Importance and Sensitivity Analysis of Multi-state Systems

PART V. Practices and Emerging Applications

4.5 Importance and Sensitivity Analysis of Multi-state Systems

Methods for evaluating the relative influence of component availability on the availability of the entire system provide useful information about the importance of these elements. Importance evaluation is a key point in tracing bottlenecks in systems and in the identification of the most important components. It is a useful tool to help the analyst find weaknesses in design and to suggest modifications for system upgrade.

Some various importance measures have been introduced previously. The first importance mea- sure was introduced by Birnbaum [29]. This in- dex characterizes the rate at which the system reliability changes with respect to changes in the reliability of a given element. So, an improvement in the reliability of an element with the highest importance causes the greatest increase in system reliability. Several other measures of elements and minimal cut sets importance in coherent systems were developed by Barlow and Proschan [30] and Vesely [31].

The above importance measures have been defined for coherent systems consisting of bi- nary components. In multi-state systems the fail- ure effect will be essentially different for system elements with different performance levels. There- fore, the performance levels of system elements should be taken into account when their impor- tance is estimated. Some extensions of importance measures for coherent MSSs have been suggested, e.g.[11, 30], and for non-coherent MSSs [32].

From Equations 4.11 and 4.14 one can see that the entire MSS reliability indices are complex functions of the demandW, which is an additional factor having a strong impact on an element’s importance in multi-state systems. Availability of a certain component may be very important for one demand level and less important for another.

For the complex system structure, where each system component can have a large number of possible performance levels and there can be a large number of demand levelsM, the importance

Multi-state System Reliability Analysis and Optimization 69

Table 4.1. Measures of system performance obtained for MSS

MSS EA EU EG W

type

MSSc 0 W−a1g1−a2g2 a1g1+a2g2 W > g1+g2

a1a2 g1a1(a2−1)+g2a2(a1−1) g2< W≤g1+g2 +W (1−a1a2)

a2 (1−a2)(W−g1a1) g1< W≤g2

a1+a2−a1a2 (1−a1)(1−a2)W 0< W≤g1

MSSs 0 W−a1g1−a2g2+a1a2g1 a1(1−a2)g1+a2g2 W > g2

a2 (1−a2)(W−g1a1) g1< W≤g2

a1+a2−a1a2 (1−a1)(1−a2)W 0< W≤g1

evaluation for each component obtained using existing methods requires an unaffordable effort.

Such a problem is quite difficult to formalize because of the great number of logical functions for the top-event description when we use the logic methods, and the great number of states when the Markov technique is used.

In this section we demonstrate the method for the Birnbaum importance calculation, based on the UGF technique. The method provides the importance evaluation for complex MSS with different physical nature of performance and also takes into account the demand.

The natural generalization of Birnbaum impor- tance for MSSs consisting of elements with total failure is the rate at which the MSS reliability index changes with respect to changes in the availability of a given elementi. For the constant demandW, the element importance can be obtained as

∂A(W )

∂ai

(4.21) where ai is the availability of the ith element at the given moment,A(W )is the availability of the entire MSS, which can be obtained for MSSs with a given structure, parameters, and demand using Equation 4.8.

For the variable demand represented by vectors W and q the sensitivity of the generalized MSS availability EA to the availability of the given elementiis

SA(i)=∂EA(W,q)

∂ai (4.22)

where index EA can be obtained using Equa- tion 4.9.

In the same way, one can obtain the sensitivity of the EG and EU indices for an MSS to the availability of the given elementias

SG(i)=∂EG

∂ai (4.23)

and

SU(i)=%%

%%∂EU(W,q)

∂ai

%%%% (4.24) SinceEUis a decreasing function ofaifor each elementi, the absolute value of the derivative is considered to estimate the degree of influence of element reliability on the unsupplied demand.

It can easily be seen that all the suggested measures of system performance (EA, EG, EU) are linear functions of elements’ availability.

Therefore, the corresponding sensitivities can easily be obtained by calculating the performance measures for two different values of availability.

The sensitivity indices for each MSS element depend strongly on the element’s place in the system, its nominal performance level, and system demand.

