Research ArticleAn Optimization Method for Condition Based Maintenance ofAircraft Fleet Considering Prognostics Uncertainty
Qiang Feng1 Yiran Chen1 Bo Sun1 and Songjie Li2
1 School of Reliability and Systems Engineering Beihang University Beijing China2 Sichuan jiuzhou Aerocont Technologies Co Ltd Mianyang China
Correspondence should be addressed to Bo Sun sunbobuaaeducn
Received 28 February 2014 Accepted 19 March 2014 Published 17 April 2014
Academic Editors N Barsoum V N Dieu P Vasant and G-W Weber
Copyright copy 2014 Qiang Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
An optimizationmethod for condition basedmaintenance (CBM) of aircraft fleet considering prognostics uncertainty is proposedTheCBMand dispatch process of aircraft fleet is analyzed first and the alternative strategy sets for single aircraft are givenThen theoptimization problem of fleet CBM with lower maintenance cost and dispatch risk is translated to the combinatorial optimizationproblem of single aircraft strategy Remain useful life (RUL) distribution of the key line replaceable Module (LRM) has beentransformed into the failure probability of the aircraft and the fleet health statusmatrix is established And the calculationmethod ofthe costs and risks for mission based on health status matrix andmaintenance matrix is given Further an optimization method forfleet dispatch and CBM under acceptable risk is proposed based on an improved genetic algorithm Finally a fleet of 10 aircrafts isstudied to verify the proposed methodThe results shows that it could realize optimization and control of the aircraft fleet orientedto mission success
1 Introduction
Prognostic and health management (PHM) technology hasa rapid development and been widely used in aeronau-tical equipment in recent years The failure position andremain useful life (RUL) of equipment could be predictedby PHM Further it can be used in aircraft condition basedmaintenance (CBM) [1] However due to the uncertaintyof prognostics there are certain risks in the maintenancedecisions based on the prediction of RUL [2 3]
The aircraft usually performs mission in fleet mannerand shares limited support resource So there will be atradeoff range for fleet CBM This means each aircraft canchoose strategy among dispatching strategy standby strategyand maintenance strategy or their combination when theRUL has been obtained and the synthetic strategy for fleet(combination of each aircraftrsquos strategy) should meet themission requirement
There are three forms of RUL in PHM and each formincludes some uncertainty First is the point value of thetime of potential failure Second is the interval value of the
time of potential failure [4ndash7] Third is the RUL distributionof the device [8ndash12] The third form has the maximuminformation and the highest availability but is the mostdifficult in acquisition and application
Two methods can be used in reducing the impact ofprognostics uncertainty on CBM decision One is to reducethe uncertainty of failure prediction directly so that thedecision risk will decrease [13ndash15] The other one is totake the prediction uncertainty into account and make theoptimum decision under acceptable risk [16ndash21] Because theuncertainty of failure prediction could not be completelyeliminated the latter is more useful in engineering applica-tions
Most research about RUL for CBM is about the lifecycle maintenance optimization decisions on single aircraftand researchers would rather consider the maintenancedecisions than think about the mission requirements and thedispatched strategy The research on fleet CBM oriented tomission successes is few Agent technology and the heuristicalgorithm are used to fleet CBM in article [22 23] but samplepoint values of RUL were used only
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 430190 8 pageshttpdxdoiorg1011552014430190
2 The Scientific World Journal
Aircraft stateStrategy
Synthesized decision making for aircraft fleet
Standbyidle Preflightpreparation
Formationdispatching
missionreturn
Keepingstandby
Postflightcheck
Repair andservicing
Repair andservicing
Prognostics
Dispatching
Mission requirementMaintenance resource Maintenance method
Prognostics of aircraft
Mission
Mission
State transition of single aircraf for mission based on prognostics
∙∙
∙ ∙
Figure 1 CBM for aircraft fleet
An optimal aircraft fleet CBM method for aviation unitmaintenance is proposed in the paper considering dispatchmission and resource constraints Moreover the RUL distri-bution of the key LRM has been transformed into the failureprobability of the aircraft and the calculation method of thecosts and risks for mission is given Then an optimizationdecision making method for fleet dispatch and CBM underacceptable risk is proposed based on an improved geneticalgorithm
2 Analysis of Aircraft Fleet CBM
21 Basic Process Analysis Consider an aircraft fleet con-taining 119898 aircrafts and 119896 integrated support stations (ISS)facing continuous combat missions (119896 lt 119898) in which asingle mission requires 119897 aircrafts (119897 le 119898) Each aircraftcontains 119901 LRMs of which RUL can be estimated Themission preparation period starts at time 119905
0 while themission
period is from time 1199051to 1199052 The basic process of the
fleet CBM decisions which is mission success oriented isgiven in Figure 1 There are two kinds of single strategies(keeping standby and dispatching) and two kinds of mixedstrategies (dispatching after maintenance and standby aftermaintenance) before making synthesized decision The fleetCBM decisions consist of these single strategies that shouldmeet the requirements of missions cost and risk
22 Assumptions The basic assumptions of the problem arelisted below in order to define the problem
(1) The aircraft fails when any key LRM fails(2) The RUL distribution of LRM which is given at the
time 1199050is 119865(119905) of which probability density function is
119891(119905)(3) Assume the maintenance method of the LRM is
renew which means the LRM will be as good as newafter maintenance considering the field maintenanceof aviation unit maintenance
(4) Only one aircraft can be repaired in each ISS simulta-neously But the total number of the aircraft mainte-nance may be more than one from the time 119905
0to 1199051
(5) Different LRM in the same aircraft can be replaced atthe same time for renew is served as a maintenancemethod
(6) The maintenance cost of the different LRM variedwhile the same LRM cost is the same The mainte-nance cost of the LRM on 119895 class is 119862
119895
(7) Each aircraftmalfunctionwill cause themission to failwhen the fleet is on missionThe consequences of theeconomic loss will not be taken into consideration
(8) Spare parts are plentiful
3 Modeling Method to Aircraft Fleet CBMConsidering Prognostics Uncertainty
31 Modeling Framework The main work of the optimiza-tion decision making method for fleet CBM consideringprognostics uncertainty includes the following steps (1)
the definition of the fleet initial health status based on theRUL prognostic (2) maintenance program generating (3)maintenance time and cost estimation and (4) mission riskassessment Based on the objects above the optimal CBMand maintenance program through the rational optimizationalgorithm is obtained in the paper The modeling frameworkis given in Figure 2
