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

Authors: Jing Cai; Yibing Yin; Li Zhang; Xi Chen
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

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. [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. Brezavšček 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 ofn 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 tk=k·Tk∈N and the inspection time can be neglected compared with the inspection interval. The observed deterioration level of the unit ii=1,2,…,n at the time tk is denoted as Xitk.(2) If Xitk is greater than Lf, CM for the unit i would be implemented. If Xitk is between Lp and Lf, 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 Xitk is less than Lp, the unit i keeps on operating and its RUL (RULitk) needs to be predicted, and if RULitk is less than tb, one spare part is appointed for the unit i from the stock, so the number (Na-sparetk) 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 tk+, and the corresponding deterioration level of the unit i is denoted as Xitk+.(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 Co and the order lead time is tl=l·Tl∈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 cd.(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 ck.A hypothetical system with two critical units as an example is given to illustrate the joint strategy, as shown in Figure1, where S=3, s=1, and tl=3T. The joint decision process of two- unit system is as follows:(1)
At the timet1, both X1t1 and X2t1 are less than Lp, and both RUL1t1 and RUL2t1 are more than tb, so no appointment or maintenance is needed.(2)
At the timet2, RUL2t2 is less than tb, and one spare part needs to be appointed for the unit 2 from the stock, so Nappt2=1 and Na-sparet2 reduce from 3 to 2, but the total number of spare parts in the stock remains unchanged; that is, Nstockt2 remains unchanged.(3)
At the timet3, RUL1t3 is less than tb, and one spare part needs to be appointed for the unit 1 from the stock, so Nappt3=Nappt2+1=2 and Na-sparet3=1. Then an order for (Nordert3=2) spare parts is placed.(4)
At the timet4, X2t4 is greater than Lf, and CM for the unit 2 needs to be implemented, so Nstockt4 reduces from 3 to 2; then Nappt4=Nappt3-1=1 and Na-sparet4=Nstockt4-Nappt4=1.(5)
At the timet5, X1t5 is between Lp and Lf, so PM for the unit 1 needs to be implemented, and Nstockt5 reduces from 2 to 1, Nappt5=0, and Na-sparet5=1.(6)
At the timet6, the ordered spare units have been delivered, so both Nstockt6=Na-sparet6=3.Figure 1
Joint decision process of two-unit system.The rest can be done in the same manner.


## 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 ofn 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 tk=k·Tk∈N and the inspection time can be neglected compared with the inspection interval. The observed deterioration level of the unit ii=1,2,…,n at the time tk is denoted as Xitk.(2) If Xitk is greater than Lf, CM for the unit i would be implemented. If Xitk is between Lp and Lf, 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 Xitk is less than Lp, the unit i keeps on operating and its RUL (RULitk) needs to be predicted, and if RULitk is less than tb, one spare part is appointed for the unit i from the stock, so the number (Na-sparetk) 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 tk+, and the corresponding deterioration level of the unit i is denoted as Xitk+.(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 Co and the order lead time is tl=l·Tl∈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 cd.(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 ck.A hypothetical system with two critical units as an example is given to illustrate the joint strategy, as shown in Figure1, where S=3, s=1, and tl=3T. The joint decision process of two- unit system is as follows:(1)
At the timet1, both X1t1 and X2t1 are less than Lp, and both RUL1t1 and RUL2t1 are more than tb, so no appointment or maintenance is needed.(2)
At the timet2, RUL2t2 is less than tb, and one spare part needs to be appointed for the unit 2 from the stock, so Nappt2=1 and Na-sparet2 reduce from 3 to 2, but the total number of spare parts in the stock remains unchanged; that is, Nstockt2 remains unchanged.(3)
At the timet3, RUL1t3 is less than tb, and one spare part needs to be appointed for the unit 1 from the stock, so Nappt3=Nappt2+1=2 and Na-sparet3=1. Then an order for (Nordert3=2) spare parts is placed.(4)
At the timet4, X2t4 is greater than Lf, and CM for the unit 2 needs to be implemented, so Nstockt4 reduces from 3 to 2; then Nappt4=Nappt3-1=1 and Na-sparet4=Nstockt4-Nappt4=1.(5)
At the timet5, X1t5 is between Lp and Lf, so PM for the unit 1 needs to be implemented, and Nstockt5 reduces from 2 to 1, Nappt5=0, and Na-sparet5=1.(6)
At the timet6, the ordered spare units have been delivered, so both Nstockt6=Na-sparet6=3.Figure 1
Joint decision process of two-unit system.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,Xitk can be characterized by the following:(1)Xitk=X0+μtk+σWtk,where X(0) is the initial state, μ and σ are the drift coefficient and diffusion coefficient, respectively, and Wtk is a standard Brownian motion; that is, Wtk~N0,tk.SoΔXitk is subject to normal distribution and can be described as(2)ΔXitk~NμT,σ2T.


