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Title: Fundamentals of Model Development

1
Fundamentals of Model Development
Margaret L. Loper Georgia Tech Research
Institute Atlanta, GA margaret.loper_at_gtri.gatech.e
du
2
3-Spoke Wheel of Simulation
• Simulation is a tightly coupled and interactive
component process composed of
• Model Design
• High level specification often grounded in
predicate logic or mathematical theory
• Model Execution
• One model can be executed in many ways
• Execution Analysis
• Perform tests on the data generated from the
model by running specific analysis

Conceptual Models Model Theory Mathematical Models
Model Design
Model Execution
Execution Analysis
Serial Algorithms Parallel Algorithms
I/O Analysis Experimental Design Surface Response
Techniques Visualization of Data Verification
Validation
3
Course Outline
Part I The Nature of Modeling - Philosophy,
Roles, Uses Part II The Science of Modeling -
Theory, Assumptions, Scope Part III Problem
Formulation - Objectives Part IV Model
Foundations - Components, Variables,
Interactions Part V Model Engineering -
Techniques, Formalisms Part VI Lessons in
Modeling
4
Part I The Nature of Modeling
5
What is a Model?
• If I were to ask several people for examples of
models, I would get a variety of responses
• Mathematical equations
• Toy trains
• Prototype cars
• Fashion models
• What these very different things have in common
is that they are representations of reality
• Equations may represent growth of a population
• Toy train is a representation of a real train
• Prototype is a representation of a future car
• Fashion model is a representation of how clothes
will look when worn

Modeling in its broadest sense is the
cost-effective use of something in place of
something else for some cognitive purpose
6
Models as Purposeful Representations
• A model represents reality for the given purpose
• The model is an abstraction in the sense that it
cannot represent all aspects of reality
• This allows us to deal with the world in a
simplified manner, avoiding the complexity,
danger, and irreversibility of reality

We cannot build a model if we do not know why we
are building it and we cannot criticize or
discuss a model except in terms of its purpose
• Example A street map is a representation of
reality for navigating streets in a particular
city it is useless for driving across country or
even locating traffic jams or construction within
a city other models are needed for those purposes

7
Models as Problem Solving
• Modeling is one of the most essential activities
of the human mind
• Underlies our ability to think and imagine, to
use signs and language, to communicate, to
generalize from experience, to deal with the
unexpected, and to make sense out of the raw
bombardment of our sensations
• It allows us to see patterns, to appreciate,
predict, and manipulate processes and things, and
to express meaning and purpose

Learning to model is closely related to the more
general skills of problem solving to solve
problems is to think imaginatively and
purposefully
• In engineering terms, a model is used because it
enables predictions or calculations or in some
other way makes the design process more
convenient
• Models are used to assist the designers
thinking, analyze potential designs, realize what
is known or unknown, predict behavior, identify
connections, etc.

8
Model Representations
• Verbal Models
• Model couched in the language of everyday
discourse
• He is tall with red hair and green eyes, his
cheeks are pale and his nose is pimpled. His
left ear is larger than the right and one of his
front teeth is missing.
• Visual Models
• Diagrammatic, a picture is worth a thousand words
• Diagrams, drawings or images
• Can exclude some of the uncertainties in a verbal
description, e.g. is it the persons left ear or
the persons left ear as seen by the observer?
• Physical, a reconstruction of the real object at
a smaller scale
• Mannequins used in car crash tests
• Airplane model in a wind tunnel
• Formal Models
• Model that uses equations and formulas to
reproduce the physical object
• Q mC(t1-t2) a model of heat emitted by a body
of mass m, when cooling from temperature t1 to
temperature t2. C is the heat capacity parameter.

9
Modeling Approaches
• The issue in modeling is not complexity as much
as it is expressiveness and orientation
• Its not enough to learn one method of modeling
any more than there can be one way to model with
clay or paint a picture
• With a single modeling method, you can model a
certain way however, that method might block out
an understanding of a subsystem that is better
understood with a different modeling method
• The block model and the equation provide exactly
the same semantics either model can be
translated into the other
• However, these are different models because they
reflect different ways of looking at the physical
system
• The way you like to view systems will influence
your decision as to which model to choose

