AN ABSTRACT OF THE THESIS OF
Terrence A.
Smith for the degree of Master of Science in
Industrial Engineering presented February 26, 1993.
An Experimental Investigation of Scheduling NonIdentical Parallel Processors with Sequence-Dependent Set-up
Title:
Times and Due Dates.
Redacted for Privacy
Abstract Approved:
Dr. Sabah Randhawa
An
experimental
investigation
of
factors
effecting
scheduling a system of parallel, non-identical processors
using a series of experimental designs was carried out.
System
variables
included
were
processor
capacities
relationships, sequencing and assignment rules, job size, and
product demand distributions. The effect of the variables was
measured by comparing mean flow times, proportion of jobs
tardy, and processor utilization spread.
Results of the study found that system loading and set-up
times play a major role in system performance.
Grouping jobs
by product will minimize set-up times and hence mean flow time
and tardiness at the expense of controlling individual
processor usage.
Factors involving processor capacities and
assignment rules tend to have no affect on any of the system
performance measures.
Variability in job size and product
demand tended to give flexibility in controlling individual
processor utilization.
cCopyright by Terrence A. Smith
February 26, 1993
All Rights Reserved
An Experimental Investigation of Scheduling Non-Identical
Parallel Processors with Sequence-Dependent Set-up Times
and Due Dates
by
Terrence A. Smith
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed February 26, 1993
Commencement June 1993
APPROVED:
Redacted for Privacy
Associate
Professor
of
Industrial
Engineering in charge of major
and
Manufacturing
Redacted for Privacy
Head of Department of Industrial and Manufacturing Engineering
Redacted for Privacy
Dean of Graduat
0
chool
Date thesis is presented
Prepared by
id
February 26, 1993
Terrence A. Smith
ACKNOWLEDGEMENTS
would like
to thank the members of my graduate
Dr. Christopher Biermann, Dr. Eldon Olsen, and Dr.
Katherine Hunter-Zaworski, for their guidance and support
during my academic career at Oregon State University.
The
I
committee,
giving of their time and advice were instrumental
success.
to my
I would especially like to thank my major advisor,
Dr. Sabah Randhawa for his wisdom, patience, and support in
guiding me through the course of study and the preparation of
this thesis.
And lastly I must thank my wife, Janice for without her
constant support and encouragement, this thesis could not have
been possible.
TABLE OF CONTENTS
I.
INTRODUCTION
1
Research Objectives
Nomenclature
Literature Review
EXPERIMENTAL
........
METHODOLOGY.
.........
.
.
System Definition
Performance Measurement
Experiment Variables and Settings
Experiment Design
Job Sets
Statistical Evaluations
Simulation Description
13
15
16
16
17
Experiment One
Experiment Two
Selection of Effects for Experiments Three
and Four
Experiment Three
Experiment Four
DISCUSSION AND CONCLUSIONS
7
7
8
8
III. RESULTS
IV.
1
2
3
........
Mean Flow Time
Proportion Jobs Tardy
Processor Utilization Spread
Relationship Between Performance
Measures
Summary of Conclusions
Future Research / Enhancements
18
22
29
31
36
.
.41
42
44
47
51
51
52
V.
REFERENCES
54
VI.
BIBLIOGRAPHY
55
VII. APPENDICES
A
B
C
D
E
Experiment Data
Experiment Results
Set-up Time Matrices
Experiment Job Sets
Spreadsheet Explanation
59
60
64
68
70
78
LIST OF FIGURES
Figure
1
2
3
Page
Proportion Jobs Tardy Interaction:
Order Size Distribution versus Loading
46
Processor Utilization Spread Interaction:
Set-up Time versus Loading
49
Spreadsheet Illustration
80
LIST OF TABLES
Table
Page
20
21
Experimental Factors and Settings Used in
Experiments 1-4
Estimated Effects for Mean Flow
Exp 1
ANOVA for Mean Flow
Exp 1
Regression Coeffs. for Mean Flow- Exp 1
Estimated Effects for Proportion Jobs Tardy
Exp 1
ANOVA for Proportion Jobs Tardy
Exp 1
Regression Coeffs. for Proportion Jobs Tardy
Exp 1
Estimated Effects for Processors Utilization
Spread
Exp 1
ANOVA for Processor Utilization Spread
Exp 1.
Regression Coeffs. for Processor Utilization
Exp 1
Spread
Estimated Effects for Mean Flow
Exp 2
ANOVA for Mean Flow
Exp 2
Estimated Effects for Proportion Jobs Tardy
Exp 2
ANOVA for Proportion Jobs Tardy
Exp 2
Estimated Effects for Processor Utilization
Spread
Exp 2
ANOVA for Processor Utilization Spread
Exp 2
Summary of Results
Experiments One and Two
Significant Effects
Estimated Effects for Mean Flow
Exp 3
ANOVA for Mean Flow
Exp 3
Regression Coeffs. for Mean Flow
Exp 3.
Estimated Effects for Proportion Jobs Tardy
22
23
ANOVA for Proportion Jobs Tardy
Exp 3
Regression Coeffs. for Proportion Jobs Tardy
24
Estimated Effects for Processors Utilization
Spread
Exp 3
ANOVA for Processor Utilization Spread
Exp 3.
Regression Coeffs. for Processor Utilization
Spread
Exp 3
Estimated Effects for Mean Flow
Exp 4
ANOVA for Mean Flow
Exp 4
Regression Coeffs. for Mean Flow
Exp 4
Estimated Effects for Proportion Jobs Tardy
Exp 4
ANOVA for Proportion Jobs Tardy
Exp 4
Regression Coeffs. for Proportion Jobs Tardy
Exp 4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
.
.
.
.
14
19
19
20
.
21
.
23
24
25
26
27
28
30
33
.
31
32
33
33
34
34
Exp 3
27
28
29
30
21
21
21
Exp 3
25
26
20
20
34
.
35
35
35
37
37
37
38
38
38
Table
33
34
35
36
Page
Estimated Effects for Processors Utilization
Exp 4
Spread
ANOVA for Processor Utilization Spread
Exp 4.
Regression Coeffs. for Processor Utilization
Exp 4
Spread
Summary of Results
Experiments Three and Four
Significant Effects and Two-Factor
Interactions
39
39
39
40
An Experimental Investigation of Scheduling Non-Identical
Parallel Processors with Sequence Dependent Set-up Times
and Due Dates
I.
INTRODUCTION
A common situation in manufacturing and service industries is that of assigning tasks (work) to machines or workers
(processors) that do not have equal capabilities and capaciThree objectives of interest in assigning work may
ties.
include:
insuring that all tasks,
or as many as possible, are
competed on time;
to spread the work load equally among the processors; and
to minimize the average time for work to flow through the
system.
This thesis will investigate how processor, sequencing,
assignment, and product job variables can affect the achieve-
ment of the three objectives listed above.
Research Objectives
The primary objective of this project is to identify
system variables that affect the performance of schedules for
parallel, non-identical processors in terms of mean flow time,
proportion of jobs tardy, and processor utilization spread.
The results obtained will serve as a foundation for later
research to develop heuristic decision rules for scheduling
these systems.
2
A secondary objective of this research is investigate any
dependence between the
three performance measures listed
above.
Nomenclature
Many of the terms used in this thesis have unique
definitions that may not be consistent with lay usage and
merit clarification.
The purpose of this section is to
identify key terms and clarify their meaning.
Tasks
Jobs
are any work, job, or order to be carried out.
are individual, distinct, demands for a product or
In the context of this study, task, job, and order
can be used interchangeably.
service.
Processor
ing a task.
is any person or machine capable of undertak-
A parallel processor is the situation when a task
can be done by more than one processor but only one processor
can actually work on the task.
An example would be a typing
pool where any one of a number of workers could type a
Nonidentical processors are processors that do.not have the same
capacities and/or capabilities.
Other examples include an
airline assigning a type of aircraft to service a route, a
document, but only one person can be assigned the task.
textile plant assigning a job to a loom, and a paper plant
assigning products to different paper machines.
Performance Measure
is a criterion for judging the
impact of changes in the input to a system.
is synonymous with independent variable.
In
this study several effects of interest were chosen to investigate their impact on a system of interest.
Effect
3
Sequencing
is ranking of tasks, products,
processing on a processor.
etc.
for
It does not include assignment and
is not the same as scheduling.
it is, however, a part of
scheduling.
Scheduling
involves assigning tasks to processors and
sequencing them for
the most efficient use of resources
subject to system constraints and due date requirements.
Preemption
allows tasks being processed to be interrupted so that another with higher priority can be processed.
The original job is then completed at some future time.
Literature Review
The occurrence of parallel, non-identical processors is
quite common in both manufacturing and service industries.
It
is therefore surprising to find very little research has been
carried out in this area.
Marsh [1973] and Guinet [1991] have
carried out the two more notable studies.
In his
Marsh was primarily
concerned at evaluating optimum solutions to scheduling
parallel, non-identical processors to minimize total set-up
time.
1973
Ph.D.
dissertation,
As part of his investigation, he also analyzed computa-
tional time necessary to develop the optimal solution. His
findings were that the computational time requirements made
solving for an optimum solution prohibitive for all but the
simplest systems.
Marsh's work included evaluation of
different optimization techniques but focused heavily on the
branch-and-bound dynamic programming technique.
In Guinet's
study
of
scheduling
textile production
facilities, which abound in parallel, non-identical processor
systems, an attempt was made to minimize the mean flow time
4
which would in turn minimize the mean tardy time.
His
investigation, like Marsh's, included sequence-dependent setup
times.
Guinet also evaluated the computational time
requirements using an 80386-SX computer operating at 16 MHz.
He found that a 100 job, 5 machine schedule took 53 minutes to
Adding 25 jobs increased the time to 136 minutes.
compute.
And this portion of his study had not introduced the complexi-
ties of sequence-dependent set-up times. His research was
based on a linear programming approach.
A common observation of both of these studies is that
obtaining an optimum solution for all but the smallest systems
is not practical in common, everyday scheduling situations.
In general, an understanding of how relationships between the
parallel processors, scheduling system, and product and job
distributions affect system performance may lead to decision
satisfactory schedule.
rules that can give a feasible,
Although the schedule may not be an optimum, the tradeoffs in
programming and computation time to allow for more frequent,
and hence accurate, schedules will be beneficial.
Possible quantitative methods for analyzing parallel,
non-identical processors include:
1.
Linear Programming
2.
Branch-and-Bound
Queuing Theory
3.
4.
Simulation
at first glance appears to be an
As mentioned by
excellent method of analyzing the system.
Linear programming,
Marsh and others, scheduling this system appears to follow the
However, the nonclassical traveling salesman problem.
reversible, sequence-dependent set-up times foil this method
for three reasons. First, after one pair of jobs is selected
as a sequence, the reverse must be eliminated.
For example if
5
Job 4 is selected to follow job 7 on processor A, all set-up
times involving 7-4 and 4-7 on all processors must be elimi-
nated, typically by changing the set-up times to infinity.
This makes the problem non-linear.
Secondly, this approach assumes minimization of set-up
time which may not always be the objective. For instance the
objective may be to level processor utilization.
And thirdly, should the first two problems be resolved,
using linear programming on a daily basis for updating and
controlling schedules is not feasible due to both programming
and computational time requirements.
The branch-and-bound method was evaluated by both Marsh
and Guinet.
They both found that it is not an efficient
method in terms of programming and computational time requirements.
The third possibility considered was adapting queuing
theory [Winston, 1991].
Processors can be considered to be
servers with various service time levels and the jobs are
customers.
Service times are related to order quantities and
By defining priority queuing disciplines, scheduling and assignment rules could be modeled.
Grouping jobs by product could be handled by using a multiple
queue while ungrouped jobs, scheduled individually, would use
a single queue, multiple server system. However, the method
processor capacities.
breaks down in not being able to handle the sequence-dependent
set-up times.
Queuing theory is further restricted to well-
defined distribution forms for arrivals and service times.
The most common distributions encountered are poisson arrivals
and exponential service times.
6
Simulation was used because it allows complex systems to
be modeled without being limited by the assumptions inherent
analytical models.
interaction between
variables and complex distribution forms can be modeled with
relative ease. However, to be effective, simulation results
need to be carefully analyzed.
in
Furthermore,
The experimental design approach allows a systematic
examination of factors when little or no information is
available about them.
A large number of factors can be
screened quickly and accurately.
Significant factors can be
identified and singled out for further study. Results of an
experimental study can provide an excellent base for a later
validation using simulation software.
7
II.
EXPERIMENTAL METHODOLOGY
There are three main steps in this investigation.
The
first step was to screen variables for significance using two
statistically designed experiments (experiments one and two).
The next step was to analyze the results of the screening
experiments and select significant variables for detailed
study.
The third step was to run detailed, response surface
experiments using these variables (experiments three and
four).
A detailed description of the methodology follows.
System Definition
The framework of the system investigated in this study
consists of three processors and seven products in the initial
screening experiments. The screening experiments found that
product demand distribution did not affect system performance_
so the number of products was increased to ten for the final
experiments. There were multiple tasks for each product with
varying quantities and due times.
Distributions for job and
product quantities were varied as an experimental variable.
The
system
is
static with all
processing at time zero.
tasks
available
for
Preemption was not allowed.
$et -up time matrices were generated using a random number
generator.
The matrices were the same for each processor.
