# A New and Practical Reliability Allocation Method for a Complex System of NC Turrets

Authors: Wei Hu; Fei Chen; Yankun Wang; Qunya Xie
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

Existing methods cannot satisfy the reliability allocation demands of the early design phase for the modern complex system of NC turrets. Motivated by the need of practical application, this paper proposes a new and practical reliability allocation method in the early design stage for NC turrets considering failure mode and system complexity, information inaccessibility, and expert knowledge limitation. First, the fault tree of a NC turret is quickly built to clear the relationship between the system’s compositions and failure modes up. Second, the happening probability of each failure event in the fault tree is firstly calculated by fuzzy expert evaluation to provide the reliability allocation with complete information. Third, by discussing the practical meaning of every layer in the fault tree, the proposed allocation strategy is within the experts’ knowledge scope for evaluating accurately. Eventually, the application result of the AK36100 A turret is presented and compared with some existing allocation methods, illustrating the rationality of the proposed allocation method.


---

## Body

## 1. Introduction

Reliability allocation is an important work in the early design stage of a modern complex system for assigning an achievable reliability goal of a system to its constituent subsystems to guarantee the reasonable reliability of the targeted system with the minimum costs [1, 2]. Numerical controlled (NC) turret is one of the advanced and complicated electromechanical systems, which is the most key function part of the NC lathe [3]. Therefore, the work of reliability allocation in the design cycle for turrets to perform well is meaningful to the quality and economics of manufacturing.Various reliability allocation methods have been widely discussed and developed over the last several decades. One existing approach combines one or several criteria in different combination ways for obtaining an allocation weight and allocating reliability in proportion of the weight [4]. For example, the Advisory Group on Reliability of Electronic Equipment (AGREE) method combines complexity into allocation weight [5], and the Aeronautical Radio Inc. (ARINC) method considers failure rate as allocation weight [6]. Another conventional reliability allocation method focuses on multiobjective optimization [4], including cost minimization [7] and redundancy allocation [8]. In addition, some smart methods like event-triggered-based [9] and Newton–Raphson-based [10] distributed algorithm and small-signal model [11] are proposed to improve multienergy systems’ reliability [9, 10, 12] and increase converters’ stability. Nevertheless, the aforementioned methods still cannot satisfy the reliability allocation demands of the early design phase for the modern complex system of turrets.Firstly, the existing methods fail to access the complete information, like the reliability and the cost of the complex system and subsystems. Although the reliability of turrets is lower than other parts of NC machine tools such as screws and guide rails, it is still higher than normal products. Therefore, there is not enough available information not only in the early design stage, but also in the field-processing stage of similar products. To address this problem, expert evaluation [13] and fuzzy theory [14] have been commonly studied. AHP as a traditional expert evaluation method has been applied in many complex products’ performance assessments such as maintainability and cost effectiveness [15, 16]. Risk priority number (RPN) is another common method to evaluate the product failure effect from the experts’ perspective of severity, occurrence, and detectability [17, 18]. Fuzzy theory decreases the subjective preference of decision makers and has been used in expert evaluation like fuzzy AHP [19] and fuzzy expert systems [20]. On the contrary, Wang et al. [21] and Yang et al. [17] demonstrated that collecting field data are time-consuming, and they took several years to collect the number of field data from NC machine tools. Hence, more emphasis should be put on fuzzy expert evaluation to obtain objective, effective, and sufficient data.Secondly, most existing methods fail to consider the complexity of the system’s compositions and failure modes. However, it is necessary to understand this complexity because the effect of failure modes is one of the important factors in determining the allocation weight [18]. In a complex system, different functional components have different modules and various failure modes, bringing different influences to the system. Experts are unable to accurately evaluate the performance of all components by themselves. Failure mode and effects analysis (FMEA) [22] and fault tree analysis (FTA) [23] as a usual failure analysis tool help experts understand the relationship between components and failure modes in a complex system. Most research studies calculated the allocation weight according to the RPN of their all possible failure modes obtained by FMEA [1, 2, 4]. Compared with a bottom-up method of FMEA, FTA is a top-down method and requires less expert knowledge than FMEA [24], but FTA is seldom utilized in reliability allocation [25]. Furthermore, Peeters et al. indicate that both methods are time-consuming to apply singly than jointly [24]. It is difficult to figure out every starting point of FMEA and each branch of FTA, as for such a large complex system. Hence, the combination of FMEA and FTA can not only clarify the relationship between components and failure modes, but also improve the efficiency of allocation.Thirdly, the current methods fail to manage a variety of information with different practical meanings. In views of the complexity of the system, the contents of its failure analysis are abundant which contain at least functional modules, functional failure modes, and failure mechanisms. The expert’s knowledge about these contents is limited. For example, compared with functional modules, failure mechanisms are more realistic and easier to be scored in cost, while the former seems more likely to be evaluated through functional importance and complexity due to the fact that the cost of functional modules is confusing and is decided by the degree of maintenance. Thus, it is unreasonable to assess different contents by the same criterion, even if this criterion integrates many performance factors. In other words, the criterion selected considering the experts’ knowledge limitation to different contents has a positive impact on the practicality of the allocation weight. However, most existing allocation methods give priority to a perfect and comprehensive allocation weight [16, 17, 21, 22] rather than a practicable weight.In view of the aforementioned problems, this paper proposes a new and practical reliability allocation method in the early design stage for a complex system of NC turrets considering failure mode and system complexity, information inaccessibility, and expert knowledge limitation. The major contributions of this paper are as follows:(1)
The fault tree of a NC turret is quickly built by FMEA-FTA not only to clear the relationship between the system’s compositions and failure modes up, but also to provide guidelines for rapidly locating the failure mechanism when such a key function part of the NC lathe breaks down.(2)
The happening probability of each failure event in the fault tree is firstly calculated by fuzzy expert evaluation to provide the reliability allocation with complete information. The accuracy of the proposed evaluation method is verified by comparing the mean time between failure (MTBF) of similar turrets from field data. In addition, another allocation weight of probability-importance-based failure importance becomes available due to the fault tree and happening probabilities.(3)
By discussing the practical meanings of every layer in the fault tree, the selection of allocation weight in each layer is within the experts’ knowledge scope. The proposed allocation strategy not only caters for the experts’ evaluating preference, but also satisfies the requirement of allocation effectiveness. Furthermore, the comparison of allocation results with some existing allocation methods illustrates the rationality of the proposed allocation strategy.The remainder of this paper is organized as follows. Section2 first builds the fault tree of the AK36100 A turret based on FMEA-FTA. Then, the proposed calculation method of happening probabilities of each failure event in the fault tree based on fuzzy expert evaluation is presented. Section 3 presents a new and practical allocation strategy. In Section 4, a case application is introduced to show the rationality of the proposed allocation method. The last section concludes this paper.


