Reliability allocation of rotary ultrasonic vibration-assisted EDM machine tool based on maximum entropy ordered weighted average and constraint under the index of overall cost

In this paper, the reliability allocation and index of overall cost of rotary ultrasonic vibration-assisted EDM machine tool were studied. The failure rate of each module of the rotary ultrasonic vibration-assisted EDM machine tool can be accurately predicted by the maximum entropy ordered weighted average algorithm so that the corrected maintenance cost can be predicted. And effective control of preventive maintenance costs can be achieved by selecting the best maintenance times under the premise of reliability. The cost of corrective maintenance can be reduced by improving modules with a high failure rate, so as to realize the purpose of cost optimization under the premise of determining the maintenance time and reliability. The reliability function of the rotary ultrasonic vibration-assisted EDM machine tool is established according to the series-parallel relationship of each module. The results showed that the reliability of rotary ultrasonic vibration-assisted EDM machine tool is stabler at the non-fixed maintenance period according to the comparison between the non-fixed and the fixed maintenance period.


Introduction
The principle of electrical discharge machining [1] is to use the electro-corrosion phenomenon generated by the pulse discharge between the two poles to process the material, which belongs to the category of electro-physical machining academically. It has a wide processing range (suitable for a wide range of difficult-to-process materials) and the machining process is non-contact and free of significant mechanical force. It is suitable for precision micro processing and the accuracy can reach the micron level. It also has the advantages of a high degree of automation and a high degree of intelligence and so on. However, there are also disadvantages such as the production efficiency is lower than the common cutting process, the forming accuracy during the machining process is affected by the loss of the electrode, and the minimum corner radius is limited. In order to overcome this limitation, the rotary ultrasonic vibration-assisted EDM technology [2,3] has been studied based on various theories. Yinghuai Dong and Jianbao Song et al. [4] have proposed a non-contact power supply technology using the principle of electromagnetic induction to solve the concentration of sparks and abnormal arcs generated by the use of carbon brushes and circulating power supply, and an ultrasonic vibration-assisted EDM machine was also be designed. Yan Wang and Zhiqiang Liu et al. [5] have found that the quality of the machined surface can be altered by adjusting the performance of ultrasonic vibration energy and EDM energy. Atsutoshi Hirao and Hiromitsu Gotoh et al. [6] have studied the influence of different amplitudes on the characteristics of EDM by using the auxiliary ultrasonic vibration method of the tool electrode. Sanjay Kumar and Sandeep Grover et al. [7] have established the performance index system model of ultrasonic vibration-assisted EDM by using graph theory and obtained the performance index of ultrasonicassisted EDM from matrix model and digraph to understand the ability of influencing subjective factors. Yan-Cherng Lin and Fang-Pin Chuang et al. [8] have used the combination of ultrasonic vibration and auxiliary magnetic force with EDM technology to overcome the bottleneck of processing large-area applications. Mohammad T. Shervani-Tabar et al. [9] have studied the bubble generation and material removal rate through the influence of the simultaneous vibration of the tool and the workpiece on the hydrodynamic behavior of the liquid flow around the bubble. Reza Teimouri and Hamid Baseri [10] have studied the influence of rotating magnetic field and ultrasonic vibration of the work piece on the performance of EDM process. and the results show that the magnetic field has a positive effect on MRI and SR.
The reliability allocation of the individual modules of the rotary ultrasonic vibration-assisted EDM machine is one of the most important factors that need to be considered when determining the overall reliability and competitiveness. It is also a crucial analytical tool that can be used to improve the reliability of the system. In recent years, the rational allocation of system weights based on the maximum entropy ordered weighted average method has been applied in various fields. People have continuously researched new algorithms for the optimization of system costs to achieve cost savings under the premise of reliability. Qiang Cheng and Hao Wang et al. [11] have combined the fuzzy allocation method and the maximum entropy ordered weighted average to realize the reliable and flexible allocation of the CNC