10
The Golden Rule of Modeling
• The Golden Rule of modeling is that no model,
no matter how accurate, has any inherent value of
its own
• The value of every model is based entirely upon
the degree to which it solves someones real
world problem
• We never build a perfect model a model is built
for a specific purpose and its accuracy is judged
with respect to that purpose
• Project Take a minute and determine how many MM
candies will fit in this jar
• Why do you need to know how many candies fit in
the jar?
• To determine if one bag would fill the jar? The
accuracy of the model may be more or less one
bag.
• To determine how many bags are needed to fill the
jar? The accuracy of the model may be to within a
couple of handfuls.
• To win a contest? The accuracy of the model may
be the exact maximum number of candies.
• Purpose is a vital part of any model that you
build

11
Is a Model Different From Theory?
• A model is related to, but different from, a
theoretical description of the object
• The model may be based on theory
• But may include non-ideal behaviors observed in
experiments that are not well explained by theory
• Theory may predict certain trends, but empirical
numbers from experiments are included to get the
calculated results to agree with the real results
• The key difference is that a model must behave as
nearly as possible the same way as the real thing
• It is not directly important whether the models
behavior is well predicted by theory it is the
results that count
• However, a good theoretical basis is good,
because it will likely expand the range of
conditions over which the model will work

12
Part II The Science of Modeling
13
Definitions
• System
• A collection of entities that act and interact
together toward the accomplishment of some
logical end
• Theory
• A structured body of knowledge about some
phenomenon that allows us to make meaningful
explanatory or predictive statements about it
• Some theories can be proven mathematically, while
others require empirical validation through
observation, collection, and analysis of data
• Model
• A formalized theory, a stylistic interpretation
of a body of propositions that a theory
represents
• Answers a certain class of questions
• Simulation
• The manipulation of a model of a system in such a
way that properties of the system can be studied
• The use of a mathematical model to study the
behavior of a system as it operates over time

14
Modeling Relationships
Theory of Operation
Simulation Language (ACSL, Arena, Simscript)
15
Formal Modeling
• The logicians definition a possible realization
in which all valid sentences of a theory T are
satisfied is called a model of T
• In this context, this suggests that we might
distinguish between
• The prototype (physical entity or system being
modeled)
• The theory of the model (precise statement of the
assumptions of axioms)
• The model itself (the scheme of equations)
• From the logical point of view, the prototype is
a realization in which the valid sentences of the
mathematical model are to some degree satisfied
• There may be several sets of assumptions which
lead to the same model, since in general there is
an equivalence class of assumptions which
connects the phenomenon with the mathematical
model
• We then use the model to draw conclusions (i.e.
make predictions) this is the deductive process
if the assumptions are true, then the conclusions
must also be true
• Discrete math refresher..If P ? Q

A general method in proofs of undecidability A.
Tarski 1953
16
Assumptions
• Making assumptions is an integral part of
deciding what to take from the real to the model
world
• Assumptions one makes depend on both the purpose
of the model and the constraints under which it
is built
• Definition of the variables and their
interrelations constitute the assumptions of the
model
• Take a minute and determine how many ping-pong
balls could you fit into this room
• The room is shaped like a box?
• A ping-pong is assumed to be a cube?
• Furniture in the room can be ignored?
• Door spaces and other nooks and crannies can be
ignored?

17
Model Assumptions
• Just as theories rest on assumptions, so do
models

Assumptions
• As assumptions bound a theory and make it
possible, they structure a model and make it
viable

Model
Assumptions
Assumptions
• The more highly structured a model, the more
numerous its assumptions

Assumptions
Assumptions should be clearly stated, for it is
the assumptions that determine the purpose of a
model, and the credibility that can be ascribed
to its predictions
18
Scope of a Model
• Scope is determined by the purpose of a model
• The more complex and structured a model becomes,
the less able it is to answer new and unexpected
questions

Assumptions
Low
High
Highly structured High empirical
content Carefully phrased, narrow questions with
more certainty
Simple structure Low empirical content Broader
classes of questions, but with less certainty
19
The Value of Detail
• A model should only be as detailed as is
necessary to answer the questions at hand
• A model designer must consider, from the points
of view of structure and data, the statistical
aspects of the system being modeled
• Structural Uncertainties arise from our
imprecise knowledge of how systems function
• Inability to separate real and apparent causes
• Confusing correlation and causation
• Data Uncertainties due to difficulties in
parameter estimation
• Input data errors (estimation)
• Statistical nature of simulation models
(randomness)
• Any model that is predicted on a certain
structure that has been arrived at through
observation rather than theory runs the risk of
making predictions that must be qualified by
confidence statement
• Accepting an uncertain structure is equivalent to
making an assumption, which limits a models
scope and utility