32 The Definition of the Fleet Initial Health Status Based onthe RUL Prognostic Assume the distribution of the 119895th keyLRU119894119895on the 119894th aircraft is119865
119894119895(119905) In themission period (119905
1-1199052)
the probability of failure can be got by
119901119894119895 (
119886) = int
1199052
1199051
119891119894119895 (
119905) 119889119905 119894 = 1 2 119898 119895 = 1 2 119899 (1)
where 119891119894119895(119905) is the probability density function of the 119865
119894119895(119905)
Considering an aircraft fleet containing 119898 aircrafts andeach include 119899 LRMs the initial health status matrix of allLRM is 119875(119886) that is given by
119875 (119886) =[
[
11990111
(119886) sdot sdot sdot 1199011119899
(119886)
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199011198981 (
119886) sdot sdot sdot 119901119898119899 (
119886)
]
]119898times119899
(2)
The Scientific World Journal 3
Predict all RULij of key LRMij (i = 1 2 m j = 1 2 n)
Calculate the initial failure probability Pij(a) ofthe key LRMij at the mission period by the
distribution Fij(t) of the RULij at the time t0 inthe mission preparation period
Defne the maintenance program of the keyLRM and describe whether the key LRM will
be maintained with uij while siq stand formaintenance site
Update the failure probability Pij(b) of the keyLRM in the mission period
Calculate the total cost C and time of allLRM maintenance costs by uij
Calculate the total mission risk of proposedaircraf for dispatching in typical tasks
Evaluate and optimize the program of the CBM
Figure 2 Modeling framework for CBM of aircraft fleet
33 Maintenance Program Generating Maintenance pro-gram considers whether a certain LRM should bemaintainedand the selection of the ISSs
The maintenance matrix 119880 of fleet can be described as
119880 =[
[
11990611
sdot sdot sdot 1199061119899
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199061198981
sdot sdot sdot 119906119898119899
]
]119898times119899
(3)
where 119906119894119895= 1means that the 119895th LRMof the 119894th aircraft needs
to be repaired otherwise 119906119894119895= 0
According to the assumption (4) the LRM can be main-tained at the same place however many the LRMs failsTherefore the ISS matrix 119878 of the fleet is shown as
119878 =[
[
11990411
sdot sdot sdot 1199041119896
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199041198981
sdot sdot sdot 119904119898119896
]
]119898times119896
(4)
where 119904119894119902
= 1means that the 119894th aircraft should bemaintainedat the 119902th ISS otherwise 119906
119894119895= 0
34 Maintenance Time and Cost Estimation Assume repair-ing the 119895th LRM spends time119879
119895and needs cost119862
119895 According
to the assumption (5) the total maintenance time 119879119898119894of the
119894th aircraft is given as
119879119898119894= max (119906
119894119895times 119879119895) 119895 = 1 2 119899 (5)
There may be more than one aircraft that should berepaired at ISS 119902 so the total maintenance time of all aircraftscan be calcluated as
119898
sum
119894=1
119904119894119902times 119879119898119894
119894 = 1 2 119898 (6)
The total maintenance cost of all aircrafts can be got as
119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895] 119894 = 1 2 119898 119895 = 1 2 119899 (7)
35 Mission Risk Assessment
Step 1 (modify the health matrix of the fleet) Whether theaircraft is ldquodispatchingrdquo or ldquodispatching after maintenancerdquoshould be taken into consideration when calculating themission risk of the fleet The status of the aircraft shouldbe updated if the single strategy of the fleet is ldquodispatchingafter maintenancerdquo Then the modified health status matrix119875(119887) of the fleet can be built according to the Assumption(3) Consider 119901
119894119895(119887) = 0 after the LRM
119894119895on the 119894th aircraft
was renewed otherwise 119901119894119895(119887) = 119901
119894119895(119886) without renew The
elements in the matrix can be obtained by
119901119894119895(119887) = 119901
119894119895(119886) times (1 minus 119906
119894119895) (8)
Step 2 (estimate the failure probability of the single aircraftand rank) The failure probability of the single aircraft couldbe estimated after modifying the health matrix of the fleetAccording to the first assumption ldquothe aircraft fails when anykey LRM failsrdquo the failure probability 119875
119894(119886) of the 119894th aircraft
can be given as
119875119894 (119886) = 1 minus
119899
prod
119895=1
[1 minus 119901119894119895 (
119887)] (9)
Formula (10) can be obtained according to (1) (8) and(9)
119875119894(119886) = 1 minus
119899
prod
119895=1
[1 minus int
1199052
1199051
119891119894119895(119905) (1 minus 119906
119894119895)] 119894 = 1 2 119898
(10)
Then Pro(119894) = 1 2 119898 can be obtained by sorting thefailure probability of single aircraft in ascending order Theordered failure probability of the aircraft is given as
119875pro(119894) (119887) = 119875119894(119886) (11)
Suppose 1198752(119886) is the smallest Pro(119894) Then set Pro(2) = 1
and let 1198751(119887) = 119875
2(119886) after reordering Pick up aircrafts of
which Pro(119894) = 1 2 119897 when the mission needs dispatch 119897
aircrafts
4 The Scientific World Journal
Step 3 (calculate the mission risk of the fleet) Assume theserious consequences of the mission that failed are similarwithout taking the economic losses into account The failureprobability of the fleet mission can be calculated by thefollowing according to (7)
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] (12)
where 119875119865is the mission risk
4 Optimization Problem andAlgorithms Design
41 Problem Description The optimization problem in thepaper is to find a fleet CBM strategy with acceptable risk andlowest cost considering prognostics uncertainty
Thus describe the objective of the optimization asMin119862 = sum
119898
119894=1sum119899
119895=1[119906119894119895times 119862119895]
The constraints that should be considered about involvethe maintenance ability constraint 119877
119860of the site the time
constraint119877119861 the security risk constraint119877
119862 themission risk
constraint 119877119863 and the variable constraint 119877
119864
For first constraint 119877119860 set 119904
119894119902= 0 (119902 = 1 2 119896) and
119906119894119895
= 0 (119895 = 1 2 119899) if none of LRMs need maintenanceElse if any LRM
119894119895requires maintenance then 119906
119894119895= 1 and the
corresponding 119904119894119902
= 1 while the other 119904119894119902
= 0 Thus the 119877119860
can be described as
119877119860
119896
sum
119902=1
119904119894119902+
119899
prod
119895=1
(1 minus 119906119894119895) = 1 (13)
All maintenance of site 119902 should be finished before themission starts Thus the 119877
119861can be given as
119877119861 119879119898119894le 1199051minus 1199050| 119904119894119902
= 1 119894 = 1 2 119898 119902 = 1 2 119896
(14)
Assume that the total number of the aircraft whichmaintained at site 119902 is sum
119898
119894=1119904119894119902
= 119909 gt 1 (where the numberof aircraft is 1 2 119909) Ifsum119909minus1
119894=1119904119894119902times119879119898119894le 1199051minus1199050lt sum119909
119894=1119904119894119902times
119879119898119894 which means the maintenance for the 119909th aircraft could
not be finished before the mission start time this aircraft willnot be taken into account when the maintenance decisions isldquodispatching after maintenancerdquo