machine tool system. Sebastián Maldonado and José Merigó et al. [12] have redefined the soft-edge support vector machine (SVM) formula by introducing the ordered weighting (OWA) operator. The results show that this method has the best overall performance compared with the standard support vector machine. Alireza Chaji et al. [13] have determined the maximum Bayesian entropy model of ordered incremental average (OWA) operation related weights and determined the analytical form of OWA weights through the Lagrangian multiplier method. Golam Kabir et al. [14] have used Bayesian linear regression and uncertainty analysis of the method of sequential weighted averaging for predicting water pipe failure. G. Yari and A.R. Chaji [15] have obtained the maximum Bayesian entropy of OWA operator weights and solved the new model with a specific Orness level based on the prior OWA vector. Byeong Seok Ahn [16] has studied the metric developed from the OWA method that can attach some semantics to the approximate weight and generate the weight by the maximum entropy method. Savin Treanţă [17] has studied the optimal solution of a class of constrained interval-valued optimization problems with path-independent curvilinear integral cost function control, and established the minimum criteria of the optimal solution through the constrained variable Lu optimal solution and global Lu optimal solution of PDE and PDI. Hong Xiao and Rongyue Zhang et al. [18] have adopted the system optimization maintenance cost as the objective function and optimized the system maintenance cycle through time and maintenance interval subject to the minimum system reliability constraints. Jikai Liu and Qian Chen et al. [19] have proposed a costconstrained optimization algorithm that considers the laser powder bed additive manufacturing process, which optimizes metal additive manufacturing parts under the condition of strictly limiting the manufacturing cost. Rok Cajzek and Uroš Klanšek [20] have proposed a MINLP model for project schedule and cost optimization under resource and application constraints, and realize the optimization of the production process while resource-constrained scheduling. Saúl Domínguez-Isidro et al. [21] have proposed a multi-mode differential local search coordination algorithm for solving the constrained numerical problems, which is suitable for local search operators that are fully coordinated in a limited search space.
The above research results showed that the theory under the maximum entropy ordered weighted average and overall cost index has not been applied to rotary ultrasonic vibrationassisted EDM machine tool. In this paper, a design method of rotary ultrasonic vibration-assisted EDM machine tool based on maximum entropy ordered average and reliability allocation under the total cost constraint is proposed. Our algorithm successfully predicts the reliability of the overall structure of the machine tool and achieves the purpose of cost optimization.
2 Design of rotary ultrasonic vibration-assisted EDM machine tool 2.1 Control system of the rotary ultrasonic vibrationassisted EDM machine tool The rotary ultrasonic vibration-assisted EDM is a composite machining method which uses EDM technology to process metal surface and adds rotation and ultrasonic vibration. The design scheme of the rotary ultrasonic vibration-assisted EDM machine tool is shown in Fig. 1. The main task of the servo control of the rotary ultrasonic vibration-assisted EDM is ensuring stable and effective electrical discharge between the tool electrode and the workpiece. The rotary ultrasonic vibration-assisted EDM control system adopts an open architecture mode of upper computer (PC industrial computer) and lower computer (IMAC400 motion controller). The two are connected through the Ethernet bus to realize information transmission. The upper computer sends instructions to the IMAC400 motion control card, which drives the ball screw through the servo motor to control the tool electrode to move. The schematic diagram of the control system of the rotary ultrasonic vibration-assisted EDM machine tool is shown in Fig. 2. Figure 3 shows the 3D modeling diagram of the selfdeveloped rotary ultrasonic vibration-assisted EDM machine tool. The main body of the platform is an X-Y-Z three-axis workbench. The X and Y axes are placed in a cross shape and the three linear axes are all driven by guide screws and servo motor. A working fluid tank and a workpiece clamping device are installed on the X-axis sliding plate. The Z-axis stands vertically on the processing platform and drives the rotating spindle to move in the vertical direction. The rotating ultrasonic spindle is composed of a motor, an ultrasonic bracket, a rotary ultrasonic vibration device, and a tool electrode. The rotary ultrasonic vibration-assisted EDM machine tool is shown in Fig. 4.

