1 Introduction

Reliability distribution takes great role in designing which mainly leads to the level of reliability for products, so it requires more attention for technicians. In regular way, failure rate C_i¹ or MTBF is regarded as the index of Reliability distribution^[1-3] in numerical control lathe, and failure frequency, failure fatality, parts maintenance, parts complexity, manufacturing technique level, environment condition, expenditure are all taken as elements of distribution^[4-9], and regular distribution or fuzzy comprehensive evaluation is adopted to make Reliability distribution for lathe subsystems, which is only suit for primary designing and is the most simply method when system function is not clear. To find out lower reliability subsystem is so important that designing idea can be realized, and then the whole reliability of system can be improved by corresponding measures. In this paper, the complete machine reliability^[10] affected by subsystems, correlation among subsystems, maintenance, complexity of machine parts, technical level, spending cost, all these elements have been taken to decide how to distribute reliability, and the information entropy theory^[11] has been adopted to calculate the weight of every index, and the TOPSIS method has been presented to make a sequence of all the subsystems, and the Copula theory has been introduced to build Reliability distribution model, and all the reliability index range of subsystems will be acquired by means of Matlab. This paper will be providing with theory basis^[12] and technical support for reliability design of numerical control lathe.

2 Calculation of Influence Factors for Reliability Distribution 2.1 Reliability Influence Degree from Subsystem 2.1.1 Reliability static influence degree

Failure times of system depend on failure times of subsystems^[13], whose influence degree (C_i¹) can be viewed as one of static analyzing indexes. C_i¹ is to be a relative index, and it can express the percentage of failure times of system triggered by subsystems failure times, and its model can be represented by:

$C_i^1 = \frac{{{n_i}}}{N}$

(1)

Where n_i stands for failure times of the complete machine triggered by the ith subsystem failure during the time between (0, T); N stands for total failure times of the complete machine during the time between (0, T); T stands for total time of evaluation^[14].

The duration of subsystem failure also has the effect on the complete machine, therefore the influence degree of system quitting (C_i²) can be viewed as another index of reliability static analyzing^[15], and C_i² is also to be a relative index, representing the percentage of downtime of the complete machine induced by subsystem failure, and its model can be expressed by:

$C_i^2 = \frac{{{t_i}}}{{{T_D}}}$

(2)

where t_i stands for downtime of the complete machine triggered by the ith subsystem down during the time between (0, T); T_D stands for total downtime of the complete machine during the time between (0, T); T stands for total time of evaluation.

2.1.2 Reliability dynamic core influence degree

Numerical control lathe can be viewed as a series system made by subsystems, because any one of subsystems failure will lead to the complete machine failure in direct. According to this kind of structure correlation, the relationship can be expressed by the product of reliability function R_i of all subsystems, namely the reliability function R_sys of the complete machine can be expressed in the following Eq. (3):

${R_{sys}} = {R_1}{R_2} \cdots {R_N} = \prod\limits_{i = 1}^n {{R_i}} $

(3)

As for the complete machine, dynamic core influence degree of all the subsystems in this structure correlation can be expressed as:

$C_i^3 = \frac{{\partial {R_{{\text{sys}}}}}}{{\partial {R_i}}}/{R_i} = \frac{{{R_{{\text{sys}}}}}}{{R_i^2}}$

(4)

where each R_i has its own meanings, one reflects the series structure correlation between the complete machine and subsystems, the other reflects improving the reliability value and potential of the subsystems^[16].

Failure times, failure downtime, dynamic core influence degree, the three indexes can be taken to measure the reliability influence degree of the complete machine imposed by subsystems. Weighting the three indexes will give birth to the reliability comprehensive influence degree C_i^(Sub→Sys):

$C_i^{\left( {Sub \to Sys} \right)} = \sum\limits_{k = 1}^3 {w_i^kC_i^k} $

(5)

Analyzing by experts, Failure time, failure downtime, dynamic core influence degree, the weight coefficients of three indexes respectively are shown:

w₁=0.331 4, w₂=0.314 3, w₃=0.354 3

2.2 Influence Degree Among Subsystem

Seeking out the failure reason by analyzing the failure data, finding the correlation failure and building correlation matrixes among the correlation subsystems; and then calculating subsystem failure frequency matrix, and setting up influence degree models among subsystems. Finally, counting the values of influence degree and making their sequence.