Set-up times between tasks for the same product line are zero.
The set-up times are sequence dependent and not reversible,
i.e.
the set-up time changing from product A to B is not
necessarily the same as going from product B to A.
8
Performance Measurements
The three performance measures included in the scope of
this study are Mean Flow Time, Proportion of Jobs Tardy, and
Processor Utilization Spread.
The Mean Flow Time is defined as the amount of time a job
spends in the system from the time
is available for
processing until it is completed.
For the purpose of this
study, the flow time for each job will be equal to its
completion time since all tasks are available at time zero.
In general, a smaller mean flow time is desirable as it
indicates work is flowing through the system in a faster
manner.
it
This is consistent with current business and manufac-
turing philosophies, such as Just-in-Time, of minimizing workin-process.
The Proportion of Jobs Tardy measures the proportion of
jobs that are completed after their due date. The goal of any
operation is to have zero late tasks.
The Process Utilization Spread measures how evenly jobs
are distributed among the processors.
It is simply the
difference in the rate of usage
percent capacity used
between the heaviest and least scheduled processors.
Depend-
ing upon an organization's goals, it may be desirable to use
all processors equally or to try to run a minimum number of
processors so that unneeded capacity can be shutdown to reduce
costs.
Experiment Variables and Settings
System related
9
Processor Spread (abbreviated A:Pr_Spread in experiments) is the range of capacities of the processors under study.
This variable will identify
if the differences in capabilities between process-
ors will affect
the system performance.
This
effect was only studied in experiments one and two.
For this study, the processor capacity spread, in
units per hour, is 20 and 80, low and high, respec-
Using a mean processor capacity of 100
units per hour, the low setting will set the slow
tively.
processor at 90 units per hour and the fast processor at 110 units.
The difference is 20 units.
Processor Distribution (B:Middle_Pr) investigates
whether the location of the middle processor,
either closer to one end of the Processor Spread
described above, or in the middle of the spread
affects the system performance.
This effect was
only studied in experiments one and two.
The
settings used in this study are 50% and 75% of the
spread.
At 50% of spread, the processor will have
a capacity equal to the mean of 100 units per hour.
At 75% of spread,
the middle processor capacity
will be 105 or 120 units per hour for spreads of 20
and 80 units, respectively.
Loading, or per cent of capacity scheduled (C:Percent_Ca) will evaluate how system performance
changes as the system loading approaches capacity.
Loading is determined by summing the capacities of
the processors for the scheduling period.
Jobs are
accepted for scheduling until the sum of the job
quantities is equal to the desired loading (percent
of total system capacity).
Loading was studied in
all four experiments run under this study. The low
10
setting was 75% and the high was 90%.
Lower loadings were not used because they would not challenge
the proportion of jobs tardy criteria while loadings heavier than 90% would have a high likelihood
of all jobs being tardy.
Set-up Time (D:Set_up_Tim)
is the amount of time
needed to change over from one product line to
another.
The set-up time distributions will be
discrete uniform.
Distributions will be varied to
analyze the effect on performance.
In experiments
one and two the settings used were uniform distri-
butions of U[0,3] and U[0,12].
In experiments
three and four the settings used were either 0
(zero) set-up time or U[0,5].
Sequencing and assignment rules
Grouping (E:Grouping)
is when all tasks for each
product are grouped together and run as one "super"
This technique is often used in the chemical
industry where it is referred to as campaigning.
job.
Grouping jobs by product family is known to reduce
total set-up time.
The significance of the set-up
time reduction was evaluated with this variable.
The variable setting was either no grouping or all
jobs grouped by product.
Ranking for Processor Assignment (F:Pr_Assign)
is
the rule
for ranking jobs for assignment to a
processor.
Three common rules for ranking are
shortest
processing
time
(SPT)
first,
longest
processing time (LPT) first, and earliest due date
(EDD) first.
11
Basic scheduling theory for a one machine
system maintains that the mean flow time is minimized by sequencing jobs in non-decreasing order of
processing time,
first.
shortest processing time (SPT)
Reversing the order to rank tasks in de-
creasing order of processing time, longest process-
ing time
(LPT)
tends to minimize differences in
processor utilization.
Mean tardiness is minimized
by sequencing jobs in non-decreasing order of due
dates, earliest due date (EDD) first. By reducing
the mean tardiness, the proportion of tardy jobs
should decrease.
When jobs are grouped by product, the sum of
the job quantities for each product are analogous
to processing time. Ranking by SPT, for grouping,
is simply ordering products in increasing order of
demand.
The grouped equivalent of EDD is referred to
as Weighted Average Due Date (WADD) [Smith, 1992].
The WADD is calculated by summing the product of
the quantity and due date for every job in the
group, then dividing by the sum of the quantities
for the jobs in the group.
For this study the two settings used were to
rank jobs/groups by either LPT or EDD.
Processor
Assignment
(G:Pr_Ass_Ord)
determines
which processor a task (or product) is assigned to.
The most common rule calls for assigning tasks to
the next available processor. If more than one is
available, then the rule should specify whether it
goes to the fastest or slowest processor.
In this
12
experiment, the variable is whether to assign the
job/group to the fastest or slowest available
processor.
Processor Sequencing (H:Pr_Grp_Seq) determines how
jobs/groups (products) will be sequenced after they
have been assigned to a processor.
The two conditions for this effect are either to run in the same
sequence as it was assigned to the processor, or to
optimize the sequence to reduce set-up times based
on the set-up time matrix.
Job Sequence (I:Job_Seq) is how tasks are sequenced
within a product group,
when appropriate.
variable settings are SPT or EDD.
The
Job related
Product Demand Distribution (J:Product_Di) investi-
gates how the relative demand for individual prod-
ucts affects system performance.
Two situations
will be evaluated.
The first assumes the demand
for each product is equal so each product will have
tasks requiring the same amount of processor capacity.
The second setting assumes the Pareto effect
where 20% of the products are represented by 80% of
the demand.
Job Size Distribution
Job_Dis) will evaluate
whether fixed job sizes or a distribution of job
(K:
sizes improves system performance.
A normal distribution will be used to generate the variablequantity tasks. In experiments one and two the two
settings used were job quantities with a normal
distribution mean of 1000 units and standard devia-
13
tions of either 50 or 300 units.
In experiments
three and four the contrast was increased by using
one level with a fixed quantity of 1000 units and
the other with the N[1000,300] distribution.
Other
Set-up Considered (L:Setup_Con)
explores whether
including set-up times when assigning tasks to
processors affects system performance.
The alternative is to ignore set-up times until after all
tasks have been assigned to processors.
Set-up
times would then be included in the analysis of the
system performance.
The experimental variable (or effects) and their settings
are summarized, by experiment, in Table 1 on the following
page.
Experiment Design
The experimental designs were generated by a commercial
software package (Statgraphics 5.1).
The first two experi-
ments were designed to screen the main effects and interactions for significance.
Plackett-Burmann screening matrices
were used as they are powerful, statistically derived designs
to evaluate main effects in a minimal number of runs.
Experiment one
folded Plackett-Burmann
design of resolution IV to evaluate main effects only.
A
is
a
24
run,
folded Plackett-Burmann design is constructed by doubling a
resolution III design.
The original matrix is doubled by
adding the negative of the original design matrix.
The high
number of degrees of freedom (13) allows for a very powerful
test of the main effects at the expense of evaluating effect
14
Table 1
Experimental Factors and Settings Used in Experiments 1-4
Effect
Experiment
1
A:Pr_Spread
Low (-)
2
x
x
High (+)
20
80
20
80
B:Middle_Pr
Low
(-)
x
x
50
50
High (+)
75
75
C:Percent_Ca
Low
(-)
High (+)
D:Set_up_Tim
Low
(-)
High (+)
E:Grouping
Low (-)
High (+)
F:Pr_assign
Low (-)
High (+)
G:Pr_Ass_Ord
Low (-)
High (+)
H:Pr_Grp_Seq
Low (-)
High (+)
I:Job_Seq
Low (-)
High (+)
J:Product Di
Low
(-)
High (+)
K:Job_Dis
Low (-)
High (+)
x
x
75
90
75
90
x
U[0,3]
U[0,12]
x
U[0,3]
U[0,12]
x
No
Yes
x
No
Yes
x
LPT
EDD
x
LPT
EDD
x
Slowest
Fastest
x
Slowest
Fastest
x
Chrono
Optimi
Chrono
Optimi
x
75
90
90
x
x
U[0,5]
x
SPT
EDD
x
Equal
Pareto
x
Equal
Pareto
x
N[1000,50]
N[1000,300]
0
U[0,5]
Yes,
not a
variable
x
Chrono
Optimi
x
N[1000,0]
N[1000,300]
x
No
Yes
High (+)
Chrono
Optimi
x
75
0
L:Setup_Con
Low (-)
U
N
4
x
x
SPT
EDD
x
N[1000,50]
N[1000,300]
3
uniform distribution
normal distribution
chronological order
order based on set-up optimization
x
N[1000,0]
N[1000,300]
15
interactions [Plackett, 1946].
Experiment two is a 32 run, sixty-fourth fractional
factorial design, also of resolution IV, that gives up eight
error degrees of freedom in order to evaluate confounded
second order interactions.
Since the two-factor interactions are confounded, they can not be evaluated individually.
However, they will indicate whether there are any
significant interactions that should be planned for in
subsequent experiments.
After experiments one and two were run and the significant effects identified, 2' full-factorial experiments were
designed that would explore the significant effects in
detail.
Full-factorial experiments have a Resolution of V+
which means all combinations of the effects are examined.
Interactions up n-factors, where n is the number of effects
under study, are available for study. However, including
all possible interactions requires the use of all 2' degrees
of freedom which eliminates the possibility of correcting
for experimental error. To overcome this shortcoming, all
interactions greater than second order were ignored. The
three-, and four-factor interactions were pooled to provide
an estimate of the error term. This procedure is commonly
accepted as the probability of three- and four-factor interactions being significant is very remote.
Job Sets
The job sets used to evaluate the effects are listed in
Appendix D.
They were constructed using a commercial
spreadsheet program (Quattro Pro 3.0).
Product type, quan-
tity, and due date were all generated from a built-in random
number generator and Normal Distribution Tables.
16
The mean job quantity for jobs in all four experiments
was 1000 units and the processor capacity mean was about 100
units per hour. The scheduling window was 156 hours in
experiments one and two and 100 hours for experiments three
and four. This led to each experiment running an average of
about 40 jobs.
Statistical Evaluations
Statgraphics 5.1 was used to generate the experimental
designs and calculate the necessary statistics.
The calculations included main effects, two-factor interactions,
analysis of variance tables, and regression calculations.
All graphs necessary to clarify and interpret data where
also generated by the Statgraphics software.
Simulation Explanation
The simulation was carried out using a spreadsheet
program. An example of one run and an explanation of the
spreadsheet mechanics is contained in Appendix E.
The simulation was run by inserting the appropriate job
set into the spreadsheet. The ranking of the jobs was then
manipulated manually per the settings specified in the
experiment design.
The spreadsheet was programmed to calculate all performance measurements.
These measurements were
transferred into Statgraphics for generation of the necessary statistics.
17
RESULTS
III.
The worksheets showing the conditions for each run and
the response variables are in Appendix A.
The estimated
effects and analysis of variance (ANOVA) tables for each
experiment are listed below.
Discussion is reserved for the
next section.
The
estimated
effects
are
calculated
using
Yate's
Algorithm [Box, 1978].
The effect is interpreted as the
change in the performance measurement as a independent
variable is changed from its "low" level to its "high" level.
The +/value indicates the confidence interval for the
effect.
For the purpose of this research all confidence
levels are 95% unless specifically stated otherwise.
The ANOVA tables were also calculated using Yate's
Algorithm.
The Sum of Squares (SS) measures deviations from
the predicted values obtained using the estimated effects.
The key parameter in the ANOVA table is the P-value.
Values
less than 1 minus the confidence level (in this case 1
.95
= 0.05) indicate significant effects and interactions.
In experiments two, three, and four, interactions greater
than second order were assumed to be insignificant so that
their Sum of Squares could be pooled to estimate the error.
This is a commonly used assumption in experimental designs.
Regression coefficients can be calculated from the raw
data to determine a line of best fit [Walpole, 1989].
These
coefficients are only meaningful for significant effects so
only significant effects were calculated.
The general form of
the regression equation for each response is
y = bo + b1
* x1 + b22
*
x2
18
where bo is a constant,
bi is the coefficient for each independent variable i,
and
xi is the value of independent variable i.
When xi is a qualitative effect, it takes the value -1 or
+1 depending upon the setting of the variable.
Regression coefficients are used to define an equation
for predicting a response given the input variables (effect
values).
Two measurements used to evaluate the accuracy of
the equation are the correlation coefficient and the coefficient of determination.
The multiple correlation coefficient
measures how
well the equation fits the data on a scale of -1 to 1. Zero
indicates there is no fit, -1 indicates a perfect, negative
correlation, and 1 indicates a perfect fit.
The multiple
coefficient of determination (R2) is the square of the
correlation coefficient.
(r)
It indicates what proportion of the
variation in the response is accounted for in the regression
equation.
Experiment One
The calculated statistics for experiment 1 are contained
in Tables 2
10.
Since this experiment was of a PlackettBurmann design, no interactions were examined.
