A New Maintenance Optimization Model Based on Three-Stage Time Delay for Series Intelligent System with Intermediate Buffer

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Introduction
Intelligent series system is an important part of modern industrial manufacturing system. Due to the variety of machines and complex layout and structure, any failure may lead to the shutdown of the entire production line and cause huge economic losses for enterprises. For the continuous series production line, the preventive maintenance can also cause downtime.
Reasonably adding buffer between two machines can improve the flexibility of the production line, reduce the production dependence between upstream and downstream machine, and reduce the impact on the stability of series system due to machine downtime. e performance of the intelligent series system is closely related to its preventive maintenance plan and buffer setting. Preventive maintenance is related to the buffer stock allocation. In order to improve the production line stability and reduce the cost, it is very necessary to jointly optimize the series system buffer stock allocation and maintenance plan of machines.
Machines in the intelligent series system are closely connected, and the failure of one machine will lead to the shutdown of the entire intelligent series system. Recently, there are many literature works on the optimal preventive maintenance strategy of the series system. Wu et al. [1] established an optimized maintenance cost model and determined the optimal condition monitoring interval and the degradation level after imperfect preventive maintenance. e authors in [2] developed a dynamic maintenance strategy joint optimization problem that integrates production and opportunity maintenance. Rooeinfar et al. [3] studied the scheduling problem of uncertain flexible pipeline with finite buffer. Zhang et al. [4] investigated the incorporation of balance and preventive maintenance in U-shaped assembly line. Two metaheuristic algorithms including elitist nondominated sorting genetic algorithm and multiobjective simulated annealing algorithm were designed to solve this problem. e authors in [5] studied the integrated control of dynamic maintenance and production in a deteriorating manufacturing system and proposed a dynamic maintenance strategy that included corrective, preventive, and opportunistic maintenance. Bouslah et al. [6] discussed the integrated production, quality, and maintenance control of the production line. Motlagh et al. [7] developed an expert system for the unreliable unbalanced production line in reality, in which all time-based parameters were random. Considering a series production system with random degradation, Wang et al. [8] proposed a predictive maintenance strategy based on the predicted failure probability of each machine and a production control strategy based on the target service level, so as to meet the dynamic stochastic demand in each period. Wang et al. [9] proposed two maintenance strategies for the series production line.
e first was based on the cost rate of the machine under long-term operation, and the second was based on the single-piece machine maintenance strategy considered in the production line. e authors in [10] proposed an alternative scheduling model for railway production lines and proposed a time-based flexible displacement (FTBR) method in combination with the artificial bee colony (ABC) algorithm. Based on the concept of "energysaving opportunity window," [11] modeled the continuous deterioration process of each machine and regarded the energy-saving opportunity window of the production system as the opportunity window of preventive maintenance. However, the above literature only proposes preventive maintenance strategies for series system not combined with buffer zones.
Buffers have been used in maintenance of machines for a long time. e authors in [12] first proposed buffer stock, considering the impact of the interstage buffer on production at different production speeds, different failure rates, and different repair rates. e method of using regeneration point for analysis and treatment was given. Currently, a large number of scholars study the optimal allocation of buffer. e authors in [13] presents a model to determine the optimal length of continuous production periods between maintenance actions and the optimal buffer inventory to satisfy demand during preventive maintenance or repair of a manufacturing facility. On the basis of this model, the authors in [14] considers that the opportunities for the fabrication of the buffer inventory and opportunities to carry out a maintenance action to the production facility are random. e authors in [15] proposed a multiobjective mathematical formula and hybrid method, which can simultaneously solve the buffer size and machine allocation problems on unreliable production lines and assembly lines with general distributed time-varying parameters. e authors in [16] proposed a tabu search algorithm to find the optimal buffer allocation plan for a serial production line composed of unreliable machines. e authors in [17] considered an imperfect production system with preventive maintenance activities in order to obtain the optimal buffer stock and minimum warranty inspection policy for sold products. e production system has the probability of changing from the normal state to the out-of-control state at any time. Under the normal state and out-of-control state, the production system will produce a certain proportion of defect items. e authors in [18] developed a method to analyze the complex tradeoff between the preventive maintenance and the buffer's contribution to system performance, considering a two-machine continuous manufacturing system with a finite capacity buffer. e authors in [19] analyzed the tradeoff between buffer capacity, spare parts inventory, and throughput for a two-stage production system with buffers and established a discrete-time Markov chain for two different situations. e numerical examples showed that the effect of a spare part on the efficiency of a transfer line was much greater than the effect of additional buffer places. All of the above research is aimed at two-stage system and does not combine buffer with intelligent series system.
ere are few studies on the combination of maintenance and buffer with series system. Nahas [20] considered an unreliable serial production line.