### 3.2. Parameter Estimation

μ and σ can be estimated using maximum likelihood estimate and the likelihood function is(3)Lμ,σ=∏i=1n∏k=1mi12σ2πTexp⁡-ΔXitk-μT22σ2T.Solve the following equations:(4)∂ln⁡Lμ,σ∂μ=0,∂ln⁡Lμ,σ∂σ=0.μ
^ and σ^2 can be obtained:(5)μ^=∑i=1n∑k=1miΔXitk∑i=1nmiT,σ^2=1∑i=1nmi∑i=1n∑k=1miΔXitk2T-∑i=1n∑k=1miΔXitk2∑i=1nmiT.


### 3.3. RUL Prediction

Functional failure state threshold isLf, so the RULitk is(6)RULitk=inf⁡t:Xitk+t≥Lf∣Xitk<Lf,t≥0=inf⁡t:Xitk+t-Xitk≥Lf-Xitk,t≥0=inf⁡t:Xit≥Lf-Xitk,t≥0.According to [27], the cumulative distribution function and probability density function of RULitk can be obtained as follows: (7)Ft=Φμt-Lf-Xitkσt+exp⁡2μLf-Xitkσ2Φ-Lf-Xitk-μtσt,(8)ft=Lf-Xitk2πσ2t3exp⁡-Lf-Xitk-μt22σ2t.Ifα=Lf-Xitk/μ, λ=Lf-Xitk2/σ2, (8) can be transferred to(9)ft=λ2πt3exp⁡-λt-α22α2t.Soft is an inverse Gaussian distribution, and based on its definition, the mean and variance of RULitk can be obtained:(10)ERULitk=Lf-Xitkμ,varRULitk=Lf-Xitk·σ2μ3.


## 3.1. The Basic Model of Wiener Process

According to Wiener process,Xitk can be characterized by the following:(1)Xitk=X0+μtk+σWtk,where X(0) is the initial state, μ and σ are the drift coefficient and diffusion coefficient, respectively, and Wtk is a standard Brownian motion; that is, Wtk~N0,tk.SoΔXitk is subject to normal distribution and can be described as(2)ΔXitk~NμT,σ2T.


## 3.2. Parameter Estimation

μ and σ can be estimated using maximum likelihood estimate and the likelihood function is(3)Lμ,σ=∏i=1n∏k=1mi12σ2πTexp⁡-ΔXitk-μT22σ2T.Solve the following equations:(4)∂ln⁡Lμ,σ∂μ=0,∂ln⁡Lμ,σ∂σ=0.μ
^ and σ^2 can be obtained:(5)μ^=∑i=1n∑k=1miΔXitk∑i=1nmiT,σ^2=1∑i=1nmi∑i=1n∑k=1miΔXitk2T-∑i=1n∑k=1miΔXitk2∑i=1nmiT.


## 3.3. RUL Prediction

Functional failure state threshold isLf, so the RULitk is(6)RULitk=inf⁡t:Xitk+t≥Lf∣Xitk<Lf,t≥0=inf⁡t:Xitk+t-Xitk≥Lf-Xitk,t≥0=inf⁡t:Xit≥Lf-Xitk,t≥0.According to [27], the cumulative distribution function and probability density function of RULitk can be obtained as follows: (7)Ft=Φμt-Lf-Xitkσt+exp⁡2μLf-Xitkσ2Φ-Lf-Xitk-μtσt,(8)ft=Lf-Xitk2πσ2t3exp⁡-Lf-Xitk-μt22σ2t.Ifα=Lf-Xitk/μ, λ=Lf-Xitk2/σ2, (8) can be transferred to(9)ft=λ2πt3exp⁡-λt-α22α2t.Soft is an inverse Gaussian distribution, and based on its definition, the mean and variance of RULitk can be obtained:(10)ERULitk=Lf-Xitkμ,varRULitk=Lf-Xitk·σ2μ3.