It will not be allowed to dispatch if the failure probabilityis too high for security risk existing in single aircraft Con-sider a mission need 119897 aircrafts and the 119877
119862can be described
as (15)Moreover themissionwill fail if (15) could not bemetWe have
119877119862 119875119897(119887) lt 119875
119904119897 (15)
According to (12) 119877119863can be written as (16) considering
the mission risk for fleet We have
119877119863
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] lt 119875
119898 (16)
where 119875119898is the objective of the mission risk
The variable constraint which means that the variablesshould be in a certain range is described as
119877119864 119862119895gt 0 119906
119894119895isin 0 1 119904
119894119902isin 0 1
119894 = 1 2 119898 119895 = 1 2 119899 119902 = 1 2 119896
(17)
The conceptual model for aircraft fleet condition basedmaintenance and dispatch is given as follows
Min 119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895]
st 119877119860sim119864
is satisfied
(18)
42 Optimization AlgorithmsDesign Theoptimization prob-lem cannot meet the KKT (Karush-Kuhn-Tucker) conditionsand the dimension of decisionmaking variables which can bewritten as 119898 times 119899 + 119898 times 119896 is relatively large So an improvedgenetic algorithmwas proposed in this paper for the probleminstead of traditional mathematical methods
The optimization model can be simplified as
min 119862 (119880 119878)
st
119892 (119880 119878) le 0
ℎ (119880 119878) = 0
119906119894119895isin 0 1
119904119894119895isin 0 1
(19)
where 119880 is the maintenance matrix while the 119878 is the ISSmatrix
The problem has more variables and constraints so thesolution quality of problem and the convergence rate couldnot be satisfied Therefore the improvement strategy of thegenetic algorithm is given in Figure 3
421 Define the Initial Population of the Maintenance Matrix119880 According to the multifactor and 2-level orthogonalexperimental design in order to cover widely define theinitial population of the maintenance matrix 119880 The initialpopulation should be filtered so as to make the convergencefaster Moreover the number of the aircraft needs to berepaired in the population which should be less than 119897
considering the dispatched requirements and the cost of themaintenance The relationship among those factors is shownas
119898
sum
119894=1
[
[
119899
prod
119895=1
(1 minus 119906119894119895)]
]
ge 119898 minus 119897 (20)
The Scientific World Journal 5
Yes
Yes
No
YesNo
No
Initializeupdate the U Heuristic rule
Solve the SAdjust the U
Meet themaintenance
resourcesconstrains
Meet the missionrisk constrains
Apply penalty function
Selection crossover and mutation
Meet therequirement for ending
the iterationOutput result
Figure 3 Improved strategies for genetic algorithm
422 Solve the ISS Matrix 119878 It is necessary to find a set offeasible solutions which meet the constraint of the ability ofISS 119877119860and the maintenance time 119877
119861on the basis of a certain
119880 The following heuristic rules can be used in order toreduce the amount of computation increasing the efficiencyof solving
Step 1 According to formula (13) an initial value of the 119878
can be given with the certain 119880 If prod119899119895=1
(1 minus 119906119894119895) = 1 the
aircraft 119894 need not be repaired and all 119904119894119902(119902 = 1 2 119896) =
0 otherwise the aircraft 119894 needs to be repaired Then thedetermining condition is described as sum119896
119902=1119904119894119902
= 1 and 119904119894119902
isin
0 1
Step 2 Consider that there are119910 aircrafts need not berepaired Remove 119910 rows which stand for these aircraftsThen a new (119898 minus 119910) times 119896 matrix 119878
1015840 which represents the newrelationship between the ISS and the aircraft that needs to berepaired can be built as the reduced cycle matrix of 119878
Step 3 Themaintenance time 119879119898119894of the aircraft needs to be
repaired in matrix 1198781015840 which can be obtained by formula (5)
Then the average maintenance time AMT of the ISSs is givenby AMT = sum
119898minus119910
119894=1119879119898119894119896 It can be determined not meet the
time constrain if max(119879119898119894) gt 1199051minus 1199050orAMT gt 119905
1minus 1199050 then
turn to Step 6 Otherwise turn to Step 4
Step 4 Initialize thematrix 1198781015840 and set 119904119894119902
= 0 (119894 = 1 2 119898minus
119910 119902 = 1 2 119896)
Step 5 Set the value of the matrix 1198781015840 from the first row to the
119896th row The method of the 119902th is described as follows
(a) Calculate the value of |119879119898119894
minus AMT| (119894 =
1 2 119898-119910) If the aircraft 119911 makes the|119879119898119911minus AMT| = min |119879119898
119894minus AMT| then 119904
119911119902= 1
Furthermore the aircraft can be selected in randomif there is more than one aircraft that meets thisformula
(b) Remove the line in which the aircraft 119911 is in to builda new reduced cycle matrix 119878
1015840 Update the remainingmaintenance time 119879119866
119902= 1199051minus 1199050minus 119879119898119911of the site 119902
(c) Compare the 119879119866119902and the 119879119898
119894for the 119878
1015840 If theformula ldquomin(119879119898
119894) | 119894 = 119900 le 119879119866
119902rdquo can be met by a
parameter 119900 form a new reduced cycle matrix and set119904119900119902
= 1 Moreover the remaining available referencetime should be updated as 119879119866
119902= 119879119866119902(119887) minus119879119898
119900 This
work should be repeated until the min (119879119898119894) gt 119879119866119902
then turn to the (119902 + 1)th row
Step 6 The matrix 119880 should be adjusted if the constraintsof resource maintenance cannot be met Consider that themaintenance cost should be as low as possible and therequirement of the mission risk should be satisfied theelements which 119906
119894119895= 1 should find 119901
119894119895(119886) corresponded and
the min119901119894119895(119886) then set 119906
119894119895= 0 Return to Step 1 and repeat
after finishing the update for the 119880 until meeting constrains119877119860and 119877
119861
423 Deal with the Constrain of the Mission Risk Somematrix 119880 which is initial or got by adjusting crossover andmutation may not meet the requirement of the mission riskconstrain The penalty method can be used in the methodfollowed to solve this problem
The energy function for every 119880 can be written as
119864 (119880 119878) = 119862 (119880 119878) + 119865 (119880 119878) sdot 119872119879 (21)
where 119865(119880 119878) is the vector of the penalty function and the119865119894(119880 119878) = max0 119892
119894(119880 119878) while 119872 which is the penalty
factor vector is a large positive number
Step 1 (fitness function design) The fitness function is givenas follows in order to minimize the objective function
119891 (119880 119878) = 1 minus
119864 (119880 119878) minus 119864min119864max minus 119864min
(22)
where 119864max and 119864min are the maximum and the minimumvalues of the energy function in the population
Step 2 (selection crossover and mutation) Proportionalselection single-point crossover and the basic alleles can beused in solving this problem
