A comprehensive reliability allocation method for series systems based on failure mode and effects analysis transformed functions

Abstract

In order to address the reliability allocation problem of the series system, a comprehensive reliability allocation method based on a transformed function of the failure mode and effects analysis is introduced. First, considering multiple factors that affect the reliability allocation, an allocation matrix is established by employing significance factors. Subsequently, to overcome the limitations of the conventional failure mode and effects analysis–based allocation method, non-linear transform laws of failure severity and occurrence, known as transformed functions, are established. The reliability allocation results could be adjusted appropriately by choosing transform coefficients according to the desired allocation results of the system. Then, the transformed failure mode and effects analysis and the comprehensive allocation matrix are combined to give an allocation vector. Finally, a computerized numerical controlled lathe and a spindle system are used as examples. The allocation results of the transformed method, the conventional failure mode and effects analysis method and those in the correlated references are compared to emphasize the significance of the proposed allocation method in engineering practice.

Introduction

A system is usually designed as an assembly of subsystems, each with its own reliability attributes. Reliability allocation is a process to allocate the system reliability to subsystems according to a determined principle or method. ¹ Generally, the real reliability of a mechanical system in practice, to some extent, depends on the rational pre-allocation process of subsystems.

Computerized numerical controlled (CNC) lathes are important to the manufacturing industry; thus, the reliability of CNC lathes plays an important role in the development of the manufacturing industry.^2,3 However, because of the complex structures and the various failure modes, the pre-allocation of the system often appears irrationality and lacks in credibility, which will affect the efficiency and the service life of lathes. ⁴ Therefore, it is necessary to discuss the allocation and re-allocation of CNC lathes.

Recently, researchers have developed a considerable number of approaches to allocate the system reliability to each subsystem or component. There are mainly two categories of techniques.^5,6 One is the optimal reliability allocation which focuses on the relationship between the system performance and resources. Several optimization techniques^7–13 were formulated to deal with reliability optimization problems, for example, the multi-state system optimization, the reliability redundancy allocation, the cost optimization and the multi-objective optimization.

Another category is the weighting method that the result of the reliability allocation depends on weights which are allocated to multiple factors. The equal apportionment approach^14,15 assigns equally the failure rate goal to the subsystems as the simplest method. However, one weakness of the method is that it ignores the actual difference of each subsystem. The Aeronautical Radio Inc. (ARINC) method ¹⁶ employs previous experiences of one similar system to allocate the failure rates to subsystems with a proportional allocation method, which depends on the previous similar system with basic data. The Advisory Group of Reliability of Electronic Equipment (AGREE) method ¹⁷ quantifies the relationship between system failure and component failure according to the complexity and importance of components. However, the allocation weight is only related to the number of components in each subsystem. Besides the complexity and the importance, more influential factors which affected the reliability of the subsystem are evaluated to derive reliability allocation weights in the feasibility of objectives (FOO) technique ¹⁴ and other researches.^18,19 Noting that these conventional weighting methods^14–19 give equal importance to all factors, approaches studied by Chang et al., ²⁰ Liaw et al. ²¹ and Wang et al. ²² consider weighted average method to overcome this drawback. Specifically, Wang et al. ²² introduced the relative ratio between different subsystems for each reliability allocation factor, which provides more realistic basic for the reliability allocation.

Taking the failure effects (consequences of the failure on system, which could be stated in terms of system performance, that is, hazardous, high, moderate and low) into consideration, a few recent papers^23–26 employed failure analysis information (Risk Priority Number or Criticality Number) from failure mode and effects analysis (FMEA), warranty data (failure rate or probability of failure) of systems or field failure data from a prior similar product to solve the reliability problems. The failure effect is mainly related to the failure probability and the failure severity.^27–29 FMEA could identify potential failure modes; assess the risk which is associated with the failure modes, effects and causes; and prioritize issues for corrective actions. All of these merits are helpful to measure the failure effect. The failure analysis information of these researches^23–26 is obtained under the assumption that the ratings are linear. However, the linear rating relationship has many issues in allocating reliability as discussed in section “Transformed functions of FMEA” of this study, as well as Yadav et al. ³⁰ Consequently, a non-linear relationship between the rating and failure effects in reality is required to reduce the occurrence of the unacceptable failure effects in a newly designed system. ²⁷

Just like most of mechanical systems, a CNC lathe is designed in a series system that works normally if and only if all of its subsystems (components) work normally.^31,32 To perform reliability allocation for CNC lathes, comprehensive factors should be assessed or measured. Wang et al. ²² took seven criteria to determine the allocation weight in CNC machine reliability allocation. The criteria include the failure occurrence, severity of failure, maintainability, complexity, manufacturing technology, working condition and cost. Considering the relationship between the failure occurrence, as well as the failure severity, and failure effect should be non-linear; the proposed method of this article introduces the transformed severity and occurrence ratings into the comprehensive method. The reliability allocation results could be adjusted appropriately by choosing transform coefficients according to the allocation goal of the system. Namely, the proposed method takes into account the intrinsic associations among the subsystems and could be adjusted flexibly to apportion the weights according to the need of the systems’ requirement. Two examples are applied to discuss the properties of this method.

In what follows, the comprehensive allocation method is reviewed in section “The comprehensive allocation method.” In section “Transformed functions of FMEA,” transformed functions of failure severity and failure occurrence in FMEA are introduced. Two examples are employed in section “Examples and discussion.” One is about the comparison of conventional comprehensive allocation method and the proposed method, and the other one is about the spindle system of a CNC lathe. Finally, the properties of the proposed method are summarized in section “Conclusion.”

