# Joint Optimization of Preventive Maintenance and Spare Parts Inventory with Appointment Policy.

1. IntroductionThe 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. [1]. Based on the procedure and criterion, the air cycle machine (ACM) is judged as the critical unit of the environmental control system (ECS) used in aircraft, so more preventive maintenances (PM) should be implemented to enhance the safety and availability of ACM. More PM mean more spare parts consumption [2]; therefore, a higher inventory level is required and then more capital fund would be tied up for a long time. In order to reduce the inventory level with the requirement of service level, there has been a lot of research on the joint optimization models of maintenance and inventory. Van Horenbeek et al. [3] reviewed the pertinent literatures and concluded that the joint optimization of maintenance and inventory seemed to be more beneficial than the separate optimization.

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 [4-9]. And the periodic/block policy is widely adopted in the joint optimization models; for example, Acharva et al. [10] found a jointly optimal block preventive replacement and spare provisioning policy for a system consisting of several like units. Brezavscek and Hudoklin [11] developed a stochastic mathematical model to determine the jointly optimal "block replacement" and "periodic review spare provisioning policy." Ali Ilgin and Tunali [12] proposed a simulation optimization approach using genetic algorithm (GA) in joint models about block replacement and continuous review inventory policies. Xie and Wang [13] adopted a continuous review ordering policy (s, S) combined with an inspection period to obtain the optimal (T, s, S). Regattieri et al. [14] proposed an approach that integrated the failure and reparation processes, such as modeling, optimization algorithms, and simulation methods, were proposed to define the best maintenance strategies for complex systems. With CBM widely applied in maintenance practice, monitoring information is more and more integrated into joint optimization models. Wang et al. [15,16] presented a condition-based order-replacement policy for a single-unit system and then proposed a condition-based replacement and spare provisioning policy for deteriorating systems with a number of identical units to optimize inspection interval, maximum stock level, the reorder level, and the preventive replacement threshold. Li and Ryan [17] developed a framework for incorporating real-time condition monitoring information into inventory decisions for spare parts. Romeijnders et al. [18] proposed a two-step method for forecasting spare parts demand using information of component repairs. Louit et al. [19] presented a model to determine the ordering decision for a spare part and assumed that a lead time for spares is random. Tracht et al. [20] developed an enhanced forecast model with the information of both supervisory control and data acquisition to more accurately forecast spare part demand. Wang et al. [21] proposed a prognostics-based spare part ordering and system replacement policy based on the real-time health condition of a deteriorating system subjected to a random lead time. Wang et al. [22] utilized in situ sensor data to predict mechanism of the remaining useful lifetime (RUL) to update the integrated decisions. In addition to the above studies, from the perspective of the whole logistics, repair capacity and deterioration inventory are considered in joint optimization [23-25].

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, (s, S) policy and (s, Q) policy. Because the demand of spare parts can be forecasted through prediction of remaining useful life (RUL) based on the monitoring information, an order for spare parts can be placed correspondingly in advance instead of just when the stock drops to the safety inventory level in the existing inventory policies. So an appointment policy of spare parts based on (s, S) policy is first proposed in this paper to place an order in advance. And then a corresponding joint optimization model of preventive maintenance and spare parts inventory is established. GA has strong robustness and fast convergence, and it is easy for GA to combine with Monte Carlo (MC) method, so the combination method of GA and MC is presented to solve the joint optimization model. The rest of this paper is organized as follows: Section 2 describes the joint strategy with the appointment policy in detail. Section 3 estimates parameters and predicts RUL. Section 4 presents the joint optimization model and its algorithm. And then a case study is developed and the sensitivity to the optimal result is analyzed in Section 5. Finally, the conclusions from the work presented in this study and suggestions for future research are given in Section 6.

2. Joint Strategy with Appointment Policy

2.1. Notion. The main notations that will be used throughout the paper are summarized in Notations.

2.2. Description of Joint Strategy. A system consists of n identical critical units that are subject to Wiener deterioration process. In order to minimize the cost rate of maintenance and spare parts inventory, the proposed joint strategy with the appointment policy based on the prediction of RUL is as follows.

(1) Except for the units that have been known as being in the functional/potential failure state through previous inspections, each unit should be inspected periodically at the times [t.sub.k] = k x T (k [member of] N) and the inspection time can be neglected compared with the inspection interval. The observed deterioration level of the unit i (i = 1, 2, ..., n) at the time [t.sub.k] is denoted as [mathematical expression not reproducible].

