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Sensitivity analysis of reliability of Systems with Complex Interconnections
Journal of Loss Prevention in the Process Industries 32 (2014) 436e442
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Journal of Loss Prevention in the Process Industries journal homepage: www.elsevier.com/locate/jlp
Sensitivity analysis of reliability of Systems with Complex Interconnections szlo Pokora di La t Ba nki Faculty of Mechanical and Safety Engineering, Obuda Dona University, Hungary
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 June 2014 Received in revised form 25 August 2014 Accepted 9 September 2014 Available online 30 October 2014
Theoretical and practical investigation of complex systems has become important in several fields of technical and social sciences. One of the most important questions is the sensitivity of these systems. The main aim of this paper is to show adaptation of mathematical diagnostic methodology of aircraft systems and gas turbine engines to determine sensitivity of reliability of finite Systems with Complex Interconnections. These proposed analogous methods are named Linear Sensitivity Model of system reliability and Linear Sensitivity Model of System Unreliability. The paper shows the proposed methods theoretically and their applicability to investigate Systems with Complex Interconnections sensitivity by a simple example. © 2014 Elsevier Ltd. All rights reserved.
Keywords: System reliability Sensitivity analysis System with Complex Interconnection
1. Introduction The study of complex systems has become important in several fields of technical science. The most real multichannel systems are not simply interconnected. The system having no simple interconnections is called System with Complex Interconnection (SwCI) or complex system. These systems cannot be simplified to single equivalent elements or blocks by a combination of series and parallel reductions. The Fault Tree Analysis (FTA) and Reliability Block Diagram (RBD) methods cannot be used to determine reliability of the SwCI. One approach to computing the correct reliability parameters for the SwCI is the summation of the probabilities of all the operating states of the system. A table listing the probability of each possible state and its consequences for a system is frequently referred to as the truth table. The information within the truth table is used to determine the System reliability or unreliability. In the engineering literature, there are many studies books as well as papers dealing with complex systems, networks and their reliability theory from theoretical and practical points of view. For ska-Wojciechowska (2013) investigated time instance Werbin resource problem in logistic systems operating under an increasingly complex and diverse system environment. She established that it is necessary to take into account the possible unreliability of logistic system elements, which may lead to a possible decrease of the system availability level. Myers (2010) focused on reliability modeling of complex multichannel systems, such as the digital fly-by-wire aircraft control
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jlp.2014.09.017 0950-4230/© 2014 Elsevier Ltd. All rights reserved.
system. In his book, Iordache (2011) focused on modeling multi-level complex systems. According to Iordache (2011), a complex system is composed of subsystems characterized by an emergent behavior from nonlinear interactions among the subsystems resulting in th (2013) multiple levels of organization. Additionally, Horva described some useful measures of the properties of the complex networks using the Cxnet Complex Network Analyzer Software with some examples, and measures he collected through academic r & Neszveda (2013) paper was to experience. The aim of Lama propose a new investigative method aligned with the standards, as well as taking the features of a periodic operation into account. Paska (2013) remarked that reliability is one of the most important criteria, which should be taken into consideration during planning and operating a power system. A method based on the graph theory and the Boolean function for assessing reliability of mechanical systems was proposed by Tang. The procedure for approach of Tang (2001) consists of two parts utilizing the graph theory, generating the formula to test the reliability of a mechanical system that considers the interrelations of subsystems or elements. The main scope of Fan et al. (2009) is to investigate possible failures of a coffee maker. Torras et al. (2010) introduced a new reliability analysis technique based on conditional probabilities ranging from SPR-MP in the worst case, getting faster as the number of reconvergence sources in the circuit increases. This approach allows handling independent of the circuit size, for is only limited to the number of sources reconverging in a same sink instead of the whole circuit size. It is important to mention that this proposed method can model the reliability of the SwCI. Mi et al. (2013) presented the traditional reliability analysis, such as the truth table method, based on the assumption that events are binary, i.e. success and complete failure.
di / Journal of Loss Prevention in the Process Industries 32 (2014) 436e442 L. Pokora
Reliability analysis considering multiple possible states is known as multi-state reliability analysis. Multi-state system reliability models allow both the system and its components to assume more than two levels of performance. In paper of Yingkuia and Jing (2012), they bring a new and systematic review on multistate system reliability. Daneshkhah and Bedford (2013) present a new approach to study the sensitivity analysis of availability. This alternative sensitivity analysis of the quantities of interest in the reliability analysis, such as the availability/unavailability function, with respect to the changes of uncertain parameters has been presented. This method is originally introduced by Oakkley (2004) to examine sensitivity analysis of a complex model with respect to changes in its inputs based on an emulator which is built to approximate the model. di (2011) showed the adaptation of linear mathematical Pokora diagnostic modeling methodology for setting-up of Linear Fault Tree Sensitivity Model (LFTSM). The sensitivity analysis shows how sensitive the output parameter is while changing in any elements of the input parameters. The LFTSM is a modular approach tool that uses matrix-algebraic method based upon the mathematical diagnostic methodology of aircraft systems and gas turbine engines shown by Pokor adi (2008). The disadvantage of LFTSM is that it cannot depict correctly sensitivity of reliabilities of SwCI. The aims of this theoretical investigation are the followings: to modify the LFTSM to determine sensitivity of reliability of SwCIs; The proposed methods are named Linear Sensitivity Model of System Reliability (LSMoSR) and Linear Sensitivity Model of System Unreliability (LSMoSU). to show the methods generally; to show their possibilities of use to investigate SwCI sensitivity by a simple example; to compare results of these methods. to draw a conclusions from results. The theoretical obtained consequents and experiences can be used to investigate system reliability in a wide context. It is important to mention that, users should use only few equations, not all ones of this paper. The outline of the paper is as follows: Section 2 shows determination of reliability of SwCI using truth table. Section 3 presents the proposed modular approach algorithm for setting-up of Linear Sensitivity Model of System Reliability and Linear Sensitivity Model of System Unreliability theoretically and practically. Section 4 interprets conclusions about proposed methods and sensitivity of investigated system reliability. Section 5 summarizes the paper, outlines the prospective scientific work of the Author.