This problem can be dealt with by some method writtenin the article [24ndash26] in order to avoid the premature and thestalling that appear in the genetic algorithms
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
2 The Scientific World Journal
Aircraft stateStrategy
Synthesized decision making for aircraft fleet
Standbyidle Preflightpreparation
Formationdispatching
missionreturn
Keepingstandby
Postflightcheck
Repair andservicing
Repair andservicing
Prognostics
Dispatching
Mission requirementMaintenance resource Maintenance method
Prognostics of aircraft
Mission
Mission
State transition of single aircraf for mission based on prognostics
∙∙
∙ ∙
Figure 1 CBM for aircraft fleet
An optimal aircraft fleet CBM method for aviation unitmaintenance is proposed in the paper considering dispatchmission and resource constraints Moreover the RUL distri-bution of the key LRM has been transformed into the failureprobability of the aircraft and the calculation method of thecosts and risks for mission is given Then an optimizationdecision making method for fleet dispatch and CBM underacceptable risk is proposed based on an improved geneticalgorithm
2 Analysis of Aircraft Fleet CBM
21 Basic Process Analysis Consider an aircraft fleet con-taining 119898 aircrafts and 119896 integrated support stations (ISS)facing continuous combat missions (119896 lt 119898) in which asingle mission requires 119897 aircrafts (119897 le 119898) Each aircraftcontains 119901 LRMs of which RUL can be estimated Themission preparation period starts at time 119905
0 while themission
period is from time 1199051to 1199052 The basic process of the
fleet CBM decisions which is mission success oriented isgiven in Figure 1 There are two kinds of single strategies(keeping standby and dispatching) and two kinds of mixedstrategies (dispatching after maintenance and standby aftermaintenance) before making synthesized decision The fleetCBM decisions consist of these single strategies that shouldmeet the requirements of missions cost and risk
22 Assumptions The basic assumptions of the problem arelisted below in order to define the problem
(1) The aircraft fails when any key LRM fails(2) The RUL distribution of LRM which is given at the
time 1199050is 119865(119905) of which probability density function is
119891(119905)(3) Assume the maintenance method of the LRM is
renew which means the LRM will be as good as newafter maintenance considering the field maintenanceof aviation unit maintenance
(4) Only one aircraft can be repaired in each ISS simulta-neously But the total number of the aircraft mainte-nance may be more than one from the time 119905
0to 1199051
(5) Different LRM in the same aircraft can be replaced atthe same time for renew is served as a maintenancemethod
(6) The maintenance cost of the different LRM variedwhile the same LRM cost is the same The mainte-nance cost of the LRM on 119895 class is 119862
119895
(7) Each aircraftmalfunctionwill cause themission to failwhen the fleet is on missionThe consequences of theeconomic loss will not be taken into consideration
(8) Spare parts are plentiful
3 Modeling Method to Aircraft Fleet CBMConsidering Prognostics Uncertainty
31 Modeling Framework The main work of the optimiza-tion decision making method for fleet CBM consideringprognostics uncertainty includes the following steps (1)
the definition of the fleet initial health status based on theRUL prognostic (2) maintenance program generating (3)maintenance time and cost estimation and (4) mission riskassessment Based on the objects above the optimal CBMand maintenance program through the rational optimizationalgorithm is obtained in the paper The modeling frameworkis given in Figure 2
32 The Definition of the Fleet Initial Health Status Based onthe RUL Prognostic Assume the distribution of the 119895th keyLRU119894119895on the 119894th aircraft is119865
119894119895(119905) In themission period (119905
1-1199052)
the probability of failure can be got by
119901119894119895 (
119886) = int
1199052
1199051
119891119894119895 (
119905) 119889119905 119894 = 1 2 119898 119895 = 1 2 119899 (1)
where 119891119894119895(119905) is the probability density function of the 119865
119894119895(119905)
Considering an aircraft fleet containing 119898 aircrafts andeach include 119899 LRMs the initial health status matrix of allLRM is 119875(119886) that is given by
119875 (119886) =[
[
11990111
(119886) sdot sdot sdot 1199011119899
(119886)
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199011198981 (
119886) sdot sdot sdot 119901119898119899 (
119886)
]
]119898times119899
(2)
The Scientific World Journal 3
Predict all RULij of key LRMij (i = 1 2 m j = 1 2 n)
Calculate the initial failure probability Pij(a) ofthe key LRMij at the mission period by the
distribution Fij(t) of the RULij at the time t0 inthe mission preparation period
Defne the maintenance program of the keyLRM and describe whether the key LRM will
be maintained with uij while siq stand formaintenance site
Update the failure probability Pij(b) of the keyLRM in the mission period
Calculate the total cost C and time of allLRM maintenance costs by uij
Calculate the total mission risk of proposedaircraf for dispatching in typical tasks
Evaluate and optimize the program of the CBM
Figure 2 Modeling framework for CBM of aircraft fleet
33 Maintenance Program Generating Maintenance pro-gram considers whether a certain LRM should bemaintainedand the selection of the ISSs
The maintenance matrix 119880 of fleet can be described as
119880 =[
[
11990611
sdot sdot sdot 1199061119899
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199061198981
sdot sdot sdot 119906119898119899
]
]119898times119899
(3)
where 119906119894119895= 1means that the 119895th LRMof the 119894th aircraft needs
to be repaired otherwise 119906119894119895= 0
According to the assumption (4) the LRM can be main-tained at the same place however many the LRMs failsTherefore the ISS matrix 119878 of the fleet is shown as
119878 =[
[
11990411
sdot sdot sdot 1199041119896
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199041198981
sdot sdot sdot 119904119898119896
]
]119898times119896
(4)
where 119904119894119902
= 1means that the 119894th aircraft should bemaintainedat the 119902th ISS otherwise 119906
119894119895= 0
34 Maintenance Time and Cost Estimation Assume repair-ing the 119895th LRM spends time119879
119895and needs cost119862
119895 According
to the assumption (5) the total maintenance time 119879119898119894of the
119894th aircraft is given as
119879119898119894= max (119906
119894119895times 119879119895) 119895 = 1 2 119899 (5)
There may be more than one aircraft that should berepaired at ISS 119902 so the total maintenance time of all aircraftscan be calcluated as
119898
sum
119894=1
119904119894119902times 119879119898119894
119894 = 1 2 119898 (6)
The total maintenance cost of all aircrafts can be got as
119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895] 119894 = 1 2 119898 119895 = 1 2 119899 (7)
35 Mission Risk Assessment
Step 1 (modify the health matrix of the fleet) Whether theaircraft is ldquodispatchingrdquo or ldquodispatching after maintenancerdquoshould be taken into consideration when calculating themission risk of the fleet The status of the aircraft shouldbe updated if the single strategy of the fleet is ldquodispatchingafter maintenancerdquo Then the modified health status matrix119875(119887) of the fleet can be built according to the Assumption(3) Consider 119901