## 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, Lp, and tb, that is, fECT,S,s,Lp,tb, as follows:(11)EC=minfECT,S,s,Lp,tbs.t.0<T<tb;0<Lp<Lf;0≤s<S.Due to the complexity of the joint strategy, it is difficult to derive the analytical formulation of the functionfECT,S,s,Lp,tb. However, the average cost rate of the maintenance and inventory over an infinite time span can also be simply represented as(12)fECT,S,s,Lp,tb=limtm→∞⁡Cp·Np+Cc·Nc+Cins·Nins+Co·No+ck·∑k=1mNstocktk+cd·∑k=1mNdtktm·n,where Np, Nc, Nins, No, Nstocktk, and Ndtk can be obtained as follows by finishing the simulation of all events over the simulation time span tm.(1
)
N
p
,
N
c. According to the definitions, Np and Nc can be expressed as (13)Np=∑k=1m∑i=1nIPRik,Nc=∑k=1m∑i=1nICRik.The value ofICRik and IPRik 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 Xitk (2) whether there is a spare part or not in the stock. So ICRik and IPRik can be identified, respectively:(14)ICRik=1,Xitk≥Lf∩Nstocktk>00,others,IPRik=1,Lp≤Xitk<Lf∩Nstocktk>00,others.(2
)
N
i
n
s. According to the definition, Nins can be expressed as(15)Nins=∑k=1m∑i=1nIik.As described in the joint strategy, if the uniti 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 Iik can be identified:(16)Iik=0,Xitk-1+≥Lp1,others.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 levelXitk+ can be obtained: (17)Xitk-1++=0,IPRik-1=1∪ICRik-1=1Xitk-1,others.(3
)
N
o
,
N
s
t
o
c
k
t
k. According to the definition, No can be expressed as(18)No=∑k=1mOCtk.O
C
t
k depends on whether an order is placed at the time tk, so OCtk can be identified:(19)OCtk=1,Nordertk≥10,others.According to the joint strategy,Nordertk is equal to S-Na-sparetk. However, only one order is permitted at the same time; that is, one order is permitted when there is no undelivered order, so Nordertk can be obtained:(20)Nordertk=S-Na-sparetk,Na-sparetk≤s∩Undel=00,others.SinceNa-sparetk=Nstocktk-Napptk, Nstocktk and Napptk should be obtained first.(a
)
N
s
t
o
c
k
t
k. Nstocktk depends on two things, the number of remaining spare parts (Nstocktk-1+) and the number of the delivered spare parts (Nordertk-tl), so Nstocktk can be obtained:(21)Nstocktk=Nstocktk-1++Nordertk-tl,OCtk-tl=1Nstocktk-1+,others.The number of the units whose deterioration levels are greater thanLp is ∑i=1nICRik-1+IPRik-1 at the time tk-1, so Nstocktk-1+ is equal to Nstocktk-1-∑i=1nICRik-1+IPRik-1. However, Nstocktk-1+ does not allow being negative value, so Nstocktk-1+ should be expressed as(22)Nstocktk-1+=maxNstocktk-1-∑i=1nICRik-1+IPRik-1,0.(b
)
N
a
p
p
t
k. Napptk can be obtained:(23)Napptk=∑i=1nappik.At the timetk, even if RULitk is less than tb and Xitk is less than Lp, no new appointment could be made for the unit i if an appointment for the unit i has been made at the time tk-1; that is, only one appointment for the unit i is permitted at the same time. So appik can be identified:(24)appik=1,Xitk<Lp∩RULitk<tb∩appik-1=00,others.(4
)
N
d
t
k. According to the definition, Ndtk can be expressed as(25)Ndtk=∑i=1nFitk-1+.BecauseXitk-1+≥Lf means that Fitk-1+=1, Fitk-1+ can be identified:(26)Fitk-1+=1,Xitk-1+≥Lf0,others.


