CN106447051A - System selective maintenance decision-making method for multitask stage - Google Patents

Abstract

The invention discloses a system selective maintenance decision-making method for a multitask stage, and the method comprises the steps: determining the state and service age of each part of a system at the beginning of each task; calculating the reliability of each part of the system in each task; calculating the normal operation probability of each part of the system in each task; calculating the normal operation probability of the system in each task; taking the task completing probability of each task of the system as a target function, taking a limited maintenance resource feasible set as a constraint condition, taking the maintenance behavior of each part of the system in each task as a decision-making variable, building a maintenance decision-making optimization model, and solving an optimal maintenance strategy. Through the comprehensive consideration of the task time uncertainty and failure time uncertainty of the multitask stage, a system decision maker can configure the limited maintenance resources to each maintenance interval more reasonably and effectively, so as to maximize the normal operation probability of the system in each task.

Description

System selective maintenance decision method for multitask stage

Technical Field

The invention belongs to the field of equipment maintenance optimization management, and particularly relates to a system selective maintenance decision method for a multitask stage.

Technical Field

Maintenance is a behavioral activity that aims to restore or improve the performance of a system, enabling it to meet the needs of subsequent tasks. In practical engineering, maintenance decision optimization problems often need to be researched under limited maintenance resources (such as limited maintenance period, limited maintenance expenditure, maintenance personnel and the like). The research of maintenance decision problems under limited maintenance resources is an important problem which is commonly concerned by academic and industrial circles in recent years, and the basic content of the research is to research how to reasonably and effectively optimize and configure the limited maintenance resources into each element forming the system according to the structure of the system, the working state and the performance evolution rule of the components, so that the whole system has the maximum reliability index.

Systems are generally made up of multiple components and perform a series of defined tasks, with the system being serviced during the interval between tasks. Furthermore, the task length of each task is often uncertain, such as: one military aircraft performs patrol tasks at 9 points every day, and the return time of the aircraft cannot be determined due to weather and other reasons. During the execution of a series of tasks, a component may fail during the execution of a certain task, in which case the failure time of the component is right-truncated data. The following situations often exist in engineering: the system is to execute N tasks, and the lengths of the tasks may not be equal and there is uncertainty. There are N-1 maintenance intervals during which the components of the system are maintained, and it is not reasonable to have all the maintenance resources (e.g., maintenance costs) in N-1 maintenance intervals.

Disclosure of Invention

The invention aims to provide a selective maintenance decision method for maximizing the completion probability of a system in each task stage under the condition of multi-task stage and limited maintenance resources.

The technical scheme of the invention is as follows: a system selective maintenance decision-making method facing to a multitask stage specifically comprises the following steps:

step 1: defining the state and working age of each part of the system at the beginning of each task;

step 2: calculating the reliability of each part of the system in each task;

and step 3: calculating the normal working probability of each part of the system in each task;

and 4, step 4: calculating the normal working probability of the system in each task;

and 5: and establishing a maintenance decision optimization model and solving an optimal maintenance strategy by taking the task completion probability in each task of the system as an objective function, a limited feasible set of maintenance resources as a constraint condition and the maintenance behaviors of all parts of the system in each task as decision variables.

Further, the state of the component at the beginning of each task in step 1 can be represented as:

wherein k ═ {1,2, …, N } represents the kth task, and N represents the total number of tasks to be executed by the system;

accordingly, the state of the component at the end of each task may be represented as:

the way of repair of the part is non-sound repair, the non-sound repair model being a Kijima I model (working age model), the effective working age of part I at the start of the kth task being denoted x_i,kThe time for which the component i normally operates in the k-th task is set to u_i,k∈[0,t_k]，t_kIs the task length of the kth task, if X_i,k1, the effective service life of component i at the end of the k-th task is:

y_i,k＝x_i,k+u_i,k(1)

after maintenance, the effective service life of part i at the start of the k +1 th task is obtained

x_i,k+1＝x_i,k+b_i,ku_i,k(2)

Wherein, b_i,k∈[0,1]Representing a working age back factor, in particular if Y_i,k＝1(u_i,k＝t_k) Then y is_i,k＝x_i,k+t_i,k(ii) a If component i fails on the kth task (i.e., u_i,k<t_k) And no maintenance is performed, the part remains in a failed state (X) for the k +1 th task_i,k+10 and Y_i,k+10), in which case x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,k。

