Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 3832123 | https://doi.org/10.1155/2020/3832123

Lü Xiong, Hong Wang, Zuhua Jiang, "Tristage Bargaining Dynamic Game-Based Preventive Maintenance for Electric Multiple Unit", Mathematical Problems in Engineering, vol. 2020, Article ID 3832123, 10 pages, 2020. https://doi.org/10.1155/2020/3832123

Tristage Bargaining Dynamic Game-Based Preventive Maintenance for Electric Multiple Unit

Academic Editor: Neale R. Smith
Received29 Dec 2019
Accepted11 Jun 2020
Published04 Jul 2020

Abstract

The operation and maintenance sectors of electric multiple unit (EMU) are considered as game participants to optimize the preventive maintenance (PM) schedule of EMU components. The total cost of a component over a life cycle includes failure risk and maintenance cost. The failure risk of EMU components is assessed quantitatively by using an analytic hierarchy process and scoring and weighting the factors that affect the failure risk of such components. The operation sector expects failure risk to be minimized, whereas the maintenance sector expects maintenance costs to be low, and their interests interfere with each other to some extent. This study establishes a tristage bargaining dynamic game model of the operation sector priority bid and maintenance sector priority bid considering the PM reliability threshold R as the bargaining object. A numerical example demonstrates that the result is more beneficial to the sector that bids first, and the operation priority bid allows the component to maintain a higher reliability level over a life cycle, especially in the latter half of the component life cycle.

1. Introduction

Cost and interest allocation problems arise in many real-life situations [1, 2], where individuals, who may have different purposes, decide to work together. In such situations, the problem occurs when the times come to divide the joint costs and interests resulting from cooperation among participants. Eltoukhy et al. [3] developed a coordinated configuration of a scenario-based stochastic operational aircraft maintenance routing problem and maintenance staffing problem as a Stackelberg game to reduce costs for airlines and maintenance companies. The operation of production and utility systems is commonly optimized separately without feedback between them. In general, this separated planning increases total costs. Conceptually, a joint optimization of production and utility systems is desirable, but it is often not possible in practice. Leenders et al. [4] overcame such practical limitations by proposing a repeated Stackelberg game. Xiao et al. [5] considered the effects of differences between plan products and actual products. The two heterogeneous players always adopt suitable strategies that can improve their benefits most, and a nonlinear duopoly Stackelberg competition model on output between heterogeneous players is developed.

As for PM issues [611], many models have been developed and implemented to improve system reliability, prevent system failures, and reduce maintenance costs [1216]. In fact, the PM issues of a system are ultimately economic issues. How to reduce the PM cost or increase the availability of system under the premise of ensuring the necessary system reliability is a contended topic among scholars. In recent years, scholars and maintenance engineers have introduced game theory into the field of PM [17, 18] and have achieved numerous significant advancements. Hu et al. [19] adopted a game theory model to handle the interactive effect between the production plan and PM schedule. An extended imperfect PM model is proposed for a system running with a time-varying production rate. Tayeb et al. [20] proposed a novel game theory approach to the problem of integrating a periodic and flexible PM schedule and production scheduling for permutation flow shops. Pourahmadi et al. [21] considered that electric power system operators would have to be able to manage the system operation expenses more effectively, as waves of maintenance costs and equipment investments would be anticipated within a few years. A cost-effective reliability-centered maintenance approach based on game theory is proposed to assess the component criticality for overall system reliability and further maintenance focus and for reducing the maintenance cost of the electric power system.