The comprehensive allocation method

Review of the method

Assuming a series system, U can be divided into n subsystems, which can be denoted as $\mathit{U}=\{U_1,U_2,\dots ,U_n\}$ . V is the allocation criterion, assuming that there are m reliability allocation criteria: $\mathit{V}=\{V_1,V_2,\dots ,V_m\}$ . The relationship between U and V is shown in Figure 1.

Figure 1.

Reliability allocation criteria and a series system.

Considering each criterion $V_k(1\le k\le m)$ , the relative ratio for failure rate allocation between the $i\mathrm{th}$ subsystem and the $j\mathrm{th}$ subsystem is denoted as ${\beta }_{ij}^{(k)}(0<{\beta }_{ij}^{(k)}<1)$ , and ²²

$${\beta }_{ij}^{(k)}=1-{\beta }_{\mathrm{ji}}^{(k)}(i,j=1,2,\dots ,n;k=1,2,\dots ,m)$$

(1)

where if ${\beta }_{ij}^{(k)}>0.5$ , it means that the subsystem i should have higher failure rate than the subsystem j, and obviously, ${\beta }_{\mathrm{ii}}^{(k)}=0.5$ .

The failure rate allocation ratio matrix ${\mathit{B}}^{(k)}$ of the $k\mathrm{th}$ allocation criterion can be represented as follows

$${\mathit{B}}^{(k)}={\left[\begin{array}{cccc}\hfill {\beta }_{11}^{(k)}\hfill & \hfill {\beta }_{12}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {\beta }_{1n}^{(k)}\hfill \\ \multicolumn{1}{c}{{\beta }_{21}^{(k)}}& \hfill {\beta }_{22}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {\beta }_{2n}^{(k)}\hfill \\ \multicolumn{1}{c}{\cdots }& \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ \multicolumn{1}{c}{{\beta }_{n1}^{(k)}}& \hfill {\beta }_{n2}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {\beta }_{nn}^{(k)}\hfill \end{array}\right]}_{n\times n},k=1,2,\dots ,m$$

(2)

Each row of the matrix ${\mathit{B}}^{(k)}$ denotes the failure rate allocation ratio between a specific subsystem and every other subsystem. The average value of the $i\mathrm{th}$ subsystem, which represents the relative allocation coefficient of the $i\mathrm{th}$ subsystem under the $k\mathrm{th}$ criterion, is expressed as follows

$${\gamma }_{ki}=\frac{1}{n}\sum _{j=1}^n{\beta }_{ij}^{(k)},k=1,2,\dots ,m;i=1,2,\dots n$$

(3)

Considering there are m allocation criteria in a system, according to equation (3), a new allocation matrix $\mathbf{\Gamma }$ can be obtained as follows

$$\mathbf{\Gamma }={\left[\begin{array}{cccc}\hfill {\gamma }_{11}\hfill & \hfill {\gamma }_{12}\hfill & \hfill \cdots \hfill & \hfill {\gamma }_{1n}\hfill \\ \multicolumn{1}{c}{{\gamma }_{21}}& \hfill {\gamma }_{22}\hfill & \hfill \cdots \hfill & \hfill {\gamma }_{2n}\hfill \\ \multicolumn{1}{c}{\cdots }& \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ \multicolumn{1}{c}{{\gamma }_{m1}}& \hfill {\gamma }_{m2}\hfill & \hfill \cdots \hfill & \hfill {\gamma }_{mn}\hfill \end{array}\right]}_{m\times n}$$

(4)

Assuming that $a_i$ represents the comprehensive failure rate allocation coefficients, the allocation vector of the system, A , is defined as follows

$$A=\{a_1,a_2,\dots ,a_n\}$$

(5)

The allocation vector A can be written as follows

$$\mathit{A}=\mathit{W}·\mathbf{\Gamma }$$

(6)

where W is the weighting vector, and $\mathit{W}=\{w_1,w_2,\dots ,w_m\}$ . $w_k(1\le k\le m)$ denotes the weight of the $k\mathrm{th}$ allocation criterion.

The weight vector W can be obtained by the expert rating method. Assuming that N experts are involved, the weight of the $k\mathrm{th}$ criterion is as follows

$$w{\prime }_k=\frac{1}{N}\sum _{r=1}^N{w}_{kr},k=1,2,\dots ,m$$

(7)

where $w_{kr}$ is the rating value for the $k\mathrm{th}$ criterion from the $r\mathrm{th}$ expert.

Then, the elements of the weight vector W are as follows

$$w_k=\frac{{w\prime }_k}{{\displaystyle \sum }_{j=1}^m{w\prime }_j},k=1,2,\dots ,m$$

(8)

Substituting equations (8) and (4) into equation (6), the allocation vector A can be obtained. To each subsystem, $U_i,i=1,2,\dots ,n$ , the allocation of failure rate satisfies the following relationship

$${\lambda }_1:{\lambda }_2:\cdots :{\lambda }_n=a_1:a_2:\cdots :a_n$$

(9)

Therefore, the coefficients of each subsystem’s failure rate are $a_i/\sum _{j=1}^n{a}_j$ , then

$${\lambda }_i=\frac{a_i}{{\displaystyle \sum }_{j=1}^n{a}_j}{\lambda }_s^{*}$$

(10)

where ${\lambda }_s^{*}$ is the desired failure rate.

Here, we assume that each subsystem fails independently. For mission time t, the reliability of the series system could be given as follows ²³

$$R_s=\underset{i=1}{\overset{n}{\Pi }}{R}_i=\underset{i=1}{\overset{n}{\Pi }}{e}^{-{\lambda }_{i}t}$$

(11)

where $R_i$ is the reliability of subsystems.