Example 3. Analytical example. Consider the sys- tem consisting of two elements with total failures connected in parallel. The availabilities of the ele- ments area1anda2and the nominal performance rates are g1 and g2 (g1< g2). The analytically obtained measures of the system output perfor- mance for MSSs of both types are presented in

70 System Reliability and Optimization

Table 4.2. Sensitivity indices obtained for MSS

MSS SA(1) SA(2) SU(1) SU(2) SG(1) SG(2) W

type

MSSc 0 0 −g1 −g2 g1 g2 W > g1+g2

a2 a1 (a2−1)g1 (a1−1)g2 g2< W≤g1+g2

−a2(W−g2) −a1(W−g1)

0 1 (a2−1)g1 a1g1−W g1< W≤g2

1−a2 1−a1 (a2−1)W (a1−1)W 0< W≤g1

MSSs 0 0 (a2−1)g1 a1g1−g2 g1(1−a2) g2−a1g1 W > g2

0 1 (a2−1)g1 a1g1−W g1< W≤g2

1−a2 1−a1 (a2−1)W (a1−1)W 0< W≤g1

Figure 4.2. Example of series–parallel MSS

the Table 4.1. The sensitivity indices can also be obtained analytically. These indices for MSScand MSSsare presented in Table 4.2. Note that the sen- sitivity indices are different for MSSs of different types, even in this simplest case.

Numerical example. The series–parallel system presented in Figure 4.2 consists of ten elements of six different types. The nominal performance rate gi and the availabilityai of each type of element are presented in Table 4.3.

We will consider the example of the system with the given structure and parameters as MSSc

and MSSs separately. The cumulative demand curvesq1,W1for MSScandq2,W2for MSSsare presented in Table 4.4 (q1=q2=q).

The sensitivity indices estimated for both types of system are presented in Table 4.5. These indices depend strongly on the element’s place in the system, on the element’s nominal performance, and on the system demand.

Table 4.3. Parameters of system elements

No. of Nominal Availability

elementi performance ai

rategi

1 0.40 0.977

2 0.60 0.945

3 0.30 0.918

4 1.30 0.983

5 0.85 0.997

6 0.25 0.967

Table 4.4. Cumulative demand curves

q 0.15 0.25 0.35 0.25

W1(ton h−1) 1.00 0.90 0.70 0.50

W2(kbytes s−1) 0.30 0.27 0.21 0.15

The dependencies of sensitivity index SA on the system demand are presented in Figures 4.3 and 4.4. Here, system demand variation is de- fined as vector kW, where W is the initial de- mand vector given for each system in Table 4.4 andkis the demand variation coefficient. As can be seen from Figures 4.3 and 4.4, the SA(k)are complicated non-monotonic piecewise continu- ous functions for both types of system. The or- der of elements according to their SA indices changes when the demand varies. By using the graphs, all the MSS elements can be ordered ac- cording to their importance for any required de- mand level. On comparing graphs in Figures 4.3

Multi-state System Reliability Analysis and Optimization 71

Table 4.5. Sensitivity indices obtained for MSS (numerical example)

Element no. MSSc MSSs

SA SG SU SA SG SU

1 0.0230 0.1649 0.0035 0.1224 0.0194 0.0013 2 0.7333 0.4976 0.2064 0.4080 0.0730 0.0193 3 0.1802 0.2027 0.0389 0.1504 0.0292 0.0048 4 0.9150 1.0282 0.7273 0.9338 0.3073 0.2218 5 0.9021 0.7755 0.4788 0.7006 0.1533 0.0709 6 0.3460 0.2022 0.0307 0.1104 0.0243 0.0023

Figure 4.3. SensitivitySAas a function of demand for MSSc

and 4.4, one can notice that the physical na- ture of MSS performance has a great impact on the relative importance of elements. For exam- ple, for 0.6≤k≤1.0, SA(4) and SA(5) are al- most equal for MSSc and essentially different for MSSs.

The fourth element is the most important one for both types of system and for all the perfor- mance criteria. The order of elements according to their sensitivity indices changes when different

Figure 4.4. SensitivitySAas a function of demand for MSSs

system reliability measures are considered. For example, the sixth element is more important than the third one when SA is considered for MSSc, but is less important whenSGandSUare consid- ered.

The UGF approach makes it possible to evaluate the dependencies of the sensitivity indices on the nominal performance rates of elements easily.

The sensitivities SG(5), which are increasing functions of g5 and decreasing functions of

72 System Reliability and Optimization

Figure 4.5. SensitivitySGas a function of nominal performance rates of elements 5 and 6 for MSSc

Figure 4.6. SensitivitySGas a function of nominal performance rates of elements 5 and 6 for MSSs

g6 for both types of system, are presented in Figures 4.5 and 4.6. As elements 5 and 6 share their work, the importance of each element is proportional to its share in the total component performance.

4.6 Multi-state System