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Photo, caption follows:

The Walt Disney Concert Hall in Los Angeles, Ca. This structure was designed by the architectural firm Gehry Partners through the extensive use of computer-aided design (CAD) tools.
Credit: © Gehry Partners, LLP

 

Deciding on the Best Choice
Many basic questions and valuable applications of the mathematical sciences focus on ways to find the most economical or efficient choice among many alternatives. Examples abound.

  • Optimization. Consider the delivery problems of a concrete company with a number of trucks that haul wet, mixed concrete from one or more mixing stations to a number of construction sites in different locations. The big trucks are expensive to operate so owners want to minimize fuel use. They want to sell as much wet mix as possible to maximize revenue, and a truck needs to dump its load within a short time before the concrete hardens in the truck's drum. Combining these factors into a good solution is tough enough for highly stressed dispatchers, and today's solution probably won't work for tomorrow's new jobs.


  • Scheduling and planning scenarios like this have been studied intently since World War II, and optimization software for this kind of work is still being improved on the basis of new mathematical ingredients. Oil refineries, express package shippers, just-in-time manufacturers, airlines and many other businesses depend on tools of this sort, whose economic impact is reportedly worth billions of dollars. And optimization plans can be critical to the success of military operations.
  • Design. Computer-aided design (CAD) tools are famously used to describe shapes and manufacturing processes for structures as varied as aircraft, surgical implants and concert halls.


  • But typically CAD tools do not address many important design and engineering issues because they work differently from computer programs that perform structural analysis (will the building fall down?), model fluid flow (will the aircraft wing generate enough lift?), or control the machining of components for the structure (will the ball of the replacement femur actually fit the socket?). Mathematicians, computer scientists and engineers are investigating several approaches to computationally useful geometric descriptions that may combine the strengths of both kinds of calculations, making the CAD results simultaneously efficient and accurate.
  • Financial Mathematics. Institutional investors allocate components of their portfolios among different stocks, bonds or other instruments according to computational models that depend upon sophisticated probability theory and other techniques. The future of many pension funds, among other critical investments, depends on the validity of those calculations. In many cases, those allocations are guided by the principle of minimizing risk, and the development of ever more sophisticated mathematical tools for describing risk, estimating reward and making portfolio decisions, are a matter of intense interest in both the math and financial communities.

Pattern Hunting and Statistical Learning [Next]