An improved opportunity maintenance model of complex system

Abstract

To solve the problems of multiple parts, high cost of disassembly, and imperfect maintenance in multi-component complex systems, an opportunistic maintenance model of multi-component complex systems based on a time threshold is proposed in this article. The influence of the disassembly sequence and hybrid evolution factors on maintenance and learning-forgetting effects are considered in the proposed model. In this article, the learning-forgetting effect of the field strip in the components on the same level or components on different levels is considered in addition to the shortest disassembly path of multi-component complex systems. Imperfect and preventive maintenance is applied in a complex system, and random failure involves minor repairs. Multiple objectives that minimize maintenance costs and maximize operation time rate are proposed. Monte Carlo simulation is used to solve the model and obtain the time threshold of optimal opportunistic maintenance. The results show the feasibility of the opportunistic maintenance model.

Introduction

In general, systems can be divided into three categories: simple systems, stochastic systems, and complex systems. ¹ Simple systems are characterized by a particularly small number of elements. A stochastic system has many elements and variables, but the coupling between them is weak or random. Complex systems are featured by a large number of elements with strong coupling between them, such as in spacecraft and computer numerical control (CNC) machine tools.^2,3 Traditional maintenance methods include breakdown maintenance and preventive maintenance. Breakdown maintenance is defined as maintenance after equipment failure. Enterprises are liable to suffer losses when using this maintenance method. Preventive maintenance can prevent or delay the occurrence of dimensional failures to a certain extent. Opportunistic maintenance is a combination of breakdown maintenance and preventive maintenance: when a component is maintained during system shutdown, whether or not other parts meet the maintenance requirements is simultaneously judged. If so, the components can be maintained together, minimizing the cost of disassembly and disassembly maintenance. ⁴ The opportunistic maintenance model has a wide range of applications. Many scholars have studied the opportunistic maintenance in-depth and many kinds of opportunistic maintenance models have been introduced. A detailed discussion of opportunity maintenance is presented in detail in section “Literature review.”

Literature review

Opportunistic maintenance model

Usually, due to the complexity of the system, multi-component systems require overall shutdown and disassembly during the maintenance process, which requires opportunistic multi-component system maintenance. Ding and Tian ⁵ studied the opportunistic maintenance strategy of wind-generated equipment and reduced maintenance costs by adopting an imperfect and opportunistic maintenance strategy. Cavalcante and Lopes ⁶ defined opportunistic maintenance in thermoelectric system as a basis for decision-making with minimum maintenance, opportunistic maintenance, and preventive maintenance. They considered the benefits and maintenance costs in a complex system and illustrated the multi-criteria model that immediately benefits energy companies. Hu and Zhang^7,8 proposed a global optimization opportunistic maintenance model of advance fault defense to judge the potential for opportunistic maintenance using the timing and method of advanced defense as well as other factors. Due to the randomness of equipment degradation, Tao and Zhou ⁹ established an analysis model of cost maintenance completed under limited time with the total cost as the optimization target. Hou et al. ¹⁰ established an opportunistic maintenance model for multi-device hybrid production systems with maintenance cost as the optimized objective based on reliability. Shao and Zhou ¹¹ proposed a modeling method of preventive and opportunistic equipment maintenance based on capacity-constrained resources. Zhang et al.^12,13 established a prophylactic maintenance model for serial-parallel production systems with reliability and minimum total cost as optimization targets. They then considered the constraint theory and established an opportunistic maintenance model that optimizes the production capacity of a string/parallel production line and maintenance costs. Xia et al. ¹⁴ studied the characteristics of equipment’s degradation based on the production-driven opportunistic maintenance strategy and constructed the multi-attribute model and advance-postpone balancing (MAM-APB). Zhou et al. ¹⁵ established a dynamic opportunistic maintenance strategy and decision optimization model for three device serial systems based on device reliability. This kind of model mainly uses the serial system as the research object. To guarantee the production plan, the opportunistic maintenance strategy is chosen to reduce the maintenance costs by merging the maintenance operation. Hou et al. ¹⁶ established an opportunistic maintenance model of multi-component system based on reliability. They introduced a reliability recovery factor that overcomes the non-linear shortcomings of the traditional model based on run-time modeling while neglecting the relationship between deterioration process and time. Zhang et al. ¹⁷ proposed a comprehensive decay and evolution rule for independent equipment. Combined with multi-objective value theory and dynamic cycle decision-making model, they established a multi-objective predictive maintenance planning model for a device layer. Zhou et al. ¹⁸ studied a dynamic decision-making and optimized model for multi-device serial system opportunity maintenance based on equipment reliability and introduced the age reduction factor and the failure rate increase factor. This maintenance strategy uses the opportunistic maintenance strategy to reduce maintenance costs by combining maintenance operations. However, component similarity was not considered in this model.

Learning-forgetting effects

The learning-forgetting effect is widely used in education, supply chain management, and other fields. ¹⁹ Enterprises that consider the learning-forgetting effect can better develop mass production. ²⁰ Tarakci ²¹ studied the learning-forgetting effect on maintenance outsourcing. Considering the employee’ learning-forgetting effect in the process of group production, Wang et al. ²² proposed three models to express the actual processing time and validated the model. Liao and Zhang ²³ studied the learning-forgetting effects of employees on the group production process. They established a multi-objective model that considers the degradation effects of equipment to minimize costs and completion time. This model illustrates that the learning-forgetting effect is the objective in the production process. However, the field strip is different from the production of the product, which is characterized by the disassembly of all the components in a system not necessarily occurring in all cases. In most cases, only part of the system is disassembled while other products continue to be created in the process, so it is the production order that is studied. Due to the similarity between the finished products in group production, there is a learning-forgetting effect in the production process. Therefore, if there is similarity between the components in the multi-component system, there will be a learning-forgetting effect in the system disassembly process.

Time threshold of opportunity maintenance

Hu et al. ²⁴ introduced opportunity preventive maintenance strategies into the maintenance of the entire system and considered the repair, unloading, and installation costs. Guiras and Turki ²⁵ proposed a maintenance strategy for two—stage disassembly and assembly system integration. Wang and coworkers ²⁶ proposed a time-based threshold and demountable multi-component system opportunistic maintenance strategy. Considering the impact of the field strip on maintenance, they introduced the N-ary tree and disassembly sequence planning. This type of strategy reduces the maintenance costs by merging the disassembly paths. However, during the disassembly of the multi-component system, the maintenance staff are affected by learning-forgetting due to the similarity between the components. The disassembly time changes with the learning-forgetting effects. Therefore, the strategy above lacks consideration of the learning-forgetting effects of the maintenance staff. Other authors studied the situation where the multi-component complex system does not implement opportunistic maintenance after failure. There are many ways of determining opportunistic maintenance thresholds. Bedford et al. ²⁷ proposed a system reliability model based on fault distribution that releases a risk signal, which was used for decision-making for an opportunistic maintenance model. Huang et al. ²⁶ considered the elimination of the weight matrix multi-correlation and proposed an intelligent fault diagnosis method. Huang et al. ²⁶ proposed a method of using the Monte Carlo simulation. Tambe et al. ²⁸ considered the impact of opportunistic maintenance on the equipment in the production planning model and solved the model using a genetic algorithm.