With the development of high-speed railway technology, high-speed EMU is becoming increasingly used around the world. The PM strategy of high-speed EMU that followed has also received more attention. With regard to the operation and maintenance sectors of EMU, owing to their respective interests, they present a potential competitive relationship. Both the operation and maintenance sectors want to achieve higher benefits and maximize their own interests. The operation sector expects the EMU to transport passengers to their destination safely and comfortably at the lowest risk, whereas the maintenance sector expects to reduce maintenance costs as much as possible while maintaining the reliability of the EMU at a reasonable level. However, it is not difficult to understand that the higher the PM reliability threshold R is, the more PM measures will be adopted, the higher the PM cost will be, and the lower the failure times and risks will be. On the contrary, the lower the PM reliability threshold R is, the fewer PM measures will be carried out, the lower the maintenance cost will be, and the more failure times and risks will be. However, both maintenance sector and operation sector want the EMU to operate safely and smoothly, and they have some common interests to some extent. Hence, they present a cooperative relationship as well. To balance the competitive and cooperative nature of the relationship between the operation and maintenance sectors, a tristage bargaining dynamic game model is established to study the game behavior of these sectors.

In this study, a key EMU component is considered as the study object. Based on the tristage bargaining dynamic game model under symmetric information, considering the EMU operation and maintenance sectors as game participants, we develop a tristage bargaining dynamic game model to obtain the PM schedule of the EMU component. The following assumptions are made for the model:(1)The initial component reliability is “1.”(2)The failure rate function of the component follows a Weibull distribution.(3)The imperfect PM measure of the component in a life cycle is only junior maintenance and senior maintenance.(4)During each PM interval, minimal repair is carried out when failure occurs. Minimal repair only restores the machine to working conditions, as bad as it was before, which has been applied in numerous studies.(5)Different imperfect PM measures consistently restore the performance of nonfailure components.

2. Model Establishment

2.1. Failure Rate Evolution Rule

Malik [22] introduced the concept of age reduction factor to describe the system function. Age after the ith PM is reduced to aiλ (t) when it is λ (t) in period i of PM, the failure rate function after the ith PM becomes λi (t + aiTi) for t∈ (0, Ti + 1), where 0 < ai < 1 is the age reduction factor due to the imperfect PM action. This implies that each imperfect PM changes the initial failure rate value immediately after the PM to λi (aiTi), but not to zero. Nakagawa [23] proposed another model based on the failure rate increase factor. According to Nakagawa, the failure rate function becomes biλi (t) for t∈ (0, Ti + 1) after the ith PM, where bi > 1 is the failure rate increase factor. This indicates that each PM makes the failure rate increase at a higher rate. Combining the advantages of the age reduction factor and failure rate increase factor, Lin et al. [15] established a hybrid failure rate evolution model. The age after the ith PM is reduced to aiλ (t) when it is λ (t) in period i of PM, and the failure rate becomes biλi (t) after the ith PM, where 0 < ai < 1 and bi > 1. The hybrid failure rate evolution function can be defined as

Scheduled PM is performed whenever the system reaches the reliability threshold R. Based on this policy, a reliability equation can be constructed as

According to the bilevel imperfect PM strategy, there are two maintenance measures to choose at the scheduled PM time.(1)Junior maintenance: the failure rate of the system after the (i − 1)th PM (junior maintenance) becomes when it was after (i − 2)th PM (senior maintenance). The failure rate of the system after the ith PM (junior maintenance) becomes when it was after (i − 1)th PM, and . This indicates that regardless of the last scheduled imperfect PM, and whether junior maintenance or senior maintenance is adopted, the failure rate of the system after junior maintenance is higher than that of the last scheduled imperfect PM, as depicted in Figure 1(a).(2)Senior maintenance: the failure rate of the system after the (i − 1)th PM (junior maintenance) becomes when it was after (i − 2)th PM (senior maintenance); the failure rate of the system after the ith PM (senior maintenance) becomes when it was after the (i − 1)th PM, and . This implies that the failure rate after senior maintenance is only higher than that of the last senior maintenance and may be lower than that of the last junior maintenance . Hence, after senior maintenance, the failure rate of the system is only higher than that of the last senior maintenance, and it is not related to the last junior maintenance, as depicted in Figure 1(b).

