Time-varying comprehensive evaluation technology of CNC machine tool RMS based on improved ADC model

The quality of CNC (computer numerical control) machine tool is the key to the survival of manufacturing enterprises. How to comprehensively and accurately evaluate its quality is very important for the timely discovery and improvement of machine tool defects. The current research either only evaluates a certain quality characteristic of CNC machine tool, or ignores the dynamic characteristics of CNC machine tool quality, which reduces the accuracy of quality analysis results. To address these problems, firstly, RMS (reliability, maintainability, supportability) was taken as the comprehensive evaluation indexes of CNC machine tool quality characteristics to improve the comprehensiveness of the analysis. Secondly, considering the dynamic characteristics of machine tool reliability, the ADC (availability, dependability, capability) model was improved to obtain a comprehensive quality evaluation index function ERMS with time-varying characteristics. Finally, the characteristics of ERMS were analyzed, and its evaluation criteria were also given. Several CNC machine tools made in China were tracked, and the collected fault data were analyzed by the proposed method. The ERMS of the machine tools was obtained, and its characteristics were analyzed. The results verify the correctness of the proposed method, which provides theoretical basis and technical support for the improvement of the quality of CNC machine tool and the market competitiveness of enterprises.


Introduction
CNC machine tool is an industrial tool, which represents the level of a country's manufacturing industry. At present, the machine tool industry in China presents a situation of insufficient high-end, surplus low-end, and controlled core technology. The high-end manufacturing equipment required for the national economy and important national projects is heavily dependent on imports. The low quality of CNC machine tool is the main problem faced by China's major equipment [1]. Among the numerous quality characteristic indexes of CNC machine tool, manufacturing enterprises often pay the most attention to reliability, but users pay attention to not only the reliability of machine tool, but also its maintainability and supportability, which are often easily ignored by enterprises in the process of product manufacturing. Reliability, maintainability, and supportability are the key to enhance the processing capacity of machine tool and reduce its maintenance cost. Reasonable RMS evaluation method plays an important role in improving the quality of domestic CNC machine tool and increasing the international market competitiveness of enterprises.
At present, there are many researches on the machine tool's single quality. For example, the machine tool's reliability is studied in literature [2][3][4][5][6][7][8][9][10], the machine tool's maintainability is studied in literature [11][12][13][14][15], and the machine tool's availability is studied in literature [16][17][18][19]. Zhang et al. studied the grinding quality of the machine tool with POS-SQP algorithm and Monte Carlo method, and gave a general method for the high-precision machining quality evaluation [20]. Xu et al. established the fault model of EDM machine tool, and optimized the parameters affecting the machining accuracy of the machine tool by genetic algorithm [21]. However, these studies cannot fully demonstrate the quality properties of machine tool, and the accuracy of the analysis results remains to be discussed, so the multi-quality characteristic analysis method of CNC machine tool should be studied. RMS is a comprehensive reflection of system availability, credibility, and inherent capability [22]. The research on RMS was initially oriented to the equipment industry, and then gradually introduced to other industries. There are many relevant studies, such as literature [23][24][25][26][27][28]. At present, there are few studies that apply RMS to the quality evaluation of CNC machine tool. Considering the importance of RMS to the quality characteristics of CNC machine tool, it is necessary to deeply study the RMS evaluation method of CNC machine tool.
The RMS parameters of the CNC machine tools made in China are analyzed in detail, and the ADC model is improved on the basis of considering the time-varying characteristics of RMS parameters. The RMS comprehensive evaluation model which can reflect the quality dynamic characteristics of CNC machine tool is obtained, and the analysis criteria of the model is given. The applicability and accuracy of the proposed model are verified by an engineering example. The results show that the new method can better evaluate the comprehensive quality of machine tool, which provides the theoretical support for the subsequent quality improvement of CNC machine tool.