20
Summary
• Before a model is designed, two important
• What is the purpose of the model?
• What are the requirements for accuracy and
precision?
• Answers to these questions determine the
structure of a model, as they demand certain
assumptions to be made, that certain boundaries
be imposed and respected, that certain types of
questions can and cannot be asked, that certain
territories cannot be explored, and that certain
realities cannot be predicted
• Without a statement of purpose, there can be no
theory, no model
• Since a models value lies in the way in can be
used, detail is necessary only to the extent that
it contributes to the precision of the model
predictions or estimates without limiting the
variety of questions that can be asked

21
Part III Model Formulation
22
Building a Model
• Model building involves imagination and skill
giving rules for doing it is like listing rules
for being an artist
• At best we can provide a framework around which
to build skills and develop imagination
• Formulate the problem
• What do you wish to know? What do you intend to
do with it?
• Outline the model
• Things whose effects are neglected or ignored
• Things that affect the model but whose behavior
the model is not designed to study
(exogenous/input/independent variables)
• Things the model is designed to study the
behavior of (endogenous/output/dependent
variables)
• Specify the interrelations among variables
• Is it useful?
• Can you obtain the needed data and then use it in
the model to make the predictions you want?
• Test the model
• Use the model to make predictions that can be
checked against data or common sense.

23
Problem Formulation
• Every study should begin with a statement of the
problem
• Problem formulation is concerned with defining
the questions one is trying to answer with the
simulation study
• Typically sets of questions are defined by policy
makers who intend to use the simulation results
to make decisions four perspectives on the use
of models in decision making Dutton 1987
• Rational
• Models are seen as tools of scientific management
that provide better information for policy making
• Partisan
• Models are tools of propaganda and persuasion
rather than information models are used to
legitimate decisions made for other reasons
• Technocratic
• Modelers use information technology to baffle, to
impress and to promote their own positions
technocrats try to gain political power by the
authority of their expertise, i.e. models are
just complicated representations of the modelers
personal theory and biases
• Consensual
• Models are used as potential tools of interactive
decision-making and negotiation modeling is a
political process but as one that can be useful
in achieving consensus

24
Model Formulation
• The problems that modelers wish to solve exist in
the real world
• First step is to simplify the real world to
create a model world
• The model world leaves out much of the complexity
of the real world

Occams Razor
Interpreting and Testing
• The original question gets translated into a
question involving the model world

Model World
• Next, construct a model of the problem in the
model world

Formulating Model World Problem
Model Results
• The final step is to interpret the answer found
for the model world problem back in the real world

Model
Mathematical Analysis
25
Occams Razor
• William of Ockham, 14th Century Philosopher and
Franciscan monk (1285-1349)
• "Pluralitas non est ponenda sine neccesitate" or
"plurality should not
be posited without necessity"
• Other translations
• things should not be multiplied without good
reason
• entities must not be multiplied beyond what is
necessary
• the simplest theory that fits the facts of a
problem is the one that should be selected
• In other words, dont make things harder than
they have to be!
• Eliminate all unnecessary information relating to
the problem being analyzed
• Exclude details that are irrelevant given the
purpose, or can not be handled given the
constraints
• Cut down the world to a manageable size
• Cut too much and the model solutions have nothing
to do with reality
• Cut to little and the problem is too difficult to
solve with the available resources

26
Heuristics
• Asking questions is a crucial part of modeling
and problem solving
• The most useful questions are those which are
pragmatic that ask for information in a useful
form or ask what you would do with it if you had
it
• Heuristics tend to be most useful in the early
stages of solving a problem because they
encourage you to think fruitfully about the
problem
• Try to feel the problem (Imagine yourself inside
the system and ask what is going on around you)
• Anticipate what a solution would look like
(number, graph, table, etc)
• Think about how you would solve a simpler version
of the same problem
• Rephrase the problem to make sure you understand
it
• Draw a simple diagram of what is happening
• Are there any physical laws to consider
(conservation of matter or energy)
• Look for simplifying assumptions
• Look for bounds (simple models that would
definitely underestimate or overestimate the

27
Is the Model Useful?
• Establish a clear statement of the deductive
objectives
• Objectives indicate the questions to be answered
by simulation and may include metrics to be
collected by the simulation
• Throughput, Delay, Efficiency, etc
• Do you want the model to predict the consequences
of various policies or suggest an optimal policy?
• Provide the criterion for determining the
deductive viability or tractability of the model
however, it may prove unachievable or different
objectives may emerge as the model develops
• Model Viability
• If the wrong things are ignored, the model is no
good if too much is taken into consideration,
the resulting model will be hopelessly complex
and probably require incredible amounts of data
• Identify all significant variables and their
relative importance
• Recognize the risk of leaving out significant
factors
• Recognize the level of uncertainty associated
with each variable
• If a tractable model is obtained, enrich it.
Otherwise, simplify.