Relative ratios for failure rate

According to the comprehensive allocation method, the key point of using this method is to calculate the relative ratio ${\beta }_{ij}^{(k)}(0<{\beta }_{ij}^{(k)}<1)$ . In order to properly compare the proposed method with the previous method in Example 1 of section “Examples and discussion,” seven criteria are chosen according to Wang et al. ²² in this article. They include the failure occurrence $(k=1)$ , the failure severity $(k=2)$ , the maintainability $(k=3)$ , the complexity $(k=4)$ , the manufacturing technology $(k=5)$ , the working condition $(k=6)$ and the cost $(k=7)$ . The approach for calculating these criteria is directly listed as follows:

• When $k=1,4,5,6,7$ , the relative ratio is as follows

$${\beta }_{ij}^{(k)}=\frac{{\mathrm{\Delta }}_i}{{\mathrm{\Delta }}_i+{\mathrm{\Delta }}_j}$$

(12a)

where ${\mathrm{\Delta }}_i,{\mathrm{\Delta }}_j$ represent the failure occurrence, the complexity, the manufacturing technology, the working condition or the cost of the $i\mathrm{th}$ and the $j\mathrm{th}$ subsystem, respectively.

• When $k=2$ , the relative ratio is as follows

$${\beta }_{ij}^{(2)}=\frac{{\mathrm{\Delta }}_j}{{\mathrm{\Delta }}_i+{\mathrm{\Delta }}_j}$$

(12b)

where ${\mathrm{\Delta }}_i,{\mathrm{\Delta }}_j$ represent the failure severity of the $i\mathrm{th}$ and the $j\mathrm{th}$ subsystem, respectively.

• When $k=3$ , the relative ratio of the maintainability ${\beta }_{ij}^{(3)}$ is as follows

$${\beta }_{ij}^{(3)}=\frac{T_{Dj}+T_{Hj}}{T_{Di}+T_{Hi}+T_{Dj}+T_{Hj}}$$

(12c)

where $T_{Di},T_{Hi}$ stand for the mean down time and the mean repair cost of the $i\mathrm{th}$ subsystem, respectively; and $T_{Dj},T_{Hj}$ denote the mean down time and the mean repair cost of the $j\mathrm{th}$ subsystem, respectively.

According to equation (12a), in order to reduce the machining and maintenance cost, the failure rate of the subsystems with high failure occurrence, high complexity, advanced manufacturing technology, good working condition and high cost should be allocated a higher value than other subsystems. Meanwhile, in equations (12b) and (12c), for the subsystems with high failure severity and poor maintainability, the failure rate should be allocated less to ensure the system running smoothly.

Transformed functions of FMEA

FMEA is a systematic method designed to identify the potential failure modes, the failure causes and the failure effects on the system or the given product. ³³ In FMEA, failure modes are classified using the severity rating and the occurrence rating. ³⁴ In this section, transformed functions are employed to modify the ratings, such that a more rational and realistic allocation result is obtained.

Transformed function of failure severity

According to equation (12a), the relative ratios for failure severity of subsystems are as follows

$${\beta }_{ij}^{(2)}=\frac{S_j}{S_i+S_j},(i,j=1,2,\dots n)$$

(13)

where $S_i,S_j$ are the failure severity of the $i\mathrm{th}$ and the $j\mathrm{th}$ subsystem, respectively.

In the conventional FMEA, the failure severity S is a linear function, which means that the failure severity is divided into 10 ranks according to the failure effects, and the score of each rank increases in a linear trend. However, there are some limitations in the reliability allocation of series systems with this method. ³⁵ For example, in a series system with failure modes of $f(f=1,2,\dots ,R)$ , the severity of the $i_1\mathrm{th}$ and the $j_1\mathrm{th}$ subsystems is $S_{fi1}=1$ and $S_{fi1}=2$ , respectively. Meantime, the severity of the $i_2\mathrm{th}$ and the $j_2\mathrm{th}$ subsystems is $S_{fi2}=5$ and $S_{fj2}=10$ , respectively. If $R=1$ , namely, there is only one failure mode, then the relative ratio in the two subsystems will be the same

$${\beta }_{i1j1}^{(2)}={\beta }_{i2j2}^{(2)}=\frac{2}{3};{\beta }_{j1i1}^{(2)}={\beta }_{j2i2}^{(2)}=\frac{1}{3}$$

(14)

According to equation (14), higher failure rate (or lower reliability) will be allocated to the subsystem with low severity. Although the difference, of the subsystem $i_1$ and $j_1$ , is far less than that of the subsystem $i_2$ and $j_2(|S_{i1}-S_{j1}|<<|S_{i2}-S_{j2}|)$ , the allocation ratios in the two subsystem pairs are the same. This phenomenon seems not very reasonable and logical. Because, in general, the subsystems $i_1$ and $j_1$ should have the similar allocation ratio, while the ratio of the subsystem $i_2$ should be larger than that of the subsystem $j_2$ . To address this issue, Kim et al. ²⁷ presented a transformed function of failure severity

$${\tilde{S}}_{fi}=e^{\left(\alpha S_{fi}\right)}$$

(15)

where ${\tilde{S}}_{fi}$ represents the transformed failure severity, and $\alpha$ is the severity modified coefficient.

In FMEA, the severity and occurrence of a failure mode are ranked based on an ordinal scale from 1 to 10. Figure 2 depicts the curves of transformed failure severity ${\tilde{S}}_{fi}$ under different $\alpha (\alpha >0)$ and the failure severity $S_{fi}$ , with only one failure mode. Moreover, the failure effect is illustrated in horizontal axis, corresponding to the failure severity.

Figure 2.

The transformed severity and the severity.

