Int. J. Computer Integrated Manufacturing, Vol. 18, No. 2–3, March–May 2005, 236 – 250
Determination of efficient simulation model fidelity for flexible
manufacturing systems
MONISH MADAN{, YOUNG-JUN SON{*, HYUNBO CHO{ and BOONSERM KULVATUNYOU§
{Systems and Industrial Engineering Department, The University of Arizona, Tucson, AZ 85721-0020, USA
{Department of Industrial Engineering, Pohang University of Science and Technology, Pohang 790-784,
Republic of Korea
§MSI Division, National Institute of Standards & Technology, Gaithersburg, MD 20899, USA
(Received 5 August 2003; in final form 9 February 2004)
This paper presents a framework for the determination of an efficient level of simulation
model fidelity for flexible manufacturing systems, which will achieve acceptable output
accuracy with minimum resources and thereby reduce model building effort and
computation time. To this end, we first formally define different levels of model fidelity
using building blocks available in object–oriented (O–O) modelling, where an operation
at a higher level is either decomposed into more detailed operations or subjected to more
constraints at a lower level. In this paper, five models with different fidelities are defined.
Then, simulation models that conform to these O–O models are constructed. Using these
simulation models, intensive experiments are conducted to examine how the factors that
characterize an FMS contribute to the relative errors of outputs from different models.
Since no actual systems are considered, the results generated from the most detailed
simulation model are used as references. The experimental results are then summarized by
regression-based meta-models. In the proposed framework, the most efficient model for a
new FMS is identified so that the relative error of a model estimated from the meta-model
is closest to the threshold value provided by users. This framework is tested by two
sample FMSs, and the initial results look quite promising.
Keywords: Model abstraction; Multi-resolution; Hierarchical model; Simulation; UML;
FMS
1. Introduction
As markets become unexpectedly turbulent due to shortened product life cycles and power shifts toward the
customer, the need for systems that rapidly and costeffectively develop products has become pressing (Kidd
1994). A flexible manufacturing system (FMS) is one such
system that meets these needs. An FMS is an embellished
job shop equipped with CNC machine tools and supporting
workstations that are connected by an automated material
handling system (Askin and Standridge 1993, Park et al.
2001). The flexibilities offered by FMSs include machine
centre versatility to perform a variety of operations on a
work piece, quick changeover to new products by a change
in software, and rerouting of work pieces in-process to
avoid machines in repair or those with relatively long queues
(Wilhelm et al. 1985). Using these flexibilities, FMSs mainly
aim to improve production rates, work in progress levels,
lead times, and direct and indirect costs (Carrie 1988).
The design and operation of FMSs involve intricate and
interconnected decisions that result in maximum benefits
from the system. For example, decisions to be made in the
design stage include part-type to be produced, the type and
the size of buffers, and the number of pallets and the
*Corresponding author. Email: [email protected]
International Journal of Computer Integrated Manufacturing
ISSN 0951-192X print/ISSN 1362-3052 online # 2005 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/0951192052000288143
Model fidelity for flexible manufacturing systems
number and design of fixtures (Park et al. 2001). Decisions
to be made in the operation stage include the input
sequence of parts into the system, the schedule of parts to
machines based upon alternative routings, the sequence of
parts on a machine, and the scheduling of material
handling devices (Park et al. 2001). Both design and
operation-related decision parameters should be simultaneously considered to achieve global optimization in the
development of FMSs. Due to the intricate and interconnected nature of decisions involved in FMSs, valid
models or modelling techniques must be carefully chosen/
developed and used to make those decisions.
Broad classes of modelling techniques have been used in
the design and operation of FMSs including network
design, artificial intelligence, heuristics, mathematical programming and simulation. Of these, discrete event
simulation has been one of the most widely used due to
its rich modelling expressiveness and flexibility for representing complex and uncertain interdependencies between
entities in FMSs. In addition, simulation can provide useful
data that are otherwise difficult to obtain about the
modeled system (Persson 2002). However, modelling every
aspect of a system is seldom required for effective decisionmaking. In addition, complex dynamics associated with
material processing and handling systems may cause
considerable simulation execution time. Therefore, the
appropriate level of detail (fidelity) of a simulation model
must be carefully determined, to balance the cost involved
with model development and execution against the required
usefulness (accuracy) of the model.
Some researchers have discussed model fidelities and their
impact on the analysis of manufacturing systems. Law and
Kelton (2000) provide a general guideline for determining
the level of detail required for a simulation model. Eldabi
and Paul (2001) investigate the existing manufacturing
simulation software environments that may offer variable
detail modelling, classify model entities according to the
level of detail, and develop mechanisms to increase the level
of detail of a model effectively. Persson (2002) investigates
the impact of varying levels of modelling detail for a mobile
communication application. The experiments performed
with those models have aimed at finding differences between
the outputs from different models. Benjamin et al. (1998)
characterize the problem of multiple levels of abstraction
associated with simulation modelling, and propose a
heuristic for determining the appropriate level of abstraction and an architecture that implements the approach.
Some other researchers have discussed model fidelity
issues using the O–O modelling technique. An O–O
approach to designing manufacturing models is frequently
selected, mainly due to the complex nature of the problem
(Jacobson et al. 1992, Booch 1994). Kovacs et al. (1999) use
an O–O modelling methodology to provide methods and
tools to build FMS simulation models and control
237
strategies more efficiently. The authors emphasize the
importance of reusing O–O design techniques to facilitate
fast and efficient construction of FMS simulation models.
Anglani et al. (2002) present a procedure to develop FMS
simulation models in the ARENA1 simulation language
based on the O–O analysis/design tools, where the goal is to
improve software development efficiency through a rulebased approach and to add some fundamental objectoriented features to the ARENA1 simulation environment.
Chen and Lu (1997) have presented an O–O methodology
to design a production system software-model by means of
Petri-nets, the entity relation diagram and IDEF0.