## 2. Fault Tree Analysis and Calculation of Turret

### 2.1. A Key Fault Tree of Turret

FTA is a useful method to identify the effect of low-level events like failure mechanisms happening on high-level events like system breaking down in a complex system. In order to reduce the time of fault tree construction, we draw upon the idea of Peeters et al. [24] and use the FMEA-FTA to explore the key fault tree of the turret. We choose an example of the AK36100 A turret so as to explain the method more easily and directly.First, Figure1 divides an AK36100 A turret into five function modules according to its composition and working principle. Then, functional failure modes (FFMs) are defined in the fault tree (see Figure 2), which describe the principal failure mode of each function module. These FFMs have a major effect on the system’s reliability.Figure 1
Function modules of the AK36100 A turret. (a) Outside view. (b) Inside view. Meanings of function modules are listed in Appendix A.
(a)
(b)Figure 2
The system level’s fault tree of the AK36100 A turret. Meanings of FFMs are listed in Appendix B. SF: system failure; M: function module; FFM: functional failure mode.Second, the assessment of criticality on aforementioned FFMs is performed by FMEA. We invite five field experts to determine severity (S), occurrence (O), and detectability (D) for each FFM and to calculate the RPN (where PRN = S•O•D), as shown in Table 1. Meanwhile, it is true that the field experience and statistical knowledge can influence the experts’ performance [14]. Therefore, we use AHP to calculate the weight of experts according to their jobs, experience, and education.Table 1
RPN estimated by experts.
FFM
Exp.1 weight = 0.3
Exp.2 weight = 0.3
Exp.3 weight = 0.2
Exp.4 weight = 0.1
Exp.5 weight = 0.1
RPN
Rank
S
O
D
S
O
D
S
O
D
S
O
D
S
O
D
1
6
5
5
6
5
4
6
5
4
5
5
3
6
5
5
128
7
2
5
4
5
5
5
4
5
4
4
4
4
5
5
4
5
94
9
3
4
4
3
3
6
5
2
4
5
4
4
5
5
5
4
67
10
4
6
5
4
6
5
4
6
6
4
5
5
5
5
6
5
128
7
5
7
5
6
7
5
5
7
7
6
6
6
4
7
6
6
214
3
6
5
6
6
6
5
6
6
5
6
6
5
5
6
5
6
177
6
7
6
6
5
6
6
6
7
6
6
6
6
6
6
6
7
216
2
8
8
7
8
8
8
7
8
8
8
8
7
7
8
7
8
455
1
9
5
6
7
6
5
6
4
7
7
5
6
4
5
6
5
183
4
10
5
6
7
6
5
6
4
7
7
5
6
4
5
6
5
183
4
Exp. = expert.We find the top five key FFMs are FFM5, FFM7, FFM8, FFM9, and FFM10. Again, a fault tree for key FFMs is performed (see Figure3).Figure 3
Key FFMs’ fault tree of the AK36100 A turret. Meanings of CFMs are listed in Appendix C, and meanings of FMs are listed in Appendix D. SF: system failure; FFM: functional failure mode; CFM: complex failure mode; FM: failure mechanism.The top event represents system failure (SF) which divides into five typical FFMs. After analysing the reasons of FFMs happening again based on FMEA, ten component failure modes (CFMs) are presented. Then, the failure mechanism (FM) of each CFM is identified as a basic event. Typically, these FMs are also key FMs of the turret system.


### 2.2. Happening Probability of the Fault Tree

Traditional allocation weights requiring field data often hinder their applicability. In other words, if data are sufficient, these allocation weights can be available. Because of the difficulty of field data acquiring, it is inevitable to listen to the expert. Hence, we first choose a fuzzy expert evaluation method to evaluate happening probabilities of FMs of the AK36100 A turret and then calculate the happening probability of residual events of the fault tree based on their relationship with FMs.First, two kinds of membership functions are set to quantify expert judgment. For example, an expert measures the happening probability in seven language levels including very small (VS) level, small (S) level, relatively small (RS) level, medium (M) level, relatively large (RL) level, large (L) level, and very large (VL) level. S, M, and L are quantified by the triangular membership function, while VS, RS, RL, and VL are quantified by the trapezoidal membership function (see Figure4).Figure 4
Membership functions. (a) Triangular membership function. (b) Trapezoidal membership function.
(a)
(b)The triangular membership function is expressed by(1)fx=x−ab−a,a≤x≤b,c−xc−b,b≤x≤c,0,others,where x is the fuzzy number and represents the fuzzy happening probability of an event and fx is the membership function and means the probability of x that belongs to the expert judging language level.The trapezoidal membership function is expressed by(2)fx=x−ab−a,a≤x≤b,1,b≤x≤c,d−xd−c,c≤x≤d,0,others.Second, assuming the value offx equals α, the interval of the fuzzy number of the triangular and trapezoidal membership function is calculated by(3)triangular:xi∈b−aα+a,c−c−bα,trapezoidal:xi∈b−aα+a,d−d−cα,where xi is the fuzzy number of language level i.If there are several experts, the average fuzzy numberz of their judgment results is obtained by experts’ weights:(4)z=1n∑k=1nwkxi=Aα+B,Cα+D,where wk is the weight of expert k and can be calculated by AHP and A, B, C, and D are the average coefficient of z interval.Then, the average membership functionfwz of several experts’ judgments is expressed by(5)fwx=z−BA,B≤z≤A+B,1,A+B≤z≤D,z−DC,D≤z≤C+D,0,others.We use the left-and-right fuzzy ranking method to determine the final fuzzy happening probability (FHP):(6)FHP=1102.301×1−FPS/FPS−1/3,0FPS≠0,where FPS is the left-and-right fuzzy probability value calculated by(7)FPSRW=supfWx∧fmaxx,FPSLW=supfWx∧fminx,FPS=FPSRW+1−FPSLW2,where fmaxx and fminx are linear functions expressed as(8)fmaxx=x,0<x<1,0,others,fminx=1−x,0<x<1,0,others.Through the aforementioned fuzzy expert evaluation method, the happening probability of FMs can be calculated by FHP.In addition, because of the serial relationship between high-level events and FMs, the happening probability of high-level events is calculated by(9)Fj=1−∏i=ai=b1−FHPi,where Fj is the happening probability of the jth high-level event which includes several FMs from ath FM to bth FM.