28
Part IV Model Foundations
29
Definitions Revisited
System Part of some potential reality where we
are concerned with space-time effects and causal
relationships among parts of the system
Model Abstract from reality a description of a
system
Modeling A way of thinking and reasoning about
systems A goal is to come up with a
representation that is easy to use in describing
systems in a consistent manner
30
The Concepts of Modeling Nance
• A Model is a representation and abstraction of
anything such as a real system, a proposed
system, a futuristic system design, an entity, a
phenomena, or an idea

31
Model Definition (Formal)
• A unifying formalism that serves to represent a
wide variety of system models comes from
classical systems theory and defines a
deterministic system as ?T, U, Y, Q, ?, ?, ??
• T is the time set. For continuous systems T ?
(real numbers), and for discrete time systems, T
? (integer numbers).
• U is the input set containing the possible values
of the input to the system.
• Y is the output set.
• Q is the state set.
• ? is the set of admissible (or acceptable) input
functions. This contains a set of input
functions to use during system operation. Often,
due to physical limitations, ? is a subset of all
the possible input functions (T ? U).
• ? is the transition function. It is defined as ?
Q x T x T x ? ? Q. If the system is time
invariant, ? Q x ? ? Q
• ? is the output function, ? Q ? Y.

System
Model
Output Observable of some part of system
Input Controlling influence on system
32
System Modeling
• The task of deriving a model of the system may be
• Establish Model Structure
• Components
• The parts from which the model is constructed
• Descriptive Variables
• Describe the conditions of the components at
points in time
• Component Interactions
• Rules by which components exert influence on each
other, altering their conditions and so
determining the evolution of the models behavior
over time
• Supply Data
• Provide the values the attributes can have and
defines the relationships involved in the
activities

33
Model Structure Components
• The parts from which the model is constructed
• States
• State is a collection of variables that describe
the physical system for an interval of time
• Discrete State variables assume a discrete set
of values
• Continuous State ranges can be represented by
the real numbers
• Mixed State both kinds of variables are present
• Changes in the physical system are realized by
updating one or more of the state variables
• A state is defined as a simple tuple ?S1,Sn?,
where n is the number of components in the state
vector and Si are the states components
• Events
• An event is an instantaneous occurrence that
changes the state of the system (special kind of
state that has no time duration)
• Each event has a time associated with it
indicating when the event occurred an event is a
time tagged state ?t, S1,Sn?
• Events are relative to the level of abstraction
for which the system has been defined

34
Model Structure Components
• Input
• Input is a state that has a controlling influence
on a system which does not contain the input
state
• Input is just another kind of state except that
it permits us to place boundaries around what is
considered to be inside and outside a system
• An input that is constant (non-time varying) is a
parameter
• Output
• An output is a function of the system state and
input
• Time
• Time is denoted by either an integer or a real
number (quantitative) or a nominal variable such
as early, late, or before-lunch (qualitative)
• In quantitative systems, time can be represented
as continuous or discrete
• Discrete - time changes in increment steps (1
minute, 1 day, 1 year, etc)
• State variables change instantaneously at
distinct points in simulation time
• Continuous time is specified to flow
continuously through the real numbers
• Discrete Class time flows continuously but state
changes occur in discontinuous jumps
• Differential Equation Class continuous time and
continuous state, thus the time derivatives are
governed by differential equations

35
Model Structure Descriptive Variables
• Describe the conditions of the components at
points in time
• Quantitative Data values are counts or numerical
measurements
• Continuous data
• Measurements can be any value, usually within
some range, e.g. weight
• Discrete data
• Measurements are integers, e.g. number of people
in a household
• Qualitative Data values are non-numeric
categories
• Also described as discrete since there are a
finite number of categories observations may fall
• Color of eyes blue, green, brown etc
• Socio-economic status low, middle or high
• Nominal No natural order between the data, e.g.
eye color
• Ordinal An ordering exists among data values,
e.g. exam results pass or fail
• Random Variables Data values defined according
to a probability distribution
• Deterministic No random variables appear
• Probabilistic or Stochastic Contains at least
one random variable
• Probability distributions
• Uniform, Normal, Poisson, etc

