Condition-based maintenance decision based on inverse gaussian deterioration process under the condition of regular detection and maintenance

Abstract

As the existing research of condition-based Maintenance (CBM) decision-making neglects the influence of regular detection and maintenance (RDM) on the recovery of equipment performance, it is impossible to accurately describe the state degradation characteristics and life distribution law in this case, which is not helpful to formulate reasonable and effective maintenance strategies. Aimed at this problem, a maintenance strategy combining RDM and CBM is proposed in this paper, and the performance degradation modeling and maintenance optimization model under this strategy are studied deeply. Considering the discontinuous and catastrophic performance degradation characteristics of equipment under this condition, a performance degradation model is established by using the Inverse Gaussian process from the failure mechanism. On this basis, a combined maintenance decision model constrained by risk function is constructed. The optimal maintenance cycle and preventive maintenance threshold are obtained by optimizing the equipment maintenance cost under long-term operation conditions. The relationship between the cost rate and the maintenance strategy value is obtained through the example analysis of the equipment components, and it is proved that the joint maintenance strategy can not only prolong the service life and maintenance interval of equipment, but also reduce the maintenance risk and cost.

Keywords

Regular detection and maintenance condition-based maintenance inverse gaussian process cost rate maintenance strategy

1 Introduction

With the development of advanced sensors and detection technology, Condition-Based Maintenance (CBM) has become a new trend of equipment maintenance support [1, 2]. According to the current testing state and historical information of the equipment, CBM implements the corresponding preventive maintenance activities with a certain optimization target to avoid the military and economic losses caused by equipment failure. Therefore, it is an important goal of military support task to formulate reasonable maintenance strategy and to reduce maintenance cost in life cycle. Maintenance scheduling have always been the research hotspot in the field of maintenance. Garg et al. presented periodic preventive maintenance (PM) of a system with deteriorated components and found the best interval for providing the necessary PM actions [3]. Meanwhile, a methodology for analyzing the system performance of an industrial system by utilizing uncertain data and an alternative method for computing the membership function of the system under intuitionistic fuzzy set environment were presented by Garg [4, 5]. Also, in order to reduce the uncertainty level for decision makers, the fuzzy set theory has been implemented during the analysis for handling the uncertainties in the collected data [6]. Liao et al. proposed a multi-objective optimization approach for parallel machines to deals with the joint problem of production scheduling and maintenance planning [7].

Modeling the deterioration law of the system is a necessary prerequisite for rational optimization of maintenance scheduling strategy. The current research on the description of deterioration law is mainly focused on the stochastic process theory, such as [8, 9] describe the trend of system state deterioration through Gamma process and determine the value of maintenance strategy. Guo [10] and Li [11] use Wiener process to model deterioration and optimize inspection intervals and maintenance thresholds for tasks and storage stages respectively. In addition, many achievements have been made in establishing maintenance decision model based on Markov process [12, 13].

In the maintenance support work of the grass-roots units, in order to maintain the technical state of the equipment, the technical means of Regular detection and maintenance (RDM) are usually adopted. For example, the missile storage box would be regularly measured and maintained for air tightness, and the launcher hydraulic frame would be regularly lubricated and maintained. However, the existing research only considers the deterioration process of performance from the state test itself and neglects the effect of RDM activities on the restoration of equipment state in engineering practice. The performance degradation characteristics of discontinuity and catastrophe are also not taken into account in the model. It is obvious that the maintenance modeling based on the above theory is not in accordance with the actual situation. Therefore, it is an unsolved problem to reasonably formulate the maintenance strategy of the combination of RDM and CBM.

Compared to the exiting works on maintenance optimization, the unique contributions of the present work are threefold: (1) The maintenance strategy requirements of the combination of RDM and CBM are described, and the performance degradation characteristics under this condition are analyzed. (2) The performance degradation model is established by inverse Gaussian process under the condition of joint strategy. (3) A failure risk function is defined, and the life cycle expectation of equipment is calculated in different situations. On the basis of them, a decision optimization model with the lowest average maintenance cost rate under the constraint of failure risk is established. Finally, the inspection period and state maintenance threshold are determined by a numerical example of an equipment component, and the relationship between the cost rate and the maintenance strategy is analyzed, which proves the superiority of the joint maintenance strategy.