(2) If [mathematical expression not reproducible] is greater than [L.sub.f], CM for the unit i would be implemented. If [mathematical expression not reproducible] is between [L.sub.p] and [L.sub.f], PM for the unit i would be implemented. The unit is as good as new after PM/CM, and the PM/CM time can also be neglected compared with the inspection interval. If [mathematical expression not reproducible] is less than [L.sub.p], the unit i keeps on operating and its RUL ([mathematical expression not reproducible]) needs to be predicted, and if [mathematical expression not reproducible] is less than [t.sub.b], one spare part is appointed for the unit i from the stock, so the number ([mathematical expression not reproducible]) of the available spare parts reduces by one. After the inspections and PM/CM, the time when the units are put into operation again is denoted as [t.sup.+.sub.k], and the corresponding deterioration level of the unit i is denoted as [mathematical expression not reproducible].

(3) An order is placed when the number of the available spare parts is less than or equal to s. However, if the ordered spares have not been delivered, no new order could be made. The ordering cost per order is [C.sub.o] and the order lead time is [t.sub.l] = l x T (l [member of] N).

(4) If no spare unit is available in the stock, the unit in potential failure state would keep on operating and the unit in functional failure state would stop operating. The shortage cost per unit time per unit is [c.sub.d].

(5) When the ordered spare units have been delivered, the unit in functional failure state has priority in maintenance. And the remaining spare parts should be put into the stock, and the holding cost and capital charge per unit time per spare part are [c.sub.k].

A hypothetical system with two critical units as an example is given to illustrate the joint strategy, as shown in Figure 1, where S = 3, s = 1, and [t.sub.l] = 3T. The joint decision process of two- unit system is as follows:

(1) At the time [t.sub.1], both [mathematical expression not reproducible] and [mathematical expression not reproducible] are less than [L.sub.p], and both [mathematical expression not reproducible] and [mathematical expression not reproducible] are more than [t.sub.b], so no appointment or maintenance is needed.

(2) At the time [t.sub.2], [mathematical expression not reproducible] is less than [t.sub.b], and one spare part needs to be appointed for the unit 2 from the stock, so [mathematical expression not reproducible] and [mathematical expression not reproducible] reduce from 3 to 2, but the total number of spare parts in the stock remains unchanged; that is, [mathematical expression not reproducible] stock remains unchanged.

(3) At the time [t.sub.3], [mathematical expression not reproducible] is less than [t.sub.b], and one spare part needs to be appointed for the unit 1 from the stock, so [mathematical expression not reproducible] and [mathematical expression not reproducible]. Then an order for ([mathematical expression not reproducible]) spare parts is placed.

(4) At the time [t.sub.4], [mathematical expression not reproducible] is greater than [L.sub.f], and CM for the unit 2 needs to be implemented, so [mathematical expression not reproducible] reduces from 3 to 2; then [mathematical expression not reproducible] and [mathematical expression not reproducible].

(5) At the time [t.sub.5], [mathematical expression not reproducible] is between [L.sub.p] and [L.sub.f], so PM for the unit 1 needs to be implemented, and [mathematical expression not reproducible] reduces from 2 to 1, [mathematical expression not reproducible], and [mathematical expression not reproducible].

(6) At the time [t.sub.6], the ordered spare units have been delivered, so both [mathematical expression not reproducible].

The rest can be done in the same manner.

3. Parameter Estimation and RUL Prediction

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 [26]. So it is assumed that the unit is subject to Wiener deterioration process in this paper.

3.1. The Basic Model of Wiener Process. According to Wiener process, [mathematical expression not reproducible] can be characterized by the following:

[mathematical expression not reproducible], (1)

where X(0) is the initial state, [mu] and [sigma] are the drift coefficient and diffusion coefficient, respectively, and W([t.sub.k]) is a standard Brownian motion; that is, W([t.sub.k]) ~ N(0, [t.sub.k]).