36
Qualitative Simulation
• There are many situations in which it is not
possible to quantify the attributes in a way that
has any meaning or validity or even if it can be
quantified, it is not appropriate for the
particular study
• Higher pay must increase incentives, and so
increase productivity and profits
• Mixes actual quantities (e.g., pay) with
quantities such as incentives, which cannot
actually be quantified, but for which at least
more or less can be imagined
• Qualitative simulation was developed to describe
complex physical phenomena in the absence of good
quantitative information
• Modeling a robot that makes a cup of coffee what
variable to use to describe the cup?
• A distinguishing characteristic is coverage
• Qualitative simulations simulate all possible
• When it determines the next possible state it can
determine that there are several next possible
states due to the imprecise nature of the data
• Executes each of these possible next states
• Resulting envisionments include all possible
event sequences

Qualitative Physics Past, Present, and
Future, Exploring Artificial Intelligence, K.D.
Forbus, 1988
37
Model Structure Component Interactions
• Rules by which components exert influence on each
other, altering their conditions and so
determining the evolution of the models behavior
over time
• Rules of Interaction
• An interaction is an explicit action taken by an
object that may have some effect or impact on
another object
• Rules specifying object interaction determine the
manner in which the state variables change over
time
• Time Invariant rules of interaction are stated
entirely in terms of the values that the
descriptive variables can assume
• Time Varying time is an argument of the rules
of interaction and may thus appear to be
different at different times
• Interactions do not persist in time, however they
can affect the state of a persistent object
• History
• Whether responses are influenced by past history
• Models that make use of such remembered
information are called adaptive in recognition of
their ability to learn from previous experience

38
Model Structure Component Interactions
• Environment
• The environment in which the object exists and
operates has impact upon the outcomes of every
operation
• Autonomous
• The system is cut off from all influences of its
environment (closed)
• Data collected under specific conditions
• Integrated into the objects and interactions
Simulated environment conditions under which
data collected
• Non-Autonomous
• The system is influenced by the environment via
input variables (open)
• Need to collect and manage large volume of data
terrain surface, natural and cultural features,
atmosphere, sea surface, sub-surface, and ocean
floor

39
Model Data
• The next step to building a model of a system is
to gather data associated with that system
• The real or virtual system that we are interested
in modeling is viewed as a source of observable
data
• Systems differ with regard to how much data is
available to populate the system database
• Data-rich, data is abundant from prior
experiments or obtained from measurements
• Data-poor, meager amounts of historical data or
low-quality data
• In some cases it is impossible to acquire better
data (e.g., combat) in others it is too expensive
(topography and vegetation of a forest)
• Types of data
• Numeric Form
• Physical Sensors (thermocouple, stress gauge)
• Human Sensors (eye sight, tactile feedback)
• Nominal Data (interviewing methods, knowledge
acquisition techniques developed to obtain
qualitative knowledge)
• Symbolic Form
• Extends standard data to the case of variables
not restricted to be numerical
• Includes data having internal variations such as
value distributions background knowledge can be
added as input such as ontology, taxonomies,
• Language

40
Levels of Description
• To build a model there are many decisions that
must be made, either explicitly or implicitly
each is a continuum rather than a discrete choice

These are meta-modeling questions there is no
rigorous ways to make these choices, but there
are rigorous ways to use them once the decision
41
Classifications of Models
• Abstract or Physical Models
• Abstract Model is one in which symbols and logic
constitute the model. The symbolism used can be
a language or mathematical notation.
• A verbal or written description in English is an
abstract model
• A mathematical model is described in the
language of mathematical symbols
• Physical Model takes the form of a physical
replica, often on a reduced scale, of the system
it represents. A physical model looks like the
object it represents and is also called an Iconic
Model.
• A model of an airplane (scaled down), a model of
the atom (scaled up), a map, a model car
• Static or Dynamic Models
• Static Model describes relationships that do not
change with respect to time
• An architectural model of a house which helps us
visualize floor plans and space relationships, A
fractal structures, An equation relating the
lengths and weights on each side of a playground
seesaw
• Dynamic Model describes time-varying
relationships
• A wind tunnel which shows the aerodynamic
characteristics of proposed aircraft designs,
Prediction of the trajectory of a space craft,
Prediction of the effect of a tax cut on an
economy, Equations of motion of the planets
around the sun constitute a dynamic mathematical
model of the solar system

