سوال ۲۲ کنکور ارشد ریاضی ۱۴۰۰

PASSAGE 2: Motivation

During the past twenty years mathematics and engineering have been increasingly directed towards problems of decision making in physical or organizational systems. This trend has been inspired primarily by the significant economic benefits which often result from a proper decision concerning the distribution of expensive resources, and by the repeated demonstration that such problems can be realistically formulated and mathematically analyzed to obtain good decisions.

The arrival of high-speed digital computers has also played a major role in the development of the science of decision making. Computers have inspired the development of larger systems and the coupling of previously separate systems, thereby resulting in decision and control problems of correspondingly increased complexity. At the same time, however, computers have revolutionized applied mathematics and solved many of the complex problems they generated.

It is perhaps natural that the concept of best or optimal decisions should emerge as the fundamental approach for formulating decision problems. In this approach a single real quantity, summarizing the performance or value of a decision, is isolated and optimized (i.e., either maximized or minimized depending on the situation) by proper selection among available alternatives. The resulting optimal decision is taken as the solution to the decision problem. This approach to decision problems has the virtues of simplicity, preciseness, elegance, and, in many cases, mathematical tractability. It also has obvious limitations due to the necessity of selecting a single objective by which to measure results. But optimization has proved its utility as a mode of analysis and is firmly entrenched in the field of decision making.

Much of the classical theory of optimization, motivated primarily by problems of physics, is associated with great mathematicians: Gauss, Lagrange, Euler, the Bernoulis, etc. During the recent development of optimization in decision problems, the classical techniques have been reexamined, extended, sometimes rediscovered, and applied to problems having quite different origins than those responsible for their earlier development. New insights have been obtained and new techniques have been discovered. The computer has rendered many techniques obsolete while making other previously impractical methods feasible and efficient. These recent developments in optimization have been made by mathematicians, system engineers, economists, operations researchers, statisticians, numerical analysts, and others in a host of different fields.

22- The availability of computers created ------------.