A first course in mathematical modeling

ereface To facilitate an early initiation of the modeling experience, the first edition of this text was designed to be taught concurrently or immediately after an introductory business or engineering calculus course In the second edition, we added chapters treating discrete dynamical systems, linear programming and numerical search methods, and an introduction to probabilistic modeling Additionally, we expanded our introduction of simulation In this edition we have included solution methods to some simple dynamical systems to reveal their long-term behavior We have also added basic numerical solution methods to the chapters covering modeling with differential equations The text has been reorganized into two parts: Part One, Discrete Modeling (Chapters 1-8), and Part Two, Continuous Modeling (Chapters 9-12) This organizational structure allows for teaching an entire modeling course based on Part One and which does not require the calculus Part Two then addresses continuous models based on optimization and differential equations which can be presented concurrently with freshman calculus The text gives students an opportunity to cover all phases of the mathematical modeling process The new CD-ROM accompanying the text contains software, additional modeling scenarios and projects, and a link to past problems from the Mathematical Contest in Modeling We thank Sol Garfunkel and the COMAP staff for preparing the CD and for their support of modeling activities that we refer to under Resource Materials below Goals and Orientation The course continues to be a bridge between the study of mathematics and the applications of mathematics to various fields The course affords the student an early opportunity to see how the pieces of an applied problem fit together The student investigates meaningful and practical problems chosen from common experiences encompassing many academic disciplines, including the mathematical sciences, operations research, engineering, and the management and life sciences This text provides an introduction to the entire modeling process The student will have occasions to practice the following facets of modeling and enhance their problem-solving capabilities: Creative and Empirical Model Construction: Given a real-world scenario, the student learns to identify a problem, make assumptions and collect data, propose a model, test the assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze the underlying mathematical structure of the model to appraise the sensitivity of the conclusions when the assumptions are not precisely met ix x Preface Model Analysis: Given a model, the student learns to work backward to uncover the implicit underlying assumptions, assess critically how well those assumptions fit the scenario at hand, and estimate the sensitivity of the conclusions when the assumptions are not precisely met Model Research: The student investigates a specific area to gain a deeper understanding of some behavior and learns to use what has already been created or discovered Student Background and ~ourse Content Because our desire is to initiate the modeling experience as early as possible in the student's program, the only prerequisite for Chapters 9, 10, and 11 is a basic understanding of single-variable differential and integral calculus Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics already known by the students after completing high school This emphasis is especially true in Part One The modeling course will then motivate students to study the more advanced courses such as linear algebra, differential equations, optimization and linear programming, numerical analysis, probability, and statistics The power and utility of these subjects are intimated throughout the text Further, the scenarios and problems in the text are not designed for the application of a particular mathematical technique Instead, they demand thoughtful ingenuity in using fundamental concepts to find reasonable solutions to "open-ended" problems Certain mathematical techniques (such as Monte Carlo simulation, curve fitting, and dimensional analysis) are presented because often they are not formally