Nowadays, the elevator has become an indispensable tool in people’s daily life with the development of economy and urbanization. Although elevators bring convenience to people in life, the safety problems of elevators have aroused the attention of all parties due to frequent swallowing or fall incident in elevators, as shown in Table 1. Therefore, the elevator needs regular maintenance in order to ensure that the elevators can operate stably, efficiently and safely. However, there are many problems in the actual maintenance of elevator equipment which make it difficult to guarantee the safe operation of the elevator, such as improper maintenance methods, improper updating of elevator equipment, high maintenance costs and so on. So, it is the primary problem for the safe operation
Table 1. The number of elevator accidents (occurred in China).
of the elevator to take which scientific management approach to reduce the equipment failure rate.
There are many scholars to study the preventive maintenance of equipment. Lim et al. constructed a periodic preventive maintenance plan model for the equipment by setting the number of preventive maintenance times as the optimization variable  . Zhuo et al. considered the equipment maintenance costs, equipment age, learning effects and other factors into the improvement factor model and established a periodic preventive maintenance model of reliability limit device  . Qin et al. introduced the service age decline factor and the inefficiencies increment factor for the recession system, established a single equipment prevention and repair non-new model, and optimized the dynamic sequential maintenance plan based on reliability  . Dohi sets the expected cost of per unit time as the threshold of the restricted maintenance costs strategy. If the equipment maintenance cost is less than the set threshold, the equipment only needs to be maintained and the equipment must be replaced when this threshold is reached  . Based on previous studies      , this paper attempts to establish a model that satisfies certain reliability and minimizes the average maintenance cost. Finally, the optimal preventive maintenance number and preventive maintenance interval of the elevator can be obtained by solving this model.
2. Problem Description
Elevator equipment will inevitably fail because of the accumulation of running time and service age. Preventive maintenance can reduce the occurrence of equipment failure. However, this maintenance process requires a certain cost (such as labor costs, spare parts costs, outage costs). The cost of the entire system will increase if the equipment is frequently maintained (causing over- maintenance). If preventive maintenance activities are not enough, the frequency of equipment failure will be greatly increased and post-maintenance costs and downtime costs of the entire system have also increased. This paper explores the elevator equipment optimal maintenance strategy to minimize the average main- tenance cost. This strategy is mainly based on the effect of preventive maintenance, which is “repair non-new”.
The problems of elevator equipment can be described as follows: there are three maintenance methods of preventive maintenance, minor repairs and equipment replacement under this preventive maintenance. The equipment is maintained by preventive maintenance in the whole life cycle, assuming that the optimal preventive maintenance cycle of the equipment is N. When the equipment system reliability in the N-th preventive maintenance reaches the threshold R set in advance, the device will be replaced; In other N − 1 times preventive maintenance cycle, the equipment fails and cannot run, unplanned maintenance is taken (minor repairs) to restore equipment to function, but minor repairs cannot change equipment failure rate and reliability.
3. Establishment of Preventive Maintenance Model for Elevator Equipment
3.1. Model Assumptions
In order to analyze conveniently, this paper makes the following assumptions:
The model satisfies the following assumptions:
1) The initial state of the equipment is completely new;
2) The state of the equipment can be monitored and diagnosed in real time, and the occurrence of equipment failure is random;
3) Minor repairs (non-planned maintenance) are taken, when the equipment fails during preventive maintenance;
4) During the limited service life of the equipment, the maintenance mode of the equipment is only preventive maintenance and non-planned maintenance, without equipment replacement;
5) The failure of the system can be described by mathematical function, and the distribution of weibull is adopted in this paper;
6) The system has sufficient maintenance resources and the failure can be maintained in time.
3.2. Symbols Description
Firstly, the paper defines the symbols to be used in the modeling, as shown in Table 2.
3.3. Model Establishment
1) The determination of the failure rate function
Table 2. Symbols and its meanings.
As the equipment age and the number of maintenance increases, the possibility of a device failure also increases. In other words, the equipment failure rate gradually increases. However, equipment maintenance work does not make it as a new unit.
This paper introduces the mixed adjustment factor based on service age decreasing factor and failure rate increasing factor and gets the failure rate function before and after the elevator equipment preventive optimization, which is shown below    :
The approximate function of failure rate is
Among them, ai and bi are judged or estimated by expert experience, which is generally the result of historical data analysis of elevator equipment operation.
2) Optimization of non-planned maintenance cost
The improvement factor is introduced to describe the improvement degree of the equipment after preventive maintenance, which reflects the functional relationship between the equipment unplanned maintenance cost and the maintenance effect. The improvement factor is set as follows  :
Among them, is preventive maintenance cost, is replacement cost, and “ ”, is cost regulation coefficient, v is time adjustment coefficient, w is learning effect adjustment coefficient, , θ is percentage of empirical curves, which is according to the experience judgment or estimation.