119894119895(119887) = 0 after the LRM
119894119895on the 119894th aircraft
was renewed otherwise 119901119894119895(119887) = 119901
119894119895(119886) without renew The
elements in the matrix can be obtained by
119901119894119895(119887) = 119901
119894119895(119886) times (1 minus 119906
119894119895) (8)
Step 2 (estimate the failure probability of the single aircraftand rank) The failure probability of the single aircraft couldbe estimated after modifying the health matrix of the fleetAccording to the first assumption ldquothe aircraft fails when anykey LRM failsrdquo the failure probability 119875
119894(119886) of the 119894th aircraft
can be given as
119875119894 (119886) = 1 minus
119899
prod
119895=1
[1 minus 119901119894119895 (
119887)] (9)
Formula (10) can be obtained according to (1) (8) and(9)
119875119894(119886) = 1 minus
119899
prod
119895=1
[1 minus int
1199052
1199051
119891119894119895(119905) (1 minus 119906
119894119895)] 119894 = 1 2 119898
(10)
Then Pro(119894) = 1 2 119898 can be obtained by sorting thefailure probability of single aircraft in ascending order Theordered failure probability of the aircraft is given as
119875pro(119894) (119887) = 119875119894(119886) (11)
Suppose 1198752(119886) is the smallest Pro(119894) Then set Pro(2) = 1
and let 1198751(119887) = 119875
2(119886) after reordering Pick up aircrafts of
which Pro(119894) = 1 2 119897 when the mission needs dispatch 119897
aircrafts
4 The Scientific World Journal
Step 3 (calculate the mission risk of the fleet) Assume theserious consequences of the mission that failed are similarwithout taking the economic losses into account The failureprobability of the fleet mission can be calculated by thefollowing according to (7)
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] (12)
where 119875119865is the mission risk
4 Optimization Problem andAlgorithms Design
41 Problem Description The optimization problem in thepaper is to find a fleet CBM strategy with acceptable risk andlowest cost considering prognostics uncertainty
Thus describe the objective of the optimization asMin119862 = sum
119898
119894=1sum119899
119895=1[119906119894119895times 119862119895]
The constraints that should be considered about involvethe maintenance ability constraint 119877
119860of the site the time
constraint119877119861 the security risk constraint119877
119862 themission risk
constraint 119877119863 and the variable constraint 119877
119864
For first constraint 119877119860 set 119904
119894119902= 0 (119902 = 1 2 119896) and
119906119894119895
= 0 (119895 = 1 2 119899) if none of LRMs need maintenanceElse if any LRM
119894119895requires maintenance then 119906
119894119895= 1 and the
corresponding 119904119894119902
= 1 while the other 119904119894119902
= 0 Thus the 119877119860
can be described as
119877119860
119896
sum
119902=1
119904119894119902+
119899
prod
119895=1
(1 minus 119906119894119895) = 1 (13)
All maintenance of site 119902 should be finished before themission starts Thus the 119877
119861can be given as
119877119861 119879119898119894le 1199051minus 1199050| 119904119894119902
= 1 119894 = 1 2 119898 119902 = 1 2 119896
(14)
Assume that the total number of the aircraft whichmaintained at site 119902 is sum
119898
119894=1119904119894119902
= 119909 gt 1 (where the numberof aircraft is 1 2 119909) Ifsum119909minus1
119894=1119904119894119902times119879119898119894le 1199051minus1199050lt sum119909
119894=1119904119894119902times
119879119898119894 which means the maintenance for the 119909th aircraft could
not be finished before the mission start time this aircraft willnot be taken into account when the maintenance decisions isldquodispatching after maintenancerdquo
It will not be allowed to dispatch if the failure probabilityis too high for security risk existing in single aircraft Con-sider a mission need 119897 aircrafts and the 119877
119862can be described
as (15)Moreover themissionwill fail if (15) could not bemetWe have
119877119862 119875119897(119887) lt 119875
119904119897 (15)
According to (12) 119877119863can be written as (16) considering
the mission risk for fleet We have
119877119863
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] lt 119875
119898 (16)
where 119875119898is the objective of the mission risk
The variable constraint which means that the variablesshould be in a certain range is described as
119877119864 119862119895gt 0 119906
119894119895isin 0 1 119904
119894119902isin 0 1
119894 = 1 2 119898 119895 = 1 2 119899 119902 = 1 2 119896
(17)
The conceptual model for aircraft fleet condition basedmaintenance and dispatch is given as follows
Min 119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895]
st 119877119860sim119864
is satisfied
(18)
42 Optimization AlgorithmsDesign Theoptimization prob-lem cannot meet the KKT (Karush-Kuhn-Tucker) conditionsand the dimension of decisionmaking variables which can bewritten as 119898 times 119899 + 119898 times 119896 is relatively large So an improvedgenetic algorithmwas proposed in this paper for the probleminstead of traditional mathematical methods
The optimization model can be simplified as
min 119862 (119880 119878)
st
119892 (119880 119878) le 0
ℎ (119880 119878) = 0
119906119894119895isin 0 1
119904119894119895isin 0 1
(19)
where 119880 is the maintenance matrix while the 119878 is the ISSmatrix
The problem has more variables and constraints so thesolution quality of problem and the convergence rate couldnot be satisfied Therefore the improvement strategy of thegenetic algorithm is given in Figure 3
421 Define the Initial Population of the Maintenance Matrix119880 According to the multifactor and 2-level orthogonalexperimental design in order to cover widely define theinitial population of the maintenance matrix 119880 The initialpopulation should be filtered so as to make the convergencefaster Moreover the number of the aircraft needs to berepaired in the population which should be less than 119897
considering the dispatched requirements and the cost of themaintenance The relationship among those factors is shownas
119898
sum
119894=1
[
[
119899
prod
119895=1
(1 minus 119906119894119895)]
]
ge 119898 minus 119897 (20)
The Scientific World Journal 5
Yes
Yes
No
YesNo
No
Initializeupdate the U Heuristic rule
Solve the SAdjust the U
Meet themaintenance
resourcesconstrains
Meet the missionrisk constrains
Apply penalty function
Selection crossover and mutation
Meet therequirement for ending
the iterationOutput result
Figure 3 Improved strategies for genetic algorithm
422 Solve the ISS Matrix 119878 It is necessary to find a set offeasible solutions which meet the constraint of the ability ofISS 119877119860and the maintenance time 119877
119861on the basis of a certain
119880 The following heuristic rules can be used in order toreduce the amount of computation increasing the efficiencyof solving
Step 1 According to formula (13) an initial value of the 119878
can be given with the certain 119880 If prod119899119895=1
(1 minus 119906119894119895) = 1 the
aircraft 119894 need not be repaired and all 119904119894119902(119902 = 1 2 119896) =
0 otherwise the aircraft 119894 needs to be repaired Then thedetermining condition is described as sum119896
119902=1119904119894119902
= 1 and 119904119894119902
isin
0 1
