RELIABILITY:STRUCTURE FUNCTION OF A SYSTEM

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STRUCTURE FUNCTION OF A SYSTEM

Suppose an automobile can be divided into n components (subsystems). The performance of each component can be denoted by a random variable, Xi, that takes on the value xi = 1 if the component performs satisfactorily for the desired time and xi = 0 if the component fails during this time. In general, then, Xi is a binary random variable defined by

INTRODUCTION TO OPERATIONS RESEARCH-0603INTRODUCTION TO OPERATIONS RESEARCH-0602The function cp is called the structure function of the system and is just a function of the n-component random variables. Thus, the performance of the automobile is a function of its n components and takes on the value 1 if the automobile functions properly for the de- sired time and 0 if it does not. Because the performance of each component in the auto- mobile takes on the value 1 or 0, the function cp is defined over 2n points, with each point resulting in a 1 if the automobile performs satisfactorily and a 0 if the automobile fails.

There are several important structure functions to consider, depending upon how the components are assembled. Three structure functions will be discussed in detail.

Series System

The series system is the simplest and most common of all the configurations. For a series system, the system fails if any component of the system fails; i.e., it performs satisfacto- rily if and only if all the components perform satisfactorily. The structure function for a series system is given by

INTRODUCTION TO OPERATIONS RESEARCH-0604

This equation holds because each Xi is either 1 or 0. Hence, the structure function takes on the value 1 if each Xi equals 1 or, equivalently, if the minimum of the Xi equals 1. For example, suppose the automobile is divided into only two components: the engine (X1) and the transmission (X2). Then it is reasonable to assume that the automobile will per- form satisfactorily for the desired time period if and only if the engine and the transmis- sion both perform satisfactorily. Hence,

INTRODUCTION TO OPERATIONS RESEARCH-0605Parallel System

A parallel system of n components is defined to be a system that fails if all components fail, or alternatively, a system that performs satisfactorily if at least one of the n compo- nents performs satisfactorily (with all n components operating simultaneously). This prop- erty of parallel systems is often called redundancy (i.e, there are alternative components, existing within the system, to help the system operate successfully in case of failure of one or more components). The structure function for a parallel system is given by

INTRODUCTION TO OPERATIONS RESEARCH-0606This equation again follows because each Xi is either 1 or 0. The structure function takes on the value 1 if at least one of the Xi equals 1 or, equivalently, if the largest Xi equals 1. In the automobile example, the car is equipped with front disk (X1) and rear drum (X2) brakes.

1Note that Xi and cp are functions of the time t, but t will be suppressed for each of notation.

25.2 SYSTEM RELIABILITY 25-3

The automobile will perform successfully if either the front or rear brakes operate properly.1 If one is concerned with the structure function of the brake subsystem, then

INTRODUCTION TO OPERATIONS RESEARCH-0607

In the automobile example, consider a large truck equipped with eight tires. The structure function for the tire system is an example of a four-out-of-eight system. (Although the system’s performance may be degraded if fewer than eight tires are operating, rearrange- ment of the tire configuration will result in adequate performance as long as at least four tires are usable.)

It is reasonable to expect the performance of an automobile to improve if the per- formance of one or more components is improved. This improvement can be reflected in the characterization of the structure function, where, for example, one would expect cp(1, 0, 0, 1) to be no less than cp(1, 0, 0, 0). Hence, it will be assumed that if xi < yi, for i = 1, 2, . . . n, then

cp( y1, y2, . . . , yn) > cp(x1, x2, . . . , xn).

A system possessing this property (cp is an increasing function of x) is called a coherent (or monotone) system.

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