Fig. 1. Investigated (example) system (source: Myers (2010)).
437
2. Reliability of the System with Complex Interconnections To demonstrate the proposed sensitivity determination method, firstly let us determine the reliability parameters of the system shown by Fig. 1. The system shown in Fig. 1 has five elements, A B C D E. Their reliability can be characterized by: reliability (the probability that the element accomplishes its assigned task): ri i2L; probability of failure: pi i2L, where: L e set of Latin characters A B C D E. The elements have two operating and non-operating states. In case of operating state an item is performing a required function. If an item is in non-operating state, it is not performing a required function Sum of probabilities of operating and non-operating should be unity:
ri þ pi ¼ 1;
pi ¼ 1 ri ;
ri ¼ 1 pi :
(1)
For farther application, the reliability and the probability of the failure of the elements should be arranged into vectors:
rT ¼ ½ rA
rB
rC
rE ;
rD
(2)
and
pT ¼ ½ pA
pB
pC
pD
pE :
(3)
One of the approaches to correctly compute the reliability parameters for the SwCI is the summation of the probabilities of all the states of the investigated system. A table listing the probability of each possible state for a system is frequently referred to a truth Table 1 Possible states of the system in Fig. 1. j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Element A
B
C
D
E
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
System state
Qj
0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1
pApBpCpDpE rApBpCpDpE pArBpCpDpE rArBpCpDpE pApBrCpDpE rApBrCpDpE pArBrCpDpE rArBrCpDpE pApBpCrDpE rApBpCrDpE pArBpCrDpE rArBpCrDpE pApBrCrDpE rApBrCrDpE pArBrCrDpE rArBrCrDpE pApBpCpDrE rApBpCpDre pArBpCpDrE rArBpCpDrE pApBrCpDrE rApBrCpDrE pArBrCpDrE rArBrCpDrE pApBpCrDrE rApBpCrDrE pArBpCrDrE rArBpCrDrE pApBrCrDrE rApBrCrDrE pArBrCrDrE rArBrCrDrE
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table. The elements can be operating (designated as the 1 state) or non-operating ones (designated as the 0 state). Therefore a common, n-element system has 2n (in our case: 25 ¼ 32) possible states. The possible system states are summarized in the form of a truth table, shown in Table 1, with each element being assigned either an operating or a non-operating state. The Qj j2 אcolumn lists the probabilities of each of the system states. Since the table covers all of the possible combinations, the sum of all of the state probabilities should be 1. The state probabilities resulting in an operating system are included in the rows 10; 11; 12; 14; 15; 16; 19; 20; 21; 22; 23; 24; 26; 27; 28; 29; 30; 31 and 32.
Rsys ¼ Q10 þ Q11 þ Q12 þ Q14 þ Q15 þ Q16 þ Q19 þ Q20 þ Q21 þ Q22 þ Q23 þ þQ24 þ Q26 þ Q27 þ Q28 þ Q29 þ Q30 þ Q31 þ Q32 (4) The sum of the operating system state probabilities included in this column is the reliability of the system. The system unreliability could be calculated by summing of the 1; 2; 3; 4; 5; 6; 7; 8; 9; 13; 17; 18 and 25 non-operating state (rows) probabilities:
Psys ¼ Q1 þ Q2 þ Q3 þ Q3 þ Q4 þ Q5 þ Q6 þ Q7 þ Q8 þ Q9 þ Q13 þ Q17 þ Q18 þ Q25 : (5) For this particular case, the calculation of the system unreliability involves summing fewer terms (13 system states) than does the calculation of the system reliability (19 system states). This is not generally the case, however. 3. Creating sensitivity model For the purpose of investigation, the sensitivity model of the discussed system should be created. In this section, we demonstrate the method of the modular approach sensitivity model by using the sample SwCI mentioned above (Fig.1.). In our study we will investigate:
Using these coefficients, the relative changing of output parameter y depends on relative changing of elements of the vector x can be determined by equation,
dyj ¼ Kj1 dxj2 þ … þ Kjn dxn :
where: d signs the relative changing of parameters. Next task is to separate the system parameters into vectors y dependent and x independent ones. In our study, the dependent parameters are the probabilistic parameter of system and the possible system states. Independent parameters are the probabilistic parameters of the elements. Then, the connection between the independent and the dependent parameters can be described by equation,
Ady ¼ Bdx;
D ¼ A1 B
system unreliability Psys, depending on probabilities of failures of elements:
(7)
These parameters will be called generally probabilistic parameters in Chapter 3.1, showing the general methodology of proposed method.
relative sensitivity coefficient matrix of the investigated system, the equation
dy ¼ Ddx
(12)
can be used for relative sensitivity investigations. Vector of the relative changing of the independent parameters can be determined by:
dx ¼ X1 nom Dx;
(13)
where: Xnom2
Xnom
x1nom 6 0 ¼6 4 « 0
0 x2nom …
… 1 0
0 0 «
3 7 7: 5
(8)
(14)
xnnom
Dx2
During modular sensitivity analysis firstly the system should be separated into modules. In our case, these modules are the possible system states. Investigating these modules, their (probabilistic) models should be determined by yj ¼ fj(x1,x2,…,xn) fj :
vyj xi vf ðx1 ; x2 ; …xn Þ xi ¼ vxi f ðx1 ; x2 ; …xn Þ vxi yj
(11)
(15)
can be determined. Applying the
3.1. General solution
Kyj ;xi ¼ Kji ¼
D2
2
(6)
Psys ¼ fp ðp Þ;
(10)
where A2
system reliability Rsys, depending on reliability of elements:
Rsys ¼ fr r ;
(9)
Ynom
y1nom 6 0 ¼6 4 « 0
0 y2nom …
… 1 0
0 0 «
3 7 7: 5
(16)
ymnom
nominal value matrix of the dependent parameters Y nom 2
Dy ¼ Y nom dy ¼ Ynom DX1 nom Dx:
(17)
Introducing the so called measured sensitivity coefficient matrix of the investigated system.