## 2.1. A Key Fault Tree of Turret

FTA is a useful method to identify the effect of low-level events like failure mechanisms happening on high-level events like system breaking down in a complex system. In order to reduce the time of fault tree construction, we draw upon the idea of Peeters et al. [24] and use the FMEA-FTA to explore the key fault tree of the turret. We choose an example of the AK36100 A turret so as to explain the method more easily and directly.First, Figure1 divides an AK36100 A turret into five function modules according to its composition and working principle. Then, functional failure modes (FFMs) are defined in the fault tree (see Figure 2), which describe the principal failure mode of each function module. These FFMs have a major effect on the system’s reliability.Figure 1
Function modules of the AK36100 A turret. (a) Outside view. (b) Inside view. Meanings of function modules are listed in Appendix A.
(a)
(b)Figure 2
The system level’s fault tree of the AK36100 A turret. Meanings of FFMs are listed in Appendix B. SF: system failure; M: function module; FFM: functional failure mode.Second, the assessment of criticality on aforementioned FFMs is performed by FMEA. We invite five field experts to determine severity (S), occurrence (O), and detectability (D) for each FFM and to calculate the RPN (where PRN = S•O•D), as shown in Table 1. Meanwhile, it is true that the field experience and statistical knowledge can influence the experts’ performance [14]. Therefore, we use AHP to calculate the weight of experts according to their jobs, experience, and education.Table 1
RPN estimated by experts.
FFM
Exp.1 weight = 0.3
Exp.2 weight = 0.3
Exp.3 weight = 0.2
Exp.4 weight = 0.1
Exp.5 weight = 0.1
RPN
Rank
S
O
D
S
O
D
S
O
D
S
O
D
S
O
D
1
6
5
5
6
5
4
6
5
4
5
5
3
6
5
5
128
7
2
5
4
5
5
5
4
5
4
4
4
4
5
5
4
5
94
9
3
4
4
3
3
6
5
2
4
5
4
4
5
5
5
4
67
10
4
6
5
4
6
5
4
6
6
4
5
5
5
5
6
5
128
7
5
7
5
6
7
5
5
7
7
6
6
6
4
7
6
6
214
3
6
5
6
6
6
5
6
6
5
6
6
5
5
6
5
6
177
6
7
6
6
5
6
6
6
7
6
6
6
6
6
6
6
7
216
2
8
8
7
8
8
8
7
8
8
8
8
7
7
8
7
8
455
1
9
5
6
7
6
5
6
4
7
7
5
6
4
5
6
5
183
4
10
5
6
7
6
5
6
4
7
7
5
6
4
5
6
5
183
4
Exp. = expert.We find the top five key FFMs are FFM5, FFM7, FFM8, FFM9, and FFM10. Again, a fault tree for key FFMs is performed (see Figure3).Figure 3
Key FFMs’ fault tree of the AK36100 A turret. Meanings of CFMs are listed in Appendix C, and meanings of FMs are listed in Appendix D. SF: system failure; FFM: functional failure mode; CFM: complex failure mode; FM: failure mechanism.The top event represents system failure (SF) which divides into five typical FFMs. After analysing the reasons of FFMs happening again based on FMEA, ten component failure modes (CFMs) are presented. Then, the failure mechanism (FM) of each CFM is identified as a basic event. Typically, these FMs are also key FMs of the turret system.


