still
by machine
2.
the effect of
downstream machines on
therefore, consider only the first
equation (3.17),
i.e.,
we
the job transfer
at the first
mechanism
the release time delay
two machines
is
is
machine due
free flow rather than no-wait.
difficult to incorporate.
in defining the i-to-j
delay parameters
(0,
P|2-Pj,
for
}
all
i,j
=
1
compute
Lj;
interval time
4.
time units
Xj (using
We.
L,| in
if it
follows job
equations (3.18)) and
lower bound
i.
As
^^(j,']')
in the
(3.20)
n, i^^j.
Since the NIS system does not have any storage capacity between machines
wait at least
to blocking
set
= max
L,j
experience a delay
NIS/NW case, we
1
and
use these
2,
job
L,.
j
must
values to
(using equation (3.19), thus improving the
Ij^.
Assignment-Patching heuristic for flowshop scheduling
The slack-based lower bounds provide benchmarks
to evaluate the quality
of heuristic
solutions for various flowshop scheduling models. Instead of implementing different
specialized heuristics for IIS/FR, NlS/FR,
NlS/NW
and NIS/SYN, we apply
a single
optimization-based method called the assignment-patching heuristic to generate a promising job
sequence o, and then evaluate the objective value for the a-optimal schedule corresponding
the
model of
Subsequently,
interest
procedure.
The assignment-patching
NIS/NW/C
models.
time objectives;
4.1
we
It
we improve
this solution
heuristic exploits the
using a local interchange
TSP
strucmre of the
NIS/NW/M
and
uses slighdy different arc length parameters for the makespan and cycle
describe the makespan version
first.
Assignment-Patching heuristic for minimum makespan models
Reddi and Ramamoorthy [1972] transfonned the
NIS/NW/M model
into an equivalent
directed traveUng salesman problem defined over a network containing (n+1
....
to
n correspond to the n original jobs, and node (n+1)
22-
is
a
dummy job
)
nodes. Nodes
1
representing the starting
and completion of the schedule. The length of arc
time
Ljj,
dummy
computed using
Arc
zero.
~
i,
=
j
(i,n+l), for all
has length equal to the post-1 processing time for job
^(n+l)j
for all
All the arcs (n+lj) for
(3.17).
node (n+1) have length
(i,j)
i
...,
n,
1, ...,
n,
1
i
^^ j, is
the delay
emanating from the
n,
incident to node (n+1)
i.e.,
for all
0'
1.
=
=
i, j
j
=
1,
....
n,
i
=
1,
...,
n.
and
(4.1a)
m
H(n+1)
Any
hamiltonian tour on
=
X ^'k
k=2
this
for
network corresponds
contains arcs (n+lj) and (j'.n+l) then jobs
and
if it
all
to a feasible
and j' are the
j
first
contains arc (p,q), for p,q <
n,
then job q follows job
NIS/NW
is
the interval time
job sequence (for
for the last job
systems)
on machines 2 through
tour plus the total processing time
TPj
n, the length
for
all
job sequence.
and
last
jobs
If the
in the
tour
sequence.
Since the makespan of an\
p.
on machine
plus the processing time
1
of the corresponding traveling salesman
jobs on machine
Minimizing the length of the traveling salesman
(4.1b)
I
equals the makespan.
tour, therefore, solves the
NIS/NW/M model
optimally.
Instead of solving the equivalent
patching (AF) heuristic.
all
i, j
=
1,
...,
The formulation has assignment
the sequence,
apply the following assignment-
We can formulate the TSP as an integer program using the
sequencing variables Y^: for
otherwise.
TSP optimally, we
n+1,
i
*].
Y-
is
1
if
job
i
precedes job
binar\
j.
and
constraints (to assign each node to one position in
and one node to each of the n positions) and subtour breaking constraints
(Dantzig, Fulkerson, and Johnson [1954]).
If
get the following assignment relaxation of the
Assignment Relaxation
minimize
of
we
ignore the subtour breaking constraints,
TSP:
TSP:
n+1
n+1
J^
Pl
X
pi
^'^
ij
'*'^^
'J
subject to
23-
*^~'
ue
n+1
2^ Y,j
=
1
for
=
1
for alii
all j
=
1
n+1.
(4.3)
n+1. and
(4.4)
i=l
n+1
y Y,j
forallij =
Y.j€ {0,1)
If the
assignment solution
Otherwise,
we must
Lowe
TSP
tour,
it
1, ...,
n+1,
i
NIS/NW/M
provides an optimal
^j.
(4.5)
sequence.
patch together the subtours in the optimal assignment in order to construct
a feasible job sequence.
and
a
is
=1
We
use the following subtour patching method (adapted from Piante
[1987]).