ADC model
In the RMS research of this paper, reliability refers to the ability of machine tool to complete a specified task under specified conditions and time, and the commonly used indicators about it are mainly mean time between failures, failure rate, and time between serious failures. Maintainability refers to the probability that the machine tool can maintain or recover to the specified state under the specified conditions and time according to the specified procedures and methods, and the commonly used indicators about it mainly include average repair time, average preventive maintenance time, maximum repair time, and maintenance man-hour rate. Supportability is the design characteristic of machine tool and the ability of planned support resources to meet its normal use needs, and the commonly used indicators about it are mainly average support time, re-use preparation time, and no maintenance standby time.
Effectiveness is a measure of the degree to which a product satisfies a set of specific task requirements, and it is a comprehensive function of product availability, reliability, and capability [29]. System effectiveness uses comprehensive analysis technology to comprehensively analyze various factors that affect equipment effectiveness from the system perspective, so as to obtain a single measure. Its comprehensive degree is high, and the result is a single value, which is convenient for decision analysis, and it is the most commonly used effectiveness measure. WSEIAC model is a classical model for evaluating system effectiveness, also known as ADC model, proposed by the USA Weapon System Effectiveness Industry Advisory Committee in the mid-1960s. Therefore, this paper uses the ADC model to comprehensively analyze the RMS of CNC machine tool, and its general expression is shown in Formula (1).
where E, A, D, and C are the effectiveness indicator vector, availability matrix, dependability matrix, and inherent capability matrix of the machine tool, respectively; C refers to the ability of the machine tool to meet the given requirements under given internal conditions.
A can be obtained by Formula (2).
where a i represents the probability that the machine tool is in state i at the beginning of a random task in terms of availability, and n is the number of possible states of the machine tool. D can be obtained by Formula (3).
where d ij represents the probability that the machine tool is in the i state at the beginning of the task and in the j state during the expected task time. C can be obtained by Formula (4).
where c ij represents the probability of the machine tool reaching the task j(j = 1, 2, …, m) in the possible state i(i = 1, 2, …, n).

Model assumptions
To accurately describe the effectiveness of machine tool's RMS, the following assumptions need to be made.
1. In the construction of availability matrix A, only basic reliability, basic maintainability, and supportability are considered. 2. In the construction of dependability matrix D, only task reliability and task maintainability are considered. 3. In the construction of inherent capability matrix C, it is assumed that the machine tool in good condition can (4) C = c ij n×m complete the specified task; otherwise, it cannot complete the task. 4. There are only two states of the machine tool at the time of task preparation, that is, normal or fault. 5. The machine tool must complete the tasks in sequence, and the next task cannot be performed until the previous task is completed. 6. The tasks to be completed by the machine tool in different stages are independent of each other.

Model improvement
Based on Formula (1), the RMS effectiveness of CNC machine tool can be expressed as Formula (5).
where E RMS is the RMS effectiveness of the machine tool; is the availability matrix of the machine tool, and R b , M b , and S are the basic reliability, basic maintainability, and supportability of the machine tool respectively; is the dependability conditional probability matrix of the machine tool, and R t and M t are the task reliability and task maintainability of the machine tool respectively; C' is the inherent capability matrix of the machine tool.

Availability matrix A'
From assumption (4), it can be known that n = 2 in Formula (2). Then, A' = [a 1 , a 2 ], a 1 represents the probability that the machine tool is in a normal state when the task starts, a 2 represents the probability that the machine tool is in a fault state when the task starts, and a 1 + a 2 = 1. The value of a 1 can be obtained by Formula (6).
where MTBF (mean time between failure) is used to reflect the basic reliability parameter R b of the machine tool; MTTR (mean time to repair) is used to reflect the basic maintainability parameter M b of the machine tool; and MLDT (mean logistic delay time) refers to the delay time waiting for repairing faults except MTTR , which is used to reflect the supportability parameter S of the machine tool.

Dependability matrix D'
The fault of the machine tool during the task can be repaired, so its dependability matrix can be shown in Formula (9).
where d 11 represents the probability that the machine tool is in a normal state (state 1) both at startup and at work; d 12 represents the probability that the machine is in a normal state (state 1) at startup and in a faulty state (state 2) within the expected task time; d 21 represents the probability that the machine is in a faulty state (state 2) at startup and in a normal state (state 1) within the expected task time; and d 22 represents the probability that the machine tool is in a faulty state (state 2) both at startup and at work.
(I) Time-varying task reliability R t (a) Model primaries CNC machine tool is a repairable system [30], and the maintenance after machine tool fault belongs to the process of "repair as old" [31], so the traditional machine tool fault modeling method developed on the basis of the assumption of "repair as new" is unreasonable [32][33][34]. In fact, for the same CNC machine tool with the same failure time interval, when the failure sequence is inconsistent, even at the same time, its reliability is not the same [35]. The task reliability R t is a random function of time.
Considering the "bathtub curve" characteristic of the CNC machine tool fault intensity function, a superimposed NHPP (nonhomogeneous Poisson process) is suitable for machine tool modeling [18,36]. Then the fault intensity function of CNC machine tool is shown as Formula (10).
where λ 1 (t) and λ 2 (t) are the early fault process and wear fault process of CNC machine tool respectively.
Literature [37] shows that among the current NHPP modeling methods for CNC machine tool, the most suitable and accurate method is BBIP (bounded bathtub intensity process) method, so this method is selected to model CNC machine tool. So Formula (10) is rewritten as Formula (11).

