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Importance analysis deals with the investigation of influence of individual system component on system operation. This paper mainly focuses on dynamic important analysis of components in a multistate system. Assuming that failure probabilities of system components are independent, a new time integral-based importance measure approach (TIIM) is proposed to measure the loss of system performance that is caused by each individual component. Reversely the importance of a component can be evaluated according to the magnitude of performance loss of the system caused by it. Moreover, the dynamic varying curve of importance of a component with time can be described by calculating criticality of the component at each state. On the other hand, in the proposed approach, the importance probability curve of a component is fitted by using the failure data from all components of system excluding that of the component itself so as to solve the problem of inaccurate fitting caused by small sample data. The approach has been verified by probability analysis of failure data of CNC machines.

Importance measures have been widely used for identifying system weaknesses and supporting system improvement activities from the design viewpoint. With the importance values of all components, proper actions can be taken on the weakest component to improve system reliability at the minimum cost or effort [

Every component of a system has different contribution to the system performance. Investigation of this influence is the major objective of importance analysis, which can be qualitative or quantitative. From the viewpoint of classical reliability, a system is always regarded as binary-state. That is, all components only have two states of “working perfect” or “completely failed.” This representation is suitable for systems where any deviation from perfect functioning may cause a disaster, for example, nuclear power plants [

Unlike binary-state systems that are either perfectly working or completely failed, newer approaches show that typical components of a system may work at many performance loss levels from zero to one, each level with a certain probability. Despite a system like CNC machine tools can be called a multistate system (MSS) that has a changing situation over time, many of the results for the binary case can be computed for multistate systems using the binary structure and reliability function concepts [

Birnbaum [

In recent years, many researchers have made contributions to multistate system IM analysis. They focused on developing different approaches to permit setting limitations of the calculation methods. For example, the IM for complex systems with multiple components was proposed in [

This paper aims to seek an efficient method for evaluating component importance of complex system as a multistate system. With the loss of system performance of complex system, one of the open questions is how to accurately calculate the dynamic importance of a component. Traditional importance measures mainly concern the change of system performance caused by the reliability change of components, but they seldom consider the joint effect of probability distributions, transition intensities of the object component states, and the loss of system performance. As a matter of fact, the expected loss of system performance from failures is related to the expected number of component failures and the effect of system structure. It means that some new mathematical approaches for known IM need to be studied. In this paper, considering how the transition of component states affects the system’ mean time to failure (MTTF), we study the time integral importance measure (TIIM) to evaluate the importance of components of system.

The remainder of this paper is organized as follows. The traditional importance measures are analyzed in Section

Comparison of traditional methods is shown in Table

Comparison of traditional measures.

Refs. | Up to year | Method | Formula | Dynamic characteristics |
---|---|---|---|---|

Birnbaum [ | 1969 | Birnbaum importance measure | | Dynamic |

Barlow and Proschan [ | 1975 | Barlow-Proschan importance measure | | Dynamic |

Griffith [ | 1980 | Griffith importance measure | | Dynamic |

Natvig [ | 1982 | Natvig importance measure | | Static with the mean lifetime and repair time |

Wu [ | 2005 | Wu importance measure | | Static |

Si et al. [ | 2012 | Integrated importance measure | | Dynamic |

From Table

Some assumptions are described as follows:

All components and a system under consideration have the set of reliability states

The reliability states are ordered: the state 0 is the worst and the state

The component and the system reliability states degrade with time

The above assumptions mean that the reliability states of the ageing system and component may change over time only from better to worse.

A vector

A vector

It is clear that from Definitions

Under the above definitions, the mean time to failure (MTTF) of the system

Moreover, the mean time to failure (MTTF) of the component

Let

In order to avoid the inaccurate evaluation caused by the few failure data, we wiped off the failure data of component

Assume that

Zhang et al. [

From (

By normalizing the parameters,

Thus, the importance of the component

This paper collected the failure data of CNC machine tools for fifteen months. Table

Goodness-of-fit.