covered at the undergraduate level Instructors should find great flexibility in adapting the text to meet the particular needs of students through the problem assignments and student projects We have used this material to teach courses to both undergraduate and graduate students, and even as a basis for faculty seminars Organization of the Text The organization of the text is best understood with the aid of Figure The first eight chapters constitute Part One and require only precalculus mathematics as a prerequisite We begin with the idea of modeling change using simple finite difference equations This approach is quite intuitive to the student and provides us with several concrete models to support our discussion of the modeling process in Chapter There we classify models, analyze the modeling process, and construct several proportionality models or submodels which are then revisited in the next two chapters In Chapter the student is presented with three criteria for fitting a specific curve-type to a collected data set, with emphasis on the least-squares criterion Chapter addresses the problem of capturing the trend of a collected set of data In this empirical construction process, we begin with fitting simple oneterm models approximating collected data sets and progress to more sophisticated interpolating models, including polynomial smoothing models and cubic splines APPENDIX B An ElevatorSimulationAlgorithm 525 queue total length of current queue of customers waiting for an elevator to become available startque clock time at which the (possibly updated) current queue commences to form stop} total number of stops made by elevator j during the entire simulation eldel} total time elevator j spends in delivering its current load of passengers operate} total time elevator j operates during the entire simulation limit customer number of the last person to enter an available elevator before it commence~ transport max largest index of a nonzero entry in the array selvec} (highest floor selected) remain number of customers left in the queue after loading next available elevator quetotal total number of customers who spent time waiting TIME current clock time in seconds, starting at t = DELTIME ELEVTIME MAXDEL MAXELEV average delivery time of a customer to reach destination floor from time of arrival, including any waiting time average time a person spends in an elevator maximum time required for a customer to reach his or her floor of destination from time of arrival maximum time a customer spends in an elevator QUELEN number of customers waiting in the longest queue QUETIME average time a customer who must wait spends in a queue MAXQUE longest time a customer spends in a queue Elevator System Simulation Algorithm Input None required 'utput Number of passengers serviced, DEL TIME, ELEVTIME, MAXDEL, MAXELEV, QUELEN, QUETIME, MAXQUE, stop}, and the percentage time each elevator is in use STEP Initially set the following parameters to zero: DELTIME, ELEVTIME, MAXDEL, MAXELEV, QUELEN, QUETIME, MAXQUE, quetotal, remain STEP For the first customer, generate time between successive arrivals and floor destination and initialize delivery time: i =1 Generate betweeni and floori deliveryi = 15 APPENDIX B An Elevator Simulation Algorithm 531 e Answer Problems a-d for direction How would you use your simulation to determine the switching period for which the total waiting time in both directions is as small as possible? (You will have to modify it to account for the waiting times in direction 2.) Modify Steps 20-32 in the elevator simulation algorithm so that loading of the first available elevator commences immediately upon its return Thus, if TIME > return j so that elevator j is available for loading, then loading commences at time return j rather than TIME Consider how you will now process customer i in Step 20 who has not yet quite arrived on the scene B.1 Project Find a building in your local area that has from to 12 floors that are serviced by 1-4 elevators Collect data for the interarrival times (and, possibly, floor destinations) of the customers during a busy hour (e.g., the morning rush hour), and build the interarrival and destination