Summary of literature review

Although the above authors have completed considerable research on the opportunistic maintenance model, there are still some shortcomings: (1) no clear description is provided about whether opportunistic maintenance occurs after the multi-component complex system malfunctioned; (2) in the maintenance of complex systems with high disassembly costs, the similarity between parts is accompanied by the learning-forgetting effect, which results in changes in maintenance time. This influence factor exists objectively but is often ignored; (3) under normal circumstances, the equipment cannot be restored to as new condition after being used for a certain period of time, even with preventive maintenance (Figure 1). In this article, an opportunistic maintenance model for a complex system based on a time threshold is proposed, which introduces the effects of disassembly sequence and hybrid evolution factors on maintenance. The effect of learning-forgetting on demolition time was considered and the time threshold of opportunistic maintenance was scientifically and effectively deter-mined, providing a more cost-effective maintenance program for maintenance activities. To address the problems of multiple parts, high cost of disassembly, and imperfect maintenance in multiple component complex recession systems, a scheme was given in accordance with practical production.

Figure 1.

Summary of relevant literature review.

Methodology

Problem description

Introducing the effect of disassembly sequence and hybrid evolution factors on maintenance, we studied the integration problem of equipment maintenance in a class of complex fading system equipment maintenance. Considering the effect of learning-forgetting on maintenance time, the advanced maintenance threshold with the lowest equipment maintenance cost and maximized equipment running time was studied. The basic assumptions were as follows:

1. The study object is a detachable complex system.

2. When the system is maintained, the final source of the fault involves the least amount of disassembly of the part, which is called the component of root node.

3. The multi-component system maintenance activities fall into three types: random failure maintenance, preventive maintenance, and opportunistic maintenance. Among them, preventive maintenance and opportunistic maintenance must consider the impact of hybrid evolution factors on maintenance and involves “repair non-new.” Random failure maintenance takes the approach of “recovery as old.”

4. Serial logic that exists between the components of a multi-component system and the system can process something or perform maintenance operations in the same normal operation time. The whole system is shut down for random failure maintenance, preventive maintenance, and opportunistic maintenance.

5. Maintenance of the shortest path disassembly operation should apply when any component must stop for maintenance.

6. Considering the multi-component system’s decision-making algorithm for disassembly cost, each component requires a preventive maintenance plan. Preventive maintenance should be carried out when the reliability of the component drops below a certain threshold.

7. Only one component can be disassembled at a time when the system is being maintained and maintenance cannot be interrupted. The total disassembly time is related to the similarity between the components.

8. The complex systems have many types of subsystems and hierarchical structures. The connections between them are complex and coupled.

9. The maintenance time that is closely related to the similarity between components includes maintenance setting time and maintenance processing time.

10. The possible benefit of the advanced maintenance of other parts that require no maintenance activities is considered when a part in the system is destroyed, and the model determines whether other components have opportunistic maintenance based on threshold policy.

The tree traversal algorithm can be used to find the optimal disassembly path of a specific part in a multi-component complex system. Based on the standard processing time and unit maintenance cost of each component, the actual dismantling time is calculated and the actual disassembly cost is obtained by multiplying the demolition cost while considering the influence of the learning-forgetting effect given the similarity between components. Incorporating the disassembly sequence and hybrid evolution factors into the influence of maintenance, we adopted reparative non-new preventive maintenance for the system. With the goals of minimizing maintenance costs and maximizing operational time, we applied minor repair when a random failure occurs to build an opportunistic maintenance model that considers disassembly sequence and the learning-forgetting effect. Monte Carlo simulation was used to solve the model and determine the time threshold for opportunistic maintenance.

Influencing factors and model

Disassembly tree model

The connection structure between parts includes a tree or a ring, and the tree structure can effectively express the disassembly model. ²⁵ Gao et al. ²⁹ proposed the concept of disassembly priority and verified the effectiveness of the disassembly tree method. Cao and Zhang ³⁰ proposed a generation algorithm for disassembly tree that conforms to the disassembly rules and disassembly relations table. This algorithm was implemented on the virtual maintenance platform using examples that verified the feasibility of the disassembly tree structure. In this article, the complex decay system disassembly process is expressed by a tree disassembly structure.

As shown in Figure 2, the disassembly structure includes three factors: the disassembled parent node, the disassembled child node, and the disassembly relation. The disassembly relationship is divided into four types: no direct disassembly, parent-child disassembly, sequential disassembly, and synchronous disassembly. The parent node represents the structure before the disassembly and the child node represents the structure obtained after disassembly. Taking Figure 2 as an example, j₁ is the parent node of j₂, and j₂ is the child node of j₁. At the same time, j₂ is the parent node of j₆, and j₆ is the child node of j₂. The parent-child disassembly relationship indicates the relationship between the disassembled parent node and child node obtained after disassembly, denoted by j₁ and j₂ in Figure 2. The sequential disassembly relation means that other nodes must first remove the non-parent node when removing a node, as shown by j₂ and j₆ in Figure 2. Synchronous disassembly means that when a node needs to be removed, another node must be removed at the same time, as shown by j₄ and j₅ in Figure 2. The disassembly relationship outside the disassembly relationship between father and son, the disassembly relation in sequence, and the synchronous disassembly relation have no direct disassembly relationship, as shown by j₂ and j₃ in Figure 2.

Figure 2.

Disassembly structure model.

Definition 1

For each node j, the disassembly time $T_j^d$ includes two parts, the disassembly preset time $T_j^{\mathrm{dy}}$ and the disassembly operation time $T_j^{\mathrm{dz}}$ . The unit operation cost rate $c_j^{\mathrm{dy}}$ represents the disassembly preset cost in unit time and unit operation cost rate $c_j^{\mathrm{dz}}$ represents disassembly operation cost in unit time. The disassembly cost formula of the node is

$$D{C}_j=T_j^{\mathrm{dy}}\times c_j^{\mathrm{dy}}+T_j^{\mathrm{dz}}\times c_j^{\mathrm{dz}}$$

(1)

With the basic disassembly structure model shown in Figure 2, we constructed an N-ary disassembly tree model and found the shortest path path(j) for disassembly by disassembly sequence planning. The shortest path’s acquisition steps are as follows:

• Step 1. Set the root node j and set path(a), path(b), and path(c) to be empty.

