Open access peer-reviewed chapter

Complex System Reliability Analysis Method: Goal‐Oriented Methodology

By Yi Xiao‐Jian, Shi Jian and Hou Peng

Submitted: November 26th 2016Reviewed: May 5th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.69610

Downloaded: 483

Abstract

Goal‐oriented (GO) methodology is a success‐oriented method for complex system reliability analysis based on modeling the normal operating sequence of a system and all possible system states. Recently, GO method has been applied in reliability and safety analysis of a number of systems, spanning defense, transportation, and power systems. This chapter provides a new approach for reliability analysis of complex systems, first, by providing its development history, its engineering applications, and the future directions. Then, the basic theory of GO method is expounded. Finally, the comparison of GO method, fault tree analysis and Monte‐Carlo simulation is discussed.

Keywords

  • system reliability analysis
  • complex system
  • GO method
  • reliability model

1. Introduction

Quality and reliability are key attributes of economic success of a system because they result in an increase in productivity at low cost and vital for business growth and enhanced competitive position. The recent advances in electronics, computing, communication, control, and networking have resulted in integrated systems that are: (i) complex in structure, (ii) large in scope and scale, (iii) characterized by multimode operation, (iii) capable of working in varied working conditions, and (iv) hierarchically organized. The reliability of such complex systems is a critical factor of their fitness for their intended use and hence is vital for in design and manufacturing. Reliability analysis method of complex systems is conducive to prevent defects in the first place in all aspects when they do occur in operation, in order to improve their reliability and reduce their life‐cycle cost.

Fault tree analysis (FTA) and Monte‐Carlo simulation (MCS) are now the standard reliability and safety analysis methods. Different from them, goal‐oriented (GO) methodology [1] is a success‐oriented method for system reliability analysis based on modeling the normal operating sequence of a system and all possible system states. It is especially suitable for complex systems having time‐sequence of operation, multiple states, and so on, and the quantitative analysis and the qualitative analysis of the GO method are conducted by the GO operation according to the GO model. The keys of the GO method are the GO model and the GO operation. Although the GO method was introduced in 1976 [1], it was largely unknown until recently. The GO method has become increasingly popular in recent years because of its advantages in terms of its ease of creating a model and of its representational and analysis power [2, 3]. This chapter is an attempt at providing a basic theory of GO method in terms of GO model, GO operation, and comparison with FTA and MCS.

1.1. Development of GO method

The major application of GO method is in establishing a system reliability model, and its quantitative and qualitative analysis. The chronological development of the GO method can be broadly divided into two periods of growth: (1) 1970s∼the early 1990s, the basic model and theory of the GO method, its comparison with FTA method, and its GO operator type and function were explained in the research reports by the Electric Power Research Institute in US [48]; (2) After the late 1990s, the GO method has attracted more attention again, in particular in the People’s Republic of China. Perhaps, the initial application of the GO method was in reliability and safety analysis of missile and weapons systems. Recently, the GO method has also been applied in the reliability analysis of defense systems, water, oil, and gas supply systems, manufacturing systems, transportation systems, power systems, and logistics management systems [2, 3]. Furthermore, the theory of GO method for a complex system with complex correlations (dependencies), closed‐loop feedback, multiple functions, multiple fault modes, etc., has been developed from three aspects, which are GO model, basic GO algorithm, and GO method for complex systems with various characteristics [917]. Literature [2, 3] gave the overview of development of GO method in detail.

1.2. Further of GO method

The GO method has been widely used. There are a number of areas where the powerful advantages of the GO model and its reliability analysis methodology can be exploited.

Although there is a great deal of interest in the design of complex systems for reliability, research on deigning these systems for both functional and structural reliability and life‐cycle cost is needed. To go further, there is still a substantial research gap in the optimal design of systems for reliability and life‐cycle cost, taking into account the issues of structure, function, behavior, and other characteristics, such as active or cold standby redundancy, and fault‐tolerant mechanisms. In terms of modeling, analysis, and software tools, there are a number of research issues to consider. These include: (i) How to integrate the product structure, behavior, and functions in a reliability model; (ii) How to conduct the reliability analysis of complex systems accurately, thoroughly, and quickly; (iii) How to optimally allocate reliability among subsystems, taking into consideration the structural and functional hierarchy, as well as redundancy management techniques under resource constraints; and (iv) How to develop software tools to support design of complex systems for reliability that are intuitive and support collaborative design. Considering the advantages of GO model and its reliability analysis method, the GO method not only can solve the existing problems above, but also it can further develop system reliability theory and application. Meanwhile, the corresponding software will also have a more extensive application prospects and important value.