di / Journal of Loss Prevention in the Process Industries 32 (2014) 436e442 L. Pokora nm S ¼ Ynom DX1 ; nom 2<
(18)
are inverse ones. If the state of the element is supplemental of the investigated system states, the intrinsic function, seeing equation (1), uj ¼ 1 xj , then sensitivity coefficient is:
(19)
Kji ¼
the equation (17) can be simplified
Dy ¼ SDx:
439
Oftentimes, during the system reliability investigation only the probabilistic system parameter is of interesting. But, the other dependent probabilistic parameters can be important too, from technical point of view. If just the probabilistic system parameter and its sensitivities are being investigated only the first row of matrix D and S should be used as vectors d2
n xi Y uk : yj
(22)
k¼1 ksi
In case of functions that determine directly the probabilistic system parameters e see equations (4) and (5) e, the sensitivity coefficients can be determined by:
Qj Qj ; or KRj ¼ : Psys Rsys
3.2. Determination of typical sensitivity coefficients
KPj ¼
The next task is to determine the sensitivity coefficients. The probabilities of system states (see Table 1) can be described by.
Following the general determinations mentioned above we can now set up the linear sensitivity models of system reliability and unreliability.
yj ¼ fj ðx1 ; x2 ; …xk Þ ¼
n Y
ui ðxi Þ;
(20) 3.3. Linear Sensitivity Model of System Reliability (LSMoSR)
i¼1
general equation, where the inner function ui(xi) can have one of the following two forms. The state of element and system state can be the same or its inverse. For example, the 1st row of Table 1, system state and states of all elements are non-operating e the states are the same. If state of element is the same as the investigated system states, the intrinsic function ui ¼ xi, then the sensitivity coefficient is:
Kji ¼ 1;
(23)
(21)
The vector of the independent parameters for the set up of LSMoSR is described by equation (2) and their relative changing vector is:
dxT ¼ ½ drA
drB
drC
drD
drE ;
and the vector of the relative changing of the dependent parameters by equation (4):
dyT ¼ dRsys dQ10 dQ11 dQ12 dQ14 dQ15 dQ16 dQ19 dQ20 dQ21 dQ22 dQ23 dQ24 dQ26 dQ27 dQ28 dQ29 dQ30 dQ31 dQ32
On the other hand, in the 2nd row of Table 1, the state of element A is operating, but system state is non-operating e these two states
(24)
(25)
The coefficient matrix of the dependent parameters:
2
3 1 KR10 KR11 KR12 KR14 KR15 KR16 KR19 KR20 KR21 KR22 KR26 KR27 KR28 KR29 K KR30 KR24 KR23 KR31 KR32 60 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 7 A¼ 6 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 7 6 7 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (26)
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the coefficient matrix of independent parameters:
2
0 6 1 6 6 K11A 6 6 1 6 6 1 6 6 K15A 6 6 1 6 6 K19A 6 6 1 6 6 K21A B¼6 6 1 6 6 K23A 6 6 1 6 6 1 6 6K 6 27A 6 1 6 6K 6 29A 6 1 6 4K 31A 1
0
0
K10B 1 1 K14B 1 1 1 1 K21B K22B 1 1 K26B 1 1 K29B K30B 1 1
K10C K11C K12C 1 1 1 K19C K20C 1 1 1 1 K26C K27C K28C 1 1 1 1
0 1 1 1 1 1 1
0
K19D K20D K21D K22D K23D K24D 1 1 1 1 1 1 1
The coefficient matrix of independent parameters:
3
2
K10E 7 7 K11E 7 7 K12E 7 7 K14E 7 7 K15E 7 7 K16E 7 7 1 7 7 1 7 7 1 7 7; 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 5 1
(27)
0 6 1 6 6 K2A 6 6 1 6 6 K4A 6 6 1 6 6 K6A B¼6 6 1 6 6 K8A 6 6 1 6 6 1 6 6 1 6 4K
18A
1
The relative and measured sensitivity coefficient vectors of system reliability in cases of different rinom nominal values of element reliabilities are shown by Table 2. Figs. 2 and 4 show the sensitivities of system reliability.
0 1 1 K3B K4B 1 1 K7B K8B 1 1 1 1 1
0 1 1 1 1 K5C K6C K7C K8C 1 K13C 1 1 1
0 1 1 1 1 1 1 1 1 K9D K13D 1 1 K25D
0 1 1 1 1 1 1 1 1 1 1
3
7 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 K17E 7 7 5 K
(31)
18E
K25E
The sensitivity coefficient vectors of system unreliability in cases of different pinom nominal values of element probabilities are shown in Table 3 and Figs. 3 and 4. During the application of the proposed method experts should determine truth table of investigated system, sensitivity coefficients using equations (21)e(23), and nominal value and coefficient matrices of independent and dependent parameters. Then, the relative and measured sensitivities of system reliability can be determined using equations (11) and (18). Knowing them, the optimal decision can be made for maximization of system reliability id. est. minimization of unreliability.