Subtour patching heuristic:
Step
I
Step
2:
:
Pick two sub-tours Tj and T2 arbitrarily.
For every pair of arcs
(i,j)
€ Tj and
(i',j')
replacing these two arcs with the arcs
minimum
Step 3:
If the
e T2, evaluate the incremental cost of
and
(i,i')
(j,j').
Select the pair of arcs with the
incremental cost, and perform the interchange.
current solution has two or
The subtour patching
feasible job sequence
heuristic requires at
which we
call the
a relaxation of the
(4.2) to (4.5)
is
a valid lower
bound on
more
subtours, repeat Step
most 0(n)
AP makespan
TSP representing
the optimal value of
iterations,
solution.
the
Since the assignment model
heuristic for cycle time minimization
For cycle time minimization,
the
m machines.
TSP
To minimize
we examine sequences
the interval time
arc lengths to represent the delay
24-
i
test
its
optimal value
we compare
is
this
problems.
models
that
reduce the interval time on each of
on any machine
between jobs
model,
In Section 5,
assignment lower bound with our slack-based lower bound for our
^2 AP
Otherwise, stop.
and terminates with a
NIS/NW/M
NIS/NW/M.
1.
and
j
k,
we must modify
the previous
on machine k (instead of machine
1
).
1
and
set Lj j^j
equal to the
processing time for job
Let us
defuie Ljj, for
i).
how
describe
first
k
all
i, j
=
processing time TP|^ on machine k (instead of the post-
total
to
compute
n, as the
1, ...,
the delay times for a
necessary delay between jobs
ensure no- wait processing. Using this notation, the value
corresponds to
job
i
Consider a sequence
L|-.
leaves machine
processing job
i
at
at
1
{ t
+
time L Then,
X
^ih
)
in
L,.,
i
and
j
in the
We
k.
on machine
k to
computed using equation
which job j immediately follows job
(3. 17).
and suppose
i,
no-wait system, machine k must complete
Furthermore, job
•
downstream machine
j
is
released into the line
at
{
t
+ L
)
.
and
^~^
reaches machine k
at {t
+ L
1
+
'J
interval
X
Pjh}- Therefore, the i-to-j slack time
h=l
i
and the
^r
'J
we
The TSP with L^ and
arrival
=
L*'
approximately using the
The
of job j
at
machine
k, is:
^^6'
(4.7.
TP,.
_^,
,
for all
i,
j
=
1, ..., n,
AP
heuristic, for
as arc lengths
NIS/NW
models
We
system.
every machine k =
the sequence with the smallest cycle time,
cycle time solution
4.3 Local
the time
h=2
minimizing the interval time on machine k for the
i.e.,
is
define
Li,n.l
sequence,
which
l^jh- l^.hb=l
Similarly,
,
k
k-l
=
k
''
between the completion of job
^iU
L
-
the
problem of
solve this
TSP
m, and choose the best
1
among
the
m TSP solutions as the AP
.
improvement procedure
AP makespan or cycle
time solution provides an
makespan or cycle time models. For
initial
candidate sequence
a given model, the initial upper
bound
is
o
for all the
the objective
value of the corresponding o-optimal schedule satisfying the model's intermediate storage and
job transfer constraints.
We
then
anempt
to
improve
25-
this
upper bound by applying a local
.
exchange
.
NIS/SYN/M
heuristic. Consider, for instance, the
makespan sequence,
for every pair of jobs
and
i
j,
we
model. Starting with the
new sequence
consider a
AP
obtained by
a'
interchanging the positions of these two jobs in the current sequence, and compute the objective
function value (makespan) for the a'-optimal NIS/S
than the current sequence, then
method stops when no
Improved
has a lower makespan
interchange and examine other job pairs. The
this
improvement
If a'
is p>ossible;
we
refer to the final job sequence as the
solution
Computational experience
5.
5.1
AP
further
we perform
YN schedule.
Test problems
To
lower bounds and heuristics, we generated random
test the
different sizes ranging
the size
from 6 jobs and 3 machines
(number of jobs and machines)
for each
to
test
problems
in
nine
50 jobs and 3 machines. Table
problem
class.
...,
n,
and k =
we mean
independent, identically distributed processing times
m. The processing times for a job are said
1
consistently smaller or greater than average on
times exhibit a trend
To
By random
we
Given
the
processing time
parameters of
they increase or decrease for
follow the method used by
all
Lageweg
On
jobs as
for
all
i
=
1.
to be correlated if they are
the other hand, job processing
we
progress
down
et al.
the line.
[1978] for their computational
number of jobs and machines, our problem generator chooses an
Pjj^
this
for each job j
on every machine k from an uniform
uniform distribution
integer job
distribution.
The
reflect the desired correlation and/or trend structure.
Problems with random processing times have
To
machines.
P:|^
generate problem instances corresponding to each of the four processing time
distributions,
tests.
if
all
shows
We considered four different
processing time distributions-random, correlated, trend, and correlated with trend.
processing times
1
introduce correlation in processing times,
P^
we
!L
Unif (1,100) for
all
jobs
j
generate n additional integers
26-
and machines
c.,
for
all
j
=
1
k.
....
n,
from the Unif
processing times,
For problems with only correlation and no trend
(1, 4) distribution.
we randomly
select
Pj,^
Problems with trend (but no correlation)
from the Unif (20c.+l. 20c.+20)
in the
distribution.
data are generated by selecting machine ks
processing times P^^ from the Unif (12.5(k-l)+l, 12.5(k-l)+100) distribution, for
...,
m. This scheme increases the average processing dme per job
processing times have an increasing trend. Finally,
correlation
and trend hy sampling from
distribution, for all
j
=
1, ...,
n,
For each problem size shown
in
1,
Table
we
at
=
all k
successive stages,
1.
i.e..
the
construct problem instances with hoih
Unif (2.5(k-l)+20c., 2.5(k-l)+20c.+20)
the
and k =
in
...,
1,
m.
we
generated four groups of three problem
instances each; the groups correspond to the four processing time distributions-random, with
correlation, with trend,
and with correlation +
trend.
We also "inverted"
by renumbering machine k as machine m+l-k for k =
positive trend
original
now have
negative trend.
m;
...,
To summarize, we
thus, original
problems with
generated 24 problem instances- 12
problems consisting of 3 random instances each for four processing time
and 12 inverted problems-for each of
models corresponding
5.2
1,
each problem instance
to
the 9
problem
sizes,
and considered
all
distribution.s.
8 flowshop
each of these 216 problem instances.
Performance comparisons
Our computational
bounds
tests
seek to address three questions:
relative to previous lower
(i)
how good
are the slack-based
bounding methods for makespan models;
assignment heuristic perform compared
to previous heuristic
methods; and,
(ii)
how does
(iii)
what
is
quality of the heuristic solution obtained by locally improving the assignment-based job
sequence?
We elaborate on each of these issues next.
27
the
the
Comparison
5.2.1
Recall that
we have
bounds:
of lower
three versions of the slack-based
bound
for the
minimum makespan
problem: the basic version that provides a lower bound for the IIS/FR/M model, and enhanced
versions for the
N1S/FRA1 and NIS/NW/M models. For
the
nS/FR/M model,
LLR bound
the
(described in Section 3.1) based on two-machine subproblem solutions has proven to be the
most
effective. This
bound
NIS/FR/M model. We,
also valid for the
is
slack-based (basic and enhanced) for the IIS/FR/M and
Note
that,
model
is
using both the slack-based and
the
same
as the
LLR
NIS/FR/M model
so
comparison of bounds for the NlS/SYN/M case. For the
the
Szwarc [1983] lower bound and
(4.5)) with the
enhanced slack-based bound. The
based solely on
5.2.2
To
its
we do
we
LLR
NIS/SYN/M
NIS/NW/M
model, we compare both
problem
((4 2) to
and Szwarc bounds do not apply
evaluate the quality of the slack-based bounds
AP heuristic, we compare
the objective values of the
solution (before local improvement) and the solution produced by the
1970]) for
all
model
types.
The
CDS
heuristic,
CDS
algorithm, and selects the best
k machines and
last
k machines
each of the two machines
minimization models,
among
in the
k'*^
we augment
the p job sequences.
in the original
auxiliary
this
The
developed for the IIS/FR/M
total
s
processing time on the
problem serve as the job processing times on
two-machine problem. For cycle time
method by applying an improvement procedure
sequencing the jobs using Johnson's algorithm.
-28
AP
heuristic (Campbell.
model, solves p ^ (m-1) auxiliary two-machine flowshop subproblems using Johnson
first
to
of heuristics:
evaluate the effectiveness of the
[
bound.
closeness to the heuristic value (see Section 5.2.3).
Comparison
Dudek and Smith
LLR
the
not perform a separate
the optimal value of the assignment
cycle time minimization problems, and so
with the
methods, the lower bound for the
NIS/FTVM lower bound, and
compare
therefore,
after
Gap between upper and
lower bounds:
assess the quality of the improved
AP
5.2.3
To
%
the
gap between the objective value of
of the slack-based lower bound and the
values of this
%
gap imply
that both the
solution (after local improvement),
this solution
LLR
with the best lower bound
or Szwarc lower bounds) for
we compute
(i.e..
the better
8 models.
all
Small
lower bounding method and the heuristic procedure are
effective.