The remainder of this paper is organized as follows. Section 2 gives the description of joint maintenance strategy of RDM and CBM. The Model of performance degradation based on Inverse Gaussian process is established in Section 3, and the joint maintenance optimization model is established in Section 4. In Section 5, an illustrative example and the comparison of maintenance strategies are given to illustrate the effectiveness the proposed method. The paper is concluded in Section 6.

2 Description of joint maintenance strategy of RDM and CBM

2.1 Notations

The following notations have been used in the entire paper.

{X (t) , t⩾ 0 }— Performance degradation function.

Δt— Detection interval.

ξ— Preventive maintenance threshold.

L— failure threshold.

f (·)— Probability distribution function.

IG (1pt · 1pt, 1pt ·)— Distribution Function of Inverse Gauss Process.

μ— Mean parameters.

λ— Scale parameters.

T— Life Cycle.

Λ (t)— increasing function.

C (Δt, ξ)— Average cost rate.

C_d— Periodic testing and maintenance costs.

C_p— Preventive replacement costs.

C_f— Replacement cost after failure.

r (kΔt)— The risk function at Time kΔt.

E (T)— Expectation function of life cycle.

2.2 Performance state analysis considering regular detection

In the process of storage or use of equipment, the performance state X (t) will continue to decline, when it exceeds a certain value (recorded as the failure threshold L), the equipment is deteriorated, that is, it cannot meet the normal work requirements. In order to improve the efficiency of the system, CBM compares the performance status with the preventive maintenance threshold ξ in time and adopts the corresponding maintenance activities before the failure of the system. This process is shown in Fig. 1.

Fig.1

Deterioration process under the condition of CBM.

In fact, in order to maintain and restore the performance of military equipment in the life cycle, many equipment systems will carry out regular inspection and maintenance, such as regular detection, mechanical alignment before use of inertial navigation system and measurement calibration of weapon system and so on. RDM not only detects the performance state of the system, but also returns the performance to the initial value, but the maintenance activity does not change the deterioration rate of the system itself. The process of deterioration of the system performance under the condition of RDM is shown in Fig. 2. It can be seen that regular maintenance activities postpone the failure time of the system, the deterioration process is no longer continuous, and the deterioration of the system is gradually increasing in each period.

Fig.2

Deterioration process under the condition of RDM.

2.3 Maintenance policy description and assumptions

The whole time that the equipment begins to deteriorate from a new state until it is restored to a new state due to maintenance activity is defined as a life cycle. Considering the limited restoration degree of the equipment performance state, CBM will be taken in the actual equipment support work, and the preventive repair or replacement is carried out before the equipment failure in order to reduce the failure loss. For equipment that combines RDM with CBM, its specific maintenance strategy and assumptions are described as follows:

The equipment periodically carries out periodic detection, the detection period is Δt, if the state value is lower than the preventive maintenance threshold ξ, only maintenance is carried out to restore the performance state to the specified zero value.

If the performance state value is higher than the preventive maintenance threshold ξ at a certain time of detection, only preventive maintenance is carried out to repair the equipment as new and end the life cycle.

if the state value exceeds the failure threshold L, the equipment has already failed, and then the equipment is replaced and repaired immediately, and the life cycle is completed at the same time.

3 Modeling of performance degradation based on Inverse Gaussian process

From the analysis of the deterioration of equipment performance under the condition of RDM in the previous section, it can be seen that the traditional performance deterioration model of continuous variation of state maintenance is no longer applicable to the joint maintenance strategy. The modeling of performance deterioration under the condition of RDM can not only objectively reflect the characteristics of performance degradation, but also serve as the theoretical basis for the next step of maintenance decision optimization.

In recent years, the degradation modeling method based on the Inverse Gaussian process has shown flexible fitting advantages in the field of deterioration modeling and reliability evaluation [14]. As the ultimate extension of the Gamma process, the Inverse Gaussian process is also applicable to the modeling of monotonous degradation processes and is the natural selection for deteriorating data [15, 16], so it has a more flexible advantage than the Gamma process. Therefore, according to the performance deterioration of a certain system under the condition of RDM, the following assumptions are made. Assuming that the stochastic performance degradation process (a is time) is an inverse Gaussian process, the following properties are satisfied: ding172X (0) = 0 and the probability is 1; ding173X (t) has independent increments, that is, for any 0 ⩽ t₁ < t₂ ⩽ t₃ < t₄, random increment X (t₂) - X (t₁) and X (t₄) - X (t₃) are independent of each other; ding174 the increment ΔX (t) = X (t + Δt) - X (t) is distributed by inverse Gaussian ΔX (t) = X (t + Δt) - X (t), i. e. ΔX (t) ∼ IG (μΔΛ (t) , λΔΛ² (t)), where μ and λ are mean parameter and scale parameter. Λ (t) is a monotone increasing function and Λ (0) = 0.