submodels based on your data (by constructing the cumulative histograms) Write a computer program incorporating your submodels into the elevator system algorithm to obtain results such as those given in Table B.l For those of you familiar with matrix algebra, we demonstrate how to accomplish a pivot using matrix techniques Any desired extreme point can be determined by first inverting a submatrix of the original tableau, followed by premultiplying the original tableau by the inverted submatrix The method is called the Revised Simplex Method and has advantages of both speed and accuracy In particular, round-off errors can be minimized in subsequent pivots because the Revised Simplex Method can use the original data to perform any desired pivot We begin by illustrating pivoting by matrix inversion, using the carpenter's problem as presented in Section 7.4 for illustration Pivoting by Matrix Inversion and Multiplication We illustrate how to use matrix inversion and multiplication to move from Tableau o to Tableau (see Section 7.4, Example 1, The Carpenter's Problem) For Tableau 1, we want the set of dependent variables to be (in order) The Revised Simplex Method Note that the columns for the matrix P are selected from the original tableau Also note that the matrix p-l premultiplies the original data Any intersection point can be enumerated in this fashion This method is known as the Revised Simplex Method and is beneficial when solving large linear programs By returning to the original data to compute p-l, the round-off error (which accumulates in successive pivots using other methods) is reduced Let's illustrate the idea by computing Tableau by premultiplying the original tableau by an appropriate pivot matrix Examining TI above, the optimality test determines Xl as the entering variable The feasibility test as demonstrated in Section 7.4 determines Y2 as the exiting variable Thus the new dependent variables are index Absolute deviations, 110 Agility in sports, 95 Anchoveta, 480 Approximation, 99 Area under a curve, 177 Monte Carlo algorithm, 178 Arms Race, 336, 444 Assumptions, 57 Asymptotic stability, 414, 423 Attrition-rate coefficient, 438 Automobile gasoline mileage, 88, 311 autonomous system, 413 Average rate of change, 369, 371 Average weight density, 92 Baleen whale, 480 Bass fishing derby, 79-83 Binary variables, 241 Biological optimum, population level, 483 Birthrate, 371 Blood volume flow-rate problem, 313 Body temperature problem, 86 Body weight: versus blood flow through the heart, 85, 86 versus height model, 91 versus pulse rate, 85,86 Bomstein, Marc and Helen, 134 Borrie, Mo So, 393 Brahe, Tycho, 67 Braking distance, 59, 68 differential equations model, 391 Brewed coffee project, 64 Buckingham, Eo, 307 theorem, 307 Buffer stock, 464 Burghes, David, 393 Buying or renting a home project, 65 Canadian lynx and hare problem, 436 Centrifugal force problem, 313 Characteristic dimension, 76, 79 Chebyshev approximation criterion, 107 compared with least-squares criterion, 119 Choosing a best model, 119 Clamped spline, 164 Classification of variables, 57 Coefficient of heat conduction, 324 Collected data, 98 Communicable disease problem, 380 Competitive hunter model, 419 trajectories, 422 Complete set: of dimensionless products, 306 of solutions to algebraic system, 306 Concave downward, 129 Concave upward, 128 Constrained optimization problem, 237 Constraints, 237 Consumption (of gasoline), 88 Cost, 55 Crater volume, 320, 323 Critical point, 396 Cubic spline interpolation, 159 Cumulative frequency, 197 Damped pendulum, 313 Dantzig, George, 263 Data analysis, 98 Deathrate, 371 Decision variables, 237 Deer population problem, 403 Delivery costs, 194 Demand curve, 361 Density, 79, 91 Dependent variables, 57 Derivative, 369 rate of change, 370 srope of tangent line, 370 Design condition, 332 Deterministic process, 177 Dichotomous, 282 Difference equation, Difference table, 149, 150 Differential equation, 368 graphing solutions, 395 systems of, 412 Dimension, 292 table of physical entities, 302 Dimensional analysis process, 304 summary, 309 Dimensional homogeneity, 297 Buckingham's theorem, 307 Dimensionless product, 295 