## 2.2. Happening Probability of the Fault Tree

Traditional allocation weights requiring field data often hinder their applicability. In other words, if data are sufficient, these allocation weights can be available. Because of the difficulty of field data acquiring, it is inevitable to listen to the expert. Hence, we first choose a fuzzy expert evaluation method to evaluate happening probabilities of FMs of the AK36100 A turret and then calculate the happening probability of residual events of the fault tree based on their relationship with FMs.First, two kinds of membership functions are set to quantify expert judgment. For example, an expert measures the happening probability in seven language levels including very small (VS) level, small (S) level, relatively small (RS) level, medium (M) level, relatively large (RL) level, large (L) level, and very large (VL) level. S, M, and L are quantified by the triangular membership function, while VS, RS, RL, and VL are quantified by the trapezoidal membership function (see Figure4).Figure 4
Membership functions. (a) Triangular membership function. (b) Trapezoidal membership function.
(a)
(b)The triangular membership function is expressed by(1)fx=x−ab−a,a≤x≤b,c−xc−b,b≤x≤c,0,others,where x is the fuzzy number and represents the fuzzy happening probability of an event and fx is the membership function and means the probability of x that belongs to the expert judging language level.The trapezoidal membership function is expressed by(2)fx=x−ab−a,a≤x≤b,1,b≤x≤c,d−xd−c,c≤x≤d,0,others.Second, assuming the value offx equals α, the interval of the fuzzy number of the triangular and trapezoidal membership function is calculated by(3)triangular:xi∈b−aα+a,c−c−bα,trapezoidal:xi∈b−aα+a,d−d−cα,where xi is the fuzzy number of language level i.If there are several experts, the average fuzzy numberz of their judgment results is obtained by experts’ weights:(4)z=1n∑k=1nwkxi=Aα+B,Cα+D,where wk is the weight of expert k and can be calculated by AHP and A, B, C, and D are the average coefficient of z interval.Then, the average membership functionfwz of several experts’ judgments is expressed by(5)fwx=z−BA,B≤z≤A+B,1,A+B≤z≤D,z−DC,D≤z≤C+D,0,others.We use the left-and-right fuzzy ranking method to determine the final fuzzy happening probability (FHP):(6)FHP=1102.301×1−FPS/FPS−1/3,0FPS≠0,where FPS is the left-and-right fuzzy probability value calculated by(7)FPSRW=supfWx∧fmaxx,FPSLW=supfWx∧fminx,FPS=FPSRW+1−FPSLW2,where fmaxx and fminx are linear functions expressed as(8)fmaxx=x,0<x<1,0,others,fminx=1−x,0<x<1,0,others.Through the aforementioned fuzzy expert evaluation method, the happening probability of FMs can be calculated by FHP.In addition, because of the serial relationship between high-level events and FMs, the happening probability of high-level events is calculated by(9)Fj=1−∏i=ai=b1−FHPi,where Fj is the happening probability of the jth high-level event which includes several FMs from ath FM to bth FM.


## 3. Proposed Reliability Allocation Strategy

Reliability allocation is a top-down method for assigning an achievable reliability goal of a system to its constituent subsystems. For the reliability allocation based on FTA, the happening probability target of the SF can be allocated to low-level events including FFMs, CFMs, and FMs. It is inevitable that experts have the knowledge limitation to failure events with different practical meanings. A reasonable reliability allocation strategy can decrease the requirement of expert knowledge. The common allocation weights consist of failure frequency, failure importance, complexity, maintainability, and cost. Not all of the allocation weights have decisive effect on the event performance evaluation and not all of them are easily calculated by experts. Thus, it is necessary to carefully select an appropriate allocation weight for every level event.


### 3.1. Reliability Allocation from SF to FFMs

FFMs are the last things anyone wants to happen, which have irreplaceable function on the system. These function modules play an important role in the system of turrets. When failure occurs in such a function module, it causes large amounts of losses and problems for the manufacturing. Thus, failure importance of function modules should have priority over other allocation weights. In addition, because the cost and maintainability of function modules are fluctuant and decided by their failure mechanisms, experts have difficulty in assessing them. Therefore, the probability-importance-based allocation weight is utilized to evaluate the failure importance during the allocation phase of happening probability from SF to FFMs. Probability importance of a function module can be determined by its influence on the system’s reliability when this function module’s reliability is changed. This allocation weight also combines failure frequency and complexity. According to the fault tree, SF has five serial FFMs. The probability importance of FFMj is obtained by(10)IFFMj=∂FSYS∂FFFMj=∏i=1,i≠j51−FFFMi,where FSYS is the happening probability of SF and FFFMi is the happening probability of FFMi.Then, the allocation formula based on the probability importance is(11)ΔFFFM1:ΔFFFM2:⋯:ΔFFFM5=IFFM1:IFFM2:⋯IFFM5,FFFMi∗=FFFMi+ΔFFFMi,FSYS∗=1−∏i=151−FFFMi∗,where ΔFFFMi is the variation of the happening probability of FFMi, FFFMi∗ is the new allocated happening probability of FFMi, and FSYS∗ is the target happening probability of SF.


### 3.2. Reliability Allocation from FFMs to CFMs

CFMs are failure models of the components of function modules. Although CFMs detail FFMs, they are so abstract that experts are unable to estimate the cost and maintainability. For another thing, the number of CFMs is more than FFMs. It is unrealistic to allocate reliability to the whole CFMs based on failure importance, due to the high developed cost. Therefore, the allocation-number-based allocation weight is selected to allocate FFMs’ target happening probability to CFMs with lower happening probability. The allocation-number-based allocation weight also combines failure frequency and complexity and believes the number of improved components is as little as possible. If the component’s reliability is worse, the more the component needs to be improved. This allocation weight is calculated as follows. First, CFMs’ happening probabilities from the same FFM are sorted from high to low:(12)FCFM1>FCFM2>⋯>FCFMk>⋯>FCFMn.Second, CFMs 1, 2, …,k with high happening probability are assigned to a lower happening probability F0 and residual CFMs k + 1, k + 2, …, n maintain the status quo. Hence, a serial FFM’s happening probability after allocation is decreased to(13)FFFM=1−1−F0k∏i=k+1n1−FCFMi.Because of the constrain of the FFM’s target happening probabilityFFFM∗, FFFM should be as follows:(14)FFFM≤FFFM∗.The CFM’s happening probability targetF0 is determined using(15)F0=1−1−FFFM∗∏i=k+1n1−FCFMi1/k,where k is the minimum number satisfying the equation (15).


### 3.3. Reliability Allocation from CFMs to FMs

FMs are fundamental causes of CFMs, FFMs, and even the SF to happen. Their performance (including failure frequency, failure importance, maintainability, and cost effectiveness) is apparent to experts, except for complexity. Because FMs are bottom events without any component, their complexity is same. Hence, AHP is applied to combine the aforementioned four allocation weights (see Figure5).Figure 5
An example form of decision elements in AHP of reliability allocation. Decision elements have three hierarchy, including goal, attributes, and alternatives. Alternatives are components of the system, and each component includes several attributes, like failure frequency, probability importance, and so on.For every FM in the same CFM branch, the happening probability and probability importance are used to describe failure frequency and failure importance, respectively, while maintainability and cost effectiveness are estimated by the nine-point numerical scale and pairwise comparison matrix from experts which is commonly used in AHP. Meanwhile, the weight among these four attributes is also determined by this way. According to the detailed calculation process of AHP [23], the allocation relative weight wFMi of FMs can be obtained. And the new happening probability of FM is(16)FFMi=FCFM/wFMi∑i=1nwFMi.In conclusion, in order to propose a practical and comprehensive reliability allocation method for the turret, this paper adopts FMEA-FTA, fuzzy expert evaluation, and some comprehensive allocation weights, respectively, to overcome the existing challenges of reliability allocation for a large complex system. The process of reliability allocation for turrets is finally concluded as follows (see Figure6).Figure 6
Flow chat of the practical and comprehensive reliability allocation method of the turret.