### 4.2. Algorithm of Model

The combination method of GA and MC is adopted to obtain an approximate optimization resultθ∗=T∗,S∗,s∗,Lp∗,tb∗. The flow diagram for the combination method is given in Figure 2.Figure 2
Flow diagram for the combination method.
(a)
Flow diagram for GA
(b)
Flow diagram for MCThe main steps of the flow diagram are as follows.Step 1.
“Initialize population (N)” to obtain N groups of initial parameters (T,S,s,Lp,tb).Step 2.
Evaluate the fitnessfECT,S,s,Lp,tb of each group of the parameters by simulation. In order to eliminate the randomness of simulation, tm 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 ∑k=1KfECkT,S,s,Lp,tb/K, where fECkT,S,s,Lp,tb is evaluated through number k simulation described in Figure 2(b).Step 3.
If the variance of all population’s fitness obtained based on Step2 is less than a sufficiently small value ε, it means that it is unnecessary to optimize further. The parameters whose fitness is the minimum are the optimal result θ∗=T∗,S∗,s∗,Lp∗,tb∗; otherwise, go to Step 4.Step 4.
The new populations are obtained through selection, crossover, and mutation based on GA method and go to Step2 if the iteration does not end.


## 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, Lp, and tb, that is, fECT,S,s,Lp,tb, as follows:(11)EC=minfECT,S,s,Lp,tbs.t.0<T<tb;0<Lp<Lf;0≤s<S.Due to the complexity of the joint strategy, it is difficult to derive the analytical formulation of the functionfECT,S,s,Lp,tb. However, the average cost rate of the maintenance and inventory over an infinite time span can also be simply represented as(12)fECT,S,s,Lp,tb=limtm→∞⁡Cp·Np+Cc·Nc+Cins·Nins+Co·No+ck·∑k=1mNstocktk+cd·∑k=1mNdtktm·n,where Np, Nc, Nins, No, Nstocktk, and Ndtk can be obtained as follows by finishing the simulation of all events over the simulation time span tm.(1
)
N
p
,
N
c. According to the definitions, Np and Nc can be expressed as (13)Np=∑k=1m∑i=1nIPRik,Nc=∑k=1m∑i=1nICRik.The value ofICRik and IPRik 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 Xitk (2) whether there is a spare part or not in the stock. So ICRik and IPRik can be identified, respectively:(14)ICRik=1,Xitk≥Lf∩Nstocktk>00,others,IPRik=1,Lp≤Xitk<Lf∩Nstocktk>00,others.(2
)
N
i
n
s. According to the definition, Nins can be expressed as(15)Nins=∑k=1m∑i=1nIik.As described in the joint strategy, if the uniti 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 Iik can be identified:(16)Iik=0,Xitk-1+≥Lp1,others.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 levelXitk+ can be obtained: (17)Xitk-1++=0,IPRik-1=1∪ICRik-1=1Xitk-1,others.(3
)
N
o
,
N
s
t
o
c
k
t
k. According to the definition, No can be expressed as(18)No=∑k=1mOCtk.O
C
t
k depends on whether an order is placed at the time tk, so OCtk can be identified:(19)OCtk=1,Nordertk≥10,others.According to the joint strategy,Nordertk is equal to S-Na-sparetk. However, only one order is permitted at the same time; that is, one order is permitted when there is no undelivered order, so Nordertk can be obtained:(20)Nordertk=S-Na-sparetk,Na-sparetk≤s∩Undel=00,others.SinceNa-sparetk=Nstocktk-Napptk, Nstocktk and Napptk should be obtained first.(a
)
N
s
t
o
c
k
t
k. Nstocktk depends on two things, the number of remaining spare parts (Nstocktk-1+) and the number of the delivered spare parts (Nordertk-tl), so Nstocktk can be obtained:(21)Nstocktk=Nstocktk-1++Nordertk-tl,OCtk-tl=1Nstocktk-1+,others.The number of the units whose deterioration levels are greater thanLp is ∑i=1nICRik-1+IPRik-1 at the time tk-1, so Nstocktk-1+ is equal to Nstocktk-1-∑i=1nICRik-1+IPRik-1. However, Nstocktk-1+ does not allow being negative value, so Nstocktk-1+ should be expressed as(22)Nstocktk-1+=maxNstocktk-1-∑i=1nICRik-1+IPRik-1,0.(b
)
N
a
p
p
t
k. Napptk can be obtained:(23)Napptk=∑i=1nappik.At the timetk, even if RULitk is less than tb and Xitk is less than Lp, no new appointment could be made for the unit i if an appointment for the unit i has been made at the time tk-1; that is, only one appointment for the unit i is permitted at the same time. So appik can be identified:(24)appik=1,Xitk<Lp∩RULitk<tb∩appik-1=00,others.(4
)
N
d
t
k. According to the definition, Ndtk can be expressed as(25)Ndtk=∑i=1nFitk-1+.BecauseXitk-1+≥Lf means that Fitk-1+=1, Fitk-1+ can be identified:(26)Fitk-1+=1,Xitk-1+≥Lf0,others.


## 4.2. Algorithm of Model

The combination method of GA and MC is adopted to obtain an approximate optimization resultθ∗=T∗,S∗,s∗,Lp∗,tb∗. The flow diagram for the combination method is given in Figure 2.Figure 2
Flow diagram for the combination method.
(a)
Flow diagram for GA
(b)
Flow diagram for MCThe main steps of the flow diagram are as follows.Step 1.
“Initialize population (N)” to obtain N groups of initial parameters (T,S,s,Lp,tb).Step 2.
Evaluate the fitnessfECT,S,s,Lp,tb of each group of the parameters by simulation. In order to eliminate the randomness of simulation, tm 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 ∑k=1KfECkT,S,s,Lp,tb/K, where fECkT,S,s,Lp,tb is evaluated through number k simulation described in Figure 2(b).Step 3.
If the variance of all population’s fitness obtained based on Step2 is less than a sufficiently small value ε, it means that it is unnecessary to optimize further. The parameters whose fitness is the minimum are the optimal result θ∗=T∗,S∗,s∗,Lp∗,tb∗; otherwise, go to Step 4.Step 4.
The new populations are obtained through selection, crossover, and mutation based on GA method and go to Step2 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 (Lf=10°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.Figure 3
The collected outlet temperature data of the ACM.According to Section3.1, 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 N0.333,0.314, and the deterioration parameters can be obtained: μ^=3.33×10-4; σ^=0.0099. Because the initial outlet temperature is usually 2°C, the outlet temperatures can be expressed as(27)Xt=2+0.000333t+0.0099Wt.Figure 4
The distribution of the deterioration increment.


### 5.2. Joint Optimization Simulation

According to the preliminary statistics,Cins=1,000, Cp=100,000, Cc=400,000, Co=5,000 (RMB), cd=100, ck=10 (RMB/FH), and tl=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 tm is 100,000 (FH) and the mean value of 50 simulation results is taken as the fitness of each population; that is, fitness=∑k=150fECkT,S,s,Lp,tb/50. The iterative process of the GA is shown in Figure 5, and the optimal result S∗,s∗,Lp∗,tb∗ is 4,1,9.17,3391, and fECS∗,s∗,Lp∗,tb∗=116.03 (RMB/FH).Figure 5
The change of the fitness with iteration times (with the appointment policy).With the optimal resultS∗,s∗,Lp∗,tb∗, 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∗=4, which is caused by applying the appointment policy. For example, at time t43=43000 FH, Nstockt43=1 and Nappt43=2, as shown with the red point, respectively, in Figures 6 and 7. Therefore, based on (20), Na-sparet43=Nstockt43-Nappt43=-1 and Nordert43=S∗-Na-sparet43=5, as shown with the red point, respectively, in Figures 8 and 9. So an order for 5 spare parts is placed.Figure 6
The change of the inventory level.Figure 7
The change of number of appointed spare parts.Figure 8
The change of number of available spare parts.Figure 9
The change of order number.The order for 5 spare parts will arrive at timet45=45000 FH, there is one PM at time t44=44000 FH as shown with the red point in Figure 10, and there is no CM at time t44=44000 FH and t45=45000 FH as shown in Figure 11, so one spare part is used and the inventory level becomes 5 at time t45=45000 FH, as shown with a blue point in Figure 6.Figure 10
The change of number of PM.Figure 11
The change of number of CM.


### 5.3. Comparison with the Separate Optimization Method

According to the data in Figure6, 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:(28)FR=∑sDD+s·e-DD+s!.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,Lp∗,tb∗=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.Figure 12
Inventory difference of whether joint optimization is adopted or not.


### 5.4. Comparison with the Traditional Joint Optimization

In the same example, without the appointment policy in the optimization model, the optimal result isS∗,s∗,Lp∗=4,1,9.10 and fECS∗,s∗,Lp∗=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.Figure 13
The change of the fitness with iteration times (without the appointment policy).Figure 14
The change of the inventory level (with the appointment policy or not).From Figure14, 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 t26=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 CostCc

Table1 shows the different optimal results under the different Cc.Table 1
The different optimal results under the differentCc.
C
c
S
s
L
p
t
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
118.98From Table1, it can be obtained that when Cc increases, Lp decreases; however tb 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 Cc=200,000 RMB; however, it is 85 when Cc=700,000 RMB. In addition, Table 1 shows that the sharp increase of Cc does not result in a substantial increase of EC, for which the reason is that the CM will be almost avoided when Cc is taken as a big value.Figure 15
Accumulated ordered number of spare parts in differentCc.Bothcd and Cc are associated with the functional failure; therefore, they have the similar influence on the optimal result. So the sensitivity of cd to the optimal result does not need to be further discussed.


#### 5.5.2. Sensitivity of Lead Timetl

Table2 shows the different optimal results under the different tl.Table 2
The different optimal results under the differenttl.
t
l
S
s
L
p
t
b
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
153.24From Table2, 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 TimesCins, Co, and Ck

Tables3 and 4 show the different optimal results under the different Cins and Co, respectively. Tables 3 and 4 show that Cins 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. Co also has almost no impact on the optimal result, for which the reason is that Co are too small compared with Cc and Cp.Table 3
The different optimal results under the differentCins.
C
i
n
s
S
s
L
p
t
b
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
161.09Table 4
The different optimal results under the differentCo.
C
o
S
s
L
p
t
b
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
123.24Table5 shows the different optimal results under the different Ck, and in Table 5, with the increase of Ck, 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.Table 5
The different optimal results under the differentCk.
C
k
S
s
L
p
t
b
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


## 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 (Lf=10°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.Figure 3
The collected outlet temperature data of the ACM.According to Section3.1, 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 N0.333,0.314, and the deterioration parameters can be obtained: μ^=3.33×10-4; σ^=0.0099. Because the initial outlet temperature is usually 2°C, the outlet temperatures can be expressed as(27)Xt=2+0.000333t+0.0099Wt.Figure 4
The distribution of the deterioration increment.


## 5.2. Joint Optimization Simulation

According to the preliminary statistics,Cins=1,000, Cp=100,000, Cc=400,000, Co=5,000 (RMB), cd=100, ck=10 (RMB/FH), and tl=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 tm is 100,000 (FH) and the mean value of 50 simulation results is taken as the fitness of each population; that is, fitness=∑k=150fECkT,S,s,Lp,tb/50. The iterative process of the GA is shown in Figure 5, and the optimal result S∗,s∗,Lp∗,tb∗ is 4,1,9.17,3391, and fECS∗,s∗,Lp∗,tb∗=116.03 (RMB/FH).Figure 5
The change of the fitness with iteration times (with the appointment policy).With the optimal resultS∗,s∗,Lp∗,tb∗, 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∗=4, which is caused by applying the appointment policy. For example, at time t43=43000 FH, Nstockt43=1 and Nappt43=2, as shown with the red point, respectively, in Figures 6 and 7. Therefore, based on (20), Na-sparet43=Nstockt43-Nappt43=-1 and Nordert43=S∗-Na-sparet43=5, as shown with the red point, respectively, in Figures 8 and 9. So an order for 5 spare parts is placed.Figure 6
The change of the inventory level.Figure 7
The change of number of appointed spare parts.Figure 8
The change of number of available spare parts.Figure 9
The change of order number.The order for 5 spare parts will arrive at timet45=45000 FH, there is one PM at time t44=44000 FH as shown with the red point in Figure 10, and there is no CM at time t44=44000 FH and t45=45000 FH as shown in Figure 11, so one spare part is used and the inventory level becomes 5 at time t45=45000 FH, as shown with a blue point in Figure 6.Figure 10
The change of number of PM.Figure 11
The change of number of CM.