e target was to minimize the total cost of the system through finding the optimal preventive maintenance strategy and optimal buffer size at a given level of the system throughput. Extend the flood algorithm was put forward in order to solve this problem. e decomposition approximation method was used to estimate the production capacity of production line. Zandieh et al. [21] studied buffer and preventive maintenance cycle allocation issues. e model was built with three objective functions: the maximization of production rate, the minimization of buffer size, and the total number of defect units. Finally, a synthetic simulation method and a metaheuristic algorithm were used to solve the model. Lopes [22] minimized the total cost of each product while considering product quality testing for production lines with buffers. According to the machine degradation stage and buffer level, Kang and Ju [23] used Markov decision model to obtain the optimal maintenance strategy of the machine in the production line. Alfieri et al. [24] used the approximated mathematical programming formulation of the Buffer Allocation Problem (BAP) simulation-optimization based on the time buffer concept. However, in these studies, the preventive maintenance strategy of the series system is considered as a whole, and it is not disassembled to study each machine in the production line. Preventive maintenance strategy may not be adapted to each machine in the production line.
In this paper, the buffer is added to the intelligent series system to jointly optimize the buffer stock and the optimal preventive maintenance cycle. e approximate decomposition method is used in this paper to decompose the intelligent series system into several virtual series systems with two machines and a buffer. e approximate decomposition method is used to study the problem of production line preventive maintenance, which was first proposed by [25]. After that, it was widely used. Based on the approximate decomposition method, [26] presented an efficient method to evaluate performance of tree-structured assembly/disassembly (AD) systems with finite buffer capacity. For mixed-model flexible transfer lines, [27] proposed a general simulation model. Compared with the approximate decomposition method, the numerical results show that the proposed method is robust for predicting the throughput of transmission lines. Li et al. [28] proposed a common model that unifies several approximate methods for the analysis of tandem queueing systems with blocking. Xia et al. [29] proposed an efficient decomposition method based on a generalized exponential distribution to analyze the homogeneous transfer lines with unreliable buffers. e authors in [30] developed three heuristic approaches to solve the formulated combinatorial optimization problem. To estimate the production line throughput, an approximate decomposition method was used. Xia et al. [31] decomposed the original long line into several small decoupled subsystems and added relation condition variables between the subsystems. Bai et al. [32] proposed a new aggregation-based iterative algorithm to calculate the performance metrics of a multimachine serial line by representing it using a group of virtual two machine lines. In this paper, based on the approximate decomposition method, the influence factors are introduced according to the importance of different machines in the series system to obtain the optimal maintenance strategy for each machine.
Another innovation of this paper is to introduce the time delay theory into the maintenance model. Many studies have presented the preventive maintenance strategies by considering buffer stock. However, the machine's degradation is not considered in this area. In this paper, the preventive maintenance strategy can be developed by introducing the time delay theory to simulate the degradation process. e time delay theory is often used to simulate the machine degradation process. e time delay model proposed by Christer and Waller [33] is the first time to extend the time delay theory to the maintenance of industrial plants. e basic model of inspection and maintenance and some change models observed in practice are presented. Later, a large number of scholars applied the time delay theory to the field of establishing the correlation between machine maintenance cost and preventive maintenance inspection interval cycle. Wang [34] proposed such a model for a serviceable onecomponent system to jointly model the effect of RS and inspection with replacement on the basis of the delay-time concept. Zhao et al. [35] developed a model to evaluate the reliability and optimized the inspection schedule for a multidefect component. Gomes da Silva and Lopes [36] simulated the preventive maintenance model based on the nonhomogeneous Poisson distribution. Mahmoudi et al. [37] studied the occurrence process of machine defects presented as homogeneous Poisson distribution, and then solved the optimal maintenance cycle of preventive maintenance strategy. Based on the traditional time delay theory, Wang et al. [38] introduces a two-level inspection policy model for a single component plant system based on a three-stage failure process. ey divided the machine failure into three states: original defect, serious defect, and fault, so as to simulate the fault random process. By applying the three-stage time delay theory to the simulation of machine degradation and renewal process, the clustering summary of different kinds of faults or defects that may occur in the machine can be performed in a more precise and quantitative manner.
In this paper, first, the intelligent series system is decomposed into n − 1 virtual series systems by approximate decomposition method, and one virtual series system includes two virtual machines. e aim is to find the relationship between machines and present the cost ratio and maintenance ratio model by considering an influence factor. en, for each two virtual machines, the buffer stock and machine monitoring time can be described as decision variables, and the total cost rate can be minimized as the objective function. A novel maintenance model based on three-stage time delay is developed to obtain the optimal preventive maintenance strategy and the buffer stock. e proposed model can be divided into four types, and it contains the whole process of machine degradation based on different machines status and monitoring time. Finally, the proposed model is compared with the maintenance model based on two-stage time delay by a case study. e overall optimal maintenance strategy and buffer stock are obtained. In the case study, this paper analyzes an intelligent series system of Shanghai Pangyuan Machinery Co. e remainder of the paper is as follows. In Section 2, a description of general problem is presented. In Section 3, the notation and assumption are presented. e maintenance rate and cost ratio model of virtual machine and the preventive maintenance model based on the three-stage time delay theory are introduced in Section 4. In Section 5, solving method is introduced. In Section 6, numerical examples are presented and analyzed. Finally, the most important results and future work are summarized in Section 7.

