Joint maintenance decision based on remaining useful lifetime prediction using accelerated degradation data

Existing joint maintenance decision research typically ignores remaining useful lifetime (RUL) predictions for the accelerated degradation of equipment. A joint maintenance decision method for the replacement and spare-parts ordering strategy based on RUL prediction using accelerated degradation data for equipment is proposed in this paper. First, an RUL prediction model under accelerated stress is built by considering the proportional relationship between the drift coefficient and diffusion coefficient in the Wiener process. Second, based on the principle of step-by-step estimation, accelerated degradation test (ADT) data of the equipment are used to estimate the a priori unknown parameters. Finally, based on the RUL prediction results, a joint optimization model for the replacement and spare-parts ordering strategy is developed. Through example verification and cost parameter sensitivity analysis, the proposed method is shown to effectively improve the accuracy of RUL prediction and the scientific value of the joint optimization plan for equipment replacement and spare-part ordering, which is important to many engineering applications.


I. INTRODUCTION
With the increasingly complex battlefield environment of modern war, the technological level of weapons and equipment has improved markedly in recent decades. As a result, stricter requirements for the reliability of equipment as well as the maintenance and support capability of troops have been proposed. To improve the reliability of equipment, as well as the maintenance and support capability of troops, prognostics and health management (PHM) have been proposed and have gained wide public attention. This technology could effectively improve the maintenance support efficiency of equipment and represents the future development direction of equipment support [1]- [4].
The essence of PHM is to obtain the status information of weapons and equipment via advanced sensor technology and then to predict their performance evolution trends and failure states, thereby obtaining RUL prediction information. Thus, scientific maintenance support methods have been developed to effectively improve the maintenance and support efficiency of equipment. Narrowly defined PHM technology primarily consists of two components: 1) predicting the RUL of equipment; and 2) making maintenance decisions for equipment based on RUL prediction information.
The specificity of the immediate task makes the weapons and equipment generally exhibit high reliabilities and long lifetimes, which makes it difficult to determine sufficient lifetimes or degradation data via conventional life and degradation testing. To address the shortcomings of traditional methods, the accelerated degradation test (ADT) was used in this study for the RUL prediction of equipment with high reliability and long lifetimes, and good results were achieved [5]- [6]. Tang et al. [7] analyzed This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3165050, IEEE Access VOLUME XX, 2017 1 the accelerated degradation data of lasers and constructed an accelerated degradation model based on the Wiener process to predict the RUL. However, the fact that the diffusion coefficient varies with stress was not considered; thus, the accuracy of RUL prediction decreased. Based on Tang's research, Liu et al. [8] believed that accelerated stress affects the drift coefficient and the diffusion coefficient to some extent. Based on this assumption, the effectiveness of the method was validated via accelerometer degradation data by Liu et al. [8]. However, this method can only predict the RUL via the accelerated degradation data of equipment, and the on-site monitoring data are more general in the real operating environment. Accelerated degradation data and on-site monitoring data can be improve the accuracy of RUL prediction. Thus, a method to update the parameters of the accelerated degradation model of equipment using the Bayes principle was proposed by Cai et al. [9], and the online RUL prediction of equipment based on the fusion data was realized. However, in this method, the effect of accelerated stress on the diffusion coefficient was neglected, and thus, the accuracy promotion of RUL prediction was limited because only the drift coefficient was updated. To synchronously update the drift and diffusion coefficients, Wang et al. [10] used Bayes' principle to online update the drift and diffusion coefficients based on the assumption that these coefficients obey a specific conjugate prior distribution. However, prediction was poor when the drift and diffusion coefficients did not meet the specific conjugate prior distribution hypothesis. Based on [10], Wang et al. [11] proposed a method in which the Bayesian statistical inference of the drift coefficient and diffusion coefficient was conducted based on a no conjugate prior distribution. In combination with the accelerated degradation data and on-site test data, the RUL of the target equipment was predicted, which extends the application scope of the method. However, to develop the degradation model, the default drift/diffusion coefficient in the Wiener process obeys the conjugate prior distribution similar to the normal-gamma distribution. However, when the drift and diffusion coefficients do not obey the distribution type, the RUL prediction result obtained by this method lacks credibility. Also, considering the acceleration factor constant principle in [12], Wang et al. [13] and Wang et al. [14] showed that the drift and diffusion coefficients both change based on stress and satisfy the proportional relationship. Based on these conclusions, Wang et al. [15] proposed a proportional relationship between the drift and diffusion coefficients under accelerated stress. Thus, an accelerated degradation model is constructed, which improves the effectiveness of the RUL prediction method. However, this model cannot be applied to accelerated-stress situations and does not consider the effects of individual differences and measurement errors on degradation modeling. Scientific maintenance decisions can effectively improve the operating state of equipment and reduce the maintenance cost