B in the formula (2)_i,kThe value of (a) is related to the maintenance behavior, the component maintenance distinguishes a plurality of different maintenance grades, the higher the maintenance grade is, the higher the consumed maintenance resources (maintenance cost) are correspondingly. The repair grade adopted during the kth repair interval for component i is denoted as l_i,k∈{1,2,…,N_iIn which N is_iRepresenting the highest repair level, the relationship between the repair cost and the repair level for component i during the kth repair interval may be expressed as:

wherein,is indicated at maintenance level of l_i,kThe cost of the repair/preventive maintenance that follows,indicating the fixed disassembly and assembly costs of the component i,indicating the cost of replacing component i, in particular l_i,k＝{1,2,N_iMeans no service, minimum service and replacement, respectively, for component i during the kth service interval. .

B is further obtained from the relationship between the maintenance cost and the maintenance grade of the equation (3)_i,kAnd maintenance costs. If component i fails before the end of the kth task, b_i,kThe relationship to maintenance costs is expressed as:

wherein,represents the maximum repair cost of the restorative repair, m_i,kIs a characteristic parameter.

If the component i is still working at the end of the k-th task, b_i,kThe relationship to maintenance costs is expressed as:

wherein,representing the maximum maintenance cost for preventive maintenance. Characteristic parameter m in formulae (4) and (5)_i,kThe average remaining life of a part i at the end of the k-th task, which may be derived from the valid work-age of the part i at the end of the k-th task, is defined as:

wherein r is_i，k(t) is the reliability of component i, as will be further explained in step 2. The characteristic parameters of equation (5) are:

further, the calculation of the reliability of each part of the system in each task in step 2 is divided into the following cases.

If X_i,kIf 0, the reliability of component i in the k-th task is 0, x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,kIf X is_i,kIf 1, the reliability of component i in the k-th task is:

wherein λ is_i(x_i,k+ t) is the failure rate function for component i, t ∈ (0, t)_k]The timing starts from the system executing the kth task. If Y is_i,kWhen the k-th task ends, the reliability of the component i is r_i,k(t_k) Before and after maintenance the effective service life is y_i,k＝x_i,k+t_i,kAnd x_i,k+1＝x_i,k+b_i,kt_i,k. If Y is_i,k＝0，u_i,k∈(0,t_k) The right truncated data represents the effective service life before maintenance as y_i,k＝x_i,k+u_i,kIt can be expressed by a conditional probability density function as:

the cumulative density function for any valid work age T at the end of the kth task is:

wherein, Pr { y_i,k<T represents the effective service life y before maintenance_i,k<The probability of T is determined by the probability of T,is u_i,kThe cumulative density function of (2). Thus, y can be obtained_i,kThe probability density function of (a) is:

in the kth maintenance interval, non-intact maintenance is performed on component i, if b_i,k∈ (0,1), the effective service life of part i at the start of the k +1 th task is x_i,k+1＝x_i,k+b_i,ku_i,k。x_i,k+1The cumulative density function of (a) is expressed as:

thus, x can be obtained_i,k+1The probability density function of (a) is:

in particular, if b_i,k1, then component i remains in the failed state, x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,k. If b is_i,kWhen 0, the component i is repaired to the working state, x_i,k+1＝x_i,kAnd isFurthermore, x_i,1Probability density function ofKnown as x_i,1The upper and lower limits of the value are respectivelyAndtaking into account the effective working life(f_k1,2, …, k), the component i accumulates a reliability at the end of the kth task as:

further, the calculation of the probability that each component of the system works normally in each task in step 3 needs to consider a plurality of state paths of the component. Each time a task ends, component i may be in a working or failure state, thus there is 2 from task 1 to task k^kDifferent paths are followed. Therein, there are 2^k-1The strip path may result in Y_i,k1, and then the component i is found at the kth task end_pThe working probability under a path is:

wherein k is_p＝1,2,…,2^k-1. Therefore, the cumulative operating probability of component i at the end of the kth task is:

further, the probability that the system normally works in each task in step 4 can be obtained from the accumulated working probability of the system structure and the components. The system state may be derived from the state of the component:

Y_s,k＝φ(Y_1,k,Y_2,k,…,Y_M,k) (17)

where φ (-) is the structural equation for the system and M is the total number of components that make up the system.