So [mathematical expression not reproducible] is subject to normal distribution and can be described as

[mathematical expression not reproducible]. (2)

3.2. Parameter Estimation. [mu] and [sigma] can be estimated using maximum likelihood estimate and the likelihood function is

[mathematical expression not reproducible]. (3)

Solve the following equations:

[partial derivative] ln (L([mu], [sigma]))/[partial derivative][mu] = 0,

[partial derivative] ln (L([mu], [sigma]))/ [partial derivative][sigma] = 0. (4)

[??] and [[??].sup.2] can be obtained:

[mathematical expression not reproducible]. (5)

3.3. RUL Prediction. Functional failure state threshold is [L.sub.f], so the [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (6)

According to [27], the cumulative distribution function and probability density function of [mathematical expression not reproducible] can be obtained as follows:

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible]. (8)

If [mathematical expression not reproducible], (8) can be transferred to

f(t) = [square root of ([lambda]/2[pi][t.sup.3])] exp[-[lambda][(t - [alpha]).sup.2]/2[[alpha].sup.2]t]. (9)

So f(t) is an inverse Gaussian distribution, and based on its definition, the mean and variance of [mathematical expression not reproducible] can be obtained:

[mathematical expression not reproducible]. (10)

4. Joint Optimization Model and Solution

4.1. Establishment of Joint Optimization Model. According to the joint strategy, the average cost rate (EC) of maintenance and inventory can be represented as a function of T, S, s, [L.sub.p], and [t.sub.b], that is, [f.sub.EC](T, S, s, [L.sub.p], [t.sub.b]), as follows:

[mathematical expression not reproducible]. (11)

Due to the complexity of the joint strategy, it is difficult to derive the analytical formulation of the function [f.sub.EC](T, S, s, [L.sub.p], [t.sub.b]). However, the average cost rate of the maintenance and inventory over an infinite time span can also be simply represented as

[mathematical expression not reproducible], (12)

where [N.sub.p], [N.sub.c], [N.sub.ins], [N.sub.o], [mathematical expression not reproducible], and [mathematical expression not reproducible] can be obtained as follows by finishing the simulation of all events over the simulation time span [t.sub.m].

(1) [N.sub.p], [N.sub.c]. According to the definitions, [N.sub.p] and [N.sub.c] can be expressed as

[N.sub.p] = [m.summation over (k=1)][n.summation over (i=1)] [IPR.sup.k.sub.i],

[N.sub.c] = [m.summation over (k=1)][n.summation over (i=1)] [ICR.sup.k.sub.i]. (13)

The value of [ICR.sup.k.sub.i] and [IPR.sup.k.sub.i] depends on whether PM/CM for the unit i is implemented or not, and then whether PM/CM is implemented depends on two factors: (1) the observed deterioration level [mathematical expression not reproducible] (2) whether there is a spare part or not in the stock. So [ICR.sup.k.sub.i] and [IPR.sup.k.sub.i] can be identified, respectively:

[mathematical expression not reproducible], (14)

(2) [N.sub.ins]. According to the definition, [N.sub.ins] can be expressed as

[N.sub.ins] = [m.summation over (k=1)][n.summation over (i=1)] [I.sup.k.sub.i]. (15)

As described in the joint strategy, if the unit i has been known as being in the functional/potential failure state, the inspection of the unit i does not need to be implemented regardless of whether there are spare parts or not. So [I.sup.k.sub.i] can be identified:

[mathematical expression not reproducible]. (16)

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 [mathematical expression not reproducible] can be obtained:

[mathematical expression not reproducible]. (17)

(3) [N.sub.o], [mathematical expression not reproducible]. According to the definition, [N.sub.o] can be expressed as

[mathematical expression not reproducible]. (18)

[mathematical expression not reproducible] depends on whether an order is placed at the time [t.sub.k], so [mathematical expression not reproducible] can be identified:

[mathematical expression not reproducible]. (19)

According to the joint strategy, [mathematical expression not reproducible] is equal to [mathematical expression not reproducible]. However, only one order is permitted at the same time; that is, one order is permitted when there is no undelivered order, so [mathematical expression not reproducible] order can be obtained:

[mathematical expression not reproducible]. (20)

Since [mathematical expression not reproducible] and [mathematical expression not reproducible] should be obtained first.