According to the above properties, the probability density function of X (t) can be deduced as follows: $f_{IG} = \sqrt{\frac{λ Λ {(t)}^{2}}{2 π x^{3}}} exp [- \frac{λ}{2 x} {(\frac{x}{μ} - Λ (t))}^{2}]$ (1)

The Maximum Likelihood Estimate (MLE) method is further used to estimate the parameters of the performance degradation model. The n samples are measured for m times respectively. then X_i,j represents the state value of the i sample at the measuring time t_i,j, where i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , m. The performance deterioration of the system between time t_i,j-1 and t_i,j is recorded as ΔX_ij = X_i,j - X_i,j-1, and the likelihood function of the deterioration of the system under the estimated parameter θ (μ, λ) is as follows: $\begin{matrix} L (θ) = \prod_{i = 1}^{n} \prod_{j = 1}^{m} f (Δ X_{ij}; θ) \\ = \prod_{i = 1}^{n} \prod_{j = 1}^{m} [\begin{matrix} \sqrt{\frac{λ {(Δ Λ_{j})}^{2}}{2 π {(Δ X_{i} (t_{j}))}^{3}} •} \\ exp [- \frac{λ {(Δ X_{ij} - μ Δ Λ_{j})}^{2}}{2 μ^{2} Δ X_{i} (t_{j})}] \end{matrix}] \end{matrix}$ (2)

Where ΔΛ_j = Λ (t_j) - Λ (t_j-1), the logarithmic likelihood function is obtained by finding the derivative on both sides of formula (2): $\begin{matrix} ln L (θ) = \\ \sum_{i = 1}^{n} \sum_{j = 1}^{m} {\begin{matrix} - \frac{1}{2} ln \frac{λ {(Δ Λ_{j})}^{2}}{2 π {[Δ X_{i} (t_{j})]}^{3}} - \\ \frac{λ {[Δ X_{ij} - μ {(Δ Λ_{j})}^{2}]}^{2}}{2 μ^{2} Δ X_{i} (t_{j})} \end{matrix}} \end{matrix}$ (3)

Considering the intermittent state of the deterioration performance under the condition of RDM, if the system carries out the detection activities at time jΔt (j = 1, 2, ⋯), combined with the properties of the Inverse Gaussian process, the life distribution function at any time t ∈ ((j - 1) Δt, jΔt] can be derived: $\begin{matrix} F (t) = P (T ⩽ t) = P (X (t) ⩾ L) = \\ Φ {\sqrt{\frac{λ}{L}} [Λ (t) - Λ ((j - 1) Δ t) - \frac{L}{μ}]} - \\ exp (\frac{2 λ Λ (t) - Λ ((j - 1) Δ t)}{μ}) \cdot \\ Φ {- \sqrt{\frac{λ}{L}} [\frac{L}{μ} + Λ (t) - Λ ((j - 1) Δ t)]} \end{matrix}$ (4)

Where Φ (•) is the Standard Normal distribution function.

However, because the p-quantile of the above formula does not have explicit solution, the system generally has a longer life cycle in the actual working environment. In order to conveniently express the law of deterioration of performance under the condition of RDM, it is known that at this time the Inverse Gaussian process can be considered to be approximately subordinate to the normal distribution of the mean value of μΛ (t) and the variance of μ³Λ (t)/ - λ [17]. The life distribution rule of the system at any time t ∈ ((j - 1) Δt, jΔt] is: $\begin{matrix} F_{T_{L}} (t) = P_{j} (T ⩽ t) = P_{j} (X (t) ⩾ L) \\ \approx 1 - Φ [\frac{L - μ (Λ (t) - Λ ((j - 1) Δ t))}{\sqrt{μ^{3} (Λ (t) - Λ ((j - 1) Δ t)) / - λ}}] \\ = Φ [\begin{matrix} \sqrt{\frac{λ (Λ (t) - Λ ((j - 1) Δ t))}{μ}} \\ - L \sqrt{\frac{λ}{μ^{3} (Λ (t) - Λ ((j - 1) Δ t))}} \end{matrix}] \end{matrix}$ (5)

Where j is an arbitrary positive integer, and when j = 1, the deterioration state of the system is consistent with that under the general condition of CBM. The analysis of the deterioration law under the condition of RDM also lays a theoretical foundation for the next maintenance decision optimization.

