most of us saw in our high school mathematics classes: Find the largest area Maximize profit What is the shortest distance Minimize the cost Calculate the shortest time How fast can we Or what is good can be a combination of goal factors, such as capacity, performance, and cost in wireless telephone systems. Sometimes the factors are harder to quantify, such 1We know few scientists who consider themselves just "biologists." They are immunologists or hematologists or in some other subfield. One of us (Rosenberg) is in the field of operations re- search, a field of mathematics specializing in constrained optimization. We will have more to say about this in Sec. 22.5. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Capacity and Quality Tradeoffs Capacity and Quality Tradeoffs as goodwill, community support, customer satisfaction, and reliability. We have to re- member that goals are important even when they are hard to quantify. What can be, what is possible, also can be a single variable or an easily described mathematical set: Where x is a real number Where n is a positive integer Where j and k are integers from 1 to n Often, figuring out the optimization choices is the hard part of the optimization, espe- cially when thousands or even millions of variables are in play. Sometimes the inter- actions of the system parameters are the important part of the problem, as we often see in complex aerodynamic systems. More often in economic problems, the system con- straints and their relationships are the interesting part of the optimization. The calculation of an optimal solution depends on the formulation of the problem. Many interesting business problem formulations have optimization calculations be- yond the realm of practical computing. More important, many interesting business problems have important factors not formulated at all in the mathemat