Two-stage Wiener process situation-based replacement decision optimization method considering random change points

Abstract

The invention provides a two-stage Wiener process situation-based replacement decision optimization method considering random change points, and relates to the field of replacement decisions. According to the method, firstly, it is defined that the degradation level of a system obeys a two-stage Wiener process with random change points; secondly, the posterior probability and the degradation level of the system are converted into a discrete state model, and the state transition probability of the system and the expected cost corresponding to execution of different actions at the next detection point are solved in a discrete state; and finally, traversing each detection point by adopting an iterative algorithm, solving the expected cost of executing different actions by each detection point, and selecting the action corresponding to the minimum expected cost as the optimal action of the detection point. The actions include no action and preventive replacement. Compared with a traditional maintenance strategy of a two-stage Wiener process, the strategy provided by the scheme can provide credible results for various types of change points; in addition, a strategy iteration algorithm is proposed to determine an optimal strategy that minimizes the long-term average expected cost.

Description

Technical Field

[0001] This invention relates to the field of replacement decision-making, and more specifically to an optimization method for situational replacement decision-making in a two-stage Wiener process that considers random variable points. Background Technology

[0002] In modern production processes, machine malfunctions can halt production for a period, resulting in costs associated with inspection, repair, replacement, and overall production losses. Therefore, improving machine reliability and extending its lifespan is of paramount importance.

[0003] Preventive maintenance is a viable method for improving reliability. It is generally divided into Time-Based Maintenance (TBM) and Condition-Based Maintenance (CBM). The former makes maintenance decisions based on technician experience or the probability distribution of failure times, while the latter uses condition information collected from condition monitoring (or inspection) for maintenance decisions. Condition-based maintenance requires additional investment in condition monitoring but is far superior to time-based maintenance in preventing future failures. With the development of the Internet of Things (IoT), the cost of advanced sensors has significantly decreased, driving the application of condition monitoring across various industries. To date, Condition-Based Maintenance (CBM) is widely recognized as the most advanced maintenance program in practice.

[0004] To make sound state-based maintenance (CBM) decisions, the degradation process of a system should be appropriately identified and described. The stochastic processes used to describe degradation characteristics can be discrete-state processes, such as discrete-time Markov chains, continuous-time Markov chains, semi-Markov processes, and hidden Markov models; or they can be continuous-state processes, including gamma processes, Wiener processes, and inverse Gaussian processes. Wiener and inverse Gaussian processes are suitable for modeling monotonically degrading paths. Wiener processes are suitable for non-monotonic degradation processes caused by slight re-pairing, self-healing, or reduced strength due to use. Compared to gamma and inverse Gaussian processes, Wiener processes have recently received more attention because they can provide a satisfactory and flexible description of the degradation signals of systems, including rotary bearings, bridge degradation, batteries, LED lamps, and carbon film resistors.

[0005] However, most existing research frameworks assume that the diffusion and drift coefficients of the Wiener process do not change abruptly, or that the distribution of the change point time is restricted to a specific form. But in many real-world situations, this assumption does not hold; due to changes in the external environment, internal mechanisms, or maintenance activities, the degradation rate may exhibit significant variations. Summary of the Invention

[0006] (a) Technical problems to be solved

[0007] To address the shortcomings of existing technologies, this invention provides a two-stage Wiener process decision optimization method that considers random change points. This method solves the technical problems of most existing research frameworks that assume the diffusion coefficient and drift coefficient of the Wiener process will not change suddenly, or that the distribution of change point time is restricted to a certain form.

[0008] (II) Technical Solution

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] A two-stage Wiener process optimization method considering random variable points and conditional replacement decision optimization includes:

[0011] The degradation level of the system is defined to follow a two-stage Wiener process with random variable points.

[0012] The posterior probability and degradation level of the system are converted into a discrete state model, and the state transition probability of the system and the expected cost of performing different actions at the next detection point are solved in the discrete state.

[0013] An iterative algorithm is used to traverse each detection point, calculate the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

[0014] Preferably, the two-stage Wiener process with random variable points is represented as X = {X...} t ,t≥0}, where:

[0015] X t This indicates the level of degradation of the system at time t;

[0016] In the first stage, the degradation process follows a degradation process with drift coefficients and diffusion coefficients of μ1 and σ1, respectively. After experiencing a change point τ, the degradation process enters the second stage, which follows a degradation process with drift coefficients and diffusion coefficients of μ2 and σ2, respectively. The change point refers to the time point at which the process parameters change abruptly.

[0017] Preferably, when the system is in stage i∈{1,2}, its degradation increment within one detection period Δ follows a normal distribution. The probability density function for stage i is:

[0018]

[0019] Where exp is an exponential function; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i.

[0020] Assuming the system's failure threshold is D>0, if the system's degradation level exceeds the threshold D within a detection cycle, then the system is assumed to have failed. When the system's degradation level is x and the degradation process at the next detection point is in stage i∈{1,2}, then the conditional reliability of the system at the next detection point is:

[0021]

[0022] Among them, R i (x) represents the reliability function of the system at stage i; Φ(·) is the cumulative distribution function of the normal distribution; Δ is the degradation increment; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i.

[0023] Considering the Bayesian-updated posterior probability π and the system's degradation level x, the conditional reliability of the system at the next detection point is corrected as follows:

[0024] R(π,x)=(1-π)R1(x)+πR2(x) (3).

[0025] Preferably, the step of converting the posterior probability and degradation level of the system into a discrete state model includes:

[0026] The posterior probability interval [0,1) of the system is discretized into K+1 states, and the first state space is defined as Ω. p ={0,1,···,K}, where K is the first discrete precision parameter, l p =1 / (K+1) is the first degree of dispersion; when the system is in the probability interval state k∈Ω p When the probability interval falls within [kl] p ,(k+1)l p When the system is in state k∈Ω p When using the midpoint r of the interval p (k)=(k+0.5)l p This represents the posterior probability of that interval;

[0027] The degradation level of the system is discretized into N+1 states, and the second state space is defined as Ω. d ={0,1,···,N}, where N is the second discrete precision parameter, l d =1 / (N+1) is the second discreteness; if the degradation level of the system is in (-∞,0), then the system is assumed to be in state 0; if the degradation level of the system is in [(n-1)l d ,nl dIf the system is in a degenerate state n∈{1,···,N}, then assume the system is in a degenerate state; for each state n∈Ω d Using the midpoint r of the interval d (n)=(n-0.5)l d Indicates its degradation level; for any n∈Ω d The upper bound of state n is B U (n)=n·l d When n≠0, the lower bound of state n is B. L (n)=(n-1)·l d Otherwise B L (n) = -∞;

[0028] Based on the first state space Ω p ={0,1,···,K} and the second state space Ω d ={0,1,···,N}, and the state space of the system is defined as S={(k,n)|k∈Ω p ,n∈Ω d}

[0029] Preferably, the step of solving the state transition probability of the system in discrete states and the expected cost corresponding to performing different actions at the next detection point includes:

[0030] The system's operating costs are defined as C1 and C2 when it is in the first and second phases, respectively, and the cost of preventative replacement is C. P >0, corrective replacement will incur additional replacement costs C K Action set A = {0, 1}, where a = 0 indicates no action is taken at the current detection point, and a = 1 indicates a preventative replacement is performed at the current detection point; p (k,n),(k′,n′) (a) represents the probability that the system transitions to state (k′,n′)∈S at the next detection point when it chooses action a∈A from its current state (k,n)∈S; c (k,n) (a) represents the expected cost of the next detection point when the current state is (k,n)∈S and action a∈A is selected;

[0031] 1) When a = 0, the system may be in state at the next detection point. The system is running; the two factors determining its state depend on the degradation increment y at the next detection point, and the state transition probability is:

[0032]

[0033] Where, f(y|r p (k) represents the probability interval falling within [kl] p ,(k+1)l p), the probability density function when the degradation increment is y;

[0034] Z1={y:k′l p ≤Π n (y|r p (k))≤(k′+1)l p} (5)

[0035] Z2=(B U (n′)-r d (n),B L (n′)-r d (n)) (6)

[0036] Among them, Π n (y|r p (k) represents a degenerate state of n, with a probability interval falling within [kl]. p ,(k+1)l p ), the posterior probability when the degradation increment is y;

[0037] When σ1=σ2=σ

[0038]

[0039] The state transition to (0,0) includes two cases:

[0040] The first scenario involves a system failure at the next detection point, with a transition probability of 1 - R(r). p (k),r d (n));

[0041] The second scenario is that the system reaches state (0,0) at the next detection point, and its transition probability is:

[0042]

[0043] The expected cost is:

[0044]

[0045] 2) When a = 1, the system will first return to the intermediate state (0,0), and then there is a transition path similar to a = 0; for any (k,n) ∈ S, the transition probability is:

[0046] p (k,n),(k′,n′) (1)=p( 0,0),(k′,n′ (0) (10)

[0047] The expected cost is:

[0048]

[0049] Preferably, arbitrary strategies are calculated by solving the following system of equations. The value functions {V(k,n)} and the average expected cost per unit time λ(ψ) are as follows:

[0050]

[0051] By selecting the action a′ that satisfies the following conditions (k,n) Construct a new policy that is no worse than policy ψ.

[0052]

[0053] Preferably, the step of using an iterative algorithm to traverse each detection point, calculating the expected cost of performing different actions at each detection point, and selecting the action corresponding to the minimum expected cost as the optimal action for that detection point includes:

[0054] STEP0, Initialization parameters include μ1, μ2, σ1, σ2, C1, C2, D, C P C K ,Δ,K,N,l p ,l d Set a (k,n) Let a be the set of policies when the system is in state (k,n), where a (k,n) =0 means no action is taken, a (k,n) =1 indicates a preventative replacement;

[0055] STEP 1: Input the initial strategy

[0056] STEP 2. For all k∈{0,···,K}, n∈{1,···,N}, calculate the value function {V} of strategy ψ using formula (12). (k,n)} and average expected cost λ(ψ);

[0057] STEP 3: Initialize a new strategy

[0058] STEP 4: Set a control limit

[0059] STEP5: Let k = K;

[0060] STEP6: If k < 0, then execute STEP12; otherwise, execute STEP7.

[0061] STEP 7, Order

[0062] STEP 8. Calculate a′ according to formula (13).(k,n) The value;

[0063] STEP 9, If a′ is calculated (k,n) The value of a is 1, and a' is set for all x = n, ..., N. (k,n) =1, and update The value of is equal to n;

[0064] STEP10: If n > N, then let k = k-1 and execute STEP6; otherwise, execute STEP11.

[0065] STEP 11: Set n = n + 1, then return to STEP 8;

[0066] STEP12: If ψ≠ψ′, let ψ=ψ′ and return to STEP2; otherwise, end the algorithm and output the optimal action set ψ.

[0067] A two-stage Wiener process-based changeover decision optimization system considering stochastic variable points includes:

[0068] Define the module to define that the degradation level of the system follows a two-stage Wiener process with random variable points;

[0069] The solution module is used to convert the posterior probability and degradation level of the system into a discrete state model, and solve the state transition probability of the system and the expected cost corresponding to performing different actions at the next detection point in the discrete state.

[0070] The decision module is used to traverse each detection point using an iterative algorithm, solve the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

[0071] A storage medium storing a computer program for situational replacement decision optimization of a two-stage Wiener process considering random variable points, wherein the computer program causes a computer to execute the situational replacement decision optimization method of the two-stage Wiener process as described above.

[0072] An electronic device, comprising:

[0073] One or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the programs including a two-stage Wiener process situational replacement decision optimization method as described above.

[0074] (III) Beneficial Effects

[0075] This invention provides a situational replacement decision optimization method for a two-stage Wiener process considering random variable points. Compared with existing technologies, it has the following advantages:

[0076] In this invention, the degradation level of the system is first defined as following a two-stage Wiener process with random variable points. Secondly, the posterior probability and degradation level of the system are converted into a discrete-state model, and the state transition probabilities and expected costs corresponding to different actions at the next detection point are solved in the discrete state. Finally, an iterative algorithm is used to traverse each detection point, solve for the expected cost of different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point. These actions include taking no action and preventative replacement. Compared with the maintenance strategy of the traditional two-stage Wiener process, the strategy proposed in this invention can provide reliable results for various types of variable points. Furthermore, an improved strategy iterative algorithm based on the discretization method is used to determine the optimal strategy that minimizes the long-term average expected cost. Attached Figure Description

[0077] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0078] Figure 1 A block diagram of a two-stage Wiener process situational replacement decision optimization method considering random variable points, provided for an embodiment of the present invention;

[0079] Figure 2 A flowchart of an iterative algorithm provided in an embodiment of the present invention;

[0080] Figure 3 This is a schematic diagram illustrating an optimal strategy provided in an embodiment of the present invention. Detailed Implementation

[0081] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention are described clearly and completely. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0082] This application provides a two-stage Wiener process decision optimization method that considers random changing points, thus solving the technical problem that most existing research frameworks assume that the diffusion coefficient and drift coefficient of the Wiener process will not change suddenly, or introduce a distribution of changing point time that is restricted to a certain form.