Although researchers have discussed model fidelities for
the analysis of manufacturing systems, common limitations
have been identified. First, multiple (usually two) models
with arbitrary abstractions are constructed for a manufacturing system, and results from those models are
statistically analysed in ad hoc manners. However, the
relations between the results obtained from those models
have not been further analysed. Second, the impact of
system characteristics on the accuracy of outputs obtained
from different model fidelities has not been analysed, and
the correlation between the factors and the accuracy has not
been studied. Third, generic and formal specification of
varying model fidelities, which will enhance the reusability
of models, is missing. Fourth, O–O modelling has been used
in some instances for formally designing FMS simulation
models, and a rule-based approach has been suggested for
conceptual translation to simulation models. However, this
approach appears incomplete because it has not considered
the accuracy obtained from different models.
This paper attempts to overcome the identified drawbacks mentioned above, and to develop a framework to
determine the efficient model fidelity for FMSs. To
accomplish this goal, we first formally define different
levels of model fidelity using the O–O modelling technique.
Five different fidelities (formal models) are defined in this
work. Second, simulation models are then constructed to
conform to these formal models. Third, using these
simulation models, structured experiments are designed
and conducted to study the contribution of several factors
to the accuracy of outputs obtained from different fidelities.
The accuracy of an output is defined as the relative error
between the metrics calculated from a model with the given
fidelity and the real system. Since such a real system is not
considered here, the results for the simulation model with
the highest fidelity are used as the references. Fourth,
results of experimentation will be summarized with a
regression-based empirical meta-model. This meta-model
will be used to determine an efficient level of model fidelity
for an FMS, which will be characterized by five input
factors from a user. Finally, we will demonstrate the
proposed framework and meta-model with two sample
FMSs.
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2. Formal specification of different model fidelities
2.1.
Object–oriented modelling and UML
The O–O modelling technique is useful for understanding
problems, communicating with application experts, modelling enterprises, preparing documentation, and designing
programs and databases (Booch et al. 1999). The important
properties of O–O modelling include inheritance, information hiding, encapsulation, abstraction and polymorphism.
Unified modeling language (UML) is a set of formal
graphical notations standardized by Object Management
Group and used to design, deploy, and develop software
systems from an O–O modelling perspective (France et al.
1998, Booch et al. 1999). UML provides suitable modelling
constructs for developing formal, configurable, reusable
multi-perspective models for a system (Kim et al. 2003).
The formalism and reusability of UML could eventually
facilitate automatic determination of an appropriate model
fidelity and composition of a simulation model, which is a
long-term goal and beyond the scope of this paper.
2.2.
Specification of different model fidelities using UML
We propose using class diagrams and functional models to
specify static, structural and transformational aspects of
FMSs. First, class diagrams represent the various classes,
their attributes and their relationships for FMSs under
consideration. In this work, a generic class diagram is
proposed (see figure 1) based on the work by Steele et al.
(2001) and Son et al. (2003) to specify the static information
needed to represent operations scheduling of FMSs.
Inheritance and abstraction, two prominent features of
the O–O paradigm, are used in the class diagram.
The class is represented in UML as a rectangle with three
compartments. The top compartment has the name of the
class, the middle has the attributes and the bottom
compartment has the operations of the class. As shown in
figure 1, a shop can have one or more than one workstation
(cells), where a ‘whole/part’ or ‘has’ relationship is used
between the Shop and the WorkStation classes. Each
workstation has parts and equipment. A ‘has’ relationship
is used between the WorkStation and the Equipment
classes and the WorkStation and the Part classes. The
pieces of equipment are classified into different types based
on their functions: MaterialProcessor, MaterialHandler,
MaterialTransporter and Buffer classes. As each of these
types is a special type of equipment, an inheritance
relationship is used between them. The buffers are subclassified as StationInputBuffer, StationOutputBuffer and
IntermediateBuffer classes using an inheritance relationship. The material processors can have a set of tools and
fixtures based on the aggregation relation between them.
Parts in material processors, material handlers, material
transporters, and buffers are represented by the classes
PartInMP, PartInMH, PartInMT, PartInStnInBuffer, PartInStnOutBuffer, PartInInterBuffer, respectively. Similarly,
parts are classified as RawMaterial class, WorkInProcess
class and FinishedProduct class using an inheritance
relationship between them with the Part class. Each part
is associated with an order (Order class) and a process plan
(ProcessPlan class). Each process plan has the details of the
operations to be performed (Operation class).
Figure 1. Class diagram for flexible manufacturing systems.
Model fidelity for flexible manufacturing systems
Functional diagrams do not belong to the standard set of
UML diagrams. Therefore, we have constructed customized Functional diagrams using Actors, Use Cases and
Class diagrams available in UML, which are similar to
IDEF0 diagrams. In this paper, notations for elements have
been partially adapted from the UML Functional models
proposed by Anglani et al. (2002). Those elements include
information input, physical input, control, mechanism,
information output and physical output. In this paper,
Functional diagrams for five different fidelities (see figures
2–4) have been developed. These fidelities are denoted as
Fidelity i (i = 1,. . .,5) where Fidelity 1 is the most abstract
model and Fidelity 5 is the most detailed model. Among
those five Functional diagrams, the ‘mechanism’ and
‘function’ varies, while the other elements are common.
Common elements are summarized in table 1, and the
mechanisms and other specifics for different fidelities are
explained in the following sections.
2.2.1. Fidelity 1. In this model, activities of the material
processors and non-material processing resources are
Figure 2.
239
abstracted to a workstation level activity. To this end, the
non-material processing resources are not explicitly considered; however, their mean times are added to the
processing times in the process plans. In addition, a cell
with n material processors pertaining to a cell is represented
as a cell with a capacity of n. In other words, different
machines pertaining to a cell are treated as the same.