Variables
identified as being significant on a 95% level for each of the
three performance measurements were as follows:
Mean Flow
Set-up Time
Grouping
Processor Sequencing
19
Job Sequence
Product Demand Distribution
Proportion of Jobs Tardy
Processor Sequencing
Processor Utilization Spread
None
Table 2
Estimated Effects for Mean Flow
mean flow
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
=
=
=
=
=
=
=
79.6094
-1.6625
-0.935833
8.11889
7.61083
11.0775
0.254167
-2.08917
12.2725
8.96083
7.18417
+/+/+/+/+/+/+/+/+/+/+/-
Exp 1
1.52285
2.8894
2.8894
3.85254
2.8894
2.8894
2.8894
2.8894
2.8894
2.8894
2.8894
Standard error estimated from total error with 13 d.f.
(t = 2.16092)
Table 3
ANOVA for Mean Flow
Effect
Sum of Squares
A:Pr_Spread
B:Middle Pr
C:Percent_Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
Total error
Total (corr.)
R-squared = 0.82405
16.583438
5.254704
222.467704
347.548704
736.266037
.387604
26.187704
903.685538
481.779204
309.673504
651.195054
3701.02920
DF
1
1
1
1
1
Exp
Mean Sq.
16.58344
5.25470
222.46770
347.54870
736.26604
1
.38760
1
26.18770
903.68554
481.77920
309.67350
50.09193
1
1
1
13
1
F-Ratio
.33
.10
4.44
6.94
14.70
.01
.52
18.04
9.62
6.18
P-value
.5809
.7546
.0551
.0206
.0021
.9322
.4900
.0010
.0084
.0273
23
R-squared (adj.
for d.f.) = 0.688704
20
Table 4
Regression Coeffs. for Mean Flow
constant
D:Set_up_Tim
E:Grouping
H:Pr_Grp_Seq
I:Job_Seq
J:ProductDi
=
=
=
=
=
=
Exp 1
78.5946
3.80542
5.53875
6.13625
4.48042
3.59208
Table 5
Estimated Effects for Proportion Jobs Tardy
mean pro.tar
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
=
=
=
=
=
=
=
=
=
=
=
20.6414
4.4275
1.3225
4.60111
7.47583
4.44917
5.16083
0.755833
12.6058
3.90083
0.1175
+/+/+/+/+/+/+/+/+/+/+/-
Exp 1
2.67438
5.07428
5.07428
6.76571
5.07428
5.07428
5.07428
5.07428
5.07428
5.07428
5.07428
Standard error estimated from total error with 13 d.f.
(t = 2.16092)
Table 6
ANOVA for Proportion Jobs Tardy
Effect
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
Total error
Total (corr.)
Sum of Squares
R-squared = 0.481053
DF
Mean Sq.
117.61654
10.49404
71.44950
335.32850
118.77050
159.80520
3.42770
953.44220
91.29900
1
1
117.61654
10.49404
71.44950
335.32850
118.77050
159.80520
3.42770
953.44220
91.29900
.08284
1
.08284
2008.37032
13
154.49002
3870.08636
23
1
1
1
1
1
1
1
Exp 1
F-Ratio
.76
.07
.46
2.17
.77
1.03
.02
6.17
.59
.00
P-value
.4079
.8012
.5155
.1645
.4057
.3277
.8854
.0274
.4638
.9821
R-squared (adj. for d.f.) = 0.0818627
21
Table 7
Regression Coeffs. for Proportion Jobs Tardy
constant
=
H:Pr_Grp_Seq =
Exp 1
20.0662
6.30292
Table 8
Estimated Effects for Processor Utilization Spread
mean pr.ut.sp
A:Pr Spread
B:Middle_Pr
C:Percent Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
=
=
=
=
=
=
=
I:JoTD Seq
J:Proauct_Di
=
21.574
-10.707
-8.82702
-2.27952
-5.48202
-18.3613
17.9604
-4.46131
9.88964
8.08964
10.7446
Exp 1
+/- 5.07299
+/9.62531
+/9.62531
+/- 12.8338
+/9.62531
+/9.62531
+/9.62531
+/9.62531
+/9.62531
+/- 9.62531
+/- 9.62531
Standard error estimated from total error with 13 d.f.
(t = 2.16092)
Table 9
ANOVA for Processor Utilization Spread
Effect
Sum of Squares DF
A:Pr_Spread
B:Middle_Pr
C:Percent Ca
D:Set_up_Tim
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
687.84231
467.49822
17.53725
180.31559
2022.82586
1935.44683
119.41963
586.83007
392.65381
692.68395
7226.43908
I:Joi;_Seq
J:Product_Di
Total error
Total (corr.)
14329.4926
R-squared = 0.495695
1
1
1
1
1
1
1
1
1
1
13
Exp 1
Mean Sq.
F-Ratio
P-value
687.8423
467.4982
17.5372
180.3156
2022.8259
1935.4468
119.4196
586.8301
392.6538
692.6839
555.8799
1.24
.2861
.3854
.8636
.5847
.0788
.0848
.6555
.3229
.4247
.2845
.84
.03
.32
3.64
3.48
.21
1.06
.71
1.25
23
R-squared (adj.
for d.f.) = 0.107768
Table 10
Regression Coeffs. for Processor Utilization Spread
No significant effects.
Exp 1
22
Experiment Two
The calculated statistics for Experiment Two are contained in Tables 11
16.
Since this experiment has confound-
ed effects, a regression equation could not be calculated.
Interactions greater than second order were used to increase
the degrees of freedom of the error estimate.
Variables
identified as being significant on a 95% level for each of the
three performance measurements were as follows:
Mean Flow
Loading (Percent Capacity)
Set-up Time
Confounded Interaction AD + CI + EJ + FG
Proportion of Jobs Tardy
Loading (Percent Capacity)
Set-up Time
Ranking for Processor Assignment
Job Size Distribution
Processor Utilization Spread
Grouping
Product Demand Distribution
Confounded Interaction AD + CI + EJ + FG
23
Table 11
Estimated Effects for Mean Flow
mean flow
A:Pr_Spread
B:Middle_pr
C:Percent_Ca
D:Set_Up_Tim
E:Grouping
F:Pr_Assign
G:Gr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
K:Job_Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
DR + CK + EG
CE + DH + GK
EF + GJ + IK
+
+
JK
HJ
FG
+
HK =
+
+
+
FK =
EH =
DE =
+
FI =
+
+
FJ =
+
=
=
=
IJ =
79.2359
3.78688
4.20688
16.3444
12.2331
7.16437
1.47062
2.22187
4.63062
3.54813
5.88937
2.39562
1.28438
0.834375
9.70812
0.210625
1.20313
0.374375
0.799375
7.34312
3.11562
0.976875
0.679375
0.596875
6.72313
2.72687
4.40937
+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/-
Exp 2
1.63831
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
3.27662
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
24
Table 12
ANOVA for Mean Flow
Effect
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_Up Ti
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_seq
J:Product Di
K:Job_Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
BH + CK + EG
CE + DH + GK
EF + GJ + IK
Total error
Sum of Squares
+ JK
+ HJ
+ FG
+ HK
+ FK
+ EH
+ DE
+ FI
+ FJ
+ IJ
Total (corr.)
R-squared = 0.93842
114.72338
141.58238
2137.10875
1197.19478
410.62615
17.30190
39.49383
171.54150
100.71353
277.47790
45.91215
13.19695
5.56945
753.98153
DF
Exp
Mean Sq.
F-Ratio
P-value
1.34
1.65
24.88
13.94
4.78
.00
.13
.01
.06
.3000
.2554
.0041
.0135
.0804
.6769
.5347
.2167
.3283
.1322
.5049
.7152
.8117
.0314
.9519
.7322
.9146
.8194
.0751
.3950
.7806
.8460
.8645
.0954
.4517
.2362
1
114.7234
141.5824
2137.1088
1197.1948
410.6262
17.3019
39.4938
171.5415
100.7135
277.4779
45.9122
13.1970
5.5695
753.9815
.35490
1
.3549
11.58008
1.12125
5.11200
431.37188
77.65695
7.63428
3.69240
2.85008
361.60328
59.48678
155.54070
429.44999
1
1
1
5
11.5801
1.1213
5.1120
431.3719
77.6570
7.6343
3.6924
2.8501
361.6033
59.4868
155.5407
85.8900
6973.87877
31
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
.20
.46
2.00
1.17
3.23
.53
.15
.06
8.78
5.02
.90
.09
.04
.03
4.21
.69
1.81
R-squared (adj. for d.f.)
=
0.618205
25
Table 13
Estimated Effects for Proportion Jobs Tardy
mean pro.tar
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_Up_Ti
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product Di
K:Job_Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
BH + CK + EG
CE + DH + GK
EF + GJ + IK
=
=
+
JK
HJ
FG
+
HK
=
+
+
+
FK
EH
DE
=
=
=
+
FI =
+
+
FJ =
IJ =
+
+
=
31.8906
5.25625
-1.63125
15.3687
9.71875
-8.74375
-10.1437
-2.25625
-6.26875
1.78125
3.36875
16.9562
5.44375
1.26875
4.21875
-3.84375
1.55625
5.96875
7.18125
5.15625
-7.35625
-1.71875
-1.24375
-1.93125
-4.95625
-5.70625
3.98125
+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/-
Exp 2
1.77382
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
3.54765
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
26
Table 14
ANOVA for Proportion Jobs Tardy
Effect
A:Pr Spread
B:Miadle_Pr
C:Percent Ca
D:Set_Up_Ti
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product Di
K:Job_Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
BH + CK + EG
CE + DH + GK
EF + GJ + IK
Total error
Total (corr.)
Sum of Squares
221.02531
21.28781
1889.58781
755.63281
611.62531
823.16531
40.72531
314.37781
25.38281
90.78781
2300.11531
237.07531
12.87781
142.38281
118.19531
19.37531
285.00781
412.56281
212.69531
432.91531
23.63281
12.37531
29.83781
196.51531
260.49031
126.80281
503.43156
+ JK
+ HJ
+ FG
+ HK
+ FK
+ EH
+ DE
+ FI
+ FJ
+ IJ
10119.8872
R-squared = 0.950253
DF
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
Exp 2
Mean Sq.
F-Ratio
P-value
221.0253
21.2878
1889.5878
755.6328
611.6253
823.1653
40.7253
314.3778
25.3828
90.7878
2300.1153
237.0753
12.8778
142.3828
118.1953
19.3753
285.0078
412.5628
212.6953
432.9153
23.6328
12.3753
29.8378
196.5153
260.4903
126.8028
100.6863
2.20
.1985
.6696
.0075
.0408
.0569
.0354
.5592
.1375
.6420
.3956
.0050
.1855
.7389
.2878
.3281
.6837
.1533
.0988
.2058
.0928
.6535
.7438
.6151
.2212
.1686
.3127
.21
18.77
7.50
6.07
8.18
.40
3.12
.25
.90
22.84
2.35
.13
1.41
1.17
.19
2.83
4.10
2.11
4.30
.23
.12
.30
1.95
2.59
1.26
31
R-squared (adj. for d.f.)
=
0.69157
27
Table 15
Estimated Effects for Processor Utilization Spread
mean pr. ut.
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_Up_Ti
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
K:Job Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
BH + CK + EG
CE + DH + GK
EF + GJ + IK
sp.
+ JK
+ HJ
+ FG
=
=
+ HK
=
+ FK
+ EH
+ DE
=
=
=
+ FI
=
+ FJ
+ IJ
=
=
=
32.7187
7.8125
-4.8125
14.0625
9.0625
39.9375
-10.3125
-15.0625
-8.3125
6.8125
33.9375
5.5625
-0.1875
2.4375
38.9375
3.5625
-16.9375
-11.9375
5.3125
-1.0625
-0.6875
-0.5625
6.3125
-4.8125
-14.3125
5.0625
-17.5625
+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/+/-
Exp 2
5.13469
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
10.2694
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
28
Table 16
ANOVA for Processor Utilization Spread
Effect
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_Up_Ti
E:Grouping
F:Pr_Assign
G:Pr_Ass_Ord
H:Pr_Grp_Seq
I:Job_Seq
J:Product_Di
K:Job_Dis
AB + CF + GI
AC + BF + DI
AD + CI + EJ
AE + DJ + HI
AF + BC + DG
AG + BI + DF
AH + CJ + EI
AI + BG + CD
AJ + BK + CH
AK + BJ + FH
BD + CG + EK
BE + DK + GH
BH + CK + EG
CE + DH + GK
EF + GJ + IK
Total error
Total (corr.)
Sum of Squares
488.2813
185.2813
1582.0312
657.0313
12760.0312
850.7812
1815.0312
552.7813
371.2812
9214.0312
247.5313
+ JK
+ HJ
+ FG
+ HK
+ FK
+ EH
+ DE
+ FI
+ FJ
+ IJ
.2813
47.5312
12129.0313
101.5313
2295.0313
1140.0312
225.7812
9.0313
3.7812
2.5313
318.7812
185.2813
1638.7813
205.0313
2467.5313
4218.4062
53712.4688
R-squared = 0.921463
DF
Exp
2
Mean Sq.