(a) [mathematical expression not reproducible] depends on two things, the number of remaining spare parts ([mathematical expression not reproducible]) and the number of the delivered spare parts ([mathematical expression not reproducible]), so [mathematical expression not reproducible] can be obtained:

[mathematical expression not reproducible]. (21)

The number of the units whose deterioration levels are greater than [L.sub.p] is [[summation].sup.n.sub.i=1]([ICR.sup.k-1.sub.i] + [IPR.sup.k-1.sub.i]) at the time [t.sub.k-1], so [mathematical expression not reproducible] is equal to [mathematical expression not reproducible]. However, [mathematical expression not reproducible] does not allow being negative value, so [mathematical expression not reproducible] should be expressed as

[mathematical expression not reproducible]. (22)

(b) [mathematical expression not reproducible] can be obtained:

[mathematical expression not reproducible]. (23)

At the time [t.sub.k], even if [mathematical expression not reproducible] is less than [t.sub.b] and [mathematical expression not reproducible] is less than [L.sub.p], no new appointment could be made for the unit i if an appointment for the unit i has been made at the time [t.sub.k-1]; that is, only one appointment for the unit i is permitted at the same time. So [app.sup.k.sub.i] can be identified:

[mathematical expression not reproducible]. (24)

(4) [mathematical expression not reproducible]. According to the definition, [mathematical expression not reproducible] can be expressed as

[mathematical expression not reproducible]. (25)

Because [mathematical expression not reproducible] means that [mathematical expression not reproducible], [mathematical expression not reproducible] can be identified:

[mathematical expression not reproducible]. (26)

4.2. Algorithm of Model. The combination method of GA and MC is adopted to obtain an approximate optimization result [[theta].sup.*] = (T, S, s, [L.sub.p], [t.sub.b]). The flow diagram for the combination method is given in Figure 2.

The main steps of the flow diagram are as follows.

Step 1. "Initialize population (N)" to obtain N groups of initial parameters (T, S, s, [L.sub.p], [t.sub.b]).

Step 2. Evaluate the fitness [f.sub.EC](T, S, s, [L.sub.p], [t.sub.b]) of each group of the parameters by simulation. In order to eliminate the randomness of simulation, [t.sub.m] is taken as a big enough value, and the mean value of K simulation results is taken as the fitness of each population; that is, the fitness of each population is equal to [[summation].sup.K.sub.k=1][f.sup.k.sub.EC] (T, S, s, [L.sub.p], [t.sub.b])/K, where [f.sup.k.sub.EC](T, S, s, [L.sub.p], [t.sub.b]) is evaluated through number k simulation described in Figure 2(b).

Step 3. If the variance of all population's fitness obtained based on Step 2 is less than a sufficiently small value [epsilon], it means that it is unnecessary to optimize further. The parameters whose fitness is the minimum are the optimal result [[theta].sup.*] = ([T.sup.*], [S.sup.*], [s.sup.*], [L.sub.p.sup.*], [t.sub.b.sup.*]); otherwise, go to Step 4.

Step 4. The new populations are obtained through selection, crossover, and mutation based on GA method and go to Step 2 if the iteration does not end.

5. Case Study

5.1. Estimation of Deterioration Parameter. 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 ([L.sub.f] = 10[degrees]C), ACM would be removed and overhauled. The outlet temperatures of 20 ACMs in AIR CHINA have been inspected very 1000 flight hours (FH), that is, T = 1000 FH, and the data of those outlet temperatures are described in Figure 3.

According to Section 3.1, the outlet temperatures can be seen as subject to Wiener deterioration process if the deterioration increment of the outlet temperatures can be proven to be subject to normal distribution. Based on the deterioration increment of the outlet temperatures of 20 ACMs, the parameters can be estimated as shown in Figure 4. With the Kolmogorov-Smirnov test, the P value is 0.2 greater than 0.05 when the level of significance is 5%, so the deterioration increment of the outlet temperatures fellows normal distribution N(0.333, 0.314), and the deterioration parameters can be obtained: [??] = 3.33 x [10.sup.-4]; [??] = 0.0099. Because the initial outlet temperature is usually 2[degrees]C, the outlet temperatures can be expressed as

X(t) = 2 + 0.000333t + 0.0099W (t). (27)

5.2. Joint Optimization Simulation. According to the preliminary statistics, [C.sub.ins] = 1,000, [C.sub.p] = 100,000, [C.sub.c] = 400,000, [C.sub.o] = 5,000 (RMB), [c.sub.d] = 100, [c.sub.k] = 10 (RMB/FH), and [t.sub.l] = 2,000 (FH). In the GA, roulette wheel selection and elite strategy are used in selection. Crossover probability is 0.8. Mutation probability is 0.05. Population size is 80.

Maximum generation is 300. In order to eliminate the randomness of simulation, the simulation time span [t.sub.m] is 100,000 (FH) and the mean value of 50 simulation results is taken as the fitness of each population; that is, fitness = [[summation].sup.50.sub.k=1][f.sup.k.sub.EC](T, S, s, [L.sub.p], [t.sub.b])/50. The iterative process of the GA is shown in Figure 5, and the optimal result [[S.sup.*], [s.sup.*], [L.sup.*.sub.p], [t.sup.*.sub.b]] is [4, 1, 9.17, 3391], and [f.sub.EC]([S.sup.*], [s.sup.*], [L.sub.p.sup.*], [t.sub.b.sup.*]) = 116.03 (RMB/FH).

With the optimal result [[S.sup.*], [s.sup.*], [L.sup.*.sub.p], [t.sup.*.sub.b]], the change of the inventory level over the simulation time span is shown in Figure 6, where the actual maximum inventory level is 5 greater than [S.sup.*] = 4, which is caused by applying the appointment policy. For example, at time [t.sub.43] = 43000 FH, [mathematical expression not reproducible] and [mathematical expression not reproducible], as shown with the red point, respectively, in Figures 6 and 7. Therefore, based on (20), [mathematical expression not reproducible], as shown with the red point, respectively, in Figures 8 and 9. So an order for 5 spare parts is placed.

The order for 5 spare parts will arrive at time [t.sub.45] = 45000 FH, there is one PM at time [t.sub.44] = 44000 FH as shown with the red point in Figure 10, and there is no CM at time [t.sub.44] = 44000 FH and [t.sub.45] = 45000 FH as shown in Figure 11, so one spare part is used and the inventory level becomes 5 at time [t.sub.45] = 45000 FH, as shown with a blue point in Figure 6.

5.3. Comparison with the Separate Optimization Method. According to the data in Figure 6, the average required number of spare parts per 1000 FH over the simulation time span is about 2.38. The ACM normally operates about 3000 FH each year; therefore, the average annual required number (D) of spare parts is about 7. In most airlines, according to the calculation method of Boeing and Airbus, it is assumed that D is subject to Poisson distribution. In this paper it is assumed that the required shortage rate (FR) of ACM is less than 0.1%; therefore, S and s can be obtained in accordance with the calculation method of Boeing and Airbus as follows:

FR = [summation over (s)] [D.sup.(D+s)] x [e.sup.-D]/(D + s)!. (28)

By (28), it can be obtained that s is equal to 3, and the maximum inventory S is equal to D + s = 7 + 3 = 10. So [S, s, [L.sup.*.sub.p]. [t.sup.*.sub.b]] = [10, 3, 9.17, 3391] is taken as an input to the simulation model, the average cost rate is 168.66 (RMB/FH) that is 45.36% higher than the above optimal result 116.03 (RMB/FH), and the average inventory level without the joint optimization is 6.5 that is far greater than 2.37. The inventory difference of whether the joint optimization is adopted or not is shown in Figure 12.

5.4. Comparison with the Traditional Joint Optimization. In the same example, without the appointment policy in the optimization model, the optimal result is [[S.sup.*], [s.sup.*], [L.sup.*.sub.p] = [4, 1, 9.10] and [f.sub.EC]([S.sup.*], [s.sup.*], [L.sub.p.sup.*] = 120.95 (RMB/FH) that is 4.24% higher than the above optimal result 116.03 (RMB/FH) with the appointment policy. The iterative process of the GA is shown in Figure 13, and the change of inventory level with the appointment policy or not over the simulation time span is shown in Figure 14.

From Figure 14, it can be found that the average required number of spare parts per 1000 FH per unit without the appointment policy is about 0.132 that is 10.92% higher than with the appointment policy. In addition, without the appointment policy, the shortage happens at time [t.sub.26] = 26000 FH; however the shortage never happens over the simulation time span with the appointment policy. In summary, the appointment policy reduces not only the average cost rate but also the probability of shortage.

5.5. Sensitivity Analysis. 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.

5.5.1. Sensitivity of CM Cost [C.sub.c]. Table 1 shows the different optimal results under the different [C.sub.c].

From Table 1, it can be obtained that when [C.sub.c] increases, [L.sub.p] decreases; however [t.sub.b] increases, which means that more PM should be implemented to avoid the functional failures. So the corresponding required number of spare parts increases, as shown in Figure 15. For example, the accumulated required number of spare parts over the simulation time span is 78 when [C.sub.c] = 200,000 RMB; however, it is 85 when [C.sub.c] = 700,000 RMB. In addition, Table 1 shows that the sharp increase of [C.sub.c] does not result in a substantial increase of EC, for which the reason is that the CM will be almost avoided when [C.sub.c] is taken as a big value.