In the example above, $R=1$ , there goes

$${\beta }_{j1i1}^{(2)}=\frac{{\tilde{S}}_{i1}}{{\tilde{S}}_{i1}+{\tilde{S}}_{j1}}=\frac{{\mathrm{e}}^{\left(\alpha S_{i1}\right)}}{{\mathrm{e}}^{\left(\alpha S_{i1}\right)}+{\mathrm{e}}^{\left(\alpha S_{j1}\right)}}=\frac{1}{1+{\mathrm{e}}^{\alpha }}$$

(16a)

$${\beta }_{j2i2}^{(2)}=\frac{{\tilde{S}}_{i2}}{{\tilde{S}}_{i2}+{\tilde{S}}_{j2}}=\frac{{\mathrm{e}}^{\left(\alpha S_{i2}\right)}}{{\mathrm{e}}^{\left(\alpha S_{i2}\right)}+{\mathrm{e}}^{\left(\alpha S_{j2}\right)}}=\frac{1}{1+{\mathrm{e}}^{5\alpha }}$$

(16b)

Because ${\beta }_{j2i2}^{(2)}<{\beta }_{j1i1}^{(2)}$ , the subsystem $j_2$ (rank = 10) will obtain lower relative failure severity than the subsystem $j_1$ (rank = 2). That is, the subsystem $j_2$ (rank = 10) will be allocated higher reliability than the subsystem $j_1$ (rank = 2). Furthermore, a suitable value of $\alpha$ should be selected flexibly in accordance with the requirement that how seriously the designer takes the failure severity. If the designer attempts to take the failure severity 10 more seriously in the example given, then the value $\alpha$ will be selected higher. Eventually, a more reasonable and appropriate reliability allocation could be obtained after $S_{fi}$ is replaced by ${\tilde{S}}_{fi}$ .

According to equation (12), in order to obtain the relative ratio of the failure severity, the transformed severities ${\tilde{S}}_i,{\tilde{S}}_j$ of the $i_2\mathrm{th}$ and the $j_2\mathrm{th}$ subsystems are required. Obviously, when $R=1$ , we have ${\tilde{S}}_i={\tilde{S}}_{fi}$ . When $R>1$ , there are two main approaches to calculate the failure severity of a subsystem: the maximum method ²² and the weighted mean method. ³⁶

The maximum method is as follows

$${\tilde{S}}_i=\max ({\tilde{S}}_{1i},{\tilde{S}}_{2i},\dots ,{\tilde{S}}_{Ri})$$

(17a)

The weighted mean method is as follows

$${\tilde{S}}_i=\sum _{f=1}^R{h}_f{\tilde{S}}_{fi}$$

(17b)

where $h_f$ is the weight of the $f\mathrm{th}$ failure modes, $\sum _{f=1}^R{h}_f$ and $h_f>0$ .

In equation (17b), when there is a lack of $h_f$ , the equation can be written as follows

$${\tilde{S}}_i=\frac{1}{R}\sum _{f=1}^R{\tilde{S}}_{fi}$$

(17c)

According to equation (17), the maximum method focuses on one failure mode’s effect, and it cannot reflect the comprehensive effect of all failure modes. Therefore, equation (17b) or (17c) is chosen in the following analysis.

The severity relative ratio ${\beta }_{ij}^{(2)}$ after transformation is as follows

$${\beta }_{ij}^{(2)}=\frac{{\tilde{S}}_j}{{\tilde{S}}_i+{\tilde{S}}_j},(i,j=1,2,\dots n)$$

(18)

Transformed function of failure occurrence

In FMEA, for the failure occurrence of the subsystem i under the $f\mathrm{th}$ failure mode, $O_{fi}$ , there also exists the same limitation as the failure severity discussed above, which means that the linear distribution of the FMEA rate cannot reflect the effect of the failure occurrence to a system’s reliability. According to equation (12a), the relative ratio of a subsystem’s failure occurrence is as follows

$${\beta }_{ij}^{(1)}=\frac{O_i}{O_i+O_j},(i,j=1,2,\dots n)$$

(19)

According to the analysis in section “Transformed function of failure severity,” the transformed failure occurrence is directly given as follows

$${\tilde{O}}_{fi}=e^{\left(\gamma O_{fi}\right)}$$

(20)

where ${\tilde{O}}_{fi}$ is the transformed failure occurrence, and $\gamma$ is the modified coefficient of failure occurrence.

Similarly, the transformed failure occurrence of a subsystem can be defined as follows

$${\tilde{O}}_i=\sum _{f=1}^R{h}_f{\tilde{O}}_{fi}$$

(21a)

When there is a lack of $h_f$ , equation (21a) can be written as follows

$${\tilde{O}}_i=\frac{1}{R}\sum _{f=1}^R{\tilde{O}}_{fi}$$

(21b)

The transformed relative ratio of failure occurrence ${\beta }_{ij}^{(1)}$ is defined as follows

$${\beta }_{ij}^{(1)}=\frac{{\tilde{O}}_i}{{\tilde{O}}_i+{\tilde{O}}_j},(i,j=1,2,\dots n)$$

(22)

Examples and discussion

Based on the reliability allocation method proposed above, two examples are employed to demonstrate the application of this method, as well as investigate the proprieties of the allocation results. The first example is the same as that in Wang et al., ²² in order to verify and analyze the proposed method. The second example is to allocate reliability to subsystems of a CNC lathe’s spindle system by the presented allocation approach.