F-Ratio
P-value
.58
.22
1
488.281
185.281
1582.031
657.031
12760.031
850.781
1815.031
552.781
371.281
9214.031
247.531
1
.281
1
47.531
12129.031
101.531
2295.031
1140.031
225.781
9.031
3.781
2.531
318.781
185.281
1638.781
205.031
2467.531
843.681
.4889
.6638
.2292
.4269
.0115
.3614
.2024
.4633
.5432
.0214
.6168
.9863
.8242
.0127
.7463
.1600
.2975
.6323
.9227
.9499
.9590
.5719
.6638
.2222
.6479
.1479
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
1.88
.78
15.12
1.01
2.15
.66
.44
10.92
.29
.00
.06
14.38
.12
2.72
1.35
.27
.01
.00
.00
.38
.22
1.94
.24
2.92
31
R-squared (adj. for d.f.)
=
0.513072
29
Selection of Effects for Study in Experiments Three and Four
The results of experiments one and two (Table 17) were
not conclusive as to which effects warranted closer study.
Grouping (E) was identified in two of the six cases as being
significant. Since it is difficult to apply sequencing rules
in a consistent fashion between grouped and ungrouped jobs, it
was decided to run two separate experiments, one not grouping
jobs (experiment three) and one grouping jobs (experiment
four).
In order to keep the number of runs reasonable, the
number of effects was limited to four so full factorial
experimental designs could be used. By using a full factorial
design two-factor interactions could be identified because
there would be no confounding. A four-factor design requires
16 runs per experiment, 32 runs total for both experiments,
while a five-factor design requires 32 runs per experiment for
a total of 64.
Loading
(C)
and Set-up Times
(D)
were found to be
significant in two and three of the six measurements so they
were to be used in both experiments.
Job Size Distribution (K) was included in both experi-
ments even though it was only found to be significant in
affecting the proportion of orders tardy.
It is worth
investigating in detail because of the debate between whether
using fixed lot sizes is better than allowing lot sizes to
vary in manufacturing operations.
The results of the first
two experiments were inconclusive.
Processor Sequencing (H) was also found to be significant
in two cases.
However it is more related to grouped jobs so
it was included in experiment four only.
discussed further in the next paragraph.
The reasoning is
30
Table 17
Summary of Results
Experiments 1 and 2
Significant Effects
Mean Flow
Effect
E 1
E 2
Prop. Tardy
E 1
E 2
Pr.Ut.Sp
E 1
E 2
A:Pr_Spread
B:Middle_Pr
C:Percent_Ca
D:Set_up_Tim
x
E:Grouping
x
x
x
x
x
x
F:Pr_Assign
x
G:Pr_Ass_Ord
H:Pr_Grp_Seci
x
I:Job_Seq
x
J:Product_Di
x
x
x
K:Job_Dis
x
L:Setup_Con
Confound.
Interact.
x
x
31
Experiment three added a new variable referred to as
"Set-up Considered".
During the running of the first two
experiments, the question arose of whether set-up times should
be included when jobs/groups were being assigned to a proces-
sor or ignored until the jobs had already been assigned and
were being sequenced.
To allow the set-up times to be
considered during assignment requires the jobs to be sequenced
on the processor in the order they are assigned since set-up
times are sequence dependent. This was another justification
for running separate experiments for grouped and ungrouped
jobs.
It is also why the processor sequencing effect was
evaluated in experiment four only.
Job sequence
(I)
was not included for further study
because it has been investigated extensively in the literature
[Johnson, 1974] and was only significant in one of the first
two experiments.
The results of the screening experiments
indicated that no new information could be expected and with
a limit of four effects, did not warrant further study.
The other effect identified as significant was Product
Demand Distribution (K).
It was not included for further
study because of the four effect limitation.
Experiment Three
Experiment three is a full 24 factorial design.
It used
the results of experiments one and two to focus on effects
that were identified as being significant.
experiment
four with
It is identical to
the exception
that experiment four
grouped all tasks by product while experiment three scheduled
each job independently.
This eliminated Grouping as a
variable.
matrices.
They both used the same job sets and set-up time
32
calculated statistics for experiment three are
contained in Tables 18
26.
Interactions greater than second
order were used to increase the degrees of freedom of the
error estimate. Variables identified as being significant on
a 95% level for each of the three performance measurements
were as follows:
The
Mean Flow
Percent Capacity
Set-up Time
Job Size Distribution
Two factor interactions involving Percent Capacity
and the other two main effects.
Proportion of Tasks Tardy
Percent Capacity
Set-up Time
Processor Utilization Spread
None
33
Table 18
Estimated Effects for MEAN FLOW
mean flow
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_Con
CD
CK
=
=
CL
DK
DL
KL
51.1506
10.1788
10.0613
0.66375
-0.05375
1.37875
0.85375
0.01375
-0.27375
-0.05375
-0.29375
+/+/+/+/+/+/+/+/+/+/+/-
Exp 3
0.146658
0.293316
0.293316
0.293316
0.293316
0.293316
0.293316
0.293316
0.293316
0.293316
0.293316
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 19
ANOVA for MEAN FLOW
Effect
Sum of Squares
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_con
CD
414.427806
404.915006
1.762256
.011556
7.603806
2.915556
.000756
.299756
.011556
.345156
1.720681
CK
CL
DK
DL
KL
Total error
Total (corr.)
834.013894
R-squared = 0.997937
Exp
3
DF
Mean Sq.
F-Ratio
P-value
1
414.42781
404.91501
1.76226
1204.26
1176.61
5.12
.0000
.0000
.0731
.8637
.0053
.0334
.9649
.4030
.8637
.3626
1
1
1
.01156
.03
1
7.60381
2.91556
22.10
8.47
.00076
.29976
.01156
.34516
.34414
.00
.87
.03
1
1
1
1
1
5
1.00
15
R-squared (adj.
for d.f.)
Table 20
Regression Coeffs. for MEAN FLOW
constant
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
CD
CK
=
=
-4.8325
0.678583
-2.5525
-5.0275
0.0919167
0.0569167
Exp 3
=
0.993811
34
Table 21
Estimated Effects for Proportion Jobs Tardy
mean pro.tar
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_Con
CD
CK
=
=
=
=
=
=
=
=
CL
DK
DL
KL
=
=
=
43.3756
35.4612
36.0862
10.1588
-0.69875
12.0663
13.8612
2.55125
-1.79625
-0.69875
-2.55125
+/+/+/+/+/+/+/+/+/+/+/-
Exp 3
3.8325
7.66501
7.66501
7.66501
7.66501
7.66501
7.66501
7.66501
7.66501
7.66501
7.66501
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 22
ANOVA for Proportion Jobs Tardy
Sum of Squares
Effect
C:Percent_ Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_Con
CD
CK
5030.00101
5208.86976
412.80081
1.95301
582.37756
768.53701
26.03551
12.90606
1.95301
26.03551
1175.04678
CL
DK
DL
KL
Total error
13246.5160
Total (corr.)
R-squared = 0.911294
DF
1
1
1
1
1
1
1
1
1
1
5
Exp 3
Mean Sq.
F-Ratio
P-value
5030.0010
5208.8698
412.8008
1.9530
582.3776
768.5370
26.0355
12.9061
1.9530
26.0355
235.0094
21.40
22.16
1.76
.0057
.0053
.2424
.9318
.1763
.1303
.7561
.8264
.9318
.7561
.01
2.48
3.27
.11
.05
.01
.11
15
R-squared (adj. for d.f.)
=
0.733882
Table 23
Regression Coeffs. for Proportion Jobs Tardy
constant
C:Percent_Ca
D:Set_Up_Tim
=
=
=
-151.661
2.36408
18.0431
Exp 3
35
Table 24
Estimated Effects for Processor Utilization Spread
mean pr.ut.sp
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_Con
CD
CK
CL
DK
DL
KL
8.65
-2.0625
6.135
-3.2125
-2.0675
0.7725
-1.245
0.215
-1.3275
-2.0675
2.04
+/+/+/+/+/+/+/+/+/+/+/-
Exp 3
1.2841
2.5682
2.5682
2.5682
2.5682
2.5682
2.5682
2.5682
2.5682
2.5682
2.5682
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 25
ANOVA for Processor Utilization spread
Effect
Sum of Squares
C:Percent_Ca
D:Set_Up_Tim
K:Job_Dis
L:Setup_Con
CD
DF
1
1
1
Total error
17.015625
150.552900
41.280625
17.098225
2.387025
6.200100
.184900
7.049025
17.098225
16.646400
131.913350
Total (corr.)
407.426400
15
CK
CL
DK
DL
KL
R-squared = 0.676228
1
1
1
Mean Sq.
17.01563
150.55290
41.28062
17.09823
2.38702
6.20010
1
.18490
1
1
7.04903
17.09823
16.64640
26.38267
1
5
Exp
3
F-Ratio
P-value
.64
.4665
.0625
2663
.4655
.7787
.6533
.9374
.6325
.4655
.4711
5.71
1.56
.65
.09
.24
.01
.27
.65
.63
R-squared (adj. for d.f.) = 0.0286833
Table 26
Regression Coeffs. for Processor Utilization Spread
No significant effects or interactions.
Exp 3
36
Experiment Four
Experiment Four is also a full 24 factorial design.
The
calculated statistics for Experiment Four are contained in
Tables 27
35.
Interactions greater than second order were
used to increase the degrees of freedom of the error estimate.
Variables identified as being significant on a 95% level
for
each of
the
three performance measurements were
as
follows:
Mean Flow
Percent Capacity
Set-up Time
Processor Sequencing
Interaction between the percent capacity and
optimizing the processor sequence.
Proportion of Jobs Tardy
Percent Capacity
Job Size Distribution
Interaction of these two main effects
Processor Utilization Spread
Percent Capacity
Set-up Time
Job Size Distribution
All two factor interactions of these three main
effects.
The results from Experiments Three and Four are summarized in Table 36.
37
Table 27
Estimated Effects for Mean Flow
mean flow
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
CH
CK
DH
DK
HK
=
=
=
=
=
=
=
=
=
=
=
47.0856
7.66375
2.01375
-0.25375
-0.67375
-0.10375
0.21375
0.12875
-0.47125
0.08875
-0.15375
+/+/+/+/+/+/+/+/+/+/+/-
Exp 4
0.0906629
0.181326
0.181326
0.181326
0.181326
0.181326
0.181326
0.181326
0.181326
0.181326
0.181326
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 28
ANOVA for Mean Flow
Effect
Sum of Squares
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
CH
CK
DH
DK
HK
Total error
DF
Mean Sq.
F-Ratio
P-value
234.932256
16.220756
1
234.93226
16.22076
.257556
1.815756
.043056
.182756
.066306
.888306
.031506
.094556
.657581
1
1786.34
123.34
1.96
13.81
.0000
.0001
.2206
.0138
.5978
.2915
.5166
.0483
.6502
.4438
1
1
1
1
1
1
1
1
5
255.190394
Total (corr.)
Exp 4
R-squared = 0.997423
.25756
1.81576
.04306
.18276
.06631
.88831
.03151
.09456
.13152
.33
1.39
.50
6.75
.24
.72
15
R-squared (adj.
for d.f.)
Table 29
Regression Coeffs. for Mean Flow
constant
H:Pr_Grp_Seq
C:Percent_Ca
D:Set_Up_Ti
K:Job_Dis
DH
=
=
=
=
=
=
4.935
-1.26875
0.510917
1.00687
-0.336875
-0.235625
Exp 4
=
0.99227
38
Table 30
Estimated Effects for Proportion Jobs Tardy
mean pro.tar
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
CH
CK
DH
DK
HK
=
=
=
=
=
=
=
=
=
=
=
Exp 4
15.3175 +/- 0.65541
14.73
+/- 1.31082
1.6
+/- 1.31082
4.1675 +/- 1.31082
-16.415 +/- 1.31082
-0.675 +/- 1.31082
4.1675 +/- 1.31082
-14.14
+/- 1.31082
0.4625 +/- 1.31082
-1.6
+/- 1.31082
1.8125 +/- 1.31082
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 31
ANOVA for Proportion Jobs Tardy
Effect
Sum of Squares
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
867.89160
10.24000
69.47222
1077.80890
1.82250
69.47223
799.75840
CH
CK
DH
DK
HK
Total error
Total (corr.)
Exp 4
DF
Mean Sq.
F-Ratio
P-value
1
867.8916
10.2400
69.4722
1077.8089
1.8225
69.4722
799.7584
126.28
1.49
10.11
156.82
.0001
.2767
.0246
.0001
.6338
.0246
.0001
.7422
.2767
.2253
1
1
1
1
1
1
.27
10.11
116.36
.85562
1
.8556
.12
10.24000
13.14063
34.36500
1
10.2400
13.1406
6.8730
1.49
1.91
2955.06710
R-squared = 0.988371
1
5
15
R-squared (adj. for d.f.)
=
0.965112
Table 32
Regression Coeffs. for Proportion Jobs Tardy
constant
C:Percent_Ca
H:Pr_Grp_Seq
K:Job_Dis
=
=
CH
CK
=
=
=
=
-65.6975
0.982
-20.8375
69.5625
0.277833
-0.942667
Exp 4
39
Table 33
Estimated Effects for Processor Utilization Spread
mean pr.ut.sp
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
=
=
CH
CK
DH
DK
HK
=
=
=
=
=
=
=
=
=
15.7425
2.205
0.68
-0.15
9.715
-2.07
0.15
-3.405
-0.125
1.37
0.35
+/+/+/+/+/+/+/+/+/+/+/-
Exp 4
0.200471
0.400943
0.400943
0.400943
0.400943
0.400943
0.400943
0.400943
0.400943
0.400943
0.400943
Standard error estimated from total error with 5 d.f.
