Research on reliability comprehensive evaluation method of five‐axis CNC machine tools based on AHP and extension theory

Currently, some five-axis CNC machine tools have almost achieved the same advanced level of the western countries in terms of functionality and performance. However, the reliability of system is much lower than similar foreign products. The reliability has become one of bottlenecks to restrict the development of domestic CNC machine tools. At present, domestic reliability research of CNC machine tools developed rapidly, but the evaluation research is limited to a single indicator, lacking overall evaluation about it. To fully evaluate the reliability of five-axis CNC machine tools, the matter-element model of reliability evaluation is established with the application of matter-element in extension theory, and the reliability evaluation indices system is established comprehensively. The correlation functions are introduced to calculate the correlation degree, and the weights of each evaluation indices are determined by using AHP. The actual example and result on two types of domestic CNC machine tools show that the proposed method is effective, reasonable, and accurate in the process of reliability evaluation. The proposed methodology provides a detailed instruction for comprehensive reliability evaluation of CNC machine tools.


Introduction
Machine tool is the foundation of equipment manufacturing, whose performance, quality and holding quantity are used as one of the important symbols to measure a nation's modernisation level, industry level, and national aggregate strengths [1]. High speed, high efficiency, high precision, and high reliability are the important modern machine tool developing trends [2]. Five-axis CNC machine tools shoulder the important task of manufacturing the key products, components, and parts in aerospace and other industries. Its' development level directly affect the industrial security and national security [3]. Currently, some five-axis CNC machine tools have almost achieved the same advanced level of the western countries in terms of functionality and performance. However, the reliability of system is much lower than similar foreign products. The reliability has become one of bottlenecks to restrict the development of domestic CNC machine tools [4]. Hence, the reliability evaluation of domestic five-axis CNC machine tools plays an important role in improving the development of machine tool industry.
In view of reliability evaluation of CNC machine tools, much work has been done by scholars and there were already many achievements in this field. Keller et al. [5] analysed field failure data which were collected over a period of 3 years on ∼35 CNC machine tools during their warranty period, and then obtained the values of mean time between failures (MTBF) and mean time to repair (MTTR). Zhu et al. [6] proposed a method of reliability comprehensive evaluation based on rough set fuzzy rule. Ni and Piao [7] constructed the comprehensive evaluation indices system of foreign numerical control lathe reliability by the mean time to first failure (MTTFF), MTBF, and equivalent failure rate (D). The weights of each evaluation indicators were distributed by using the variable coefficient method. There is currently no consensus definition of reliability evaluation indices system, and MTBF was usually used as the main evaluation indicator. In fact, the different faults have different influence degree to the machine tools. Therefore, other evaluation indices such as MTTR, MTTFF, D etc. are necessary to be considered in comprehensive evaluation process. Shen et al. [8] proposed an entropy weight method with objective weighting to evaluate the reliability of machine tools. The method avoids the shortcoming of the weighted comprehensive evaluation method which lies in the subjectivity in weight determination, but it pays more attention to the difference between indices and the variability of its results is high. Analytic hierarchy process (AHP) is a sort of method which is widely applied to establish the evaluation indices system and determine the weights of evaluation indices. Zhao [9] proposed a method which combines AHP and grey theory to evaluate the reliability of machine tools. Shen et al. [10] presented a new method of evaluating the reliability of machine tools by using extenics theory, and AHP was used to calculate the weights of evaluation indices. However, the method above which used AHP did not give a method which able to modify the no-consistency judgment matrices. To address these issues, this paper introduces a new comprehensive evaluation approach which combined with AHP and extension theory to fully evaluate the reliability of five-axis CNC machine tools. The AHP is used to establish the reliability evaluation indices system and determine the weights of evaluation indices. The reliability evaluation indices system proposed here adequately considered fail-safety indices, maintainability indices and availability indices. The extension theory is used to establish the matter-element model of reliability evaluation. Furthermore, the detailed steps for modifying the no-consistency judgment matrices are put forward in the proposed approach.
where R j is matter-element model of the jth evaluation level for machine tool, N j is the jth evaluation level of machine tool, C i is the reliability evaluation indicator of evaluation level N j , X ji is value range of N j corresponding to the evaluation indicator C i , namely, a ji , b ji .

Determination of joint domain:
The joint domain R p are value ranges of each evaluation indicator in all evaluation levels.
where p is the entire evaluation levels, x pi is value range of p corresponding to the evaluation indicator c i , namely, a pi , b pi .

Determination of the Matter element to be evaluated:
For the evaluated machine tool p MT , by using matterelement, the detailed numerical values of each evaluation indicator through analysis and calculation can be represented as follows: where x i is value range of p MT corresponding to c i , namely, the reliability indicator value of p MT .

Calculate the correlation degree between evaluation indicator and evaluation level
Assume the distance between p MT , N j (j = 1, 2, …, m) and evaluation indicator C i is ρ(x i , x ji ), the distance between p MT , N p and evaluation indicator C i is ρ(x i , x pi ), then the correlation function between evaluation indicator C i of the evaluated machine tool and the jth level is: Assume ω i is the weight of evaluation indicator C i , by using AHP, the correlation degree between evaluation indicator and the jth evaluation level can be calculated as follows:

Determination of weights by using AHP
The AHP introduced and developed by Saaty (1980) has been widely used in multi-criteria decision-making [11]. This method uses both quantitative and qualitative data (that are translated into numbers). The AHP is a theory of measurement through pairwise comparisons. The pairwise comparison method is used to compare alternatives and determine their importance over each other. The comparisons are made using a scale of absolute judgements that represents the domination measure of one element over another with respect to a given attribute. Notably, the AHP method has the advantages of yielding more precise results and verifying consistency of judgments. Therefore, the analytic hierarchy process is used to determine the weights of evaluation indices here. A five-step implementation procedure is developed for assessing the weights of evaluation indices.
i. The first step in the analytic hierarchy process is to model the problem as a hierarchy. In doing this, participants explore the aspects of the reliability indicator at levels from general to detailed, then express it in the multileveled way that the AHP requires. The reliability evaluation indices include [12][13][14] MTTFF, MTBF, equivalent failure rate (D), MTTR, and inherent availability (A), as shown in Fig. 1. ii. Develop a paired comparison matrix for evaluation indices.
Conduct the pair-wise comparisons in each cluster according to the hierarchical tree of evaluation indices. Based on the pairwise comparisons of evaluation indices, the relative importance degrees are estimated for the same cluster. A total number of n(n − 1)/2 pair-wise comparisons are evaluated for a cluster with n evaluation indices. As for measurement scale, this study uses numerical values 1-9 and their reciprocals.
The diagonal elements in the matrix A are self-compared of the evaluation indices, and thus a ij = 1, where i = j, i, j = 1, 2, …, n. The values on the left and right sides of the matrix diagonal represent the strength of the relative importance degree of the ith element compared to the jth element. Let a ij = 1/a ji , where a ij > 0, i≠j.
iv. Test the consistency of the weights of evaluation indices. To ensure that the evaluation of the pair-wise comparison matrix is reasonable and acceptable, a consistency check is performed, as described below.
i. Calculate the maximum eigenvalue λ max.
ii. With the maximal eigenvalue, the consistency index (C.I.) can be determined by iii. Calculate the consistency ratio (C.R.).
Generally, a consistency ratio (CR) can be used as a guidance to check for consistency. R.I. denotes the average random index with the value obtained by different orders of the pair-wise comparison matrices. If the value of C.R. is below than the threshold of 0.1, then the evaluation of the weights of evaluation indices are considered to be reasonable. Otherwise, the pair-wise comparison matrix should be modified. The detailed steps on how to modify the matrix were shown as follows.
Suppose the pair-wise comparison matrix is A = (a ij ) n×n , k is the number of iterations.
Step 3 Calculate the consistency index C.I. and the consistency ratio C.R.
Step 4 If C.R. < C.R.* go to Step (7), otherwise go to the next step.
Step 5 Normalise the column of A (k) , and the normalised matrix A = (a 1 , a 2 , …, a n ) can be obtained, a i (i = 1, 2, …, n) are column vectors of A (k) . Calculate the cosine value between a i (k) and ω (k) to obtain the r which makes cos θ r = min i cos θ i , the calculating formula is: Then, set A (k+1) = (a ij (k+1) ) n×n , where Step 6 Set k = k + 1, go to Step 2.
In the above steps, the weights of elements to one element at the upper level are obtained. Based on this, the overall weights of the elements at the lowest level to the objective level can be calculated.
Suppose the level B has m factors: B 1 , B 2 , …, B m , the synthetic weights are b 1 , b 2 , …, b m , respectively. The level C has n factors: C 1 , C 2 , …, C n , their weights for B j are c 1j , c 2j , …, c nj , respectively. (If there don't have relationship between C k and B j , then c kj = 0). The overall importance degrees of evaluation indices in level C are:

Results
The reliability evaluation indices include 5 indices: MTTFF, MTBF, D, MTTR and A, namely, n = 5. The reliability level is divided into four levels: excellent, good, fair, and poor. To validate the practicability and effectiveness of the proposed method, in this chapter, a study is carried out to perform the method in reliability comprehensive evaluation of two types of domestic five-axis CNC machine tools. The process of the example will be explained in the following sections.

Establishment of matter-element model
i. Determination of classical domain: For the reliability evaluation indicator MTTFF, the value ranges from 600 to 800 in level excellent, the value ranges from 400 to 600 in level good, the value ranges from 300 to 400 in level fair, and the value ranges from 200 to 300 in level poor, namely, 600,800 , 400,600 , 300,400 and 200,300 , respectively. The value ranges of the other indices in different levels are essentially the same as the previous case MTTFF, so that will not be covered again here. The detailed classical domains of each evaluation indices are shown as follows:    Table 1.
According to Table 1, the Matter element to be evaluated can be determined, as shown below:
Bringing ω and (13) and (14) into (5), the comprehensive correlation degrees between two machine tools and four evaluation levels can be obtained as follows: By the principle of maximum membership degree, the reliability level of machine 1 is good and the reliability level of machine 2 is fair.

Conclusion
i. To fully evaluate the reliability of five-axis CNC machine tools, the reliability evaluation indices which include MTTFF, MTBF, D, MTTR, and A are considered comprehensively. The results of evaluation were credible. ii. The weights of each evaluation indices are determined by using AHP, and the correlation degrees are calculated through the correlation functions. The evaluation progress is more scientific and reasonable. iii. The proposed methodology provides a detailed instruction for comprehensive reliability evaluation of CNC machine tools.