3.4. Linear Sensitivity Model of System Unreliability (LSMoSU) The vector of independent parameters for the set up of LSMoSU and their relative change vector is described by equation (3):
dxT ¼ ½ dpA
dpB
dpC
dpD
dpE ;
(28)
and the vector of the relative change of the dependent parameters by equation (5):
dyT ¼ dPsys
dQ1
dQ2
dQ3
dQ4
dQ5
dQ6
dQ7
dQ8
dQ9
4. Discussions of results The following conclusions can be deduced from the results of modeling and analysis:
4.1. Conclusions about the proposed method
dQ13
dQ17
dQ18
dQ25 ;
(29)
The coefficient matrix of dependent parameters:
2
1 60 6 60 6 60 6 60 6 60 6 60 A¼6 60 6 60 6 60 6 60 6 60 6 40 0
KP1 1 0 0 0 0 0 0 0 0 0 0 0 0
KP2 0 1 0 0 0 0 0 0 0 0 0 0 0
KP3 0 0 1 0 0 0 0 0 0 0 0 0 0
KP4 0 0 0 1 0 0 0 0 0 0 0 0 0
KP5 0 0 0 0 1 0 0 0 0 0 0 0 0
KP6 0 0 0 0 0 1 0 0 0 0 0 0 0
KP7 0 0 0 0 0 0 1 0 0 0 0 0 0
KP8 0 0 0 0 0 0 0 1 0 0 0 0 0
KP90 0 0 0 0 0 0 0 0 1 0 0 0 0
KP13 0 0 0 0 0 0 0 0 0 1 0 0 0
KP17 0 0 0 0 0 0 0 0 0 0 1 0 0
KP18 0 0 0 0 0 0 0 0 0 0 0 1 0
3 KP25 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 5 1
(30)
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Table 2 Sensitivity vectors of system reliability. rinom
dpinom
0.99
d0;99 ¼ ½ 0; 000197 s0;99 ¼ ½ 0; 000197 d0;05 ¼ ½ 0; 004631 s0;95 ¼ ½ 0; 004631 d0;9 ¼ ½ 0; 017100 s0;9 ¼ ½ 0; 017100
0.95 0.90
0; 000296 0; 000296 0; 007006 0; 007006 0; 026100 0; 026100
0; 000197 0; 000197 0; 004631 0; 004631 0; 017100 0; 017100
0; 010097 0; 010097 0; 052131 0; 052131 0; 107100 0; 107100
0; 010097 0; 010097 0; 052131 0; 052131 0; 107100 0; 107100
Fig. 4. Measured sensitivity of system.
1-4. From the two methods described above, the one which has the lower number of possible system states should be chosen. 1-5. Sensitivity parameters of the possible system states can be used in cases with separate professional meaning (for example road traffic or logistical management). In this case, the Conclusion 1-4 has to be taken into account from professional points of view.
Fig. 2. Relative sensitivity of system reliability.
1-1. The proposed method can be used to analyze sensitivity of the reliability of the SwCI. 1-2. Elements of coefficient matrices can be easily determined. 1-3. The information from the analysis of reliability and unreliability of the SwCI depicts the system sensitivity in the same way.
dpinom
0.01
d0;01 ¼ ½ 0; 019135 s0;01 ¼ ½ 0; 000197 d0;05 ¼ ½ 0; 081241 s0;05 ¼ ½ 0; 004631 d0;1 ¼ ½ 0; 135607 s0;1 ¼ ½ 0; 017100
0.05 0.1
0; 028750 0; 000296 0; 122903 0; 007006 0; 206979 0; 026100
4.2. Conclusions about the investigated system 2-1. Increasing the reliability of elements, namely a reduction their failure probabilities, reduce the sensitivity of the system (see Figs. 5 and 6).
Table 3 Sensitivity vectors of system unreliability. pinom
1-6. The disadvantage of the proposed method is that the number of possible states of the system grows by a exponential function depending on the number of elements.
0; 019135 0; 000197 0; 081241 0; 004631 0; 135607 0; 017100
0; 980672 0; 010097 0; 914483 0; 052131 0; 849326 0; 107100
Fig. 3. Relative sensitivity of system unreliability.
0; 980672 0; 010097 0; 914483 0; 052131 0; 849326 0; 107100
In each modeled case the system had the same degree of sensitivity to reliability parameters of the A and C as well as D and E components. This conclusion can be deducted from the symmetric physical structure of system. (In all modeled situation the reliability parameters of elements were equivalent. In case of different element parameters this conclusion can be false.) 2-2. The sensitivity of the system is “symmetrical”. 2-3. The system has more sensitivity to reliability parameters of “second line” D and E elements.
Fig. 5. Relative Sensitivities of the System Reliability depend on Reliabilities of Elements.
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decision making includes the study of methodologies of complex system uncertainty using other mathematical tools, for example linear interval equations, Monte-Carlo Simulation on basis of Linear Sensitivity Model of System Reliability (LSMoSR). Acknowledgment This work has been supported by the visiting scholarship of Center for Complex Networks and Systems Research (CNetS), Indiana University Bloomington, which is gratefully acknowledged. References
Fig. 6. Relative Sensitivities of the System depend on Elements Probability Failures.
2-4. In the case of the “first line” elements the system has more sensitivity to reliability parameters of B. 5. Summary, future work This paper discussed a new sensitivity investigation method of reliability of SwCI elaborated by the Author. The paper showed the adaptation of the linear mathematical diagnostic modeling methodology to sett-up the Linear Sensitivity Model of System Reliability (LSMoSR) and Linear Sensitivity Model of System Unreliability (LSMoSU). The models are analogue modular approach tools that use matrix-algebraic method based upon the mathematical diagnostics methodology of aircraft systems and gas turbine engines. In this paper their possibility of use was demonstrated to investigate the SwCI sensitivity by a simply example. Using demonstrated mathematical connections and procedure, the technical experts can get an easy-used methodology to build up sensitivity model of the given SwCI. These LSMoSR and LSMoSU models can be used to investigate system reliability and dependability in a wide context. The Author's proposed prospective scientific research related to this field of applied mathematics and maintenance management
Daneshkhah, A., Bedford, T., 2013. Probabilistic sensitivity analysis of system availability using Gaussian processes. Reliab. Eng. Syst. Saf. 112, 82e93. Fan, J., Zhao, X., Tao, C., 2009. Failure evaluation of coffee maker. Eng. Fail. Anal. 16 (6), 1948e1954. 2013. The Cxnet complex network analyser software. Acta Polytech. Horv ath, A., Hung. 10 (6), 43e58. Iordache, O., 2011. Modeling Multi-level Systems. Springer-Verlag, Berlin. Lam ar, K., Neszveda, J., 2013. Average probability of failure of aperiodically operated devices. Acta Polytech. Hung. 10 (8), 153e167. Mi, J., Li, Y., Huang, H.-Z., Liu, Y., Zhang, X.-L., 2013. Reliability analysis of multi-state system with common cause failure based on Bayesian networks. Eksploatacja i Niezawodnosc Maintenance Reliab. 15 (2), 169e175. Myers, A., 2010. Complex System Reliability. Springer-Verlag, London. Oakley, J.E., O'Hagan, A., 2004. A probabilistic sensitivity analysis of complex models: a Bayesian approach. J. R. Stat. Soc. Ser. B 66, 751e769. Paska, J., 2013. Chosen aspects of electric power system reliability optimization. Eksploatacja i Niezawodnosc Maintenance Reliab. 15 (2), 202e208. s folyamatok modelleze se. Campus Kiado , Pokor adi, L., 2008. Rendszerek e Debrecen. Pokor adi, L., 2011. Sensitivity investigation of fault tree analysis with matrixalgebraic method. Theory Appl. Math. Comput. Sci. 1 (1), 34e44. Tang, J., 2001. Mechanical system reliability analysis using a combination of graph theory and boolean function. Reliab. Eng. Syst. Saf. 72, 21e30. Torras Flaquer, J., Daveau, J.M., Naviner, L., Roche, P., 2010. Fast reliability analysis of combinatorial logic circuits using conditional probabilities. Microelectron. Reliab. 50 (9e11), 1215e1218. ska-Wojciechowska, S., 2013. Time resource problem in logistics systems Werbin dependability modelling. Eksploatacja i Niezawodnosc Maintenance Reliab. 15 (4), 427e433. Yingkuia, G., Jing, L., 2012. Multi-state system reliability: a new and systematic review. Procedia Eng. 29, 531e536.