(t = 2.57141)
Table 34
ANOVA for Processor Utilization Spread
Effect
Sum of Squares
C:Percent_Ca
D:Set_Up_Ti
H:Pr_Grp_Seq
K:Job_Dis
CD
Total error
19.448100
1.849600
.090000
377.524900
17.139600
.090000
46.376100
.062500
7.507600
.490000
3.215100
Total (corr.)
473.793500
CH
CK
DH
DK
HK
R-squared = 0.993214
Exp
4
DF
Mean Sq.
F-Ratio
P-value
1
1
19.44810
1.84960
30.24
2.88
1
1
1
.09000
.14
377.52490
17.13960
587.11
26.65
.0027
.1507
.7275
.0000
.0036
.7275
.0004
.7710
.0189
.4315
1
.09000
.14
1
46.37610
72.12
1
1
1
.06250
.10
7.50760
11.68
.49000
.64302
.76
5
15
R-squared (adj. for d.f.)
=
0.979642
Table 35
Regression Coeffs. for Processor Utilization Spread
constant
C:Percent_Ca
D:Set_Up_Ti
K:Job_Dis
CD
CK
DK
=
=
=
=
=
=
=
3.615
0.147
11.725
23.585
-0.138
-0.227
0.685
4
40
Table 36
Summary of Results
Experiments 3 and 4
Significant Effects and Two-factor Interactions
Effect
Mean Flow
Prop. Tardy
Pr.Ut.Sp
E 3
E 4
E 3
E 4
C:Percent_Ca
x
x
x
x
D:Set_up Tim
x
x
x
H:Pr_Grp_Seq
K:Job_Dis
x
E 3
E 4
x
x
x
x
x
x
x
L:Setup_Con
CD
x
x
CH
x
CK
x
x
x
CL
DH
x
DK
x
DL
(
x
indicates significant effect)
indicates effect not included)
41
IV.
CONCLUSIONS AND RECOMMENDATIONS
Before discussing the results in terms of the performance
measures, an explanation of the non-significant variables is
in order.
The system variables, "Processor Spread" (A) and "Proces-
sor Distribution" (B) were not found to be significant in any
experiment under any measurement.
As processor capacity
spread is increased, more of the workload is shifted to the
faster processors.
However, the scheduled load in terms of
capacity is not influenced by the distribution.
Factors such
as grouping and set-up time have a far greater impact that
effectively masks any influence by the processor capacity
variables.
A surprising result was that the "Ranking for Processor
Assignment" (F) was not found to be significant in any
situation.
The loading of the system has such a dominant
influence on the system that the Ranking effect cannot be
detected.
The "Processor Assignment" (G) variable was also insignificant in all cases. This is not surprising as it is only
used to break ties between two or more processors that are
available at the same time. Since the chances of two processors being available at exactly the same time, except for the
start time zero, are very remote, the variable was never
really used.
In all four experiments time was recorded to
0.001 hours.
42
Mean Flow Time
Experiments one and two indicated five main effects
significantly influence mean flow times.
The level of loading in terms of percent capacity was
significant in three of the four experiments and marginally
significant (p=0.0551) in the fourth. This result is totally
consistent with expectations. As the loading increases, the
time to complete all tasks increases and in a static system
the average completion time is equal to the mean flow.
The
effect increases the mean flow by 16 to 20% when the quantity
of tasks increases from the 75% to 90% level.
Increasing set-up time also increased the mean flow,
again consistent with expectations. Experiments one and two
used two different set-up time distributions (U[0,5] and
U[0,12]) while the last two used either no set-up times or a
U[0,5] distribution. Mean flow times increased by 10
15%
when set-up times were increased from U[0,5] to U[0,12]. When
going from no set-up time to U[0,5] when tasks are not grouped
by product increases mean flow by almost 20%.
If tasks are
grouped by product, the same set-up time distribution only
increases mean flow by about 4%.
Experiment one estimated that grouping tasks by product
decreased the mean flow time by 13.9%. The average mean flow
time was about 8% lower on the grouped experiment four than
the ungrouped experiment three.
Both experiments used the
same job sets and set-up matrices.
This is attributed to the
fewer number of set-ups required for grouped jobs.
The number
of set-ups is equal to the number of jobs/groups minus the
number of processors.
Grouped jobs require a maximum of 4
set-ups (experiments one and two) or 7 set-ups (experiment
four).
However the maximum number of set-ups required for
43
ungrouped jobs ranged from about 17 (20 jobs, 3 processors)to
Ungrouped jobs were requiring from 2.5 to almost 6 times
as many set-ups. Assuming an average set-up time of two hours
40.
per set-up, grouped orders require about 8 to 14 hours for
set-up time compared to 34 to 80 hours for jobs scheduled
independently.
Sequencing the product groups to minimize set-up times by
set-up time matrix optimization was found to reduce mean flow
times by 15.4% in the first experiment. The fourth experiment
did not confirm the significance of sequencing groups.
The first experiment also confirmed the literature that
ranking tasks by SPT also decreases mean flow time.
The final significant effect was product distribution.
As the product demand distribution goes from equal to a Pareto
distribution, i.e. 20% of the product groups are responsible
for 80% of demand, the mean flow increases by 9%.
Conclusions:
Increased loading and set-up times will increase mean flow time in a static system. To
minimize mean flow, tasks should be grouped by
product whenever set-up times are required.
The relationship between processors in terms
of capacities has no effect on the mean flow
times.
Neither does the method for ranking
groups/tasks for assignment to the processors.
When more than one processor is available to
process a job or product group, it doesn't
matter which processor the job or group is
assigned to.
44
Proportion Jobs Tardy
Experiment one only had one significant effect, processor
sequence.
Optimizing the sequence on a processor to minimize
set-up times decreased the percent of jobs tardy by 12.6%.
Experiment two indicated four effects were significant at
the 95% confidence interval: loading, set-up time, ranking for
processor assignment, and job size distribution.
factor interactions were significant.
No two-
Both loading and set-up time effects on the proportion of
tardy are as expected.
As loading increases, the
probability of jobs being late should and does increase. As
loading increases from 75 to 90% of capacity, the percentage
jobs
of tasks tardy increases by about 15%.
distributions from U[0,5]
Increasing set-up time
to U[0,12] increases the percent
jobs tardy by almost 10%.
Ranking product groups by weighted average due date
(WADD), or jobs by EDD when grouping isn't used, will result
in 10% fewer tardy jobs than ranking by LPT.
This result
follows theory reported in the literature that sequencing by
EDD will reduce the mean tardy time and hence proportion of
jobs tardy [Johnson, 1974].
Experiment three confirmed that increasing loading and
set-up times increased the proportion of jobs that were tardy.
With no grouping used, increases of about 36% in tardy jobs
were found for both loading and set-up times.
Experiment
four,
with grouping,
estimated
that
loading increase would result in 15% more tardy jobs.
the
Set-up
times didn't have a significant effect due to the reduced
number of set-ups required when jobs are grouped by product.
45
Experiment four confirmed the preliminary experiment
findings that optimizing the group sequence on a processor
was significant, as was the job size distribution.
As the job
distribution widens, the percentage of tardy jobs
decreases by 16%.
This finding is surprising and cannot be
explained at this time. Further research of this phenomenon
would be beneficial as new manufacturing philosophies are
size
stressing the use of either smaller or variable lot sizing.
The two-factor interaction between loading and job size
distribution was very significant with a p-value of 0.0001. At
75% loading the proportion of tardy tasks is the same for both
job size distributions.
However, at 90% loading the propor-
tion of tardy jobs is about 38% for the fixed job size
compared to about 8% for the variable job size.
tionship is shown
This rela-
graphically in Figure 1 on the next page.
This interaction can be responsible for up to 14% of the tardy
jobs.
The two-factor interaction between loading and processor
group sequencing is also significant but only accounts for 4%
of the tardy jobs.
Conclusions:
Loading significantly effects the proportion
of jobs that are tardy, regardless of whether
product grouping is used or not.
Set-up times are only significant when product
grouping is not used.
Job size distribution and processor group sequencing are important when tasks are grouped
46
Two-Factor Interaction
Job Size Distribution vs. Loading
t
13 25
0
--'c 20
0
.-e.
a
15
2
o_
10
75
90
System Loading ( %)
9 Job Size Fixed --00 Job Size Varied
Figure
1
Proportion Jobs Tardy Interaction:
Order Size Distribution versus Loading
47
by product.
In general, allowing job sizes to vary,
rather than using a fixed lob quantity will minimize the
proportion of jobs tardy.
This _is especially true_at
higher system loadings.
Processor Utilization Spread
The results of experiment one did not identify any
effects as significant at the 95% confidence level.
At the
90% confidence interval, grouping by product and the ranking
of groups or tasks for assignment to processors were significant. Grouping increased the difference between processor use
by about 18%.
Ranking groups or tasks by LPT reduces the
spread by 18% compared to ranking by EDD/WADD.
Experiment
identified grouping,
product demand
distribution, and a set of confounded, two-factor interactions
two
as significant.
The confounded interactions included the one
between the two main effects listed above. A comparison of
the effects indicates that the two-factor interaction between
grouping and the ranking is the likely significant interaction.
By grouping tasks by product in the second experiment,
the processor utilization spread increased by almost 40%. As
the product demand distribution moves away from equal demand
to the Pareto position, the spread increased by 34%. The
interaction between these main effects show an increase of
39%.
The average processor utilization spread in experiment
three (ungrouped tasks) is about one-half of average spread in
experiment four (grouped tasks), consistent with experiment
two.
48
The ungrouped study in experiment three had no signifi-
cant main effects or two-factor interactions at the 95%
confidence level.
Set-up times were significant at the 93%
level, however.
Experiment four, with tasks grouped by product, found
loading and job size distribution to be significant.
The two-
factor interactions involving all combinations of loading,
set-up time, and job size distribution were significant. The
main effect of set-up time must then be considered significant
since it was significant in an interaction.
Job size distribution has the biggest effect on the
processor spread, about 10% wider when the job size is allowed
to vary.
The other main effects and interactions are 4% or
less.
The two-factor interaction between loading and set-up
time is displayed in Figure 2.
When there are no set-up times
in the system,
the processor utilization spread increases
dramatically as loading increases.
When set-up times are
distributed U[0,5] there is essentially no change in the
spread.
At the lower loading, utilization is more equal
without set-up time, but as the loading increased the lack of
set-up time causes a wider spread than the set-up time
conditions.
Apparently, as loading increases, set-up times
can actually function like small jobs by filling in and
levelling the utilization.
Conclusions:
Grouping jobs by product will tend to increase
the difference in processor utilization.
This
is explained by the decreased flexibility in
going from many small jobs to, in effect, a
49
Two-Factor Interaction
1
'Set-up Time vs. System Loading
17.5
-0
17
cts
Pal 16.5
u)
c
1
"trsi.
15.5
0
D
8 14.5
--=:
1
0
8
14
Ci:
13.5
1
90
75
System Loading (%)
a No Set-up Time
a, Set-up Time U[0,5]
Figure 2
Processor Utilization Spread Interaction:
Set-up Time versus Loading
50
If seven jobs are divided
few large jobs.
between three processors then the probability
of being able to level processor usage is very
small due to the lack of fine tuning. As the
number of jobs is increased, and the size of
each job decreases, it is easier to balance
processor usage due to the higher degree of
The negative side
detail and hence control.
of the smaller job sizes is the strong possibility of more set-ups.
If the goal is to maximize usage on one
Or more processors
and minimize usage
of
others so as to shut down excess production,
it will again be desirable to use smaller jobs
for the finer degree of control.
As loading increases in systems with productgrouped jobs, set-up times tend to level pro-
In this case, set-up
cessor utilization.
times function as small jobs in order to help
level processor usage.
This finding may not
hold in all cases.
function
of
the
It may very well be a
between
relationship
the
magnitude of the set-up times, the job processing times, and the scheduling window.
Further research on these relationships could
help define them in stronger terms.
Fixed size jobs tend to reduce processor
utilization spread when tasks are grouped by
product.
This effect cannot be explained
within the scope of this project.
possible
this
result may be
It is very
explained by
evaluating the effect product demand distribu-
51
tion has on processor utilization in conjunction with the first conclusion stated above.
Relationship Between Performance Measures
Regression analysis was used to determine if any one of
the performance measurements could be used to predict another.
The three performance measurements had little if any correla-
tion indicated they are independent.
The processor utilization spread is almost totally
independent of the other two measures with a coefficient of
determination less than 0.10 for every experiment and performance measure.
Mean flow time and the proportion of tasks
tardy showed slight correlation with coefficients of determi-
nation ranging from 0.20 to 0.50.
Summary of Conclusions
The results of this study can provide the following
guidelines for system design and scheduling heuristics.
In general the relationships between processor capacities
need not be considered.
Jobs will be assigned to the next
available processor which will naturally level usage with
faster processors handling more jobs than slower processors.
It is no surprise that mean flow time and the proportion
of jobs tardy increases as system loading and set-up times
increase.
Increases in set-up times can be minimized by
grouping jobs by product.
52
Grouping jobs by product is beneficial at high loadings
as
it minimizes non-productive
time used for set-ups.
Optimizing the group processing sequence on a processor will
decrease set-up time
further.