during their life cycle, which is important in military and economic considerations. In daily maintenance, the maintenance strategy that only considers equipment cannot effectively reduce maintenance costs during equipment life cycles. A well-developed spare-parts management strategy can reduce costs, thus promoting the transformation from a single maintenance strategy to joint optimization of the maintenance strategy and a spare-parts management strategy. The joint optimization between the preventive replacement strategy and spare parts production strategy was performed to achieve the lowest maintenance cost by Aghezzaf et al. [16]. The combined optimization of the preventive replacement strategy and spare parts inventory strategy was also conducted by Zequeiraa et al. [17]. Overall, the RUL prediction information of equipment was not considered in the literature, which has reduced the scientific nature of maintenance decisions to a certain extent. Currently, few studies have investigated the joint optimization of maintenance and spare-parts ordering strategies based on RUL prediction. The RUL prediction was obtained by Elwany et al. [18] using the linear and exponential degradation model. Thus, the maintenance cost was reduced by establishing the sequential optimization model of product replacement and spare-parts inventory. To expand the applicability of this method, Wang et al. [19] constructed a degradation model based on the Wiener process, and the maintenance decision model was established according to the RUL prediction of the equipment. Thus, the optimal decision of the equipment replacement strategy and spare-parts ordering strategy was achieved. However, the spare-parts ordering decision was made based on the optimal replacement strategy, which may lead to local optima of the decision result and affect its effectiveness. Jiang et al. [20] effectively enhanced the scientific nature of the results and ensured the rationality of the maintenance scheme via the joint optimization of the equipment replacement strategy and spare parts ordering strategy. However, this method can only update the drift coefficient online during RUL prediction but fails to update the diffusion coefficient synchronously, which restricts the accuracy enhancement of RUL prediction and is not conducive to the realization of scientific maintenance decisions.
According to the problems with current joint optimization for equipment replacement and the spare-parts ordering strategy based on the RUL prediction information of accelerated degradation, the synchronous effect of accelerating stress on the drift coefficient and diffusion coefficient is analyzed in this study, where the relationship between the drift coefficient and diffusion coefficient is assumed to be proportional. Also, the drift and diffusion coefficients were updated synchronously using a Kalman filter and on-site monitoring data of target equipment, which can effectively reduce the uncertainty of prediction while ensuring RUL prediction accuracy. Thus, this study develops a joint optimization model of replacement and spare-parts ordering strategy that considers RUL prediction information based on the renewal reward theory and determines the optimal average cost ratio of an equipment operation cycle via joint optimization with the equipment replacement time and spare-parts ordering time.
In different situations, Wiener process models can be generally divided into linear drift models, logarithmic change models, time scale transformation models, and general Wiener process models [21]- [22], among which the time-scale transformation model is primarily used in the process of accelerated degradation modeling and can be expressed as: where X(t) is the performance degradation for the equipment at t time; ) is the function of t, ν is the unknown parameter;  is the drift coefficient, and is the standard Brownian motion item, and )) ( To shorten the time and reduce the cost of highly reliable product degradation tests, a high-stress level is often used to accelerate equipment degradation. Considering the effect of acceleration stress on the drift coefficient and diffusion coefficient, an accelerated degradation model can be established as follows:  [12] demonstrated that the accelerated degradation test should satisfy the principle of a constant acceleration factor, which can be expressed as follows: is the accelerated factor of 2 S to 1 S , and its value only depends on 1 S and 2 S . Equation (3) implies that: where g is a constant. Equation 4 implies that the drift and diffusion coefficients of the equipment degradation model have a fixed proportion relationship under any acceleration stress, the ratio is independent of the stress level under the circumstance that the failure mechanism does not change, and g S S Substituting this proportion relation into Equation (2), the proportional accelerated degradation model can be obtained: For ease of analysis, the Arrhenius acceleration model and step-accelerated degradation test are used as examples in this study. The combination of other acceleration models and the accelerated degradation test types is similar to the abovementioned analysis and will not be described here. The Arrhenius model can be expressed as follows: We assume that the test data of the on-site equipment are ] , , , , and its corresponding performance degradation data are ] , , , . According to Equation (5) and Equation (6), the degradation model of equipment under constant stress is: In this study, the unknown parameters in the degradation model are updated online based on the Kalman filtering principle. The state-space model of the equipment under constant stress is: Also, Equation (8) can be converted into the standard form of the Kalman filter: , and The iterative process of Kalman filtering can be expressed as: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3165050, IEEE Access where: In [23], the probability distribution function of RUL corresponding to the degradation model is as follows: . More information is available in [23].
A Kalman filter was used to update the process, where Z is a two-dimensional normal random variable, and the distribution coefficient of can be obtained. The integral representation of Equation (20) is as follows:

III. Prior parameter estimation
To determine the specific situation of the probability distribution of the RUL of the equipment, it is necessary to estimate the unknown parameters contained in Equation (21). In this paper, the unknown parameters in the RUL prediction model under constant stress were estimated using the degradation data of similar equipment under accelerated stress. Due to the complexity of the model considered in this study, the traditional prior parameter estimation method based on the maximum likelihood principle and EM principle has difficulty establishing the likelihood function and performing the iterative calculation. Therefore, a new step-by-step maximum likelihood estimation (MLE) method is proposed to estimate model parameters.
In this study, represents all the unknown parameters of the RUL prediction model. Assuming that the accelerated degradation test contains N samples, and each sample undergoes M accelerated stress, then if is the k th observed time of the i th sample under the j th stress, represents all the degradation data under the stress of j , and all the data from the accelerated degradation test can be expressed as Based on this analysis, which can be expressed as follows: Based on this analysis, the contour logarithmic likelihood function corresponding to the accelerated degradation data is as follows: , Taking the partial derivatives with respect to and for Equation (30) and making them equal to zero, we can obtain: Substituting Equations (31) and (32) The covariance matrix   Replacement and spare-parts ordering are the key factors that restrict the efficient operation of equipment. The reasonable replacement and spare-parts strategy can effectively reduce the cost in the life cycle of the equipment. Currently, the renewal reward theorem has been widely used in the study of equipment maintenance decisions [24]. With this theorem, the mathematical relationship between equipment maintenance cost per unit time and maintenance decision variables can be easily expressed, which could lay a solid foundation for the forthcoming study.
In this study, the RUL prediction information, in combination with the renewal reward theorem, was used to develop the joint optimization model of equipment replacement and spare-parts purchasing strategy to minimize the cost of one life cycle. The detailed information can be expressed as follows:

Assumption 2:
The equipment begins to run at the initial moment, and there is no spare-part inventory at the initial moment, but at most one spare part is purchased or stored at any time thereafter.
Assumption 3: There is a fixed delivery period for spare parts from the beginning of procurement to the arrival of the goods.
Assumption 4: The time required to replace the equipment can be neglected compared to the running time. If the equipment can still run properly after the spare parts arrive, preventive replacement is performed at time p t , and the replacement cost is p C . The storage cost of spare parts per unit time is 1 H . If the equipment fails before the spare parts arrive, the replacement cost is f C , and the unit time loss caused by downtime is 2 H . For Assumption 1, the Wiener process can accurately describe monotonic and nonmonotonic degradation processes, and its universal applicability makes degradation modeling more general. For Assumption 2, taking single equipment as the research object is representative, and similar ideas can be used for modeling and analysis under multiple equipment conditions. Assumptions 3 and 4 are set based on the real process of spare parts ordering and equipment maintenance support.
In this study, progressive analytical thinking was used to analyze the equipment replacement strategy model and spare parts ordering strategy model, and then, the combined optimization model was obtained.