## 3.1. Reliability Allocation from SF to FFMs

FFMs are the last things anyone wants to happen, which have irreplaceable function on the system. These function modules play an important role in the system of turrets. When failure occurs in such a function module, it causes large amounts of losses and problems for the manufacturing. Thus, failure importance of function modules should have priority over other allocation weights. In addition, because the cost and maintainability of function modules are fluctuant and decided by their failure mechanisms, experts have difficulty in assessing them. Therefore, the probability-importance-based allocation weight is utilized to evaluate the failure importance during the allocation phase of happening probability from SF to FFMs. Probability importance of a function module can be determined by its influence on the system’s reliability when this function module’s reliability is changed. This allocation weight also combines failure frequency and complexity. According to the fault tree, SF has five serial FFMs. The probability importance of FFMj is obtained by(10)IFFMj=∂FSYS∂FFFMj=∏i=1,i≠j51−FFFMi,where FSYS is the happening probability of SF and FFFMi is the happening probability of FFMi.Then, the allocation formula based on the probability importance is(11)ΔFFFM1:ΔFFFM2:⋯:ΔFFFM5=IFFM1:IFFM2:⋯IFFM5,FFFMi∗=FFFMi+ΔFFFMi,FSYS∗=1−∏i=151−FFFMi∗,where ΔFFFMi is the variation of the happening probability of FFMi, FFFMi∗ is the new allocated happening probability of FFMi, and FSYS∗ is the target happening probability of SF.


## 3.2. Reliability Allocation from FFMs to CFMs

CFMs are failure models of the components of function modules. Although CFMs detail FFMs, they are so abstract that experts are unable to estimate the cost and maintainability. For another thing, the number of CFMs is more than FFMs. It is unrealistic to allocate reliability to the whole CFMs based on failure importance, due to the high developed cost. Therefore, the allocation-number-based allocation weight is selected to allocate FFMs’ target happening probability to CFMs with lower happening probability. The allocation-number-based allocation weight also combines failure frequency and complexity and believes the number of improved components is as little as possible. If the component’s reliability is worse, the more the component needs to be improved. This allocation weight is calculated as follows. First, CFMs’ happening probabilities from the same FFM are sorted from high to low:(12)FCFM1>FCFM2>⋯>FCFMk>⋯>FCFMn.Second, CFMs 1, 2, …,k with high happening probability are assigned to a lower happening probability F0 and residual CFMs k + 1, k + 2, …, n maintain the status quo. Hence, a serial FFM’s happening probability after allocation is decreased to(13)FFFM=1−1−F0k∏i=k+1n1−FCFMi.Because of the constrain of the FFM’s target happening probabilityFFFM∗, FFFM should be as follows:(14)FFFM≤FFFM∗.The CFM’s happening probability targetF0 is determined using(15)F0=1−1−FFFM∗∏i=k+1n1−FCFMi1/k,where k is the minimum number satisfying the equation (15).


## 3.3. Reliability Allocation from CFMs to FMs

FMs are fundamental causes of CFMs, FFMs, and even the SF to happen. Their performance (including failure frequency, failure importance, maintainability, and cost effectiveness) is apparent to experts, except for complexity. Because FMs are bottom events without any component, their complexity is same. Hence, AHP is applied to combine the aforementioned four allocation weights (see Figure5).Figure 5
An example form of decision elements in AHP of reliability allocation. Decision elements have three hierarchy, including goal, attributes, and alternatives. Alternatives are components of the system, and each component includes several attributes, like failure frequency, probability importance, and so on.For every FM in the same CFM branch, the happening probability and probability importance are used to describe failure frequency and failure importance, respectively, while maintainability and cost effectiveness are estimated by the nine-point numerical scale and pairwise comparison matrix from experts which is commonly used in AHP. Meanwhile, the weight among these four attributes is also determined by this way. According to the detailed calculation process of AHP [23], the allocation relative weight wFMi of FMs can be obtained. And the new happening probability of FM is(16)FFMi=FCFM/wFMi∑i=1nwFMi.In conclusion, in order to propose a practical and comprehensive reliability allocation method for the turret, this paper adopts FMEA-FTA, fuzzy expert evaluation, and some comprehensive allocation weights, respectively, to overcome the existing challenges of reliability allocation for a large complex system. The process of reliability allocation for turrets is finally concluded as follows (see Figure6).Figure 6
Flow chat of the practical and comprehensive reliability allocation method of the turret.


## 4. Reliability Allocation for AK36100 A Turret

The proposed reliability allocation method is applied to the AK36100 A turret which is a hot sale product in China.