Mean flow times and the
proportion of jobs tardy will be decreased at the expense of
controlling the difference in processor utilization.
Scheduling
jobs
individually gives
more
control
in
scheduling individual processors, either levelling or preclud-
ing specific processors at the expense of more set-up time,
and longer mean flow times and tardy jobs (when set-up times
are required).
Sequencing individual jobs by SPT will reduce mean flow
time on a processor.
Sequencing by EDD on a processor will
reduce the mean tardy time but not necessarily the proportion
of jobs tardy.
Future Research/Enhancements
A natural extension of this study would be to propose and
validate heuristic decision rules for scheduling.
These
heuristics could then be incorporated into a software program,
possibly incorporating artificial intelligence and/or expert
systems, to schedule parallel, non-identical processors for
both services and manufacturing.
Other areas of study suggested by this study include
dynamic systems, and the possible application of queuing
theory to solve scheduling problems.
Further research would also be valuable in validating
these results.
The references used in this research, with the
exception of Smith (1992), investigated similar areas that do
53
not overlap this area.
Since this study was based on statis-
95% confidence level, it leaves a very small
possibility for error which should be investigated.
tics
at
a
54
V.
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T.A.,
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VII.
Appendices
60
Experiment Data
Appendix A:
Experiment 1
Response variables
Experimental factors
No.
A
B
C
D
E
F
G
H
I
J
Name
Pr_Spread
Middle_Pr
Percent_Ca
Set_up Tim
Grouping
Pr_Assign
Pr_Ass_Ord
Pr_Grp Seq
Job_Seq
Product_Di
Low
High
20
0.5
80
0.75
U[0,3]
Yes
LPT
Slowest
Optimize
SPT
Equal
Units
0.75
0.90
U[0,12]
No
EDD
%
%
hrs
Fastest
Chrono
EDD
Pareto
Cont.
No
Yes
Yes
No
No
No
No
No
No
No
No. Name
1
2
3
Units
Mean Flow hrs
Propor Tar %
Pr_Ut_Spre %
Effect Setting
run
A
1
2
Small
Small
Large
Small
Large
Small
Large
Large
Small
Large
Small
Small
Large
Small
Small
Small
Large
Large
Large
Large
Large
Small
Large
Small
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0.50
0.75
0.75
0.50
0.50
0.50
0.50
0.50
0.50
0.75
0.75
0.50
0.50
0.75
0.75
0.50
0.75
0.50
0.75
0.75
0.50
0.75
0.75
0.75
0.90
0.75
0.90
0.75
0.90
0.75
0.90
0.90
0.75
0.75
0.90
0.90
0.75
0.90
0.75
0.90
0.90
0.75
0.75
0.75
0.75
0.75
0.90
0.90
U[0,12]
U[0,3]
U[0,3]
U[0,12]
U(0,12]
U(0,3)
U[0,12)
U[0,3]
U[0,12]
U(0,3]
U(0,12]
U(0,3)
U(0,3)
U(0,3]
U[0,12)
U[0,3)
U[0,12]
U[0,3]
U[0,12]
U[0,12)
U[0,12]
U[0,3]
U[0,3]
U[0,12]
No
Yes
No
No
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
Yes
No
Yes
No
Yes
Yes
LPT
EDD
EDD
EDD
EDD
LPT
LPT
LPT
LPT
EDD
EDD
EDD
LPT
LPT
EDD
EDD
EDD
EDD
LPT
LPT
EDD
LPT
LPT
LPT
Fastest
Fastest
Slowest
Slowest
Slowest
Slowest
Slowest
Fastest
Fastest
Slowest
Fastest
Slowest
Slowest
Slowest
Slowest
Fastest
Fastest
Fastest
slowest
Fastest
Fastest
Fastest
Fastest
Slowest
CHRONO
CHRONO
CHRONO
CHRONO
OPTIMIZE
OPTIMIZE
OPTIMIZE
CHRONO
OPTIMIZE
OPTIMIZE.
OPTIMIZE
CHRONO
CHRONO
OPTIMIZE
OPTIMIZE
OPTIMIZE
CHRONO
OPTIMIZE
CHRONO
OPTIMIZE
CHRONO
CHRONO
OPTIMIZE
OPTIMIZE
SPT
SPT
SPT
EDD
SPT
SPT
EDD
EDD
EDD
EDD
EDD
EDD
SPT
EDD
SPT
SPT
EDD
EDD
EDD
SPT
SPT
EDD
SPT
SPT
80/20
80/20
Equal
Equal
80/20
Equal
Equal
Equal
80/20
Equal
Equal
80/20
80/20
80/20
80/20
Equal
80/20
80/20
80/20
Equal
Equal
Equal
80/20
Equal
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CD C) C) CD CD CD CD CD CD C) CD C) C) CD C) C) CD CD CD C) C) CD CD CD C) C) CD C) CD CD CD C)
Ul Ul Ul Ul O C) 0 un un C) CD 10 Ul Ul 111 Ul Ul C) C) CD C) CD C) CD Ul CD 1.11 0 Ul U1 CD CD
r- r- r- r- Ch 0) 0) Cs Cs 01
r- r- r- r- r r 01 01 01 01 01 01 CT r 01 Cs.- O)N Cs 0) 0)
un Ln Ul c) c) In Ln un Ln
U1 CD Ul CD CD CD Ul
In CD C) Ul CD CD CD C) CD CD C) LY)
C)
r- r- r- Ul Ln Cs Cs r- r- r- r- Lin r- Ul Ul Ul r- r- r- un un r- Ln un un
un 1.11 Ul r Cs Ul
CD CD CD C) C) CD CD CD CD CD CD C) CD CD CD 0 CD C) CD CD CD C) CD CD CD CD CD CD C) CD CD CD
ri
VD r- co Ch C)
Ul (.c) r- OD 0)OH N 01
U.) VD r- oo
0H
CV
ri ri H ri ri H ri rA 1-1 N N N N N N N N N N m m m
HNm
00 N 00 N 00 N CO CO N 00 00 N 00 00 N CO N N N N N N CO N CO CO N CO CO N CO N
.1 N 01.0 Ul
62
Experiment
3
Experimental factors
No. Name
C:Percent_Ca
D:Set_up_Tim
K:Job_Dis
L:Setup_Con
Response variables
Low
High
Units
75
90
%
U[0,5]
N[1000,300]
Yes
hrs
units
0
0
No
Effect Settings
run
C
1
75.
90.
75.
90.
75.
90.
75.
90.
75.
90.
75.
90.
75.
90.
75.
90.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
0
0
0
U[0,5]
U[0,5]
0
0
U[0,5]
U[0,5]
N[1000,300]
N[1000,300]
N[1000,300]
N[1000,300]
0
0
0
0
U[0,5]
U[0,5]
0
0
N[1000,300]
N[1000,300]
N[1000,300]
N[1000,300]
0
0
0
U[0,5]
U[0,5]
0
No
No
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Cont.
Yes
No
No
No
No. Name
1
2
3
Units
Mean Flow hrs
Propor_Tar %
Pr_Ut_Spre %
63
Experiment 4
Experimental factors
No. Name
C:Percent_Ca
D:Set_up_Tim
H:Pr_Grp_Seq
K:Job_Dis
Response variables
Low
High
Units
75
90
%
0
U[0,5]
Chrono
N[1000,0]
hrs
Optimize
N[1000,300]
Cont.
Yes
No
No
No
Effect Settings
run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
C
D
H
K
(%)
(hrs)
(%)
(units)
75.
90.
90.
90.
90.
90.
90.
75.
90.
75.
75.
75.
75.
75.
90.
75.
0
0
U[0,5]
0
U[0,5]
U[0,5]
0
0
0
U[0,5]
0
U[0,5]
U[0,5]
0
U[0,5]
U[0,5]
Optimize
Optimize
Chrono
Chrono
Optimize
Chrono
Optimize
Chrono
Chrono
Optimize
Optimize
Chrono
Optimize
Chrono
Optimize
Chrono
N[1000,0]
N[1000,300]
N[1000,300]
N[1000,300]
N[1000,0]
N[1000,0]
N[1000,0]
N[1000,0]
N[1000,0]
N[1000,300]
N[1000,300]
N[1000,300]
N[1000,0]
N[1000,300]
N[1000,300]
N[1000,0]
No.
1
2
3
Name
Units
Mean_Flow hrs
Propor_Tar %
Pr_Ut Sp
%
64
Appendix B:
Experiment Results
Experiment 1
run
Mean Flow
(hrs)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
106.33
67.21
79.59
96.58
73.26
59.04
83.08
74.46
69.49
66.13
74.72
92.71
79.03
78.12
76.56
76.27
103.13
75.89
99.04
71.47
62.61
83.55
65.47
72.53
Propor_Tar
(%)
46.2
21.2
14.3
33.3
25.6
9.1
37.5
20.0
0.0
11.1
26.8
15.4
31.3
0.0
0.0
2.5
31.7
18.8
34.3
16.7
9.1
35.3
17.1
24.4
Pr_Ut_Sp
(%)
5.3
84.1
14.8
57.1
53.1
11.2
13.4
24.2
46.8
32.6
6.5
84.4
3.9
2.3
1.9
1.1
19.8
10.9
143.6
8.8
2.4
11.4
12.8
12.9
65
Experiment
run
Mean Flow
(hrs)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
63.28
78.53
92.58
68.93
113.17
88.78
75.79
78.95
70.08
122.04
78.86
71.98
71.68
81.53
63.99
60.46
69.90
76.37
78.86
80.14
97.75
102.76
61.65
79.97
66.47
101.51
68.14
76.9
65.37
65.15
87.82
76.15
Propor_Tar
(%)
13.5
2.8
40.5
36.1
40.5
34.9
15.6
45.9
17.1
64.4
38.6
28.6
29.7
40.0
20.0
31.4
2.8
26.8
46.3
2.4
68.3
46.5
32.5
43.9
8.6
41.7
42.9
0.0
28.6
43.2
21.4
2
Pr_Ut_Sp
(%)
2
14
148
65
167
91
9
1
12
42
15
10
7
11
7
21
49
14
2
9
23
9
39
112
32
29
6
9
5
21
49
17
66
Experiment 3
run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Mean Flow
Pro_Tardy
(hrs)
(%)
42.63
50.00
50.86
61.96
40.81
51.04
50.08
62.04
42.63
50.00
51.16
62.62
40.81
51.04
49.51
61.22
9.09
29.62
18.18
92.59
18.18
44.44
63.63
74.07
9.09
29.62
18.18
100.00
18.18
44.44
50.63
74.07
Pr_Ut_Spre
(%)
6.46
6.59
15.60
20.59
7.54
1.74
13.69
5.26
6.46
6.59
14.14
5.62
7.54
1.74
6.02
12.82
67
Experiment 4
run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Mean Flow
Propor_Tar
(hrs)
(%)
42.64
50.07
52.11
49.22
52.06
52.14
50.28
42.64
50.28
43.17
41.76
44.74
44.51
41.74
51.18
44.83
4.54
11.11
3.70
3.70
44.44
33.33
40.74
9.09
33.33
9.09
9.09
4.54
9.09
4.54
11.11
13.64
Pr_Ut_Sp
(%)
7.08
19.70
20.30
19.70
12.00
12.00
15.38
7.08
15.38
23.40
19.40
22.50
8.08
19.50
20.30
10.08
68
Appendix C:
Set-up Time Matrices
Experiments One and Two Set-up Time Matrices
U[0,3]
FROM
de
f
g
1
1
2
3
0
1
1
3
3
3
3
0
0
3
3
2
3
TO
a
a
b
c
3
1
1
b
2
c
2
0
d
2
2
1
e
3
0
2
2
f
3
0
3
0
3
g
2
0
3
0
3
0
2
U[0,12]
a
a
b
c
d
e
f
g
6
1
11
1
6
2
7
12
11
12
6
6
10
7
12
10
1
4
2
8
b
4
c
6
4
d
2
4
7
e
6
1
7
9
f
6
9
10
2
9
g
5
10
0
4
3
9
12
69
Experiments Three and Four Set-up Time Matrix
U[0, 5]
From
To
a
b
c
d
e
f
g
h
i
j
a
0
4
3
3
4
4
1
2
4
2
b
2
0
3
3
3
1
4
3
1
2
c
5
5
0
2
4
3
1
2
2
2
d
1
4
3
0
2
4
4
3
3
5
e
5
3
3
3
0
3
5
2
3
4
f
3
3
3
5
0
0
3
4
4
2
g
4
4
2
1
5
1
0
2
3
3
h
3
3
2
3
1
4
3
0
0
2
i
1
2
4
0
1
5
5
2
0
2
j
3
4
4
3
3
4
3
3
3
0
70
Appendix D:
Experiment Job Sets
Experiments One and Two Job Sets
Pareto Product Demand Distribution
Jobs N[1000,300]
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
R.N.
0.518
0.362
0.289
0.198
0.129
0.037
0.774
0.271
0.655
0.531
0.374
0.51
0.63
0.812
0.589
0.275
0.222
0.172
0.174
0.725
0.349
0.21
0.95
0.718
0.175
0.242
0.564
0.01
0.071
0.051
0.687
0.533
0.918
0.635
0.879
0.29
0.35
0.402
0.115
0.226
0.303
0.965
0.699
Product
b
a
a
a
a
c
b
a
b
b
a
b
b
b
b
a
a
a
Due
Date
132
84
156
84
132
84
156
84
84
108
156
84
156
156
156
156
84
156
a
60
b
108
a
a
f
60
84
60
b
132
a
a
b
c
d
d
b
b
e
b
b
a
a
a
a
a
a
f
b
60
108
156
132
132
84
156
108
156
132
108
108
132
108
156
60
156
60
60
R.N.