Supposing that the matrix is the correlation matrix of subsystems, as shown:

${a_{ij}} = {n_{ij}}/N$

(6)

where n_ij stands for failure times of the jth subsystem triggered by the i th subsystem, the correlation degree is defined as 1 when subsystem failure triggered by itself^[17]; N stands for total times of correlation failures; influence degree among the subsystems when concerning about the failure correlation can be expressed:

$C_i^{Sub \to Sys} = B \cdot A$

(7)

where B stands for matrix composed by failure frequency of subsystem, B=[b_i]_1×n; b_i is failure frequency of the i th subsystem, B=[b_i]_1×n.

2.3 Maintenance

In this paper, six maintenance indexes, namely failure inspection and seclusion, accessibility, convertibility, adjustability, ergonomics and maintenance tools are selected, and the criteria of all indexes can be obtained according to Ref. [18]. As a result of dependence of among the indexes, the directed causality diagram has been introduced to analyze the maintenance of system.

Evaluation model of indexes of subsystems is shown in Fig. 1, where directed arc with arrow can represent the preceding three ranks, connection with no arc represent the irrelevance between two indexes. The correlation degree has been divided into four ranks: strong, medium, weak and none. The grades of the four ranks are shown in Table 1.

Correlation degree matrix of maintenance is defined as:

$ MF = \left\{ \begin{gathered} \left( {{I_i}} \right),i = j \hfill \\ \left( {{c_{ij}}} \right),i \ne j \hfill \\ \end{gathered} \right. $

where I_i stands for grade of maintenance index; and c_ij stands for correlation degree of the index i relative to index j.

The directed model in Fig. 1 can be described by Eq. (8), and this matrix can be expressed by Permanent function as shown in Eq. (9), which can figure out the maintenance level of all subsystems^[19].

The maintenance of every subsystem can be calculated by Eq. (9), and make normalization for every Per(MF), namely distributing the influence degree for every maintenance index of subsystems, which is written by C_i^M.

$\boldsymbol{MF} = \left[ {\begin{array}{*{20}{c}} {{I_1}}&0&0&0&{{c_{15}}{c_{16}}}&{} \\ {{c_{21}}}&{{I_2}}&{{c_{23}}}&{{c_{24}}}&{{c_{25}}}&{{c_{26}}} \\ 0&0&{{I_3}}&0&{{c_{35}}}&{{c_{36}}} \\ {{c_{41}}}&0&0&{{I_4}}&{{c_{45}}}&{{c_{46}}} \\ 0&0&0&0&{{I_5}}&{{c_{56}}} \\ 0&0&0&{{c_{64}}}&{{c_{65}}}&{{I_6}} \end{array}} \right]$

(8)

$\begin{gathered} Per\left( {MF} \right) = {I_1}{I_2}{I_3}{I_4}{I_5}{I_6} + \left( {{c_{46}}{c_{64}}{I_1}{I_2}{I_3}{I_4}{I_5} + {c_{56}}{c_{65}}{I_1}{I_2}{I_3}{I_4}} \right) \hfill \\ + {c_{16}}{c_{64}}{c_{41}}{I_2}{I_3}{I_5} + {c_{45}}{c_{56}}{c_{64}}{I_1}{I_2}{I_3} + {c_{15}}{c_{56}}{c_{64}}{c_{41}}{I_2}{I_3} \hfill \\ \end{gathered} $

(9)

2.4 Complexity

Complexity of subsystem can be described as the ratio which key parts of subsystem (the one causing the subsystem failure) compare with the key parts of the whole system.

Supposing that the number of key parts in the ith subsystem is n_i (or the score getting form expert), and the number of key parts in the jth subsystem is n_j (or the score getting form expert), and then the complexity (manufacturing technique/condition/cost) of ith subsystem relative to jth subsystem can be expressed:

${b_{ij}} = \frac{{{n_i}}}{{{n_i} + {n_j}}}$

(10)

The relative complexity of subsystems b_ij constitute a matrix of B=|b_ij|_n×n. The complexity (manufacturing technique/condition/cost) of all the subsystems will be expressed:

$C_i^{\left( k \right)} = \sum\limits_{j = 1}^n {{b_{ij}}} $

(11)

where k stands for Four evaluation factors including complexity, manufacturing technology, working condition and spending cost.

Putting all data of parts into Eqs.(10) and (11), and then influence degree of distribution affected by subsystem complexity will be obtained, and written by C_i^C.

2.5 Other Influence Factors

It is more difficult to quantify the factors of manufacturing technology, working condition and cost, because the three factors are involved into special situations of corporation itself, such as welfare, division structure, employee skills and so on. The experts are invited to make an evaluation by virtue of the three factors and the scores are obtained^[20].The higher scores explain the lower manufacturing level, the worse working conditions and the more sensitive to cost. Putting all the scores into Eqs. (10) and (11), and the influence degree of Reliability distribution imposed by the three factors, and written respectively with C_i^MT, C_i^WC and C_i^cost.