Formally, activities within a cell are represented by a single
function, ‘Execute Cell Operation (A1)’, in this model (see
figure 2). Thus, the results generated from this model are
expected to include some error. The constraint (‘Equipment_Type = = Buffer’) shown in figure 2 denotes that
only the Buffer class is further specified from the Equipment class, while other subclasses (material processors,
material transporters, and material handlers) are not
further specified. The processing times, material handling
times, material and transportation times are aggregated
into a single activity time, which corresponds to the
attribute Cell_Activity_Time of the Cell class. The buffers
are then classified into station input buffer, station output
buffer, and intermediate buffer. Since interactions within a
Functional diagram for the Fidelity 1 model.
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Figure 3.
Functional diagram for the Fidelity 3 model.
cell are not considered, issues with the finite intermediate
buffer are not considered either.
2.2.2. Fidelity 2. The Fidelity 2 model is more detailed
than the Fidelity 1 model, because material processors and
finite intermediate buffers are explicitly and separately
considered. The function ‘Execute Cell Operation (A1)’ is
further classified into sub-functions ‘Execute Material
Processor Operation (A11)’ and ‘Store and Retrieve part
Operation (A12)’ in this model. However, materialtransporters and material handlers are still not explicitly
considered. It is noted that Functional models are explicitly
shown in this paper only for Fidelities 1, 3, and 5. The
Processing_Time attribute of the MaterialProcessor class is
the sum of the processing time, material transportation
time, and material handling time. The results obtained from
this model are expected to be more accurate than those
from the Fidelity 1 model, but still to include some error.
2.2.3. Fidelity 3. In the Fidelity 3 model, unlike in the
Fidelity 2 model, material-transporters are explicitly
considered. The function ‘Execute Cell Operation (A1)’ is
further subdivided with the addition of another sub
function ‘Execute Material Transporter Operation (A13)’
in this model (see figure 3). Consequently, material
transportation times are not added to material processing
times any more. Hence, the Processing_Time attribute of
the MaterialProcessor class (see figure 3) is now the sum of
the processing times and material-handling times. The
results generated from this model are expected to be more
Model fidelity for flexible manufacturing systems
241
Figure 4. Functional diagram for the Fidelity5 model.
accurate than results from Fidelity 2 because the part
waiting times for seizing material-transporters are explicitly
modeled.
Table 1.
2.2.4. Fidelity 4. In the Fidelity 4 model, unlike in the
Fidelity 3 model, material-handlers are explicitly considered as resources, and, therefore, material handling times
are not included as part of the processing times. The
function ‘Execute Cell Operation (A1)’ includes another
sub function ‘Execute Material Handler Operation (A14)’
in this level of fidelity. The Processing_Time attribute of the
MaterialProcessor class is the actual processing time. The
results obtained from this model should be more accurate
than previous models since part contention (waiting times)
for material-transporters and material-handlers are considered.
Information The order class contains the details of a job. The
input
PartInOrder class provides the number of parts of each
part-type required by the customer and has an
aggregation relation with the Order class. The
ProcessPlan class contains the routing information of
part to be manufactured. The Operation class has a
whole/part relation with the ProcessPlan class.
Physical
The parts in the workstation are denoted by the Part
input
class. The RawMaterial class has an inheritance relation
with the Part class.
Control
Control consists of the Decision Planner actor which
evaluates a set of rules to determine the routing of the
next part
Physical
The parts in the workstation are denoted by the Part
output
class. The FinishedPart class has an inheritance relation
with the Part class.
Information The information output contains the Result class
output
denoting the performance metrics.
2.2.5. Fidelity 5. The Fidelity 5 model is the most detailed
model, considering tooling availabilities in each machine in
addition to the details considered in the Fidelity 4 model. In
the case when the tool required for a certain operation is not
available on the machine, the tool change time is taken into
Elements
Common elements of functional models for all
fidelities.
Description
account. In this level of fidelity the function ‘Execute
Material Processor Operation (A11)’ is further classified by
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the inclusion of a sub-function ‘Execute Tool Change
Operation (A111)’ (see figure 4). The results obtained from
this model are most accurate among the five models, and are
considered as the true values, which will be used to calculate
the relative errors for other models (Fidelities 1–4).
3. Experimental design and simulation modelling
This section describes the experiments conducted and the
simulation models used in the experiments. First, various
factors characterizing FMSs, which may affect the outputs
generated from models with different fidelities, have been
identified. The factors are then set at two levels. A total of
25 (five factors, two levels) test FMSs are generated, for
each of which five different simulation models are built.
Two performance metrics, throughput and average machine utilization, are calculated from these simulation runs.
It is noted that the throughput results will be discussed in
this paper. The four responses used in the experimental
analyses are the relative errors of the performances from
the models with Fidelity i (i = 1,. . .,4) compared with the
reference (performances from Fidelity 5).
3.1.
Experimental factors and conditions
Five factors that may affect the output generated from
different models have been identified. They are complexity
of FMSs, routing flexibility, stochasticity of data used, the
number of part types and batch size. A range is selected for
each factor and the experimental results in this paper are
limited to this range. The factors are set at two levels as
shown in table 2, which will provide different test FMS
scenarios. The range has been selected from a practical
perspective and from preliminary experience regarding the
range of these factors.
3.1.1. Factor A: complexity of FMSs. First, complexity
in this work denotes the number of material processors,
Table 2.
Experimental factors at two levels.