Distribution | Anderson-Darling | Rank |
---|---|---|

Weibull | 2.551 | 4 |

Lognormal | 2.327 | 2 |

Exponential | 3.450 | 8 |

Log Logistic | 2.488 | 3 |

| | |

3-Parameter Lognormal | 2.667 | 6 |

2-Parameter Exponential | 2.839 | 7 |

3-Parameter Log Logistic | 2.591 | 5 |

Smallest extreme value | 12.776 | 11 |

Normal | 7.959 | 10 |

Logistic | 6.507 | 9 |

As evaluated by Minitab, the mean time to failure (MTTF) of CNC machine tools is 876.8 h, as shown in Figure

Probability plot of MTTF of CNC machine tools.

Figure

Probability plot of MTTF of components.

Probability plot of MTTF of CNC machine tools with removal of CNC control system

Probability plot of MTTF of CNC machine tools with removal of turning tool carry

Probability plot of MTTF of CNC machine tools with removal of power supply and electrical system

Probability plot of MTTF of CNC machine tools with removal of chip-removal system

Probability plot of MTTF of CNC machine tools with removal of lubrication system

Probability plot of MTTF of CNC machine tools with removal of hydraulic system

Probability plot of MTTF of CNC machine tools with removal of main transmission system

Probability plot of MTTF of CNC machine tools with removal of clamp system

According to Figure

When a machine operates into steady state we can get the equations

Component Importance for

Component | Birnbaum importance | Rank | Barlow-Proschan importance | Rank | Time integral importance | Rank | Natvig importance | Rank | Wu importance | Rank | Integrated importance | Rank |
---|---|---|---|---|---|---|---|---|---|---|---|---|

CNC control system | 0.0401 | 6 | 0.1115 | 6 | 1.0000 | 8 | 0.1160 | 6 | 11.84 × 10^{−4} | 5 | 2.2146 × 10^{−5} | 6 |

Tool carry | 0.0524 | 2 | 0.1355 | 4 | | | | | ^{−4} | | 2.9116 × 10^{−5} | 2 |

Power supply and electrical system | | | | | 1.9501 | 2 | 0.1640 | 2 | 11.45 × 10^{−4} | 2 | ^{−5} | |

Chip-removal system | 0.0507 | 3 | 0.1405 | 3 | 1.7962 | 4 | 0.1494 | 4 | 11.62 × 10^{−4} | 4 | 2.8137 × 10^{−5} | 3 |

Lubrication system | 0.0374 | 7 | 0.1062 | 7 | 1.4474 | 6 | 0.0907 | 7 | 12.21 × 10^{−4} | 7 | 2.0562 × 10^{−5} | 7 |

Hydraulic system | 0.0268 | 8 | 0.0805 | 8 | 1.5984 | 5 | 0.0205 | 8 | 12.58 × 10^{−4} | 8 | 1.4645 × 10^{−5} | 8 |

Main transmission system | 0.0474 | 5 | 0.1330 | 5 | 1.2464 | 7 | 0.1326 | 5 | 11.92 × 10^{−4} | 6 | 2.6193 × 10^{−5} | 5 |

Clamp system | 0.0498 | 4 | 0.1447 | 2 | 1.8899 | 3 | 0.1584 | 3 | 11.50 × 10^{−4} | 3 | 2.7659 × 10^{−5} | 4 |

From Table

Although power supply or electrical system is the least reliable unit, tool carry has a smaller mean time to failure (MTTF) or a larger criticality than power supply and electrical system. Therefore, from the TIIM, Natvig importance measure, and Wu importance measure, the importance of tool carry is larger than that of the power supply and electrical system.