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Journal of Loss Prevention in the Process Industries journal homepage: www.elsevier.com/locate/jlp
Sensitivity analysis of reliability of Systems with Complex Interconnections szlo Pokora di La t Ba nki Faculty of Mechanical and Safety Engineering, Obuda Dona University, Hungary
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 June 2014 Received in revised form 25 August 2014 Accepted 9 September 2014 Available online 30 October 2014
Theoretical and practical investigation of complex systems has become important in several fields of technical and social sciences. One of the most important questions is the sensitivity of these systems. The main aim of this paper is to show adaptation of mathematical diagnostic methodology of aircraft systems and gas turbine engines to determine sensitivity of reliability of finite Systems with Complex Interconnections. These proposed analogous methods are named Linear Sensitivity Model of system reliability and Linear Sensitivity Model of System Unreliability. The paper shows the proposed methods theoretically and their applicability to investigate Systems with Complex Interconnections sensitivity by a simple example. © 2014 Elsevier Ltd. All rights reserved.
Keywords: System reliability Sensitivity analysis System with Complex Interconnection
1. Introduction The study of complex systems has become important in several fields of technical science. The most real multichannel systems are not simply interconnected. The system having no simple interconnections is called System with Complex Interconnection (SwCI) or complex system. These systems cannot be simplified to single equivalent elements or blocks by a combination of series and parallel reductions. The Fault Tree Analysis (FTA) and Reliability Block Diagram (RBD) methods cannot be used to determine reliability of the SwCI. One approach to computing the correct reliability parameters for the SwCI is the summation of the probabilities of all the operating states of the system. A table listing the probability of each possible state and its consequences for a system is frequently referred to as the truth table. The information within the truth table is used to determine the System reliability or unreliability. In the engineering literature, there are many studies books as well as papers dealing with complex systems, networks and their reliability theory from theoretical and practical points of view. For ska-Wojciechowska (2013) investigated time instance Werbin resource problem in logistic systems operating under an increasingly complex and diverse system environment. She established that it is necessary to take into account the possible unreliability of logistic system elements, which may lead to a possible decrease of the system availability level. Myers (2010) focused on reliability modeling of complex multichannel systems, such as the digital fly-by-wire aircraft control
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jlp.2014.09.017 0950-4230/© 2014 Elsevier Ltd. All rights reserved.
system. In his book, Iordache (2011) focused on modeling multi-level complex systems. According to Iordache (2011), a complex system is composed of subsystems characterized by an emergent behavior from nonlinear interactions among the subsystems resulting in th (2013) multiple levels of organization. Additionally, Horva described some useful measures of the properties of the complex networks using the Cxnet Complex Network Analyzer Software with some examples, and measures he collected through academic r & Neszveda (2013) paper was to experience. The aim of Lama propose a new investigative method aligned with the standards, as well as taking the features of a periodic operation into account. Paska (2013) remarked that reliability is one of the most important criteria, which should be taken into consideration during planning and operating a power system. A method based on the graph theory and the Boolean function for assessing reliability of mechanical systems was proposed by Tang. The procedure for approach of Tang (2001) consists of two parts utilizing the graph theory, generating the formula to test the reliability of a mechanical system that considers the interrelations of subsystems or elements. The main scope of Fan et al. (2009) is to investigate possible failures of a coffee maker. Torras et al. (2010) introduced a new reliability analysis technique based on conditional probabilities ranging from SPR-MP in the worst case, getting faster as the number of reconvergence sources in the circuit increases. This approach allows handling independent of the circuit size, for is only limited to the number of sources reconverging in a same sink instead of the whole circuit size. It is important to mention that this proposed method can model the reliability of the SwCI. Mi et al. (2013) presented the traditional reliability analysis, such as the truth table method, based on the assumption that events are binary, i.e. success and complete failure.