4 Joint maintenance optimization model

For equipment with RDM, increasing the frequency of testing and maintenance will be more conducive to understanding the deterioration of the equipment performance status and achieving the purpose of prolonging working hours and service life. However, the decrease of detection cycle Δt will lead to the increase of maintenance cost, and it does not meet the requirements of some standardized use of equipment. On the other hand, setting a lower preventive maintenance threshold ξ can reduce the risk of equipment failure, but preventive replacement maintenance can shorten the life cycle and the cost of long-term operation cannot meet the economic requirements. Therefore, it is necessary to optimize the inspection cycle Δt and the preventive maintenance threshold ξ to achieve the goal of the lowest average cost rate C (Δt, ξ) within the life cycle.

At the same time, in order to avoid economic losses and safety accidents caused by equipment deterioration and failure, the failure probability should be limited to a certain extent [18] and the risk at any time kΔt should be defined as: $\begin{matrix} r (k Δ t) = \frac{Pr {Work fails until time (k Δ t, (k + 1) Δ t]}}{Pr {Work normally until time k Δ t}}, \\ k = 1, 2, \dots \end{matrix}$ (6)

Therefore, in order to balance the maintenance cost in the life cycle, a joint maintenance optimization model of RDM and CBM is established to meet the upper bound constraints. $\begin{matrix} min & C (Δ t, ξ) \\ s . t . & r (k Δ t) ⩽ q \end{matrix}$ (7) where C (Δt, ξ) represents the average cost rate under the joint maintenance strategy, and q is the highest limit of the risk coefficient.

4.1 Expected life cycle

Based on the analysis of the performance deterioration law of the equipment system in the previous section, the life cycle expectation of the equipment is determined in combination with the specific requirements of the joint decision of RDM and CBM. According to the previous hypothesis, there are two main situations for equipment to end life cycle: ① preventive replacement; ② Replacement after failure.

1) if the equipment performs preventive maintenance at time jΔt (j = 1, 2, ⋯) and the life cycle ends, in this case it means that the deterioration status of the equipment did not exceed the preventive maintenance threshold ξ in previous j - 1 detection, when the system continues to work until time jΔt after maintenance at time (j - 1) Δt, the performance state exceeds ξ, but the failure threshold L is not reached. Therefore, the probability of preventive replacement at time jΔt is: $\begin{matrix} P (ξ ⩽ X (j Δ t) < L) = \prod_{m = 1}^{j - 1} P (X (m Δ t) < ξ) • \\ [P (X (j Δ t) < L) - P (X (j Δ t) ⩽ ξ)] = \\ \prod_{m = 1}^{j - 1} Φ [\frac{ξ - μ (Λ (m Δ t) - Λ ((m - 1) Δ t))}{\sqrt{μ^{3} (Λ (m Δ t) - Λ ((m - 1) Δ t)) / - λ}}] • \\ [Φ (\frac{L - μ (Λ (j Δ t) - Λ ((j - 1) Δ t))}{\sqrt{μ^{3} (Λ (j Δ t) - Λ ((j - 1) Δ t)) / - λ}}) - \\ Φ (\frac{ξ - μ (Λ (j Δ t) - Λ ((j - 1) Δ t))}{\sqrt{μ^{3} (Λ (j Δ t) - Λ ((j - 1) Δ t)) / - λ}})] \end{matrix}$ (8)

The expected life cycle E₁ (T) meets: $E_{1} (T) = \sum_{j = 1}^{\infty} j Δ t P (ξ ⩽ X (j Δ t) < L)$ (9)

