Performing continuous tasks is an inherent feature of military equipment and other complex systems, so to meet the required reliability of continuous tasks, it is necessary to arrange maintenance work scientifically during the mission interval of equipment [1-2]. Constrained by time, limited available resources, maintenance costs, and other factors, the amount of feasible maintenance actions set are finite [1-4]. It is obvious that when the task interval length is limited, a series of selective maintenance actions must be chosen to maximize the reliability of continuous tasks, while the time consumption can meet the constraint. Then, the continuous missions can be guaranteed to the maximum degree.

The decision content of selective maintenance mainly includes the selection of the maintenance subjects and the corresponding maintenance actions. Some research has been done on selective maintenance. Bris [1] assumed that all components of the system were subject to random failure and that replacement was the only maintenance option. Aiming at minimizing maintenance costs and meeting the system availability requirement, a selective maintenance decision-making method for complex system was put forward. Wildeman [5] and Yao [6] proposed a combinational maintenance decision making model for multi-component systems by taking replacement and minimal repair as alternative actions. Cassady [7] et al. supposed the components follow the Weibull distribution[8] and the available maintenance time is limited. They established a decision-making model aiming at the maximization of mission reliability, while considering minimal repair and replacement as optional actions. Moghaddam [9] took the running age into account and proposed a selective maintenance decision model aiming at maximizing the reliability of the system under limited maintenance costs. Pandey [10] assumed that the system components follow the exponential distribution and established maintenance decision models using different solving algorithms. However, the exponential distribution does not fit most mechanical systems, and optimal maintenance decision-making at the component level might lead to the loss of overall benefit for the system, so the model requires further research. In fact, maintenance usually cannot recover the condition of complex systems to brand-new; many maintenance actions are in fact imperfect. The effect of imperfect maintenance should be considered in the models. Modeling of imperfect maintenance is helpful for enhancing the modeling accuracy of system conditions and engineering application prospects. Moghaddam[11] analyzed the effect of imperfect maintenance on equivalent age but did not illustrate the relationship between the age and the maintenance soundly. To deal with this disadvantage, Pandey [8] established an equivalent age transition model reflecting the effect of imperfect maintenance on actual running time, and a single selective maintenance decision method was studied. Mayank [12] took imperfect maintenance into account and proposed a sequential selective maintenance decision method during a finite time span for systems composed of multiple components, aiming at the optimization of task reliability. A preventive maintenance frequency optimization model through given life spanwas also presented. However, the optimization model took the maintenance decision-making as a single maintenance decision problem, which made the model lack flexibility. Making real-time maintenance decisions according to the real-time system status was the key problem. Also, this model failed to differentiate preventive maintenance and corrective maintenance consequences, which affected the utilization of the model. Based on the fact that the system state updatesalong with the execution process of the continuous tasks, to improve the selective maintenance decision-making efficiency influenced by imperfect maintenance, a maintenance action set composed of minimal maintenance, imperfect repair, preventive replacement, and corrective replacement was proposed as a maintenance option. An optimal maintenance decision-making model was proposed for system performing mission in series, and the solving algorithm was also studied.

Complex large-scale systems usually can be regarded as complex series-parallel systems. Assume one system is composed of m subsystem connected in series, and the subsystem i(i =1,2, ⋯,m) is composed of n_i components connected in parallel. Then, the sum number of the component in the system is N=$\sum\limits_{i=1}^{m}{{{n}_{i}}}$. Assume the life of each component in the system follows the Weibull distribution:

$f(t)=\left( \left( \beta {{t}^{\beta -1}} \right)/{{\alpha }^{\beta }} \right){{e}^{-}}^{{{(t/\alpha )}^{\beta }}},\text{ }t\ge 0,\text{ }\beta >0,\text{ }\alpha >0$

The states of the components and the system are identified as functioning or failure, and maintenance can recover the state to functioning from failure. Suppose the ages of all components are 0 at the beginning. The system performs missions in sequence. The mission interval and followed task interval are supposed to be identical, and the two parameters of the k^th mission are denoted as O_k and M_k respectively. The executive process of continuous missions is given in Figure 1.