[0083] The technical solution in this application is to solve the above-mentioned technical problems, and the general idea is as follows:

[0084] Most existing research frameworks assume that the diffusion and drift coefficients of the Wiener process do not change abruptly, or that the distribution of the change point time is restricted to a specific form. However, this assumption does not hold true in many real-world situations; the degradation rate may exhibit significant variations due to changes in the external environment, internal mechanisms, or maintenance activities. Furthermore, much research on the two-stage Wiener process focuses on parameter estimation and Remaining Useful Life (RUL) prediction, but condition-based maintenance (CBM) strategies have not been thoroughly investigated, especially when the change point is not fixed. In addition, parameter estimation for the two-stage Wiener process is often challenged by small sample sizes, leading to inaccurate change point estimations. Therefore, the objectives of this invention include:

[0085] (1) This invention optimizes a state-based maintenance (CBM) strategy for a two-stage Wiener process with stochastic change points. Compared with traditional maintenance strategies for two-stage Wiener processes, this strategy does not restrict the distribution of change point times to any particular form and proves that the proposed method can provide reliable results for various types of change points. Moreover, in the second stage of the Wiener process, the optimization problem is formulated as a Markov decision process.

[0086] (2) The present invention provides a discretization method to derive the Markov decision process quantity in discrete state space.

[0087] (3) This invention provides an iterative strategy algorithm to find the optimal control constraint strategy. At each decision-making stage, if the system degradation exceeds a critical level, corrective replacement must be performed. Otherwise, it is decided to take no action or preventatively replace the system. Both corrective and preventative replacements restore the system to a completely new state.

[0088] To better understand the above technical solutions, the following will provide a detailed explanation of the technical solutions in conjunction with the accompanying drawings and specific implementation methods.

[0089] Example 1:

[0090] like Figure 1As shown, this embodiment of the invention provides a two-stage Wiener process situational replacement decision optimization method considering random variable points, including:

[0091] S1. Define the degradation level of the system as following a two-stage Wiener process with random variable points.

[0092] S2. Convert the posterior probability and degradation level of the system into a discrete state model, and solve the state transition probability of the system and the expected cost of performing different actions at the next detection point in the discrete state.

[0093] S3. Use an iterative algorithm to traverse each detection point, solve for the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

[0094] Compared with the maintenance strategy of the traditional two-stage Wiener process, the strategy proposed in this embodiment can provide reliable results for various types of change points; in addition, an improved strategy iteration algorithm based on the discretization method is proposed to determine the optimal strategy that minimizes the long-term average expected cost.

[0095] The following will detail each step of the above solution:

[0096] S1. Define the degradation level of the system as following a two-stage Wiener process with random variable points.

[0097] Consider a two-stage Wiener process X = {X t For a system with t ≥ 0, X t This represents the degradation level of the system at time t. In the first stage, the degradation process follows a degradation process with drift coefficients of μ1 and diffusion coefficients of σ1. After experiencing a change point τ, the degradation process enters the second stage, following a degradation process with drift coefficients of μ2 and diffusion coefficients of σ2. The change point refers to the time point at which the process parameters undergo abrupt changes.

[0098] When the system is in stage i∈{1,2}, its degradation increment within one detection period Δ follows a normal distribution. The probability density function for stage i is:

[0099]

[0100] Where exp is an exponential function; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i.

[0101] Assuming the system's failure threshold is D>0, if the system's degradation level exceeds the threshold D within a detection cycle, then the system is assumed to have failed. When the system's degradation level is x and the degradation process at the next detection point is in stage i∈{1,2}, then the conditional reliability of the system at the next detection point is:

[0102]

[0103] Among them, R i (x) represents the reliability function of the system at stage i; Φ(·) is the cumulative distribution function of the normal distribution; Δ is the degradation increment; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i.

[0104] It should be noted that in this embodiment of the invention, the decision-making framework is changed by updating Bayes' theorem and establishing and analyzing Markov decision processes (MDPs) to determine whether to change it.

[0105] Considering the Bayesian-updated posterior probability π and the system's degradation level x, the conditional reliability of the system at the next detection point is corrected as follows:

[0106] R(π,x)=(1-π)R1(x)+πR2(x) (3)

[0107] Specifically, define an action set A = {0, 1}, where a = 0 indicates that no action is taken in the current decision cycle, and a = 1 indicates that a preventative change is made in the current decision cycle.

[0108] A new cycle begins with a completely new state of the system and ends with either a preventative or corrective replacement, whichever comes first. The cost of each cycle is calculated based on the transition probability, and the optimal action is determined by minimizing the average cost.

[0109] In step S2, the posterior probability and degradation level of the system are converted into a discrete state model, and the state transition probability of the system and the expected cost corresponding to performing different actions at the next detection point are solved in the discrete state.