Structural design of rotary ultrasonic vibrationassisted EDM machine tool
The rotary ultrasonic vibration-assisted EDM machine tool includes five modules: X-Y axis motion system, Z-axis motion system, spindle rotation system, ultrasonic vibration system, and servo control system. The X-Y axis motion system adjusts the position of the working slot in the horizontal plane through the motors on the X and Y axes. The Z-axis motion system adjusts the position of the tool electrode in the longitudinal direction through the motor on the Z-axis to control the relative position of the work piece and the electrode in the working groove to ensure smooth processing.  Fig. 2 Schematic diagram of the control system of the rotary ultrasonic vibration-assisted EDM rotation system is achieved by a rotating motor installed at the top, which is fixed above the motor bracket and joined to the connecting shaft in the bracket through a coupling. The shaft is supported and mounted on the bracket by a pair of 7207B angular contact bearings. Inside, the protruding shaft end is connected to a tapered shaft through a coupling. The ultrasonic vibration system can realize the function of ultrasonic frequency and generate mechanical vibration, which belongs to the most important part of the whole ultrasonic vibration system. The upper end of the ultrasonic transducer is a taper shaft with Mohs 5 taper. The taper shaft is closely connected with the ultrasonic generator, which drives the central shaft of the ultrasonic generator to rotate. There is a threaded shaft under the ultrasonic generator, connected with the tool electrode, so as to realize the machining mode of rotating ultrasonic vibration EDM. The hardware of the servo control system mainly includes industrial machine, motion control card, servo motor, encoder, pulse power supply, and ultrasonic vibration generator. In this paper, the five modules are analyzed based on the ordered average of maximum entropy and the reliability allocation under the overall cost constraint to achieve the overall structural reliability and cost optimization of the machine tool.
3 Analysis of the failure rate of each module of the system based on the orderly weighted average of maximum entropy The complexity (I), technical level (S), working time (P), and working environment (E) of each module of the rotary ultrasonic vibration-assisted EDM machine tool were weighted based on the maximum entropy ordered weighted average method, and the failure rate of each module can be calculated under different measures.
Firstly, the overall structure of the rotary ultrasonic vibration-assisted EDM machine tool is evaluated to determine the reliability (R) and the working time (T) of operation and to calculate the total failure rate of the system. where: R reliability of rotary ultrasonic vibration-assisted EDM machine tool; T working hours of rotary ultrasonic vibration-assisted EDM machine tool; λ s total failure rate of rotary ultrasonic vibration-assisted EDM machine tool.
The assigned value w k of each module is: where: r Ik , r Sk , r Pk , r Ek represent the ISPE value of each module; w 1 , w 2 , w 3 , w 4 ISPE assigned weight.  The complexity C k of each module is: The failure rate λ k of each module is: The Orness measurement level is used to obtain the weight of the OWA operator based on the maximum entropy method, which is the weight vector of the MEOWA operator. The Lagrangian algorithm is used to solve the OWA polynomial to obtain the analysis of the weight function as [22,23]: where: w weight vector; α measure; n number of attributes.
Different optimal weight vectors can be obtained by calculating different reliability distribution values according to the selection of different measures. Table 1 shows the measured values (α=0.5, 0.6, 0.7, 0.8, 0.9, 1.0) for calculation. The measured values from low to high represent the technician's optimism about the reliability of the machine tool, taking n=4 and calculating the optimal weight vector according to Eqs. (5)- (7).
The Delphi method is used to determine complexity (I), technical level (S), working hours (P), and working environment (E) of each module of the rotary ultrasonic vibration EDM machine tool according to the type of hardware of the servo control system, the screw type of the X, Y, Z axis, the lead of the screw, the length of the main shaft, the speed of the ultrasonic vibration gear lever, and the development method, as shown in Table 2.
The reliability of the rotary ultrasonic vibration-assisted EDM machine tool is 0.875, and the task time is 1000h. According to formula (1), the total failure rate of the system is: The failure rate of each module (every 10 5 h) can be obtained according to formulas (2)-(4), as shown in Table 3.
In Table 3, the data of the failure rate of each module under different measures are available, and the failure rate of each module is predicted under the premise of reliability, so as to achieve the analysis of the overall reliability of the machine tool. When α=0.5, the total failure rate of each module of the rotary ultrasonic vibration-assisted EDM machine has the shortest time to occur, so the technician needs to be careful in his assessment.

Reliability allocation of revolving ultrasonic vibration-assisted EDM machine tool under overall cost index
The lifetime of the rotary ultrasonic vibration-assisted EDM machine tool is a balance between the maintenance cost and reliability of multiple modules. All modules use the same preventive maintenance period to avoid frequent machine downtime due to maintenance of a single module. Therefore, the optimization of the maintenance strategy requires reasonable planning of the preventive maintenance interval and prediction of the cost of corrective maintenance based on different measures. On this basis, the maintenance cost and reliability of the entire life cycle of the system are considered [24].