Factors
Complexity
Cells
Material-processors
Material-transporter
Material-handlers
Stochasticity
Inter-arrival time
Material-transportation time
Material-handling time
Tool change time
Number of part-types
Batch size
Routing flexibility
Level 1
Level 2
2
3
1
2
6
10
2
6
Deterministic
Stochastic
5
10
Linear (fixed)
10
20
Alternate
material transporters and material handlers. Some researchers (Baker and Dzielinkski 1960, Nanot 1963, Buffa,
1968) have observed that the number of processors does
not have a significant impact on the nature of the solution
while using different dispatching rules in job shop
scheduling. On the other hand, Smith et al. (1999) has
observed that both the number of processors and material
transporters have a significant impact on the relative
performances of two job shop scheduling approaches:
scheduling considering only material processors and
scheduling considering both material processors and
handlers. In this work, it is expected that the relative
errors of the performances generated from abstract
models increase as the system complexity increases. This
factor has been considered at two levels, i.e. simple and
complex. For the simple level, two cells (see figure 5) are
considered, where the first cell is composed of two
processors, one handler and one intermediate buffer; and
the second cell is composed of one processor, one handler
and one intermediate buffer. The intra-cell part movement
is handled by a material transporter with a capacity of
four slots. The complex level (see table 3) corresponds to
six cells each comprising 10 material processors, six
material handlers and two material transporters with a
capacity of four each.
3.1.2. Factor B: routing flexibility. Ferreira and Wysk
(2001) investigated the importance of alternative processplans as compared to linear process-plans and concluded
that alternative process-plans enable faster and more
efficient decisions. The existence of alternative routing
imparts flexibility in dynamic decision-making, which
more abstract models take into less consideration. Therefore, the existence alternative routing is expected to affect
the outputs generated from different models. Figure 6
depicts the AND/OR process-plan of a part-type considered for a complex system, where nodes correspond to
either processes or junctions for branches. In the
processing nodes, data contained in each node denote
the processing_id and the machine_id. Junction nodes
denote asynchronous AND-junctions (SA: start AND,
EA: end AND) for sequence alternatives and asynchronous OR-junctions (SO: start OR, EO: end OR) for
activity alternatives.
3.1.3. Factor C: stochasticity of data. The stochasticity
of activity times has been considered as another factor since
it imparts uncertainty, which may affect the outputs from
different models. In this work, the inter-arrival times to the
system, processing times, material-transportation times,
material-handling times and tool change times are varied
at two levels: deterministic and stochastic. The arbitrary
statistical distributions considered for the stochastic level
are shown in table 4, where material-transportation times
Model fidelity for flexible manufacturing systems
243
Figure 5. FMSs of level 1 (simple).
Table 3.
Pieces of equipment in simple (level 1) and complex
(level 2) FMSs.
Simple (Level 1)
Cell 1
Cell 2
Complex (Level 2)
Cell 1
Cell 2
Cell 3
Cell 4
Cell 5
Cell 6
Number of
material
processors
Number of
material
handlers
Number of
material
transporter
2
1
1
1
1
2
1
2
1
2
2
1
1
1
1
1
1
1
1
are calculated based on the deterministic distance and
stochastic velocity. It is noted that an arbitrary upper bound
and lower bound have been set to handle the unbounded
exponential and normal distributions, respectively.
3.1.4. Factor D: number of part types. The number of
part-types is positively correlated with routing flexibility
in FMSs. A large number of part types, each with
alternative routings, increases the routing flexibility.
Therefore, similar to the ‘routing flexibility’ factor, the
number of part-types is expected to affect the outputs
from different models. The number of part-types is varied
at two levels: 5 and 10, which have been determined
based on pilot experiments. In the pilot experiments, the
lower level of the number of part-types was fixed and
then was incremented until differences in results were
obtained.
3.1.5. Factor E: batch size. A batch consists of a specific
number of parts of each part-type to be produced. The
batch-size and formation help to ensure that the machines
are utilized to the maximum and there is no machine
starvation. More detailed models are subjected to more
constraints (material handling, transporting, tooling), and,
hence, the batch-size is expected to affect the results from
different models. The effect of an increase in the batch-size
on the increase in number of parts produced is expected to
be more significant for more abstract models. In this work,
batches have been formed based on minimal part set, where
equal numbers of part-types constitute a batch. The batch
size is set at two levels, 10 and 20, which was determined
based on pilot experiments. Similar to the case of the
number of part-types, the lower level was first fixed, and the
higher level was determined.
3.1.6. Performance metric and experimental conditions. The performance metrics considered in these
experiments are the throughput and the mean machine
utilization. The throughput is the total number of parts
produced in a time frame of one week. Finally, the
following assumptions and constraints characterize the
FMS environments for which the experimental results and
conclusions are applied:
.
.
.
.
.
Machine failures and preventive maintenance are not
considered.
No pre-emption of jobs is allowed.
All machines are assumed to have four (arbitrary)
tool slots.
Tool failures are not considered.
Machines are assumed to be at equal distances from
each other and the buffers; hence, the material-
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M. Madan et al.
Figure 6.
AND/OR process plan for a sample part-type.
Table 4.
Time
Fidelity1
Statistical distributions for stochastic data.
Fidelity2
Inter-arrival time
Processing-time
Fidelity3
Expo (250)
Expo(250)
Expo(250)
Normal
Normal
Normal
(processing-time, 0.22) (processing-time, 0.22) (processing-time, 0.22)
Material-transportation time
distance/Expo (5),
where 5 is velocity
Material-handling time
Tool-change time
.
.
.
.
.
3.2.
handling times within a cell are assumed to be equally
5 min for the deterministic level and normally
distributed with a mean of 5 min and std. dev. of
0.2 for the stochastic case.
Each material-transporter is considered to be an AGV
with a speed of 5 per min. The distance between each
cell is assumed to be equal, as with the distance
between the input buffer/output buffers of the system
and each cell.
The processing-times for the deterministic case are
generated from the uniform distribution (10, 40). For
the stochastic case, the processing-times are normally
distributed with the mean set to the same number
used in the deterministic case and a standard
deviation of 0.2.
The inter-arrival times of a batch are calculated based
on the average time required to produce a batch. In
fact, to avoid starvation, the inter-arrival time is kept
at 90% of the average time required to produce a
batch for the most abstract model.
The manufacturing system has an operation time of
7200 min (8 h production shift, 3 shifts everyday and
5 days a week).