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Reliability analysis considering multiple possible states is known as multi-state reliability analysis. Multi-state system reliability models allow both the system and its components to assume more than two levels of performance. In paper of Yingkuia and Jing (2012), they bring a new and systematic review on multistate system reliability. Daneshkhah and Bedford (2013) present a new approach to study the sensitivity analysis of availability. This alternative sensitivity analysis of the quantities of interest in the reliability analysis, such as the availability/unavailability function, with respect to the changes of uncertain parameters has been presented. This method is originally introduced by Oakkley (2004) to examine sensitivity analysis of a complex model with respect to changes in its inputs based on an emulator which is built to approximate the model. di (2011) showed the adaptation of linear mathematical Pokora diagnostic modeling methodology for setting-up of Linear Fault Tree Sensitivity Model (LFTSM). The sensitivity analysis shows how sensitive the output parameter is while changing in any elements of the input parameters. The LFTSM is a modular approach tool that uses matrix-algebraic method based upon the mathematical diagnostic methodology of aircraft systems and gas turbine engines shown by Pokor adi (2008). The disadvantage of LFTSM is that it cannot depict correctly sensitivity of reliabilities of SwCI. The aims of this theoretical investigation are the followings: to modify the LFTSM to determine sensitivity of reliability of SwCIs; The proposed methods are named Linear Sensitivity Model of System Reliability (LSMoSR) and Linear Sensitivity Model of System Unreliability (LSMoSU). to show the methods generally; to show their possibilities of use to investigate SwCI sensitivity by a simple example; to compare results of these methods. to draw a conclusions from results. The theoretical obtained consequents and experiences can be used to investigate system reliability in a wide context. It is important to mention that, users should use only few equations, not all ones of this paper. The outline of the paper is as follows: Section 2 shows determination of reliability of SwCI using truth table. Section 3 presents the proposed modular approach algorithm for setting-up of Linear Sensitivity Model of System Reliability and Linear Sensitivity Model of System Unreliability theoretically and practically. Section 4 interprets conclusions about proposed methods and sensitivity of investigated system reliability. Section 5 summarizes the paper, outlines the prospective scientific work of the Author.
Fig. 1. Investigated (example) system (source: Myers (2010)).
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2. Reliability of the System with Complex Interconnections To demonstrate the proposed sensitivity determination method, firstly let us determine the reliability parameters of the system shown by Fig. 1. The system shown in Fig. 1 has five elements, A B C D E. Their reliability can be characterized by: reliability (the probability that the element accomplishes its assigned task): ri i2L; probability of failure: pi i2L, where: L e set of Latin characters A B C D E. The elements have two operating and non-operating states. In case of operating state an item is performing a required function. If an item is in non-operating state, it is not performing a required function Sum of probabilities of operating and non-operating should be unity:
ri þ pi ¼ 1;
pi ¼ 1 ri ;
ri ¼ 1 pi :
(1)
For farther application, the reliability and the probability of the failure of the elements should be arranged into vectors:
rT ¼ ½ rA
rB
rC
rE ;
rD
(2)
and
pT ¼ ½ pA
pB
pC
pD
pE :
(3)
One of the approaches to correctly compute the reliability parameters for the SwCI is the summation of the probabilities of all the states of the investigated system. A table listing the probability of each possible state for a system is frequently referred to a truth Table 1 Possible states of the system in Fig. 1. j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Element A
B
C
D
E
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
System state
Qj
0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1
pApBpCpDpE rApBpCpDpE pArBpCpDpE rArBpCpDpE pApBrCpDpE rApBrCpDpE pArBrCpDpE rArBrCpDpE pApBpCrDpE rApBpCrDpE pArBpCrDpE rArBpCrDpE pApBrCrDpE rApBrCrDpE pArBrCrDpE rArBrCrDpE pApBpCpDrE rApBpCpDre pArBpCpDrE rArBpCpDrE pApBrCpDrE rApBrCpDrE pArBrCpDrE rArBrCpDrE pApBpCrDrE rApBpCrDrE pArBpCrDrE rArBpCrDrE pApBrCrDrE rApBrCrDrE pArBrCrDrE rArBrCrDrE
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table. The elements can be operating (designated as the 1 state) or non-operating ones (designated as the 0 state). Therefore a common, n-element system has 2n (in our case: 25 ¼ 32) possible states. The possible system states are summarized in the form of a truth table, shown in Table 1, with each element being assigned either an operating or a non-operating state. The Qj j2 אcolumn lists the probabilities of each of the system states. Since the table covers all of the possible combinations, the sum of all of the state probabilities should be 1. The state probabilities resulting in an operating system are included in the rows 10; 11; 12; 14; 15; 16; 19; 20; 21; 22; 23; 24; 26; 27; 28; 29; 30; 31 and 32.
Rsys ¼ Q10 þ Q11 þ Q12 þ Q14 þ Q15 þ Q16 þ Q19 þ Q20 þ Q21 þ Q22 þ Q23 þ þQ24 þ Q26 þ Q27 þ Q28 þ Q29 þ Q30 þ Q31 þ Q32 (4) The sum of the operating system state probabilities included in this column is the reliability of the system. The system unreliability could be calculated by summing of the 1; 2; 3; 4; 5; 6; 7; 8; 9; 13; 17; 18 and 25 non-operating state (rows) probabilities:
Psys ¼ Q1 þ Q2 þ Q3 þ Q3 þ Q4 þ Q5 þ Q6 þ Q7 þ Q8 þ Q9 þ Q13 þ Q17 þ Q18 þ Q25 : (5) For this particular case, the calculation of the system unreliability involves summing fewer terms (13 system states) than does the calculation of the system reliability (19 system states). This is not generally the case, however. 3. Creating sensitivity model For the purpose of investigation, the sensitivity model of the discussed system should be created. In this section, we demonstrate the method of the modular approach sensitivity model by using the sample SwCI mentioned above (Fig.1.). In our study we will investigate:
Using these coefficients, the relative changing of output parameter y depends on relative changing of elements of the vector x can be determined by equation,
dyj ¼ Kj1 dxj2 þ … þ Kjn dxn :
where: d signs the relative changing of parameters. Next task is to separate the system parameters into vectors y dependent and x independent ones. In our study, the dependent parameters are the probabilistic parameter of system and the possible system states. Independent parameters are the probabilistic parameters of the elements. Then, the connection between the independent and the dependent parameters can be described by equation,
Ady ¼ Bdx;
D ¼ A1 B
system unreliability Psys, depending on probabilities of failures of elements:
(7)
These parameters will be called generally probabilistic parameters in Chapter 3.1, showing the general methodology of proposed method.