Both [c.sub.d] and [C.sub.c] are associated with the functional failure; therefore, they have the similar influence on the optimal result. So the sensitivity of [c.sub.d] to the optimal result does not need to be further discussed.

5.5.2. Sensitivity of Lead Time [t.sub.l]. Table 2 shows the different optimal results under the different [t.sub.l].

From Table 2, it can be obtained that the safety inventory level is equal to 0 when the lead time is 1000 FH. With the increase of the lead time, an order for spare parts should be placed in advance to ensure a timely supply of spare parts, or a high inventory level should be kept, which is in accordance with the fact that higher inventory level is needed if spare parts cannot be delivered immediately.

5.5.3. Sensitivity of Lead Times [C.sub.ins], [C.sub.o], and [C.sub.k]. Tables 3 and 4 show the different optimal results under the different [C.sub.ins] and [C.sub.o], respectively. Tables 3 and 4 show that [C.sub.ins] has almost no impact on the optimal result, for which the reason is that the total number of orders over the simulation time span is relatively steady. [C.sub.o] also has almost no impact on the optimal result, for which the reason is that [C.sub.o] are too small compared with [C.sub.c] and [C.sub.p].

Table 5 shows the different optimal results under the different [C.sub.k], and in Table 5, with the increase of [C.sub.k], the inventory level decreases significantly, which can save the holding cost and capital charge of spare parts. But the corresponding appointment and PM need to be carried out in advance in order to make up the low inventory level.

6. Conclusion

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:

(1) Adopting the appointment policy in the optimization model reduces not only the cost rate but also the probability of shortage.

(2) 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.

(3) 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.

Notations

n: Number of identical critical units, n [member of] N

i: Number of units, i = 1, 2, ..., n

T: Inspection interval

k: The serial number of inspections, k [member of] N

[t.sub.k]: The kth inspection time

[t.sup.+.sub.k]: Time of putting into operation after the kth inspection, and between [t.sub.k] and [t.sup.+.sub.k], PM/CM may be implemented or not

[mathematical expression not reproducible]: Deterioration level of the unit i at the time [t.sub.k]

[mathematical expression not reproducible]: Deterioration level of the unit i at the time [t.sup.+.sub.k]

[mathematical expression not reproducible]: Deterioration increment of the unit i from [t.sup.+.sub.k] to [t.sub.k], [mathematical expression not reproducible]

[t.sub.l]: Order lead time

[t.sub.m]: Simulation time span, [t.sub.m] = m x T (m [member of] N)

[L.sub.f]: Functional failure threshold, and if [mathematical expression not reproducible] is greater than [L.sub.f], CM for the unit i should be implemented

[L.sub.p]: Potential failure threshold ([L.sub.p] < [L.sub.f]), and if [mathematical expression not reproducible] is between [L.sub.p] and [L.sub.f], PM for the unit i should be implemented

[mathematical expression not reproducible]: Predicted RUL of the unit i at the time [t.sub.k]

[t.sub.b]: Appointment threshold, and if [mathematical expression not reproducible] is less than [t.sub.b], one spare part needs to be appointed for the unit i from the stock

[C.sub.ins]: Cost of inspection (per unit)

[N.sub.ins]: Total number of inspections over the time span [t.sub.m]

[C.sub.p]: Cost of PM (per unit)

[N.sub.p]: Total number of PM over the time span [t.sub.m]

[C.sub.c]: Cost of corrective maintenance (CM) (per unit)

[N.sub.c]: Total number of CM over the time span [t.sub.m]

[C.sub.o]: Ordering cost per order

[N.sub.o]: Total number of orders over the time span [t.sub.m]

[mathematical expression not reproducible]: Code to signify whether an order is placed at the time [t.sub.k]("[mathematical expression not reproducible]" means that an order is placed and "[mathematical expression not reproducible]" otherwise)

[c.sub.d]: Shortage cost per unit time per unit

[mathematical expression not reproducible]: Number of the units in functional failure state between [t.sup.+.sub.k-1] and [t.sub.k]

[c.sub.k]: Holding cost and capital charge per unit time per spare part

[mathematical expression not reproducible]: Number of spare parts between [t.sup.+.sub.k-1] and [t.sub.k]