2.2. Maintenance Method Selection

Whenever the reliability of the system reaches the PM threshold R, a scheduled imperfect PM action will be performed. According to the bilevel imperfect PM strategy, there are two imperfect PM measures to adopt when the scheduled PM time arrives. However, which specific imperfect PM measure to adopt, junior maintenance or senior maintenance, still needs to be determined. Here, we adopt a cost-effectiveness analysis to address this problem. The maintenance measure selection factoris introduced:

The failure rate increase rate factor ai and age reduction rate factor bi will be different as different imperfect PM measures are applied. The functions of ai and bi under different PM measures can be defined as

The failure rate of system before the ith PM is the same as the end of the (i − 1)th PM interval, and the failure rate of system after the ith PM is equal to that at the beginning of the (i + 1)th PM interval. The functions of and can be defined as

The improvement of the failure rate after PM can be constructed as . Herein, we consider the cost-effectiveness ratio as an economic analysis indicator. and represent the junior maintenance and senior maintenance cost-effectiveness ratio, respectively, and are expressed as

As shown by (6) and (7), junior maintenance is a more cost-effective maintenance measure if , and senior maintenance is a more cost-effective maintenance measure if . In addition, can be further expressed as

2.3. Maintenance Cost Modeling

The maintenance cost of a component in a life cycle includes PM, failure maintenance, and replacement costs.(1)PM costs include junior maintenance, senior maintenance, and site occupation costs:(2)Failure maintenance costs include minimal repair and site occupation costs:(3)Replacement costs for the system include labor, waste, and site occupation costs:

In summary, the maintenance cost of a system in a life cycle can be expressed as

2.4. Failure Risk Modeling

Risk is defined as the likelihood of a particular event occurring and the severity of outcome [24]. For high-speed EMUs, the damage of unpredicted failure may not only be downtime losses but also cause a delay of EMU, affecting the normal operation of other trains in the operating range. Moreover, it affects safe operation of EMU, threatening the safety of passengers and property. For factors that influence the failure risk of EMU component, we mainly consider the following three aspects: the effect of failure on EMU safety during operation, the effect of failure on EMU-delaying, and the complexity of failure maintenance. We could then evaluate the failure risk of the EMU component by weighting and scoring these three influencing factors.

In order to establish the failure risk assessment function, the failure risk factor is introduced and expressed aswhere si is the score of the ith influence factor, represents the weight of the ith influence factor, and n indicates the number of influence factors.

Referring to the evaluation method of component importance in [2528], the scoring and weighting of each influence factor is performed. Considering that the failure risk evaluation of the EMU component is neither too cumbersome nor in line with the actual situation, the scoring criteria of each influence factor are divided into 4 to 5 levels. The scoring criteria for each influence factor are listed in Tables 13.


Serial numberSeverityScore

1General failure1–2
2Dangerous failure3–4
3Big failure5–6
4Major failure7–8
5Particularly significant failure9–10


Serial numberDelay time (h)Score

1<21–2
22–103–6
310–307–9
4>3010


Serial numberDegree of difficultyScore

1General1–2
2Medium3–4
3High5–7
4Very high8–9
5Unable to maintenance10

The weight of each influence factor is obtained using an analytic hierarchy process. First, a judgment matrix of relative risk between the influence factors is constructed as follows:where uab is the relative risk of a to b. The relative risk values are shown in Table 4.


Relative risk scalesImplications

1The same
3Slightly larger
5Quite large
7Very large
9Great large
2, 4, 6, 8Between the above

The value of relative risk is shown in Table 4.

Second, the largest eigenvalue of the judgment matrix is calculated and substituted into the homogeneous linear equations:

The feature vector corresponding to the maximum eigenvalue is the weight value of each determinant.

Finally, it is necessary to test the consistency of the judgment matrix. The test formula is as follows:where UP is the random consistency ratio of the judgment matrix; UC is the general consistency indicator of the judgment matrix, and ; and UM is the average random consistency indicator of the judgment matrix. Table 5 shows the value of order 1–7 in the judgment matrix UM.


n1234567

UM0.000.000.580.901.121.241.32

If UP < 0.1, it can be judged that UM has satisfactory consistency and its weight distribution is reasonable. Otherwise, the judgment matrix needs to be adjusted until it achieves a satisfactory consistency.

