The optimization of the maintenance policy based on the simplified system reliability model XiaotongSun, כ, Xiaomeng Guan, Shuyang Guo School of Reliability and Systems Engineering Beihang University Beijing, China firstname.lastname@example.org Abstract—Degradation of multi-component system exists widely in the field of engineering. And the degradation failure of multi-component system is due to the overall degradation of the system to a certain extent, or the degradation of component. In this paper, we focus on the reliability model of multi-component degradation system, which failure caused by degradation of component failure, was simplified and applied to the optimization model of maintenance policy. For each component of the degraded system, pure degradation and degradation increment can cause irreversible damage to it. The continuous degradation process was considered to be linear. And the degradation increment caused by the shock was considered to be cumulative. The shock was random, so that subject to Poisson distribution. The reliability model of the multi-component degradation system under the cumulative Poisson shock is complex and not easy to be applied to the maintenance model optimization model of the multicomponent system. From that, this paper simplified the reliability model of multi-component degradation system under the cumulative Poisson shock. And the simplified reliability function was used as the constraint to improve the optimization model of multi-objective maintenance policy based on the reliability theory so as to facilitate its application in engineering. The feasibility of the simplified reliability model and the correctness of the improved multi-objective maintenance policy was verified by case. The influence of the change of the reliability constraint value on the whole maintenance policy was analyzed. Keywords—reliability model;cumulative Poisson distribution; optimization; degradation system preventive maintenance strategy; I. NOTATION ܹ ሺ݆ ൌ Ͳǡͳǡʹ ǥ ሻ: The performance degradation caused by the ݆th on the ݅th component ܭ ሺ݅ ൌ ͳǡʹǡ͵ ǥ ሻ : The degradation threshold of the ݅ th component. ܶ : The time of a minor maintenance aimed at ݅ th component ܥ : The cost of a minor maintenance aimed at ݅ th component ܥ : The cost of each preventive maintenance ܥ : The update cost of the whole system 978-1-5386-0370-3/17/$31.00 ©2017 IEEE ܰ: The number of preventive maintenance in one update cycle ܨ ǣ The minor maintenance times of ݅th component in ݇th preventive maintenance cycle ܾ: The reliability constraint ݎሺݐሻǣ The failure rate of the system in the first preventive maintenance cycle ߠ: The growth coefficient of failure rate ܴே ሺܶሻ : The system reliability after ܰ th preventive maintenance cycles ܥሺܰǡ ܶሻ: The average cost of maintenance ܲሺܰǡ ܶሻ: The availability of system II. INTRODUCTION Most of the engineering systems degrade over time. When the degradation reach to a certain extent, the failure occurred. The working environment of multi-component system is complex. The degree of degradation in components and the ability to withstand the shock is also different. Pure degradation and degradation increment cause damage in various parts of the system until failure. The degradation of the system is extensive in real life, so the optimization of the maintenance policy for degradation system is what the scholars concerned. Some people have studied the degradation of the system. Ross  studied generalized Poisson shock models, which has greatly influenced the later study. Poisson shock process for non-cumulative damage was discussed by Fink . Guo  studied the degradation process of multiple subsystems subject to continuous wear and random shocks. Sanling Song  analyzed the reliability of multi-component systems subject to soft