A. Replacement strategy model
The replacement strategy model is primarily used to determine the optimal preventive replacement time to balance the costs of preventive replacement and failure replacement of the equipment. A schematic diagram of the replacement strategy is shown in Figure 1 Figure 1 shows that the running time of equipment in the replacement strategy model can be divided into two areas of 1 Q and 2 Q , where 1 Q is the area before the current running time, in which the equipment is running properly, and 2 Q is the area in the future running time, in which the equipment is running properly. Thus, the expectation of the operation period corresponding to the replacement strategy model is as follows: See Appendix A for the details on the solution process for Equation (40) and Equation (41).

B. Spare parts ordering strategy model
The model of the spare parts purchasing strategy is primarily used to determine the optimal spare-parts ordering time to balance the cost caused by spare-parts shortage and spare parts storage. Figure 2 shows a schematic diagram of the spare parts ordering strategy.
(42) Figure 2 shows that if t s +τ o ≤t p , preventive replacement is performed when the equipment is in good condition, and its operation cycle is τ p . If t s +τ o ≥t p , the operating period of the equipment is prolonged due to the waiting for spare parts, τ p+ τ -. Thus, the expectation of the spare parts ordering strategy model corresponding to the equipment operation cycle is as follows: See Appendix B for the details on the solution process for Equation (42) and Equation (43).

C. Joint optimization model
Considering the entire process of replacement and spareparts ordering during the life cycle, a joint optimization model of replacement and spare-parts ordering strategy was developed to determine the optimal spare-parts ordering time and preventive replacement time to achieve the lowest average cost rate of the equipment operation cycle.
Based on this analysis, the total cost in the joint optimization model includes the cost generated by equipment replacement and the cost generated by spare parts ordering, which can be described as follows: Considering that any replacement moment is equivalent to the end of the current running cycle or the beginning of the next running cycle, comparing the expression of equipment operation cycle expectation between the replacement strategy model and the spare parts ordering model, the expected operating cycle of the equipment in the easily obtained joint optimization model is as follows: Considering the derivation of the spare-parts ordering strategy model, the optimal replacement time (t p ) in the joint optimization model should not be less than o s t   .