0.9363
0.1241
0.5375
0.5662
0.208
0.1422
0.7476
0.4492
0.9938
0.8994
0.1833
0.9929
0.8037
0.8461
0.3808
0.4589
0.5296
0.2474
0.1038
0.6894
0.5824
0.9593
0.6257
0.1107
0.2886
0.1239
0.9493
0.9307
0.9702
0.7714
0.0524
0.8908
0.8794
0.0614
0.5538
0.6797
0.9085
0.6477
0.2486
0.8709
0.7494
0.4022
0.269
Z
1.52
-1.15
0.09
0.17
-0.81
-1.07
0.67
-0.13
2.5
1.28
-0.9
2.45
0.85
1.02
-0.3
-0.1
0.07
-0.68
-1.26
0.5
0.21
1.74
0.35
-1.22
-0.56
-1.16
1.64
1.48
1.89
0.74
-1.62
1.23
1.17
-1.54
0.14
0.47
1.33
0.38
-0.68
1.13
0.67
-0.25
-0.62
Job
QTY
CUM
TOTAL
1456
655
1027
1051
757
679
1201
961
1750
1384
730
1735
1255
1306
910
970
1021
796
622
1150
1063
1522
1105
634
832
652
1492
1444
1567
1222
514
1369
1351
538
1042
1141
1399
1114
796
1339
1201
925
814
1456
2111
3138
4189
4946
5625
6826
7787
9537
10921
11651
13386
14641
15947
16857
17827
18848
19644
20266
21416
22479
24001
25106
25740
26572
27224
28716
30160
31727
32949
33463
34832
36183
36721
37763
38904
40303
41417
42213
43552
44753
45678
46492
71
Equal Product Demand
Jobs N[1000,300]
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
R.N.
0.9323
0.3622
0.4075
0.9476
0.5250
0.1348
0.7402
0.8516
0.4804
0.1429
0.2218
0.3455
0.7173
0.9980
0.9357
0.5424
0.6487
0.1453
0.3742
0.6634
0.9928
0.5946
0.7355
0.9749
0.8342
0.2946
0.8684
0.7990
0.1668
0.6974
0.4742
0.0192
0.9064
0.8105
0.9241
0.2525
0.1040
0.7827
0.8032
0.6113
0.8461
0.8217
0.0380
Product
g
Due
Date
e
108
132
132
108
108
84
132
132
156
108
108
156
108
108
84
132
84
84
156
108
g
e
60
132
f
108
108
108
c
c
g
d
a
f
f
d
a
b
c
f
g
g
d
e
b
c
g
f
c
g
f
b
60
60
84
e
132
132
108
84
84
84
132
108
156
132
108
108
f
60
f
132
132
e
d
a
g
f
g
b
a
f
f
a
R.N.
0.022
0.124
0.538
0.566
0.208
0.142
0.748
0.449
0.9940
0.899
0.183
0.9929
0.804
0.846
0.381
0.459
0.530
0.247
0.104
0.689
0.582
0.959
0.626
0.111
0.289
0.124
0.949
0.931
0.970
0.771
0.052
0.891
0.879
0.061
0.554
0.680
0.909
0.648
0.249
0.871
0.749
0.402
0.269
Z
-2.02
-1.16
0.09
0.17
-0.81
-1.07
0.67
-0.13
2.5
1.28
-0.9
2.45
0.86
1.02
-0.3
-0.10
0.07
-0.68
-1.26
0.49
0.21
1.74
0.32
-1.22
-0.56
-1.16
1.63
1.48
1.88
0.74
-1.62
1.23
1.17
-1.54
0.14
0.47
1.34
0.38
-0.68
1.13
0.67
-0.25
-0.62
Job
QTY
CUM
TOTAL
394
652
1028
1051
757
679
1201
961
1750
1384
730
1735
1258
1306
910
969
1022
796
622
1147
1063
1522
1096
634
832
652
1491
1444
1564
1222
514
1369
1351
538
1042
1141
1402
1114
796
1339
1201
925
814
1456
2111
3138
4189
4946
5625
6826
7787
9537
10921
11651
13386
14641
15947
16857
17827
18848
19644
20266
21416
22479
24001
25106
25740
26572
27224
28716
30160
31727
32949
33463
34832
36183
36721
37763
38904
40303
41417
42213
43552
44753
45678
46492
72
Pareto Product Demand Distribution
Jobs N[1000,300]
#
R.N.
Pro
1
0.4171
0.9393
0.3592
0.8960
0.1521
0.0029
0.9738
0.7995
0.2658
0.1850
0.5394
0.8316
0.6363
0.2421
0.6883
0.4141
0.2704
0.5584
0.9755
0.7636
0.7280
0.3454
0.1813
0.5899
0.7314
0.4838
0.3417
0.0094
0.3315
0.9762
0.1089
0.5988
0.7655
0.0335
0.6968
0.3070
0.6512
0.4341
b
d
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
a
d
a
a
f
b
a
a
b
c
b
a
b
b
a
b
f
b
b
a
a
b
b
b
a
a
a
f
a
b
b
a
b
a
b
b
R.N.
6.1309
100.0220
4.3683
52.9071
75.2404
36.3717
146.7467
117.8707
94.4757
32.3206
3.7215
41.4123
82.1179
52.6815
17.2142
109.5937
88.8742
118.5565
116.1869
5.4631
64.5606
2.5806
111.4354
80.5314
107.9344
75.7539
22.3729
142.8812
70.2004
16.9191
69.2647
43.6502
104.0883
38.3114
25.2483
44.5062
111.7759
7.9879
Due
Date
60
108
60
60
84
60
156
132
108
60
60
60
84
60
60
132
108
132
132
60
84
60
132
84
108
84
60
156
84
60
84
60
108
60
60
60
132
60
R.N.
0.687
0.707
0.048
0.258
0.220
0.742
0.232
0.798
0.063
0.273
0.464
0.883
0.514
0.720
0.255
0.802
0.842
0.856
0.617
0.909
0.192
0.702
0.604
0.486
0.7269
0.773
0.866
0.915
0.963
0.831
0.894
0.392
0.683
0.953
0.916
0.847
0.973
0.792
Z
0.49
0.55
-1.66
-0.65
-0.77
0.65
-0.73
0.84
-1.53
-0.61
-0.09
1.19
0.035
0.585
-0.66
0.85
1
1.06
0.3
1.34
-0.87
0.53
0.26
-0.035
-0.6
0.75
-1.11
-1.38
-1.79
0.96
1.25
-0.27
-0.48
-1.68
1.38
1.02
1.93
0.82
Job
QTY
CUM
TOTAL
1147
1165
502
805
769
1195
781
1252
541
817
973
1357
1010.5
1175.5
802
1255
1300
1318
1090
1402
739
1159
1078
989.5
820
1225
667
586
463
1288
1375
919
856
496
1414
1306
1579
1246
1147
2312
2814
3619
4388
5583
6364
7616
8157
8974
9947
11304
12314.5
13490
14292
15547
16847
18165
19255
20657
21396
22555
23633
24622.5
25442.5
26667.5
27334.5
27920.5
28383.5
29671.5
31046.5
31965.5
32821.5
33317.5
34731.5
36037.5
37616.5
38862.5
73
Pareto Product Demand Distribution
Jobs [1000,50]
#
R.N.
Pro
1
0.2321
0.0490
0.2324
0.0190
0.4841
0.4021
0.7258
0.6375
0.3223
0.4520
0.9840
0.5869
0.3700
0.4594
0.2908
0.8197
0.8383
0.1248
0.9760
0.8908
0.8451
0.8679
0.7319
0.2205
0.2244
0.1482
0.0067
0.2669
0.9376
0;1500
0.9595
0.4032
0.9436
0.7031
0.2113
0.0497
0.1390
0.2609
0.3759
0.3539
0.7262
0.1081
0.1064
0.4685
a
a
a
a
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
b
b
b
b
a
b
f
b
a
b
a
c
c
a
f
d
c
c
b
a
a
a
a
a
d
b
e
a
e
b
a
a
a
a
a
a
b
a
a
b
R.N.
14.66
149.99
52.85
36.84
85.88
119.74
109.48
131.29
65.24
121.37
106.83
135.66
114.97
48.55
26.75
15.90
131.82
109.60
98.35
69.76
50.37
23.65
121.24
71.24
139.38
49.41
22.82
143.71
124.85
29.43
150.80
72.74
3.31
105.58
75.66
147.17
48.53
149.35
116.52
4.57
136.52
76.01
61.43
56.41
Due
Date
60
156
60
60
108
132
132
132
84
132
108
156
132
60
60
60
132
132
108
84
60
60
132
80
156
60
60
156
132
60
156
84
60
108
84
156
60
156
132
60
156
84
84
60
R.N.
0.573
0.731
0.843
0.989
0.889
0.428
0.105
0.365
0.646
0.114
0.249
0.384
0.034
0.225
0.288
0.465
0.915
0.777
0.296
0.865
0.834
0.978
0.541
0.376
0.174
0.264
0.944
0.314
0.337
0.697
0.880
0.174
0.553
0.221
0.930
0.830
0.333
0.562
0.699
0.078
0.321
0.024
0.330
0.668
Z
Job
QTY
CUM
TOTAL
0.18
0.62
1.01
2.29
1.22
-0.18
-1.25
-0.35
0.38
-1.2
-0.68
-0.29
-1.83
-0.75
-0.56
-0.09
1.37
0.76
-0.53
1.1
0.97
2.02
0.1
-0.32
-0.94
-0.63
1.59
-0.48
-0.42
0.52
1.175
-0.94
0.13
-0.77
1.48
0.96
-0.43
0.16
0.52
-1.42
-0.46
-1.97
-0.44
0.43
1009
1031
1051
1114
1061
991
938
982
1019
940
966
986
908
962
972
996
1068
1038
974
1055
1048
1101
1005
984
953
968
1080
976
979
1036
1059
953
1006
962
1074
1048
978
1008
1026
929
977
902
978
1022
1009
2040
3091
4205
5266
6257
7195
8177
9196
10136
11102
12088
12996
13958
14930
15926
16994
18032
19006
20061
21109
22210
23215
24199
25152
26120
27200
28176
29155
30191
31250
32203
33209
34171
35245
36293
37271
38279
39305
40234
41211
42113
43091
44113
74
Equal Prod. Distribution
Jobs [1000,300]
#
R.N.
Pro
1
2
0.6692
0.7576
0.0544
0.2466
0.2872
0.4679
0.2829
0.8468
0.3648
0.4070
0.7723
0.6667
0.4967
0.9184
0.3877
0.7424
0.4757
0.7576
0.2605
0.6002
0.8706
0.4870
0.9510
0.3234
0.5316
0.1807
0.5020
0.1681
0.0089
0.5399
0.4750
0.9986
0.5314
0.2673
0.8948
0.2970
0.5504
0.6313
0.4507
0.6693
0.7510
0.5183
0.3149
0.0180
e
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
f
a
b
c
d
b
f
c
c
f
e
d
g
c
f
d
f
b
e
g
d
g
c
d
b
d
b
a
d
d
g
d
b
g
c
d
e
d
e
f
d
c
a
R.N.
109.29
143.99
35.12
56.30
155.23
73.18
4.44
66.19
145.22
67.90
112.75
25.81
54.10
120.25
149.59
24.62
64.02
71.24
10.55
19.36
89.38
86.73
48.66
130.47
7.71
64.61
37.29
69.04
36.52
5.17
77.55
98.02
30.98
51.38
14.52
101.86
37.72
118.05
147.46
29.03
18.98
85.79
33.15
81.27
Due
Date
132
156
60
60
156
84
60
84
156
84
132
60
60
132
156
60
84
84
60
60
108
108
60
132
60
84
60
84
60
60
84
108
60
60
60
108
60
132
156
60
60
108
60
84
R.N.
0.915
0.564
0.899
0.179
0.335
0.622
1.000
0.834
0.356
0.303
0.627
0.244
0.351
0.206
0.085
0.866
0.102
0.624
0.658
0.623
0.005
0.057
0.719
0.637
0.169
0.361
0.218
0.638
0.514
0.495
0.507
0.992
0.850
0.550
0.889
0.620
0.629
0.949
0.909
0.931
0.045
0.078
0.485
0.571
Z
1.37
0.16
1.28
-0.92
-0.43
0.31
3.5
0.97
-0.37
-0.52
0.32
-0.69
-0.38
-0.75
-1.37
1.11
-1.27
0.32
0.41
0.31
-2.58
-1.58
0.58
0.35
-0.96
-0.35
-0.78
0.35
0.035
-0.01
0.02
2.41
1.04
0.13
1.22
0.31
0.33
1.64
1.34
1.48
-1.7
-1.42
-0.04
0.18
Job
QTY
CUM
TOTAL
1411
1048
1384
724
871
1093
2050
1291
889
844
1096
793
886
775
589
1333
619
1096
1123
1093
226
526
1174
1105
712
895
766
1105
1010.5
997
1006
1723
1312
1039
1366
1093
1099
1492
1402
1444
490
574
988
1054
1411
2459
3843
4567
5438
6531
8581
9872
10761
11605
12701
13494
14380
15155
15744
17077
17696
18792
19915
21008
21234
21760
22934
24039
24751
25646
26412
27517
28527.5
29524.5
30530.5
32253.5
33565.5
34604.5
35970.5
37063.5
38162.5
39654.5
41056.5
42500.5
42990.5
43564.5
44552.5
45606.5
75
Pareto Product Distribution Job Set
Jobs [1000,300]
#
R.N.