Another important direction for research is the application of the GO methodology to related areas of quality control, fault diagnosis and prognosis, and condition‐based maintenance.

2. GO model

The GO model is a key element of GO method. It is developed directly using product schematic diagrams, its structure, and its functional hierarchy. According to the GO model, the reliability analysis is conducted by GO operation. The GO model composed of GO operator and signal flow.

2.1. GO operator

GO operator contains function operator and logical operator, which represents the unit itself or logical relationship, respectively. Its data, type, and GO operation formula are the basic attributes of GO operator. There are 17 standard GO operators in basic GO theory, and their signs and description are shown in Figure 1 and Table 1, respectively. In Figure 1, S, C, and R are the input signal, GO operator itself, and output signal, respectively.

Figure 1.

Signs of standard GO operators and three developed GO operators.

Operator typeFunctional description
Type 1Unit with failure state and operating state
Type 2Logical relation “OR”
Type 3Unit with failure state, operating state, and operating ahead state
Type 4Multi‐signal input unit
Type 5Single input unit
Type 6Unit receiving signal to turn on
Type 7Unit receiving signal to turn off
Type 8Unit with delayed response
Type 9Output signal decided by the state of two input signal
Type 10Logical relation “AND”
Type 11Logical relation “k‐out‐of‐m”
Type 12Input signal can choose the output path
Type 13Unit with multiple input signals and output signals
Type 14Linear relation between multiple input signals and one output signal
Type 15Logical relation of output signal affected by the probability event of input signal
Type 16Unit requested to resume OFF‐state
Type 17Unit requested to resume ON‐state

Table 1.

Functional description of standard GO operators.

2.1.1. Standard GO operator

In this section, six frequently‐used standard GO operators are illustrated from aspects of description, operation rule table, and GO operation formula, respectively. In operation rule table, VS, VC, and VR are the state of the input signal, GO operator itself, and output signal, respectively, and 0,,Nrepresent their state values. In GO operation formula, PS(i), PR(i), and PC(i)are the state probability of the input signal, GO operator itself, and output signal, respectively. The state cumulative probability of input signal AS(i)and output signal AR(i)are defined as

{AS(i)=j=0iPS(j)i=0,,N1;AS(N)=1AR(i)=j=0iPR(j)i=0,,N1;AR(N)=1E1

1. Type 1 operator

  • Description: It describes the unit with two states, which are success state (enable signal flow pass) and failure state (stop signal flow passing). For example, electric resistance, switch, valve, and pipeline.

  • Operation rule table:

    VSVCVR
    0, …, N−110, …, N−1
    N1N
    0, …, N2N

  • GO operation formula:

{PR(i)=PS(i)PC(1),i=0,1,,N1PR(N)=PS(N)PC(1)+PC(2)E2

2. Type 2 operator

  • Description: It describes the logical relationship OR among some inputs signal and one output signal.

  • Operation rule table:

    VS1VS2VSMVR
    0, …, N0, …, N0, …, N0, …, NMIN{VS1, VS1,…, VSM}

  • GO operation formula:

{PR(0)=AR(0)    PR(i)=AR(i)AR(i1),i=1,,NE3

3. Type 3 operator

  • Description: It describes unit with failure state, operating state, and operating ahead state. For example, control system, contactor coil, and so on.

  • Operation rule table:

    VSVCVR
    0, …, N00
    0, …, N−110, …, N−1
    N1N
    0, …, N2N

  • GO operation formula:

{PR(0)=PC(0)+PS(0)PC(1)    PR(i)=PS(i)PC(1),i=1,,N1PR(N)=PS(N)PC(1)+PC(2)E4

4. Type 5 operator

  • Description: It describes the single input unit, which as system input. For example, battery, water source, and so on.

  • GO operation formula:

PR(i)={Pj,i=Ij0,iIj,j=1,,L,i=0,,NE5

5. Type 6 operator

  • Description: It describes the unit receiving signal to turn on. For example, electric water pump, contactor, and so on.