Buckingham's theorem, 307 complete set, 306 solutions to homogeneous system, 306 Direction field, 395 Discrete dynamical systems, approximating change, iterating solutions, 12 numerical solutions, 12 spotted owl population, 32 Discrete mathematics, 368 Distortion of data points, 104 Divided differences, 148, 151 Drag force, 97 damped pendulum, 313 on a sphere problem, 313 on a submarine, 331 Drug dosage, 382 Dynamic programs, 239 Dynamical System, Economic optimum population level, 484 Elimination constant, 384 Empirical model, 98, 126 Equilibrium values, 22, 396 stability, 29, 398 Error, 100 Euler's algorithm, 406 Exhaustible resource, 355, 480 Explosion analysis, 319 cube-root law, 320 energy set, 323 gravity scaling, 321 one-fourth root law, 320 Exponential decay, 384 Exponential growth model, 373 Extreme points, 251 Falling raindrop, 77 dimensional analysis, 309 problem, 304 Farming problem, 248 Feasible region, 250 empty, 255 unbounded, 254,255 Feasible solutions, 250 Fibonacci search, 290 Fidelity, 55 Finite mathematics, 368 First divided difference, 148 Fishing industry model, 480 Fishing regulation, 488 Fixed costs, 356 Flexibility, 55 535 536 Index Fluid mechanics, 318 flow in a channel problem, 335 Formulation error, 100 Fragile, 63 French curve, 160 Frequency of demand, 196 Froude number, 318 Fuel rating (energy), 88 Function, 336 concavity, 128 decreasing, 128 increasing, 128 piecewise linear, 241 Gause's principle, 422 Gaussian elimination, 306 Geometrically similar, 75 fish, 79 submarines, 330 Goal programs, 243 Golden ratio, 283 Golden section search, 283 Gradient method, 470 Graphical analysis: limitations of, 422 Graphical model fitting, 101 Gravity scaling, 321 Gross National Product, 394 Growth of yeast, 376 Harbor system simulation, 205 algorithm, 207 Harvest the Chesapeake Bay Problem, 127 Blue fish problem, 130 Blue crabs problem, 131 Harvesting submodel, 480 Hatton, Bob, 166 Heat loss, 85, 86 Heating of a cooled object, 14 Helicopter transportation project, 64 Histogram, 196 Homogeneous system, 305 Horelick, Brindell, 382 Hydrostatic pressure problem, 304 Hyperbola, 462 Independent solution set, 305 Independent variables, 57 Infeasible, 260 Initial value problem, 404 Instantaneous rate of change, 369 Integer program, 239 Interest compounded continuously, 410 Interpolation, 98 approximation, 99 cubic spline, 159 linear, 160 smoothing, 146 Intersection points, 260 Inventory model: deterministic, 459 probabilistic, 194 Investment in college education project, 65 Investment problem, 242 Iterated solutions (DDS), 31 Jaye, Michael, 317 Kepler, Johannes, 67 Kinetic energy, 70 Koont, Sinan, 382 Ladder of Powers, 128 Ladder of Transformations, 130 Lagrange multipliers, 474 Lagrangian form, 474 Laminar flow, 334 Lanchester, F w., 438 Lanchester combat models, 437 conventional-guerilla combat, 448 parabolic law model problem, 448 refinements, 441 square law model, 439 Largest absolute deviation, 107 Chebyshev approximation criterion, 107 Launching a satellite problem, 391 Least-squares criterion, 110 applications of, 114 compared with Chebyshev criterion, 112 power curve, 115 quadratic, 116 straight line, 114 transformed data, 102, 116 Leja, Stanley, 419 Limit cycle, 423 Limited growth model, 374 Linear combination, 306 Linear interpolation, 160 Linear program, 239, 251 e1lample, 251 Linear programming algebraic, 259 geometric, 250 revised simplex, (Appendix C), 535 sensitivity analysis, 273 simplex, 263 Linear spline, 160 demand submodel, 199 harbor system simulation, 212 Logistic curve, 375 Logistic growth, 374 Long jump problem, 65 Long-term behavior, 22 Lotka- Volterra model, 435 Lottery problem, 181 lumber yield problem, 86 Mach number, 318 Malthus, Thomas, 371 model of population growth, 373 Mansfield, Ralph, 301 Marginal cost, 357 and taxation, 360 Marginal revenue, 357 Marginal value, 279 Markov Chain, 217 Mathematical Contest in Modeling (COMAP), Appendix A problems, 491 Mathematical model, 54 construction, 57 cost, 56 deterministic, 177 fidelity, 56 flexibility, 56 fragile, 63 implementation, 58 interpretation, 58 maintenance, 58 predict versus explain, 52 probabilistic, 177 robust, 63 sensitivity, 63 types, 55, 