• Step 2. Judge whether a synchronization node of the root node j exists in the node incorporation path(a).

• Step 3. Judge whether a sequential node of the root node j exists in the node incorporation path(b).

• Step 4. Judge whether there is root node j’s parent node exists in the node incorporation path(c).

• Step 5. Merge the nodes in path(a), path(b), and path(c). The nodes are subtracted to obtain the set D. Sets each node in the collection to the root node and all nodes in D return to the first step for recursion. Proceed to the sixth step if the element in set D does not change after two consecutive recursions.

• Step 6. The shortest path consists of the path(j) and nodes in the set D.

We can obtain the shortest path through the above arrangement of tree traversal

$$\mathrm{zdpath}\left(j\right)=\sum _{j\in D}\mathrm{path}\left(j\right)$$

(2)

Without considering the similarity between the components, that is, without considering the learning-forgetting effects, the total disassembly time ${\mathrm{DT}}_j^d$ and the total disassembly cost ${\mathrm{DC}}_j^d$ of the shortest path are

$${\mathrm{DT}}_j^d=\sum _{j\in D}{T}_j^d=\sum _{j\in D}(T_j^{\mathrm{dy}}+T_j^{\mathrm{dz}})$$

(3)

$${\mathrm{DC}}_j^d=\sum _{j\in D}(T_j^d\times c_j^d)=\sum _{j\in D}(T_j^{\mathrm{dy}}\times c_j^{\mathrm{dy}}+T_j^{\mathrm{dz}}\times c_j^{\mathrm{dz}})$$

(4)

Learning-forgetting effect model

In the N-ary disassembly tree, the components of the system can be divided into different component layers. The number of components in the system depends on the root nodes in the N fork and the maximum number of parent–child disassembly relationships between the initial node plus 1. For example, all root nodes in Figure 2 are j₄, j₅, j₆, j₇ and the number of parent–child disassembly relationships between them and the initial node j₁ are 2, 2, 1, and 1, respectively. The number of layers of the N-ary disassembly tree is 3, and the area of j₁ is 0. According to the similarity between the components, each component layer includes several types of components; each of them has a certain number. We supposed that the N-ary disassembly tree includes $k+1$ component layers $H_i$ . $Z_{[i][j]}$ represents the jth component in the ith component layer and ${\mathrm{n}}_i$ represents the number of components’ species in the ith component layers, $i\in \{0,1,2,...,k\}$ , $j\in \{0,1,2,...,n_i\}$ . $q_{[i][j]}$ represents the number of component’ categories of $Z_{[i][j]}$ and $t_{[i][j]}$ represents the standard disassembly operation time of individual components $Z_{[i][j]}$ . S represents the standard disassembly preset time between the internal parts of the same component layer, and F represents the standard disassembly preset time between the component layers. $s_{[m][n]}^i$ represents the actual disassembly preset time between the mth component and the nth component in the ith component layer. $F_{[j][k]}$ represents the actual disassembly preset time between the jth component layer and the kth component layer. The relationship of the disassembly preset time and the degree of similarity between the different parts in the same component layers and the different component layers is expressed as

$$s_{[m][n]}^i=(1-r_{[m][n]}^i)\times s$$

(5)

$$F_{[j][k]}=(1-r_{[j][k]})F$$

(6)

where $r_{[m][n]}^i$ represents the degree of similarity between the mth component and the nth component in the ith component layers, and $r_{[j][k]}$ represents the degree of similarity between the jth component layer and the kth component layer.

Definition 2

The degree of similarity between the component layers is equal to the degree of similarity between disassembly operations. The number of component layers is one more than the number of disassembly operation layers, and the disassembly operation layers are one more than the dimension of similarity. The formula is as follows

$$\mathrm{Va}=\mathrm{Vo}+1=\mathrm{Vs}+2$$

(7)

where $\mathrm{Va}$ is the number of component layers, Vo is the number of disassembly operation layers, and Vs is the number of dimensions of similarity. For example, there are three component layers, two disassembly operations, and one dimension of similarity as shown in Figure 2.

In the disassembly tree model, if the jth component in the ith component layer has a certain number, the degree of similarity between the same components in the same component layer is specified as 1, meaning there is no need to have a disassembly preset time between the same components in the same component layer. The disassembly preset time is supposed to be set when considering the disassembly between the different component classes in the same component layer and the disassembly operations between the different component layers.

In the process of disassembly, it is necessary to consider the degree of similarity between the component layers and between the different component classes in the same layer. The operation time shortens with experience gained from the previous disassembly process when the worker is dismantling the same component. The disassembly time should be scheduled for switching tools when a worker moves from one component to another. Given the time interval of the disassembly operation, the learning effect of the disassembly workers decreases and the forgetting effect will be shown. The forgetting model expression in a previous report ²² is adopted in this article

$$f(r_{[m][n]}^i,s)=1-\exp (-\varphi \times (1-r_{[m][n]}^i)\times s)$$

(8)

$$g(r_{[j][k]},F)=1-\exp (-\eta \times (1-r_{[j][k]}^i)\times F)$$

(9)

where $f(r_{[m][n]}^i,s)$ represents the forgetting rate of the mth component in the ith component layer and the nth component. The $g(r_{[j][k]},F)$ represents the forgetting rate of the jth component layer and the kth component layer, and $\varphi$ and $\eta$ represent the forgetting parameter of the components and component layers, respectively.

Three kinds of learning and forgetting models were introduced in the literature, ²² as shown in Figure 3.

Figure 3.

Learning-forgetting model.

The actual disassembly operation time of each node in the N-ary disassembly tree can be calculated using the above three learning and forgetting models.

Preventive maintenance model based on effectiveness factor

Preventive maintenance and sudden failure maintenance must be considered when the N-ary disassembly tree is used to represent the equipment maintenance process in a complex system. The non-new repair model is based on the effectiveness factor, which includes the age reduction factor and failure rate increase factor. Preventive maintenance adopts the non-new and non-perfect repair maintenance methods and the failure rate function of front and rear components are ²⁹

$${{\lambda }_{(i+1)}}_j\left(t\right)={b_i}_j\times {{\lambda }_i}_j(t+{a_i}_j{T}_{\mathrm{ij}})$$

(10)

$${{\lambda }_i}_j\left(t\right)=\underset{1}{\overset{i}{\Pi }}{b}_{0j}\times {{\lambda }_0}_j(t+\sum _1^i{a_i}_j{T}_{\mathrm{ij}})$$

(11)

where ${\lambda }_{(i+1)}(t)$ represents the equipment failure rate function of the jth node in N-ary disassembly tree after the ith preventive maintenance. a_ij represents the age reduction factor of the jth node after the ith preventive maintenance, 0 <a_ij < 1; and b_ij represents the failure rate increase factor of the jth node after the ith preventive maintenance, b_ij > 1.