Problem Description
e intelligent series system L includes n machines and n − 1 buffers. It is shown in Figure 1. We assume that M 1 is never starved and the supply of raw materials needed by M 1 is continuous. M n is never blocked, and the final product produced by M n does not have a backlog state. e failure rate λ i and maintenance rate μ i of each unit are known and defined as known conditions. e approximate decomposition method is used to decompose L into n − 1 virtual series systems, and each virtual series system includes two machines and one buffer.
After L is decomposed, n − 1 virtual series systems are formed. M u and M d are virtual machines of one virtual series system after decomposition. M u is the upstream machine, and semifinished product m 1 from M u is input into downstream machine M d at the production rate β through the intermediate buffer B. M d uses m 1 as the raw material to produce product m 2 at β. B needs to be accumulated before maintenance actions adopted by M u to ensure continuity of production process of the virtual series system. e traditional time delay model divides the health status into defect state and fault state. e three-stage time delay model divides the health status into normal state, original defect state, serious defect state, and fault state, as shown in Figure 2. us, the degradation process can be divided into original defect time, serious defect time, and failure time.

Shock and Vibration 3
After the upstream machine runs a certain period, B can be added by the replenishment rate α until it reaches buffer stock level S. en, status monitoring can be executed on the upstream machine. If the monitoring result shows that it is in initial defect state or serious defect state, preventive maintenance needs to be adopted immediately. If it is stopped due to failure before monitoring, then fault repair can be performed.