Mathematical model of cost and reliability of rotary ultrasonic vibration-assisted EDM machine tool
The goal of cost optimization is to determine the optimum (N, T) within a limited life span to optimize the system maintenance cost under the premise of lowest reliability of the system life, N is the number of preventive maintenance of the machine tool, and T = { T P0 , T P1 , …, T Pk , …, T PN } is the best maintenance interval. The maintenance cost of the machine tool includes corrective maintenance (CM) cost and N preventive maintenance (PM) costs. The intervals between N preventive maintenance activities are not fixed. The measure of corrective maintenance should be taken for the machine tool failures in each maintenance period. The number of corrective maintenance is related to the reliability of the multicomponent system. The maintenance cost of the rotary ultrasonic vibrationassisted EDM machine tool is: where: the total maintenance cost of the system over its limited lifetime; c p preventive maintenance cost of machine tool; c e corrective maintenance cost.
Corrective maintenance cost is: where: c i corrective maintenance cost of each module of the machine tool; h itotal the failure time of the i-th module in its limited life.
Considering that the optimization of maintenance costs would lead to the reduction in machine reliability, the reliability of the machine tool can be used as the constraint for solving the model, and the maintenance strategy optimization model is obtained as: where: R k machine reliability; R min machine reliability constraints.
The reliability of rotary ultrasonic vibration-assisted EDM machine tools mainly follows Weibull distribution, which is applied to the description of the failure rate function and the modeling of the lifetime of each module of the machine. It is assumed that the reliability of all modules satisfies the Weibull distribution to simplify the calculation, namely: where: λ i (t) failure rate of the i-th module of the machine tool; m i shape parameter; μ i scale parameter.
Considering the influence of shape parameters and scale parameters on failure rate, a new model for the change in failure rate after preventive maintenance of equipment was developed. The function shows the relationship of failure rate before and after maintenance, which is as follows: where: life reduction factor of the i-th module during the k-th preventive maintenance; b i, k risk increase factor of the i-th module during the k-th preventive maintenance.
In this recurrence relationship, the initial value of the equipment failure rate is updated to b i, k (α i, k ,T k ) after the maintenance operation. The age reduction factor and the risk rate increase factor remain unchanged in order to simplify the calculation, which are shown as follows: The failure rate of the i-th module of the machine tool after K preventive maintenance cycles is: where: λ i, 0 the initial failure rate of the machine tool.
The corrective maintenance times of the i-th module in the k-th preventive maintenance period can be obtained according to formula (13): System reliability from the perspective of the module shows the reliability of the machine tool can be calculated by the reliability of the key modules. The reliability and failure rate of the modules are shown below: The reliability function of the i-th module after preventive maintenance can be obtained according to the recurrence law of the failure rate function: The comprehensive reliability of each module of rotary ultrasonic vibration-assisted EDM can be expressed as the reliability function of the machine tool. The reliability formula after k-th preventive maintenance activities is as follows: Taking the minimum reliability of the system as the constraint and reducing the cost as much as possible as the goal, the objective function is Minc total (N, T). N and T = {T 0 , T 1 … T k , …T N } are decision variables, and the constraint function is: where: t k time spent on preventive maintenance activities during K.
The optimization analysis costs of the five modules of the rotary ultrasonic vibration-assisted EDM machine tool, including servo control system, X-Y axis motion system, Zaxis motion system, spindle rotation system, and ultrasonic vibration system, are determined by establishing a mathematical model of total cost and reliability analysis.
The lifetime of the rotary ultrasonic vibration-assisted EDM machine tool is 10 years (87,600 h), the reliability constraint is 0.875, the preventive maintenance cost is 500, and other parameters are shown in Table 4.
As shown in Table 4, the failure time for each module of the rotary ultrasonic vibration-assisted EDM machine tool within 10 years is determined by the failure rate of each module when α=0.7 in Table 3. Substituting the parameters in Table 4 into formulas (14) and (16), the reliability of each   module after the k-th preventive maintenance can be obtained as follows: The series-parallel structure of the five modules of the rotary ultrasonic vibration-assisted EDM machine tool is shown in Fig. 5.
The reliability of the rotary ultrasonic vibration-assisted EDM machine tool is determined by the series-parallel relationship of each module. According to formula (17):

Optimal maintenance times of rotary ultrasonic vibration-assisted EDM
As shown in Table 5, given the maintenance times (N) and the total cost c total (N, T), the average system reliability can be calculated according to formula (10). Obtaining the best maintenance time can be based on the relationship between average system reliability and cost under the premise of the lowest reliability.