In particular, for any n number of components i in series₁,i₂,…,i_nThe combined relationship is expressed as:

where j ═ is (1,2, …, n).

For any n parallel components i₁,i₂,…,i_nThe combined relationship is expressed as:

where j ═ is (1,2, …, n).

The probability of the system working properly is defined as:

R_s,k＝Pr{Y_s,k＝1} (20)

considering the uncertainty of the task length, the obtained system accumulated working probability is as follows:

wherein f is_k(t_k) Is a probability density function of the task length of the k-th task,andare respectively t_kUpper and lower limits of the values.

Further, the maintenance optimization model in step 5 is:

wherein N is the total number of tasks required to be executed by the system, C⁰Representing a total limited maintenance cost. In particular, if a completely new system performs the task, the reliability of the 1 st task is not affected by the maintenance, and the objective function in equation (22) is changed to

The invention has the beneficial effects that: by comprehensively considering the uncertainty of task time and the uncertainty of failure time in the multitask stage, the system decision maker can more reasonably and effectively optimize and configure limited maintenance resources in each maintenance interval so as to maximize the probability of normal work of the system in each task.

Drawings

FIG. 1 is a schematic flow chart of a system selective maintenance decision method for a multitasking stage according to the present invention;

FIG. 2 is a block diagram of the reliability of a coal delivery system to which an embodiment of the present invention is directed.

Detailed Description

The invention will be further described with reference to the accompanying drawings and specific embodiments, which are illustrated herein as a coal delivery system.

The coal delivery system is used to power a combustion chamber. The system has 5 subsystems, and 14 components are connected in series and parallel. The parts 1-3 constitute a feeder 1 for feeding coal to the conveyor 1, the parts 4-5 constitute the conveyor 1 for feeding coal to the stacker-reclaimer, the parts 6-8 constitute the stacker-reclaimer for feeding coal to the combustion furnace, the parts 9-10 constitute a feeder 2 for feeding coal to the conveyor 2, and the parts 11-14 constitute the conveyor 2 for feeding coal to the combustion chamber. All components are treated as one unit, and the failure time follows a Weibull distribution. Parameters of the weibull distribution of the components and the maintenance cost ($1000) are shown in table 1, respectively.

TABLE 1

All parts of the system execute 3 tasks in a brand new state, and the task length of each task is uniformly distributed Is the task length average of the kth task. The task lengths of the three tasks are equally divided intoEach part is provided withA feasible maintenance class,/_i,k1 andindicating respectively two maintenance classes, from maintenance class l, no maintenance and replacement_i,k2 toMaintenance costs are divided equally intoAnd (4) grading.

Given a total repair cost budget of C₀Here, the maintenance model can be solved by various metaheuristic algorithms, such as: genetic algorithms, particle swarm algorithms, and the like. The results of the one-time optimization of the 3 tasks and the division of the maintenance costs into two equal optimizations are compared separately, as shown in table 2.

TABLE 2

^*Optimal objective function value

As can be seen from table 2, the optimal objective function value obtained by the method is higher than the result of the optimization of dividing the maintenance cost into two equal parts. In conclusion, the invention not only comprehensively considers the uncertainty of the task time and the uncertainty of the failure time, but also performs combined optimization on the maintenance decisions of a plurality of maintenance intervals, and a system decision maker can more reasonably and effectively optimize and configure limited maintenance resources into each maintenance interval so as to maximize the completion probability of the system in each task.

It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (8)

1. A system selective maintenance decision-making method facing to a multitask stage specifically comprises the following steps:

step 1: defining the state and working age of each part of the system at the beginning of each task;

step 2: calculating the reliability of each part of the system in each task;

and step 3: calculating the normal working probability of each part of the system in each task;

and 4, step 4: calculating the normal working probability of the system in each task;

and 5: and establishing a maintenance decision optimization model and solving an optimal maintenance strategy by taking the task completion probability in each task of the system as an objective function, a limited feasible set of maintenance resources as a constraint condition and the maintenance behaviors of all parts of the system in each task as decision variables.