and hard failure process. Sanling Song considered the effect of degradation of various components on the overall degradation of the system under the cumulative Poisson shock and established the reliability model. But not considered the practicality of the reliability model, so that the reliability model can not be applied directly. Many people have studied the optimization of the maintenance degradation system in the detail. Zhao  analyzed the cost function of the system by using the failure rate as the effect of the maintenance parameter. Love and Guo  discussed the optimal preventive maintenance policy under the failure rate constraint assuming that the failure process of the system follows the Weibull distribution. Barlow  describes a preventive maintenance policy by considering the relationship between the maintenance cycle and the remaining life of the device, each maintenance cycle has a different cycle. Different maintenance policies for degraded systems have been studied. H.Wang  summarized, classified and compared various maintenance policies. J.M  studied the application of gamma processes in maintenance. Guo  established an optimization model of multi-objective maintenance policy, but not considered the practical application of reliability as a constraint. In view of the complexity of multi-component reliability model system under Poisson shock, this paper simplified it and made it easy to use in mathematical analysis and engineering application. The reliability model was used to further improve the optimization model of the existing multiobjective maintenance policy. In addition, the influence of the reliability constraint value on the whole maintenance policy was analyzed. This paper is organized as follows. Section 2 introduces the reliability model of multi-component system under cumulative Poisson shock and the method how to simplify it. Section 3 describes the optimal model for periodic preventive maintenance policy and improve the model to convenient to use. An example is given in Section 4 to illustrate the correctness of the improved optimal model and practicality of the simplified function of reliability. In Section 4, the influence of the reliability value as a constraint on its overall maintenance policy is analyzed. III. RELIABILITY MODEL FOR MULTI-COMPONENT SYSTEM UNDER ACCUMULATED POISSON SHOCK AND ITS SIMPLIFICATION A. Reliability Model There are many systems subject to cumulative Poisson shock, such as Micro-electro mechanical systems (MEMS). This kind of system suffers from two degradation process, pure degradation and degradation increment. The pure degradation process of the multi-component system is Dlinear degradation process, and the degradation increases linearly with time. But occasionally it subject to external shock led to its sudden increase in the amount of degradation, and each shock on the various components of the amount of degradation is accumulated. We assume that each component is under cumulative Poisson shock, failure of any component can cause the failure of system. Only if the degradation below the respective thresholds can the system work properly. From Song  we can get the reliability model of multi-component system under cumulative Poisson shock as follows: ሺݐሻ ൌ σஶ ୀ ςୀଵ Ȱ ቌ ିቀఓഁ ௧ାఓೈ ቁ మ మ ටఙഁ ௧ మାఙೈ ቍ ή ௫ሺିఒ௧ሻሺఒ௧ሻ Ǩ (1) In this model, the system consists of ݉ parts; ݊ is the number of shocks; The number of shock occur in time ሺͲǡ ݐሻ is subject to the Poisson distribution; ܹ ሺ݆ ൌ Ͳǡͳǡʹ ǥ ሻ is the performance degradation caused by the ݆th shock on the ݅ th component, ܹ ൌ Ͳ ; The performance degradation ൛ܹ ൟ caused by each shock is independent and equally distributed. The degradation of performance of the ݅th component caused by each shock is subject to a normal distribution, which is ଶ ൯; ܭ ሺ݅ ൌ ͳǡʹǡ͵ ǥ ሻis the degradation threshold ܹ ̱ܰ൫ߤௐ ǡ ߪௐ of the ith component. The parameter vector ߚ is subject to a normal distribution, that is ߚ ̱ܰ൫ߤఉ ǡ ߪఉଶ ൯ . The reliability model has been verified by Song . B. The simplification of reliability model Since the cumulative reliability model of the multicomponent system under the cumulative Poisson shock in Section 2 is too complicated, containing the cumulative distribution function of normal distribution and the sum of the infinite terms. SR the reliability function is not conducive to the use of analysis, especially for some mathematical transformation. In the optimization of system maintenance policy, the problem is more prominent. So it is necessary to find a way to simplify the model to make it more practical. In this paper, the reliability model of multi-component system under the cumulative Poisson shock is processed linearly and simplified into a linear function to facilitate practical application. In the practice of projects, the whole reliability function is usually not used. Only the part of the reliability function is valuable. According to the requirement from maintenance policy, we can focus on the part of reliability function where the value of reliability is high or the part of the reliability function is changing rapidly. Moreover, it is very difficult to perform polynomial fitting on the whole reliability function when the concavity and convexity of the reliability function curve is complicated. It is simple and easy to operate, and it can guarantee the high degree of fit of the reliability function curve, when we choose the focus interval to simplify the fitting of the reliability function curve. The concrete simplification process is as follows: a) b) c) d) According to the actual system reliability requirements, the focus interval of the reliability function is selected. The part of reliability function with high value or the part of the reliability function with rapidly changes should be focused on. Observe the concavity and convexity of the curves in the focus interval. The points that reflects the concavity and convexity of the original reliability function curve can be given. The number of points should be as little as possible. Use software such as Matlab to draw the polynomial curve through the points which can reflect the concavity and convexity of the original reliability function curve. Compare the fit of the polynomial curve and the original reliability curve. Add a point in the part of the reliability curve with poor fit, and we can adjust the new point until the degree of fit of the polynomial curve and the original reliability curve satisfies the requirement. Sometimes, it is necessary to select the overall function of reliability. Select all the function interval can guarantee the integrity of the reliability data. But in the actual project, we always choose to focus on the interval of reliability function. Select the focus interval can guarantee the simplified curve and the original reliability function image in height fit. We always get simplified function as follow: ሺݐሻ ൌ ݐଶ ݐܤ ܥ the failure rate with its simplified reliability function. Apply it to the optimization model of multi-objective preventive maintenance policy. The mathematical expression of the system failure rate is: (2) ሺݐሻ ൌ Not only for the reliability model of multi-component under cumulative Poisson shock. But also the reliability model of system under the other condition can use this method to simplify the function of the reliability model. The use of the fitted polynomial function instead of the original reliability function which is easy to carry out various mathematical processing. IV. OPTIMIZATION MODEL OF PREVENTIVE MAINTENANCE POLICY WITH CONSTRAINT OF RELIABILITYF The