56 verification, 58 Mathematical world, 52 Maximum profit, 356 McNulty, Jim, 166 Measurement errors, 100 Medical records, 65 Mixed integer program, 239 Mixing nuts problem, 246 MLT system, 296 Model fitting, 97 analytic methods, 107 empirical model fitting, 126 Modeling process: 52 as a closed system, 54 iterative nature of, 62 model construction, 57 scientific method, 61 Model type, 53 deterministic, 177 probabilistic, 177 Monte Carlo simulation, 177 area under a curve, 177 elevator system algorithm, (Appendix B),525 harbor system algorithm, 205 inventory algorithm, 202 Morning rush hour, 213 Multiobjective program, 242 Mutualism, 412 problem, 436 Natural spline, 164 Newton's inverse square law, 303 Newton's Method, 291 Nonrenewable resources, 355 Nuclear arms race, 336 civil defense, 345 Index economic model, 444 graphical model, 343 MIRV's, 347 mobile launching pads, 346 Numerical methods, 405 Euler's algorithm, 406 Nutritional requirements problem, 245 Objective functions, 237 One-term model, 127 Optimization, 236 basic model, 237 formulation of, 236 Orbit, 413 Oscillation, 143 Outliers (in data), 133 Path,413 Pendulum: damped, 313 simple, 293, 298 Period,293 Periodic trajectories, 431 Phase plane, 413 Piecewise linear function, 241 Pinch test, 96 Pivot, 264 Polynomials, 138 advantages for interpolation, 142 Lagrangian form, 141 oscillation, 143 sensitivity of coefficients, 144 smoothing, 146 Population growth, 371 Population models, 371 Postage stamp problem, 171 Power, 89 problem, 317 Power curve, 115 Predator-prey model, 427 harvesting effects, 433 trajectories, 432 Predict, 98 Pressure coefficient, 319, 334 Pressure drop problem, 334 Principle of Competitive Exclusion, 422 Probabilistic process, 186 submodels, 186 Producing electronics equipment problem, 247 Product, 295 dimensionless, 295 Production schedule, 251 problem, 246 Projectile problem, 312 Proportional, 65 geometric interpretation, 66 modeling, 67 Propulsion force, 88 Prototype, 330 Purchasing trucks problem, 467 Random number, 182 algorithm, 185 Range (of data), 101 Ratio rule, 267 Real-world systems, 52 and modelling process, 53 Regression, 227 residual plots, 229 Relative error, 66 Reliability, 223 computer components reliability, 225 series systems, 224 parallel systems, 224 space shuttle components reliability, 225 stereo system components reliability, 225 Renewable resources, 480 Rental car problem, 35 Replacing a car project, 65 Residual plots, 229 Residuals, 121,229 Resisting force, 77 Resources: allocation, 245 nonrenewable, 355 renewable, 480 Rest point, 413, 414 Reynolds number, 318, 332 laminar flow, 334 turbulent flow, 334 Rickover, Admiral Hyman G., 337 Right-hand side, 237 Roasting a turkey, 324 Robust, 63 Round-off error, 100 Rousseau, Jean Jacques, 379 Satellite problem, 391 Scalar multiple, 305 Scatterplot, 128 Schmidt, Robert M., 319 Scientific method, 61 Search techniques, 279 Second divided difference, 150 Sensitivity, 63 of coefficients, 274 Sensitivity analysis, 273 Separation of variables, 409 Sequence (DDS), Shape factor, 332 Similitude, 330 Simple pendulum, 293, 298 Simplex method, 263 revised simplex (Appendix C), 535 Simulation, 175 Monte Carlo, 177 Simulation labs, 186 flip of a coin, 186 roll of a fair die, 187 Simulation labs (cont.) roll of an unfair die, 188 Smith, Adam, 480 Smoothing, 146 Social diffusion problem, 381 Social optimum yield, 483 Solution: algebraic system, 305 autonomous system, 413 complete set, 306 independent set, 306 scalar multiple, 305 sum, 305 trivial, 305 Space shuttle problem, 240 Space shuttle water container, 476 Species coexistence, 422 Speed of sound problem, 313 Spheres in a fluid problem, 313 Spline, 159 Spline interpolation, 160 Spotted owl population model, 41 Stable rest point, 414 Stationary point, 414 Stochastic program, 239 Stock-outs, 465 Storage costs, 194 Submarine, 331 Submode1, 57 Sum of absolute deviations, 102 minimization of, 110, 252 Sum of solutions, 305 Sum of squared deviations, 110 least-squares criterion, 110 Supply curve, 360 Survival of whales problem, 428, 480 Sustainable yield, 484 System, 52 Systems of Difference Equations, 35 Tableau, 266 Taylor, General Maxwell D., 337 