## 5.3. Comparison with the Separate Optimization Method

According to the data in Figure6, 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:(28)FR=∑sDD+s·e-DD+s!.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,Lp∗,tb∗=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.Figure 12
Inventory difference of whether joint optimization is adopted or not.


## 5.4. Comparison with the Traditional Joint Optimization

In the same example, without the appointment policy in the optimization model, the optimal result isS∗,s∗,Lp∗=4,1,9.10 and fECS∗,s∗,Lp∗=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.Figure 13
The change of the fitness with iteration times (without the appointment policy).Figure 14
The change of the inventory level (with the appointment policy or not).From Figure14, 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 t26=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 CostCc

Table1 shows the different optimal results under the different Cc.Table 1
The different optimal results under the differentCc.
C
c
S
s
L
p
t
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
118.98From Table1, it can be obtained that when Cc increases, Lp decreases; however tb 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 Cc=200,000 RMB; however, it is 85 when Cc=700,000 RMB. In addition, Table 1 shows that the sharp increase of Cc does not result in a substantial increase of EC, for which the reason is that the CM will be almost avoided when Cc is taken as a big value.Figure 15
Accumulated ordered number of spare parts in differentCc.Bothcd and Cc are associated with the functional failure; therefore, they have the similar influence on the optimal result. So the sensitivity of cd to the optimal result does not need to be further discussed.


### 5.5.2. Sensitivity of Lead Timetl

Table2 shows the different optimal results under the different tl.Table 2
The different optimal results under the differenttl.
t
l
S
s
L
p
t
b
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
153.24From Table2, 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 TimesCins, Co, and Ck

Tables3 and 4 show the different optimal results under the different Cins and Co, respectively. Tables 3 and 4 show that Cins 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. Co also has almost no impact on the optimal result, for which the reason is that Co are too small compared with Cc and Cp.Table 3
The different optimal results under the differentCins.
C
i
n
s
S
s
L
p
t
b
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
161.09Table 4
The different optimal results under the differentCo.
C
o
S
s
L
p
t
b
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
123.24Table5 shows the different optimal results under the different Ck, and in Table 5, with the increase of Ck, 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.Table 5
The different optimal results under the differentCk.
C
k
S
s
L
p
t
b
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


## 5.5.1. Sensitivity of CM CostCc

Table1 shows the different optimal results under the different Cc.Table 1
The different optimal results under the differentCc.
C
c
S
s
L
p
t
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
118.98From Table1, it can be obtained that when Cc increases, Lp decreases; however tb 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 Cc=200,000 RMB; however, it is 85 when Cc=700,000 RMB. In addition, Table 1 shows that the sharp increase of Cc does not result in a substantial increase of EC, for which the reason is that the CM will be almost avoided when Cc is taken as a big value.Figure 15
Accumulated ordered number of spare parts in differentCc.Bothcd and Cc are associated with the functional failure; therefore, they have the similar influence on the optimal result. So the sensitivity of cd to the optimal result does not need to be further discussed.


## 5.5.2. Sensitivity of Lead Timetl

Table2 shows the different optimal results under the different tl.Table 2
The different optimal results under the differenttl.
t
l
S
s
L
p
t
b
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
153.24From Table2, 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 TimesCins, Co, and Ck

Tables3 and 4 show the different optimal results under the different Cins and Co, respectively. Tables 3 and 4 show that Cins 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. Co also has almost no impact on the optimal result, for which the reason is that Co are too small compared with Cc and Cp.Table 3
The different optimal results under the differentCins.
C
i
n
s
S
s
L
p
t
b
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
161.09Table 4
The different optimal results under the differentCo.
C
o
S
s
L
p
t
b
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
123.24Table5 shows the different optimal results under the different Ck, and in Table 5, with the increase of Ck, 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.Table 5
The different optimal results under the differentCk.
C
k
S
s
L
p
t
b
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


## 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.



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*Source: 3493687-2017-03-20.xml*