The consumption of time and resources and the recovery degree are determined according to maintenance actions. Among the actions, minimal repair, one kind of corrective maintenance, consumes the least amount of time and cost to recover the state but cannot enhance the reliability. The effect of imperfect repair, regardless of whether it ispreventive repair or corrective repair, usually goes between minimal repair and replacement, and the money and time consumption amount usually goes between minimal repair and replacement. It is reasonable to suppose that the repair degree is in positive correlation with the amount of money and time invested. Preventive replacement and corrective replacement both can recover the state to brand new, but corrective maintenance requires relatively more money and time for relatively more logistic time.

During one task interval, maintenance can change the state of the components and the task reliability of the maintenance subjects. Based on the structure of the system, the components’ identification number is coded in sequence, and the s^th component is marked as s (1<s<$\sum\nolimits_{i=1}^{m}{{{n}_{i}}}$). Let k represent the accumulated task number and X_s(k) and Y_s(k)the condition of component s before and after maintenance respectively, then

${{X}_{s}}(k)=\left\{ \begin{align} & 1;\text{ component}s\text{available}\text{before}\text{the}{{k}^{\text{th}}}\text{mission} \\ & 0;\text{ component}s\text{failure}\text{before}\text{the}{{k}^{\text{th}}}\text{mission} \\ \end{align} \right.$

In the equation, 0 means failure and 1 means functional.

When the k^th mission is finished, the condition of component s can be presented as Y_s(k):

${{Y}_{s}}(k)=\left\{ \begin{align} & 1;\text{ component}s\text{available}\text{after}\text{the}{{k}^{\text{th}}}\text{mission} \\ & 0;\text{ component}s\text{failure}\text{after}\text{the}{{k}^{\text{th}}}\text{mission} \\ \end{align} \right.$

Also, the state of the subsystem and the system can be presented with the same equations.

According to section 1.1, the maintenance actions set is {minimal repair, imperfect repair, preventive replacement, corrective replacement}, and the imperfect repair can be distinguished into more specified types. The elements in the set are not always compatible, and the feasible actions rely on the state of maintenance subject. As for component k, if Y_s(k)=1, no failure happensand the feasible actions only include imperfect maintenance and preventive replacement. IfY_s(k)=0, the component fails and the feasible options are minimal repair, imperfect repair, and corrective replacement [12].

Let l_s(k) represent the maintenance action variable of component s in the k^th maintenance. Then, it is necessary to assign unique constant values to represent specified actions. To make the model easier to understand, the value assignment rule is determined and shown in Table 1.

It is shown in Table 1 that the codes of maintenance actions increase from 0 to p_s to represent different maintenance actions. If l _s(k)=1, the minimal repair is chosen, the fault is corrected with minimal cost, and the reliability before and after maintenance stays unchanged. It is necessary to point out that preventive maintenance is always better than minimal repair. By cleaning, adjusting, and correcting, the reliability of the maintenance subject can be partially promoted without replacement, and these kinds of maintenance actions all can be classified as imperfect maintenance. If the value of l _s(k) is between 2 to p_s-2, one imperfect maintenance action is chosen. The higher thel _s(k), the higher the recovery degree.

There are currently two popular imperfect maintenance models. One is the age reduction method, which modifies the equivalent age with age reduction parameterb and changes the reliability consequently. Assuming the age before repair is t, the age will reduce by bt (0≤b≤1). The higher theb, the higher the reliability. The other modelmodifies the failure rate function λ(x), and after maintenance the failure rate function will be recovered to aλ(x) (x>0, 0≤a≤1) from λ(x) (x>0). The lower thea, the higher the reliability. The two models can present the maintenance effect from different aspects.

It is necessary to note that increasing maintenance frequency does not necessarily lead to better performance. Besides the influence of imperfect maintenance, it is necessary to consider possible flaws resulting from imperfect repair.

To better describe the effect of imperfect maintenance and the chances of flawed maintenance, an integrated model is proposed. The failure rate change sketch-up map is presented in Figure 2.

As shown isFigure 2, the first and second maintenance actions can reduce the failure rate to different degrees, and the decrease scale is predetermined according to maintenance actions. At the same time, the flaw brought in by imperfect maintenance will accumulate. Judging from the trend of the curve in Figure 2, the slope of the failure rate will get relatively higher along with the increment of maintenance times. As a result, the proposed model can better describe the actual characteristics of maintenance actions.

