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CHAPTER 2

Formulation of Optimal Redundancy Problems

# 2.1 Problem Description

One of the most frequently used methods of reliability increase is the use of additional (redundant) units, circuits, and blocks. This method is especially convenient when the principal solution of the system design has already been found: the use of redundant units usually does not cause a change in the overall structure of the system, but the use of extra units entails additional expense. Naturally, a system designer always tries to find the least expensive way to improve reliability. Thus, a designer faces two problems:

(1) Direct problem of optimal redundancy: Find such allocation of redundant units among different subsystems that warrants the required level of reliability index while using the smallest amount of resources possible.
(2) Inverse problem of optimal redundancy: Find such allocation of redundant units among different subsystems that maximizes the level of chosen reliability index under some specified constraints on the total cost of the system.

The choice of constraints depends on the specific engineering problem. Of course, the cost of a set of redundant units is not a unique objective function. For instance, for submarines the most serious constraint is the total volume (or weight) of spare units.

Consider a series system composed of n independent redundant groups (or subsystems). A redundant group is not necessarily a separate part of a system. In this context, this may be a group ...

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