Mathematical Modeling Today

The applications of mathematical methods in management and economics today are so manifold that it is difficult to find a single person who is aware of their full scope. The following list can only be incomplete.

• Economists use linear and nonlinear programming, the theory of variational inequalities, optimal control theory, dynamic programming, game theory, probability choice models, utility theory, regression and factor analysis and other techniques to study equilibrium, optimal investment, competition, consumer behavior, and a host of other phenomena.
• People in operations management use statistical sampling and estimation theory, linear and integer programming, network programming, dynamic programming and optimal control theory, queuing theory, simulation, artificial intelligence techniques, and combinatorial optimization methods to solve problems in quality control, allocation of resources, logistics, project scheduling, labor and machine scheduling, job shop scheduling and assembly line balancing, and facility layout and location. The introduction of flexible manufacturing systems, robots and other automated devices has posed a whole new array of unsolved mathematical problems.
• People in finance use linear, nonlinear and integer programming, optimal control theory and dynamic programming, Markov decision theory, regression and time series to determine optimal resource allocation, multiperiod investments, capital budgeting, and investment and loan portfolio design, and to try to forecast market behavior.
• People in marketing use regression and factor analysis, time series, game theory, Markov decision theory, location theory, mathematical programming, probability choice models and utility theory to study consumer preferences, determine optimal location in product space, allocate advertising resources, design distribution systems, forecast market behavior, and study competitive strategy.
• People in information systems and decision support systems use artificial intelligence techniques, propositional and quantified logic, Bayesian methods, probabilistic logic, data structures and other computer science techniques, mathematical programming, and statistical decision theory to design expert and other knowledge-based systems, develop efficient inference and retrieval methods, and evaluate the economic and organizational effects of information systems.

The peculiar nature of mathematics unfortunately raises two obstacles to learning it. One is that students often believe that mathematics can be learned simply by studying a book. Mathematics is learned by working through problems. A second obstacle is that students often believe that they can learn to work problems by studying a book. A book can get one started, but learning to work problems is like learning to play basketball or play the piano--it requires practice, practice, practice. Mathematical skills are very athletic in this sense. (The phrase ``quant jock'' is appropriate.) That is why this course assigns a lot of homework problems.

Acknowledgements: We would like to thank John Hooker, Bill Hrusa, Rick Green and Anuj Mehrotra for their inputs.