2) if the equipment is replaced after maintenance at time jΔt (j = 1, 2, ⋯) and the life cycle ends, it is equivalent to that the performance state of the equipment system does not exceed ξ in previous j - 1 detection, and then continues to work after maintenance until time t ∈ ((j - 1) Δt, jΔt] exceeds the failure threshold L. The probability expression of this case is as follows: $\begin{matrix} P (X (t) ⩾ L) = \\ \prod_{m = 1}^{j - 1} P (X (m Δ t) < ξ) • P_{j} (X (t) ⩾ L) = \\ \prod_{m = 1}^{j - 1} Φ [\frac{ξ - μ (Λ (m Δ t) - Λ ((m - 1) Δ t))}{\sqrt{μ^{3} (Λ (m Δ t) - Λ ((m - 1) Δ t)) / - λ}}] • \\ {1 - Φ [\frac{L - μ (Λ (t) - Λ ((j - 1) Δ t))}{\sqrt{μ^{3} (Λ (t) - Λ ((j - 1) Δ t)) / - λ}}]} \end{matrix}$ (10)

The expected life cycle E₂ (T) meets: $\begin{matrix} E_{2} (T) = \sum_{j = 1}^{\infty} \int_{(j - 1) Δ t}^{j Δ t} t d [P_{j} (X (t) ⩾ L)] • \\ \prod_{m = 1}^{j - 1} P (X (m Δ t) < ξ) \end{matrix}$ (11)

Combined with the above analysis, the average expected life cycle of CBM under the condition of RDM is expected to meet: $E (T) = E_{1} (T) + E_{2} (T)$ (12)

4.2 Maintenance cost analysis

Under the influence of the joint maintenance strategy, maintenance costs mainly include periodic test maintenance costs C_d, preventive replacement costs C_p, and post-fault repair and replacement and related ancillary expenses C_f. The value of the maintenance cost of the equipment is related to the maintenance activity during the life cycle. Based on the probability analysis of each situation in above Section 4.1, the expected cost of the equipment is: $\begin{matrix} E (C) = \sum_{j = 1}^{\infty} [(j - 1) C_{d} + C_{p}] • \\ P (ξ ⩽ X (j Δ t) < L) + \\ \sum_{j = 1}^{\infty} [(j - 1) C_{d} + C_{f}] P_{j} (X (t) ⩾ L) \end{matrix}$ (13)

In order to measure the economics and effectiveness of the maintenance strategy, the average cost of the product life cycle is: $\begin{matrix} C (t) = lim_{t \to \infty} \frac{Maintenance costs during the period of (0, t]}{t} \\ = \frac{E (C)}{E (T)} \end{matrix}$ (14)

The joint maintenance decision for periodic inspection and maintenance is the optimal maintenance strategy to achieve the lowest average cost rate as the optimization goal, to obtain the inspection cycle and replace the threshold joint decision.

4.3 Failure risk

$r (j Δ t) = \frac{\sum_{j = 0}^{\infty} \prod_{m = 1}^{j} P (X (m Δ t) < ξ) • P (X_{j + 1} ((j + 1) Δ t) > L)}{\sum_{j = 0}^{\infty} \prod_{m = 1}^{j} P (X (m Δ t) < L)}$ (15)

4.4 Model solving

In summary, the decision-making optimization model that satisfies the constraint (15) can be established as: $\begin{matrix} min C (Δ t, ξ) = \frac{\sum_{j = 1}^{\infty} [(j - 1) C_{d} + C_{p}] • P (ξ ⩽ X (j Δ t) < L) + \sum_{j = 1}^{\infty} [(j - 1) C_{d} + C_{f}] P_{j} (X (t) ⩾ L)}{\sum_{j = 1}^{\infty} j Δ t P (ξ ⩽ X (j Δ t) < L) + \sum_{j = 1}^{\infty} \int_{(j - 1) Δ t}^{j Δ t} t d [P_{j} (X (t) ⩾ L)] • \prod_{m = 1}^{j - 1} P (X (m Δ t) < ξ)} \\ s . t . r (k Δ t) ⩽ q \end{matrix}$ (16)

There are many typical solving methods successfully applied in the target optimization model of the above formula [19, 20], but this paper uses the iterative method to obtain the global optimal solution [21]. The details are as follows:

Considering the actual situation of the project, the inspection period Δt takes a positive integer and increases from small to large, and respectively calculates the objective function values corresponding to different ξ values under the constraint conditions.