Example 1

In general, one CNC lathe can be divided into 15 subsystems. They are the turret, the clamping accessory, the electric and electronic system, the main transmission, the X feed system, the Z feed system, the CNC system, the power supply, the hydraulic system, the servo system, the cooling system, the swarf conveyors, the lubricant system, the spindle assembly and the guard. For the lack of the original FMEA data of each failure mode, we approximately evaluate the failure severity $S_i$ using the severity of failure ${\overline{S}}_i$ given by Wang et al. ²² and assume that $\alpha =0.8$ , then the transformed failure severity ${\tilde{S}}_i$ is obtained. ${\overline{S}}_i,S_i\mathrm{and}{\tilde{S}}_i$ are shown in Table 1. Let $\gamma =0.8$ , the relative failure occurrence ${\overline{O}}_i$ , FMEA failure occurrence $O_i$ and the transformed failure occurrence ${\tilde{O}}_i$ are shown in Table 2. In addition, the mean down time $T_D$ and the mean repair cost $T_H$ , the complexity $\hat{C}$ , the manufacturing technology $\hat{M}$ , the working condition $\hat{E}$ and the cost $C_0$ are listed in Tables 3–5. The cost $C_0$ is defined as the ratio ${\delta }_C/{\delta }_R$ of the cost increment ${\delta }_C$ and the reliability increment ${\delta }_R$ , and the rating scores of $\hat{M},\hat{E}\mathrm{and}{C}_0$ should be in the range of 0–1. ²²

Table 1.

The failure severities of subsystems.

Symbol	Subsystems	${\overline{S}}_i/\times {10}^{-4}$ 22	$S_i$	${\tilde{S}}_i/\times {10}^3$
M	Turret	3.177	10	2.9810
J	Clamping accessory	2.813	9	1.3394
V	Electric and electronic systems	2.092	8	0.6018
S1	Main transmission	1.693	7	0.2704
X	X feed system	1.402	5	0.0546
Z	Z feed system	1.427	5	0.0546
NC	CNC system	1.575	6	0.1215
E	Power supply	1.418	5	0.0546
D	Hydraulic system	0.834	3	0.0110
F	Servo system	1.368	4	0.0245
W	Cooling system	1.219	4	0.0245
K	Swarf conveyors	0.7905	3	0.0110
L	Lubricant system	0.48	2	0.0050
S2	Spindle assembly	0.4308	2	0.0050
Q	Guard	0.3148	1	0.0022

NC: numerical control; CNC: computerized numerical control.

Table 2.

The failure occurrences of subsystems.

Symbol	Subsystems	${\overline{O}}_i$ 22	$O_i$	${\tilde{O}}_i/\times {10}^3$
M	Turret	0.269	10	2.9810
J	Clamping accessory	0.099	9	1.3394
V	Electric and electronic systems	0.096	9	1.3394
S1	Main transmission	0.084	8	0.6018
X	X feed system	0.075	7	0.2704
Z	Z feed system	0.065	6	0.1215
NC	CNC system	0.063	6	0.1215
E	Power supply	0.049	5	0.0546
D	Hydraulic system	0.041	4	0.0245
F	Servo system	0.038	4	0.0245
W	Cooling system	0.035	4	0.0245
K	Swarf conveyors	0.030	3	0.0110
L	Lubricant system	0.029	3	0.0110
S2	Spindle assembly	0.013	2	0.0050
Q	Guard	0.008	1	0.0022

NC: numerical control; CNC: computerized numerical control.

Table 3.

The mean down time and the mean repair cost of subsystems.

Symbol	Subsystems	$T_D$	$T_H$
M	Turret	2.72	1.97
J	Clamping accessory	1.99	1.16
V	Electric and electronic system	0.75	0.50
S1	Main transmission	2.64	1.03
X	X feed system	2.38	1.35
Z	Z feed system	2.00	1.92
NC	CNC system	1.03	1.01
E	Power supply	0.76	0.55
D	Hydraulic system	1.82	0.98
F	Servo system	0.89	2.42
W	Cooling system	3.05	2.76
K	Swarf conveyors	4.04	1.71
L	Lubricant system	2.33	1.17
S2	Spindle assembly	2.89	1.37
Q	Guard	4.79	2.36

NC: numerical control; CNC: computerized numerical control.

Table 4.

The complexities $\hat{C}$ of subsystems.

Subsystems	M	J	V	S1	X	Z	NC	E	D	F	W	K	L	S2	Q
Number of components	300	150	250	80	100	100	30	50	80	50	40	50	30	60	10

NC: numerical control.

Table 5.

The manufacturing technology $\hat{M}$ , the working condition $\hat{E}$ and the cost $C_0$ .

Subsystems	M	J	V	S1	X	Z	NC	E	D	F	W	K	L	S2	Q
Manufacturing technology $\hat{M}(k=5)$	0.98	0.70	0.26	0.34	0.40	0.40	0.14	0.20	0.40	0.20	0.30	0.44	0.34	0.38	0.10
Working condition $\hat{E}(k=6)$	0.96	0.76	0.20	0.44	0.40	0.40	0.10	0.24	0.34	0.14	0.30	0.56	0.26	0.44	0.14
Cost $C_0(k=7)$	0.98	0.70	0.30	0.58	0.50	0.50	0.14	0.26	0.40	0.20	0.36	0.46	0.34	0.54	0.10

NC: numerical control.

According to Tables 1–5, we can calculate an allocation matrix $\mathbf{\Gamma }$ through equations (1)–(4), where the failure severity and the failure occurrence of subsystems are replaced by ${\tilde{S}}_i$ and ${\tilde{O}}_i$

$$\mathbf{\Gamma }=\left\{\begin{array}{ccccccccccccccc}\hfill 0.90\hfill & \hfill 0.84\hfill & \hfill 0.84\hfill & \hfill 0.76\hfill & \hfill 0.67\hfill & \hfill 0.58\hfill & \hfill 0.58\hfill & \hfill 0.47\hfill & \hfill 0.36\hfill & \hfill 0.36\hfill & \hfill 0.36\hfill & \hfill 0.26\hfill & \hfill 0.26\hfill & \hfill 0.17\hfill & \hfill 0.10\hfill \\ \multicolumn{1}{c}{0.08}& \hfill 0.13\hfill & \hfill 0.19\hfill & \hfill 0.27\hfill & \hfill 0.47\hfill & \hfill 0.47\hfill & \hfill 0.36\hfill & \hfill 0.47\hfill & \hfill 0.70\hfill & \hfill 0.59\hfill & \hfill 0.59\hfill & \hfill 0.70\hfill & \hfill 0.80\hfill & \hfill 0.80\hfill & \hfill 0.88\hfill \\ \multicolumn{1}{c}{0.42}& \hfill 0.52\hfill & \hfill 0.72\hfill & \hfill 0.48\hfill & \hfill 0.48\hfill & \hfill 0.47\hfill & \hfill 0.62\hfill & \hfill 0.71\hfill & \hfill 0.55\hfill & \hfill 0.51\hfill & \hfill 0.38\hfill & \hfill 0.38\hfill & \hfill 0.49\hfill & \hfill 0.45\hfill & \hfill 0.33\hfill \\ \multicolumn{1}{c}{0.79}& \hfill 0.67\hfill & \hfill 0.76\hfill & \hfill 0.54\hfill & \hfill 0.59\hfill & \hfill 0.59\hfill & \hfill 0.33\hfill & \hfill 0.44\hfill & \hfill 0.54\hfill & \hfill 0.44\hfill & \hfill 0.39\hfill & \hfill 0.44\hfill & \hfill 0.33\hfill & \hfill 0.48\hfill & \hfill 0.16\hfill \\ \multicolumn{1}{c}{0.74}& \hfill 0.68\hfill & \hfill 0.45\hfill & \hfill 0.51\hfill & \hfill 0.55\hfill & \hfill 0.55\hfill & \hfill 0.32\hfill & \hfill 0.39\hfill & \hfill 0.55\hfill & \hfill 0.39\hfill & \hfill 0.48\hfill & \hfill 0.57\hfill & \hfill 0.51\hfill & \hfill 0.54\hfill & \hfill 0.25\hfill \\ \multicolumn{1}{c}{0.74}& \hfill 0.69\hfill & \hfill 0.40\hfill & \hfill 0.57\hfill & \hfill 0.55\hfill & \hfill 0.55\hfill & \hfill 0.26\hfill & \hfill 0.43\hfill & \hfill 0.52\hfill & \hfill 0.32\hfill & \hfill 0.49\hfill & \hfill 0.63\hfill & \hfill 0.45\hfill & \hfill 0.57\hfill & \hfill 0.32\hfill \\ \multicolumn{1}{c}{0.71}& \hfill 0.65\hfill & \hfill 0.45\hfill & \hfill 0.60\hfill & \hfill 0.57\hfill & \hfill 0.57\hfill & \hfill 0.29\hfill & \hfill 0.42\hfill & \hfill 0.52\hfill & \hfill 0.36\hfill & \hfill 0.50\hfill & \hfill 0.55\hfill & \hfill 0.48\hfill & \hfill 0.59\hfill & \hfill 0.23\hfill \end{array}\right\}$$

(23)

By the method of expert rating, the weight W is obtained as follows

$$\mathit{W}=\left\{\begin{array}{ccccccc}\hfill 0.20\hfill & \hfill 0.10\hfill & \hfill 0.08\hfill & \hfill 0.14\hfill & \hfill 0.17\hfill & \hfill 0.15\hfill & \hfill 0.16\hfill \end{array}\right\}$$

(24)

According to equation (6), the allocation vector A is as follows

$$\begin{array}{c}\mathit{A}=\{\begin{array}{ccccc}\hfill 0.68\hfill & \hfill 0.64\hfill & \hfill 0.56\hfill & \hfill 0.56\hfill & \hfill 0.57\hfill \end{array}\hfill \\ \begin{array}{ccccc}\hfill 0.55\hfill & \hfill 0.39\hfill & \hfill 0.46\hfill & \hfill 0.52\hfill & \hfill 0.41\hfill \end{array}\hfill \\ \begin{array}{ccccc}\hfill 0.45\hfill & \hfill 0.49\hfill & \hfill 0.45\hfill & \hfill 0.49\hfill & \hfill 0.28\hfill \end{array}\}\hfill \end{array}$$

(25)

The desired mean time between failures (MTBF) of the CNC lathe is specified as 500 h, in other words, ${\lambda }_s^{*}=0.002$ , and the desired reliability of the system is greater than or equal to 0.998. Substituting ${\lambda }_s^{*}$ into equation (10) yields

$$\begin{array}{c}\left\{{\lambda }_i\right\}=\{\begin{array}{ccccc}\hfill 1.82\hfill & \hfill 1.70\hfill & \hfill 1.49\hfill & \hfill 1.50\hfill & \hfill 1.52\hfill \end{array}\hfill \\ \begin{array}{ccccc}\hfill 1.47\hfill & \hfill 1.03\hfill & \hfill 1.23\hfill & \hfill 1.38\hfill & \hfill 1.08\hfill \end{array}\hfill \\ \begin{array}{ccccc}\hfill 1.20\hfill & \hfill 1.32\hfill & \hfill 1.20\hfill & \hfill 1.30\hfill & \hfill 0.76\hfill \end{array}\}\times {10}^{-4}\hfill \end{array}$$

(26)

In equation (26), the reciprocal of each element is the MTBF. Table 6 shows the result of subsystems’ MTBF allocation, ²² the allocated results of the conventional FMEA $(O_i,S_i)$ and the allocated results of the proposed method $({\tilde{O}}_i,{\tilde{S}}_i)$ .