Due to the opportunistic maintenance that our proposed model adopts, we supposed the service age $\mathrm{elin}{g}_{\mathrm{ij}}$ is $T_{\mathrm{ij}}$ . The formulas are as follows

$${{\lambda }_{(i+1)}}_j\left(t\right)={b_i}_j\times {{\lambda }_i}_j(t+{a_i}_j\mathrm{elin}{g}_{\mathrm{ij}})$$

(12)

$${{\lambda }_i}_j\left(t\right)=\underset{1}{\overset{i}{\Pi }}{b}_{0j}\times {{\lambda }_0}_j(t+\sum _1^i{a_i}_j\mathrm{elin}{g}_{\mathrm{ij}})$$

(13)

Optimal opportunistic maintenance threshold model of a single component

The failure time of equipment components usually obeys the Weibull distribution. A similar error was corrected. The failure rate function ${\lambda }_{(i)}(t)$ of component j in (t₁, t₂) and the cumulative failure frequency equations are as follows

$${\lambda }_{1j}\left(t\right)=\frac{m_j}{{\eta }_j}{\left(\frac{t}{{\eta }_j}\right)}^{m_j-1}$$

(14)

$${{\lambda }_{(i+1)}}_j\left(t\right)={b_i}_j\times {{\lambda }_i}_j(t+{a_i}_j{T}_{\mathrm{ij}})$$

(15)

$$N_j(t_1,t_2)=\underset{t_1}{\overset{t_2}{\int }}{\lambda }_{\mathrm{ij}}\left(t\right)\mathrm{dt}$$

(16)

where m_j represents the shape parameter of the jth node in N-ary disassembly tree and n_j represents the dimension parameter of the jth node in N-ary disassembly tree.

A single component of the jth node in the N-ary disassembly tree undergoes minor maintenance if there is a sudden failure, and the service age of the component will not change. In this article, repair non-new preventive maintenance methods are adopted based on the preventive maintenance model of the failure factor. The preventive maintenance of components is supposed to be completed when the service age of components achieve the preventive maintenance interval T_j. The service age is a_ijT_ij.

A maintenance period for each node consists of four parts: preventive maintenance interval, disassembly time, preventive maintenance time, and minor repair time. The disassembly operation includes disassembly preset time and disassembly operation time. The cost incurred during the cycle consists of disassembly costs, preventive maintenance costs, minor repair costs, and downtime costs. The downtime includes disassembly time, preventive maintenance time, and minor repair time. The formulas are as follows

$$\begin{array}{c}\multicolumn{1}{c}{c_j(T_j)=\{c_j^c\times T_j^c\times N_j(0,T_j)+c_j^p\times T_j^p+{\mathrm{DC}}_j^d}\\ \multicolumn{1}{c}{\times [N_j(0,T_j)+1]+c_j^s\times T_j^s\}/T_j}\end{array}$$

(17)

$$T_j^s=T_j^c\times N_j(0,T_j)+T_j^p+{\mathrm{DT}}_j^d\times (N_j(0,T_j)+1)$$

(18)

where $c_j^c$ is the minor repair maintenance cost rate of component j, $T_j^c$ is the minor repair maintenance time of component j, $c_j^p$ is the preventive maintenance cost rate of component j, $T_j^p$ is the single preventive maintenance cost time of component j, $c_j^s$ is the downtime of component j, and $T_j^s$ is the total downtime of component j. Aiming to minimize costs, the optimal chance maintenance threshold $T_j^{*}$ of a function is deduced by the derivation of functions of $c_j(T_j)$ . We then obtain $T_j^{*}$ by $d{c}_j(T_j)/d{T}_j=0$ .

Opportunistic maintenance model based on threshold

When a component j (component j is an arbitrary root node in the N-fork disassembly tree) in the N-fork disassembly tree is randomly faulty or preventive maintenance is performed, all the machines need to stop and the other components can possibly be maintained. We select the components for opportunistic maintenance by setting the threshold T_w. If the difference in the optimal opportunistic maintenance threshold $T_j^{*}$ of component j and the service age $\mathrm{elin}{g}_j$ of the component is less than or equal to the threshold T_w, $0\le T_w\le max(T_j^{*})$ , the component k (the arbitrary root node of N-ary disassembly tree) will be embedded in opportunistic maintenance set $\{\mathrm{OM}\}$ . In particular, if the component j experiences a burst failure, the component will immediately receive opportunistic maintenance if it meets the condition for opportunistic maintenance.

In this opportunistic maintenance strategy, the ith downtime is composed of three operation parts: minor repair maintenance work, preventive maintenance work, and opportunistic maintenance work (all components that meet the opportunistic maintenance requirements, and j itself is also eligible opportunistic maintenance, including disassembly work of component j). The calculation formula of downtime $T_i$ and downtime cost $C_i$ are

$$\begin{array}{c}\multicolumn{1}{c}{T_i={\kappa }_i\times T_{\mathrm{ji}}^p+(1-{\kappa }_i)\times T_{\mathrm{ji}}^c+\sum _{k\in \left\{\mathrm{OM}\right\}}{T}_k^p}\\ \multicolumn{1}{c}{+\sum _{l\in \left\{\mathrm{OM}\right\}\cup \left\{j\right\}}{\mathrm{DT}}_l^d}\end{array}$$

(19)

$$\begin{array}{c}\multicolumn{1}{c}{C_i={\kappa }_i\times c_j^p\times T_{\mathrm{ji}}^p+(1-{\kappa }_i)\times c_j^c\times T_{\mathrm{ji}}^c+\sum _{k\in \left\{\mathrm{OM}\right\}}{c}_k^p\times T_k^p}\\ \multicolumn{1}{c}{+\sum _{l\in \left\{\mathrm{OM}\right\}\cup \left\{j\right\}}{\mathrm{DC}}_l^d}\end{array}$$

(20)

where

$$k_i=\{\begin{array}{l}0,foraminorrepairandmaintenance\hfill \\ 1,foranopportunisticmaintenance\hfill \end{array},k\in \left\{k|OP{T}_k-E{L}_k\le T_w\right\}$$