Terminal velocity, 77, 309 Theory of the firm, 359 Thermal conductivity, 325 Total cost, 356 Total profit, 356 Total revenue, 356 Trajectory, 413 basic Lanchester combat model, 437 competitive hunter, 419 nuclear arms race, 444 predator-prey, 427 properties, 413 Transformed least-squares, 116 Transforming data, 102 Trend (of data), 127 Trivial solution, 305 Truncation errors, 100 UMAP no 60 62, The Distribution of Resources, 354 53 538 Index UMAP no 67, Modeling the Nervous System: Reaction Time and the Central Nervous System, 391 UMAP no 69, The Digestive Process of Sheep, 411 UMAP no 79, Selection in Genetics, 390 UMAP no 74, Tracer Methods in Permeability, 390 UMAP no 75, Feldman's Model, 411 UMAP no 211, The Human Cough, 468 UMAP no 232, Kinetics of Single Reactant Reactions, 394 UMAP no 234, Radioactive Chains: Parents and Daughters, 394 UMAP no 269, Monte Carlo: The Use of Random Digits to Simulate Experiments, 185 UMAP no 270, Lagrange Multipliers: Applications to Economics, 479 UMAP no 292-293, Listening to the Earth: Controlled Source Seismology, 330 UMAP no 294, Price Discrimination and Consumer Surplus, 367 UMAP no 303, The Diffusion of Innovation in Family Planning, 17, 381 UMAP no 304, The Growth of Partisan Support I: Model and Estimation, 51, 427 UMAP no 305, The Growth of Partisan Support II: Model Analytics, 51, 427 UMAP no 308, The Richardson Arms Race Model, 354, 449 UMAP no 311, The Geometry of the Arms Race, 354 UMAP no 321, The Curve Fitting via the Criterion of Least Squares, 119 UMAP no 322, Difference Equations with Applications, 34, 381 UMAP no 327, Nuclear Deterrence, 354 UMAP no 332, The Budgetary Process: Incrementalism, 427 UMAP no 333, The Budgetary Process: Competition, 427 UMAP no 340, The Poisson Random Process, 205 UMAP no 341, Five Applications of Max-Min Theory from Calculus, 468 UMAP no 376, Differentiation, Curve Sketching, and Cost Functions, 367 UMAP no 468, Calculus of Variations with Applications in Mechanics, 249 UMAP no 506, The Relationship Between Directional Heading of an Automobile and Steering Wheel Deflection, 394 UMAP no 509, The Cobb-Douglas Production Function, 381 UMAP no 517, Lagrange Multipliers and the Design of Multistage Rockets, 479 UMAP no 518, An Application of Calculus to Economics: Oligopolistic Competition, 468 UMAP no 520, Random Walks: An Introduction to Stochastic Processes, 427 UMAP no 522, Unconstrained Optimization, 249 UMAP no 539, I Will if You Will: A Critical Mass Model, 367 UMAP no 551, The Pace of Life, An Introduction to Model Fitting, 138 UMAP no 553, Graphical Analysis for Some Difference Equations in Biology, 51, 437 UMAP no 564, Keeping Dimension Straight, 304 UMAP no 590, Random Numbers, 185 UMAP no 610, Whales and Krill: A Mathematical Model, 418 UMAP no 737, Geometric Programming, 249,418 Unconstrained optimization problem, 238 Unimodal function, 279 Unstable rest point, 413, 414 Variable costs, 356 Vehicular stopping distance, 59, 68 constructed model, 72 cubic spline interpolation, 168 differential equations submodel, 391 divided difference table, 153 empirical quadratic table, 153 least -squares fit, 122 traffic flow problem, 466 Verhulst, Pierre-Francois, 374 model of population growth, 374 Volume of a sphere in the first octant, 180 Volterra's principle, 435 Voting in an election problem, 44 Wallace, Michael, 336 Weber number, 318 Weight-lifting problem, 84, 159 Weight of a fish, 79 Whales and krill, 428 Wind force on a van, 301 Wine making project, 64 Work, 70 Yeast growth: 155, 376 divided difference table, 156 empirical quadratic model, 156 ... mathematical modeling process The new CD-ROM accompanying the text contains software, additional modeling scenarios and projects, and a link to past problems from the Mathematical Contest in Modeling. .. Often a mathematical model can help us understand a behavior better or aid us in planning for the future Let's think of a mathematical model as a mathematical construct designed to study a particular... follow, we approximate a continuous change by examining data taken at discrete time intervals Approximating a continuous change by difference equations is an example of model simplification EXAMPLE