# Ranking the Reliabilities of Systems and Processes by Using Inequalities

## Improving Reliability and Reducing Risk by Proving an Abstract Inequality Derived from the Real Physical System or Process

An important way of using inequalities to improve reliability and reduce risk is to start with the real system or a process and derive and prove an abstract inequality.

This process includes several basic steps (Figure 5.1): (i) detailed analysis of the system (e.g., by using reliability theory); (ii) conjecturing an inequality about the competing alternatives or an inequality related to the bounds of a risk-critical parameter; (iii) testing the conjectured inequality by using Monte Carlo simulation and (iv) proving the conjectured inequality rigorously.

This generic strategy can be followed, for example in comparing the reliabilities of competing systems. It starts with building the functional diagram of the system, creating the reliability network for the system, deriving expressions for the system reliability of the competing alternatives, conjecturing an inequality, testing the conjectured inequality by using Monte Carlo simulation and finishing with a rigorous proof by using some combination of the analytical techniques for proving inequalities (Figure 5.2).

## Using Algebraic Inequalities for Ranking Systems Whose Component Reliabilities Are Unknown

Often, the reliabilities of the components building the system are unknown. The epis- temic uncertainty associated with the reliabilities of the components building the system translates into epistemic uncertainty related to which system is superior. Algebraic inequalities can eliminate the uncertainty about which is the superior system or process.

The first approach related to using inequalities for reliability improvement and risk reduction can be demonstrated with comparing the reliabilities of competing systems. It starts with building the functional diagram of the system, creating the reliability network for the system, deriving expressions for the system reliability of the competing alternatives, conjecturing inequalities ranking the competing alternatives, testing the conjectured inequalities and finishing with rigorous proofs based on some of the analytical techniques for proving inequalities.

FIGURE 5.1 A generic strategy for improving reliability and reducing risk by proving an abstract inequality derived from a real physical system/process.

FIGURE 5.2 Improving reliability and reducing risk by comparing the reliabilities of competing systems.

For two competing systems (a) and (b) built on components whose reliabilities are unknown, the steps which lead to establishing the system with intrinsically superior reliability can be summarised as follows.

- • For each of the competing systems, build the reliability network from its functional diagram.
- • By using methods from system reliability analysis, determine the system reliabilities
*R*and_{a}*R*of the systems or the probabilities of system failure_{h}*F„*and*F*_{b}. - • Subtract the reliabilities of the competing systems or the probabilities of system failure and test and prove any of the inequalities:
*R*>0,_{u}- R_{h}

*R„* - *R _{h}* < 0

*.F*0.

_{a}-F_{h}>*F*0.

_{a}-F_{h}<• Select the system with the superior reliability or the system with the smaller probability of failure.

### Reliability of Systems with Components Logically Arranged in Series and Parallel

This section covers the basics of evaluating the reliability of systems with components logically arranged in series and parallel.

FIGURE 5.3 A system with components (a) logically arranged in series and (b) logically arranged in parallel.

Consider a system including *n* independently working components. Let *S* denote the event ‘the system is in working state at the end of a specified time interval’ and *C _{k}*

*(к*= 1,2,...,

*n)*denote the events ‘component

*к*is in working state at the end of the specified time interval’. For components logically arranged in series (Figure 5.3a) the system is in working state at the end of the specified time interval only if all components are in working state at the end of the time interval.

Reliability is the ability of an entity to work without failure for a specified time interval, under specified conditions and environment. The ability to work without failure within the specified time interval is measured by the probability of working without failure during the specified time interval.

According to the reliability theory (Bazovsky, 1961), the probability of system success (system in working state at the end of the specified time interval) is a product of the probabilities that the components will be in working state at the end of the specified time interval:

Denoting by *R* the probability *P(S)* that the system will be in working state at the end of the specified time interval and by *r _{k} = P(C_{k})* the probability that the kth component will be in working state at the end of the specified time interval, equation (5.1) becomes

In equation (5.2), *R* will be referred to as the reliability of the system and *r _{k}* as the reliability of the kth component related to the specified time interval.