42
Classifications of Models
• Analytical or Numerical Mathematical Models
• Analytical Model solved by using the deductive
reasoning of mathematical theory
• An M/M/1 queueing model, a Linear Programming
model, a Mixed Integer Linear Programming model,
a nonlinear optimization model
• Numerical Model solved by applying computational
procedures
• Finding the roots of a nonlinear algebraic
equation, f(x) 0, using the method of Interval
Halving or Simple Iteration
• System Simulation is considered to be a numerical
computational technique
• Stochastic or Deterministic Models
• Stochastic Model picks up the response from a
set of possible responses according to a fixed
probability distribution
• Behavior of systems under random conditions,
Gamblers Ruin
• Deterministic Model generates the response to a
given input by one fixed law
• Particle interactions of classical physics,
Positions of buildings and local land features,
Virtual circuits

43
Classifications of Models
• Individual or Aggregate Models
• Individual Models detailed representations of
individual entities or objects and their behavior
• Urban combat, Ecological systems, Social and
Economical processes
• Aggregate Models collective representation of
entities or objects in the system
• Pedestrian travel patterns, Climate change,
Economic growth, Military wargames
• Continuous or Discrete Models
• Continuous Model changes occur continuously
• Painting, smooth lines and color changes,
Differential equations in partial derivatives,
Storm water management, Variational optic flow
• Discrete Model changes occur at fixed time
intervals
• Mosaic, discrete uniform segments that change
their shape and color in a stepwise manner,
Finite elements or difference schemes,
Manufacturing

44
Classifications of Models
• Qualitative or Quantitative Models
• Qualitative Models non-numeric models
• Natural language and cognition, Biological
networks, Fuzzy systems, Molecular function
• Quantitative Models numeric models
• Supply chain management, Organic chemistry,
Communication networks
• Causal or Correlational Models
• Causal Models reflect the cause-effect
relationships between entities in the system
• Reasoning, Physical and Chemical processes, Time
• Correlational Models observed phenomena are not
the cause of others
• Weather forecasting, Fractal Gaussian noise,
International politics
• Linear or Nonlinear Mathematical Models
• Linear Model is one which describes
relationships in linear form
• The equation y 3x 4z 1 is a linear model
• Nonlinear Model is one which describes
relationships in nonlinear form
• The equation F (2x 4z 2) / (3y x) is a
nonlinear model

45
Part V Modeling Engineering
46
Concepts of Modeling Osman Balci
On the one hand, a model should not contain
unnecessary details and become needlessly complex
and difficult to analyze
Modeling is an artful balancing of opposites
On the other hand, it should not exclude the
essential details of what it represents
47
Mathematical Modeling
• What is a mathematical model?
• The formulation in mathematical terms of the
assumptions and their consequences believed to
underlie a real world problem
• An abstract, simplified, mathematical construct
related to a part of reality and created for a
particular purpose
• Fundamental steps in developing a mathematical
model

Mathematics in Nature, John Adams 2003
48
Mathematical Methods
• Three kinds of mathematical methods are discussed
in the literature
• Numerical
• Numerical methods involve the actual data that
appear in tables
• Working directly with the numbers and trying to
understand relationships
• Uses direct operation on numbers as its main tool
• Graphical
• Graphical methods use pictorial or spatial
representations to communicate relationships
• Most useful in conveying qualitative
relationships or approximate data which involve
only a few variables
• A graphical approach to a problem is most likely
to be useful when not much information is
available or when it is given in imprecise form
• Theoretical
• Theoretical methods are also called analytic or
symbolic methods
• Theoretical methods employ mathematical tools for
expressing and manipulating relationships or
patterns these tools typically involve variables
and algebra as well as an understanding of how
different objects in the mathematical universe
behave

49
More on Mathematical Methods
• Observations
• The three methods complement each other and
contribute to understanding and insight
• Each method helps us think about the problem in a
special way
• In all methods, assumptions are not exactly
correct, however we simplify the problem by
ignoring the differences between the real data
and the equation and this allows us to proceed
and make predictions
• Numerical
• More correct result, but obtained via calculator
trial-and-error (tedious)
• Graphical
• Get a rough idea of the situation limited
accuracy
• The results may only be partial correct, but
provides an overall picture of the original data
• Theoretical
• Most correct results quickly applied and yields
• The theoretical method has greater power and
generality
• Easily extended to related problems by analyzing
the steps used to solve a problem, it is possible
to formulate procedures or rules that can be
applied to a large number of closely related
problems