[0110] Specifically, this step of converting the system's posterior probability and degradation level into a discrete-state model includes:

[0111] The posterior probability interval [0,1) of the system is discretized into K+1 states, and the first state space is defined as Ω. p ={0,1,···,K}, where K is the first discrete precision parameter, l p =1 / (K+1) is the first degree of dispersion; when the system is in the probability interval state k∈Ω o When the probability interval falls within [kl]p ,(k+1)l p When the system is in state k∈Ω p When using the midpoint r of the interval p (k)=(k+0.5)l p This represents the posterior probability of that interval;

[0112] The degradation level of the system is discretized into N+1 states, and the second state space is defined as Ω. d ={0,1,···,N}, where N is the second discrete precision parameter, l d =1 / (N+1) is the second discreteness; if the degradation level of the system is in (-∞,0), then the system is assumed to be in state 0; if the degradation level of the system is in [(n-1)l d ,nl d If the system is in a degenerate state n∈{1,···,N}, then assume the system is in a degenerate state; for each state n∈Ω d Using the midpoint r of the interval d (n)=(n-0.5)l d Indicates its degradation level; for any n∈Ω d The upper bound of state n is B U (n)=n·l d When n≠0, the lower bound of state n is B. L (n)=(n-1)·l d Otherwise B L (n) = -∞;

[0113] Based on the first state space Ω p ={0,1,···,K} and the second state space Ω d ={0,1,···,N}, and the state space of the system is defined as S={(k,n)|k∈Ω p ,n∈Ω d}

[0114] Specifically, this step involves solving for the system's state transition probabilities and the expected costs corresponding to different actions performed at the next detection point in discrete states, including:

[0115] The system's operating costs are defined as C1 and C2 when it is in the first and second phases, respectively, and the cost of preventative replacement is C. P >0, corrective replacement will incur additional replacement costs C K Action set A = {0, 1}, where a = 0 indicates no action is taken at the current detection point, and a = 1 indicates a preventative replacement is performed at the current detection point; p (k,n),(k′,n′)(a) represents the probability that the system transitions to state (k′,n′)∈S at the next detection point when it chooses action a∈A from its current state (k,n)∈S; c (k,n) (a) represents the expected cost of the next detection point when the current state is (k,n)∈S and action a∈A is selected.

[0116] Next, we will calculate the two variables above in two different cases:

[0117] 1) When a = 0, the system may be in state at the next detection point. The system is running; the two factors determining its state depend on the degradation increment y at the next detection point, and the state transition probability is:

[0118]

[0119] Where, f(y|r p (k) represents the probability interval falling within [kl] p ,(k+1)l p ), the probability density function when the degradation increment is y;

[0120] Z1={y:k′l p ≤Π n (y|r p (k))≤(k′+1)l p} (5)

[0121] Z2=(B U (n′)-r d (n),B L (n′)-r d (n)) (6)

[0122] Among them, Π n (y|r p (k) represents a degenerate state of n, with a probability interval falling within [kl]. p ,(k+1)l p ), the posterior probability when the degradation increment is y;

[0123] When σ1=σ2=σ

[0124]

[0125] The state transition to (0,0) includes two cases:

[0126] The first scenario involves a system failure at the next detection point, with a transition probability of 1 - R(r). p (k),r d (n));

[0127] The second scenario is that the system reaches state (0,0) at the next detection point, and its transition probability is:

[0128]

[0129] The expected cost is:

[0130]

[0131] 2) When a = 1, the system will first return to the intermediate state (0, 0), and then there is a transition path similar to a = 0; for any (k, n) ∈ S, the transition probability is:

[0132] p (k,n),(k′,n′) (1)=p (0,0),(k′,n′) (0) (10)

[0133] The expected cost is:

[0134]

[0135] In step S3, an iterative algorithm is used to traverse each detection point, calculate the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

[0136] In an optional implementation, before executing the iterative algorithm, embodiments of the present invention first calculate the arbitrary strategy by solving the following system of equations. The value functions {V(k,n)} and the average expected cost per unit time λ(ψ) are as follows:

[0137]

[0138] Next, by selecting an action a′ that satisfies the following conditions (k,n) Construct a new policy that is no worse than policy ψ.

