Under the background of the wide application of condition-based maintenance (CBM) in maintenance practice, the joint optimization of maintenance and spare parts inventory is becoming a hot research to take full advantage of CBM and reduce the operational cost. In order to avoid both the high inventory level and the shortage of spare parts, an appointment policy of spare parts is first proposed based on the prediction of remaining useful lifetime, and then a corresponding joint optimization model of preventive maintenance and spare parts inventory is established. Due to the complexity of the model, the combination method of genetic algorithm and Monte Carlo is presented to get the optimal maximum inventory level, safety inventory level, potential failure threshold, and appointment threshold to minimize the cost rate. Finally, the proposed model is studied through a case study and compared with both the separate optimization and the joint optimization without appointment policy, and the results show that the proposed model is more effective. In addition, the sensitivity analysis shows that the proposed model is consistent with the actual situation of maintenance practices and inventory management.
The critical unit in complex systems has an important impact on the system utilization, total operating costs, and so on, and both the procedure and criterion of how to judge the critical unit are presented by Godoy et al. [
The joint optimization models are developing with the change of maintenance policies from age-based, periodic/block to condition-based maintenance (CBM). The age-based policy has been applied in the joint optimization models, such as [
To summarize, increasing studies have focused on maintenance and inventory together, and it is the tendency to integrate the monitoring information into the joint optimization, in order to optimize maintenance decision, inventory level, and so on. The joint optimization based on monitoring information makes just-in-time inventory possible for a single component; however, it almost does not change the existing inventory policies of many identical components, for instances, (
The main notations that will be used throughout the paper are summarized in Notations.
A system consists of
(
(
(
(
(
A hypothetical system with two critical units as an example is given to illustrate the joint strategy, as shown in Figure At the time At the time At the time At the time At the time At the time
Joint decision process of two-unit system.
The rest can be done in the same manner.
Wiener process can be used to describe a variety of performance degradation process of typical unit and has been applied in many fields like unit corrosion, mechanical vibration, and so forth [
According to Wiener process,
So
Solve the following equations:
Functional failure state threshold is
According to [
If
So
According to the joint strategy, the average cost rate (
Due to the complexity of the joint strategy, it is difficult to derive the analytical formulation of the function
The value of
As described in the joint strategy, if the unit
The unit becomes as good as new if PM/CM is implemented after an inspection; otherwise the deterioration level remains the same, so the deterioration level
According to the joint strategy,
Since
The number of the units whose deterioration levels are greater than
At the time
Because
The combination method of GA and MC is adopted to obtain an approximate optimization result
Flow diagram for the combination method.
Flow diagram for GA
Flow diagram for MC
The main steps of the flow diagram are as follows.
“Initialize population (
Evaluate the fitness
If the variance of all population’s fitness obtained based on Step
The new populations are obtained through selection, crossover, and mutation based on GA method and go to Step
ACM is an important refrigeration unit of the ECS used in pressurized gas turbine-powered aircraft, and the outlet temperature of ACM rises with its performance deterioration. When the outlet temperature rises up to the functional failure threshold (
The collected outlet temperature data of the ACM.
According to Section
The distribution of the deterioration increment.
According to the preliminary statistics,
The change of the fitness with iteration times (with the appointment policy).
With the optimal result
The change of the inventory level.
The change of number of appointed spare parts.
The change of number of available spare parts.
The change of order number.
The order for 5 spare parts will arrive at time
The change of number of PM.
The change of number of CM.
According to the data in Figure
By (
Inventory difference of whether joint optimization is adopted or not.
In the same example, without the appointment policy in the optimization model, the optimal result is
The change of the fitness with iteration times (without the appointment policy).
The change of the inventory level (with the appointment policy or not).
From Figure
The input parameters of the joint optimization model may not be absolutely accurate; for example, the shortage loss or CM cost of ACM is very difficult to be estimated. Therefore, it is necessary to analyze the parameter sensitivity to the optimal result.