2. The multitask stage oriented system selective maintenance decision method according to claim 1, wherein the state of the component at the beginning of each task in the step 1 can be expressed as:

wherein k ═ {1,2, …, N } represents the kth task, and N represents the total number of tasks to be executed by the system;

the state of the component at the end of each task may be represented as:

the way of repairing the part is an imperfect repair, the imperfect repair model is a Kijima I model, the effective working age of part I at the start of the kth task is denoted x_i,kThe time for which the component i normally operates in the k-th task is set to u_i,k∈[0,t_k]，t_kIs the task length of the kth task, if X_i,k1, the effective service life of component i at the end of the k-th task is:

y_i,k＝x_i,k+u_i,k(1)

after repair, the effective service life at the start of the k +1 th task of part i is obtained as:

x_i,k+1＝x_i,k+b_i,ku_i,k(2)

wherein, b_i,k∈[0,1]Represents a service age backoff factor;

if Y is_i,k＝1(u_i,k＝t_k) Then y is_i,k＝x_i,k+t_i,k(ii) a If component iFailing in the kth task (i.e., u)_i,k<t_k) And no maintenance is performed, the part remains in a failed state (X) for the k +1 th task_i,k+10 and Y_i,k+10), in which case x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,k；

The repair grade adopted during the kth repair interval for component i is denoted as l_i,k∈{1,2,…,N_iIn which N is_iRepresenting the highest repair level, the relationship between the repair cost and the repair level for component i during the kth repair interval may be expressed as:

$C_{i,k}\left(l_{i,k}\right)=\left\{\begin{array}{cc}0& l_{i,k}=1\\ c_{i,k,l_{i,k}}+c_i^0& l_{i,k}\&Element;\{2,...,N_i-1\}\\ c_i^{rp}& l_{i,k}=N_i\end{array}\right.---\left(3\right)$

wherein,is indicated at maintenance level of l_i,kThe cost of the repair/preventive maintenance that follows,indicating the fixed disassembly and assembly costs of the component i,indicating the cost of replacing component i, in particular l_i,k＝{1,2,N_iMeans no service, minimum service and replacement, respectively, for component i during the kth service interval.

B is obtained from the relationship between the maintenance cost and the maintenance grade of the equation (3)_i,kIn relation to maintenance costs, if the component i fails before the end of the kth task, b_i,kThe relationship to maintenance costs is expressed as:

wherein,represents the maximum repair cost of the restorative repair, m_i,kIs a characteristic parameter.

If the component i is still working at the end of the k-th task, b_i,kThe relationship to maintenance costs is expressed as:

wherein,represents a maximum maintenance cost for preventive maintenance;

the average remaining life of component i at the end of the kth task is defined as:

$L_{i,k}=\frac{{\&Integral;}_{u_{i,k}}^{\mathrm{\&infin;}}{r}_{i,k}\left(t\right)dt}{r_{i,k}\left(u_{i,k}\right)}---\left(6\right)$

wherein r is_i,k(t) is the reliability of the component i, and the characteristic parameters of equation (5) are:

$m_{i,k}=\frac{L_{i,k}}{y_{i,k}}=\frac{{\&Integral;}_{u_{i,k}}^{\mathrm{\&infin;}}{r}_{i,k}\left(t\right)dt}{y_{i,k}{r}_{i,k}\left(u_{i,k}\right)}---\left(7\right)$

3. the method for selective maintenance decision-making of a system facing a multitasking stage according to claim 2, characterized in that the reliability of each part of the system in each task in the step 2 is calculated as follows:

if X_i,kIf 0, the reliability of component i in the k-th task is 0, x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,kIf X is_i,kIf 1, the reliability of component i in the k-th task is:

$r_{i,k}\left(t\right)=\exp \&lsqb;-{\&Integral;}_0^t{\mathrm{\&lambda;}}_i(x_{i,k}+s)ds\&rsqb;---\left(8\right)$

wherein λ is_i(x_i,k+ t) is the failure rate function for component i, t ∈ (0, t)_k]The timing starts from the system executing the kth task. If Y is_i,kWhen the k-th task ends, the reliability of the component i is r_i,k(t_k) Before and after maintenance the effective service life is y_i,k＝x_i,k+t_i,kAnd x_i,k+1＝x_i,k+b_i,kt_i,k. If Y is_i,k＝0，u_i,k∈(0,t_k) The right truncated data represents the effective service life before maintenance as y_i,k＝x_i,k+u_i,kIt can be expressed by a conditional probability density function as:

$\begin{array}{c}{f}_{u_{i,k}}\left(u_{i,k}\right)=-\frac{{\mathrm{dr}}_{i,k}\left(u_{i,k}\right)/{\mathrm{du}}_{i,k}}{r_{i,k}\left(t_k\right)}\\ =\frac{r_{i,k}\left(u_{i,k}\right){\mathrm{\&lambda;}}_i\left(y_{i,k}\right)}{r_{i,k}\left(t_k\right)}\end{array}---\left(9\right)$

the cumulative density function for any valid work age T at the end of the kth task is:

$\begin{array}{c}{F}_{y_{i,k}}\left(T\right)=\mathrm{Pr}\{y_{i,k}<T\}\\ =\mathrm{Pr}\{y_{i,k}<T-x_{i,k}\}\\ =F_{u_{i,k}}(T-x_{i,k})\end{array}---\left(10\right)$

wherein, Pr { y_i,k<T represents the effective service life y before maintenance_i,k<The probability of T is determined by the probability of T,is u_i,kTo obtain y_i,kThe probability density function of (a) is:

$\begin{array}{c}{f}_{y_{i,k}}\left(y_{i,k}\right)={\mathrm{dF}}_{y_{i,k}}\left(y_{i,k}\right)/{\mathrm{dy}}_{i,k}\\ ={\mathrm{dF}}_{u_{i,k}}(y_{i,k}-x_{i,k})/{\mathrm{dy}}_{i,k}\\ =f_{u_{i,k}}\left(u_{i,k}\right)\end{array}---\left(11\right)$

in the kth maintenance interval, non-intact maintenance is performed on component i, if b_i,k∈ (0,1), the effective service life of part i at the start of the k +1 th task is x_i,k+1＝x_i,k+b_i,ku_i,k，

x_i,k+1The cumulative density function of (a) is expressed as:

$\begin{array}{c}{F}_{x_{i,k+1}}\left(T\right)=\mathrm{Pr}\{x_{i,k+1}<T\}\\ =\mathrm{Pr}\{u_{i,k}<(T-x_{i,k})/b_{i,k}\}\\ =F_{u_{i,k}}\left(\right(T-x_{i,k})/b_{i,k})\end{array}---\left(12\right)$

to obtain x_i,k+1The probability density function of (a) is:

$\begin{array}{c}{f}_{x_{i,k+1}}\left(x_{i,k+1}\right)={\mathrm{dF}}_{x_{i,k+1}}\left(x_{i,k+1}\right)/{\mathrm{dx}}_{i,k+1}\\ ={\mathrm{dF}}_{u_{i,k}}\left(u_{i,k}\right)/d(x_{i,k}+b_{i,k}{u}_{i,k})\\ =\frac{f_{u_{i,k}}\left(u_{i,k}\right)}{b_{i,k}}\end{array}---\left(13\right)$

if b is_i,k1, then component i remains in the failed state, x_i,k+1And u_i,k+1Are respectively set as x_i,kAnd u_i,k；

If b is_i,kWhen 0, the component i is repaired to the working state, x_i,k+1＝x_i,kAnd is

Furthermore, x_i,1Probability density function ofKnown as x_i,1The upper and lower limits of the value are respectivelyAndtaking into account the effective working lifeThe cumulative reliability of component i at the end of the kth task is:

$R_{i,k}\left(t_k\right)={\&Integral;}_{x_{i,0}^{-}}^{x_{i,0}^{+}}{\&Integral;}_{x_{i,1}}^{x_{i,1}+b_{i,1}{u}_{i,1}}...{\&Integral;}_{x_{i,k-1}}^{x_{i,k-1}+b_{i,k-1}{u}_{i,k-1}}{r}_{i,k}\left(t_k\right)f_{i,k}\left(t_k\right)...f_{i,2}\left(x_{i,2}\right)f_{i,1}\left(x_{i,1}\right){\mathrm{dx}}_{i,k}...{\mathrm{dx}}_{i,2}{\mathrm{dx}}_{i,1}---\left(14\right)$