  • Operation rule table:

    VS1VS2VCVR
    I1(0, …, N)I2(0, …, N)0I1
    I1(0, …, N)I2(0, …, N)1MAX{I1, I2}
    I1(0, …, N)I2(0, …, N)2N

  • GO operation formula:

{AR(i)=AS1(i)[PC(0)+AS2(i)PC(1),i=0,,N1    PR(N)=PS1(N)+AS1(N1)[PC(2)+PS2(N)PC(1)]PR(0)=AR(0)PR(i)=AR(i)AR(i1)i=1,,N1E6

6. Type 10 operator

  • Description: It describes the logical relationship AND among some inputs signal and one output signal.

  • Operation rule table:

    VS1VS2VSMVR
    0, …, N0, …, N0, …, N0, …, NMAX{VS1, VS1,…, VSM}

  • GO operation formula:

{AR(i)=j=1MASj(i), i=0,,N1    PR(N)=1AR(N1)PR(0)=AR(0)PR(i)=AR(i)AR(i1)i=1,,N1E7

2.1.2. Developed GO operator

In this section, three developed GO operators are illustrated from aspects of description, operation rule table, and GO operation formula, respectively.

1. Type 18 operator

  • Description: It describes the logical relation of standby mode, which is combination of primary equipment group CG and standby equipment group CBG working under condition of primary equipment group faulting. The input signals and output signal of Type 18 operator are denoted as L1, L2, and R, respectively. The signal flow L1 represents primary equipment group working. The signal flow L2 represents standby equipment group working under condition of primary equipment group faulting, and the signal flow L2 is also the output signal of GO operator, which represents the standby equipment group. And the signal flow R represents standby structure working. L1, L2, and R have two states, which are state 1: success state and state 2: fault state.

  • Operation rule table: The combination composed of a Type 18 operator and a Type 20 operator is often used to represent standby structure at any place. S1, S2, and S0 represent the input signal flows and output signal flow of Type 20 operator, respectively, and S2represents also the input signal flow of Type 18 operator, i.e., L1.

    VCGVCBGVS1VS0VS2VL2VR
    1112121
    1122222
    1212121
    1222222
    2111211
    2122222
    2211222
    2222222

  • GO operation formula:

{PR(1)=PL1(1)+PL2(1)PR(2)=1PR(1)PL1(1)=PS1(1)PCG(1)=PS1(1)PS0(1)PL2(1)=PS0(1)PCBG(1)E8

where PS0(1)and PCBG(1)are the success probability of S0 and CBG, respectively.

2. Type 19 operator

  • Description: It describes the unit turning unstable operation into normal operating. The input signals and output signals of Type 19 operator are denoted as S and R, respectively. S is a multistate signal flow, which contains an operating state, a faulting state, and m unstable operation states. And m unstable operation states are divided into two kinds, which are q unstable operation states turned into operation state by Type 19 operator, and m‐q unstable operation states. R is also a two‐state signal flow, which contains operating state and faulting state. C is the unit itself, i.e., Type 19 operator, which contains operating state and faulting state.

  • Operation rule table:

    VSVCVR
    000
    0NN
    1, …, q00
    1, …, qNN
    q+1,…, m0, NN
    N0, NN

  • GO operation formula:

{PR(0)=PS(0)PC(0)+j=1qPS(j)PC(0)=j=0qPS(j)PC(0)PR(N)=PS(0)PC(N)+j=1qPS(j)PC(N)+j=q+1mPS(j)PC(0)+j=q+1mPS(j)PC(N)+PS(N)PC(0)+PS(N)PC(N)=j=0qPS(j)PC(N)+j=q+1mPS(j)+PS(N)E9

where PS(j)is state probability of unstable operation states for S, j=1,2,,m, PS(0)and PS(N)are state probability of operating state and faulting state for S, respectively, PS(0)+PS(N)+j=1qPS(j)+j=q+1mPS(j)=1; PC(0)and PC(N)are state probability of operating state and faulting state for C, PC(0)+PC(N)=1.

3. Type 20 operator

  • Description: It describes the signal flow of conditional operating mode, which is one of its input signal faults under the condition of another input signal success. The input signals and output signal of Type 20 operator are denoted as S1, S2, and R, respectively. S1, S2, and R have two states, which are state 1: success state and state 2: fault state. It is often used in standby structure.

  • Operation rule table:

    VS1VCGVS2VR
    1112
    1221
    21, 222

  • GO operation formula:

{PR(1)=PS1(1)[1PCG(1)]=PS1(1)PS2(1)PR(2)=1PR(1)E10

where PSα(1), PCG(1), and PR(1)are the success probability of Sα, CG, and R, respectively, α=1,2; PR(2)is the fault probability of R.

2.2. Signal flow

Signal flow represents specific fluid flow, such as oil, gas, electricity, and so on, or a logical process. It describes the relationships among the GO operator, its inputs, and outputs. And its attribution includes state value and state probability. The signal flow is used to connect the GO operator and as the direction of GO operation.

2.3. GO model

GO model is developed by using signal flows to connect GO operators according to the system principle diagram, engineering drawing, and function constitute directly. And it reflects the system characteristics visually. The proper GO model should be satisfied with:

  • The operator in GO model must be labeled with type and number, especially the number which is unique. The first number in the GO operators represents the type of operator and the second number represents the numbering of the operators.

  • There is at least one input GO operator (such as Type 4 or Type 5). Generally, the number of GO operator begins with the input GO operator.

  • For each GO operator in GO model, its input signal flow must be the output signal flow of the other GO operators. Each signal flow must be labeled with the unique number, and the numbers on the signal line represent the signal flow numbering.

  • The signal flow sequence must start with any input GO operator and end with the output signal flow of system. The GO model does not allow being a cyclic model. Generally, the number of signal flow should be labeled with the output signal flow of an input GO operator.

Example:

The structure diagram of pressurized water reactor (PWR) purification system and its GO model are, respectively, shown in Figures 2 and 3, and the GO operator type of component is presented in Table 2.

Figure 2.

Structure diagram of PWR purification system.

Figure 3.

GO model of PWR purification system.

Operator numberComponentOperator type
1Coolant input5
2, 3, 4, 5, 6, 7Control signal5
8Control power5
9Control valve A6
10Clean‐up pump A6
11Check valve A1
12Clean‐up pump B6
13Check valve B1
15Control valve B6
16Control valve C6
17Regenerative heat exchanger1
18Nonregenerative heat exchanger1
19Control valve D6
20Control valve E7
21Ion exchange1
22Filter1
24Control valve F6

Table 2.

GO operator type of component for Figure 2.

3. GO operation

GO operation is defined that it begins with output signal flow of input GO operators in GO model to calculate the state probability and state value of output signal flow for the next GO operator, and it will be finished until the system output signal flow is calculated after the sequence of signal flow. GO operation contains the quantitative and qualitative analyses. And the quantitative and qualitative analyses are conducted using the GO operation based on the GO algorithm, following the reliability analysis process of GO method. GO algorithm and GO analysis process are the key elements of GO operation.

3.1. GO algorithm

The operational efficiency and accuracy of analysis are thus affected by the GO algorithm. The GO algorithm is comprised of a state combination algorithm [4] and a probability formula algorithm [1821]. The number of state combinations for a complex system is very large, and the probability of a combination state cannot be easily computed. The probability formula algorithm is faster and easier than the state combination algorithm, and it is the mainstream GO algorithm. Thus, this section illustrates two kinds of probability formula algorithms, which are direct GO algorithm and GO algorithm with shared signals.

3.1.1. Direct GO algorithm

Direct GO algorithm is based on the calculation of state probability for signal flow, and the proper direct GO algorithm should be satisfied with:

  • The state probability of input GO operator is the state probability of its output signal flow. The output signal flow of input GO operator is the input signal flow of the next GO operator.

  • The state probability of output signal flow for the next GO operator is calculated based on its GO operation formula and data, and this output signal flow will be the next GO operator’s input signal flow.

  • Based on the above rules, following the signal flow sequence, the output signal flow of every GO operator in GO model can be obtained, and the GO operation will be finished until the state probability of output signal flow represented system output is calculated.

  • When GO operator is executed with quantitative analysis, it is not necessary to list the corresponding state combination.

3.1.2. GO algorithm with shared signals

In GO model, the output signal flow of a GO operator often connects multiple GO operators, and they are the input signal flows of more than one GO operators, such output signal flow is called shared signal. If the GO operation adopts the direct GO algorithm, the quantitative analysis results will have biases. Thus, GO algorithm with shared signals was proposed in order to obtain more accurate result.