50
Types of Mathematical Models
• Analytical
• Analytical models are mathematical models that
have a closed form solution, i.e. the solution to
the equations used to describe changes in a
system can be expressed as a mathematical
analytic function
• Can deduce everything there is to know about a
system the cost is limited applicability much
of the world is too complicated to be described
this way
• Ordinary differential and difference equations,
partial differential equations, variational
methods and stochastic processes
• Numerical
• Numerical models are mathematical models that use
some sort of numerical time-stepping procedure to
obtain the models behavior over time
• Numerical models use approximations to solve
differential equations
• Require the model domain and time be discretized
• Model domain is represented by a network of grid
cells or elements, and the time of the simulation
is represented by time steps
• Finite differences for ODES and PDES, finite
elements, cellular automata
• Observational
• Model inference based on observations and
measured data
• Used to characterize and classify data,
generalize from measurements in order to make
predictions, or learn about the rules underlying
the observed behavior
• Function fitting, data transformations, search
techniques, density estimation, filtering and
state estimation, linear and nonlinear time series

51
The Sound of a Violin
• There are many levels of description choosing
among them depends on your goal and on available
tools
• Analytical You could try to use an analytical
(pencil-and-paper) solution to the governing
equations in return for some large
approximations you may be able to find a useful
explicit solution, but it might not sound very
good
• Numerical You could use a numerical model based
on first-principles description you match your
model parameters to measurements on a real
instrument, and change parameters between a
Stradivarius and Guarneri however running it in
real time would require a super computer and the
effort to find good parameters for the model is
almost as much work as building the real violin
• Observational You could forget about the
underlying governing equations entirely and
experimentally try to find an effective
description of how the players actions are
related to the sound made by the instrument

52
Modeling Formalisms
• Perhaps the hardest general problem is
determining the exact method that one should use
to create a model
• There are many modeling formalisms, but which
technique is best under what conditions?
• Differential Equations
• Difference Equations
• Finite Elements
• Linear Programming
• Nonlinear Programming
• Algebraic Equations
• Predicate Calculus
• Statistical Models
• Geometry
• System Dynamics
• Fuzzy Logic
• Graph Theory
• Monte Carlo Methods
• Case-Based Reasoning
• Neural Networks
• Cellular Automata
• Petri Nets
• Finite State Automata
• Queueing Models
• Markov Chains
• Genetic and Evolutionary Algorithms
• Qualitative Modeling

53
Part VI Lessons in Modeling
54
Approximations
• It should be remembered at all times that models
and simulations are all approximations of reality
• They may use simplifying assumptions
• Unknown effects can not be included
• Equations may be solved by numerical methods,
which do not yield exact results
• Often, models are only valid over a specific
range of conditions, especially if they are
semi-empirical (use measured data)
• The engineer must understand
• The theory, models, and techniques on which the
solution is based
• Nature of the approximations used in the model
• The situations for which the technique is valid
• There is no substitute to experience with a
particular modeling tool
• Often engineers know when a particular tool gives

55
Problems with Models
• There seems to be three broad sets of problems
• Modeling can be done more or less badly, so much
so that it can become dangerous
• The modeling of situations involving human choice
raises all sorts of difficulties that do not
arise in modeling the behavior of inanimate
objects
• Models once created can be used in dubious ways
• Problems arise from a confusion between types of
models and the grounds we have for believing in
them
• Some models can be checked for consistency
(Newtonian mechanics) while others are based on
purely ad hoc assumptions, the only check being
against empirical observations (economic models
or risk analysis)

Models based on good theory can compensate for
lack of data, and models based on broad evidence
can compensate for lack of theory, but models
alone can hardly compensate for the lack of both!
56
Examples from History
Bad designs result from errors of judgment, which
is not reducible to science or mathematics
• Tacoma Narrows Bridge collapse, November 7, 1940
• The designer omitted the vertical stiffening
trusses recommended for the deck of his
suspension-bridge.
• the unwisdom of allowing a particular
profession to become to inward looking and so
screened from relevant knowledge growing up in
other fields around it.