Therefore, the numbers of failure of the equipment in the i-th and i-1th preventive maintenance cycles can be obtained:
The cost of non-planned maintenance of equipment between the i-th and i-1th preventive maintenance cycle is:
3) Shutdown cost
The maintenance of the elevator equipment is bound to affect the continuous operation of the equipment, resulting in the loss of downtime. According to the above parameter, the cost of shutdown of equipment maintenance during the i-1th and i-th preventive maintenance cycles is obtained as below:
where, is a single preventive maintenance time, is a single unscheduled maintenance time, and is the improvement factor in the i-1th and i-th preventive maintenance cycles, is the unit time downtime loss.
When i = N, at this time
4) Objective function construction
According to the above analysis, preventive maintenance cost , total unplanned maintenance cost and shutdown cost constitute the total maintenance cost of the whole service life period.
When , the total maintenance cost of the i-th preventive maintenance cycle is
When i = N, the total maintenance cost is:
Combine formula (2-6) and (2-7), the equipment maintenance cost rate in the whole life is obtained：
To ensure the reliability of the elevator equipment is a prerequisite for preventive maintenance. It is assumed that the failure rate function of elevator
equipment obeys Weibull distribution, that is . According to the failure rate, we define , , where represents
the failure distribution function, and is the reliability of the elevator equipment. The unreliability of the elevator equipment is its failure situation, , so, we can deduce the relationship between the failure rate of elevator equipment and its reliability, .
The paper supposes that the upper limit of the number of maintenance cycles is , and the lower limit of reliability is 0.7, so the constraint is:
So, the objective function is:
4. Case Study
This paper takes the elevator traction machine as an example and assumes that the equipment failure rate of the system obeys the weibull distribution with shape parameter and scale parameter . Assume that the initial state of the device is completely new, that is, the initial service age of the device is 0, and the minimum reliability is , and all values of parameters are shown in Table 3.
Therefore, the failure function is
The run-time of the equipment is 3000, a single preventive maintenance costs , a single preventive maintenance time ; a single unplanned minor repair cost , a single unplanned minor repair time ; unit downtime cost of loss .
For the improvement factor , among them, the equipment
preventive maintenance cost , the equipment replacement cost , the cost adjustment parameter , the time adjustment parameter , the empirical curve parameter percentage , learning effect adjustment parameters , so
Figure 1 shows the relationship between learning effect and improvement factors. From the graph, we can see that considering learning effect the
Table 3. List of parameter values.
improvement factor is more than it when without considering learning effect at the same number of maintenance. That is said the improvement factor is decreased without considering the learning effect.
The optimal solution is (0.83, 7) by using genetic algorithm to solve based on Matlab, which means that when the traction machine for the seventh preventive maintenance, the reliability threshold is 0.83, the lowest maintenance cost of the equipment is 279.3710, the optimal number of maintenance N is 7, and this time is to replace the traction machine. Table 4 shows the (R, N) combinations of the partial maintenance strategies and the average maintenance costs EC under the combination.
Figure 1. Comparison of improvement factors with or without learning effect.
Table 4. EC values for some maintenance strategies (R, N).
The optimization results are shown in Figure 2:
As can be seen from Figure 2, the average maintenance cost decreases first and then increases with the improvement of reliability, and the average maintenance cost is the smallest when the number of maintenance is seven times. In addition, as the system reliability requirements increase, the optimal number of preventive maintenance cycles also increases, which is in line with the actual maintenance of the elevator traction machine.
When the optimal number of maintenance N is 7, preventive maintenance intervals is shown in Table 5.
From Table 5, it can be seen that the preventive maintenance intervals of the traction machine are decreasing under the optimal maintenance strategy (0.83, 7), which accords with the degradation characteristics of the equipment itself. In other words, as the running time of the traction machine increases, it needs to be carried out more frequent maintenance; the results are consistent with the actual situation, which reflects the validity and practicality of the model.
Firstly, this paper presents a mathematical description of the maintenance of elevator equipment, and then the system failure rate function is optimized by introducing the age-dependent decrement factor and failure rate increment factor. This paper introduces improvement factors in order to break the traditional
Figure 2. The relationship between average maintenance costs and system reliability.
Table 5. Optimal maintenance intervals under the optimal preventive maintenance intervals.
assumption of periodic maintenance strategy, which is expressed as “repair as new”. The equipment condition maintenance model is constructed on the basis of reconsidering the repair costs and downtime losses of the elevator equipment. Finally, the model combined with a numerical example is solved by genetic algorithm with the aim of minimizing the average maintenance cost. The optimal equipment maintenance strategy is found and the practicality and validity of the model are verified. However, the relationship between one equipment and another is not discussed in this article, which needs to be studied in the future.