Maintenance usually plays a key role in controlling a multi-component production system within normal operations. Furthermore, the failure of components in the production system will also cause large economic losses for users due to the shutdown. Meanwhile, manufacturers of the production system will be confronted with the challenges of the warranty cost. Therefore, it is of great significance to optimize the maintenance strategy to reduce the downtime and warranty cost of the system. Opportunistic maintenance (OM) is a quite important solution to reduce the maintenance cost and improve the system performance. This paper studies the OM problem for multi-component systems with economic dependence under base warranty (BW). The irregular imperfect preventive maintenance (PM) is performed to reduce the failure rate of components at a certain PM reliability threshold. Moreover, the OM optimization model is developed to minimize the maintenance cost under the optimal OM reliability threshold of each component. A simulated annealing (SA) algorithm is proposed to determine the optimal maintenance cost of the system and the optimal OM threshold under BW. Finally, a numerical example of a belt conveyor drive device in a port is introduced to demonstrate the feasibility and advantages of the proposed model in maintenance cost optimization.

With the rapid development of modern production technology, manufacturers gradually switch their attention from product-oriented to service-oriented to highlight the brand value and capture more market share [

In terms of modern production systems such as wind turbines and assembly line equipment, they usually consist of several interdependent components or subsystems. The maintenance actions of these systems play a key role in their sufficient usage in terms of cost, reliability, and availability [

For the maintenance actions of multi-component systems, most of which are carried out in the after-sales warranty period of the systems. Therefore, optimizing maintenance strategies to reduce warranty costs is of great significance to manufacturers and consumers, which can improve the profits of manufacturers and product reliability, thereby reducing shutdown loss for consumers. In the maintenance optimization framework of multi-component systems, opportunistic maintenance (OM) has been introduced, developed and successfully applied to various multi-component systems. The main idea of OM is to take the advantage of positive economic dependence, which means that group maintenance actions are cheaper than implementing maintenance actions on components separately [

However, in the above studies, there is no literature to carry out BW decision-making researches based on OM strategy for multi-component systems with economic dependence. In traditional maintenance models, most of the modeling studies on BW assume that the components are independent of each other in the system. Model construction and solution are less difficult, but there is a big error between the maintenance process and the actual situation. Therefore, in BW studies, simply treating the components in the system as independent of each other is not in line with the actual situation. It is necessary to consider the economic dependence between the components in the system to make the maintenance process more consistent with the actual situation. This paper intends to fill this gap.

In the current study, an OM strategy is introduced to minimize the maintenance cost for multi-component systems with economic dependence during the BW period. Due to the advanced materials, technology, and complex structure of the multi-component system, maintenance actions usually cannot achieve the perfect update of the system [

Therefore, the remainder of this paper is organized as follows: Section

The paper studies a series multi-component system composed of Q components with different failure rates, and the OM strategy is implemented during the BW period

To facilitate the research, the system modeling is based on the following assumptions:

Each component of the series multi-component system is repairable, and the failure distribution between the components is independent of each other obeying the two-parameter Weibull distribution.

If a component fails during the PM interval, the minimal repair will be implemented on it immediately and the minimal repair will not bring maintenance opportunities to other components.

OM includes opportunistic PM and opportunistic replacement.

In the series multi-component system, the failure of any component will result in the shutdown of the whole system.

Before the reliability of component

The following and notations are used in this paper.

In practice, most components in the multi-component system are composed of the parts with different failure forms, and the failure rate function generally presents an increasing trend in the operation cycle. The bathtub curve can better describe the failure law, as shown in Figure

Bathtub curve of failure rate.

The bathtub curve describes the three stages of decreasing, stabilizing, and increasing the failure rate of the component during the service life. Weibull distribution is used to describe the failure distribution of the component, which is suitable for each stage of the bathtub curve. It is assumed that the component life in the multi-component system obeys the two-parameter Weibull distribution. The failure rate function of component

Among them,

In terms of the failure model for imperfect repair, Malik [

Nakagawa [

Among them,

Equation (

Failure rate change of imperfect repair.

The key to the OM strategy is to determine the OM reliability threshold

The curve of system maintenance cost.

The OM strategy needs to adjust the PM interval of the single component, but the extension of the PM interval will make the reliability of the component lower than its PM reliability threshold, increasing the probability of component failure; however, shortening the PM interval of the component will cause the overrepair and increase the maintenance cost and downtime. Therefore, only the OM for the component that meets a certain control condition can effectively reduce the maintenance cost of the system, thereby reducing the BW cost of the system. Since the imperfect PM is implemented, the components cannot be restored as good as new after the PM, and the failure rate will increase to a certain extent. Hence, after N times of PM for the component, the failure rate will increase to an unbearable range. At this time, preventive replacement of the component is a more economical maintenance method. Suppose that component

Preventive maintenance (PM) process for component

At the moment

If

If

The OM reliability threshold of component u.

