# Using Inequalities to Rank Systems with the Same Topology and Different Component Arrangements

Consider the functional diagrams of three systems built with pipes and four valves (А, *В, C* and *D*) with different ages (Figure 5.6). Valve A is a new valve, followed by valves *В* and C with intermediate age. Valve *D* is an old valve. The reliabilities of the valves are denoted by *a, b, c* and *d.* With respect to the function ‘valve closure on demand’, depending on their age, the reliabilities of the valves can be ranked as follows: *a>b> c> d.* The valves are working independently from one another, and initially, all of them are open. The question of interest is which system is more reliable with respect to the function ‘stopping the fluid flow in the pipeline on command’. The signal for closing is issued to all valves simultaneously.

It has been conjectured that the arrangement in Figure 5.6c is superior to the arrangements in Figure 5.6a and b.

The reliability networks (reliability block diagrams) of the systems with respect to the function ‘stopping the fluid in the pipeline on command’ are given in Figure 5.7. The reliability networks show the logical arrangement of the valves with respect to the function ‘stopping the fluid in the pipeline on command’.

FICURE 5.6 Three possible arrangements involving four different types of valves: (a) valves A and *В* in the same branch and the rest of the valves in the other branch; (b) valves A and *C* in the same branch and the rest of the valves in the other branch; (c) valves A and *D *in the same branch and the rest of the valves in the other branch.

FIGURE 5.7 Reliability networks (a), (b) and (c) corresponding to the three systems in Figure 5.6a-c with respect to the function ‘stopping the fluid in the pipeline on command’.

The reliabilities of systems (a), (b) and (c) in Figure 5.7 are given by

Proving that the configuration in Figure 5.7c is superior reduces to proving the inequalities

which are equivalent to the inequalities Manipulating the left side of inequality (5.8) results in

Considering that *a>b> c> d,* it follows that (*a* - *c)(b -d)>* 0. Therefore, the configuration in Figure 5.7c is more reliable than the configuration in Figure 5.7a. Manipulating the left-hand side of inequality (5.9) results in

FIGURE 5.8 The same system topology, built with new valves *A* and old valves *B.* in four distinct permutations (a), (b), (c) and (d).

Considering that *a>b>c> d,* it follows that *(a* - *b)(c -d)>* 0. Therefore, the configuration in Figure 5.7c is more reliable than the configuration in Figure 5.7b. Consequently, the configuration in Figure 5.7c is the most reliable configuration.

Consider now the functional diagrams of four possible configurations built with pipes and valves of the same type: new valves *A* and old valves *В* (Figure 5.8). With respect to the function ‘valve closure on command’, a new valve *A* is more reliable than an old valve *B.* If the reliabilities of the valves are denoted by *a* and *b*, the inequality *a > b* holds.

The valves are working independently from one another and all of them are initially open. The question of interest is which configuration is most reliable with respect to the function ‘stopping the fluid in the pipeline on command’. The signal for closure is issued to all valves simultaneously.

The reliability block diagrams of the systems with respect to the function ‘stopping the fluid in the pipeline’, are given in Figure 5.9.

The reliabilities of the systems in Figure 5.9a-d are given by

FIGURE 5.9 Reliability networks (a), (b), (c) and (d) corresponding to the systems in Figure 5.8a-d, with respect to the function ‘stopping the fluid in the pipeline on command’.

It is conjectured that the system in Figure 5.9b is the most reliable system. For the differences of the reliability of configuration (Figure 5.9b) and the reliabilities of the rest of the configurations, the following relationships hold:

The configuration in Figure 5.9b is indeed characterised by the highest reliability.

In the examples considered in this section, the algebraic inequalities helped to reveal the intrinsic reliability of the competing design solutions and rank the systems in terms of reliability in the presence of significant uncertainty related to the reliabilities of their building parts.

# Using Inequalities to Rank Systems with Different Topologies Built with the Same Types of Components

Two systems with different topologies including the same types of valves (denoted by *X, Y* and Z) are shown in Figure 5.10 whose reliabilities (denoted by *x, у* and *z) *are unknown. The valves are working independently from one another and all of them are initially open. The question of interest is which system is more reliable with respect to the function ‘stopping the flow of fluid in the pipeline’. The signal for closing is sent to all valves simultaneously.

The reliability networks of the systems from Figure 5.10 are shown in Figure 5.11. The reliability values *x, у* and *z* characterising the separate valves are unknown. The only available information about the reliabilities of the valves are the obvious constraints: 0

FIGURE 5.10 Two competing systems (a) and (b) with different topology, built with the same type and number of components.

FIGURE 5.11 The reliability networks (a) and (b), corresponding to the systems from Figure 5.10a and b.

Expressing the probabilities of failure characterising the competing systems as a function of the unknown reliabilities of the valves yields

Ranking the system reliabilities consists of proving *F _{a}(x,y,z)~F_{b}(x,y,z)<*0 or

*F*0. Proving

_{a}{x,y,z)~ F_{h}(x,y,z)>*F*0. for example, is equivalent to proving the inequality

_{a}(x,y,z)~ F_{h}(x,y,z) <

To prove inequality (5.17), it suffices to prove the equivalent inequality ^(1 *x~*)(1 y^{2})(l *z~) < (l-x)’z)* or the equivalent inequality

Indeed, if inequality (5.18) is true, inequality (5.17) follows from it by squaring both sides of the inequality ^(l-jr^{2})(l->^{,2})(l-z^{2})

To prove inequality (5.18), a combination of the ‘substitution’ technique and proving a simpler, intermediate inequality will be used.

Because the reliability r, of a component is a number between zero and unity, the trigonometric substitutions *r _{t}* = since, where

*a*e (0,я / 2) are appropriate.

Making the substitutions: *x =* since; *у =* sin*(5* and z=siny transforms the left side of inequality (5.18) into

Next, the positive quantity cos a x cos/} x cos у + sin ax sin/}x sin у is replaced by the larger quantity cosaxcos/?+sinaxsin)3. Indeed, because 0

holds.

If the intermediate inequality cos *a* x cos *[5 +* since x sin *ft <* 1 could be proved, this would imply the inequality

Since cosce xcos/I +since xsin/I =cos(ee *- ft),* and cos(ce *-P)<* 1, we finally get

Inequality (5.18) has been proved and from it, inequality (5.17) follows. The system in Figure 5.10a is characterised by a smaller probability of failure than the system in Figure 5.10b; therefore, the system in Figure 5.10a is the more reliable system.

In the example considered in this section, the algebraic inequalities helped to reveal the intrinsic reliability of the competing design solutions and rank the systems in terms of reliability in the absence of any knowledge related to the reliabilities of their building parts.