The failure risk of component during the ith PM interval can be expressed as

2.5. Total Cost Modeling

The total cost of a component in a life cycle can be divided into two parts: maintenance cost and failure risk. The total cost can be expressed as

The weighting factors 1 and 2 (1 + 2 = 1) are introduced to determine the propensity of PM strategies:where 1 and 2 are the maintenance cost and the failure risk weighting factors, respectively.

3. Tristage Bargaining Dynamic Game Model

Bargaining is a game process in which participants with common interests try to reach an agreement when confronted with conflicts. It is a typical negotiation activity. In the negotiation process, when sector A puts forward an offer to sector B, together with the main contract terms, sector B analyzes all its contents, judges its intention through sector A’s offer, and gives a reoffer and other responses to make the transaction develop in a direction that is beneficial to itself and satisfies certain requirements of the other sector, so as to facilitate the exchange of interests in the negotiation. Because of negotiation costs and time loss, the cost function of both sectors should add an additional loss δ (0 < δ < 1) to the original basis. Assuming that the game process only involves three rounds, it is referred to as a tristage bargaining dynamic game model.

First, sector A proposes a PM plan, and sector B chooses to accept or reject it. If sector B rejects, it proposes another PM plan for sector A to choose to accept or reject, and so it goes on like this. Once a condition is rejected, it is no longer binding and it is no longer relevant to the latter game. In this cycle, the game ends when either sector accepts the other sector’s PM plan. Each time a sector proposes a PM plan and the other sector chooses whether to accept it can be considered as a round. According to which sector A or sector B has the priority to bid, the tristage bargaining dynamic game can be divided into Models 1 and 2.

3.1. Model 1: Operation Sector Priority Bids

The game process is as follows:(1)The operation sector takes min (r) as the decision-making goal and obtains PM reliability threshold R1 by solving the function of min (r). R1 is the solution that minimizes r.(2)The maintenance sector chooses to accept or reject the conditions put forward by the operation sector. If the maintenance sector accepts, the game is over. The maintenance cost of the EMU component is then , the risk cost is , and the total cost is . If the maintenance sector rejects, another PM reliability threshold R2 will be provided by the maintenance sector.(3)The operation sector chooses to accept or reject the conditions put forward by the maintenance sector. If the operation sector accepts, the game is over. The maintenance cost of the EMU component is then , the risk cost is , and the total cost is . If the operation sector rejects, another PM reliability threshold will be provided by the operation sector. The maintenance sector must then accept the conditions put forward by the operation sector, and the game is over. The maintenance cost of the EMU component is, the risk cost is , and the total cost is . The game process is shown in Figure 2.

3.2. Model 2: Maintenance Sector Priority Bids

The game between the two sectors is as follows:(1)The maintenance sector takes min (Cx) as the decision-making goal and obtains the PM reliability threshold by solving the function of min (Cx). is the solution that minimizes Cx.(2)The operation sector chooses to accept or reject the conditions put forward by the maintenance sector. If the operation sector accepts, the game is over. The maintenance cost of the EMU component is then , the failure risk is , and the total cost is . If the operation sector rejects, another PM reliability threshold will be provided by the operation sector.(3)The maintenance sector chooses to accept or reject the conditions proposed by the operation sector. If the maintenance sector accepts, the game is over. The maintenance cost of the EMU component is then , the failure risk is , and the total cost is . If the maintenance sector rejects, another PM reliability threshold will be provided by the maintenance sector. The operation sector must then accept the conditions put forward by the maintenance sector, and the game is over. The maintenance cost of the EMU component is then , the failure risk is , and the total cost is . The game process is shown in Figure 3.

3.3. Model Solving Method
3.3.1. Solution Method of Model 1

(1) First Round. Operation sector obtains the PM reliability threshold R1 by optimizing the objective function min (r). Maintenance sector chooses to accept or reject the conditions proposed by the operation sector. If maintenance sector accepts, the game is over. Otherwise, the maintenance sector will put forward another PM reliability threshold, R2.