According to the change trend of the expense rate and Δt t and ξ, the lowest cost value under different Δt is obtained as the local optimal solution.

Select the minimum value of the local optimal solution as the optimal objective function value. At this time, (Δt, ξ) is the global optimal decision solution.

5 Examples and results analysis

5.1 Examples analysis

Taking a type of radar seeker transmitter magnetron equipment as an example, the equipment needs regular maintenance during storage to improve the vacuum inside the tube and prevent aging. As a typical component that needs regular inspection, the Inverse Gaussian process can be used to describe the degradation law of a certain performance state [22], assuming that the component degradation law satisfies X (t) ∼ IG (μΔΛ (t) , λΔΛ² (t)), and Λ (t) = t^1.215; μ = 1.847 × 10^-3; λ = 4.966 × 10^-3; failure threshold is L = 0.2.

Assume that the periodic inspection cost of the component is C_d = 5000 RMB, the preventive replacement cost is C_p = 20000 RMB, the repair and replacement after the failure and the related cost is C_f = 60000 RMB. Under the condition that the failure risk degree is less than 0.2, the model solution method under periodic inspection conditions can be used to obtain the change relationship between the average maintenance cost rate and the maintenance strategy, as shown in Fig. 3.

Fig.3

Relationship between cost rate and maintenance strategy.

The calculation analysis shows that when the inspection period is Δt = 10 and the preventive maintenance threshold is ξ = 0.12, the average maintenance cost rate of the components is the lowest, that is min C = 1160.4 RMB /month. This requires the maintenance personnel to inspect and repair the equipment every 10 m. If the performance index exceeds 0.12, the maintenance act is required to restore the parts to a new state.

5.2 Comparative analysis

Further compare the changes in the average expected maintenance cost and the inspection cycle Δt. The optimal average maintenance cost rate for each cycle is shown in Fig. 4.

Fig.4

The relationship between cost rate and detection period.

As can be seen from the above figure, the average cost rate shows a trend of increasing first and then decreasing with the increase of the inspection cycle. When Δty is small, the inspection activities are more frequent and the maintenance cost is increased. However, regular inspection as an effective means to improve the performance status of equipment, when the inspection period increases, the life-saving effect of the inspection activities on the equipment cannot be fully reflected, which will shorten the life expectancy of the equipment. Therefore, the maintenance cost will be corresponding increase. During the inspection period Δt = 10, the average maintenance cost and the life-saving effect of the inspection on the equipment reached a balance, which is the optimal period decision value.

The relationship between the maintenance cost and the preventive maintenance threshold ξ when the inspection cycle is Δt = 10 will be analyzed below. As shown in Fig. 5. It can be seen that the average cost rate shows a decreasing trend with the increase of the state maintenance threshold. This is because the smaller maintenance threshold will lead to the advancement of the state maintenance activities, which will affect the life cycle expectation and eventually increase the expense rate. However, in combination with Figs. 3 and 5, the maintenance threshold has a certain range of values. When it is increased to a certain value, it will not change. This is due to the constraint of the risk of failure. This rule is in line with the general situation of the actual project.

Fig.5

The relationship between the cost rate and the preventive maintenance threshold.

5.3 Maintenance strategy comparison

In order to compare the maintenance effect of equipment with RDM during the life cycle, Table 1 gives the corresponding indexes of RDM period, failure risk degree, life cycle expectation and average cost rate of each maintenance strategy. Strategy 1 denotes non-RDM activities, while strategy 2 indicates that only RDM activity is carried out, and preventive CBM activities is not carried out. Strategy 3 is a joint maintenance strategy of RDM and CBM.