Pro
1
0.4171
0.9393
0.3592
0.8960
0.1521
0.0029
0.9738
0.7995
0.2658
0.1850
0.5394
0.8316
0.6363
0.2421
0.6883
0.4141
0.2704
0.5584
0.9755
0.7636
0.7280
0.3454
0.1813
0.5899
0.7314
0.4838
0.3417
0.0094
0.3315
0.9762
0.1089
0.5988
0.7655
0.0335
0.6968
0.3070
0.6512
0.4341
0.0236
0.4759
0.4973
0.7640
0.4756
0.9530
b
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
d
a
d
a
a
f
b
a
a
b
c
b
a
b
b
a
b
f
b
b
a
a
b
b
b
a
a
a
f
a
b
b
a
b
a
b
b
a
b
b
b
b
e
R.N.
6.13
100.02
4.36
52.90
75.24
36.37
146.74
117.87
94.47
32.32
3.72
41.41
82.11
52.68
17.21
109.59
88.87
118.55
116.18
5.46
64.56
2.58
111.43
80.53
107.93
75.75
22.37
142.88
70.20
16.91
69.26
43.65
104.08
38.31
25.24
44.50
111.77
7.98
87.35
5.10
74.80
105.31
65.59
146.40
Due
Date
60
108
60
60
84
60
156
132
108
60
60
60
84
60
60
132
108
132
132
60
84
60
132
84
108
84
60
156
84
60
84
60
108
60
60
60
132
60
108
60
84
108
84
156
R.N.
0.687
0.707
0.048
0.258
0.220
0.742
0.232
0.798
0.063
0.273
0.464
0.883
0.514
0.720
0.255
0.802
0.842
0.856
0.617
0.909
0.192
0.702
0.604
0.486
0.273
0.773
0.133
0.084
0.037
0.831
0.894
0.392
0.316
0.046
0.916
0.847
0.973
0.792
0.807
0.844
0.379
0.658
0.784
0.182
Z
0.49
0.55
-1.66
-0.65
-0.77
0.65
-0.73
0.84
-1.53
-0.61
-0.09
1.19
0.035
0.585
-0.66
0.85
1
1.06
0.3
1.34
-0.87
0.53
0.26
-0.035
-0.6
0.75
-1.11
-1.38
-1.79
0.96
1.25
-0.27
-0.48
-1.68
1.38
1.02
1.93
0.82
0.87
1.02
-0.31
0.41
0.8
-0.91
Job
QTY
CUM
TOTAL
1147
1165
502
805
769
1195
781
1252
541
817
1147
2312
2814
3619
4388
5583
6364
7616
8157
8974
9947
11304
973
1357
1010.5
1175.5
802
1255
1300
1318
1090
1402
739
1159
1078
989.5
820
1225
667
586
463
1288
1375
919
856
496
1414
1306
1579
1246
1261
1306
907
1123
1240
727
12314.5
13490
14292
15547
16847
18165
19255
20657
21396
22555
23633
24622.5
25442.5
26667.5
27334.5
27920.5
28383.5
29671.5
31046.5
31965.5
32821.5
33317.5
34731.5
36037.5
37616.5
38862.5
40123.5
41429.5
42336.5
43459.5
44699.5
45426.5
76
Experiments Three and Four Job Sets
Equal Product Demand Distribution
Jobs N[1000,300]
No.
RN
1
0.822953
0.215139
0.379618
0.65652
0.373074
0.478204
0.136756
0.38025
0.382113
0.314183
0.593435
0.519263
0.588362
0.221106
0.843583
0.899246
0.719384
0.755867
0.305574
0.704404
0.94239
0.781723
0.730705
0.80845
0.223795
0.316375
0.175556
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
PROD
8
2
4
7
4
5
1
4
4
3
6
5
6
2
8
9
7
8
3
7
9
8
7
8
2
3
2
RN
DUE
RAND
Z
QTY
0.667126
0.394835
0.840016
0.610077
0.006617
0.470106
0.672707
0.5833
0.292239
0.51591
0.780473
0.118545
0.711778
0.115287
0.119477
0.759983
0.15785
0.226594
0.377659
0.442456
0.443661
0.982512
0.723211
0.606399
0.736538
0.72637
0.919262
67
0.729765
0.61
0.943612
1.59
0.332635 -0.43
0.914145
1.37
0.198815 -0.85
0.534022
0.09
0.608462
0.28
0.766637
0.73
0.189644 -0.88
0.032019 -1.85
0.323178 0.46
0.51552
0.04
0.982336 2.1
0.165338 -0.97
0.162563 -0.98
0.35286 -0.38
0.879596 1.17
0.403229 -0.25
0.654815 0.4
0.210376 -0.8
0.651763 0.39
0.368504 -0.34
0.865072 1.1
0.618264 0.3
0.180057 -0.92
0.357978 -0.36
0.405357-0.24
1183
1477
871
1411
745
1027
1084
1219
736
445
1138
1012
1630
709
706
886
1351
925
1120
760
1117
898
1330
1090
724
892
514
29
84
66
11
47
67
58
29
52
78
12
71
12
12
76
16
23
38
44
44
98
72
61
74
73
92
CUM
QTY
1183
2660
3531
4942
5687
6714
7798
9017
9753
10198
11336
12348
13978
14687
15393
16279
17630
18555
19675
20435
21552
22450
23780
24870
25594
26486
27000
77
Equal Product Demand Distribution
Jobs N[1000,0]
No.
RN
1
0.843188
0.43597
0.276138
0.637592
0.929091
0.634809
0.103389
0.866971
0.611064
0.369952
0.428488
0.08407
0.912183
0.090206
0.634567
0.956134
0.990847
0.741294
0.145327
0.160351
0.169527
0.881515
0.563405
0.372075
0.432023
0.038613
0.600733
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
PROD
8
4
3
6
9
6
1
9
6
4
4
1
9
1
6
10
10
7
1
2
2
9
6
4
4
1
6
RN
0.986991
0.612727
0.666013
0.788185
0.266987
0.053027
0.690956
0.919715
0.164006
0.164736
0.782431
0.070333
0.932348
0.433852
0.294393
0.982291
0.991461
0.824742
0.405255
0.562704
0.30361
0.845276
0.622711
0.190381
0.473905
0.982063
0.131331
DUE
99
61
67
79
27
5
69
92
16
16
78
7
93
43
29
98
99
82
41
56
30
85
62
19
47
98
13
QTY
CUM
QTY
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
12000
13000
14000
15000
16000
17000
18000
19000
20000
21000
22000
23000
24000
25000
26000
27000
78
Spreadsheet Explanation
Appendix E:
The spreadsheet used to evaluate system performance in
An example
this study was developed using Quattro Pro 3.0.
follows at the end of this section.
The paragraph headings,
below, refer to designated sections on the example.
A
The first four columns are general job data: the
job number in the order it was generated, product
(a
j), due date (hour),
and the quantity (in
units).
B
The next set of columns are partitioned for each of
The rate is the processing time. It is calculated by dividing the job
quantity by the processing rate (capacity) of the
The set-up time is the time
specific processor.
needed to change from the previous product to the
The completion time is the comcurrent product.
pletion time of the previous job plus the processing time for the current job.
When required, the
set-up time is added.
the processors A,
C
B, and C.
The third set of columns calculates the completion
time, lateness, and tardiness for each order. The
mean and standard deviation for each of these
measures is calculated at the bottom of the columns.
Although lateness was not considered in this
study
it
was
useful
in
calculating
tardiness.
Lateness is calculated by subtracting the comple-
tion time from the due date
(time).
lateness indicates an early completion.
A negative
Tardiness
is then calculated as the maximum of lateness or
zero.
79
D
The area under the processor columns calculates the
The
amount of capacity used on each processor.
calculation for processor utilization is the completion time of the last order run on the processor
divided by the length of
the scheduling window
(either 156 or 100 hours depending upon the experiment).
The Processor Utilization Spread was calcu-
lated by subtracting the smallest processor utilization from the largest.
E
These two figures count the number of jobs and the
total quantity of the jobs.
F
The figures in this area represent the mean flow,
mean lateness, and mean tardiness and the standard
deviation for each.
G
This section analyzes the proportion of capacity
used for set-up, processing and total use.
The
processor utilization spread is calculated from
this data.
Note:
In this example the extremely high set-up
time requirements have made the schedule impossible
All three processors require more
to complete.
all processor utilizations
time than is available
are greater than 100%.
H
The proportion of jobs tardy was calculated by
counting the number of tardy jobs and dividing it
by the total number of jobs scheduled.
80
(60
(CI)
Capacity =
Due
MACH A
QTY
RATE
PrDate City
#
Capacity =
70
Set
:
COMPLET:
Up
MACH B
5 d
11
745
0.00
15 h
12
706
0.00
14 b
12
709
10.13
10.13
12 e
12
1012
0.00
10.13
0.00
17 g
16
1351
0.00
10.13
13.51
18 h
23
925
13.21
26.34
9 d
29
736
0.00
3
Set
RATE
Capacity =
100
::MACH C
COMPLET:: RATE
Up
10.13
-1.87
0.00
7.06
:,
7.78
15.52
15.52
3.52
3.52
23.57
::
0.00
15.52
23.57
7.57
7.57
0.00
23.57
:,
0.00
15.52
26.34
3.34
3.34
26.34
0.00
23.57
::
5.66
24.18
24.18
-4.82
0.00
24.18
42.34
13.34
13.34
35.79
35.79
-2.21
0.00
35.79
40.20
-3.80
0.00
46.38
46.38
2.38
2.38
46.38
59.87
12.87
12.87
46.38
49.79
-2.21
0.00
55.76
55.76
-2.24
0.00
55.76
64.90
3.90
3.90
67.15
2.15
2.15
80.36
13.36
13.36
3
26.34
14.77
42.34
;,
0.00
26.34
0.00
42.34
,:
8.62
20 g
44
760
10.86
40.20
0.00
42.34
::
0.00
40.20
0.00
42.34
,:
8.59
59.87
0.00
42.34
:;
0.00
49.79
::
0.00
49.79
::
9.28
64.90
:,
0.00
8.38
4
44
1117
0.00
6 e
47
1027
14.67
10 c
52
445
0.00
59.87
4.45
8 d
58
1219
0.00
59.87
0.00
4 g
61
1411
0.00
59.87
14.11
24 h
65
1090
0.00
59.87
0.00
64.90
:!
80.36
0.00
64.90
:,
80.36
11.83
78.73
::
80.36
0.00
1084
15.49
1
h
67
1183
0.00
13 f
71
5
0.00
1630
23 g
75
1330
0.00
16
0.00
5.73
0.00
67
0.00
-4.94
:,
0.00
a
-5.27
7.06
7.06
1120
7
5.73
5.73
5.73
1477
'
TARDY
;:
38
5
LATE
:,
29
i
COMPLE
COMPLET:
Up
7.06
2 b
21
Set
0.00
7.06
19 c
3
130
,
80.36 ::
3
1
2
78.73
13.30
3
,
.
5.73
2
3
3
2
0
3
0.00
0.00
12.54
95.03
::
0.00
4
67.15
:
'
,
67.15
::
67.15
::
78.73
11.73
11.73
83.68
::
83.68
12.68
12.68
20.0?
83.68
:,
95.03
20.03
83.68
i
76
886
12.66
97.01
::
0.00
95.03
:,
0.00
::
97.01
21.01
21.01
26 c
76
892
0.00
97.01
::
0.00
95.03
::
6.86
3
93.55
:,
93.55
17.55
17.55
25 b
77
724
0.00
97.01
::
0.00
95.03
:,
5.57
5 104.12
,:
104.12
27.12
27.12
11
f
78
1138
0.00
97.01
:,
11.38
107.41
:,
0.00
104.12
107.41
29.41
29.41
3 d
84
871
12.44
0 109.46
!:
0.00
107.41 H
0.00
104.12
109.46
25.46
25.46
27 b
93
514
0.00
109.46
::
0.00
107.41
::
3.95
0 108.07
108.07
15.07
15.07
22 h
98
948
0,00
109.46
::
9.48
4 120.89
::
0.00
108.07
120.89
22.89
22.89
89.46
20 109.46
99.89
21 120.89
83.07
25 108.07
127
27050
,
::
4
0.89 0.20
1.09
1
1.00 0.21
1.21
0.83 0.25
1.08
:
:
:
,;
8.82
61.22
M.Flow
M.Late
35.12
HRS USE
HRS AVAIL
USED
120.89%
272.42
300.00
90.8%
108.07%
SET UP HRS
% USED
66.00
9.83
M.Tard
9.56
10.69
s
s
U
12.82 %= PROCESSOR UTILIZATION
SPREAD
22,0%
TOTAL USED ,
338.42
No. Tardy:
% CAPACITY=
112.81%
Pro. Tardy:
Figure 3
Spreadsheet Illustration
19.00
1-4
70.37*'