Notation
replenishment rate of buffer β: production rate of L X: machine normal operation phase Y: machine original defect operation phase Z: machine serious defect operation phase f x (x): probability density function from initial state of one machine to occurrence of original defect f y (y): probability density function from original defect state of one machine to the occurrence of serious defect f z (z): probability density function from serious defect state of one machine to the occurrence of failure

Assumptions
(1) e machine status needs to be monitored after T, and monitoring time can be ignored.   Shock and Vibration (2) All the states can be accurately monitored. e corresponding maintenance actions can be adopted immediately after monitoring. Production can be resumed immediately after completing maintenance.
(3) Extra production capacity is always available in order to produce buffer stock. (4) If there is buffer stock after completing maintenance, the next production cycle firstly consumes buffer stock.

The Model
4.1. Decomposing Production Line. L is decomposed into n − 1 virtual series systems by approximate decomposition method, as shown in Figure 3. Each virtual series system has only two virtual machines and a buffer (where M u is never starved and M d is never blocked). Buffer in Line 1 corresponds to B 1 , and buffer in Line n − 1 corresponds to B n−1 . For Line i, failure rates λ u (i), λ d (i) and maintenance rates μ u (i), μ d (i) are unknown. us, the next step is to solve the failure rate and maintenance rate of virtual machine.

Failure Rate and Maintenance Rate Model of Virtual
where t u (i) and t i are the average maintenance time of M u (i) and M i respectively. r u (i − 1) is the average remaining maintenance time of M u (i − 1) when M i is starved. It is the maintenance time after consuming inventory in B i−1 .

Cost Ratio Model.
For each cycle, the machine can be monitored immediately after completing buffer stock replenishment. Different maintenance strategies can be adopted based on machine monitoring status. e possible occurrence time of original defect state, serious defect state, and fault state is T x , T y , T z , respectively. ere are four different situations for adopting preventive maintenance based on monitoring status, T and S.
e state monitoring time of machine occurs before the original defect time. Machine status is in a nondefective state, and it is unnecessary to execute any maintenance action. After that, in order to prevent the failure and to detect the defect in time, it is necessary to execute a state monitoring every day until the original defect is detected. Buffer stock has been replenished from first status monitoring. Buffer stock level change during a running cycle is shown in Figure 4 and preventive maintenance of machine is executed under the original defect state.
(1) e probability of Twithin [0, T x ]: (2) Operation cycle of machine for 0 < T < T x e operational cycle of machine includes monitoring time and maintenance time. One cycle is from the end of the above maintenance to the end of the next maintenance. For this situation, operational time is T x , and maintenance time is W 1 . en, e total cost includes inventory holding cost, shortage cost, and maintenance cost within a cycle.
(3) Inventory holding cost for 0 < T < T x e inventory holding cost can be generated from beginning to produce buffer stocks, and it can be increased with the increasing of buffer stock. From the beginning of buffer stock replenishment to the end of maintenance actions, buffer is always occupied. us, (4) Shortage cost for 0 < T < T x A shortage occurs when buffer stocks are depleted and maintenance actions have not ended. erefore, there will be a shortage of cost. e shortage time is from the end of buffer stock depletion to the end of maintenance action. en, Maintenance cost includes the expected cost of preventive maintenance and all testing costs. In this case, preventive maintenance of the original defect state is carried out. x − T + 1 tests were conducted before the maintenance activity. us, maintenance cost in a cycle is e expected cost in a cycle for 0 < T < T x is the summation of inventory holding cost, shortage cost, and maintenance cost. e expected cost C 1 (S, T)is obtained: Shock and Vibration 4.3.2. T x < T < T y . e state monitoring time of machine occurs after the original defect time and before the serious defect time. Machine status is in an original defect operation state, and it needs to execute preventive maintenance of the original defect state. After the buffer stock is replenished, machine status is monitored immediately. Buffer stock level change during a running cycle is shown in Figure 5.
Line i Line n -1 (2) Operation cycle of machine for T x < T < T y e operational cycle of machine includes operational time and maintenance time. One cycle is from the end of the above maintenance to the end of the next maintenance. For this situation, operational time of machine is T, and maintenance time is W 1 . en, (3) Inventory holding cost for T x < T < T y e inventory holding cost can be generated from beginning to produce buffer stocks, and it can be increased with the increasing of buffer stock. From the beginning of buffer stock replenishment to the end of maintenance actions, buffer is always occupied. us, (4) Shortage cost for T x < T < T y e shortage cost in one cycle is the same as in Section 4.3.1.
e expected cost C 2 (S, T) in a cycle for T x < T < T y is obtained as 4.3.3. T y < T < T z . e state monitoring time of machine occurs after the serious defect time and before the failure time. Machine status is in a serious defect operation state, and it needs to execute preventive maintenance of the serious defect state. After the buffer stock is replenished, the machine status is monitored immediately. Buffer stock level change during a running cycle is shown in Figure 6.
(1) e probability of Twithin [T y , T z ] Shock and Vibration 7 (2) Operation cycle of machine for T y < T < T z For this situation, operational time of machine is T, and maintenance time is W 2 . en, (3) Inventory holding cost for T y < T < T z For this situation, from the beginning of buffer stock replenishment to the end of maintenance activities, there will be inventory occupation. erefore, the inventory holding cost is as follows: (4) Shortage cost for T y < T < T z In this case, preventive maintenance of machine in the serious defect state is required. us, the shortage cost in a cycle is

(5) Maintenance cost for T y < T < T z
In this case, the machine makes one state monitoring process. e test result is the serious defect state, so the preventive maintenance of the serious defect state is carried out. us, maintenance cost in a cycle is e expected cost C 3 (S, T) in a cycle for T y < T < T z is obtained.

T > T z .
In this case, the machine fails before the state detection is carried out. e planned state detection takes place after the failure of machine. At this time, the corrective maintenance action is executed. Buffer stock has not been replenished, or the production of buffer stock has not started before the failure shutdown. us, the buffer stock change during a running cycle is shown in Figure 7.
(1) e probability of T within [T z , ∞] (2) Operation cycle of machine for T > T z For this situation, before machine status is monitored, the failure has already occurred. Operational time of machine is T z , and maintenance time is W 3 . en, (3) Inventory holding cost for T > T z In this case, from the beginning of buffer stock replenishment to the failure occurrence, there will be inventory occupation. en,

Time
Buffer S T T x T y T z Figure 6: Buffer stock change diagram in a cycle for T y < T < T z .
(4) Shortage cost for T > T z In this case, the machine fails before the buffer stock is replenished. e machine breaks down, so corrective maintenance is carried out. us, the shortage cost in a cycle is

(5) Maintenance cost for T > T z
In this case, the status of the machine has not been detected, and the machine breaks down. erefore, corrective maintenance is executed. us, the maintenance cost of machine in a cycle is the maintenance cost under the state of failure, and there is no detection cost. en, e expected cost C 4 (S, T) in a cycle for T > T z is obtained:

Cost Ratio
Model. e cost rate in a cycle is expressed as the total cost within a cycle divided by the cycle time. en, us, the maintenance cost ratio model is

Shock and Vibration
λ u (1) is equal to λ 1 . Similarly, (2) is solved by mathematical induction: Get the average maintenance time of virtual machine M u (i), and then get its maintenance rate μ u (i).

Solving the Optimal Maintenance Cycle and Buffer Stock.
In this paper, discrete iterative algorithm is used to solve the optimal solution. e specific steps are as follows: Step 1. To assign S � S min Step 1.6. To assign S � S + ΔS, to judge if S < S max . If so, it goes to Step 1.1; otherwise, the program ends.
rough Step 1, we can obtain the optimal operating cycle T * under different stock allocation amounts S, as well as all the cost rates TCR (T * , S). Record all the TCR (T * , S) that we get. After sorting, it is easy to find the system's minimum average cost rateTCR (T * , S * ) � min S min ≤ S ≤ S max TCR (T * , S) { } and the most joint strategy (T * , S * ). e flow chart of discrete iteration algorithm is shown in Figure 8.

Case Study
In this numerical example, the specific parameters and data of the intelligent series system were obtained from Shanghai Pangyuan Machinery Co.. e workshop has a lathe production line consisting of four machines and three buffers. By monitoring the equipment history fault record, the equipment fault parameters are summarized as follows. e original defect stage, serious defect stage, and failure stage of machine M i are subject to exponential distribution independently. f x (x), f y (y), f z (z) are used to represent the probability density functions of machine deterioration in each stage, respectively. e definition of the exponential distribution function is given as follows: λ i1 , λ i2 , λ i3 are used to represent the parameters in the exponential distribution of the f x (x), f y (y), f z (z) which are shown in Table 1.
e productivity of production line β is 30000 units per year. e buffer replenishment rate α is 6000 units per year. Shortage cost ρ � $200 per unit. e cost of each machine monitoring process is $800. e unit cost of corrective repair is $15000, the unit cost of serious defect repair is $7000, and the unit cost of original defect repair is $4000. e maintenance time of each machine in the original defect state is supposed to be uniformly distributed between 0.5 and 1 day. e maintenance time of each machine in the serious defect state is supposed to be uniformly distributed between 2 and 5 days. e maintenance time of corrective maintenance is supposed to be uniformly distributed from 3 to 7 days. S varies from 0 to 211 units. T ranges from 0 to 105 days.
Using approximate decomposition method, the original production line is decomposed into three virtual series systems with two machines and one buffer. In the specific solution, a � 0.2, a � 0.5, a � 0.8 are, respectively, taken into the solution.
Since the maintenance rate of each machine is the same, the maintenance rate of the decomposed virtual machine is the same as that of the original machine, so only the failure rate of the decomposed virtual machine needs to be solved.
In the case of a � 0.2, the failure rate λ u (i) of the virtual machine M u (i) solved is shown in Table 2.
In order to simplify the difficulty of solving and relate to the actual situation, only the case where period T and buffer stock S are integers is considered in this paper. e cost ratio model is a double integer parameter nonlinear programming problem. One discrete iteration algorithm is used to solve the model. e optimal monitoring time T 1 , T 2 , T 3 of the machines M u (1), M u (2), M u (3) is 29, 27, 27 days. e optimal stock allocation amounts S 1 , S 2 , S 3 of buffers B 1 , B 2 , B 3 are 79, 80, 79 units. Figure 9 shows the change of the cost rate of machines M u (1), M u (2), M u (3) with S, T.
When a � 0.2, the operating cycle of each machine in the intelligent series system, the buffer stock allocation amount, and the corresponding lowest cost rate are shown in Table 3.
Similarly, in the case of a � 0.5, the failure rate λ u (i) of the virtual machine M u (i) solved is shown in Table 4.When a � 0.5, the operating cycle of each machine in the production line, the buffer stock allocation amount, and the corresponding lowest cost rate are shown in Table 5.
Similarly, in the case of a � 0.5, the failure rate λ u (i) of the virtual machine M u (i) solved is shown in Table 6.When a � 0.8, the monitoring time of each machine in the intelligent series system, the buffer stock allocation amount, and the corresponding lowest cost rate are shown in Table 7.