Regarding AND/OR graph-based alternative routings, only single nested AND/OR process-plans are
considered.
Simulation modelling specifics
To conduct the experiments discussed earlier, a total of 25
(five factors, two levels) test FMSs have been generated, for
each of which five different simulation models have been
built in Arena1 conforming to the UML diagrams. In this
section, we will discuss four technical issues involved in
Fidelity4
Fidelity5
Expo(250)
Normal
(processing-time, 0.22)
distance/Expo (5)
Expo(250)
Normal
(processing-time, 0.22)
distance/Expo (5)
Normal (5, 0.22)
Normal (5, 0.22)
Expo (2)
simulation modelling and analysis, including deadlock
handling, interface with the database containing AND/
OR process plans, number of replications and verification
of simulation models.
First, researchers have identified three standard approaches to manage deadlock situations: prevention,
avoidance, and detection and resolution (Venkatesh et al.
1998). The approach adopted in this work is deadlock
prevention. The deadlock prevention method averts a
deadlock by imposing constraints on how resource requests
are made, ensuring that the necessary conditions for a
deadlock cannot occur (Venkatesh et al. 1998). To prevent
deadlock, the material handler and processors are seized
together. Furthermore, when the intermediate buffer reaches
its maximum capacity, the parts are placed in the system
input buffer, thus restricting the occurrence of deadlock.
Second, Visual Basic Application1 embedded in ARENA1 has been used to write custom code to read in the
process plans from the MS Access Database1 and then
control the ARENA1 simulation model. The SIMAN1
code is an ActiveX1 object, which can be used to query or
modify the model’s runtime data, such as resource states,
variable values and entity attributes. The SIMAN1 code in
this work has been used in setting/reading the resource
states, tooling availabilities, and the attributes of the
entities based on the process plans retrieved from the MS
Access Database1. Machine selection problems in Fidelities 1–4 are resolved by examining processing times of
alternate routings and machine availability. In Fidelity 5,
tooling availability in machines is also considered. Parts are
selected from the input/output and intermediate buffers
based on the FIFO rule.
Third, multiple replications are required to perform the
necessary statistical analysis and achieve accurate results.
Model fidelity for flexible manufacturing systems
Point estimates give little information on how close the
estimate of the unknown parameter is to its true value (Law
and Kelton 2002). An interval estimate using a point
estimate and variance, referred to as a confidence interval
estimate, gives more descriptive estimates. A confidence
interval for a mean is a range within which the true mean
falls, with a certain
level of confidence, represented as
qffiffiffiffiffiffiffiffiffiffiffiffi
xðnÞ tn1;1a =2 s2 ðnÞ =n , where x is the mean and s is the
standard deviation of n observations. a represents the level
of confidence. The number of replications, n, is calculated
using the approximation presented in the following
2
equation (Law and Kelton 2000): n ffi z21a = hs 2 , where s is
2
the sample standard deviation from an initial set of
replications, n0. The desired half-width is represented by
h, and h0 is the actual half-width. The complex stochastic
model was first run for five replications and the half width
was obtained using the output analyzer. Since the desired
and the obtained half width were found to be close, all of
the models were run for five replications.
Finally, the simulation models developed were verified by
allowing a single entity into the system and following the
entity through the system until the entity leaves the system.
This was accomplished by creating an entity of each parttype and studying the routing through the system using
tracks and animation. Model verification was performed in
parallel throughout the simulation model development
stage. Each model built was verified before being used for
analysis.
4. Experimental results and analyses
This section describes the results obtained from the
experiments. First, we provide insights about the impact
of the factors characterizing FMSs on the output generated
from different models. Based on the experimental results,
we then construct regression-based empirical meta-models.
Finally, a methodology based on the meta-models is
proposed to determine an efficient level of model fidelity
for FMSs.
4.1.
Impact of factors on outputs from different models
Intensive experiments (see Section 3) were conducted, and
the results obtained from these experiments were analyzed
using ANOVA to study the effect of the factors along with
their interactions on the response, i.e. the relative error in
the performance outputs from a model (Fidelities 1–4).
Four responses (Y1, Y2, Y3 and Y4) corresponding to
Fidelities 1–4 were calculated based on the following
equation:
Response ðYiÞ ¼ðperformance from Fidelity i
true value from Fidelity 5Þ=true value
245
Table 5 summarizes the factorial fit for the response 1
(Y1) versus the factors for the throughput from MinitabTM
Release 13.31. A full factorial fit was carried out resulting
in 160 combinations. In table 5, the effect column depicts
the relative strength of the effects. A positive sign indicates
that the high factor setting resulted in a higher response
than the low setting (Minitab Stat Guide 2000). For
example, regarding factor A, the higher factor (complex
system) resulted in a higher relative error than the lower
factor (simple system). The coefficients column in table 5 is
used to construct an equation representing the relationship
between the response and the factors. The SE Coef in table
5 is the standard error of the coefficients. The t-value of the
predictor equals the coefficient of the predictor divided by
the standard error of the coefficient. A larger calculated tvalue corresponds to a smaller p-value. The p-values
determine which of the effects in the model are statistically
significant. For example, the factor A was found significant
for Response Y1.
The analysis of variance (ANOVA) table in table 5 was
used to determine which of the effects in the model were
statistically significant. The F statistic was used to test
whether the effect of a term in the model (factor or
interaction) was significant (Minitab Stat Guide 2000). For
example, the main factors (see table 5) were all found
significant.
The factorial fits for other responses (Y2, Y3 and Y4)
were constructed, and table 6 summarizes the significant
factors for each response (Y1–Y4) that were determined
based on the p-values obtained from the ANOVA tables
with a value of 0.1. The effects of the main factors on
different relative errors (Y1, Y2, Y3 and Y4) are discussed
in the following sections.