V. Case Study
Micro-electro mechanical system (MEMS) gyroscopes are the core equipment of modern navigation and positioning systems; have high service reliabilities and long effective life cycles; and have been widely used in aviation, aerospace, and equipment. In this study, the RUL of a target piece of equipment is predicted based on the stepstress accelerated degradation data and field monitoring data of a certain MEMS gyroscope, and the joint optimization decision of equipment replacement and spare parts ordering strategy is made accordingly. The step-stress accelerated degradation test included 4 samples and 3 groups of stress levels (S 1 =40℃, , and 50 samples were taken at an interval of 10 h under each group of stress conditions. Fifty samples were taken at an interval of 10 h. The field monitoring data included all degradation data of the target equipment operating under normal stress (S 0 =25 ℃) for 180 days, and the specific degradation process is shown in Figures 3 and 4. To verify the accuracy of the parameter estimation method, we simulate the degradation data of the equipment under normal stress (S 0 ) for different Θ and calculate the estimated value Θ using the proposed step-by-step MLE estimation method. The parameter estimation results are shown in TABLE I. TABLE I shows that the relative errors between the estimated and real values of the different parameters are all less than 30%, which indicates that the step-by-step MLE estimation method can accurately estimate the parameters.
The degradation process of the MEMS gyroscope is nonlinear under normal stress conditions. We let 1 in this study. Using the acceleration degradation data of the MEMS gyroscope in Figure 4 and the a priori parameter estimation method proposed above, the a priori parameter estimates can be described as follows.
The prior parameter estimation results in TABLE II are calculated from accelerated degradation data (hour to day). Also, on-site monitoring data were used to update the state based on the Kalman filter method. The detailed update process is shown in Figure 5.
Generally, when the zero offset increment of the MEMS gyroscope exceeds 2.5% of the initial value, it can be considered to fail, and the failure threshold value . Therefore, the target equipment fails at 180 days; thus, the real life of the target piece of equipment is 180 days. Based on this analysis, the RUL online prediction method was used to perform the online prediction of RUL of target equipment under the constant stress condition. For the ease of comparative analysis, the joint optimization model proposed in this paper is referred to as M0, and the joint optimization model proposed in [14] is referred to as M1. The online prediction method of RUL proposed in [11] is applied to the joint optimization model of replacement and spare-parts ordering, which can be marked as M2. The RUL prediction result, mean squared error (MSE) and 95% confidence interval of RUL prediction corresponding to different methods are shown in TABLE III.
The MSE of RUL can be calculated by Equation (46): where T is the lifetime of the piece of equipment.    III shows that M0 has more accurate RUL prediction results and smaller MSE than M1 and M2 at different state monitoring times. The 95% confidence interval of RUL corresponding to M0 can completely contain the real RUL of the target equipment, which indicates that the M0 model can predict the RUL of equipment more accurately and will have a positive impact on the subsequent joint optimization decision of the replacement and spare parts ordering strategy. Further analysis of TABLE III shows that M1 has the narrowest confidence interval width compared to M0 and M2, which shows that the uncertainty of RUL prediction of M1 is small and the prediction accuracy is high. The primary cause of this result is that M1 only updated the drift coefficient as a random variable online, while M0 and M2 updated the drift and diffusion coefficients synchronously, increasing prediction uncertainty. However, M1 can achieve a high prediction accuracy based on the loss of precision, which keeps the confidence interval from covering the real RUL of the target piece of equipment, which is not conducive to scientific maintenance decisions.
Comparing M0 and M2, the prediction results of M0 and M2 are found to be similar, but the width of the confidence interval under M0 is narrower than that under M2, which shows that M0 has a lower uncertainty and better performance based on ensuring the accuracy of the RUL prediction.
To compare and analyze the advantages and disadvantages of M0, M1 and M2 in the joint optimization decision of replacement and spare-parts ordering, the variation curves of the average cost ratio of the equipment operation cycle corresponding to different models are calculated when it is 40, 80 and 120 days, as shown in Figure 6, where     Figure 8 shows that the average cost ratio of the equipment operation cycle increases with increasing failure replacement cost, but this increase is relatively small, which indicates that the average cost ratio of the equipment operating cycle is not sensitive to the change. This result demonstrates that the probability of failure replacement will markedly decrease, thus reducing the impact of the failure replacement fee on the total operating cost when the RUL of equipment can be accurately predicted. Similarly, the variance of f C has a negligible effect on the optimal spare parts ordering time and preventive replacement time.   Figure 10 shows that the average cost ratio in one operation cycle changes marginally with the change in the spare parts shortage loss, and s t and p t remain stable.
Therefore, the sensitivity of the optimal joint decision pairs is low.  which demonstrates that when the spare parts ordering strategy perfectly fits the equipment maintenance strategy, the optimal solution of the joint optimization model can be realized.

VI. Conclusion
In this study, the online prediction of RUL is achieved by integrating accelerated degradation data and on-site monitoring data. The optimal replacement and spare parts ordering strategies are obtained using the RUL prediction information.
1) To predict the RUL of the accelerated degradation equipment, it is necessary to update the drift coefficient and diffusion coefficient of the equipment simultaneously, which can effectively improve RUL prediction accuracy.
2) The accelerated degradation model with the proportional relationship can more intuitively describe the true degenerate features of the equipment, achieves better model fitting, and reduces the uncertainty of RUL prediction.
3) The higher the accuracy and precision of the RUL prediction method are, the more helpful it is to obtain the optimal maintenance strategy. Higher accuracies can also achieve lower cost consumptions and effectively improve maintenance efficiencies.
In this study, the time-scale transformation model is used, and the application of this model has certain limitations. In future research, we intend to test a more general nonlinear Wiener degradation model, such as in [25], to expand the applicability of the proposed method.

APPENDIX A
Based on the basic principles of probability theory, we can obtain: From the basic property of opposite events, we know that: