ENM 503 Lesson 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 2

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. 3

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 4

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 models are such great fun. 5

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 6

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. 7

A Conceptual Model of Modeling System/process Model variables, parameters Model solution / prediction Model evaluation - verification & validation Model implementation problem formulation 8

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. 9

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 10

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 11

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 12

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. 13

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 14

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 16

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! 17