(2) Second Round. The condition for game to enter the second round is that the maintenance cost of the second round should be lower than that of the first round, and the total cost of the second round should be lower than that of the first round. Thus, the game enters the second round, where R2 should meet the following condition:

(3) Third Round. The condition for game to enter the third round is that the failure risk of the third round must be less than that of the second round, and the total cost of the third round should be lower than that of the second round. Therefore, game enters the third round, where R should meet the following condition:

3.3.2. Solution Method of Model 2

(1) First Round. The maintenance sector obtains the PM reliability threshold by optimizing the objective function min (Cx). The operation sector chooses to accept or reject the conditions proposed by the maintenance sector through calculation. If the operation sector accepts, the game is over. Otherwise, the operation sector will propose another PM reliability threshold, .

(2) Second Round. The condition for the game to enter the second round is that the failure risk of the second round must be less than that of the first round, and the total cost of the second round is lower than that of the first round. Thus, the game enters the second round, where should meet the following condition:

(3) Third Round. The condition for the game to enter the third round is that the maintenance cost of the third round must be less than that of the second round, and the total cost of the third round is lower than that of the second round. Therefore, the game enters the third round, where should meet the following condition:

4. Numerical Example

The failure rate distribution function of component is described by Weibull distribution. The Weibull distribution is expressed aswhere m is the shape parameter and η is the characteristic life parameter. In order to obtain the values of m and η, the actual maintenance records of a key component of the CRH3C EMU are investigated and collated, and a total of 216 life data points are obtained as follows (unit: 103 km):

Using MATLAB software (MathWorks Inc., Natick, MA, USA) and the maximum likelihood estimation method, the above 216 life data points were analyzed. Finally, the shape parameter m = 3 and characteristic life parameter η = 100 were obtained. To compute the optimal reliability threshold R, the adjustment factors (ak, bk), cost parameters (cj, cs, cm, cr, , ψ, δ), and time parameters (tj, ts, tm, tr) need to be known. Usually, maintenance engineers are responsible for the determination of these parameters, which are listed in Table 6.


ParameterValue

cj (yuan)300
cs (yuan)600
cm (yuan)600
cr (yuan)100
(yuan)1000
Δ0.04
Aji/3i + 1
Asi/5i + 1
tj (h)8
ts (h)10
tm (h)10
tr (h)5
cd (yuan)2000
lmax/104 km240
bj12i + 1/8i + 1
bs12i + 1/11i + 1

The judgment matrix of relative risk between various influence factors is as follows:

According to Tables 13, the influence factors of failure risk are scored, and the failure risk factor of component can be obtained.

Table 7 presents the results of PM optimization of component with different weight coefficients.


12RCx (yuan)R (yuan)C (yuan)

100.5447991461319412
0.90.10.6349411316718108
0.80.20.6349411316718108
0.70.30.7656031077516378
0.60.40.826405940715812
0.50.50.846696905215748
0.40.60.877590834915939
0.30.70.898199804716246
0.20.80.898199804716246
0.10.90.9210125765917784
010.9210125765917784

As observed in Table 7(1)The PM reliability threshold R decreases while the maintenance cost weight 1 increases, and it increases as the failure risk weight increases as well.(2)The maintenance cost Cx decreases while 1 increases, and the failure risk r decreases as 2 increases. This reveals the impact of 1 and 2 on maintenance cost and failure risk.

Figures 4 and 5 reveal the relationship between C and R under conditions of operation sector priority bid and maintenance sector priority bid, respectively. These figures indicate that the total cost of the system over a life cycle steadily declines at first and rises rapidly as R increases. Additionally, the total cost is lower at a certain value of R, which represents a best-case scenario.

Tables 8 and 9 present the optimization results and PM schedules of the two game models, respectively. Figure 6 shows the reliability evolution comparison between the two models.