The normal working time of system is $T_{\mathrm{gz}}$ . Assuming _W times of downtime occur during the period, the total maintenance time ZT, the total maintenance cost ZC, the total flow time ZL, and the operation time rate LV are calculated as follows

$$\mathrm{ZT}=\sum _{i=0}^W{T}_i$$

(21)

$$\mathrm{ZC}=\sum _{i=0}^W{C}_i$$

(22)

$$\mathrm{ZL}=T_{\mathrm{gz}}+\mathrm{ZT}=T_{\mathrm{gz}}+\sum _{i=0}^W{T}_i$$

(23)

$$\mathrm{LV}=\frac{T_{\mathrm{gz}}}{\mathrm{ZL}}=\frac{T_{\mathrm{gz}}}{T_{\mathrm{gz}}+\sum _{i=0}^W{T}_i}$$

(24)

Aiming at minimizing the total maintenance costs ZC and maximum operation time rate, the optimal opportunistic maintenance threshold $T_w^{*}$ can be solved, which is equivalent to the aim of minimizing total maintenance costs ZC and minimizing total maintenance operation time ZT.

Simulation solution

Fusing the N-ary disassembly tree model, the learning-forgetting model of similarity between components, and the failure-based device prevention repair non-new preventive maintenance model, and considering the single-piece optimal opportunistic maintenance threshold model and opportunistic maintenance model of optimal maintenance interval, we constructed the opportunistic maintenance model for a complex recessive system considering the disassembly sequence and learning-forgetting effect.

The opportunistic maintenance model is a dynamic process that uses the Monte Carlo simulation to solve the questions of the model effectively. The process of solving the model is shown in Figure 4. The shortest disassembly path and disassembly time of each node obtained first discretizes the normal working time. Compare the relationship between the maintenance threshold and the disassembly time. If the conditions are met, the next calculation and selection, such as the comparison between CS and B and the comparison between t and $T_{\mathrm{gz}}$ , compare the random number with the number of failures to determine the maintenance when the condition is met again. As such, the statistics of the disassembly time and cost are calculated using the learning-forgetting effect model in accordance with the conditional dimensions, and the total costs and total time are determined.

Figure 4.

Flow chart followed for solving the model.

Case study

Parameter setting

In order to verify the effectiveness of the proposed model, we design a suitable complex fading system. Suppose the normal working time $T_{\mathrm{ge}}$ of system is 4000 days, the unit downtime cost $c_s$ is 8 CNY, and the disassembly preset cost $c^{\mathrm{dy}}$ unit time is 10 CNY. The system uses N-ary disassembly tree which is divided into four layers. The N-ary disassembly tree includes 21 nodes: 15 root nodes is 15 and 6 middle nodes, shown in Figure 5. The relevant parameters of disassembly are shown in Table 1.

Figure 5.

N-ary disassembly tree of complex fading systems.

Table 1.

Parameters related to a N-ary disassembly tree node in a complex fading system.

Node j	Cost rate of disassembly(CNY/day)	Time of standard disassembly operation(days)	The shortest disassemblypath (j)
I	0	0	{I}
II	100	4	{I, II}
III	100	4	{I, III}
IV	125	3	{I, IV}
V	150	2	{I, II, V}
VI	250	2	{I, III, IV, VI}
1	200	0.3	{I, 1}
2	200	0.3	{I, 3, 2}
3	200	0.3	{I, 2, 3}
4	200	0.4	{I, II, 4}
5	200	0.4	{I, II, 5}
6	200	0.4	{I, III, 6}
7	200	0.4	{I, III, 7}
8	200	0.3	{I, II, V, 8}
9	200	0.3	{I, II, V, 9}
10	200	0.3	{I, II, V, 11, 10}
11	200	0.3	{I, II, V, 10, 11}
12	200	0.3	{I, III, IV, VI, 12}
13	200	0.3	{I, III, IV, VI, 13}
14	200	0.3	{I, III, IV, VI, 15, 14}
15	200	0.3	{I, III, IV, VI, 14, 15}

The nodes in each rectangle in Figure 5 are the same component in the same layer, which includes 4 layers and 12 components in Figure 4. For example, j₁, j₂, and j₃ are the first component in the second layer, denoted as $Z_{[2][1]}$ . Because of the similarity between the components, the learning-forgetting effect must be considered in the process of disassembly. We supposed that the fixed learning rate $\epsilon$ is 0.8, the forgetting parameters between components $\varphi$ is 0.1, the forgetting parameters between component layers $\eta$ is 0.2, the basic disassembly preset time $s$ of the component is 0.2 days, the basic disassembly preset time $F$ between component layers is 0.5 days, the initial disassembly preset time $F_0$ between component layers is 0.4 days, and the similarity $r_{[j][m]}^i$ between components within a component layer and the similarity $r_{[i][n]}$ between component layers are as follows, respectively

$$\begin{array}{c}\multicolumn{1}{c}{r_{[j][m]}^1=\left[\begin{array}{ccc}1\hfill & 0.50\hfill & 0.40\hfill \\ \multicolumn{1}{c}{0.50}& 1\hfill & 0.25\hfill \\ \multicolumn{1}{c}{0.40}& 0.25\hfill & 1\hfill \end{array}\right],}\\ \multicolumn{1}{c}{r_{[j][m]}^2=\left[\begin{array}{cccc}1\hfill & 0.50\hfill & 1.00\hfill & 0.50\hfill \\ \multicolumn{1}{c}{0.50}& 1\hfill & 0.50\hfill & 1.00\hfill \\ \multicolumn{1}{c}{1.00}& 0.50\hfill & 1\hfill & 0.50\hfill \\ \multicolumn{1}{c}{0.50}& 1.00\hfill & 0.50\hfill & 1\hfill \end{array}\right]}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{r_{[j][m]}^3=\left[\begin{array}{cccc}1\hfill & 0.20\hfill & 1.00\hfill & 0.20\hfill \\ \multicolumn{1}{c}{0.20}& 1\hfill & 0.20\hfill & 1.00\hfill \\ \multicolumn{1}{c}{1.00}& 0.20\hfill & 1\hfill & 0.20\hfill \\ \multicolumn{1}{c}{0.20}& 1.00\hfill & 0.20\hfill & 1\hfill \end{array}\right],}\\ \multicolumn{1}{c}{r_{[i][n]}=\left[\begin{array}{ccc}1\hfill & 0.25\hfill & 0.20\hfill \\ \multicolumn{1}{c}{0.25}& 1\hfill & 0.10\hfill \\ \multicolumn{1}{c}{0.20}& 0.10\hfill & 1\hfill \end{array}\right]}\end{array}$$

${\mathrm{DC}}_j^d$ and ${\mathrm{DT}}_j^d$ can be calculated after considering the learning-forgetting effect according to above relevant parameters, and the calculation results are shown in Table 2.

Table 2.

Calculation results of ${\mathrm{DC}}_j^d$ and ${\mathrm{DT}}_j^d$ .

Without considering learning-forgetting effect		Considering learning-forgettingeffect
${\mathrm{DC}}_j^d(\mathrm{CNY})$	${\mathrm{DT}}_j^d(\mathrm{days})$	${\mathrm{DC}}_j^d(\mathrm{CNY})$	${\mathrm{DT}}_j^d(\mathrm{days})$
64	0.7	64	0.7
126	1.2	112	0.9
126	1.2	112	0.9
485.3	4.9	485.3	4.9
485.3	4.9	485.3	4.9
485.3	4.9	485.3	4.9
485.3	4.9	485.3	4.9
765.8	6.9	765.8	6.9
765.8	6.9	765.8	6.9
827.7	7.4	813.7	7.1
827.7	7.4	813.7	7.1
1342.8	10.1	1342.2	10
1342.8	10.1	1342.2	10
1404.7	10.6	1390.3	10.3
1404.7	10.6	1390.3	10.3

We set the shape parameter $m_j$ , dimension parameter ${\eta }_j$ , age reduction factor $a_j$ , hazard rate increase factor $b_j$ , cost rate of minor repair and maintenance $c_j^c$ , time of minor repair and maintenance $T_j^c$ , the cost rate $c_j^p$ of preventive maintenance, and single preventive maintenance cost time $T_j^p$ of the jth component on this basis as outlined in Table 2. As shown in Table 3, the optimal maintenance interval of each root component is obtained according to the optimal maintenance time interval formula.

Table 3.

Optimal maintenance time interval of root node considering learning-forgetting effect.

$j$	$m_j$	${\eta }_j$	$a_j$	$b_j$	$c_j^c(\mathrm{CNY})$	$T_j^c(\mathrm{days})$	$c_j^p(\mathrm{CNY})$	$T_j^p(\mathrm{days})$	${\mathrm{DC}}_j^d(\mathrm{CNY})$	${\mathrm{DT}}_j^d(\mathrm{days})$	$T_j^{*}(\mathrm{days})$
1	1000	2.5	0.001	1.04	600	0.32	300	0.21	64	0.7	649
2	1000	2.5	0.001	1.04	600	0.32	300	0.21	112	0.9	687
3	1000	2.5	0.001	1.04	600	0.32	300	0.21	112	0.9	687
4	800	2.0	0.0008	1.03	800	0.46	420	0.27	485.3	4.9	676
5	800	2.0	0.0008	1.03	800	0.46	420	0.27	485.3	4.9	676
6	800	2.0	0.0008	1.03	800	0.46	420	0.27	485.3	4.9	676
7	800	2.0	0.0008	1.03	800	0.46	420	0.27	485.3	4.9	676
8	600	1.8	0.0006	1.05	700	0.62	260	0.15	765.8	6.9	550
9	600	1.8	0.0006	1.05	700	0.62	260	0.15	765.8	6.9	550
10	500	2.2	0.0008	1.03	900	0.37	500	0.16	813.7	7.1	413
11	500	2.2	0.0008	1.03	900	0.37	500	0.16	813.7	7.1	413
12	600	1.8	0.0006	1.05	700	0.62	260	0.15	1342.2	10	594
13	600	1.8	0.0006	1.05	700	0.62	260	0.15	1342.2	10	594
14	500	2.2	0.0008	1.03	900	0.37	500	0.16	1390.3	10.3	430
15	500	2.2	0.0008	1.03	900	0.37	500	0.16	1390.3	10.3	430

By combining all the analysis results, it is helpful to determine the disassembly cost, disassembly time, and opportunity maintenance threshold through considering learning-forgetting effects. Therefore, it is feasible to consider the learning-forgetting effects in the model.

Monte Carlo simulation

On the MATLABR2015a platform, Monte Carlo simulation was used to simulate and example. Taking 1 day as a unit, the normal working time of the system can be divided into 4000d time units. Monte Carlo simulation of a component based on service age was run for each time unit to judge whether a random fault occurs within the time unit by generating a random number. The core step involves calculating the number of random failures for the service age of a unit component. Compared with the random number that occurs in [0,1], a random failure occurs if the random number is less than or equal to the number of random failures. This situation requires a minor repair. Otherwise, it is assumed that no random failure occurs. Other components make a judgment as to whether or not to perform opportunistic maintenance using the opportunistic maintenance model based on threshold T_w when there is a random failure in a part of the system or the optimal maintenance interval met for preventive maintenance. After preventive maintenance and opportunistic maintenance, the components are not as good as new. The service age of the component is not zero due to the age reduction factor and the failure rate increment factor. The similarity between components should be considered for the learning-forgetting effect in the process of component maintenance. In particular, when the equipment experiences a random failure and meets the conditions for opportunistic maintenance, the component should receive opportunistic maintenance immediately after a minor repair. The total maintenance operation time ZT and total maintenance cost ZC can be obtained by the $T_{\mathrm{ge}}$ after being given the decision variable $T_w$ according to the above strategy. Owing to the range in decision variables, $T_w$ is $0\le T_w\le max(T_j^{*})$ , the interval of $T_w$ is 25d for simulation in this article. Under normal circumstances, a relatively good result can be obtained using Monte Carlo simulation 1000 times. ³¹ According to the existing literature, the time and cost of Monte Carlo simulation increases with the number of run times. In order to ensure the accuracy of the simulation without excessively increasing the simulation time and cost, the number of simulations was set to 2000. The curves of ZT and ZC that change with the decision variable $T_w$ were obtained as shown in Figure 6.

Figure 6.

Monte Carlo simulation result.

As shown in Figure 6, the curve is smooth and there is a unique minimum point after 2000 simulations. The threshold $T_w$ is 300 days, ZT is 625.80 days, and ZC is 109,200 CNY. The variation trends in ZT and ZC are the same and their extreme points are similar, but their pattern curves are not exactly the same as in Figure 5. Opportunistic maintenance can effectively reduce maintenance costs, but excessive opportunistic maintenance thresholds can lead to exorbitant costs. In the literature, ²⁵ the influence of the threshold on the disassembly cost was systematically and hierarchically analyzed, and this article is not repeat. The main problems in this article are (1) the difference in preventive maintenance and without preventive maintenance for components that have random faults and satisfy the conditions for opportunistic maintenance, (2) the influence of the decreasing age reduction factor and hazard rate increase factor, (3) the influence of learning-forgetting effects, and (4) the combined effects of the above three factors.

Analysis of impact of preventive maintenance

When a random failure occurs and the opportunistic maintenance condition is satisfied, the differences in the impact on ZC and ZT with and without preventive maintenance is shown in Figure 7(a) and (b), respectively. Preventive maintenance is considered for components that have random breakdowns and meet the conditions for opportunistic maintenance. The opportunistic maintenance has less influence on the threshold of optimal opportunistic maintenance and the curve shifts to the right. $T_w$ adds a step size of 25–325 days. The curve of the opportunistic maintenance is similar to the ZC curve of the non-opportunistic maintenance and the minimum total maintenance cost ZC of opportunistic maintenance is less than that of non-opportunistic maintenance by 7860 CNY. The curve of the opportunistic maintenance is similar to that of ZT for the non-opportunistic maintenance and the minimum total maintenance operation time of opportunistic maintenance is less than that for non-opportunistic maintenance by 55.10d. The results show that preventive maintenance is superior to non-preventive maintenance in terms of cost and time for a component experiencing random failure and meets the conditions for opportunistic maintenance, so opportunistic maintenance will be implemented more conservatively.

Figure 7.

Influence analysis chart of preventive maintenance.

Analysis on the effect of hybrid evolution factors

Impact analysis of age reduction factor ${a_i}_j$

Taking the simulation results of the original parameters of the model as the control group, the value of the age reduction factor was adjusted by multiplying six different adjustment magnifications of $\omega (\omega =0.00,0.01,0.50,1.00,10.00,20.00)$ under the condition that the other parameters are constant. We performed 2000 Monte Carlo simulations and the results are shown in Figure 8 and Table 4.

Figure 8.

Effect of age reduction factor.

Table 4.

Simulation results of changes in age reduction factor.

$\omega$	$T_{w1}(\mathrm{days})$	$\mathrm{ZC}(\mathrm{CNY})$	$T_{w2}(\mathrm{days})$	$\mathrm{ZT}(\mathrm{days})$
0	325	105,370	325	608.9201
0.01	325	107,090	325	610.5975
0.1	300	108,530	300	612.3917
1	300	109,200	300	625.8006
10	300	115,860	300	663.8
20	225	124,200	225	720.7

The results show that the minimum values of the ZC and ZT curves shift to the left, the threshold of optimal opportunistic maintenance decrease, the minimum total maintenance cost ZC and the minimum total operation time ZT increase, the range of the increase continually expands, and is obvious especially after multiples of greater than 1 with multiple increase in the age reduction factor. However, other influencing factor parameters remain unchanged. The results show that the optimal opportunistic maintenance threshold $T_w$ reduces and the system tends to perform opportunistic maintenance conservatively when the age reduction factor of the component increases.

Analysis of the effect of failure rate increase factor ${b_i}_j$

The simulation results of the primary model parameters were used as the control group. The incremental factor of the failure rate was adjusted by adding to the numerical value $\lambda (\lambda =-0.03,-0.02,-0.01,$ $+0.00,+0.01,+0.02)$ , under the condition that the other parameters remain unchanged. Then, 2000 times of Monte Carlo simulation were performed and the results are shown in Figure 9 and Table 5.

Figure 9.

Influence diagram of failure rate increase factor.

Table 5.

Simulation results for failure rate increase factor’ changes.

$\omega$	$T_{w1}(\mathrm{days})$	$\mathrm{ZC}(\mathrm{CNY})$	$T_{w2}(\mathrm{days})$	$\mathrm{ZT}(\mathrm{days})$
b–0.03	300	94,860	675	478.6909
b–0.02	325	100,510	650	530.1687
b–0.01	275	104,260	550	572.7318
b–0.00	300	109,200	300	625.8006
b+0.01	300	114,020	300	654.8837
b+0.02	275	118,820	300	684.7

The analysis showed that the minimum points of the ZC and ZT curves shifted to the left, the optimal opportunistic maintenance thresholds $T_w$ decrease, the minimum maintenance cost and minimum maintenance time increase with multiple increases in the failure rate increment factor, but other influence factor parameters remain unchanged. The research showed that the threshold T_w is small and the components that require opportunistic maintenance decreases progressively when the failure rate increase factor of the component is large. So, the system tends to complete opportunistic maintenance conservatively at this time.

Analysis of influence for learning-forgetting effect

Analysis of influence of fixed learning rate

Regarding the simulation results produced by the model parameters as the control group, the fixed learning rate $\epsilon (\epsilon =0.50,0.60,0.70,0.80,0.90,1.00)$ should be adjusted under the condition that the other parameters remain unchanged. We performed 2000 times of Monte Carlo simulation and the results are shown in Figure 10 and Table 6.

Figure 10.

Influence diagram of fixed learning rate.

Table 6.

Statistical table of simulation results for different fixed learning rates.

$\epsilon$	$\gamma$	$T_{w1}$ (days)	$\mathrm{ZC}$ (CNY)	$T_{w2}$ (days)	$\mathrm{ZT}$ (days)
0.50	1.00	350	88,840	675	416.1333
0.60	0.74	325	92,850	675	455.6187
0.70	0.51	325	102,640	675	526.2002
0.80	0.32	300	109,200	300	625.8006
0.90	0.15	300	116,750	300	672.9289
1.00	0.00	275	123,450	275	716.5746

The analysis showed that (1) the learning coefficient $\gamma$ is 0 when the fixed learning rate $\epsilon$ is 1 under the condition of the other influence factor parameter remaining unchanged. There is no influence of the learning effect and the curves of ZC and ZT are all over the other curves without considering the cost and time of the learning effect being higher than the factor’s value. (2) The learning coefficient $\gamma$ decreases, the minimum total maintenance cost ZC and the minimum total maintenance time ZT increase, and minimum point of ZC and ZT curves shift to the right and the threshold of optimal opportunistic maintenance decreases progressively with the increase in the fixed learning rate $\epsilon$ . (3) The minimum value of the ZT curve is 675 days when the learning coefficient $\gamma$ is too big because the reduction in the disassembly time caused by the too large learning coefficient results the time difference between the n components and the n + 1 components that are machined is too small. The minimum total maintenance operation time ZT appears at the end point of 675 days. The research showed that the learning coefficient $\gamma$ and the threshold $T_w$ of the optimal opportunistic maintenance are large, too many components require opportunistic maintenance, and the system tends to actively complete opportunistic maintenance when the fixed learning rate of the component is low.

Analysis of influence of forgetting parameters

Regarding the simulation results produced using the model parameters as the control group, the forgetting parameters $\varphi$ and $\eta$ should be adjusted by multiplying by six different rates of $\psi (\psi =0.00,0.25,0.50,1.00,2.00,4.00)$ and completing 2000 Monte Carlo simulation with the other parameters remain unchanged. The results are shown in Figure 11 and Table 7.

Figure 11.

Influence of forgetting parameters.

Table 7.

Statistical table of simulation results for change of forgetting parameters.

$\psi$	$T_{w1}(\mathrm{days})$	$\mathrm{ZC}(\mathrm{CNY})$	$T_{w2}(\mathrm{days})$	$\mathrm{ZT}(\mathrm{days})$
0.00	300	103,890	300	597.5489
0.25	300	105,630	300	606.7213
0.50	300	106,260	300	610.0515
1.00	300	109,200	300	625.8006
2.00	300	110,450	300	633.4452
4.00	300	112,340	300	642.938

The analysis showed that the minimum point $T_w$ of the ZC and ZT curves and the threshold $T_w$ of the optimal opportunistic maintenance remain unchanged, and the minimum maintenance costs and minimum maintenance times increase but not by much. The multiple of the forgetting parameter increases, but the other influencing factor parameters remain unchanged. The research showed that the threshold $T_w$ is unchanged and the change in the forgetting parameters has little effect on the opportunistic maintenance of the system when the forgetting parameters of component are large.

Comprehensive factor influence analysis

A special case for comparative analysis was examined where the opportunistic maintenance is completed with and without considering three factors. These three factors were considered in the original parameter model in this article. Under the condition that the other parameters remain unchanged and without considering the model of the three factors, the parameters were as follows: (1) preventive maintenance is not performed for the component when it experiences a random failure and meets the conditions of opportunistic maintenance, (2) the age reduction factor $a_j$ is 0 and the increasing factor $b_j$ of failure rate is 1, and (3) the fixed learning rate $\epsilon$ is 0.5 and the forgetting parameters $\varphi$ and $\eta$ are both 1.

We performed 2000 Monte Carlo simulations and the results are shown in Figure 12 and Table 8.

Figure 12.

Comprehensive impact of three factors.

Table 8.

Statistics table of simulation results with and without considering three factors.

	$T_{w1}(\mathrm{days})$	$\mathrm{ZC}(\mathrm{CNY})$	$T_{w2}(\mathrm{days})$	$\mathrm{ZT}(\mathrm{days})$
Considering three factors	300	109,200	300	625.8006
Without considering three factors	500	95,970	500	518.1953

By comparing the graphs without considering three factors, the minimum points of the ZC and ZT curves in the graph that considers three factors, and with a small threshold $T_w$ of optimal opportunistic maintenance, the minimum maintenance cost and minimum maintenance time are large and the systems tends to perform opportunistic maintenance. If the three factors are not considered, the decision value point is 500 days. However, the three factors exist objectively and cannot be neglected. The point is compared with the decision point at 300 days which considers the three factors as shown in Table 9. The cost is 14,070 CNY and the time loss 54.20 days. As a result, considering the three factors to determine the threshold of optimal opportunistic maintenance can provide more accurate information for decision makers.

Table 9.

Decision loss without considering the three factors.

Items	$T_{w1}(\mathrm{days})$	$\mathrm{ZC}(\mathrm{CNY})$	$T_{w2}(\mathrm{days})$	$\mathrm{ZT}(\mathrm{days})$
The best decision point without considering three factors	500	123,270	500	680.00
The best decision point considering three factors	300	109,200	300	625.80
Difference value	–	14,070	–	54.20

Conclusion

Considering the influence of the learning-forgetting effect and the hybrid evolution factors during maintenance operation, we proposed an opportunistic maintenance model for a complex fading system based on a time threshold by introducing a learning-forgetting effect model into complex system maintenance planning. We quantitatively analyzed the influence of the choice to perform preventive maintenance on the thresholds for components that have suffered random failures and met the conditions for opportunistic maintenance. We also examined the influence of the age reduction factor and the failure rate increase factor on the threshold, and the influence of the learning-forgetting effect on the threshold. The research showed that the variation trend in ZT is same as ZC and their extreme points are similar, but their pattern curves are not exactly the same. The numerical values of the multiple targets of minimizing maintenance cost and maximizing operation time are same in terms of results. Performing preventive maintenance for components that have random failure and meet the condition of opportunistic maintenance is superior to not performing preventive maintenance in terms of maintenance cost and maintenance time. The system tends to perform the opportunistic maintenance conservatively. The optimal opportunistic maintenance threshold $T_w$ decreases and the system tends to perform the opportunistic maintenance conservatively by increasing the age reduction factor or the failure rate increase factor of component. When the fixed learning rate of component is low, the learning coefficient $\gamma$ and the threshold $T_w$ of optimal opportunistic maintenance are high, the number of components that receive opportunistic maintenance is large, and the system tends to actively perform opportunistic maintenance. Changing the forgetting parameter has little influence on the frequency of performing opportunistic maintenance. Considering the impact of the three factors of the components that experience random failure and meet the condition for opportunistic maintenance on preventive maintenance, the hybrid evolution factors, and learning-forgetting effects, the threshold of optimal opportunistic maintenance provides more accurate information for decision makers. The advantages of the proposed model in this article are as follows: (1) an opportunistic maintenance model for complex recession system based on time threshold was proposed; it can solve the problems of multiple parts, high cost of disassembly, and imperfect maintenance in multiple component complex recession systems. (2) As equipment maintenance time can be affected by equipment disassembly process, the influence of disassembly sequence and hybrid evolution factors on maintenance is considered in this article. The threshold of opportunity maintenance can be determined more reasonably by considering this factor. (3) The similarity between parts is accompanied by learning-forgetting effect, which results in the changes of maintenance time. The effect of learning-forgetting effect on disassembly time is considered in article and taking this factor into account can make the opportunity maintenance model more relevant to actual production.

In this article, an opportunistic maintenance model for complex system based on time threshold was proposed, which introduces the effect of disassembly sequence and hybrid evolution factors on maintenance. At the same time, the effect of learning-forgetting effect on demolition time was considered and the time threshold of opportunistic maintenance was determined scientifically and effectively, which provides a more cost-effective maintenance program for maintenance activities.

However, the limitation of the research is that the proposed model ignores the correlation between components in complex system. In future research, the correlation between components of the system will be discussed.