During the execution of sequential missions, assume the finish time of the first mission is t₁ and the start time of the second mission is t₂. Then, during the second mission, the failure rate function of component s can be calculated by Equation (1)[13]:

In Equation (1), a is the correction parameter, b is the age reduction factor, λ_s,0(·) is the failure rate function of component s during the first episode, λ_s,1(·)the second episode, and so on. It is clear that the age of component s can be reduced to bt₁ from t₁ after imperfect maintenance, and a≥1can make the slope of the curve increase in Figure 1 to represent the potential negative influence of maintenance.

Before replacement, the failure rate function is updated with every maintenance. Suppose the correction parameters and age reduction factors are all recorded along with the failure rate function updates, and the records until the k^th maintenance respectively are(a_s,1, a_s,2, ⋯, a_s,k) and (b_s,1, b_s,2, ⋯, b_s,k). Then, the failure rate function of the (k+1)^th mission is:

In the equation, t_k+1 is the start time of the (k+1)^th mission and ${{A}_{s}}_{,k}=\prod\limits_{p=1}^{k}{{{a}_{s,p}}}$ is the accumulated correction parameter of the failure rate function. The more maintenances are done, the higher the A_s,k, so redundant maintenance are not necessary; B_s,k is the equivalent age after the k^th maintenance, and the influence on the equivalent age is also accumulated. The equivalent age is calculated as follows:

As shown in Figure 3, after the (k-1)^th maintenance, the equivalent age is reduced to b_s,k-1B_s,k-1 from B_s,k-1. It is obvious that O₁, O₂, ⋯, O_k-1 are all affected by b_s,k-1. The model can depict the influence of all kinds of maintenance actions more precisely.

The value of b mainly is determined by the ageand the repair expenses. The more money and time invested in maintenance, the smaller the b and the better the condition after maintenance. For a given maintenance action, the lower the age, the smaller the band the better the condition after maintenance. The calculation measure of reference [12] is opted, and the age reduction parameter of the k^th maintenance is calculated as follows:

In the equation, ${{c}_{s}}_{,k}({{l}_{s}})$ is the cost determined by maintenance action l_s,${{c}_{s}}_{,R}$is the replacement cost of component s, and$m({{B}_{s,k}})$ is the characteristic index of the age. The value of $m({{B}_{s,k}})$ can be calculated by:

In the equation, ${{B}_{s}}_{,k}$is the equivalent age and${{R}_{s}}_{,k}\left( {{B}_{s}}_{,k} \right)$ is the corresponding reliability. If the reliability of the component follows the Weibull distribution:

To substitute Equation (6) into Equation (5):

It is shown in Equation (4) that $m({{B}_{s,k}})>0$. The smaller the $m({{B}_{s,k}})$, the smaller the equivalent age after maintenance.The higher the $m({{B}_{s,k}})$, the higher the equivalent age.

Similar to the age reduction factor, the definition of the correction parameter is denoted as ${{a}_{s}}_{,k}({{B}_{s}}_{,k},{{l}_{s}})$:

In the equation, q is a constant value predetermined by experience. The higher the happening rate of flawed maintenance, the smaller the q.

To analyze from the reliability aspect, all maintenance actions can be taken as imperfect maintenance. In Equation (1), if a=1 and b=1, the age stays unchanged and the maintenance can be taken as minimal repair. If a=1 and b=0, the condition after maintenance can be taken as brand new, like replacement.

The aim of maintenance is to maximize the task reliability of the next mission, so it is vital to make the most optimistic and feasible selective maintenance decision to meet the requirement. The recovery effect is straight determined by the action, and the failure rate function during the k^th mission is:

In the equation, l_s,k-1 is the code of the (k-1)^th maintenance action. If ${{l}_{s,k-1}}\in \{0,1\}$, minimal repair or no repairis doneand the failure rate stay unchanged; if ${{l}_{s,k-1}}\in \{2,\cdots ,{{p}_{s}}-2\}$, imperfect maintenance is done; if ${{l}_{s,k-1}}\in \{{{p}_{s}}-1,{{p}_{s}}\}$, the component is replaced and the equivalent age is turned to 0 as a result.

According to Equation (9), the task reliabilityof components during the k^th task is:

The system reliability is:

During continuous missions, the decision-making object is to minimize the cost while satisfying the constraint related to time and reliability. Judging from Equations (10) and (11), the reliability during the first task is determined only by the length of mission interval given the reliability model.

As for complex systems performing tasks in series, time is often the major constraint. Therefore, based on the practical need, a maintenance decision-making object model wasbuilt.