### 4.1. Happening Probability Calculation

Figure3 has built the fault tree of key FFMs for the AK36100 A turret; then, the happening probability of FMs can be calculated as follows:First, seven membership functions of an event’s happening probability are established (see Figure7). Second, six engineers are invited at the AK36100 A turret manufacturing factory to express their ideas about the happening probability language levels of FMs (see Table 2). The happening probability of FMs is also calculated through equation (1) to equation (8).Figure 7
Membership function of FMs’ happening probability.Table 2
Happening probabilities of FMs estimated by experts.
Event
Exp.1
Exp.2
Exp.3
Exp.4
Exp.5
Exp.6
Happening probability
FM1
VS
VS
VS
VS
S
VS
1.13E − 07
FM2
VS
VS
VS
VS
S
S
2.10E − 07
FM3
VS
S
VS
VS
VS
S
3.04E − 07
FM4
S
S
VS
VS
VS
S
9.56E − 07
FM5
S
VS
VS
VS
S
VS
4.69E − 07
FM6
VS
S
VS
VS
VS
VS
1.73E − 07
FM7
S
VS
VS
VS
VS
S
3.90E − 07
FM8
S
S
VS
VS
VS
VS
6.35E − 07
FM9
S
S
S
S
S
S
4.24E − 06
FM10
S
S
S
RS
RS
S
8.33E − 06
FM11
S
S
VS
VS
S
S
1.58E − 06
FM12
VS
S
VS
VS
VS
S
3.04E − 07
FM13
S
VS
VS
VS
VS
S
3.90E − 07
FM14
S
VS
VS
VS
S
S
7.30E − 07
FM15
RS
S
RS
S
S
S
1.13E − 05
FM16
RS
RS
S
S
S
S
1.17E − 05
FM17
RS
S
RS
S
S
S
1.13E − 05
FM18
S
VS
VS
VS
S
S
7.30E − 07
FM19
S
VS
VS
VS
VS
S
3.90E − 07
FM20
S
VS
VS
VS
VS
S
3.90E − 07
FM21
S
RS
S
S
S
S
6.92E − 06
FM22
S
S
S
S
RS
S
5.94E − 06
FM23
S
RS
S
S
S
RS
8.59E − 06
FM24
RS
RS
S
S
S
S
1.17E − 05
FM25
S
S
VS
VS
VS
VS
6.35E − 07
FM26
S
S
VS
VS
VS
S
9.56E − 07
FM27
S
S
S
VS
VS
VS
1.32E − 06
FM28
VS
VS
S
VS
VS
VS
1.55E − 07
FM29
VS
VS
VS
VS
S
VS
1.13E − 07
FM30
S
S
S
VS
VS
VS
1.32E − 06
FM31
S
S
S
VS
VS
VS
1.32E − 06
FM32
VS
VS
S
VS
VS
VS
1.55E − 07
FM33
VS
S
VS
VS
VS
VS
1.73E − 07
FM34
S
S
VS
VS
VS
VS
6.35E − 07
FM35
RS
RS
S
S
S
S
1.17E − 05
FM36
S
RS
S
S
S
S
6.92E − 06
FM37
S
S
VS
VS
VS
VS
6.35E − 07
FM38
S
S
VS
VS
S
VS
1.11E − 06
Exp. = expert. The weight of each expert is 0.2452, 0.1980, 0.1804, 0.1471, 0.1332, and 0.0962.According to the serial relationship between high-level events and FMs, the happening probabilities of CFMs, FFMs, and SF are listed in Table3 through equation (9).Table 3
Happening probabilities of CFMs, FFMs, and SF.
Event
Happening probability
CFM1
1.58E − 06
CFM2
1.67E − 06
CFM3
1.41E − 05
CFM4
3.64E − 05
CFM5
1.47E − 05
CFM6
2.09E − 05
CFM7
2.43E − 06
CFM8
1.66E − 06
CFM9
3.61E − 05
CFM10
2.80E − 06
CFM11
1.44E − 06
CFM12
1.34E − 05
CFM13
3.97E − 05
CFM14
2.00E − 05
FFM8
1.74E − 05
FFM5
7.44E − 05
FFM7
4.06E − 05
FFM9
1.48E − 05
FFM10
5.97E − 05
SF
2.068E − 04Assuming the reliability function of the turret is exponentially distributed, the MTBF evaluated by the proposed method equals 4835 h. We tracked the reliability of 22 CNC lathes that used AK36 series turrets from July 2015 to December 2015 in a factory. The MTBF of these CNC lathes tested was 1532.3 h, and the MTBF of turrets evaluated was 5082 h using Monte Carlo simulation. By comparison, the MTBF obtained by fuzzy expert evaluation is consistent with the MTBF obtained from field data, thereby proving the rationality of the proposed method.


### 4.2. Result of Reliability Allocation

Assuming the turret’s target failure happening probability is 1.50E − 04, we allocate this target happening probability to FFMs, CFMs, and FMs through the proposed allocation strategy. Furthermore, the allocation results are compared with the results from other existing allocation weights.Firstly, FFMs are allocated diverse happening probabilities under different allocation weights, including recommended failure importance, failure frequency, complexity, and allocation number (see Figure8). The failure-frequency-based allocation weight is calculated by the conventional ARINC method, where(17)wFFMi=FFFMi∑i=15FFFMi,and the complexity-based allocation weight is calculated by the conventional AGREE method, where(18)wFFMi=nFFMi∑i=15nFFMi,where nFFMi is the number of FMs in FFMi.Figure 8
Happening probability allocation results for FFMs.As shown in Figure8, FFM5 (impossible turn of the cutter head) is given the highest happening probability in the recommended weight, which is the same as the results of other allocation weights. The happening probabilities of FFMs are declined by a similar degree based on the failure importance weight, due to the fact that these five FFMs are as important as each other, which is also verified by the RPN of FMEA in Table 1 again. The result of the allocation-number-based weight shows that only the happening probabilities of FFM5, FFM7, and FFM10 need to be declined, while the happening probability of FFM8 with the highest RPN still stays the unreasonably same. Meanwhile, the result of complexity-based weight has the similar problem, which even increases the happening probability of FFM8. The failure-frequency-based allocation method has a close result to the proposed method, but the former method is considered less than the proposed method in terms of complexity.Secondly, according to the allocation target of FFMs under the aforementioned proposed weight of failure importance, CFMs are allocated diverse happening probabilities under different allocation weights (see Figure9).Figure 9
Happening probability allocation results for CFMs.As shown in Figure9, the happening probabilities of CFM3, CFM4, CFM9, CFM12, and CFM13 are reduced in the allocation-number-based allocation method which conforms to the allocation rule that the higher happening probability the event has, the easier it declines. On the contrary, there is no doubt that the lower happening probability the event has, the more difficult it declines. However, the failure-importance-based allocation method assigns negative happening probabilities to the CFMs which have lower happening probabilities, and the failure-frequency-based allocation method assigns the teeny happening probabilities. Obviously, it is incorrect. In addition, the unreasonable allocation result that the CFM14 having high happening probability is still assigned a higher happening probability appears in the complexity-based allocation method. Furthermore, there are only five CFMs’ happening probabilities to be developed under the proposed allocation weight, while fourteen CFMs’ happening probabilities need to be done in the other weights which are more likely to be time-consuming and costly for improving the turret.Thirdly, according to the allocation target of CFMs under the aforementioned proposed weight of allocation number, FMs are allocated diverse happening probabilities under different allocation weights (see Figure10).Figure 10
Happening probability allocation results for FMs.As shown in Figure10, after carefully evaluating four performances of each event, the comprehensive weight-based allocation method not only reduces the occurrence of some FMs with high happening probabilities, but also encourages some FMs with low happening probabilities to occur more times, and this strategy is rule abiding and cost saving. As mentioned above, the lower happening probability the event has, the more difficult it declines. It is obviously an incorrect result obtained by the failure-importance-based allocation method and the failure-frequency-based allocation method because the former one assigns negative happening probabilities to the FMs with lower happening probabilities, and the latter one assigns the teeny happening probabilities. Although the allocation-number-based allocation method has a close result of happening probability reduction to the proposed method, the latter method points out some FMs to increase happening probabilities for saving cost.Hence, the proposed allocation strategy brings the most reasonable and efficient allocation scheme. Finally, it is found that the FMs especially requiring happening probability reduction are FM9 (substandard quality of sensor), FM10 (loosening of sensor connection screw), FM15 (unclean oil), FM16 (introducing impurities during assembly), FM17 (substandard quality of sealing ring), and FM35 (loosening connection turret fastener). Meanwhile, this paper advises that the happening probability of FM12 (wear of main rear piston connection pin), FM13 (substandard quality of main piston), FM14 (poorly assembled piston), and FM18 (substandard quality of pipe joint) is no need to remain so low.