4. the multitask stage oriented system selective maintenance decision method according to claim 3, wherein the calculation of the probability that each component of the system will work properly in each task in step 3 requires consideration of multiple state paths of the component. Each time a task ends, component i may be in a working or failure state, thus there is 2 from task 1 to task k^kA different path, wherein there is 2^k-1The strip path may result in Y_i,k1, and then the component i is found at the kth task end_pThe working probability under a path is:

$R_{i,k}^{k_p}=\underset{j=1}{\overset{k}{\&Pi;}}\&lsqb;Y_{i,j}{R}_{i,j}\left(t_j\right)+(1-Y_{i,j})(1-R_{i,j}(t_j\left)\right)\&rsqb;---\left(15\right)$

wherein k is_p＝1,2,…,2^k-1；

The cumulative working probability of component i at the end of the kth task is:

$R_{i,k}^C=\underset{k_p=1}{\overset{2^{k-1}}{\&Sigma;}}{R}_{i,k}^{k_p}---\left(16\right)$

5. the multitask stage oriented system selective maintenance decision method according to claim 4, wherein the probability of the system working normally in each task in the step 4 can be obtained from the accumulated working probability of the system structure and the components. The system state may be derived from the state of the component:

Y_s,k＝φ(Y_1,k,Y_2,k,…,Y_M,k) (17)

where φ (-) is the structural equation for the system and M is the total number of components that make up the system.

6. The multitask stage oriented system selective maintenance decision method according to claim 5, wherein for any n number of cascaded components { i } i₁,i₂,…,i_nThe combined relationship is expressed as:

$\mathrm{\&phi;}(Y_{i_1,k},Y_{i_2,k},...,Y_{i_n,k})=\left\{\begin{array}{cc}1& \&ForAll;Y_{i_j,k}=1\\ 0& \&Exists;Y_{i_j,k}=0\end{array}\right.---\left(18\right)$

where j is (1,2, …, n),

for any n parallel components i₁,i₂,…,i_nThe combined relationship is expressed as:

$\mathrm{\&phi;}(Y_{i_1,k},Y_{i_2,k},...,Y_{i_n,k})=\left\{\begin{array}{cc}1& \&Exists;Y_{i_j,k}=1\\ 0& \&ForAll;Y_{i_j,k}=0\end{array}\right.---\left(19\right)$

the probability of the system working properly is defined as:

R_s,k＝Pr{Y_s,k＝1} (20)

the obtained system accumulated working probability is:

$R_{s,k}^C={\&Integral;}_{{\mathrm{\&tau;}}_k^{-}}^{{\mathrm{\&tau;}}_k^{+}}...{\&Integral;}_{{\mathrm{\&tau;}}_2^{-}}^{{\mathrm{\&tau;}}_2^{+}}{\&Integral;}_{{\mathrm{\&tau;}}_1^{-}}^{{\mathrm{\&tau;}}_1^{+}}{R}_{s,k}{f}_1\left(t_1\right)f_2\left(t_2\right)...f_k\left(t_k\right){\mathrm{dt}}_1{\mathrm{dt}}_2...{\mathrm{dt}}_k---\left(21\right)$

wherein f is_k(t_k) Is a probability density function of the task length of the k-th task,andare respectively t_kUpper and lower limits of the values.

7. The multitask stage oriented system selective maintenance decision method according to claim 5 or 6, characterized in that the maintenance optimization model in the step 5 is:

$\begin{array}{c}\max \min \{R_{s,1}^C,R_{s,2}^C,R_{s,3}^C,...,R_{s,N}^C\}\\ \begin{array}{cc}s.t.& \underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}{c}_{i,k,l_{i,k}}\&le;C^0\\ & c_{i,k,l_{i,k}}\&le;c_i^{rp}\\ & -c_{i,k,l_{i,k}}\&le;0\\ & i=1,2,...,M;k=1,2,...,N\end{array}\end{array}---\left(22\right)$

wherein N is the total number of tasks required to be executed by the system, C⁰Representing a total limited maintenance cost.

8. The method as claimed in claim 7, wherein if a new system performs the task, the reliability of the 1 st task is not affected by the maintenance, and the objective function in equation (22) is recorded