Table 6.

The comparisons of the reliability allocation methods.

Symbol	Subsystems	Observed MTBF/h	Allocated MTBF/h 22	Conventional FMEA $O_i,S_i/h$	Transformed FMEA ${\tilde{O}}_i,{\tilde{S}}_i/h$
M	Turret	752.14	5523.56	5680.28	5489.43
J	Clamping accessory	2044.06	6140.60	6049.73	5882.70
V	Electric and electronic system	2118.84	7067.28	6967.94	6700.56
S1	Main transmission	2413.13	6890.52	6848.95	6654.85
X	X feed system	2714.77	6318.62	6776.49	6575.59
Z	Z feed system	3102.59	6398.46	6875.05	6816.43
NC	CNC system	3217.50	9679.27	9657.02	9705.08
E	Power supply	4136.79	8154.97	8029.57	8166.12
D	Hydraulic system	4964.14	7230.30	7151.62	7261.30
F	Servo system	5265.00	9222.45	8953.74	9236.96
W	Cooling system	5791.50	8305.45	8086.74	8317.32
K	Swarf conveyors	6682.50	7412.56	7362.13	7601.24
L	Lubricant system	6949.80	8346.89	8060.15	8323.71
S2	Spindle assembly	15795.00	7645.18	7437.08	7673.18
Q	Guard	24820.71	13099.09	12741.44	13187.74

NC: numerical control; CNC: computerized numerical control; MTBF: mean time between failures; FMEA: failure mode and effects analysis.

Some conclusions indicated by the example results are listed as follows:

1. It can be seen from Table 6 that the allocated results obtained by the conventional FMEA method and transformed FMEA are similar to the reference, which indicates the applicability of the two methods.

2. In Table 2, the FMEA rates of the subsystem M, J, V and S1 are all over 8, which indicates that the failure occurrences of these subsystems are relatively higher. Comparing the fifth and the sixth column data of Table 6, the transformed FMEA method apportions lower MTBF to the four subsystems than the Wang et al. ²² and conventional FMEA method. It shows that the proposed method emphasizes the effect of failure occurrence $O_i$ in reliability allocation.

3. Consider L, S2 and Q. The failure severities and the failure occurrences are relatively low, ranking with 3, 2 and 1. In Table 6, the MTBF obtained by the transformed FMEA method is closer to the reference than the conventional FMEA method, while they are also all higher than the conventional FMEA method. It reflects that the effect of low severity and occurrence is decreased by proposed method.

4. Based on allocated results of the transformed FMEA method, the original and actual reliability of the allocated system $(t=1)$ could be obtained according to equation (11)

$$R_s=\underset{i=1}{\overset{15}{\Pi }}{R}_i=\underset{i=1}{\overset{15}{\Pi }}{e}^{-{\lambda }_{i}t}=0.998002>0.998$$

(27)

Obviously, the original and actual reliability of the system meets the desired requirement.

Example 2

The second example is focused on a spindle system of a CNC lathe. In this study, six criteria are under consideration: the failure severity $(k=1)$ , the failure occurrence $(k=2)$ , the complexity $(k=3)$ , the manufacturing technology $(k=4)$ , the working condition $(k=5)$ and the cost $(k=6)$ .The failure severity and occurrence of each failure modes are listed in Table 7, while the transformed severity and occurrence of subsystems are calculated according to equations (15), (17c), (20) and (21b). The values of the other four criteria are shown in Table 8, which are derived from expert ratings.

Table 7.

The FMEA of the spindle system.

	Subsystems	Failure modes	Occurrence, $O_{fi}$	Transformed occurrence, ${\tilde{O}}_{fi}$ $(\gamma =0.8)$	Subsystem’s failure occurrence, ${\tilde{O}}_i$	Severity, $S_{fi}$	Transformed severity, ${\tilde{S}}_{fi}$ $(\alpha =0.6)$	Subsystem’s failure severity, ${\tilde{S}}_i$
1	Support	FM11	4	24.53	39.57	3	6.05	4.685
		FM12	5	54.60		2	3.32
2	Measuring assembly	FM21	7	270.43	270.43	8	121.51	121.51
3	Front support	FM31	2	4.95	43.80	4	11.02	16.98
		FM32	6	121.51		2	3.32
		FM33	2	4.95		6	36.60
4	Back support	FM41	2	4.95	43.80	4	11.02	16.98
		FM42	6	121.51		2	3.32
		FM43	2	4.95		6	36.60
5	Front Seal	FM51	3	11.02	11.02	6	36.60	36.60
6	Back seal	FM61	3	11.02	11.02	6	36.60	36.60
7	Driving assembly	FM71	8	601.45	312.99	3	6.05	13.07
		FM72	4	24.53		5	20.09
8	Braking assembly	FM81	5	54.60	29.78	9	221.41	144.05
		FM82	2		4.95	7	66.69

Table 8.

Criterion values of subsystems from $k=3$ to $k=6$ .