## 4.1. Happening Probability Calculation

Figure3 has built the fault tree of key FFMs for the AK36100 A turret; then, the happening probability of FMs can be calculated as follows:First, seven membership functions of an event’s happening probability are established (see Figure7). Second, six engineers are invited at the AK36100 A turret manufacturing factory to express their ideas about the happening probability language levels of FMs (see Table 2). The happening probability of FMs is also calculated through equation (1) to equation (8).Figure 7
Membership function of FMs’ happening probability.Table 2
Happening probabilities of FMs estimated by experts.
Event
Exp.1
Exp.2
Exp.3
Exp.4
Exp.5
Exp.6
Happening probability
FM1
VS
VS
VS
VS
S
VS
1.13E − 07
FM2
VS
VS
VS
VS
S
S
2.10E − 07
FM3
VS
S
VS
VS
VS
S
3.04E − 07
FM4
S
S
VS
VS
VS
S
9.56E − 07
FM5
S
VS
VS
VS
S
VS
4.69E − 07
FM6
VS
S
VS
VS
VS
VS
1.73E − 07
FM7
S
VS
VS
VS
VS
S
3.90E − 07
FM8
S
S
VS
VS
VS
VS
6.35E − 07
FM9
S
S
S
S
S
S
4.24E − 06
FM10
S
S
S
RS
RS
S
8.33E − 06
FM11
S
S
VS
VS
S
S
1.58E − 06
FM12
VS
S
VS
VS
VS
S
3.04E − 07
FM13
S
VS
VS
VS
VS
S
3.90E − 07
FM14
S
VS
VS
VS
S
S
7.30E − 07
FM15
RS
S
RS
S
S
S
1.13E − 05
FM16
RS
RS
S
S
S
S
1.17E − 05
FM17
RS
S
RS
S
S
S
1.13E − 05
FM18
S
VS
VS
VS
S
S
7.30E − 07
FM19
S
VS
VS
VS
VS
S
3.90E − 07
FM20
S
VS
VS
VS
VS
S
3.90E − 07
FM21
S
RS
S
S
S
S
6.92E − 06
FM22
S
S
S
S
RS
S
5.94E − 06
FM23
S
RS
S
S
S
RS
8.59E − 06
FM24
RS
RS
S
S
S
S
1.17E − 05
FM25
S
S
VS
VS
VS
VS
6.35E − 07
FM26
S
S
VS
VS
VS
S
9.56E − 07
FM27
S
S
S
VS
VS
VS
1.32E − 06
FM28
VS
VS
S
VS
VS
VS
1.55E − 07
FM29
VS
VS
VS
VS
S
VS
1.13E − 07
FM30
S
S
S
VS
VS
VS
1.32E − 06
FM31
S
S
S
VS
VS
VS
1.32E − 06
FM32
VS
VS
S
VS
VS
VS
1.55E − 07
FM33
VS
S
VS
VS
VS
VS
1.73E − 07
FM34
S
S
VS
VS
VS
VS
6.35E − 07
FM35
RS
RS
S
S
S
S
1.17E − 05
FM36
S
RS
S
S
S
S
6.92E − 06
FM37
S
S
VS
VS
VS
VS
6.35E − 07
FM38
S
S
VS
VS
S
VS
1.11E − 06
Exp. = expert. The weight of each expert is 0.2452, 0.1980, 0.1804, 0.1471, 0.1332, and 0.0962.According to the serial relationship between high-level events and FMs, the happening probabilities of CFMs, FFMs, and SF are listed in Table3 through equation (9).Table 3
Happening probabilities of CFMs, FFMs, and SF.
Event
Happening probability
CFM1
1.58E − 06
CFM2
1.67E − 06
CFM3
1.41E − 05
CFM4
3.64E − 05
CFM5
1.47E − 05
CFM6
2.09E − 05
CFM7
2.43E − 06
CFM8
1.66E − 06
CFM9
3.61E − 05
CFM10
2.80E − 06
CFM11
1.44E − 06
CFM12
1.34E − 05
CFM13
3.97E − 05
CFM14
2.00E − 05
FFM8
1.74E − 05
FFM5
7.44E − 05
FFM7
4.06E − 05
FFM9
1.48E − 05
FFM10
5.97E − 05
SF
2.068E − 04Assuming the reliability function of the turret is exponentially distributed, the MTBF evaluated by the proposed method equals 4835 h. We tracked the reliability of 22 CNC lathes that used AK36 series turrets from July 2015 to December 2015 in a factory. The MTBF of these CNC lathes tested was 1532.3 h, and the MTBF of turrets evaluated was 5082 h using Monte Carlo simulation. By comparison, the MTBF obtained by fuzzy expert evaluation is consistent with the MTBF obtained from field data, thereby proving the rationality of the proposed method.