All maintenance actions cost a certain amount of resources and time, and the detailed mapping relationship should be analyzed respectively. Assume the replacement time is constant. Compared withother maintenance actions, replacement usually costs a certain amount of logistic delay time, so it is reasonable to assume that replacement time is higher than other types of actions. Corrective maintenance normallycosts relatively more time than preventive maintenance. The maintenance time isseparated into fixed maintenance time and maintenance time related to maintenance depth. Fixed maintenance time is the time cost by basicactions such as disassembling, installing, lubricating, and adjusting, and the length of fixed time is assumed to be constant. During one maintenance, the more money and time that arespent, the better the maintenance effect.

The time cost for component s during the k^th maintenance is ${{T}_{s}}_{,k}\left( {{l}_{s}}_{,k} \right)$:

In the equation, ${{l}_{s}}_{,k}$ is the code of the k^th maintenance action for component s. If ${{l}_{s}}_{,k}=0$, no maintenance is selected and no time will be spent; if $1\le {{l}_{s}}_{,k}\le {{p}_{s}}-2$, minimal repair or imperfect maintenance is selected, and ${{t}_{s}}_{,k}({{l}_{s}}_{,k})$ is the maintenance time determined by maintenance depth; if ${{l}_{s}}_{,k}={{p}_{s}}-1$, preventive replacement is chosen and ${{t}_{s,pr}}$ is the time consumed; if ${{l}_{s}}_{,k}={{p}_{s}}$, corrective maintenance is chosen and t_s,cr is the corresponding time.

During one task interval, the overall maintenance time can be calculated with Equation (13), and the k^th maintenance time T(k) is:

In the equation, N=$\sum\nolimits_{i=1}^{m}{{{n}_{i}}}$, so N is the sum of the maintenance time spent on each component referred.

Assuming that all components are newand according to the aim determined before, the objective is to maximize the task reliability during the following episode. To denoted the gap time left for the k^th maintenance by M_k, the objective function is:

Equation (14)aims to maximize the reliability at the beginning of the next mission after the k^th maintenance, and Equation (15) is set to make sure the summed maintenance time is less than M_k.

At the same time, the chosen action for each maintenance subject should be a feasible solution related to the observed real-time condition. The constraints about the feasible solutions are listed in Table 2.

In Table 2, each available value of l_s(k) is feasible only when the value of Y_s(k) belongs to the set listed in the table. Taking the second column forexample, l_s(k)=1 is feasible only when Y_s(k)=0. It is easy to understand that minimal repair can be selected only if the component is in fault. After maintenance, the component can be turned into a functional state. The constraints shown in Table 2 are necessary to avoid unreasonable decisions.

According to the model proposed above, the regular linear programming method is running short of dealing with multiple real-time conditions and constraints, so it is necessary to prefer intelligent optimization algorithms thatare more adaptive and have stronger portability. The common intelligent optimization algorithms are the genetic algorithm and the particle swarm algorithm. The genetic algorithm is selected to solve the model proposed above.

As illustrated in section 1, the maintenance action set for every component is {0,1,2, ⋯, p_s}, and each time the decision-making chooses one code from the set. According to the feature, a genetic chromosome with length of N is constructed, and each position stores the maintenance action code for one appointed component in the system. Therefore, the range of each position in the chromosome is the set {0,1,2, ⋯, p_s}. According to the definition, the binary or real number coding measures are not feasible, so it is necessary to design a special code to perform crossover and mutation of the chromosome, so that the chromosomes generated from the last generation can be feasible [14].

The solving flow of the decision-making optimization modelis presented in Figure 3.

As shown in Figure 3, the components’ age are all 0 and the system is in perfect condition at the beginning. To push forward simulation time until the end of the current task, the equivalent age of all the components can be updated and the state of each component is determined randomly. By inputting the characteristic parameters into the genetic algorithm model, the best solution for the system can be solved. To perform the scheduled best maintenance plans and update the state of components, the simulation time can be pushed forward, and the decision-making process can be performed again. When the preset conditionis met or the simulation time is reached, the simulation should be terminated.

To verify the sequential optimal selective maintenance decision-making model, a mechanical system is taken as the subject for case study. The structure of the system is shown as Figure 4.

As shown in Figure 4, the system is composed of three subsystems connected in series, and each subsystem has different components. The reliability function and parameters related to maintenance are listed in Table 3.