[0139]

[0140] Based on this, this step uses an iterative algorithm to traverse each detection point, calculate the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point. Figure 2 The relevant steps shown include:

[0141] STEP0, Initialization parameters include μ1, μ2, σ1, σ2, C1, C2, D, C P C K ,Δ,K,N,lp ,l d Set a (k,n) Let a be the set of policies when the system is in state (k, n), where a (k，n) =0 means no action is taken, a (k，n) =1 indicates a preventative replacement;

[0142] STEP 1: Input the initial strategy

[0143] STEP 2. For all k∈{0,···,K}, n∈{1,···,N}, calculate the value function {V} of strategy ψ using formula (12). (k,n)} and average expected cost λ(ψ);

[0144] STEP 3: Initialize a new strategy

[0145] STEP 4: Set a control limit

[0146] STEP5: Let k = K;

[0147] STEP6: If k < 0, then execute STEP12; otherwise, execute STEP7.

[0148] STEP 7, Order

[0149] STEP 8. Calculate a′ according to formula (13). (k,n) The value;

[0150] STEP 9, If a′ is calculated (k,n) The value is 1, and a' is set for all x = n, ..., N. (k,n) =1, and update The value of is equal to n;

[0151] STEP10: If n > N, then let k = k-1 and execute STEP6; otherwise, execute STEP11.

[0152] STEP 11: Set n = n + 1, then return to STEP 8;

[0153] STEP12: If ψ≠ψ′, let ψ=ψ′ and return to STEP2; otherwise, end the algorithm and output the optimal action set ψ.

[0154] To help understand the embodiments of the present invention, a specific implementation example is provided below:

[0155] The initial degradation level of the system is x0 = 0. Drift coefficients are given in two cases: (1) μ1 = 0.8 and μ2 = 3; (2) μ1 = 3 and μ2 = 0.8. For better comparison, the diffusion coefficient is σ1 = σ2 = 0.64, and the operating cost parameter is set to C1 = C2 = 0. The failure threshold is D = 20. The preventative replacement cost is C. P =100, corrective replacement cost is C K =300. One detection cycle is Δ=2. Algorithm parameters are set as follows: K=49 (i.e., l p =0.02) and N=20 (i.e., l d =1).

[0156] Figure 3 The optimal strategies for two scenarios are shown. In each scenario, a control limit is used to determine the optimal action. For a given posterior probability, if the degradation exceeds a threshold, preventative replacement is recommended; otherwise, doing nothing is the better decision. The difference is that in Case 1, the control limit decreases as the posterior probability increases, but in Case 2, it increases. That is, for a fixed degradation level, in Case 1, the optimal action if the posterior probability exceeds a certain level is preventative replacement, while in Case 2, the opposite is true.

[0157] This invention has detailed a replacement strategy for a two-stage Wiener process with stochastic change points. This strategy can provide reliable results for various types of change points. Secondly, this invention transforms the posterior probability and degradation level of the system into a discrete-state model and solves for the state transition probabilities of the system in the discrete states. Then, compared with traditional Wiener process replacement strategies for three types of change points, the replacement strategy for a two-stage Wiener process with stochastic change points proposed in this invention performs better when the change point occurs early. If checks are performed frequently enough, this strategy outperforms non-Bayesian decision-making even when the system enters the second stage at the end. Finally, this invention improves a strategy iteration algorithm based on discretization methods to determine the optimal strategy that minimizes the long-term expected average cost per unit of time.

[0158] Example 2:

[0159] This invention provides a two-stage Wiener process situational replacement decision optimization system considering random variable points, comprising:

[0160] Define the module to define that the degradation level of the system follows a two-stage Wiener process with random variable points;

[0161] The solution module is used to convert the posterior probability and degradation level of the system into a discrete state model, and solve the state transition probability of the system and the expected cost corresponding to performing different actions at the next detection point in the discrete state.

[0162] The decision module is used to traverse each detection point using an iterative algorithm, solve the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

[0163] Example 3:

[0164] This invention provides a storage medium storing a computer program for situational replacement decision optimization of a two-stage Wiener process considering random variable points, wherein the computer program causes a computer to execute the situational replacement decision optimization method of the two-stage Wiener process as described in Embodiment 1.

[0165] Example 4:

[0166] This invention provides an electronic device, comprising:

[0167] One or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the programs including methods for performing a two-stage Wiener process situational replacement decision optimization method as described in Example 1.

[0168] It is understood that the two-stage Wiener process situational replacement decision optimization system, storage medium and electronic device considering random variable points provided in the embodiments of the present invention correspond to the two-stage Wiener process situational replacement decision optimization method considering random variable points provided in the embodiments of the present invention. The explanation, examples and beneficial effects of the relevant contents can be referred to the corresponding parts of the method, and will not be repeated here.