How Engineers Lose Touch, Eugene Ferguson,
Invention Technology, Vol. 8 No. 3, 1993
57
Examples from History contd
• Hartford Civic Center roof collapse, January 18,
1978
• The modern space-frame roof collapsed under a
• To design a space frame with a slide rule or
mechanical calculator was a laborious process
with too many uncertainties for nearly any
engineer, so they were seldom built before
computer programs were available
• The computers apparent precision (to six or
seven significant figures) can give engineers an
unwarranted confidence in the validity of the
resulting numbers.
• The roof design was extremely susceptible to
buckling which was a mode of failure not
considered in the computer analysis and,
therefore, left undiscovered

58
Examples from History contd
• NASA Hubble space telescope, April 24, 1990
• Controlling computer program had been based on an
outdated star chart introducing a pointing error
that prevented it from being pointed accurately
at stars and planets
• An unanticipated cycle of expansion and
contraction of the solar panel supports caused
the panels to sway. This confused the the
program which stabilized the spacecraft and
caused it to take corrective measures that
exacerbated the vibration
• These blunders resulted not from mistaken
calculations but from the inability to visualize
realistic conditions. Although a great deal of
hard thinking may have been done, the ability to
imagine the mundane things that can go wrong
remained deficient
• Aegis Air Defense System, USS Vincennes, July 3,
1988
• Shot down an Iranian civilian airliner
• Designers underestimated the demands that their
designs would place on operators, who often lack
the knowledge of the idiosyncrasies and
limitations built into the system
• Hubris and an absence of common sense in the
design process set the conditions that produce
the confusingly overcomplicated tasks that the
equipment demands of operators

Students have been taught to rely far too
completely on computer models, and their lack of
old-fashioned, direct hands-on experience can be
disastrous
59
Causes of Simulation Failure
• The 10 most frequent causes of simulation
analysis failure
• Failure to define an achievable goal
• Incomplete mix of essential skills
• Inadequate level of user participation
• Inappropriate level of detail
• Poor communication
• Wrong computer language
• Obsolete or nonexistent documentation
• Using an unverified or invalid model
• Failure to use modern tools and techniques to
manage the development of a large complex
computer program
• Poor presentation of results

Anini and Russell, INTERFACES 11(3)59-63, 1981
60
• In the very nature of a model, it is a restricted
and simplified representation of reality
• This makes the problem of what a model does in
circumstances for which it was not designed or
which were not foreseen much more problematic
• Therefore it is necessarily open to surprise a
surprise can kill
• Computational models can help us resolve some
complex questions, but by no means all
• Where data and relationships are clear cut, but
are too extensive and complex for our minds to
manage, they are at their best
• Where complexity arises from an uncertainty as to
how to analyze the problem at all, models may at
best be suggestive
• Computational models have taught us that
complexity itself is more complicated than we
thought
• Very simple models can generate very complex and
unpredictable behavior simplicity is not the
same as determinism

61
References
• Simulation Model Design and Execution, Paul
Fishwick, Prentice Hall, 1995.
• Theory of Modeling and Simulation, Bernard
Zeigler, 1976.
• Learning with Artificial Worlds Computer Based
Modelling in the Curriculum, Mellar (editor), The
Farmer Press, 1994.
• A Course in Mathematical Modeling, Douglas Mooney
and Randall Swift, The Mathematical Association
of America, 1999.
• Elementary Mathematical Modeling, Dan Kalman, The
Mathematical Association of America, 1997.
• Mathematical Modelling Techniques, Rutherford
Aris, Dover Publications, 1994.
• How to Model It Problem Solving for the Computer
Age, Anthony Starfield, Karl Smith, and Andrew
Bleloch, Interaction Book Company, 1994.
• Qualitative Simulation and Analysis, Paul
Fishwick and Paul Luke (editors),
Springer-Verlag, 1991.
• The Nature of Mathematical Modeling, Neil
Gershenfeld, Cambridge University Press, 1999.
• Simulation Modeling Analysis, Law and Kelton,
McGraw Hill, 1991.

62
References
• Digital Computer Simulation Modeling Concepts,
P.J. Kiviat, RAND Report, 1967.
• On the Art of Modeling, William Morris,
Management Science, Vol. 13, No. 12, August 1967.
• The Nature of Modeling, Jeff Rothenberg, RAND
Report, November 1989.
• CS 4214 Course Notes, Richard Nance and Osman
Balci, VA Tech, http//courses.cs.vt.edu/cs4214/