## 4.2. Result of Reliability Allocation

Assuming the turret’s target failure happening probability is 1.50E − 04, we allocate this target happening probability to FFMs, CFMs, and FMs through the proposed allocation strategy. Furthermore, the allocation results are compared with the results from other existing allocation weights.Firstly, FFMs are allocated diverse happening probabilities under different allocation weights, including recommended failure importance, failure frequency, complexity, and allocation number (see Figure8). The failure-frequency-based allocation weight is calculated by the conventional ARINC method, where(17)wFFMi=FFFMi∑i=15FFFMi,and the complexity-based allocation weight is calculated by the conventional AGREE method, where(18)wFFMi=nFFMi∑i=15nFFMi,where nFFMi is the number of FMs in FFMi.Figure 8
Happening probability allocation results for FFMs.As shown in Figure8, FFM5 (impossible turn of the cutter head) is given the highest happening probability in the recommended weight, which is the same as the results of other allocation weights. The happening probabilities of FFMs are declined by a similar degree based on the failure importance weight, due to the fact that these five FFMs are as important as each other, which is also verified by the RPN of FMEA in Table 1 again. The result of the allocation-number-based weight shows that only the happening probabilities of FFM5, FFM7, and FFM10 need to be declined, while the happening probability of FFM8 with the highest RPN still stays the unreasonably same. Meanwhile, the result of complexity-based weight has the similar problem, which even increases the happening probability of FFM8. The failure-frequency-based allocation method has a close result to the proposed method, but the former method is considered less than the proposed method in terms of complexity.Secondly, according to the allocation target of FFMs under the aforementioned proposed weight of failure importance, CFMs are allocated diverse happening probabilities under different allocation weights (see Figure9).Figure 9
Happening probability allocation results for CFMs.As shown in Figure9, the happening probabilities of CFM3, CFM4, CFM9, CFM12, and CFM13 are reduced in the allocation-number-based allocation method which conforms to the allocation rule that the higher happening probability the event has, the easier it declines. On the contrary, there is no doubt that the lower happening probability the event has, the more difficult it declines. However, the failure-importance-based allocation method assigns negative happening probabilities to the CFMs which have lower happening probabilities, and the failure-frequency-based allocation method assigns the teeny happening probabilities. Obviously, it is incorrect. In addition, the unreasonable allocation result that the CFM14 having high happening probability is still assigned a higher happening probability appears in the complexity-based allocation method. Furthermore, there are only five CFMs’ happening probabilities to be developed under the proposed allocation weight, while fourteen CFMs’ happening probabilities need to be done in the other weights which are more likely to be time-consuming and costly for improving the turret.Thirdly, according to the allocation target of CFMs under the aforementioned proposed weight of allocation number, FMs are allocated diverse happening probabilities under different allocation weights (see Figure10).Figure 10
Happening probability allocation results for FMs.As shown in Figure10, after carefully evaluating four performances of each event, the comprehensive weight-based allocation method not only reduces the occurrence of some FMs with high happening probabilities, but also encourages some FMs with low happening probabilities to occur more times, and this strategy is rule abiding and cost saving. As mentioned above, the lower happening probability the event has, the more difficult it declines. It is obviously an incorrect result obtained by the failure-importance-based allocation method and the failure-frequency-based allocation method because the former one assigns negative happening probabilities to the FMs with lower happening probabilities, and the latter one assigns the teeny happening probabilities. Although the allocation-number-based allocation method has a close result of happening probability reduction to the proposed method, the latter method points out some FMs to increase happening probabilities for saving cost.Hence, the proposed allocation strategy brings the most reasonable and efficient allocation scheme. Finally, it is found that the FMs especially requiring happening probability reduction are FM9 (substandard quality of sensor), FM10 (loosening of sensor connection screw), FM15 (unclean oil), FM16 (introducing impurities during assembly), FM17 (substandard quality of sealing ring), and FM35 (loosening connection turret fastener). Meanwhile, this paper advises that the happening probability of FM12 (wear of main rear piston connection pin), FM13 (substandard quality of main piston), FM14 (poorly assembled piston), and FM18 (substandard quality of pipe joint) is no need to remain so low.


## 5. Conclusions

The proposed method is more useful in solving reliability allocation issues for NC turrets in the design phase. In this method, happening probabilities of the whole failure events based on the fault tree and fuzzy expert evaluation accommodate to the early design phase of a complex system, where information for the reliability allocation is incomplete and confused. The accuracy of evaluated happening probabilities is verified by comparing the MTBF of similar turrets from field data. And three factors for reliability allocation including failure importance, allocation number, and comprehensive weight based on AHP enable the proposed allocation strategy to not only cater for the experts’ evaluating preference but also satisfy the requirement of allocation effectiveness. The validity of this proposed method is demonstrated by the case application. Compared with existing allocation methods, the proposed method is successful because it can create valid information for reliability allocation, define the complex relationship of failure events, and consider experts’ knowledge limitation. Finally, the proposed method successfully provides AK36100 A turrets with ten detailed optimizing points. It can help engineers quickly and accurately understand the turret reliability and optimizing events without field data. The engineers at the AK36100 A turret manufacturing factory consider the method to be efficient and effective.



---

*Source: 1036729-2019-10-24.xml*