Step 2 Consider that there are119910 aircrafts need not berepaired Remove 119910 rows which stand for these aircraftsThen a new (119898 minus 119910) times 119896 matrix 119878
1015840 which represents the newrelationship between the ISS and the aircraft that needs to berepaired can be built as the reduced cycle matrix of 119878
Step 3 Themaintenance time 119879119898119894of the aircraft needs to be
repaired in matrix 1198781015840 which can be obtained by formula (5)
Then the average maintenance time AMT of the ISSs is givenby AMT = sum
119898minus119910
119894=1119879119898119894119896 It can be determined not meet the
time constrain if max(119879119898119894) gt 1199051minus 1199050orAMT gt 119905
1minus 1199050 then
turn to Step 6 Otherwise turn to Step 4
Step 4 Initialize thematrix 1198781015840 and set 119904119894119902
= 0 (119894 = 1 2 119898minus
119910 119902 = 1 2 119896)
Step 5 Set the value of the matrix 1198781015840 from the first row to the
119896th row The method of the 119902th is described as follows
(a) Calculate the value of |119879119898119894
minus AMT| (119894 =
1 2 119898-119910) If the aircraft 119911 makes the|119879119898119911minus AMT| = min |119879119898
119894minus AMT| then 119904
119911119902= 1
Furthermore the aircraft can be selected in randomif there is more than one aircraft that meets thisformula
(b) Remove the line in which the aircraft 119911 is in to builda new reduced cycle matrix 119878
1015840 Update the remainingmaintenance time 119879119866
119902= 1199051minus 1199050minus 119879119898119911of the site 119902
(c) Compare the 119879119866119902and the 119879119898
119894for the 119878
1015840 If theformula ldquomin(119879119898
119894) | 119894 = 119900 le 119879119866
119902rdquo can be met by a
parameter 119900 form a new reduced cycle matrix and set119904119900119902
= 1 Moreover the remaining available referencetime should be updated as 119879119866
119902= 119879119866119902(119887) minus119879119898
119900 This
work should be repeated until the min (119879119898119894) gt 119879119866119902
then turn to the (119902 + 1)th row
Step 6 The matrix 119880 should be adjusted if the constraintsof resource maintenance cannot be met Consider that themaintenance cost should be as low as possible and therequirement of the mission risk should be satisfied theelements which 119906
119894119895= 1 should find 119901
119894119895(119886) corresponded and
the min119901119894119895(119886) then set 119906
119894119895= 0 Return to Step 1 and repeat
after finishing the update for the 119880 until meeting constrains119877119860and 119877
119861
423 Deal with the Constrain of the Mission Risk Somematrix 119880 which is initial or got by adjusting crossover andmutation may not meet the requirement of the mission riskconstrain The penalty method can be used in the methodfollowed to solve this problem
The energy function for every 119880 can be written as
119864 (119880 119878) = 119862 (119880 119878) + 119865 (119880 119878) sdot 119872119879 (21)
where 119865(119880 119878) is the vector of the penalty function and the119865119894(119880 119878) = max0 119892
119894(119880 119878) while 119872 which is the penalty
factor vector is a large positive number
Step 1 (fitness function design) The fitness function is givenas follows in order to minimize the objective function
119891 (119880 119878) = 1 minus
119864 (119880 119878) minus 119864min119864max minus 119864min
(22)
where 119864max and 119864min are the maximum and the minimumvalues of the energy function in the population
Step 2 (selection crossover and mutation) Proportionalselection single-point crossover and the basic alleles can beused in solving this problem
This problem can be dealt with by some method writtenin the article [24ndash26] in order to avoid the premature and thestalling that appear in the genetic algorithms
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 3
Predict all RULij of key LRMij (i = 1 2 m j = 1 2 n)
Calculate the initial failure probability Pij(a) ofthe key LRMij at the mission period by the
distribution Fij(t) of the RULij at the time t0 inthe mission preparation period
Defne the maintenance program of the keyLRM and describe whether the key LRM will
be maintained with uij while siq stand formaintenance site
Update the failure probability Pij(b) of the keyLRM in the mission period
Calculate the total cost C and time of allLRM maintenance costs by uij
Calculate the total mission risk of proposedaircraf for dispatching in typical tasks
Evaluate and optimize the program of the CBM
Figure 2 Modeling framework for CBM of aircraft fleet
33 Maintenance Program Generating Maintenance pro-gram considers whether a certain LRM should bemaintainedand the selection of the ISSs
The maintenance matrix 119880 of fleet can be described as
119880 =[
[
11990611
sdot sdot sdot 1199061119899
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199061198981
sdot sdot sdot 119906119898119899
]
]119898times119899
(3)
where 119906119894119895= 1means that the 119895th LRMof the 119894th aircraft needs
to be repaired otherwise 119906119894119895= 0
According to the assumption (4) the LRM can be main-tained at the same place however many the LRMs failsTherefore the ISS matrix 119878 of the fleet is shown as
119878 =[
[
11990411
sdot sdot sdot 1199041119896
sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199041198981
sdot sdot sdot 119904119898119896
]
]119898times119896
(4)
where 119904119894119902
= 1means that the 119894th aircraft should bemaintainedat the 119902th ISS otherwise 119906
119894119895= 0
34 Maintenance Time and Cost Estimation Assume repair-ing the 119895th LRM spends time119879
119895and needs cost119862
119895 According
to the assumption (5) the total maintenance time 119879119898119894of the
119894th aircraft is given as
119879119898119894= max (119906
119894119895times 119879119895) 119895 = 1 2 119899 (5)
There may be more than one aircraft that should berepaired at ISS 119902 so the total maintenance time of all aircraftscan be calcluated as
119898
sum
119894=1
119904119894119902times 119879119898119894
119894 = 1 2 119898 (6)
The total maintenance cost of all aircrafts can be got as
119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895] 119894 = 1 2 119898 119895 = 1 2 119899 (7)
35 Mission Risk Assessment
Step 1 (modify the health matrix of the fleet) Whether theaircraft is ldquodispatchingrdquo or ldquodispatching after maintenancerdquoshould be taken into consideration when calculating themission risk of the fleet The status of the aircraft shouldbe updated if the single strategy of the fleet is ldquodispatchingafter maintenancerdquo Then the modified health status matrix119875(119887) of the fleet can be built according to the Assumption(3) Consider 119901
119894119895(119887) = 0 after the LRM
119894119895on the 119894th aircraft
was renewed otherwise 119901119894119895(119887) = 119901
119894119895(119886) without renew The
elements in the matrix can be obtained by
119901119894119895(119887) = 119901
119894119895(119886) times (1 minus 119906
119894119895) (8)
Step 2 (estimate the failure probability of the single aircraftand rank) The failure probability of the single aircraft couldbe estimated after modifying the health matrix of the fleetAccording to the first assumption ldquothe aircraft fails when anykey LRM failsrdquo the failure probability 119875
119894(119886) of the 119894th aircraft
can be given as
119875119894 (119886) = 1 minus
119899
prod
119895=1
[1 minus 119901119894119895 (
119887)] (9)
Formula (10) can be obtained according to (1) (8) and(9)
119875119894(119886) = 1 minus
119899
prod
119895=1
[1 minus int
1199052
1199051
119891119894119895(119905) (1 minus 119906
119894119895)] 119894 = 1 2 119898
(10)
Then Pro(119894) = 1 2 119898 can be obtained by sorting thefailure probability of single aircraft in ascending order Theordered failure probability of the aircraft is given as
119875pro(119894) (119887) = 119875119894(119886) (11)
Suppose 1198752(119886) is the smallest Pro(119894) Then set Pro(2) = 1
and let 1198751(119887) = 119875
2(119886) after reordering Pick up aircrafts of
which Pro(119894) = 1 2 119897 when the mission needs dispatch 119897
aircrafts
4 The Scientific World Journal
Step 3 (calculate the mission risk of the fleet) Assume theserious consequences of the mission that failed are similarwithout taking the economic losses into account The failureprobability of the fleet mission can be calculated by thefollowing according to (7)
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] (12)
where 119875119865is the mission risk
4 Optimization Problem andAlgorithms Design
41 Problem Description The optimization problem in thepaper is to find a fleet CBM strategy with acceptable risk andlowest cost considering prognostics uncertainty
Thus describe the objective of the optimization asMin119862 = sum
119898
119894=1sum119899
119895=1[119906119894119895times 119862119895]
The constraints that should be considered about involvethe maintenance ability constraint 119877
119860of the site the time
constraint119877119861 the security risk constraint119877
119862 themission risk
constraint 119877119863 and the variable constraint 119877
119864
For first constraint 119877119860 set 119904
119894119902= 0 (119902 = 1 2 119896) and
119906119894119895
= 0 (119895 = 1 2 119899) if none of LRMs need maintenanceElse if any LRM
119894119895requires maintenance then 119906
119894119895= 1 and the
corresponding 119904119894119902
= 1 while the other 119904119894119902
= 0 Thus the 119877119860
can be described as
119877119860
119896
sum
119902=1
119904119894119902+
119899
prod
119895=1
(1 minus 119906119894119895) = 1 (13)
All maintenance of site 119902 should be finished before themission starts Thus the 119877
119861can be given as
119877119861 119879119898119894le 1199051minus 1199050| 119904119894119902
= 1 119894 = 1 2 119898 119902 = 1 2 119896
(14)
Assume that the total number of the aircraft whichmaintained at site 119902 is sum
119898
119894=1119904119894119902
= 119909 gt 1 (where the numberof aircraft is 1 2 119909) Ifsum119909minus1
119894=1119904119894119902times119879119898119894le 1199051minus1199050lt sum119909
119894=1119904119894119902times
119879119898119894 which means the maintenance for the 119909th aircraft could
not be finished before the mission start time this aircraft willnot be taken into account when the maintenance decisions isldquodispatching after maintenancerdquo
It will not be allowed to dispatch if the failure probabilityis too high for security risk existing in single aircraft Con-sider a mission need 119897 aircrafts and the 119877
119862can be described
as (15)Moreover themissionwill fail if (15) could not bemetWe have
119877119862 119875119897(119887) lt 119875
119904119897 (15)
According to (12) 119877119863can be written as (16) considering
the mission risk for fleet We have
119877119863
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] lt 119875
119898 (16)
where 119875119898is the objective of the mission risk
The variable constraint which means that the variablesshould be in a certain range is described as
119877119864 119862119895gt 0 119906
119894119895isin 0 1 119904
119894119902isin 0 1
119894 = 1 2 119898 119895 = 1 2 119899 119902 = 1 2 119896
(17)
The conceptual model for aircraft fleet condition basedmaintenance and dispatch is given as follows
Min 119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895]
st 119877119860sim119864
is satisfied
(18)
42 Optimization AlgorithmsDesign Theoptimization prob-lem cannot meet the KKT (Karush-Kuhn-Tucker) conditionsand the dimension of decisionmaking variables which can bewritten as 119898 times 119899 + 119898 times 119896 is relatively large So an improvedgenetic algorithmwas proposed in this paper for the probleminstead of traditional mathematical methods
The optimization model can be simplified as
min 119862 (119880 119878)
st
119892 (119880 119878) le 0
ℎ (119880 119878) = 0
119906119894119895isin 0 1
119904119894119895isin 0 1
(19)
where 119880 is the maintenance matrix while the 119878 is the ISSmatrix
The problem has more variables and constraints so thesolution quality of problem and the convergence rate couldnot be satisfied Therefore the improvement strategy of thegenetic algorithm is given in Figure 3
421 Define the Initial Population of the Maintenance Matrix119880 According to the multifactor and 2-level orthogonalexperimental design in order to cover widely define theinitial population of the maintenance matrix 119880 The initialpopulation should be filtered so as to make the convergencefaster Moreover the number of the aircraft needs to berepaired in the population which should be less than 119897
considering the dispatched requirements and the cost of themaintenance The relationship among those factors is shownas
119898
sum
119894=1
[
[
119899
prod
119895=1
(1 minus 119906119894119895)]
]
ge 119898 minus 119897 (20)
The Scientific World Journal 5
Yes
Yes
No
YesNo
No
Initializeupdate the U Heuristic rule
Solve the SAdjust the U
Meet themaintenance
resourcesconstrains
Meet the missionrisk constrains
Apply penalty function
Selection crossover and mutation
Meet therequirement for ending
the iterationOutput result
Figure 3 Improved strategies for genetic algorithm
422 Solve the ISS Matrix 119878 It is necessary to find a set offeasible solutions which meet the constraint of the ability ofISS 119877119860and the maintenance time 119877
119861on the basis of a certain
119880 The following heuristic rules can be used in order toreduce the amount of computation increasing the efficiencyof solving
Step 1 According to formula (13) an initial value of the 119878
can be given with the certain 119880 If prod119899119895=1
(1 minus 119906119894119895) = 1 the
aircraft 119894 need not be repaired and all 119904119894119902(119902 = 1 2 119896) =
0 otherwise the aircraft 119894 needs to be repaired Then thedetermining condition is described as sum119896
119902=1119904119894119902
= 1 and 119904119894119902
isin
0 1
Step 2 Consider that there are119910 aircrafts need not berepaired Remove 119910 rows which stand for these aircraftsThen a new (119898 minus 119910) times 119896 matrix 119878
1015840 which represents the newrelationship between the ISS and the aircraft that needs to berepaired can be built as the reduced cycle matrix of 119878
Step 3 Themaintenance time 119879119898119894of the aircraft needs to be
repaired in matrix 1198781015840 which can be obtained by formula (5)
Then the average maintenance time AMT of the ISSs is givenby AMT = sum
119898minus119910
119894=1119879119898119894119896 It can be determined not meet the
time constrain if max(119879119898119894) gt 1199051minus 1199050orAMT gt 119905