Table
The different optimal results under the different
|
|
|
|
|
EC |
---|---|---|---|---|---|
200000 | 3 | 1 | 9.27 | 2943 | 114.57 |
300000 | 3 | 1 | 9.22 | 3254 | 115.82 |
400000 | 4 | 1 | 9.17 | 3391 | 116.03 |
500000 | 4 | 1 | 9.11 | 3721 | 117.08 |
700000 | 4 | 1 | 9.02 | 3982 | 118.98 |
From Table
Accumulated ordered number of spare parts in different
Both
Table
The different optimal results under the different
|
|
|
|
|
EC∞ |
---|---|---|---|---|---|
1000 | 1 | 0 | 9.37 | 3482 | 110.22 |
2000 | 4 | 1 | 9.17 | 3391 | 116.03 |
3000 | 5 | 2 | 9.13 | 3132 | 126.13 |
4000 | 7 | 2 | 9.06 | 3721 | 131.93 |
6000 | 9 | 3 | 9.16 | 3693 | 153.24 |
From Table
Tables
The different optimal results under the different
|
|
|
|
|
EC∞ |
---|---|---|---|---|---|
200 | 4 | 1 | 9.03 | 3576 | 98.66 |
500 | 4 | 1 | 9.10 | 3422 | 103.34 |
1000 | 4 | 1 | 9.17 | 3391 | 116.03 |
2000 | 4 | 1 | 9.21 | 3292 | 131.90 |
4000 | 4 | 1 | 9.11 | 3591 | 161.09 |
The different optimal results under the different
|
|
|
|
|
EC∞ |
---|---|---|---|---|---|
1000 | 4 | 1 | 9.17 | 3382 | 115.22 |
2000 | 4 | 1 | 9.19 | 3366 | 115.43 |
5000 | 4 | 1 | 9.17 | 3391 | 116.03 |
7000 | 4 | 1 | 9.16 | 3421 | 116.93 |
10000 | 4 | 1 | 9.12 | 3439 | 123.24 |
Table
The different optimal results under the different
|
|
|
|
|
EC∞ |
---|---|---|---|---|---|
1 | 6 | 2 | 8.91 | 3692 | 94.78 |
5 | 4 | 2 | 9.21 | 3497 | 107.82 |
10 | 4 | 1 | 9.17 | 3391 | 116.03 |
20 | 3 | 0 | 9.23 | 3543 | 132.88 |
40 | 2 | 0 | 9.27 | 3738 | 151.63 |
In this study, a joint optimization model with the appointment policy is first proposed based on the prediction of RUL in order to place an order for spare parts in advance and minimize the cost rate of maintenance and inventory, and the algorithm has been developed and described in detail. In the case study, the proposed model and its optimal results are analyzed, compared with both the model without joint optimization and the joint optimization without the appointment policy. Finally the parameter sensitivity to the optimal result is analyzed.
Through the case study, the conclusions are as follows: Adopting the appointment policy in the optimization model reduces not only the cost rate but also the probability of shortage. The proposed optimization model saves 45.36% of the cost compared with the model without joint optimization and saves 4.24% of the cost compared with the joint optimization without the appointment policy, which means that the proposed optimization model is effective. The results of sensitivity analysis show that the proposed optimization model is consistent with the actual situation of maintenance practices and inventory management.
In reality the inventory management is always classified into the initial provisioning phase and ongoing provisioning phase. The initial provisioning phase is called a “maintenance honeymoon” with limited demand for spare parts, differing from the ongoing provisioning phase, so the different spare provisioning phases needed to be considered in the joint optimization model in the further research.
Number of identical critical units,
Number of units,
Inspection interval
The serial number of inspections,
The
Time of putting into operation after the
Deterioration level of the unit
Deterioration level of the unit
Deterioration increment of the unit
Order lead time
Simulation time span,
Functional failure threshold, and if
Potential failure threshold (
Predicted RUL of the unit
Appointment threshold, and if
Cost of inspection (per unit)
Total number of inspections over the time span
Cost of PM (per unit)
Total number of PM over the time span
Cost of corrective maintenance (CM) (per unit)
Total number of CM over the time span
Ordering cost per order
Total number of orders over the time span
Code to signify whether an order is placed at the time
Shortage cost per unit time per unit
Number of the units in functional failure state between
Holding cost and capital charge per unit time per spare part
Number of spare parts between
Code to signify whether the unit
Code to signify whether PM of the unit
Code to signify whether CM of the unit
Code to signify whether an inspection of the unit
Code to signify whether all orders until the current time has been delivered
Maximum inventory level
Safety inventory level
Code to signify whether an appointment for the unit
Total number of appointed parts at the time
Actual inventory level at time
Number of the available spare parts at the time
Number of the ordered spare parts at time
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was cosupported by the Fundamental Research Funds for the Central Universities (no. NS2015072) and the National Natural Science Foundation of China (nos. 61079013 and U1233114).