Assume that the system is performing four tasks in series with time gap between tasks. Regardless of the mission number, O_k=100 days and M_k=6 days, and the value of p is also the same for every component,p=6. The condition updating process of the components in the system is simulated by the Monte Carlo simulation method. Along with the simulation steps, the optimal decisions are made for maintenance, and the conditions are updated accordingly. Due to the random properties of failures, selective maintenance should comply with the real-time situation. The state transition and corresponding maintenance decision-making result during one simulation process is shown in Table 4.

In the table, k is the number of tasks, Y_s(k) is the state vector after the k^th task, L(k)is the optimal maintenance decision code vector, T(k) is the maintenance time used for the k^th maintenance, X_s(k+1) is the state of component s at the beginning of the (k+1)^th task, and R(k +1) is the expected task reliability in the (k+1)^thtask.

As shown in Table 4, given the real-time age and state of the system, the optimized maintenance decisions L(k)are made accordingly using the model proposed in the paper, and the task reliability of the next mission episode can reach 0.9365, 0.9470, and 0.9276 respectively. Given the task interval as 6 days, each maintenance cost 5.6 days, 5.5 days, and 5.4 days, so the constraints are all met.

Theoretically, imperfect maintenance can offermore options for decision makers by supplying more combined maintenance decisions at the component level, so the reliability can be promoted. To analyze the effect of imperfect maintenance on the whole system, the imperfect maintenance actions are eliminated from the action set, so the available set is changed to {minimal repair, preventive replacement, corrective maintenance}. The simulation process is run while the other situations are the same. Set the task cycle number to 15 and the simulation repentant times 40. The result is shown in Figure 5.

As shown in Figure 5, maintenance with imperfect maintenance pushes the reliability relative higher, and when the gap between the two curves turns stable, the precise distance is near 0.03. The comparison proved that imperfect maintenance can enrich the decision-making spaces and make the task reliability higher.

If the time and cost constraints do not exist, all the components can be replaced so that the reliability can be kept at the ideal level. In the case, the reliability can reach 0.9558, which means that all task reliability can be kept to 0.9588 regardless of the cycle number. If the gap time between tasks is shortened, the time left for maintenance will be limited and the decision-making space will be constrained. The less time that is left for repair, the lower the chance to perform deeper maintenance, and the task reliability will be lowered as a result. To verify the hypothesis, the related parameter is changed and the simulation process is run multiple times. The task cycle numberis set to 10 and simulation times 40. The reliability of the system is shown in Figure 6.

It is shown in Figure 6 that along with the shortened maintenance interval, the descend trend of reliability will be more and more obvious. When M_k∈{6, 7, 8}, the task reliability can turn stable gradually above 0.86 with the accumulation of task cycles. When M_k≤5, the reliability will drop more rapidly, and when M_k=3, the 10^th task reliability will drop to 0.5275 and the task will be difficult to perform. The simulation result can offer information from a different aspect. If the reliability threshold is 0.85, then the maintenance interval length should be at least above 6.

The age reduction factor and correction parameter introduced in Equation (1) can help reflect the real situation with more accuracy. To analyze the effect of the two factors, eliminate one of the two factors. An age reduction model is gained by eliminating the correction parameter, and a risk correction model is gained by elimination of the age reduction factor. The comparison result is shown in Figure 7.

The combined model proposed in the paper does not reduce the age back to zero, and the flawed maintenance chance is also depicted by the correction parameter. Compared with the other two models, this model should be lower at a given time node. The curves in Figure 6 verified the hypothesis, and in this case, the gap between the proposed model and the other two are between 0.02 and 0.04, so it is reasonable to believe that the contribution of the two factors is considerable. In practice, if the two factors do exist, the proposed model in the paper should be applied so that the reliability would not be optimistically estimated in excess, or else the execution of the task might be affected due to the lack of preparation.

To better demonstrate the effect of maintenance and help make more practical selective maintenance decisions, the influence of imperfect maintenance on the system is analyzed. A decision-making model aiming at the maximization of task reliability is proposed considering the time constraints. The case study result showed that the introduction of imperfect maintenance is helpful for offering more feasible solutions so that the reliability can be promoted.

As for complex systems such as military equipment, during the execution process of missions, the selective maintenance decision-making is constrained not only by time but also by spares, tools, and so on, so more complicated cases will be further researched.