1. Shared signal

There are two situations in GO model on shared signal, which are as follows:

  • Completely contain. It means all items in formula of a signal flow’s state probability have state probability of shared signal.

  • Not completely contain. It means some items in formula of a signal flow’s state probability have not state probability of shared signal.

The rules of the processing shared signals in GO algorithm with shared signals are as follows:

  • Behind a shared signal, all of signal flows in sequence of GO model still have this shared signal, and there are two situations, which are completely contain and not completely contain.

  • For multiple signal flows with the same shared signal, their joint state probability cannot be obtained by using the product of their state probabilities directly.

  • For two signal flows completely contain the same one shared signal, their joint state probability can be obtained by using the product of their state probabilities to divide the state probability of shared signal.

  • For multiple signal flows with the same one shared signal, their joint state probability formula can be obtained by turning a high stage term of shared signal in the formula into a one stage term of shared signal.

2. Probability formula

For a system with L shared signals, the probability formula of GO algorithm with shared signals is given by Eq. (10).

PR=K1=01K2=01Kl=01PRK1K2Kll=1L[(1PSl)(1Kl)+PSlKl]E11

where PRK1K2KLis the cumulative probability of system output under a combination of all shared signals, and Slis a shared signal in the system. Kl=0and Kl=1are the failure and success states of the shared signal l, respectively. PSland PRare success probability of shared signal l and system output, respectively. The item of l=1L[(1PSl)(1Kl)+PSlKl]is the state probability for each combination of shared signals.

3. Calculating form

It is difficult and complex to derive mathematical formulae for a complex system with a large number of shared signals. A new form of Eq. (10) involves probabilistic weighting of shared signals. The probabilistic weighting improves operation efficiency greatly and avoids the need for complex mathematical formulae. The calculation process is shown in Table 3. In Table 3, numbers 1 and 0 represent success state and failure state of a shared signal Sl, respectively.

State combination of shared signalState probability of combinationSuccess probability of system
S1S2SL
0000A1B1
0001A2B2
1111A2LB2L
Success probability of systemPR

Table 3.

Calculation form of GO algorithm with shared signal.

The success probability of system can be obtained by Eq. (11).

PR=j=12LAjBjE12

where Aj is the state probability for each combination of shared signals, and Bj can be obtained by the GO operation of the system, which sets success and failure probability of a shared signal to 1 and 0 depending on the state of shared signal for each combination of shared signals.

3.2. Reliability analysis process of GO method

The reliability analysis process of GO method is the criterion and prerequisite for conducting quantitative analysis and qualitative analysis. Generally, the steps of GO analysis process are as follows:

  1. Step 1. Conducting system analysis. The system analysis is the base of GO method, and it directly affects the developing GO model and conducting GO operation. First, to analyze system structure and system function constitutes according to the principle diagram, engineering drawing, or function flowchart of system. Second, to determine system characteristics, such as correlations, multistate, and so on. Then, to determine the interfaces, inputs, and output of system. Finally, to define the success rule of system according to system analysis result.

  2. Step 2. Developing GO model. First, to select GO operator according to system analysis results, and then to establish GO model through the signal flow to connect GO operator.

  3. Step 3. Processing data of GO operator. According to engineering practice, to obtain the state probabilities of GO operator.

  4. Step 4. Operating quantitative analysis. If the GO model does not contain shared signal, the direct algorithm can be selected to conduct GO operation. If the GO model contains shared signal, the GO algorithm with shared signal should be selected to conduct GO operation. Otherwise, it will cause a big error.

  5. Step 5. Operating qualitative analysis. Setting the reliability of a function GO operator in GO model is 0, and the reliabilities of other GO operators are kept constant; in this case, if system reliability is 0 by GO operation, this GO operator will be a one‐order minimum cut set. Setting the reliabilities of two function GO operators in GO model is 0 except one‐order minimum cut set, and the reliabilities of other GO operators are kept constant; in this case, if system reliability is 0 by GO operation, the two GO operators will be a two‐order minimum cut set. In the same way, the higher‐order minimum cut sets of system can be obtained.