First, the effect of the main factor A on different relative
errors (Y1, Y2, Y3 and Y4) is summarized in table 7. As
shown in table 7, the complexity of the system affected the
relative error significantly for responses 1 and 2, confirming
the intuition that the relative errors of performances
generated from abstract models increase as the system
complexity increases (see Section 3.1). In other words,
results from abstract simulation models for complex
systems could be misleading. However, the complexity of
the system did not affect responses 3 and 4, which indicates
that the system complexity does not affect the results of
somewhat more detailed simulation models (Fidelities 3
and 4).
Second, the effect of the main factor B on different
relative errors (Y1, Y2, Y3 and Y4) is summarized in table
8. As shown in table 8, the routing flexibility affected the
relative errors significantly for responses 1, 2 and 3. This
result is expected since more detailed models take flexibility
in part-routing available via alternative routings into more
account. However, the routing flexibility did not affect
response 4, which indicates that it does not affect the results
246
M. Madan et al.
Table 5.
Fractional factorial fit, Y1 versus A, B, C, D, and E
Estimated effects and coefficients for Y1 [coded units].
Term
Constant
A
B
C
D
E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
A*B*C
A*B*D
A*B*E
A*C*D
A*C*E
A*D*E
B*C*D
B*C*E
B*D*E
C*D*E
A*B*C*D
A*B*C*E
A*B*D*E
A*C*D*E
B*C*D*E
A*B*C*D*E
Effect
Coef
5.775
3.637
8.066
1.366
0.367
3.352
5.642
0.122
7 0.852
3.618
1.812
7 1.524
1.476
7 0.081
7 3.837
3.093
1.476
7 1.860
7 0.018
7 1.210
7 4.845
1.679
7 1.639
7 4.790
7 3.781
1.441
7 1.728
7 5.491
7 4.958
7 5.010
7 5.283
4.623
2.888
1.819
4.033
0.683
0.183
1.676
2.821
0.061
7 0.426
1.809
0.906
7 0.762
0.738
7 0.041
7 1.919
1.546
0.738
7 0.930
7 0.009
7 0.605
7 2.422
0.840
7 0.820
7 2.395
7 1.891
0.720
7 0.864
7 2.746
7 2.479
7 2.505
7 2.642
SE Coef
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
1.095
T
4.22
2.64
1.66
3.68
0.62
0.17
1.53
2.58
0.06
7 0.39
1.65
0.83
7 0.70
0.67
7 0.04
7 1.75
1.41
0.67
7 0.85
7 0.01
7 0.55
7 2.21
0.77
7 0.75
7 2.19
7 1.73
0.66
7 0.79
7 2.51
7 2.26
7 2.29
7 2.41
p
0.000
0.009
0.099
0.000
0.534
0.867
0.128
0.011
0.956
0.698
0.101
0.410
0.488
0.501
0.971
0.082
0.160
0.502
0.397
0.993
0.581
0.029
0.445
0.455
0.031
0.087
0.512
0.431
0.013
0.025
0.024
0.017
Analysis of Variance for Y1 (coded units)
Source
DF
Seq SS
Adj SS
Adj MS
F
p
Main effects
2-Way interactions
3-Way interactions
4-Way interactions
5-Way interactions
Residual error
Pure error
Total
5
10
10
5
1
122
122
153
4379.4
3299.0
2965.3
3079.0
1066.5
22347.6
22347.6
37136.8
4245.8
3076.4
3187.4
3182.1
1066.5
22347.6
22347.6
849.2
307.6
318.7
636.4
1066.5
183.2
183.2
4.64
1.68
1.74
3.47
5.82
0.001
0.093
0.079
0.006
0.017
of very detailed simulation models (Fidelity 4). In
summary, for adequate modelling of systems with alternative routings, the corresponding simulation must be
detailed (Fidelity 4).
Third, the effect of the main factor C on different relative
errors (Y1, Y2, Y3 and Y4) is summarized in table 9. As
shown in table 9, stochasticity significantly affected the
relative error in throughput for all fidelities, which seems to
be attributable to the variation in data. In fact, as the
model fidelity increases, the model contains a greater
number of stochastic data, and, therefore, the amount of
variation increases. Interestingly, regardless of model
fidelity, stochasticity is observed to affect the results a lot.
This indicates that results from deterministic simulation
models for stochastic systems can be misleading. In other
words, stochasticity should not be neglected, regardless of
the model fidelity selected.
Fourth, the effect of the main factor D on different
relative errors (Y1, Y2, Y3 and Y4) is summarized in table
10. As shown in table 10, the number of part-types did not
247
Model fidelity for flexible manufacturing systems
Table 6.
Significant factors for response for throughput.
Table 10.
Terms
Y1 Y2 Y3 Y4 Terms
Y1 Y2 Y3 Y4
A
B
C
D
E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
A*B*C
S
S
S
NS
NS
NS
S
NS
NS
NS
NS
NS
NS
NS
S
NS
NS
NS
NS
NS
NS
NS
NS
S
S
NS
NS
S
S
S
S
S
S
S
NS
NS
NS
S
NS
NS
S
NS
NS
NS
NS
NS
NS
NS
S
S
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
S
NS
S
NS
NS
S
NS
NS
NS
NS
NS
S
NS
NS
A*B*D
A*B*E
A*C*D
A*C*E
A*D*E
B*C*D
B*C*E
B*D*E
C*D*E
A*B*C*D
A*B*C*E
A*B*D*E
A*C*D*E
B*C*D*E
A*B*C*D*E
NS
NS
NS
NS
S
NS
NS
NS
NS
NS
NS
S
S
S
S
NS
NS
NS
NS
S
NS
NS
NS
NS
NS
NS
S
S
S
S
NS
NS
S
NS
NS
NS
S
NS
NS
NS
NS
NS
NS
NS
NS
Notes: S: significant, NS: non-significant.
Table 7.