ModelR2Cx (yuan)R (yuan)C (yuan)

10.840.50.56696905215748
20.760.70.356031077516378


ModelRPM mileage interval (104 km)Maintenance measureFailure maintenance times

10.8452−92−114−128−160−176−186−208−218–2360-0-0-1-0-0-1-0-1-01.91
20.7666−108−134−150−186−204−2160-0-0-1-0-0-12.27

As shown in Tables 8 and 9 and Figure 6,(1)Under the tristage bargaining dynamic game model, the PM reliability threshold R of Model 1 (operation sector priority bid) is higher than that of Model 2 (maintenance sector priority bid), and the failure risk of Model 1 is significantly lower than that of Model 2. The maintenance cost of Model 1 is significantly higher than that of Model 2, and the total cost of the two models is similar.(2)Model 1 allows components to maintain a higher reliability level than that of Model 2 during a life cycle, especially in the latter half of the component life cycle (1.2 million km to 2.4 million km).(3)Model 1 adopted two more junior maintenance measures and one more senior maintenance measure than Model 2, which results in the number of failure maintenance time being reduced by 15.9%.(4)Maintenance sector is stronger than operation sector. Irrespective of the operation sector priority bid or maintenance sector priority bid, game results show that maintenance cost weight 1 is always equal to or greater than that of failure risk weight 2.

5. Conclusion

The PM schedule of the EMU component is jointly developed by the maintenance and operation sectors. This study established a tristage bargaining dynamic game model between these EMU sectors to balance the allocation of resources between them. The numerical example shows that operation sector priority bid allows components to maintain a higher level of reliability during a life cycle, especially in the latter half of the component life cycle, and its failure probability is reduced by nearly 1/6 than that under maintenance sector priority bid. In the tristage bargaining dynamic game model, the game outcome is more beneficial to the sector that priority bids. Therefore, if the status of the two game participants is equal, the participants should attempt to obtain the right to bid first and take the initiative in the bargaining process.

The imperfect PM method applied in this paper is a bilevel strategy where there are only two types of imperfect PM measures available for EMU components, namely, senior maintenance and junior maintenance. However, in the field of EMU PM, engineers may adopt a multi-level imperfect PM measure strategy for components according to its actual situation, i.e., there are multiple types of imperfect PM measures for EMU components at scheduled PM time. Further studies will attempt to develop such multi-level imperfect PM strategies for the EMU component to adopt according to their actual PM situation.

Notation

cj:Junior maintenance cost
λi (t):Failure rate function within the ith PM
cs:Senior maintenance cost
:Failure rate before the ith PM
τ:Site occupation rate
:Failure rate after the ith PM
tj:Junior maintenance time
ai:Age reduction factor
ts:Senior maintenance time
bi:Failure rate increase factor
R:Reliability threshold for scheduled PM
Ti:Time interval between ti and ti−1
Cf:Failure maintenance cost
:Failure rate after junior maintenance
cm:Minimal repair cost
:Failure rate after senior maintenance
tm:Minimal repair time
:Failure rate after last senior maintenance
Cr:Replacement cost
:Maintenance method selection factor
cl:Labor cost
:Age reduction factor after junior maintenance
:Waste cost
:Age reduction factor after senior maintenance
tr:Replacement time
:Failure rate increase factor after junior maintenance
Cx:Maintenance cost
:Failure rate increase factor after senior maintenance
ψ:Failure risk factor
Cp:PM cost
cd:Unit failure risk cost
:Maintenance (junior and senior) cost
C:Total cost
:Site occupation cost
δ:Negotiation loss factor.

Data Availability

The maintenance record data used to support the findings of this study are included within the supplementary information file.

Conflicts of Interest

The authors declare that there is no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (no. 71361019) and Youth Innovative Talents Project of Universities in Guangdong Province (no. 2018GkQNCX075).

Supplementary Materials

Maintenance record data. The records the actual maintenance data of CRH3C Electric Multiple Unit (EMU) brake pads, a total of 216 items. And the shape parameter (m) and scale parameter (ƞ) in numerical illustration are obtained by analyzing these data. (Supplementary Materials)

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Copyright © 2020 Lü Xiong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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