[mathematical expression not reproducible]: Code to signify whether the unit i is in functional failure state the time [t.sub.k]("[mathematical expression not reproducible]" means that the unit i is in functional failure state and "[mathematical expression not reproducible]" otherwise)

[IPR.sup.k.sub.i]: Code to signify whether PM of the unit i is implemented at the time [t.sub.k] ("[IPR.sup.k.sub.i] = 1" means that PM is implemented and "[IPR.sup.k.sub.i] = 0" otherwise)

[ICR.sup.k.sub.i]: Code to signify whether CM of the unit i is implemented at the time [t.sub.k] ("[ICR.sup.k.sub.i] = 1" means that CM is implemented and "[ICR.sup.k.sub.i] = 0" otherwise)

[I.sup.k.sub.i]: Code to signify whether an inspection of the unit i is implemented at the time [t.sub.k] ("[I.sup.k.sub.i] = 1" means that an inspection is implemented and "[I.sup.k.sub.i] = 0" otherwise)

Undel: Code to signify whether all orders until the current time has been delivered [t.sub.k] ("Undel = 0" means that all orders have been delivered and "Undel = 1" otherwise)

S: Maximum inventory level

s: Safety inventory level

[app.sup.k.sub.i]: Code to signify whether an appointment for the unit i is made at the time [t.sub.k] ("[app.sup.k.sub.i] = 1" means that an appointment is made and "[app.sup.k.sub.i] = 0" otherwise)

[mathematical expression not reproducible]: Total number of appointed parts at the time [t.sub.k]

[mathematical expression not reproducible]: Actual inventory level at time [t.sub.k]

[mathematical expression not reproducible]: Number of the available spare parts at the time [t.sub.k], [mathematical expression not reproducible]

[mathematical expression not reproducible]: Number of the ordered spare parts at time [t.sub.k], [mathematical expression not reproducible].

Conflicts of Interest

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

https://doi.org/10.1155/2017/3493687

Acknowledgments

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).

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Jing Cai, (1) Yibing Yin, (1) Li Zhang, (1) and Xi Chen (2)

(1) College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

(2) Shanghai Aircraft Customer Service Co., Ltd., Shanghai 200241, China

Correspondence should be addressed to Jing Cai; caijing@nuaa.edu.cn

Received 2 September 2016; Accepted 12 February 2017; Published 20 March 2017

Academic Editor: Inmaculada T. Castro

Caption: Figure 1: Joint decision process of two-unit system.

Caption: Figure 2: Flow diagram for the combination method.

Caption: Figure 3: The collected outlet temperature data of the ACM.

Caption: Figure 4: The distribution of the deterioration increment.

Caption: Figure 5: The change of the fitness with iteration times (with the appointment policy).

Caption: Figure 6: The change of the inventory level.

Caption: Figure 7: The change of number of appointed spare parts.

Caption: Figure 8: The change of number of available spare parts.

Caption: Figure 9: The change of order number.

Caption: Figure 10: The change of number of PM.

Caption: Figure 11: The change of number of CM.

Caption: Figure 12: Inventory difference of whether joint optimization is adopted or not.

Caption: Figure 13: The change of the fitness with iteration times (without the appointment policy).

Caption: Figure 14: The change of the inventory level (with the appointment policy or not).

Caption: Figure 15: Accumulated ordered number of spare parts in different [C.sub.c].

Table 1: The different optimal results under the different [C.sub.c]. [C.sub.c] S s [L.sub.p] [t.sub.b] 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 Table 2: The different optimal results under the different [t.sub.l]. [t.sub.l] S s [L.sub.p] [t.sub.b] [EC.sub.[infinity]] 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 Table 3: The different optimal results under the different [C.sub.ins]. [C.sub.ins] S s [L.sub.p] [t.sub.b] [EC.sub.[infinity]] 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 Table 4: The different optimal results under the different [C.sub.o]. [C.sub.o] S s [L.sub.p] [t.sub.b] [EC.sub.[infinity]] 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 5: The different optimal results under the different [C.sub.k]. [C.sub.k] S s [L.sub.p] [t.sub.b] [EC.sub.[infinity]] 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

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Title Annotation: | Research Article |
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Author: | Cai, Jing; Yin, Yibing; Zhang, Li; Chen, Xi |

Publication: | Mathematical Problems in Engineering |

Date: | Jan 1, 2017 |

Words: | 7139 |

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