Reliability of rotary ultrasonic vibration-assisted EDM under the fixed maintenance intervals and the non-fixed maintenance period
As shown in Fig. 6, when N is set to 3, the average reliability of the system is close to 0.875 and the maintenance cost is high beyond our expectation, meanwhile its cost performance present low level. Then, we set N as 5, calculating the R i, k of the fixed and the non-fixed maintenance period of the rotary ultrasonic vibration-assisted EDM machine tool according to formulas (19)- (20), in order to ensure that the factor R i, k is greater than 0.875. Data are shown in Table 6 and Table 7.
The reliability comparison of the rotary ultrasonic vibration-assisted EDM machine tool after the i-th maintenance under the fixed and the non-fixed maintenance period is shown in Fig. 7. The reliability of rotary ultrasonic vibration-assisted EDM machine tool declines rapidly within the fixed maintenance period as the number of maintenance increases, and its reliability no longer meets the requirements after the fifth maintenance. The reliability of rotary ultrasonic vibration-assisted EDM machine tool slowly declines within the non-fixed maintenance period as the number of maintenance increases; meanwhile, its reliability still meets the requirements after the fifth maintenance.  Fig. 6 The relationship between average system reliability and cost Cost optimization can be achieved by improving the modules with high failure rate and thus changing the expert's evaluation of the measurement of rotary ultrasonic vibration EDM machine tool under the premise of determining the maintenance times and reliability. When α is taken as 0.5, the total maintenance cost of the rotary ultrasonic vibration-assisted EDM machine tool is 17135.145 RMB, and when α is taken as 0.5, the total maintenance cost of the rotary ultrasonic vibration-assisted EDM machine tool is 15306.07 RMB, reduced by 1829.075 RMB compared to before.

Conclusions
The maximum entropy ordered weighted average algorithm was used to accurately predict the failure rate of each module of the rotary ultrasonic vibration-assisted EDM machine tool. And the corrective maintenance cost of each module was determined by this algorithm and combined with the preventive maintenance cost to establish a cost model. This method was applied to rotary ultrasonic vibration-assisted EDM machine tools (as shown in Fig. 4) and it predicted the cost of five preventive maintenance in 10 years according to both the non-fixed maintenance period and the fixed maintenance period. The results showed that the preventive maintenance cost of 2500 RMB can be determined by the optimal maintenance time of the rotary ultrasonic vibration-assisted EDM machine tool. Theoretically, the reliability of rotary ultrasonic vibration-assisted EDM machine tool was reduced 19.33% through five preventive maintenance according to the fixed maintenance period and it was reduced 7.77% through five preventive maintenance according to the non-fixed maintenance period. It was verified that the reliability of the machine tool is better under the non-fixed maintenance period than under the fixed maintenance period. The cost of corrective maintenance was reduced 10.67% by improving the modules with high failure the premise of determining the best maintenance time and reliability. Therefore, the reliability and the cost optimization of rotary ultrasonic vibration-assisted EDM machine tool can be achieved through the algorithm of maximum entropy ordered weighted average combined with overall cost index.
Availability of data and materials The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability Not applicable
Author contribution Minggang Xu and Hao Fu: validation, analysis, investigation, writing of the original draft. Wang Tian and Binbin lyu: data calculation, analysis, investigation, writing review.
Funding National Natural Science Foundation of China (51205005)

Declarations
Ethics approval This paper is our original unpublished work, and it has not been submitted to any other journal for reviews.
Consent to participate All authors were fully involved in the study and preparation of the manuscript; each of the authors has read and concurs with the content in the final manuscript.
Consent for publication All authors consent to publish the content in the final manuscript.

Competing interests
The authors declare no competing interests.