Now consider independently working components logically arranged in parallel (Figure 5.3b). According to the system reliability theory (Bazovsky, 1961; Hoyland and Rausand, 1994), the probability of system success (system in working state at the end of the specified time interval) is equal to the probability that at least a single component will be in working state at the end of the specified time interval.

The event ‘at least a single component will be in working state at the end of the specified time interval’ and the event ‘none of the components will be in working state at the end of the specified time interval’ are complementary events. From probability theory (DeGroot, 1989), the probabilities of complementary events add up to unity. Therefore, the probability that at least a single component will be in working state at the end of the specified time interval can be evaluated by subtracting from unity the probability that none of the components will be in working state at the end of the specified time interval. The advantage offered by this inverse-thinking approach is that the probability that none of the components will be in working state at the end of the specified time interval is very easy to calculate.

Indeed, if *r _{u}r_{2},...,r„* denote the reliabilities of the separate components (the probabilities that the components will be in working state at the end of the specified time interval), the probability

*P(S)*that none of the components will be in working state at the end of the specified time interval (the probability of system failure) is given by

Consequently, the probability that the system will be in working state at the end of the specified time interval (the probability of system’s success) is given by

Note that for a logical arrangement of the components in series, the system reliability is a product of the reliabilities of the components, while for a logical arrangement of the components in parallel, the probability of system failure is a product of the probabilities of failure of the components.

A system with components logically arranged in series and parallel can be reduced in complexity in stages, as shown in Figure 5.4. In the first stage, the components in Figure 5.4a, logically arranged in series, with reliabilities *R _{t}* and

*R*are reduced to an equivalent component with reliability

_{2},*R*The components logically arranged in parallel with reliabilities

_{i2}= R_{t}R_{2}.*R*and /?

_{4}_{5}are reduced to an equivalent component with reliability /?

_{45}= 1 — (1 —

*R*

_{4})(1 -

*R*and the components in parallel with reliabilities

_{5})*R*and

_{(},*R*are reduced to an equivalent component with reliability

_{7}*R*= 1 - (1 - /4)(1 -

_{(>1}*R-).*The resultant equivalent reliability network is shown in Figure 5.4b.

In the second stage, the components in parallel, with reliabilities *R _{l2}* and Z?

_{3}in Figure 5.4b, are reduced to an equivalent component with reliability

*Rm*= 1 — (1 — /?i

_{2})(l — /?з) and the components in series with reliabilities

*R*and

_{hl}*R$*are reduced to an equivalent component with reliability

*R*/?

_{f},n =_{67}x

*R*The resultant equivalent network is shown in Figure 5.4c. Next, the reliability network in Figure 5.4c is further simplified by reducing the equivalent components with reliabilities

_{4}.*R*and

_{v23}*Rif,*to a single equivalent component with reliability

*Rims = Rm*x

*R*The final result is the reliability network in Figure 5.4d, whose reliability is

_{4}s-*R =*1 - (1 - /?i234s)(l - /?

_{6}?»)•

It needs to be pointed out that there is a critical difference between a physical arrangement of components in a system and their logical arrangement. Thus, the valves in Figure 5.5a are physically arranged in series. If initially both valves are open, and the production fluid passes through the pipeline, with respect to stopping the production fluid through the pipeline, the valves are logically arranged in parallel (Figure 5.5b).

FIGURE 5.4 Network reduction method for determining the reliability of a system including components logically arranged in series and parallel; (a), (b), (c) and (d) - stages of the network reduction method.

FIGURE 5.5 Difference between a physical arrangement (diagrams [a] and [c]) and logical arrangement (diagrams [b] and [d]) for valves on a pipeline with respect to the function ‘stopping the production fluid on command’.

This is because at least one of the valves is necessary to work on command for the flow of production fluid through the pipeline to be stopped.

In Figure 5.5c, the valves are physically arranged in parallel. If initially both valves are open, the production fluid passes through both valves. With respect to stopping the production fluid through the pipeline, the valves are now logically arranged in series (Figure 5.5d). This is because both valves must stop the flow in their branches for the flow of production fluid through the pipeline to be stopped. Methods for determining the reliability of systems with reliability networks different from networks with series- parallel arrangement of the components are discussed in detail in (Todinov, 2016).