  6. Step 6. Evaluating system. The quantitative analysis result and qualitative analysis result can be used as a guidance and theoretical basis for improving system and fault diagnosis of system, and so on.

Above all, the reliability analysis of GO method is formulated, as shown in Figure 4.

Figure 4.

Reliability analysis process of GO method.

4. Example

In this section, taken a hydraulic oil supply system (HOSS) of an armored vehicle as a case, its reliability analysis is respectively conducted by GO method, FTA, and MCS in order to illustrate the usage and characteristics of GO method.

4.1. Reliability analysis of HOSS based on GO method, FTA, and MCS

4.1.1. Reliability analysis of HOSS based on GO method

1. Conducting system analysis

  • To analyze system principle, function, and structure: HOSS supplies oil for an armored vehicle, pump‐motor system, pump‐motor control system, hydraulic torque converter, and lubrication system. HOSS consists of pressure oil tank, pump P1, P2, P3, and P4, oil filter LF1, LF2, and LF3, pressure relay, bypass valve LF2B and LF3B, one‐way valve CV1 and CV2, constant pressure valve RV1, RV2, and RV3, hydraulic torque converter TC, radiator HE, and so on, as shown in Figure 5. Oil is extracted by P1 from oil pan via LF1, and then oil is injected into pressure oil tank via LF2 and case inner passage. When LF2 group is obstructed and pressure between input and output becomes more than 0.5 mega Pascal, oil will be injected into pressure oil tank via LF2B. Oil is extracted by P2 from pressure oil tank; then oil is injected into CV2 via LF3 and then it is injected into hydraulic manifold block as the pressure oil provided for oil cylinder of variable speed control system and pump‐motor control system by P4. LF3 and LF3B are another parallel structure, and they are same as LF2 group and LF2B. Because of the pressure of control oil which decreases a little at the situation of high speed, ingress oil of P2 can meet requirements of system. In addition, oil is extracted by P3 from pressure oil tank via DRV to TC. Then, ingress of oil is injected into lubrication system via HE. TC and TCB, and HE and HEB are same as LF2 group and LF2B. RV1, RV2, and RV3 are constant pressure valves of variable speed control and pump‐motor system, lubrication system, and pump‐motor control system, respectively.

  • To determine characteristics of system: According to analysis of HOSS, the LF2 group and LF2B, LF3 and LF3B, TC and TCB, and HE and HEB are standby structures in HOSS. The standby equipments haven't the change-over switch.

  • To determine interfaces, input, and output of system: According to analysis of HOSS, the oil from oil pan and pump group power are system inputs, and oil supply of variable speed control system, pump‐motor system, pump‐motor control system, hydraulic torque converter, and lubrication system are system outputs.

  • To define system success rule: According to analysis of HOSS, success rule can be defined as a system that can provide oil to variable speed control system, pump‐motor system, pump‐motor control system, hydraulic torque converter, and lubrication system of an armored vehicle under high speed condition at steering situation without considering overload protection.

Figure 5.

Diagram of HOSS.

2. Developing GO model

  • To select GO operator: According to the system analysis result and the types of GO operator, the functional GO operators and logical GO operators are selected to describe the units itself and logical relationships in HOSS, respectively, as presented in Table 4.

  • To establish GO model: According to diagram of HOSS and analysis result of HOSS, the GO model of HOSS is developed from system input to system output, as shown in Figure 6.

Figure 6.

GO model of HOSS.

No.ComponentOperator typeNo.ComponentOperator type
1Oil pan524RV31
2, 3LF1125P36
5, 6P1626DRV1
7Pump group power528TC1
9, 10LF2130TCB1
13LF2B132HE1
15Pressure oil tank134HEB1
16P2636RV21
17LF314, 8, 11, 272
18CV213710
20LF3B114, 21,31, 3518A
22RV1112, 19, 29, 3320
23P46

Table 4.

GO operator in GO model of HOSS.

3. Processing data of GO operator

According to statistical results of data from engineering, the success state probabilities of component in HOSS are presented in Table 5.

No.Success state probabilityNo.Success state probabilityNo.Success state probability
10.99975006256130.99867457268240.99949819446
20.99413847887150.99990821563250.99867457268
30.99413847887160.99950052524260.99923302636
50.99950052524170.99987532520280.97865595028
60.99950052524180.99891901058300.99867457268
70.98676571976200.99867457268320.99548611722
90.99987532520220.99949819446340.99966858815
100.99987532520230.99950052524360.99949819446

Table 5.

GO model of HOSS.

4. Operating quantitative analysis

It is shown in Figure 6 that signal flow S1, S4, S7, S8, S15, S16, S17, and S22 are shared signals; the GO algorithm with shared signals should be adopted to conduct GO operation, and the calculation form is as presented in Table 6.

Figure 7.

Brief fault tree model of HOSS.

State of shared signalState probability of combinationSuccess probability of system
S1S4S7S8S15S16S17S22
000000008.154e‐320
000000011.620e‐280
000000111.290e‐240
0
111111110.985282728610.99846792597
Success probability of HOSS (P37)0.9837732025

Table 6.

Quantitative analysis result by calculation form of GO algorithm with shared signal for Figure 6.

5. Operating qualitative analysis

According to the Step 5 in Section 3.2, all minimum cut sets of HOSS can be obtained by multiple GO operations, as presented in Table 7.

OrderNo.Minimum cut setsOrderNo.Minimum cut sets
11Oil pan22, 3LF1 group
7Pump group power5, 6P1 group
15Pressure oil tank18, 20CV2, LF3B
16P228, 30TC, TCB
22RV132, 34HE, HEB
23P439, 10, 13LF2 group, LF2B
24RV3
36RV2
17LF3

Table 7.

Qualitative analysis result by GO method for Figure 6.

4.1.2. Reliability analysis of HOSS based on FTA and MCS

1. Reliability analysis of HOSS based on FTA

The reliability analysis process of FTA mainly contains developing fault tree model of system, obtaining all minimum cut sets of system by using Fussell‐Vesely method, and obtaining the system success probability according to the minimum cut sets of system. In this case, the brief fault tree model of HOSS is shown in Figure 7, and the brief analysis processes are presented in Table 8.

Brief analysis processesMinimum cut setsOrder
A0A1B11111
C22, 32, 32
C39, 10, 139, 10, 133
1515151
C45, 65, 62
7771
161616161
77771
B2C1, 2017, 20
18, 2018, 202
222222221
A2B1B1B1
161616161
77771
171717171
A3A2A2A2
232323231
77771
242424241
A4A1, B3A1, B1A1, B1
A1, 25A1, 25
A1, 7A1, 7
A1, 26A1, 26
A5A4A4A4
B428, 3028, 3028, 302
B532, 3432, 3432, 342
363636361

Table 8.

Brief analysis processes of FTA for HOSS.

2. Reliability analysis of HOSS based on MCS

The reliability analysis process of MCS mainly contains generating random numbers of success probability of GO operators, establishing simulation model, and obtaining success probability of system by operating a specified number of simulation times according to simulation model. In this case, the quantitative analysis result of HOSS by simulating 1 million times is 0.9838136000.

4.2. Comparison with FTA and MCS

According to reliability analysis process and analysis results of GO method, FTA, and MCS, the qualitative analysis results by the GO method are consistent with FTA, and the quantitative analysis result by the GO method is very close to the result by MCS. It illustrates that both the accurate quantitative analysis result and qualitative analysis result can be obtained by multiple GO operations based on GO method, and the comparisons of GO method, FTA, and MCS are presented in Table 9.

FeatureGOFTAMCS
Modeling orientedSuccessFailureLogical
Modeling methodDecision‐making treeFault treeLogical language description
Model consistencyHighPoorHigh
Model structureSimilar with schematic diagramHierarchy logical graphLogical relationship
Model sizeSmall and compactHierarchy and largeLarge
Model elementComponents and logical gate, characteristicsFailure event and logical gateComponents and logical language
Model descriptionReflect system naturalReflect cause and effect of failureReflect logical relation in system
Model signVarious types and typicalFew types and not typicalFew types and not typical
Quantitative analysisAccurate, stableApproximativeAccurate, unstable
Qualitative analysisEasyComplex

Table 9.

Comparisons of GO method, FTA, and MCS.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Yi Xiao‐Jian, Shi Jian and Hou Peng (December 20th 2017). Complex System Reliability Analysis Method: Goal‐Oriented Methodology, System Reliability, Constantin Volosencu, IntechOpen, DOI: 10.5772/intechopen.69610. Available from:

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