Because of the performance loss of CNC machine tools, the importance of components may be changing. Therefore, the case of

For

Component importance for

Component | Birnbaum importance | Rank | Barlow-Proschan importance | Rank | Time integral importance | Rank | Natvig importance | Rank | Wu importance | Rank | Integrated importance | Rank |
---|---|---|---|---|---|---|---|---|---|---|---|---|

CNC control system | 0.0299 | 6 | 0.1085 | 6 | 1.0255 | 6 | 0.1160 | 6 | 10.33 × 10^{−4} | 5 | 1.6513 × 10^{−5} | 6 |

Tool carry | 0.0388 | 3 | 0.1355 | 4 | | | | | ^{−4} | | 2.1559 × 10^{−5} | 3 |

Power supply and electrical system | | | | | 1.3634 | 2 | 0.1640 | 2 | 10.09 × 10^{−4} | 2 | ^{−5} | |

Chip-removal system | 0.0380 | 4 | 0.1365 | 3 | 1.2495 | 4 | 0.1494 | 4 | 10.23 × 10^{−4} | 4 | 2.1089 × 10^{−5} | 4 |

Lubrication system | 0.0290 | 7 | 0.1072 | 7 | 1.0176 | 7 | 0.0907 | 7 | 10.65 × 10^{−4} | 7 | 1.5944 × 10^{−5} | 7 |

Hydraulic system | 0.0261 | 8 | 0.0972 | 8 | 1.0000 | 8 | 0.0205 | 8 | 10.82 × 10^{−4} | 8 | 1.4263 × 10^{−5} | 8 |

Main transmission system | 0.0352 | 5 | 0.1289 | 5 | 1.0793 | 5 | 0.1326 | 5 | 10.49 × 10^{−4} | 6 | 1.9452 × 10^{−5} | 5 |

Clamp system | 0.0395 | 2 | 0.1412 | 2 | 1.2620 | 3 | 0.1584 | 3 | 10.13 × 10^{−4} | 3 | 2.1938 × 10^{−5} | 2 |

As shown in Tables

In the case the reliability loss of a CNC machine tool with time is shown in Figure

Reliability change of CNC machine tools.

TIIM change of components for CNC machine tools.

Known from the definition of TIIM, the proposed method can be used for importance evaluations of major components of a machine in binary and multiple states, respectively. Therefore, the conclusions derived from multistate systems can also be used for binary-state ones.

This paper has discussed the TIIM of component states based on the loss of system performance. First, we present the definition of time integral importance measure (TIIM) of component states. Then the proposed method is compared with the existing approaches, such as Birnbaum importance, Wu importance, and Natvig importance. Finally, the proposed method is verified by a type of CNC machine tools. The major conclusions obtained are summarized as follows:

To evaluate the influence of a component of complex mechanical system on system’ mean time to failure (MTTF), the component reliability fits with the failure data which is removal of the component fault data in the whole failure data; in this way, TIIM can avoid the problem of inaccurate fitting due to small sample data. Then, the integral difference in reliabilities between the system and a component is measured for the evaluation purpose. In comparison with the existing method, the coupling relations among components is not required and the computational complexity is reduced greatly.

TIIM is a new dynamic importance measure. In comparison with traditional importance measures, criticality of a component is taken into account by the computation of the component failure probability, and the calculation results are more practical.

Although a component at a specific state has great influence on system performance, the influence may be little at the other states. Therefore, TIIM can evaluate the component importance in its whole lifetime and find out the most responsible component for system performance loss. Then, the component could be monitored to improve system performance.

In addition, the proposed approach in this study is only the first step of the importance measure of components of complex system. Future researches are needed to confirm the component states

Computer numerical control

Mean time to failure

Multistate systems

Importance measure

Structural importance

Birnbaum importance

Universal generating function

Time integral importance measure

Number of components in a system

Index of component

The performance level corresponding to state

The expectation

The expected performance of a system

The reliability of component

The mean lifetime of component

The mean repair time of component

Birnbaum importance of component

Griffith importance of component

Wu importance of component

Natvig importance of component

Time integral importance of component

The authors declare that they have no conflicts of interest.

This work is supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Grant no. 2017ZDJC-21) and Open Project for Chinese Institute for Quality Research, SJTU (Grant no. 2016-05).