Term
D
D
D
D
A
A
A
A
B
B
B
B
C
C
C
C
SE Coef
t
SE Coef
t
p
1.366
2.253
1.366
0.01757
0.683
1.126
0.683
0.00879
1.095
1.116
0.4494
0.01258
0.62
1.01
1.52
0.70
0.534
0.315
0.131
0.486
Response
Response
Response
Response
1
2
3
4
1.095
2.64
1.116
2.19
0.4494
0.54
0.01258 7 0.57
Regression-based meta-models for responses and factors
p
0.009
0.030
0.593
0.570
Response
Response
Response
Response
1
2
3
4
Effect of factor B on different relative errors
(Y1, Y2, Y3, and Y4).
Effect
Coef
SE Coef
t
p
3.637
5.178
1.979
0.01089
1.819
2.589
0.990
0.00544
1.095
1.116
0.4494
0.01258
1.66
2.32
2.20
0.43
0.099
0.022
0.030
0.666
Table 9.
Term
Coef
5.775
2.888
4.895
2.447
0.481
0.241
7 0.01433 7 0.00716
Term
Coef
Effect of factor A on different relative errors
(Y1, Y2, Y3, and Y4).
Effect
Table 8.
Effect
models. Thus, it can be concluded that this factor does
not play an important role in deciding the appropriate
model fidelity and, therefore, can be ignored.
Finally, the effect of the main factor E on different
relative errors (Y1, Y2, Y3 and Y4) is summarized in table
11. As shown in table 11, the batch size did not affect the
relative error for throughput for responses 1, 2 and 3. This
result does not agree with our intuition (see Section 3.1)
that the effect of an increase in the batch-size on the
increase in number of parts produced is expected to be
more significant for more abstract models.
4.2.
Term
Effect of factor D on different relative errors
(Y1, Y2, Y3, and Y4).
Response
Response
Response
Response
1
2
3
4
Effect of factor C on different relative errors
(Y1, Y2, Y3, and Y4).
Effect
Coef
SE Coef
t
p
8.066
8.441
2.670
0.07421
4.033
4.221
1.335
0.03710
1.095
1.116
0.4494
0.01258
3.68
3.78
2.97
2.95
0.000
0.000
0.004
0.004
Response
Response
Response
Response
1
2
3
4
affect the relative error for throughput for any of the
responses. This result does not agree with our intuition (see
Section 3.1) that a large number of part types, each with
alternative routings, increases the routing flexibility and,
therefore, affects the outputs generated from different
A regression equation is an algebraic representation of the
regression line used to describe the relationship between the
response and predictor variables (Minitab Stat Guide
2000). The coefficient column (see table 5) lists the
estimated coefficients for the predictors. Based on the
experimental results, regression-models are constructed
considering only the significant factors identified. These
regression models will be used for the proposed framework
(see Section 4.5) to determine the most efficient model for
FMSs. Table 12 depicts the regression model of Y1, where
four-way and five-way interactions among the factors could
not be neglected due to sparsity of effect principle as they
were found to be significant. The R2 value is a measure of
the amount of reduction in the variability of Y1 obtained
by using regression variables A, B, C, D and E in the
model. However, a large value of R2 does not necessarily
imply that the regression model is a good one (Montgomery 2000). In this case, it was also seen that the R2 value did
not increase when a quadratic or cubic regression equation
was fitted in place of a linear equation. The R2 value may
be improved by (1) increasing the number of levels of
significant factors and conducting the regression analysis or
(2) by transforming the response variable and conducting
the regression analysis, which is beyond the scope of this
paper. Regression equations for Y2, Y3 and Y4 have been
similarly calculated and are shown in table 13.
4.3.
Validation of regression models
The constructed regression models were validated by
comparing the responses calculated from the regression
248
M. Madan et al.
Table 11.
Term
E
E
E
E
Effect of factor E on different relative errors
(Y1, Y2, Y3, and Y4).
Effect
Coef
SE Coef
t
p
0.367
0.720
0.936
0.06567
0.183
0.360
0.468
0.03283
1.095
1.116
0.4494
0.01258
0.17
0.32
1.04
2.61
0.867
0.747
0.300
0.010
Table 12.
Response
Response
Response
Response
1
2
3
4
Regression equation: Y1 versus A, C, AC, BDE,
ABDE, ACDE, BCDE, ABCDE.
The regression equation is
Y1 = 4.67 + 2.94 A + 4.08 C + 2.88 AC 7 2.46 BDE 7 2.81 ABDE
7 2.39 ACDE 7 2.57 BCDE 7 2.70 ABCDE
Predictor
Constant
A
C
AC
BDE
ABDE
ACDE
BCDE
ABCDE
S = 8.72
Coef
SE Coef
4.665 1.107
2.944 1.107
4.075 1.107
2.878 1.107
7 2.462 1.107
7 2.807 1.106
7 2.385 1.107
7 2.572 1.107
7 2.703 1.106
R2 = 62.5%
t
p
4.22
0.000
2.66
0.009
3.68
0.000
2.60
0.010
7 2.22
0.028
7 2.54
0.012
7 2.16
0.033
7 2.32
0.022
7 2.44
0.016
R2 (adj) = 60.5%
Analysis of Variance
Source
DF
SS
MS
F
p
Regression
Residual error
Total
8
145
153
9852.2
27284.5
37136.8
1231.5
188.2
6.54
0.000
Source
DF
Seq SS
1
1
1
1
1
1
1
1
1214.2
2543.5
1302.7
764.5
1094.1
823.8
986.4
1123.0
distributed about zero and less concentrated as they move
away from zero. This residual plot is used to test model
assumptions and identify unusual observations, such as: (1)
linear relationships; (2) constant variance; and (3) outliers
(Minitab Stat Guide 2000). It was observed that the
residual plot does not contain significant outliers, and
therefore, the regression model seems valid. Other regression equations (for Y2, Y3 and Y4) can be validated
similarly.
4.4.