development of preventive maintenance policies can reduce and prevent failure of the system. The optimization of maintenance policy can give the best combination results of preventive maintenance cycle and equipment update cycle. It can bring the maximum profits to the enterprise. Guo  has established an optimization model for periodic preventive maintenance policies, which considered three objects: failure rate, maintenance cost rate and availability. The model of optimization is as follows: ሺܰǡ ܶሻ ൌ ଵିఠ ఊమ ቆെ ఠ ఊభ ൭ భషഇಿ ሺ௧ሻௗ௧ భషഇ బ ಿ భషഇ ே்ା் ሺ௧ሻௗ௧ భషഇ బ ା ሺேିଵሻା ே் ே்ା்ή భషഇಿ ሺ௧ሻௗ௧ భషഇ బ ߙቇ ߙ൱ (3) subject to ܴே ሺܶሻ ܾǡ ܰ ͳǡ ܶ Ͳ The time of a minor maintenance aimed at ݅th component is ܶ ; The expense of a minor maintenance aimed at ݅ th component is ܥ . The expense of each preventive maintenance is ܥ ; The update cost of the whole system is ܥ . The number of preventive maintenance in one update cycle is ܰ (update the system at the last time), and the minor maintenance times of ݅th component in ݇th preventive maintenance cycle is ܨ . The reliability constraint is ܾ. The failure rate of the system in the first preventive maintenance cycle is ݎሺݐሻ. Once the system after a preventive maintenance, the failure rate of growth faster than the last time, the growth coefficient of failure rate is ߠ. ܴே ሺܶሻ is the system reliability after ܰ preventive maintenance cycles. Į is a positive number in the dimensionless process, taking Į=1. It assume that the failure rate of equipment would be reduced to the initial value after a preventive maintenance and the growth coefficient of failure rate ߠ will be faster than before. The failure rate of the system after the Nth preventive maintenance cycle is ݎ ሺݐሻ ൌ ߠ ିଵ ݎሺݐሻሺͲ ൏ ݐ ܶሻ . The minor maintenance times of ݅ th component in ݇ th preventive ் maintenance cycle is ܨ ൌ ݎ ሺݐሻ ݀ ݐ. It use these two parameters to describe the recovery of system reliability In this paper, the reliability of multi-component system under cumulative Poisson shock is taken as the constraint. Since the failure rate ݎሺݐሻ of the multi-component system under cumulative Poisson shock is not easy to obtain, we can replace ሺ௧ሻ ோሺ௧ሻ ൌെ ோ ǡ ሺ௧ሻ (4) ோሺ௧ሻ Calculate the integral on the both side of the function: ் ் න ݎሺݐሻ݀ ݐൌ െ න ܴ ǡ ሺݐሻ ݀ ݐൌ െ൫݈ܴ݊ሺܶሻ െ ݈ܴ݊ሺͲሻ൯ ܴሺݐሻ ൌ െ݈ܴ݊ሺܶሻ (5) It can be seen that the relationship between the reliability and the failure rate of the system is as shown in (6). ் ܴሺܶሻ ൌ ݁ ݔቄെ ݎሺݐሻ݀ݐቅ (6) After ܰ th preventive maintenance cycles, the system reliability is: ் ܴே ሺܶሻ ൌ ݁ ݔቄെ ߠ ேିଵ ݎሺݐሻ݀ݐቅ ൌ ൫ܴሺܶሻ൯ ఏ ಿషభ (7) The whole cost of maintenance mainly comprised the cost of update, the cost of a minor maintenance and the cost of preventive maintenance. Furthermore, the average cost of maintenance can be deduced as follow: ܥ ܥ ሺܰ െ ͳሻ ܥ σே ୀଵ ܨ ே ܰܶ ܶ σୀଵ ܨ ͳ െ ߠே ் ܥ ܥ ሺܰ െ ͳሻ ܥ ݎሺݐሻ݀ݐ ͳ െ ߠ ൌ ே ͳെߠ ் ݎሺݐሻ݀ݐ ܰܶ ܶ ͳ െ ߠ ͳ െ ߠே ܥ ܥ ሺܰ െ ͳሻ െ ܥ ሺܶሻ ͳെߠ ൌ ሺͺሻ ே ͳെߠ ሺܶሻ ܰܶ െ ܶ ͳെߠ The availability of system can be deduced as follow: ܥሺܰǡ ܶሻ ൌ ܰܶ ͳ െ ߠே ் ܰܶ ܶ ή ݎሺݐሻ݀ݐ ͳ െ ߠ ܰܶ ൌ ሺͻሻ ͳ െ ߠே ሺܶሻ ܰܶ െ ܶ ή ͳെߠ A multi-objective optimization can be established. Using linear weighting method in evaluation function , the optimization model of the single-objective preventive maintenance policy is further calculated: ܲሺܰǡ ܶሻ ൌ ሺܰǡ ܶሻ ൌ ଵିఠ ఊమ ቆെ ఠ ఊభ ൭ ே்ି் ே் భషഇಿ ே்ି்ή ோሺ்ሻ భషഇ subject to ൫ܴሺܶሻ൯ భషഇಿ ோሺ்ሻ భషഇ భషഇಿ ା ሺேିଵሻି ఏ ಿషభ భషഇ ோሺ்ሻ ߙቇ ܾǡ ܰ ͳǡ ܶ Ͳ ߙ൱ (10) Take the simplified curve of reliability model into the optimized model for periodic preventive maintenance policy. We can get the function as follow: ሺܰǡ ܶሻ ൌ ଵିఠ ఊమ ቆെ ఠ ఊభ ൭ భషഇಿ ൫௧ మା௧ା൯ భషഇ భషഇಿ ሺ௧ మ ା௧ାሻ ே்ି் భషഇ ା ሺேିଵሻି ே் భషഇಿ ሺ௧ మା௧ାሻ ே்ି்ή భషഇ subject to ሺ ݐଶ ݐܤ ܥሻఏ ಿషభ Part 1 Part 2 Part 3 ܭ ͲǤͲͳʹͷ݀݉ଷ ͲǤͲͲͳʹ݀݉ଷ ͲǤͳͳͷ݀݉ଷ Ⱦଵ ̱൫Ɋஒభ ǡ ɐଶஒభ ൯ Ⱦଶ ̱൫Ɋஒమ ǡ ɐଶஒమ ൯ Ⱦ୧ ̱൫Ɋஒయ ǡ ɐଶஒయ ൯ ߤఉయ ൌ ͺǤ ൈ ͳͲି ݉ଷ. ߪఉయ ൌ Ǥʹ ൈ ͳͲି଼ ݉ଷ. ߤఉభ ൌ ͺǤͶͺʹ͵ ൈ ͳͲି଼ ݉ଷ. ߪఉభ ൌ ǤͲͲͳ ൈ ͳͲିଽ ݉ଷ . ୧୨ ̱൫Ɋభ ǡ ɐଶభ ൯ ߤఉభ ൌ ͳ ൈ ͳͲିଷ ݉ଷ . ߪఉభ ൌ ʹ ൈ ͳͲିସ ݉ଷ. ߚ (11) In the actual application, the simplified reliability curve instead of the reliability model can be used directly. And it is easy to obtain the best combination of preventive maintenance cycle and equipment update cycle under the constraint of reliability. Taking reliability of system after ܰ preventive maintenance cycles as the constraint and integrating cost of maintenance and availability of system, the optimization model was established in (10). By using software, we can get the threedimensional surface of the interval of maintenance, number of preventive maintenance in one update cycle and multi-objective optimization function:ܹሺܰǡ ܶሻ. Through the software we can find the lowest point of the three-dimensional surface using the steepest descent method. And the lowest point is the optimal value of the objective function. By changing the value of the reliability constraint condition value b, the influence of the reliability as the constraint on the whole maintenance policy is analyzed. We can determine the importance of the accuracy of the reliability function curve to the maintenance policy THE PARAMETERS OF SYSTEM COMPONENTS Parameters ߙ൱ ߙቇ. ܾǡ ܰ ͳǡ ܶ Ͳ TABLE I. ܹ ߤఉమ ൌ ͺǤͶͻ͵ ൈ ͳͲିଽ ݉ଷ . ߪఉమ ൌ ͷǤͻͲͳͳ ൈ ͳͲିଵ ݉ଷ. ୧୨ ̱൫Ɋమ ǡ ɐଶమ ൯ ߤఉమ ൌ ͲǤͻ ൈ ͳͲିସ ݉ଷ . ߪఉమ ൌ ʹǤͳ ൈ ͳͲିହ ݉ଷ.. ୧୨ ̱൫Ɋయ ǡ ɐଶయ ൯ ߤఉయ ൌ ͲǤͲͳͳ݉ଷ. ߪఉయ ൌ ʹǤʹ ൈ ͳͲିଷ ݉ଷ. ߣ ൌ ʹǤͷ ൈ ͳͲିହ ߣ ߠ ߠ ൌ ͳǤͳ ߱ ߱ ൌ ͲǤͺ ܾ ܾ ൌ ͲǤͻ ܶ ܶ ൌ Ͷͺ݄ ߙ ߙൌͳ ܥ ܥ ൌ ͷͲͲǡͲͲͲ݊ܽݑݕ ܥ ܥ ൌ ͲǡͲͲͲ݊ܽݑݕ ܥ ܥ ൌ ʹͲǡͲͲͲ݊ܽݑݕ 1 0.9 Where ݅ܭis the degradation threshold of ݅ th component; ܹ݆݅ is the performance degradation caused by ݆th shock on ݅th component; ߣ is the parameter in Poisson distribution, which indicates the random event per unit time of average incidence rate; ߱ is the weight coefficient. Take the data into (1), through software to obtain its approximate function curve. We can get the picture as shown in Fig. 1. 0.8 0.7 0.6 reliability V. CASE STUDY In this section, we use a case to demonstrate the correctness that the simplified reliability model is applied to the improved maintenance policy. A type of mechanical system consists of three components, the type of degradation in parts 1, 2 is wearing, in which the performance degradation of component 1 is the greater the amount of wear and the worse the braking performance. The performance degradation of component 2 is the greater the amount of wear and the worse the sealing performance. The degradation of part 3 is expressed as the length of the crack, and the crack length has an effect on the strength of the mechanical part. The longer the crack is, the lower the strength of the part is. Assuming that the relevant parameters in the degenerate model of each component are estimated by the parameters as shown in the following table, the parameters in the table are derived from the relevant literature  0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 time/h 1.2 1.4 1.6 1.8 2 5 x 10 Fig. 1. Reliability curve of multi-component system under cumulative Poisson shock According to the requirement from maintenance policy, in this case, we focus on the interval of the reliability model image but not the overall reliability curve. But in other actual project, we select all the reliability curve if it is necessary. It can be seen in Fig. 1, the curve is smoothly in (0.4,1) and it meet to the requirement of this case. So we can only focus on the part of ܴ ͲǤͶ. In this interval, the value of reliability is high and the change of the curve is rapidly. So, it is only necessary to sample points in the range of ܴ߳ሺͲǤͶǡͳሻ. Moreover, the complexity of the concave and convex parts of the partial function segment is lower than that of