relative sensitivity coefficient matrix of the investigated system, the equation
dy ¼ Ddx
(12)
can be used for relative sensitivity investigations. Vector of the relative changing of the independent parameters can be determined by:
dx ¼ X1 nom Dx;
(13)
where: Xnom2
Xnom
x1nom 6 0 ¼6 4 « 0
0 x2nom …
… 1 0
0 0 «
3 7 7: 5
(8)
(14)
xnnom
Dx2
During modular sensitivity analysis firstly the system should be separated into modules. In our case, these modules are the possible system states. Investigating these modules, their (probabilistic) models should be determined by yj ¼ fj(x1,x2,…,xn) fj :
vyj xi vf ðx1 ; x2 ; …xn Þ xi ¼ vxi f ðx1 ; x2 ; …xn Þ vxi yj
(11)
(15)
can be determined. Applying the
3.1. General solution
Kyj ;xi ¼ Kji ¼
D2
2
(6)
Psys ¼ fp ðp Þ;
(10)
where A2
system reliability Rsys, depending on reliability of elements:
Rsys ¼ fr r ;
(9)
Ynom
y1nom 6 0 ¼6 4 « 0
0 y2nom …
… 1 0
0 0 «
3 7 7: 5
(16)
ymnom
nominal value matrix of the dependent parameters Y nom 2
Dy ¼ Y nom dy ¼ Ynom DX1 nom Dx:
(17)
Introducing the so called measured sensitivity coefficient matrix of the investigated system.
di / Journal of Loss Prevention in the Process Industries 32 (2014) 436e442 L. Pokora nm S ¼ Ynom DX1 ; nom 2<
(18)
are inverse ones. If the state of the element is supplemental of the investigated system states, the intrinsic function, seeing equation (1), uj ¼ 1 xj , then sensitivity coefficient is:
(19)
Kji ¼
the equation (17) can be simplified
Dy ¼ SDx:
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Oftentimes, during the system reliability investigation only the probabilistic system parameter is of interesting. But, the other dependent probabilistic parameters can be important too, from technical point of view. If just the probabilistic system parameter and its sensitivities are being investigated only the first row of matrix D and S should be used as vectors d2
n xi Y uk : yj
(22)
k¼1 ksi
In case of functions that determine directly the probabilistic system parameters e see equations (4) and (5) e, the sensitivity coefficients can be determined by:
Qj Qj ; or KRj ¼ : Psys Rsys
3.2. Determination of typical sensitivity coefficients
KPj ¼
The next task is to determine the sensitivity coefficients. The probabilities of system states (see Table 1) can be described by.
Following the general determinations mentioned above we can now set up the linear sensitivity models of system reliability and unreliability.
yj ¼ fj ðx1 ; x2 ; …xk Þ ¼
n Y
ui ðxi Þ;
(20) 3.3. Linear Sensitivity Model of System Reliability (LSMoSR)
i¼1
general equation, where the inner function ui(xi) can have one of the following two forms. The state of element and system state can be the same or its inverse. For example, the 1st row of Table 1, system state and states of all elements are non-operating e the states are the same. If state of element is the same as the investigated system states, the intrinsic function ui ¼ xi, then the sensitivity coefficient is:
Kji ¼ 1;
(23)
(21)
The vector of the independent parameters for the set up of LSMoSR is described by equation (2) and their relative changing vector is:
dxT ¼ ½ drA
drB
drC
drD
drE ;
and the vector of the relative changing of the dependent parameters by equation (4):
dyT ¼ dRsys dQ10 dQ11 dQ12 dQ14 dQ15 dQ16 dQ19 dQ20 dQ21 dQ22 dQ23 dQ24 dQ26 dQ27 dQ28 dQ29 dQ30 dQ31 dQ32
On the other hand, in the 2nd row of Table 1, the state of element A is operating, but system state is non-operating e these two states
(24)
(25)
The coefficient matrix of the dependent parameters:
2
3 1 KR10 KR11 KR12 KR14 KR15 KR16 KR19 KR20 KR21 KR22 KR26 KR27 KR28 KR29 K KR30 KR24 KR23 KR31 KR32 60 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 7 A¼ 6 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 7 6 7 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (26)
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the coefficient matrix of independent parameters:
2
0 6 1 6 6 K11A 6 6 1 6 6 1 6 6 K15A 6 6 1 6 6 K19A 6 6 1 6 6 K21A B¼6 6 1 6 6 K23A 6 6 1 6 6 1 6 6K 6 27A 6 1 6 6K 6 29A 6 1 6 4K 31A 1
0
0
K10B 1 1 K14B 1 1 1 1 K21B K22B 1 1 K26B 1 1 K29B K30B 1 1
K10C K11C K12C 1 1 1 K19C K20C 1 1 1 1 K26C K27C K28C 1 1 1 1
0 1 1 1 1 1 1
0
K19D K20D K21D K22D K23D K24D 1 1 1 1 1 1 1
The coefficient matrix of independent parameters:
3
2
K10E 7 7 K11E 7 7 K12E 7 7 K14E 7 7 K15E 7 7 K16E 7 7 1 7 7 1 7 7 1 7 7; 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 7 7 1 5 1
(27)
0 6 1 6 6 K2A 6 6 1 6 6 K4A 6 6 1 6 6 K6A B¼6 6 1 6 6 K8A 6 6 1 6 6 1 6 6 1 6 4K
18A
1
The relative and measured sensitivity coefficient vectors of system reliability in cases of different rinom nominal values of element reliabilities are shown by Table 2. Figs. 2 and 4 show the sensitivities of system reliability.
0 1 1 K3B K4B 1 1 K7B K8B 1 1 1 1 1
0 1 1 1 1 K5C K6C K7C K8C 1 K13C 1 1 1
0 1 1 1 1 1 1 1 1 K9D K13D 1 1 K25D
0 1 1 1 1 1 1 1 1 1 1
3
7 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 K17E 7 7 5 K
(31)
18E
K25E
The sensitivity coefficient vectors of system unreliability in cases of different pinom nominal values of element probabilities are shown in Table 3 and Figs. 3 and 4. During the application of the proposed method experts should determine truth table of investigated system, sensitivity coefficients using equations (21)e(23), and nominal value and coefficient matrices of independent and dependent parameters. Then, the relative and measured sensitivities of system reliability can be determined using equations (11) and (18). Knowing them, the optimal decision can be made for maximization of system reliability id. est. minimization of unreliability.
3.4. Linear Sensitivity Model of System Unreliability (LSMoSU) The vector of independent parameters for the set up of LSMoSU and their relative change vector is described by equation (3):
dxT ¼ ½ dpA
dpB
dpC
dpD
dpE ;
(28)
and the vector of the relative change of the dependent parameters by equation (5):
dyT ¼ dPsys
dQ1
dQ2
dQ3
dQ4
dQ5
dQ6
dQ7
dQ8
dQ9
4. Discussions of results The following conclusions can be deduced from the results of modeling and analysis:
4.1. Conclusions about the proposed method
dQ13
dQ17
dQ18
dQ25 ;
(29)
The coefficient matrix of dependent parameters:
2
1 60 6 60 6 60 6 60 6 60 6 60 A¼6 60 6 60 6 60 6 60 6 60 6 40 0
KP1 1 0 0 0 0 0 0 0 0 0 0 0 0
KP2 0 1 0 0 0 0 0 0 0 0 0 0 0
KP3 0 0 1 0 0 0 0 0 0 0 0 0 0
KP4 0 0 0 1 0 0 0 0 0 0 0 0 0
KP5 0 0 0 0 1 0 0 0 0 0 0 0 0
KP6 0 0 0 0 0 1 0 0 0 0 0 0 0
KP7 0 0 0 0 0 0 1 0 0 0 0 0 0
KP8 0 0 0 0 0 0 0 1 0 0 0 0 0
KP90 0 0 0 0 0 0 0 0 1 0 0 0 0
KP13 0 0 0 0 0 0 0 0 0 1 0 0 0
KP17 0 0 0 0 0 0 0 0 0 0 1 0 0
KP18 0 0 0 0 0 0 0 0 0 0 0 1 0
3 KP25 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 5 1
(30)
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Table 2 Sensitivity vectors of system reliability. rinom
dpinom
0.99
d0;99 ¼ ½ 0; 000197 s0;99 ¼ ½ 0; 000197 d0;05 ¼ ½ 0; 004631 s0;95 ¼ ½ 0; 004631 d0;9 ¼ ½ 0; 017100 s0;9 ¼ ½ 0; 017100
0.95 0.90
0; 000296 0; 000296 0; 007006 0; 007006 0; 026100 0; 026100
0; 000197 0; 000197 0; 004631 0; 004631 0; 017100 0; 017100
0; 010097 0; 010097 0; 052131 0; 052131 0; 107100 0; 107100
0; 010097 0; 010097 0; 052131 0; 052131 0; 107100 0; 107100
Fig. 4. Measured sensitivity of system.
1-4. From the two methods described above, the one which has the lower number of possible system states should be chosen. 1-5. Sensitivity parameters of the possible system states can be used in cases with separate professional meaning (for example road traffic or logistical management). In this case, the Conclusion 1-4 has to be taken into account from professional points of view.
Fig. 2. Relative sensitivity of system reliability.
1-1. The proposed method can be used to analyze sensitivity of the reliability of the SwCI. 1-2. Elements of coefficient matrices can be easily determined. 1-3. The information from the analysis of reliability and unreliability of the SwCI depicts the system sensitivity in the same way.
dpinom
0.01
d0;01 ¼ ½ 0; 019135 s0;01 ¼ ½ 0; 000197 d0;05 ¼ ½ 0; 081241 s0;05 ¼ ½ 0; 004631 d0;1 ¼ ½ 0; 135607 s0;1 ¼ ½ 0; 017100
0.05 0.1
0; 028750 0; 000296 0; 122903 0; 007006 0; 206979 0; 026100
4.2. Conclusions about the investigated system 2-1. Increasing the reliability of elements, namely a reduction their failure probabilities, reduce the sensitivity of the system (see Figs. 5 and 6).
Table 3 Sensitivity vectors of system unreliability. pinom
1-6. The disadvantage of the proposed method is that the number of possible states of the system grows by a exponential function depending on the number of elements.
0; 019135 0; 000197 0; 081241 0; 004631 0; 135607 0; 017100
0; 980672 0; 010097 0; 914483 0; 052131 0; 849326 0; 107100
Fig. 3. Relative sensitivity of system unreliability.
0; 980672 0; 010097 0; 914483 0; 052131 0; 849326 0; 107100
In each modeled case the system had the same degree of sensitivity to reliability parameters of the A and C as well as D and E components. This conclusion can be deducted from the symmetric physical structure of system. (In all modeled situation the reliability parameters of elements were equivalent. In case of different element parameters this conclusion can be false.) 2-2. The sensitivity of the system is “symmetrical”. 2-3. The system has more sensitivity to reliability parameters of “second line” D and E elements.
Fig. 5. Relative Sensitivities of the System Reliability depend on Reliabilities of Elements.
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decision making includes the study of methodologies of complex system uncertainty using other mathematical tools, for example linear interval equations, Monte-Carlo Simulation on basis of Linear Sensitivity Model of System Reliability (LSMoSR). Acknowledgment This work has been supported by the visiting scholarship of Center for Complex Networks and Systems Research (CNetS), Indiana University Bloomington, which is gratefully acknowledged. References
Fig. 6. Relative Sensitivities of the System depend on Elements Probability Failures.
2-4. In the case of the “first line” elements the system has more sensitivity to reliability parameters of B. 5. Summary, future work This paper discussed a new sensitivity investigation method of reliability of SwCI elaborated by the Author. The paper showed the adaptation of the linear mathematical diagnostic modeling methodology to sett-up the Linear Sensitivity Model of System Reliability (LSMoSR) and Linear Sensitivity Model of System Unreliability (LSMoSU). The models are analogue modular approach tools that use matrix-algebraic method based upon the mathematical diagnostics methodology of aircraft systems and gas turbine engines. In this paper their possibility of use was demonstrated to investigate the SwCI sensitivity by a simply example. Using demonstrated mathematical connections and procedure, the technical experts can get an easy-used methodology to build up sensitivity model of the given SwCI. These LSMoSR and LSMoSU models can be used to investigate system reliability and dependability in a wide context. The Author's proposed prospective scientific research related to this field of applied mathematics and maintenance management
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