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Fundamentals of Model Development


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Title: Fundamentals of Model Development

Fundamentals of Model Development
Margaret L. Loper Georgia Tech Research
Institute Atlanta, GA margaret.loper_at_gtri.gatech.e
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
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
Part I The Nature of Modeling
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
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

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
  • In engineering terms, a model is used because it
    enables predictions or calculations or in some
    other way makes the design process more
  • Models are used to assist the designers
    thinking, analyze potential designs, realize what
    is known or unknown, predict behavior, identify
    connections, etc.

Model Representations
  • Verbal Models
  • Model couched in the language of everyday
  • 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.

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
  • The way you like to view systems will influence
    your decision as to which model to choose

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
  • 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

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

Part II The Science of Modeling
  • 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
  • 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

Modeling Relationships
Theory of Operation
Simulation Language (ACSL, Arena, Simscript)
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
  • 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
  • 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
  • Making assumptions is an integral part of
    deciding what to take from the real to the model
  • 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
  • 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

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

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

  • The more highly structured a model, the more
    numerous its 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
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

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
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
  • 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

  • Before a model is designed, two important
    questions must be asked and answered
  • What is the purpose of the model?
  • What are the requirements for accuracy and
  • 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

Part III Model Formulation
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
  • 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.

Problem Formulation
  • Every study should begin with a statement of the
  • 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

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

Mathematical Analysis
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
  • entities must not be multiplied beyond what is
  • 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
  • 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

  • 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
  • Heuristics tend to be most useful in the early
    stages of solving a problem because they
    encourage you to think fruitfully about the
  • 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
  • Draw a simple diagram of what is happening
  • Are there any physical laws to consider
    (conservation of matter or energy)
  • Look for ready-made formula for the answer
  • Look for simplifying assumptions
  • Look for bounds (simple models that would
    definitely underestimate or overestimate the

Is the Model Useful?
  • Establish a clear statement of the deductive
  • 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
  • Recognize the level of uncertainty associated
    with each variable
  • If a tractable model is obtained, enrich it.
    Otherwise, simplify.

Part IV Model Foundations
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
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
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

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.

Output Observable of some part of system
Input Controlling influence on system
System Modeling
  • The task of deriving a model of the system may be
    divided broadly into two subtasks
  • 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

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

Model Structure Components
  • Input
  • Input is a state that has a controlling influence
    on a system which does not contain the input
  • 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
  • Output
  • An output is a function of the system state and
  • 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

Model Structure Descriptive Variables
  • Describe the conditions of the components at
    points in time
  • Quantitative Data values are counts or numerical
  • 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
  • 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

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
    threads or environments
  • 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
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 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

Model Structure Component Interactions
  • Environment
  • The environment in which the object exists and
    operates has impact upon the outcomes of every
  • 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

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
  • 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,
    rules, metadata
  • Language

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
has been made
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
  • 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
    business profit-and-loss spreadsheet, Modeling of
    fractal structures, An equation relating the
    lengths and weights on each side of a playground
  • Dynamic Model describes time-varying
  • 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

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
  • 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

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
  • Mosaic, discrete uniform segments that change
    their shape and color in a stepwise manner,
    Finite elements or difference schemes,

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

Part V Modeling Engineering
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
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

Mathematics in Nature, John Adams 2003
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

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
  • The results may only be partial correct, but
    provides an overall picture of the original data
  • Theoretical
  • Most correct results quickly applied and yields
    answer with perfect accuracy
  • The theoretical method has greater power and
  • 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

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

The Sound of a Violin
  • There are many levels of description choosing
    among them depends on your goal and on available
  • 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
  • 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

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

Part VI Lessons in Modeling
  • 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
    good or bad results

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
  • 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
  • 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!
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
  • 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
Examples from History contd
  • Hartford Civic Center roof collapse, January 18,
  • The modern space-frame roof collapsed under a
    heavy snow load
  • 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

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,
  • 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
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
Morals About Modeling
  • 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
  • Very simple models can generate very complex and
    unpredictable behavior simplicity is not the
    same as determinism

  • 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.

  • 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//