the whole function, so that the fitting process can be made simply. At the same time it can increase the success rate of fit. Observe the concavity and convexity of the reliability curve in the interval:ܴ߳ሺͲǤͶǡͳሻ. Then choose three points that can reflect the curvature of the curve : ሺͲͲͲͲǡͲǤͶሻǡ ሺʹͲͲͲǡͲǤሻǡ ሺͲǡͳሻ. If the fit is poor, we can add a point in the interval of reliability curve. The graph of the fitted polynomial function and the original reliability curve is shown in Fig. 2. It can be seen that the fitting degree is high in ܴ߳ሺͲǤͶǡͳሻ. The graph of the fitted polynomial function and the original reliability curve is shown in Fig. 2. The polynomial function through these three points is a quadratic polynomial function, shown as in (12). ܴሺݐሻ ൌ ǤͲͲ ൈ ͳͲିଵଵ ݐଶ െ ͳǤͶʹͶʹͶʹͶ ൈ ͳͲିହ ݐ ͳ. (12) It can be seen from Fig. 3. that when the system reliability is not less than 0.96, the image of optimal function of the maintenance expense rate and the system availability is concave, and the lowest point is obtained at (4,1439). In this point the maintenance strategy achieve the average cost rate and system availability to integrate optimal. The optimal maintenance policy at this time is described as: Conduct preventive maintenance at 1439h every time, and replace the system before the fourth preventive service arrives. In order to analyze the influence of the constraint of reliability on the whole maintenance policy. Reduce the reliability constraint value, such as ܾ ൌ ͲǤͻͶ. The graph of optimized function of the average maintenance cost rate and the availability is shown in Fig. 4. The fitting degree in the interval is very high, so the simplified reliability function can be used instead of the original curve. Then take (12) and related parameters into (11). We can get Fig. 3. as follow: 1 original reliability function simplified reliability function 0.9 reliability 0.8 0.7 0.6 0.5 0.4 0 1 2 3 time/h 4 5 6 4 x 10 Fig. 2. Comparison between the curve of original reliability and the curve of simplified reliability Fig. 3. Three-Dimensional Function Graph of Optimal Maintenance Strategy ܾ ൌ ͲǤͻ Fig. 4. Three-Dimensional Function Graph of Optimal Maintenance Strategy ܾ ൌ ͲǤͻͶ As can be seen from Fig. 4, ሺͷǡͳͺͶሻis the lowest point. Comparing with Fig. 3.and Fig. 4, we can see that reducing the reliability constraint value will significantly extend the update cycle of equipment and maintenance cycle of system. The difference is obvious in these two graph. It is very important to select the exact and reliable function curve of reliability, which is the constraint in the optimization model of maintenance policy. VI. CONCLUSION In this paper, the reliability model of multi-component system under cumulative Poisson shock was simplified. By selecting the focus interval of the reliability function, the high degree of fit of reliability curve can be obtained. In addition, the optimization model of the maintenance policy of multicomponent system was improved. Furthermore, the simplified curve of reliability function was used as the constraint in the model of multi-objective maintenance policy. The correctness of the improved optimization model of the maintenance policy of multi-component system and the convenience of the simplified reliability curve was proved in the case. At the same time, the influence of the reliability value as a constraint on its overall preventive maintenance policy was analyzed with the case. This paper provides support and help to the practical application of optimization model of maintenance policy of multi-component degradation system under the cumulative Poisson shock. REFERENCES      S. M. Ross, “Generalized Poisson Shock Models,” Annals of Probability, vol. 9, pp. 896-898, 1981. 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