Based on Formula (11), the cumulative fault intensity function ω(t) about λ(t) is shown in Formula (12).
Then the time-varying task reliability R t can be obtained from Formula (13).
The fault data collection of multiple CNC machine tools is carried out and the specific method is shown in Fig. 1.
According to Formula (13), the likelihood function of the joint probability density of multiple machine tools is expressed as the following.
Let n represents the total number of m machine tools faults within T, and then Formula (15) can be obtained.
It can be known from Formula (11) that α, β > 0. Considering Formula (15), Formula (16) can be obtained as the following.  (14) and (15), and carry out a series of transformations; Formula (17) can be obtained. (17); the obtained parameter ( ′ , ′ , ′ , ′ ) is the estimated value of the BBIP model parameters of the CNC machine tool, which can be seen as Formula (18).

(c) Model test
The test index K of BBIP model of CNC machine tool is shown in Formula (19) [38].
where N t is the number of CNC machine tool faults at time t, N t is the expected number of faults at time t, and T is the total failure observation time. When K > 0.9, the selected model can be considered suitable.

(II) Task maintainability M t
The task maintainability M t herein refers to the probability that the CNC machine tool can maintain or recover to the specified state after a fatal failure maintenance. CNC machine tool is a complex repairable system. Considering the maintenance characteristics of CNC machine tool, M t can be replaced by the proportion η% of repair times that can be repaired to the specified state in every thousand fatal faults, as shown in Formula (20) [39].

Inherent capability matrix C'
Generally speaking, CNC machine tool performs a single task. According to assumption (3), the dimension of the inherent capability matrix is 2 × 1; then C' can be shown in Formula (22). c 11 represents the probability that the machine tool can complete the 1-th task under normal state (state 1); c 21 indicates the probability that the machine tool can complete the 1-th task under the fault state (state 2).

Application and analysis
In this paper, 3 CNC machine tools' models in China have been tracked for 2 years, and a total of 25 effective fault data have been collected, as shown in Table 1.

(ii) Time-varying task maintainability
It is known that the task maintenance degree of the selected machine tool is η% = 0.87; then M t is as shown in Formula (26).

Time-varying effectiveness analysis
According to Formulas (22), (23), (24), and (27), the time-varying effectiveness E RMS is obtained by Formula (28).  The simulation analysis of E RMS is shown in Fig. 2. From Fig. 2, we can know that the E RMS is a variable that changes with time. It is mainly composed of two parts, the first half of which can be regarded as a function that decreases gradually with time, and the second half is a straight line that tends to a stable value, which is greatly affected by the machine tool's maintenance degree.
E RMS reflects the comprehensive influence of reliability, maintainability, and supportability. It is a comprehensive measure of these three factors. Therefore, E RMS can be used to represent the comprehensive performance of CNC machine tool. Two analysis indexes e 1 and e 2 of the machine tool E RMS are given herein. e 1 is the absolute value of the descending slope of the first half of the ERMS curve, and e 2 is the stable value of the second half of the E RMS curve. When comparing E RMS of different machine tools, the following criteria should be followed. The performance of CNC machine tools with larger e 1 and e 2 is better. When there is little difference between the two e 1 values, the CNC machine tool with a larger e 2 value has better performance. When there is little difference between the two e 2 values, the CNC machine tool with a smaller e 1 value has better performance.

Conclusion
Quality evaluation has a great influence on the improvement of CNC machine tool quality. The current research often focuses on a certain aspect of the quality characteristics of machine tool (such as reliability and maintainability), so the comprehensiveness and accuracy of the evaluation results need to be discussed. There are few studies on machine tool RMS, and the time-varying characteristics of RMS of machine tool are also ignored. This will affect the accuracy of the machine tool quality assessment. To address these problems, this paper takes the reliability, maintainability, and supportability as comprehensive evaluation factors; improves the ADC model on the basis of considering the time-varying characteristics of machine tool task reliability; and finally obtains a time-varying quality comprehensive evaluation function E RMS of CNC machine tool that can reflect the reliability, maintainability, and supportability of the whole life cycle. At the same time, the characteristics of ERMS are analyzed, and the criteria that should be followed in ERMS analysis are given.
The study in this paper provides a theoretical basis for the precise evaluation of the quality for CNC machine tool, which is also the premise for the subsequent improvement and analysis of machine tool. Meanwhile, the study enriches the current CNC machine tool quality evaluation and analysis methods, which provides the possibility for the improvement of CNC machine tool's quality and the increase of the international market competitiveness of machine tool enterprises.