Table 1
Comparison of maintenance strategies

Maintenance strategy Detection period /month Failure risk degree Life cycle /month Cost rate /RMB · month^-1

Strategy 1 – 1 45 1333.3

Strategy 2 10 1 75 1266.7

Strategy 3 10 0.16 53 1160.4

Maintenance strategy	Detection period /month	Failure risk degree	Life cycle /month	Cost rate /RMB · month^-1
Strategy 1	–	1	45	1333.3
Strategy 2	10	1	75	1266.7
Strategy 3	10	0.16	53	1160.4

Comparing the strategy 1 and 2 of the above table, we can see that the maintenance of equipment with regular inspection can effectively prolong the life cycle expectation, which is precisely due to the increase in the maintenance interval. The average maintenance cost rate of strategy 2 decreased from 1333.1 RMB/month to 1266.7 RMB /month compared with strategy 1. Compared with strategy 3, it is found that the failure risk degree can be reduced to 0.16, and the life cycle of the former two strategies will inevitably end with after-maintenance activities. When the joint maintenance strategy of RDM and CBM is adopted, the equipment status is adjusted through RDM, so the life cycle expectation is increased compared with strategy 1, and the preventive maintenance means of CBM are taken in time. The high cost of repair maintenance is avoided, so the average cost rate is lower.

The optimal maintenance cost of the model is further compared with the existing methods, and the results are shown in Table 2.

Table 2

Cost rate comparison of maintenance models

Maintenance model	detection period /month	cost rate /RMB · month^-1
Li Ling [9]	5	1824.4
Li Mingfu [11]	3	2532.5
Phuc Do [8]	–	1577.8
Chen Nan [14]	4	1603.2
Our model	10	1160.4

From Table 2, we can see that compared with several typical monotone control limit policies which use Gamma process [8, 9] and wiener process [11] to describe equipment performance degradation, the average cost of the joint maintenance model established in this paper is the lowest, which is 1160.4 RMB/month, and can save 1372.1 RMB/month at most. At the other hand, comparing with the maintenance method which only uses inverse Gauss process for degenerate modeling [14], it can be seen that because of the performance recovery factors of RDM activities, our model not only optimizes the detection interval for the longest time, avoids frequent maintenance work, but also reduces the cost budget by 442.8 RMB/month. The modeling process is more consistent with the actual situation.

Therefore, on the one hand, the joint maintenance strategy can delay the time of equipment failure, reduce the times of condition maintenance and afterwards maintenance, and thus save the cost of maintenance. On the other hand, the strategy can ensure that some equipment components which are easy to cause heavy losses and difficult to maintain after failure can meet the requirements of a certain combat readiness rate, so as to complete the established tasks.

6 Conclusions

This paper starts from the actual situation of equipment support for RDM, taking into account the recovery effect of RDM on the performance state of complex systems. The performance degradation law is described by Inverse Gaussian process, and the life distribution under conditions of RDM is analyzed. On this basis, the joint maintenance strategy of RDM and CBM is proposed. Under the constraints of the degree of failure risk, a maintenance decision model with the goal of the minimum cost rate is established. The optimal maintenance strategy value is obtained by jointly optimizing the detection period and the condition-based maintenance threshold. The relationship between the cost rate and the decision quantity and the influence of RDM on the equipment performance are analyzed and compared. The research shows that the joint maintenance strategy can prolong the service life and reduce the maintenance cost in the life cycle, which is of great significance to the support work of weapon equipment.

However, the performance degradation model constructed in this paper does not consider the impact factors of sudden failure caused by random stocks of environmental stresses. In fact, component failure of equipment is the result of competition between degradation failure and sudden failure. Future research should focus on the establishment of correlative degradation model and explore the impact of component degradation on sudden failure probability. At the same time, aiming at the equipment with incomplete maintenance actions, the establishment of joint optimization model of maintenance decision-making and spare parts ordering is also our future research direction.

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Condition-based maintenance decision based on inverse gaussian deterioration process under the condition of regular detection and maintenance

Abstract

Keywords

1 Introduction

2 Description of joint maintenance strategy of RDM and CBM

2.1 Notations

2.2 Performance state analysis considering regular detection

3 Modeling of performance degradation based on Inverse Gaussian process

5.1 Examples analysis

Table 1 Comparison of maintenance strategies Maintenance strategy Detection period /month Failure risk degree Life cycle /month Cost rate /RMB · month-1 Strategy 1 – 1 45 1333.3 Strategy 2 10 1 75 1266.7 Strategy 3 10 0.16 53 1160.4

References

Table 1
Comparison of maintenance strategies

Maintenance strategy Detection period /month Failure risk degree Life cycle /month Cost rate /RMB · month^-1

Strategy 1 – 1 45 1333.3

Strategy 2 10 1 75 1266.7

Strategy 3 10 0.16 53 1160.4