[0169] In summary, compared with existing technologies, it has the following beneficial effects:

[0170] 1. This invention develops a replacement decision framework for a two-stage Wiener process with random variable points. It utilizes Bayes' theorem to update the posterior probability of the system and employs a Markov decision process to solve the optimization problem.

[0171] 2. A discretization method is introduced to ensure the discreteness and finiteness of the system's state space and action set. Based on this, the Markov decision process quantities in the discrete state space are derived.

[0172] 3. Compared with the traditional two-stage Wiener process maintenance strategy, our proposed strategy can provide reliable results for various types of change points.

[0173] 4. An improved policy iteration algorithm based on the discretization method is proposed to determine the optimal policy that minimizes the long-run average expected cost.

[0174] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0175] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A two-stage Wiener process optimization method considering random variable points and conditional replacement decision-making, characterized in that, include: The degradation level of the system is defined to follow a two-stage Wiener process with random variable points. The posterior probability and degradation level of the system are converted into a discrete state model, and the state transition probability of the system and the expected cost of performing different actions at the next detection point are solved in the discrete state. An iterative algorithm is used to traverse each detection point, calculate the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

2. The two-stage Wiener process situational replacement decision optimization method as described in claim 1, characterized in that, The two-stage Wiener process with random variable points is represented as X = {X} t ,t≥0}, where: X t This indicates the level of degradation of the system at time t; In the first stage, the degradation process follows a degradation process with drift coefficients and diffusion coefficients of μ1 and σ1, respectively. After experiencing a change point τ, the degradation process enters the second stage, which follows a degradation process with drift coefficients and diffusion coefficients of μ2 and σ2, respectively. The change point refers to the time point at which the process parameters change abruptly.

3. The two-stage Wiener process situation-based replacement decision optimization method as described in claim 2, characterized in that, When the system is in stage i∈{1,2}, its degradation increment within one detection period Δ follows a normal distribution. The probability density function for stage i is: Where exp is an exponential function; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i. Assuming the system's failure threshold is D>0, if the system's degradation level exceeds the threshold D within a detection cycle, then the system is assumed to have failed. When the system's degradation level is x and the degradation process at the next detection point is in stage i∈{1,2}, then the conditional reliability of the system at the next detection point is: Among them, R i (x) represents the reliability function of the system at stage i; Φ(·) is the cumulative distribution function of the normal distribution; Δ is the degradation increment; μ i σ represents the drift coefficient of the system at stage i; i This represents the diffusion coefficient when the system is in stage i. Considering the Bayesian-updated posterior probability π and the system's degradation level x, the conditional reliability of the system at the next detection point is corrected as follows: R(π,x)=(1-π)R1(x)+πR2(x) (3).

4. The two-stage Wiener process situational replacement decision optimization method as described in claim 3, characterized in that, The process of converting the system's posterior probability and degradation level into a discrete-state model includes: Discretize the posterior probability interval [0,1) of the system into K+1 states, and define the first state space as Ω. p ={0,1,···,K}, where K is the first discrete precision parameter, l p =1 / (K+1) is the first degree of dispersion; when the system is in the probability interval state k∈Ω p When the probability interval falls within [kl] p ,(k+1)l p When the system is in state k∈Ω p When using the midpoint r of the interval p (k)=(k+0.5)l p This represents the posterior probability of that interval; The degradation level of the system is discretized into N+1 states, and the second state space is defined as Ω. d ={0,1,···,N}, where N is the second discrete precision parameter, l d =1 / (N+1) is the second discreteness; if the degradation level of the system is in (-∞,0), then the system is assumed to be in state 0; if the degradation level of the system is in [(n-1)l d ,nl d If the system is in a degenerate state n∈{1,···,N}, then assume the system is in a degenerate state; for each state n∈Ω d Using the midpoint r of the interval d (n)=(n-0.5)l d Indicates its degradation level; for any n∈Ω d The upper bound of state n is B U (n)=n·l d When n≠0, the lower bound of state n is B. L (n)=(n-1)·l d Otherwise B L (n) = -∞; Based on the first state space Ω p ={0,1,···,K} and the second state space Ω d ={0, 1, ..., N}, and the state space of the system is defined as S = {(k, n) | k ∈ Ω}. p ,n∈Ω d } 5. The two-stage Wiener process situational replacement decision optimization method as described in claim 4, characterized in that, The process of solving the state transition probabilities of the system in discrete states and the expected costs corresponding to different actions performed at the next detection point includes: The system's operating costs are defined as C1 and C2 when it is in the first and second phases, respectively, and the cost of preventative replacement is C. P >0, corrective replacement will incur additional replacement costs C K Action set A = {0, 1}, where a = 0 indicates no action is taken at the current detection point, and a = 1 indicates a preventative replacement is performed at the current detection point; p (k,n),(k′,n′) (a) represents the probability that the system transitions to state (k′,n′)∈S at the next detection point when it chooses action a∈A from its current state (k,n)∈S; c (k,n) (a) represents the expected cost of the next detection point when the current state is (k,n)∈S and action a∈A is selected; 1) When a = 0, the system may be in state at the next detection point. The system is running; the two factors determining its state depend on the degradation increment y at the next detection point, and the state transition probability is: Where, f(y|r p (k) represents the probability interval falling within [kl] p ,(k+1)l p ), the probability density function when the degradation increment is y; Z1={y:k′l p ≤Π n (y|r p (k))≤(k′+1)l p } (5) Z2=(B U (n′)-r d (n),B L (n′)-r d (n)) (6) Among them, Π n (y|r p (k) represents a degenerate state of n, with a probability interval falling within [kl]. p ,(k+1)l p ), the posterior probability when the degradation increment is y; When σ1=σ2=σ, The state transition to (0,0) includes two cases: The first scenario involves a system failure at the next detection point, with a transition probability of 1 - R(r). p (k),r d (n)); The second scenario is that the system reaches state (0,0) at the next detection point, and its transition probability is: The expected cost is: 2) When a = 1, the system will first return to the intermediate state (0,0), and then there is a transition path similar to a = 0; for any (k,n) ∈ S, the transition probability is: p (k,n),(k′,n′) (1)=p (0,0),(k′,n′) (0) (10) The expected cost is:

6. The two-stage Wiener process situational replacement decision optimization method as described in claim 5, characterized in that, Compute arbitrary strategies by solving the following system of equations. The value functions {V(k,n)} and the average expected cost per unit time λ(ψ) are as follows: By selecting the action a′ that satisfies the following conditions (k,n) Construct a new policy that is no worse than policy ψ.

7. The two-stage Wiener process situational replacement decision optimization method as described in claim 6, characterized in that, The step of using an iterative algorithm to traverse each detection point, calculate the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point includes: STEP0, Initialization parameters include μ1, μ2, σ1, σ2, C1, C2, D, C P C K ,Δ,K,N,l p ,l d Set a (k,n) Let a be the set of policies when the system is in state (k,n), where a (k,n) =0 means no action is taken, a (k,n) =1 indicates a preventative replacement; STEP 1: Input the initial strategy STEP 2. For all k∈{0,···,K}, n∈{1,···,N}, calculate the value function {V} of strategy ψ using formula (12). (k,n) } and average expected cost λ(ψ); STEP 3: Initialize a new strategy STEP 4: Set a control limit STEP5: Let k = K; STEP6: If k < 0, then execute STEP12; otherwise, execute STEP7. STEP 7, Order STEP 8. Calculate a′ according to formula (13). (k,n) The value; STEP 9, If a′ is calculated (k,n) The value is 1, and a' is set for all x = n, ..., N. (k,n) =1, and update The value of is equal to n; STEP10: If n > N, then let k = k-1 and execute STEP6; otherwise, execute STEP11. STEP 11: Set n = n + 1, then return to STEP 8; STEP12: If ψ≠ψ′, let ψ=ψ′ and return to STEP2; otherwise, end the algorithm and output the optimal action set ψ.

8. A two-stage Wiener process situation-dependent replacement decision optimization system considering stochastic variable points, characterized in that, include: Define the module to define that the degradation level of the system follows a two-stage Wiener process with random variable points; The solution module is used to convert the posterior probability and degradation level of the system into a discrete state model, and solve the state transition probability of the system and the expected cost corresponding to performing different actions at the next detection point in the discrete state. The decision module is used to traverse each detection point using an iterative algorithm, solve the expected cost of performing different actions at each detection point, and select the action corresponding to the minimum expected cost as the optimal action for that detection point; the actions include not taking any action and preventive replacement.

9. A storage medium, characterized in that, It stores a computer program for situational replacement decision optimization of a two-stage Wiener process considering random variable points, wherein the computer program causes the computer to execute the situational replacement decision optimization method of the two-stage Wiener process as described in any one of claims 1 to 7.

10. An electronic device, characterized in that, include: One or more processors; Memory; And one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the programs including methods for performing the two-stage Wiener process situational replacement decision optimization method as described in any one of claims 1 to 7.

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