5.3
Computational results
We implemented the problem generator and the
an
IBM
4381 computer, and the
lower bounding methods
heuristic procedures in
3
and 4 summarize our computational
6
f)ertain to
results for the
FORTRAN on
a Macintosh computer. Tables
minimum makespan
2.
models; Tables 5 and
cycle time minimization nxxlels.
Table 2 compares the slack-based bound with the
NIS/FR/M models, and
For the
C on
in
LLR
bound,
the
LLR
bound
Szwarc and assignment lower bounds
we followed Lageweg
machine flowshop subproblems,
for
k2 =
selecting the best lower bound. Table 2
et al.'s
for the
IIS/FR/M and
NIS/NW/M
case
[1978] strategy of solving (m-1) two-
m and kl
shows
for the
=
1, ....,
m-1
(see Section 3.1), and
the average values of the ratio of lower
bounds
(averaged over 54 problem instances) for each of the four processing time distributions. As
expected, the slack-based lower bound performs well for problems with trend
times;
it
also appears to be as tight as the
random processing
times.
in total processing times
as effective.
on adjacent machines
LLR
bound when
processing
and the Szwarc bounds for problems with
However, when job processing times
We observed that, even
approaches the
LLR
in
is
are correlated, the difference
small, and hence the slack-based
bound
for problems with correlated data, the slack-based
the
problem, the assignment lower bound
number of jobs
(i.e.,
is
very large. For the
is
not
bound
NIS/NW/M
the optimal value of the assignment subproblem)
outperforms the slack-based lower bound.
-29-
)
Tables 3 and 5 compare the performance of the
and the
CDS
Although
solutions for
well as the
all
CDS
heuristic
random processing
CDS
AP heuristic
is
based upon the
(which was developed for
when
times,the
this
NIS/NW
model,
AP
model); the
the job processing times are
AP
solution for the
solution has, on average,
improvement
cycle time models,
other models as well. For the IIS/FR/M model, the
other models especially
than the
the
heuristic (without local
minimum makespan and minimum
heuristic for the
respectively.
AP
AP
it
produces good
heuristic performs as
solution
random or have
is
a trend.
supenor for
With
12%, 8%, and 67o lower makespan
NIS/NW/M, NIS/FR/M, and NIS/SYN/M models,
respectively.
Similar performance differentials hold for the cycle time minimization models.
Tables 4 and 6 compare the values of the improved
for the
the
makespan and cycle time
nxxlels, respectively.
AP solution
with the best lower bounds
The average
%
gaps are
3%
or less for
NIS/NW/M, and IIS/FR/C
IIS/FR/M, NlS/FR/M (except with random processing times),
models, indicating that the heuristic generates near-optimal solutions for these cases, and the
lower bounds are
tight.
The gaps
correlated processing times;
these cases, but the lower
6.
we
are large for the
NIS/FR/C and NIS/NW/C models w
suspect that the heuristics performance
bound
is
is
quite
good
ith
e\ en for
weak.
Conclusions
This paper has presented a new slack-based lower bound and an assignment-patching
heuristic that apply to a variety of
flowshop scheduling contexts. The slack-based bound
improves upon the machine-based bound, and applies
cycle time optimization problems.
lower bound
is
as
good
as the
trend in processing times.
LLR
The
The computational
bound
for
AP algorithm,
to
makespan,
results
show
total interval time,
that the
and
performance of the
problems with random processing times or with
together with heuristic local improvement.
30-
performs well for
all
scenarios, producing schedules that are within
10%
of optimal
in
most
cases.
Scheduling flowshops with synchronized job transfers, and with cycle time and interval
time objectives are promising topics to investigate further. Although the slack-based method
provides lower bounds for these models as well,
we might
be able to improve performance by
exploiting the models' special structure. Similarly, instead of using a general sequencing
algorithm based on no-wait principles,
we might
consider alternative heuristic methods that are
specially adapted to cyclic and synchronous scheduling.
-31
Figure
Machine k-1
Machine k
1
:
Slack time on adjacent machines
Table
Problem Group «
1:
Sizes of Test Problems
Table
3:
Performance
for
Model
of
Assignment-Patching Heuristic
Minimum Makespan Modelst
Table
4:
Model
Gap between Upper and Lower Bounds
for Minimum Makespan Modelst
Table
5:
Performance
for
Model
of
Assignment-Patching Heuristic
Minimum Cycle Time Modelst
Table
6:
Model
Gap between Upper and Lower Bounds
for Minimum Cycle Time Modelst
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in
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