ENM 503Lesson 1 – Methods and Models

The why’s, how’s, and what’s of mathematical modeling

A model is a representation in

mathematical terms of some real system or

process.

Narrator: Charles Ebeling 1

Methods

The mathematics used to model and solve problems

For example, Set theory Algebra Matrix Algebra Combinatorics Calculus

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Models

A model is an object or concept that is used to represent something else. It is realty scaled down and converted to a more comprehensible form.

A mathematical model is a model whose parts are mathematical concepts such as constants, variables, functions, equations, inequalities, etc.

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Why Mathematical Models?

Provides increased precision Is concise Makes available an entire

mathematical system consisting of definitions, concepts, notation, and theorems

Includes solution techniques, algorithms, and computer applications

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What Mathematical Models Do…

Describes, predicts, or explains the behavior of a system or process (descriptive models)

Prescribes the behavior of a system or process (prescriptive models)

Mathematical modelsare such great fun.

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Our very first mathematical model

The cost of the material used in the construction of a rectangular shaped container is $20 a square foot for the top and $10 a square foot for the sides and bottom.

Let x = the length, y = the width, and h = the height, and z = the total material cost

Then z = (20) xy + (10)(2) xh + (10)(2)yh +(10) xy

---x---

--y--

h

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Modeling

A verb, the process of creating a mathematical model for a real-world situation. Modeling includes representing quantities by appropriate variables and constants and writing statements (i.e. equations, inequalities, functions, etc.) relating the variables.

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A Conceptual Model of Modeling

System/process

Modelvariables, parameters

Model solution / prediction

Model evaluation -verification & validation

Model implementation

problem formulation

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Problem Formulation

The process of translating a verbal description of a problem into a set of mathematical relationships for the purpose of finding a solution to the problem. The primary justification for making this translation is that the mathematical relationships will be far easier to manipulate than would be the verbal description. The difficulty in carrying out this process lies in the fact that problem formulation is more art than science and cannot be learned by memorizing a few steps or observing a number of examples.

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Procedures for Problem Formulation

Understand problem summarize what is known define some variables write down obvious relationships characterize formulation (taxonomy) construct and solve prototype model

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Types of Models

Algebraic - variables related by functions and equations Linear

all relationships represented by linear equations, inequalities, and functions

Discrete variable values are integer valued only

Nonlinear one or more relationships among variables is nonlinear

Optimization Maximize or minimize a given function

Stochastic (probabilistic) one or more variables depend on random outcomes

Dynamic relationships are changing with respect to time

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Modeling Procedures

divide the problem into smaller problems research the literature for similar problems and models seek analogies with other models create a specific example with a small set of data establish some symbols, define variables and constants write down the obvious

conservation laws and input-output relations enrich or simplify

make variables constants and vice-versa eliminate or add variables make nonlinear relationships linear (and vice-versa) modify assumptions

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Modeling Pitfalls

Do not build a complicated model when a simple one will do.

Beware of molding the problem to fit the technique. A model should never be taken too literally. A model should not be pressed to do that for which

it was never intended. Beware of overselling a model. A model is no better than the information that goes

in it. Models cannot replace decision-makers.

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Solving the Model

Abstract solution symbolic, optimal solution obtained mathematically

Numerical solution computer algorithm or iterative procedure requiring

actual numerical values Experimental solution

replicate actual process or its numerical (state) values simulation

Heuristic solution set of decision rules which generate “good” solutions

A bear of a solution

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Model Verification vs. Validation

Verification Is the model

performing as it was designed to?

Is the formulation correct?

Is the algorithm and solution correct?

Internal process

Building the model right!

Validation Process of building

an acceptable level of confidence that the model and its solution are correct for solving the actual problem

External process

Build the right model!15

What could go wrong? Sources of error

Invalid modeling assumptions (wrong model) e.g. assume linear relationship when it is nonlinear

Observational errors e.g. measuring time to failure in clock hours rather than

operating hours Inaccurate or incorrect model solution

e.g. using a heuristic that generates a solution that is not very close to the optimal solution

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The End of the Beginning Lecture

That was a mighty fine discussion. I am sure

that you are now eager to get started with this

course.

O’boy, let’s get started!

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