After completing an OM, component

Before implementing the OM strategy for the multi-component system, the optimal PM strategy for each single-component needs to be formulated. On the basis of the maintenance plan for each single component, the OM for other components is considered when PM actions are carried out for a component in the system. The normal operation of the series multi-component system is usually closely related to the reliability of each single component. When the reliability of a component is lower than the reliability threshold, there will be great hidden trouble in the operation of the system. At this time, it is necessary to implement PM or replacement on the component. During the BW period

Then, the cumulative failure distribution function of component

Therefore, the reliability equation of component

Taking the logarithm of both sides at the same time, it can be derived as

It can be seen from equation (

By minimizing the objective function, the optimal number

During the BW period

During the whole PM cycle from the nth PM at

Therefore, the direct maintenance cost of the multi-component system during the BW period

The system shutdown loss is usually proportional to downtime. The system downtime

Thus, the PM shutdown loss of the system during the BW period

Therefore, the minimal repair cost of the system during the BW period

To sum up, the total maintenance cost of the series multi-component system during the BW period is the sum of the direct maintenance cost of each component and the total shutdown loss of the system:

Hence, the optimal OM reliability threshold

For the objective function

Algorithm flowchart of objective function.

Set the values of the relevant parameters, and obtain the optimal number

Let

Determine the reliability threshold

Therefore, the time required for the maintenance action of component

Then, the downtime of the system in the nth PM should be the longest maintenance time of all the components in this maintenance action, which is

According to equation (

After the nth PM of the system, determine the next PM moment of the system. According to the method of Step

Port machinery equipment is the material basis for the production and operation of port enterprises. This study takes the drive device of the BH45-1 belt conveyor in a certain port as the research object. The drive device is the power source for the operation of the belt conveyor. Its failure will cause the entire belt conveyor to stop and significant downtime loss. The OM strategy is implemented during the BW period. When a component in the drive device reaches its PM reliability threshold, the system will shut down and the imperfect PM or preventive replacement will be implemented; in the meanwhile, the OM conditions of other components in the drive device are judged, and the opportunistic imperfect PM or opportunistic preventive replacement will be performed on the components whose reliability reaches the OM reliability threshold. The belt conveyor drive device is mainly composed of five key components, such as electric motor, hydraulic coupler, reducer, low-speed coupling, and transmission drum. Five components form the series multi-component system, and its reliability block diagram is shown in Figure

Reliability block diagram of the driving device.

Assuming that the BW period of the drive device is 730 days, the shutdown loss of each component in the drive device is equal to the shutdown loss of the entire belt conveyor, that is, 50000 CNY/day. In practice, the values of

The values of parameters in the presented numerical experiment.

Component | CNY | Day | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 4.42 | 87.13 | 1740 | 480 | 250000 | 0.38 | 0.13 | 0.92 | 0.60 |

2 | 3.13 | 76.52 | 1080 | 300 | 116500 | 0.29 | 0.08 | 0.42 | 0.50 |

3 | 3.08 | 60.19 | 2160 | 620 | 178500 | 0.42 | 0.17 | 1.10 | 0.60 |

4 | 3.72 | 138.00 | 820 | 260 | 16100 | 0.20 | 0.10 | 0.55 | 0.50 |

5 | 2.69 | 55.96 | 2800 | 500 | 62600 | 0.46 | 0.15 | 1.00 | 0.55 |

According to the optimization model of PM interval for a single component based on reliability, the optimal PM interval

The optimal PM schedule for each component of the drive device.

Component | Days | CNY (day) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 74.8 | 66.7 | 59.3 | 52.6 | 46.5 | 41.2 | 36.4 | 32.1 | 28.4 | 25.1 | 22.2 | 19.7 | 1109.8 | 11 |

2 | 68.1 | 60.1 | 53.0 | 46.7 | 41.1 | 36.2 | 32.0 | 28.3 | 25.1 | 22.3 | — | — | 738.7 | 9 |

3 | 48.4 | 42.7 | 37.6 | 33.1 | 29.2 | 25.7 | 22.7 | 20.1 | 17.8 | 15.8 | — | — | 1634.1 | 9 |

4 | 125.1 | 110.9 | 98.2 | 86.8 | 76.6 | 67.6 | — | — | — | — | — | — | 210.9 | 5 |

5 | 46.0 | 40.4 | 35.5 | 31.2 | 27.5 | 24.3 | 21.5 | — | — | — | — | — | 1304.0 | 6 |

As can be seen from Table

The optimal PM plan for each component of the drive device.

It should be noted that, for each component, each point represents the PM or replacement moment in Figure

SA was firstly put forward by Metropolis [

The basic framework of the SA algorithm is shown in Figure

Flowchart of the proposed SA algorithm.

More specifically, the procedure of the proposed optimal OM method of the multi-component system is summarized in Algorithm

Input: Initial solution

Output: Best solution

Generate a random neighbor

Output

Calculate the optimized maintenance cost of the drive device considering the OM strategy.

