ve a serious stretch of the imagination to see what is flowing through what kind of system, but the mathematical solution is just valid as the maximization of water flow through a network of pipes. 22.5 Constrained Optimization We have seen optimization in mathematics class. Usually there is some variable, x, and some function, f(x), and the object is to find the value of x that gives the maximum value of f(x). The function we are optimizing is called the objective function. The usual tactic is to use calculus to find the derivative f'(x) and to set that derivative equal to zero and to solve for X. The math teacher usually makes some passing remark about checking the end points if x has a limited range in case the maximum of f(x) is at a boundary point instead of a nice hump on the (x, y) graph of f(x). 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 In our business optimization problems, there are multiple objectives, so there are many variables to be traded off. We go back to the feasible region in Fig. 22.1 and re- alize that the character of the solution depends on the shape of the frontier. While both x and y are good things, the decision as to whether Q or R is a better solution depends entirely on the relative values of increasing x and y. In our business optimization problems, the constraints are typically very important. The character of the feasible region determines what solutions are available to be traded off in the solution process. In the network flow example, the feasible region is all the possible flows in the network, all the possible allocations of resources in the system. The role of the decision maker, or the decision support system supporting the deci- sion maker, is somehow to select the frontier's best point, which will be the best feasi- ble point. This is