When improving reliability of whole system, increased costs for improving reliability should be considered, increased costs and reliability should be weighted in comprehensive way. Ratio of cost effectiveness refers to the ratio of the cost of investment and the increase of the reliability. The cost efficiency ratio is bigger, and the cost of improving the reliability is greater. Principle of considering the cost-effectiveness ratio distribution is that smaller subsystem should be assigned a higher reliability. Cost-effectiveness ratio is difficult to get, only experience of experts in related fields can provide the corresponding weight.

3 Making Comprehensive Sequence for Subsystems Based on TOPSIS Method

TOPSIS(Technique for Order Preference by Similarity to an Ideal Solution) was firstly put forward by Hwang and Yoon in 1981, making the sequence in terms of the extent that evaluated object approach the idealized object, and making a relative evaluation in the sequence, which is also called the sequence preference of approaching ideal scheme, belonging to the algorithm^[21] of “no best, only better”.

Algorithm of TOPSIS acts as follows：

Step one is to normalize of decision making matrix:

As for decision making matrix A=|Y_ij| including n evaluation objects and m evaluation indexes, the different physical dimensions require the theory to normalize all the indexes, namely putting all the indexes into the range from 0 to 1, turning into normalized matrix B, and the matrix element B_ij can be obtained by

${B_{ij}} = {Y_{ij}}/\sqrt {\sum\limits_{i = 1}^n {Y_{ij}^2} } ,i = 1,2, \cdots ,n,j = 1,2, \cdots ,m$

(12)

Step two is to get weight vector W of all indexes and weight normalization matrix X:

Weight vector W can be acquired by subjective weighting or objective weighting, and counting weighted normalization matrix X.

Step three is to calculate ideal solution X⁺ and negative ideal solution X^-:

$\begin{gathered} {X^ + }\left\{ {\left( {\mathop {\max }\limits_i {X_{ij}}|j \in J'} \right)|i = 1,2, \cdots ,n} \right\} = \hfill \\ \left\{ {X_1^ + ,X_2^ + , \cdots ,X_m^ + } \right\} \hfill \\ {X^ - }\left\{ {\left( {\mathop {\max }\limits_i {X_{ij}}|j \in J} \right)|i = 1,2, \cdots ,n} \right\} = \hfill \\ \left\{ {X_1^ - ,X_2^ - , \cdots ,X_m^ - } \right\} \hfill \\ \end{gathered} $

(13)

where J stands for attribute set of benefit; J stands for attribute set of cost.

Step four is to calculate distance scale:

Calculating the distance from the assessed object to ideal solution and the distance from the assessed object to negative ideal solution, Euclid distance with n dimension is adopted to make measurement and the distance D_i⁺ of every ideal solution is

$D_i^ + = \sqrt {\sum\limits_{j = 1}^m {{{\left( {{X_{ij}} - X_j^ + } \right)}^2}} } ,i = 1,2, \cdots ,n$

(14)

The distance D_i^- of every negative ideal solution is

$D_i^ - = \sqrt {\sum\limits_{j = 1}^m {{{\left( {{X_{ij}} - X_j^ - } \right)}^2}} } ,i = 1,2, \cdots ,n$

(15)

Step five is to calculate relative closeness:

$E_i^ + = \frac{{D_i^ + }}{{D_i^ + + D_i^ - }},i = 1,2, \cdots ,n$

(16)

where 0≤E_i⁺≤1, the smaller value of E_i⁺ is, the closer approaching the ideal solution for the evaluated object is, when D_i=D⁺ comes, the optimized target will be acquired for the object. The bigger value of E_i⁺ is, the farther deviating the ideal solution for the evaluated object is, when D_i=D^- comes, the optimized target is opposite for the object. To finding out the two extreme situations seems impossible as usual, the two solutions are hardly to confirm when facing decision making for multi-targets.

Step six is to make a sequence according to the relative closeness of E_i⁺:

According to the relative closeness of E_i⁺, making a sequence for the evaluated object to find the optimized solution, namely the smallest E_i⁺ is regarded as the optimum and the biggest E_i⁺ is regarded as the worst.

4 Mixed Reliability Distribution Based on Copula Function

Copula function belongs to a family of functions, dozens of expression for different researching object. As for numerical control lathe, conceiving the common impact of loading and the common outer working conditions, and the relationship between subsystem failures represents the positive correlation. According to Ref. [22], Gumbel Copula function can be well describing this kind of relationship which reflects the reliability level about failures, and it belongs to the family of Archimedes Copula functions with single parameter and binary association. Therefore, Gumbel Copula function has been selected to set up the Reliability distribution model.

Binary Gumbel Copula function can be expressed

$C\left( {u,v,\alpha } \right) = \exp \left( { - {{\left[ {{{\left( { - \ln \;u} \right)}^{1/\alpha }} + {{\left( { - \ln \;v} \right)}^{1/\alpha }}} \right]}^\alpha }} \right)$

(17)

where u, v stands for variables; α stands for correlation parameter.

Gumbel Copula function with n parameters can be expressed as

$\begin{gathered} C\left( {{u_1},{u_2}, \cdots ,{u_n};\alpha } \right) = \exp \left( { - {{\left[ {\sum\limits_{i = 1}^n {{{\left( { - \ln \;{u_1}} \right)}^{1/\alpha }}} } \right]}^\alpha }} \right), \hfill \\ \alpha \in \left( {0,1} \right] \hfill \\ \end{gathered} $

(18)

Gumbel Copula function is qualified with the following properties: when α tends to be more approaching 0, the stronger relation between variables is, when α equals to 0, showing the complete linear correlation of variables; when α equals to 1, showing the independence among variables^[23].

CopulaC_α(u₁, u₂, u_n) is taken to express the correlation structure of series system composed by subsystems of X₁, X₂, …，X_n, and α stands for correlation parameter. Considering the different working time and different working conditions, correlation structure and correlation parameter vary from time to time, at a certain moment the reliability model of the system can be expressed:

$\begin{gathered} {R_s}\left( t \right) = P\left( {{X_1} > t,{X_2} > t, \cdots ,{X_n} > t} \right) = \Delta _{{F_1}\left( t \right)}^1\Delta _{{F_2}\left( t \right)}^1 \hfill \\ \cdots \Delta _{{F_n}\left( t \right)}^1{C_\alpha }\left( {{u_1},{u_2}, \cdots ,{u_n}} \right) \hfill \\ \end{gathered} $

(19)

where “△” is the difference, namely △_x1^x₂f(x)=f(x₂)-f(x₁).

TOPSIS can provide us with the sequence and the proportion of uncertainty for the subsystems, and the uncertainty of key parts or weak parts is taken as benchmark^[24], and the proportions of the other eleven subsystems with respect to the benchmark will be attained. R_SD can represent the aimed value of Reliability allocation, and putting the proportions into Eq. (19), and then the Reliability distribution model will be obtained.

5 Case Analysis

According to the structure and working process, the numerical control lathe will be divided into twelve subsystems.

In terms of failure data on the spot and by use of Eqs. (1)-(11), the influence factor value of reliable distribution will be obtained as shown in Table 2.

Decision making matrix A will be obtained by chart 2, applying Eq.(12) into matrix A to make normalization, getting normalized decision making matrix B, acquiring weight vector w of seven indexes by use of information entropy, and decision making matrix X will be acquired.

$ B = \left[ {\begin{array}{*{20}{c}} {0.2353}&{0.4755}&{{\text{0}}{\text{.405 9}}}&{{\text{0}}{\text{.350 5}}}&{0.3457}&{0.3282}&{0.2798}&{0.2580}&{0.1456}&{0.1285}&{0.1069}&{0.0987} \\ {0.1276}&{0.0299}&{0.4864}&{0.5501}&{0.4305}&{0.4305}&{0.1824}&{0.1076}&{0.0837}&{0.0837}&{0.0837}&{0.0837} \\ {0.1598}&{0.5127}&{0.3840}&{0.3847}&{0.2906}&{0.4497}&{0.1453}&{0.2110}&{0.2082}&{0.0969}&{0.0768}&{0.0726} \\ {0.1739}&{0.4112}&{0.3741}&{0.4112}&{0.2777}&{0.3381}&{0.0850}&{0.2264}&{0.3452}&{0.2029}&{0.1739}&{0.2264} \\ {0.1757}&{0.2504}&{03057}&{0.4107}&{0.3057}&{0.2991}&{0.1404}&{0.2176}&{0.3742}&{0.2687}&{0.2848}&{0.3178} \\ {0.1449}&{0.2242}&{0.3081}&{0.4082}&{0.2878}&{0.3199}&{0.1803}&{0.1803}&{0.3838}&{0.2720}&{0.2541}&{0.3490} \\ {0.1732}&{0.2562}&{0.3212}&{0.4001}&{0.2929}&{0.3397}&{0.1319}&{0.2061}&{0.3623}&{0.2795}&{0.2722}&{0.3107} \end{array}} \right] $

w=[0.145 0 0.072 7 0.122 8 0.155 9 0.168 8 0.166 9 0.168 0]

$ \boldsymbol{X} = \left[ {\begin{array}{*{20}{c}} {0.0341}&{0.0689}&{0.0588}&{0.0508}&{0.0501}&{0.0476}&{0.0406}&{0.0374}&{0.0211}&{0.0186}&{0.0155}&{0.01432} \\ {0.0093}&{0.0022}&{0.0353}&{0.0400}&{0.0313}&{0.0313}&{0.0133}&{0.0078}&{0.0061}&{0.0061}&{0.0061}&{0.0061} \\ {0.0196}&{0.0630}&{0.0472}&{0.0472}&{0.0357}&{0.0552}&{0.0178}&{0.0259}&{0.0256}&{0.0119}&{0.0094}&{0.0089} \\ {0.0271}&{0.0541}&{0.0583}&{0.0641}&{0.0433}&{0.0527}&{0.0133}&{0.0353}&{0.0538}&{0.0316}&{0.0271}&{0.0353} \\ {0.0297}&{0.0423}&{0.0516}&{0.0693}&{0.0516}&{0.0505}&{0.0237}&{0.0367}&{0.0632}&{0.0454}&{0.0481}&{0.0536} \\ {0.0242}&{0.0407}&{0.0514}&{0.0681}&{0.0480}&{0.0534}&{0.0301}&{0.0301}&{0.0640}&{0.0454}&{0.0424}&{0.0582} \\ {0.0291}&{0.0430}&{0.0540}&{0.0672}&{0.0492}&{0.0571}&{0.0222}&{0.0346}&{0.0609}&{0.0469}&{0.0457}&{0.0522} \end{array}} \right] $

Oriented by high level reliability of system, and subsystems pay little influence on the reliability of system, better maintenance, little influence among the subsystems, lower complexity, higher manufacturing technique, greater working conditions, lower sensitivity of cost on reliability, and all these properties would be the target of ideal solutions.To calculate the ideal solution and negative ideal solution.

Ideal solution：

X⁺={0.014 3, 0.002 2, 0.008 9, 0.013 3, 0.023 7, 0.022 2}

Negative ideal solution：

X^-={0.068 9, 0.040 0, 0.064 1, 0.069 3, 0.068 1, 0.067 2}

The distance from solution to ideal solution:

D_i⁺={0.028 9, 0.097 7, 0.095 4, 0.113 4, 0.076 3, 0.091 9, 0.030 4, 0.041 2, 0.081 4, 0.043 8, 0.041 1, 0.058 8}, i=1, 2, …, 12

The distance from solution to negative ideal solution:

D_i^-={0.101 8, 0.059 1, 0.034 3, 0.024 0, 0.051 5, 0.037 4, 0.108 2, 0.088 3, 0.070 9, 0.094 0, 0.099 1, 0.091 9}, i=1, 2, …, 12

According to Eq. (16) to calculate relative closeness E_i⁺(i=1, 2, …, 12), making a comprehensive sequence of subsystems by E_i⁺ from big to small, as shown in Table 3.

As a typical electromechanical product, correlation degree parameter α=0.3 is taken for numerical control lathe^[25-26], and the target of reliability is taken as MTBF=900 h, and putting a certain allowance for the target, 17% allowance is adopted in this paper, namely MTBF_SD=1 053 h is taken as target. In terms of the whole life period of numerical control lathe, reliability distribution is the original stage, combined with rules of reliability experiments, R_SD=(0.984 5, 0.999 5) on the time of t∈(1, 24) is taken as the range of reliability allocation, bringing this value into Eq. (19), the uncertainty range F_M=(0.000 3, 0.010 7) of turret system will be attained by Matlab, and the reliability range R_M=(0.999 7, 0.989 3) will be also attained, and then the reliability range and MTBF range of the other subsystems will be attained, as shown in Table 4.

6 Conclusions

In this paper, firstly all reliability distribution factors are calculated, and secondly twelve subsystems have been made a sequence under the seven assessed factors by means of TOPSIS, getting the proportional relationship of uncertainty that subsystems account for the whole system. As for numerical control lathe, turret system, feed system and transmission system are the key parts or weak parks of the whole system, which is in coincidence with the practical situations. Thirdly, through considering the correlation in comprehensive way, applying the Copula function to set up the reliability distribution model, getting the results by TOPSIS, and making a resolution by Matlab, the reliability range of subsystems and MTBF range are obtained, To compare the analyzed results with the real results of working machine tools, there are some differences between them, according to these differences, reliability designing methods would be chosen to make reliability distribution come true, which will provide with the theoretical and technical support for reliability design and reliability distribution.References