Result Analysis.
For different influence factors a, the monitoring time and buffer stock allocation are obtained. Table 8 is a comparison of the optimal monitoring time for each machine under different a. Table 9 is a comparison of the best stocks for each buffer under different a.
As can be seen from Table 8, with the increase of a, the operational cycle of the same machine gradually increases, and the inventory of the same buffer gradually decreases. Considering the actual situation, the smaller the a is, the higher the importance of machine M i will be. erefore, the shorter the operation cycle is, the shorter the monitoring time is, the higher the maintenance frequency is, and the higher the inventory allocated by the corresponding buffer will be. erefore, enterprises can choose the value of impact factor a according to the importance of the machine in the production line, so as to obtain more accurate preventive maintenance strategy and buffer stock allocation strategy.
a � 0.2 is fixed. For machine M 1 , the optimal inventory and minimum cost rate under different monitoring time T and the optimal monitoring time and minimum cost rate under different inventory S were obtained by solving the problem, as shown in Table 9. As can be seen from Table 9, the increase or decrease of T and the increase or decrease of S will lead to the increase of the cost rate. If T is too small, the number of monitoring and maintenance processes will increase, which will lead to the increase of maintenance cost and the frequent shutdown of the machine. On the contrary, if T is too large, the possibility of machine failure shutdown will be greater, and the shortage cost will also increase. If S is     Table 3: e lowest cost rate with the optimal monitoring time of the machine of the intelligent series system and the optimal buffer stock for a � 0.2 λ u (13) � 1.5 λ u (23) � 1.65 λ u (33) � 1.725 Table 5: e lowest cost rate with the optimal monitoring time of the machine of the intelligent series system and the optimal buffer stock for a � 0.5. too small, it will be more likely to be out of stock, which will lead to the increase of shortage cost. If S is too large, it will inevitably lead to an increase in inventory cost. From a practical point of view, the model results are consistent with the reality. If the monitoring time is 9 days, this means that if the detection is carried out every 9 days, the maintenance cost will be too high. On the other hand, if the buffer stock is replenished to 200 pieces, the inventory cost is high. e cost rates are highest in these extremes.

Result Comparison.
e maintenance cost rate model established in this paper is combined with the three-stage time delay theory. According to the concept of three-stage fault process, the states of the system include normal, original defect, serious defect, and fault state. Compared with the traditional two-stage time delay theory, if the machine failure can be detected in the original defect state, not only the money cost but also the time cost can be saved. In this section, a � 0.2 is fixed. For machine M 1 , the maintenance strategy proposed in this paper is compared with the maintenance strategy without buffer stock and the maintenance strategy based on two-stage time delay.

Comparison with a Maintenance Strategy without
Buffer Stock. Buffer was added to the maintenance system in this paper. In order to illustrate the effectiveness of the model, and Table 10 compares the optimal monitoring time and the minimum cost rate of machines M 1 , M 2 , M 3 with and without buffer stock in the case of a � 0.2, a � 0.5, a � 0.8. Not taking buffer stock into account means that the buffer stock is 0. Table 10 shows that the cost ratio is smaller when buffer stock is taken into account than when buffer stock is not taken into account. It shows that the preventive maintenance strategy considering buffer stock is optimal, feasible, and effective.

Comparison with the Maintenance Strategy
λ u (13) � 1.5 λ u (23) � 1.56 λ u (33) � 1.608 Table 7: e lowest cost rate with the optimal monitoring time of the machine of the intelligent series system and the optimal buffer stock with different a.   (1) 0 < T < T y . e state monitoring time of the machine occurs before the defect time. Machine status is in a nondefective state, and it is unnecessary to execute any maintenance action. After that, in order to prevent the failure to detect the defect in time, it is necessary to execute a state monitoring process on the machine every day until the original defect is detected. Buffer stock level change during a running cycle is shown in Figure 10. Preventive maintenance of machine is executed under the defect state. e probability of Twithin [10, T y ]: Operation cycle of machine for 0 < T < T y : Inventory holding cost in a cycle: Shortage cost in a cycle: Maintenance cost in a cycle: e expected cost in a cycle for 0 < T < T y is the sum of inventory holding cost, shortage cost, and maintenance cost.
e expected cost C 1 (S, T) is obtained as (2) Ty < T < T z . e state monitoring time of the machine occurs after the defect time and before the breakdown time. Machine status is in a defect operation state, and it is necessary to execute preventive maintenance of the defect state. After the buffer stock is replenished, the machine status is monitored immediately. Buffer stock level change during a running cycle is shown in Figure 11. e probability of T within [T y , T z ]:

Shock and Vibration
Operating cycle of the machine for T y < T < T z : Inventory holding cost in a cycle: Shortage cost in a cycle: Maintenance cost in a cycle: e expected cost in a cycle for T y < T < T z is the sum of inventory holding cost, shortage cost, and maintenance cost.
e expected cost C 2 (S, T) is obtained as (3) T > T z . In this case, the machine fails before the state detection is carried out. e planned state detection takes place after the failure of the machine. At this time, the corrective maintenance action is executed. e planned state detection takes place after the failure of the machine. Buffer stock has not been replenished, or the production of buffer stock has not started before the failure shutdown.
us, the buffer stock change during a running cycle is shown in Figure 12. e probability of T within [T z , ∞]: Operating cycle of the machine for T > T z : Inventory holding cost in a cycle: Shortage cost in a cycle: Maintenance cost in a cycle: e expected cost in a cycle for T > T z is the sum of inventory holding cost, shortage cost, and maintenance cost. e expected cost C 3 (S, T) is obtained as e cost rate in a cycle is expressed as follows: Based on the traditional two-stage time delay theory, the machine preventive maintenance model considering buffer stock is established as follows: e a � 0.2 is fixed. For M 1 , all parameters in the solution remained unchanged, and the model was solved. e optimal monitoring time of machine M 1 is 3 days, the optimal stock of buffer is 211 units, and the minimum maintenance cost of machine in one year is $318378.7, as shown on the left of Figure 12. Considering that the buffer stock has reached the upper limit previously given, the value of S is adjusted, and 0 < S < 400 is set to solve the problem. e optimal monitoring time T of machine M 1 is 3 days, the optimal stock S of buffer is 387 units, and the minimum maintenance cost of machine in one year is $304570.2, as shown on the right of Figure 13. As can be seen from the results, based on the traditional two-stage time delay model, the monitoring time is short and the buffer stock is high. is is because there is no distinction between the original defects and the serious defects of the machine, and the machine status cannot be accurately detected. In order to prevent the machine from being shut down, it is necessary to carry out regular monitoring, which is consistent with the actual situation. e maintenance strategy based on three-stage time delay theory can save $22432.4 per year compared with the maintenance strategy based on two-stage time delay theory. e specific comparison is shown in Table 11.

Conclusion
In this paper, a new method is proposed to solve the preventive maintenance problem of intelligent series system with buffer stock. For the intelligent series system with inventory buffer, the series system is decomposed into several virtual series systems with two machines and one buffer by approximate decomposition method. e failure rate and maintenance rate of the decomposed virtual machine are calculated by mathematical induction. e influence factor is introduced here, and the enterprise can determine the value of the influence factor according to the importance of different machines in the series system. For each virtual series system, a preventive maintenance model was built with the lowest cost rate as the objective function and the monitoring time and buffer stock as independent variables. e preventive maintenance model is combined with the three-stage time delay theory to better simulate the equipment degradation process. Finally, a case is used to verify the validity of the model. e maintenance strategy in this paper is compared with the maintenance strategy without buffer stock and the maintenance strategy based on the two-stage time delay. It is proved that the proposed maintenance strategy based on the three-stage time delay theory is optimal. Taking the impact factor a � 0.2 as an example, compared with no buffer stock maintenance strategy, the annual maintenance cost of machine M 1 can save $16072.4. Compared with the traditional two-stage time delay maintenance strategy, the annual maintenance cost of machine M 1 can save $22432.4. erefore, the maintenance strategy proposed in this paper can be well used in the maintenance of the series system, which can save a lot of money for enterprises e value of impact factor a should be determined according to the importance of the machine on the production line. On the basis of the research in this study, the effective method can be adopted in following research, and the most accurate influence factor can be obtained for different machines in the series system.

Data Availability
e underlying data supporting the results of our study can be found in the article, including, where applicable, hyperlinks to publicly archived datasets analyzed or generated during the study

Conflicts of Interest
e authors declare that they have no conflicts of interest.