The adequacy of the regression models has been checked
with a simple FMS. A test model has been built where the
factors of the levels were set as (1, 1, 7 1, 1, 0) for (A, B, C,
D, E), respectively. The batch size for the zero level was
assumed to be 15. The relative errors obtained from the
regression models were Y1 (0.65), Y2 (0.91), Y3 (1.53) and
Y4 (0.063), and the relative errors obtained from the test
models were Y1 (0.85), Y2 (0.94), Y3 (1.635) and Y4
(0.0456). The relative errors obtained from the regression
equations and the models have been found to be close.
However, testing with an FMS is insufficient to conclude
that the relative errors obtained from the regression
equations will always be accurate and linear for the zero
level. If the relative errors had been found to be
significantly different, new regression equations would have
to be calculated by including the various observed values at
the zero level. This is also the case for the levels lying
beyond the region of the original observation.
4.5.
A
C
AC
BDE
ABDE
ACDE
BCDE
ABCDE
models and raw data obtained from the experiments.
According to the underlying assumptions in the regression
models, the differences are assumed to follow a normal
distribution N (0, s2). It was observed from the normal plot
for response Y1 that the residuals (differences between
calculated and observed values) follow a normal distribution as they roughly form a straight line and the ‘mean’ of
the residual is approximately equal to zero. Also, the
residual plots (residuals versus the fitted values from
regression models [Y1]) have been examined. If valid, the
residuals plotted against the fits should be randomly
Model adequacy checking
Method to determine an efficient model fidelity
Tables 12 and 13 summarize the results obtained from the
factorial fit and regression equations for four responses in
terms of the factors. Based on the relative errors estimated
from the regression models, we propose a method to
determine an efficient model fidelity composed of the
following five steps:
.
.
.
Step 1: a user provides three types of inputs: (1)
characteristics of FMSs to be modelled, (u1, u2, u3, u4,
u5), where ui is the level of ith factor; (2) performance
metrics; and (3) the threshold value for the relative
error.
Step 2: for a given performance metric, significant
factors and corresponding regression models with
varying fidelities are identified from tables 12 and 13.
It is noted that the contents of tables 12 and 13 will be
significantly expanded as a future work.
Step 3: the levels of significant factors, which were
provided by the user in Step 1, are assigned to the
regression equations, and the relative errors for each
response are estimated.
Model fidelity for flexible manufacturing systems
Table 13.
249
Regression equations for Y2, Y3, and Y4.
The regression equation is
Y2 = 4.91 + 2.57 A + 2.66 B + 4.27 C + 2.57AC + 2.39 BC 7 2.70 ADE 7 3.07 ABDE 7 2.64 ACDE 7 2.17 BCDE 7 2.92 ABCDE
The regression equation is
Y3 = 1.93 + 0.983 B + 1.32 C 7 1.13 ADE 7 1.12 ABDE 7 1.07 ACDE 7 0.979 ABCDE
The regression equation is
Y4 = 0.0580 + 0.0381 C + 0.9329 E 7 0.0304 + 0.0274 CE 7 0.0335 ACD
.
.
.
.
.
.
.
.
Step 4: the most abstract model (Fidelity i, where i is
smallest) among those models whose errors are
smaller than the threshold value is selected as the
most efficient model.
Step 5: the Functional model (see figures 1–4)
corresponding to the selected model formally provides
the user with the entities to be considered in the
simulation model.
For demonstration purposes, the proposed method is
tested with two sample FMSs.
Step 1: it is supposed that a user wants to build
simulation models of two sample FMSs to estimate
the throughput. Those two FMSs are characterized as
(+ 1, + 1, + 1, + 1, + 1) and (0, 71, 71, 0, 0),
respectively. Refer to Section 3.1 for details about
these factors. The user wants to estimate the results
whose relative errors are within +/7 0.1 (10%).
Step 2: the significant factors for the throughput and
corresponding regression models are identified from
tables 12 and 13.
Step 3: the levels of significant factors, which were
provided by the user in Step 1, are assigned to the
regression equations (see tables 12 and 13) to calculate
the relative errors for each response. The relative
errors estimated for the first FMS, (+ 1, + 1, + 1,
+ 1, + 1), are Y1 (1.640), Y2 (5.870), Y3 (7 0.066)
and Y4 (0.092). Similarly, the relative errors estimated
for the second FMS, (0, 7 1, 7 1, 0, 0), are Y1
(0.590), Y2 (0.370), Y3 (7 0.373) and Y4 (0.019).
Step 4: Fidelity 3 model and Fidelity 4 model have
been found to be the most efficient simulation model
for two FMSs since the expected relative errors are
closest and smaller than the threshold value.
Step 5: Figures 3.4 and 3.5 are the corresponding
functional models for the Fidelity 3 and Fidelity 4
models. Automatic simulation model generation from
this formal model has also been studied, but is beyond
the scope of this work.
5. Conclusion and Future Work
In this research, a framework has been proposed to
determine an efficient level of simulation model fidelity
for flexible manufacturing systems. This framework has
been based on O–O modelling, simulation, intensive
experiments and regression-based meta-models. Five different models have been defined using O–O modelling,
based on which simulation models have been developed.
Using these simulation models, intensive experiments have
been conducted with detailed factorial designs. Experimental results indicate that, under the test conditions, the
complexity of FMSs, the routing flexibility, and the
stochasticity of data in FMSs affect the throughput
generated from simulation models with varying fidelities.
On the other hand, the number of part-types and the batch
size have been found to be insignificant to the outputs.
These experimental results have been summarized with the
regression models, based on which a method to determine
the most efficient model fidelity for FMSs has been
proposed. The proposed method has been demonstrated
with two sample FMSs. While this work presented a
framework, further research efforts will focus on extending
the scope of the proposed method by studying more
performance metrics, more diverse FMSs (stable, unstable),
and more intensive validation of regression models.
Acknowledgment
The work was supported by National Institute of Standards
and Technology Grant and State of Arizona Information
Technology and Commerce Institute Grant.
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