In the meantime, the parameter initialization of the SA algorithm is set as in Table

SA parameters.

Parameter | Value |
---|---|

100 | |

Number of iterations to stop | 100 |

1 | |

Cooling rate | 0.9 |

Value range of the variable | |

0.1 |

According to the parameter values, in our observation, after a hundred iterations, SA converges to a stable state quickly. Moreover, the current temperature is close to zero. Then, we can draw the conclusion that the SA has found the global optimal value. The convergence of SA algorithm is shown in Figure

The convergence of SA algorithm for the maintenance cost optimization.

From Figure

Based on the above analysis and data, the optimal total maintenance cost of the drive device system is 3,242,080 CNY, in which, the direct maintenance cost is 1,578,614 CNY and the shutdown loss cost is 1,663,466 CNY, accounting for 48.69% and 51.31% of the total maintenance cost, respectively. Meanwhile, the number of shutdowns of the drive device is up to 22 times due to PM or replacement of each component. Hence, during the BW period, the effective operation time of the driving device is 696.73 days; however, the total shutdown time is 33.27 days. Moreover, the downtime due to PM or replacement and minimal repair is 13.49 days and 19.78 days, respectively. Therefore, the availability of the drive device during the BW period can be derived as 0.954. Then, the optimal OM schedule for the drive device is presented in Table

The optimal OM schedule for the belt conveyor drive device.

Maintenance strategy | |||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

46.0 | O | O | O | B | Y |

86.9 | O | O | O | O | Y |

122.9 | O | O | O | B | Y |

154.5 | O | O | O | B | Y |

182.5 | O | O | O | O | Y |

207.2 | O | O | O | B | G |

242.0 | O | O | Y | O | O |

262.5 | O | O | Y | B | O |

280.8 | O | R | G | B | O |

330.2 | O | O | Y | O | O |

373.4 | R | O | Y | R | O |

411.4 | O | O | Y | B | R |

445.0 | O | O | Y | O | O |

474.6 | O | O | Y | B | O |

500.7 | O | O | Y | O | O |

523.9 | O | O | Y | B | O |

544.4 | O | O | Y | O | O |

562.6 | O | R | G | B | R |

612.1 | O | O | Y | O | O |

655.2 | O | O | Y | R | O |

693.3 | O | O | Y | O | O |

726.8 | R | O | Y | O | O |

— | — | — | — | — | — |

— | — | — | — | — | — |

Y: PM; G: preventive replacement; O: opportunistic PM; R: opportunistic preventive replacement; B: no maintenance actions.

Based on the data in Table

OM schedule of the belt conveyor drive device.

To reflect the optimization degree of the OM strategy for the maintenance cost of the multi-component system with economic dependence, the results of various aspects of the drive device before and after considering the OM strategy are shown in Table

The comparison before and after considering the OM strategy on some important indicators.

Indicator | Not considering the OM | Considering the OM | Optimization degree (%) |
---|---|---|---|

Total maintenance cost | 3,648,175 CNY | 3,242,080 CNY | 11.13 |

Availability | 0.9297 | 0.9544 | 2.66 |

Direct maintenance cost | 1,081,138 CNY | 1,578,614 CNY | −46.01 |

Shutdown loss cost | 2,567,037 CNY | 1,663,466 CNY | 35.20 |

Number of shutdowns | 77 | 22 | 76.71 |

By analyzing Table

According to the PM strategy for single components based on reliability, when the reliability of component reaches the PM reliability threshold, it is necessary to carry out PM or replacement on it. After considering the OM strategy, the opportunistic PM or replacement may be performed before the PM reliability threshold of the component is reached. In the OM model of the belt conveyor drive device, the optimal OM reliability threshold of each component is

The chart of the component reliability varying with the operating time.

As can be seen from Figure

In this paper, we investigate the reliability-based OM strategy for the belt conveyor drive device in a port, taking into account the impact of imperfect repair on the failure rate of the drive device. For this purpose, a hybrid failure rate model that combines the age reduction factor and failure rate increase factor is employed to construct the failure rate function of the components. The reliability-based PM threshold and OM threshold are then proposed to develop the PM model for single components and the OM model of the multi-component system. This study also proposed an irregular imperfect PM strategy in order to better fit the actual situation where the overall trend of component failure rate is rising. The presented model is finally applied for maintenance cost optimization of a belt conveyor drive device in a port. The optimized maintenance schedule and the information on the maintenance cost are determined based on the proposed method. The numerical results demonstrate that the OM strategy can significantly reduce the maintenance cost compared with separate PM and improve the availability of the drive device. It can be concluded that the developed model provides an economically efficient method of the belt conveyor drive device maintenance planning.

All data generated or analyzed during this study are included within this article.

The authors declare that they have no conflicts of interest.

The research work was financially supported by the National Natural Science Foundation of China (no. 71871219).