Subsystems	1	2	3	4	5	6	7	8
Number of components, $\hat{N}(k=3)$	1	4	8	5	5	3	4	3
Manufacturing technology, $\hat{M}(k=4)$	0.45	0.54	0.54	0.73	0.39	0.31	0.67	0.48
Working condition, $\hat{E}(k=5)$	0.44	0.58	0.55	0.52	0.32	0.26	0.69	0.49
Cost, $C_0(k=6)$	0.46	0.61	0.58	0.53	0.37	0.31	0.65	0.50

According to the process of Example 1, we can obtain the allocation matrix as follows

$$\mathbf{\Gamma }=\left\{\begin{array}{cccccccc}\hfill 0.478\hfill & \hfill 0.797\hfill & \hfill 0.498\hfill & \hfill 0.498\hfill & \hfill 0.245\hfill & \hfill 0.245\hfill & \hfill 0.816\hfill & \hfill 0.423\hfill \\ \multicolumn{1}{c}{0.814}& \hfill 0.236\hfill & \hfill 0.599\hfill & \hfill 0.599\hfill & \hfill 0.447\hfill & \hfill 0.447\hfill & \hfill 0.648\hfill & \hfill 0.211\hfill \\ \multicolumn{1}{c}{0.231}& \hfill 0.521\hfill & \hfill 0.676\hfill & \hfill 0.572\hfill & \hfill 0.572\hfill & \hfill 0.454\hfill & \hfill 0.521\hfill & \hfill 0.454\hfill \\ \multicolumn{1}{c}{0.476}& \hfill 0.520\hfill & \hfill 0.520\hfill & \hfill 0.593\hfill & \hfill 0.441\hfill & \hfill 0.386\hfill & \hfill 0.573\hfill & \hfill 0.491\hfill \\ \multicolumn{1}{c}{0.488}& \hfill 0.556\hfill & \hfill 0.543\hfill & \hfill 0.529\hfill & \hfill 0.411\hfill & \hfill 0.363\hfill & \hfill 0.597\hfill & \hfill 0.514\hfill \\ \multicolumn{1}{c}{0.485}& \hfill 0.555\hfill & \hfill 0.542\hfill & \hfill 0.520\hfill & \hfill 0.432\hfill & \hfill 0.390\hfill & \hfill 0.570\hfill & \hfill 0.506\hfill \end{array}\right\}$$

(28)

The reliability allocation weight vector W is given as follows

$$\mathit{W}=\left\{\begin{array}{cccccc}\hfill 0.22\hfill & \hfill 0.11\hfill & \hfill 0.15\hfill & \hfill 0.18\hfill & \hfill 0.16\hfill & \hfill 0.18\hfill \end{array}\right\}$$

(29)

According to equation (6), the allocation vector is as follows

$$\begin{array}{c}\mathit{A}=\{\begin{array}{cccc}\hfill 0.480\hfill & \hfill 0.562\hfill & \hfill 0.555\hfill & \hfill 0.546\hfill \end{array}\hfill \\ \begin{array}{cccc}\hfill 0.412\hfill & \hfill 0.369\hfill & \hfill 0.630\hfill & \hfill 0.446\hfill \end{array}\}\hfill \end{array}$$

(30)

Here, we specify that the desired MTBF is 1500 h, which means ${\lambda }_s^{*}=0.00067$ , and the desired reliability of the system is greater than or equal to 0.99923. Substituting ${\lambda }_s^{*}$ into equation (10) yields

$$\begin{array}{c}\left\{{\lambda }_i\right\}=\{\begin{array}{cccc}\hfill 0.402\hfill & \hfill 0.471\hfill & \hfill 0.465\hfill & \hfill 0.458\hfill \end{array}\hfill \\ \begin{array}{cccc}\hfill 0.345\hfill & \hfill 0.309\hfill & \hfill 0.528\hfill & \hfill 0.374\hfill \end{array}\}\times {10}^{-4}\hfill \end{array}$$

(31)

The MTBF of subsystems are as follows

$$\begin{array}{c}\left\{\mathrm{MTB}{\mathrm{F}}_i\right\}=\{\begin{array}{ccc}\hfill 24862.73\hfill & \hfill 21253.99\hfill & \hfill 21522.65\hfill \end{array}\hfill \\ \begin{array}{ccc}\hfill 21859.68\hfill & \hfill 28991.00\hfill & \hfill 323.6549\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill 18951.67\hfill & \hfill 267.6387\hfill \end{array}\}/h\hfill \end{array}$$

(32)

Based on allocated results, the original and actual reliability of the system $(t=1)$ can be obtained according to equation (11) as follows

$$R_s=\underset{i=1}{\overset{8}{\Pi }}{R}_i=\underset{i=1}{\overset{8}{\Pi }}{e}^{-{\lambda }_{i}t}=0.99967>0.99923$$

(33)

It is shown that the result satisfies the desired requirement of the spindle system reliability, and then, the allocated results are acceptable.

Conclusion

This study establishes an improved comprehensive allocation method for series systems based on the transformed functions of FMEA. In two examples, one concerned with the CNC lathe system and one with the spindle system, the applicability and the properties of this proposed method are discussed. Specific conclusions are listed as follows:

1. The transformed functions of the failure severity and the failure occurrence in FMEA are established. The transformed functions are defined as exponential functions: ${\tilde{S}}_{fi}={\mathrm{e}}^{(\alpha S_{fi})}$ , ${\tilde{O}}_{fi}={\mathrm{e}}^{(\gamma O_{fi})}$ .

2. Based on the transformed functions of FMEA, a new allocation weight matrix of series systems is established by considering multiple criteria, as shown in equation (4). The matrix could be adjusted by changing the coefficients $\alpha$ and $\gamma$ to assign the reliability properly.

3. Two examples are employed to verify the applicability of the proposed method. In Example 1, the new allocation results and those in the given reference are compared, indicating that the transformed method could emphasize the effect of high failure occurrence and minimize the effect of low failure severity and occurrence.

4. As shown in Example 2, the proposed method is applied to the spindle system of a CNC lathe. The reliability allocation results which could be adjusted appropriately by choosing the transform coefficients based on the demand of the systems are satisfied. The results indicate the advantages of the proposed approach and emphasize its significance in engineering practice.