1minus 1199050 then
turn to Step 6 Otherwise turn to Step 4
Step 4 Initialize thematrix 1198781015840 and set 119904119894119902
= 0 (119894 = 1 2 119898minus
119910 119902 = 1 2 119896)
Step 5 Set the value of the matrix 1198781015840 from the first row to the
119896th row The method of the 119902th is described as follows
(a) Calculate the value of |119879119898119894
minus AMT| (119894 =
1 2 119898-119910) If the aircraft 119911 makes the|119879119898119911minus AMT| = min |119879119898
119894minus AMT| then 119904
119911119902= 1
Furthermore the aircraft can be selected in randomif there is more than one aircraft that meets thisformula
(b) Remove the line in which the aircraft 119911 is in to builda new reduced cycle matrix 119878
1015840 Update the remainingmaintenance time 119879119866
119902= 1199051minus 1199050minus 119879119898119911of the site 119902
(c) Compare the 119879119866119902and the 119879119898
119894for the 119878
1015840 If theformula ldquomin(119879119898
119894) | 119894 = 119900 le 119879119866
119902rdquo can be met by a
parameter 119900 form a new reduced cycle matrix and set119904119900119902
= 1 Moreover the remaining available referencetime should be updated as 119879119866
119902= 119879119866119902(119887) minus119879119898
119900 This
work should be repeated until the min (119879119898119894) gt 119879119866119902
then turn to the (119902 + 1)th row
Step 6 The matrix 119880 should be adjusted if the constraintsof resource maintenance cannot be met Consider that themaintenance cost should be as low as possible and therequirement of the mission risk should be satisfied theelements which 119906
119894119895= 1 should find 119901
119894119895(119886) corresponded and
the min119901119894119895(119886) then set 119906
119894119895= 0 Return to Step 1 and repeat
after finishing the update for the 119880 until meeting constrains119877119860and 119877
119861
423 Deal with the Constrain of the Mission Risk Somematrix 119880 which is initial or got by adjusting crossover andmutation may not meet the requirement of the mission riskconstrain The penalty method can be used in the methodfollowed to solve this problem
The energy function for every 119880 can be written as
119864 (119880 119878) = 119862 (119880 119878) + 119865 (119880 119878) sdot 119872119879 (21)
where 119865(119880 119878) is the vector of the penalty function and the119865119894(119880 119878) = max0 119892
119894(119880 119878) while 119872 which is the penalty
factor vector is a large positive number
Step 1 (fitness function design) The fitness function is givenas follows in order to minimize the objective function
119891 (119880 119878) = 1 minus
119864 (119880 119878) minus 119864min119864max minus 119864min
(22)
where 119864max and 119864min are the maximum and the minimumvalues of the energy function in the population
Step 2 (selection crossover and mutation) Proportionalselection single-point crossover and the basic alleles can beused in solving this problem
This problem can be dealt with by some method writtenin the article [24ndash26] in order to avoid the premature and thestalling that appear in the genetic algorithms
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
4 The Scientific World Journal
Step 3 (calculate the mission risk of the fleet) Assume theserious consequences of the mission that failed are similarwithout taking the economic losses into account The failureprobability of the fleet mission can be calculated by thefollowing according to (7)
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] (12)
where 119875119865is the mission risk
4 Optimization Problem andAlgorithms Design
41 Problem Description The optimization problem in thepaper is to find a fleet CBM strategy with acceptable risk andlowest cost considering prognostics uncertainty
Thus describe the objective of the optimization asMin119862 = sum
119898
119894=1sum119899
119895=1[119906119894119895times 119862119895]
The constraints that should be considered about involvethe maintenance ability constraint 119877
119860of the site the time
constraint119877119861 the security risk constraint119877
119862 themission risk
constraint 119877119863 and the variable constraint 119877
119864
For first constraint 119877119860 set 119904
119894119902= 0 (119902 = 1 2 119896) and
119906119894119895
= 0 (119895 = 1 2 119899) if none of LRMs need maintenanceElse if any LRM
119894119895requires maintenance then 119906
119894119895= 1 and the
corresponding 119904119894119902
= 1 while the other 119904119894119902
= 0 Thus the 119877119860
can be described as
119877119860
119896
sum
119902=1
119904119894119902+
119899
prod
119895=1
(1 minus 119906119894119895) = 1 (13)
All maintenance of site 119902 should be finished before themission starts Thus the 119877
119861can be given as
119877119861 119879119898119894le 1199051minus 1199050| 119904119894119902
= 1 119894 = 1 2 119898 119902 = 1 2 119896
(14)
Assume that the total number of the aircraft whichmaintained at site 119902 is sum
119898
119894=1119904119894119902
= 119909 gt 1 (where the numberof aircraft is 1 2 119909) Ifsum119909minus1
119894=1119904119894119902times119879119898119894le 1199051minus1199050lt sum119909
119894=1119904119894119902times
119879119898119894 which means the maintenance for the 119909th aircraft could
not be finished before the mission start time this aircraft willnot be taken into account when the maintenance decisions isldquodispatching after maintenancerdquo
It will not be allowed to dispatch if the failure probabilityis too high for security risk existing in single aircraft Con-sider a mission need 119897 aircrafts and the 119877
119862can be described
as (15)Moreover themissionwill fail if (15) could not bemetWe have
119877119862 119875119897(119887) lt 119875
119904119897 (15)
According to (12) 119877119863can be written as (16) considering
the mission risk for fleet We have
119877119863
119875119865= 1 minus
119897
prod
119894=1
[1 minus 119875119894 (119887)] lt 119875
119898 (16)
where 119875119898is the objective of the mission risk
The variable constraint which means that the variablesshould be in a certain range is described as
119877119864 119862119895gt 0 119906
119894119895isin 0 1 119904
119894119902isin 0 1
119894 = 1 2 119898 119895 = 1 2 119899 119902 = 1 2 119896
(17)
The conceptual model for aircraft fleet condition basedmaintenance and dispatch is given as follows
Min 119862 =
119898
sum
119894=1
119899
sum
119895=1
[119906119894119895times 119862119895]
st 119877119860sim119864
is satisfied
(18)
42 Optimization AlgorithmsDesign Theoptimization prob-lem cannot meet the KKT (Karush-Kuhn-Tucker) conditionsand the dimension of decisionmaking variables which can bewritten as 119898 times 119899 + 119898 times 119896 is relatively large So an improvedgenetic algorithmwas proposed in this paper for the probleminstead of traditional mathematical methods
The optimization model can be simplified as
min 119862 (119880 119878)
st
119892 (119880 119878) le 0
ℎ (119880 119878) = 0
119906119894119895isin 0 1
119904119894119895isin 0 1
(19)
where 119880 is the maintenance matrix while the 119878 is the ISSmatrix
The problem has more variables and constraints so thesolution quality of problem and the convergence rate couldnot be satisfied Therefore the improvement strategy of thegenetic algorithm is given in Figure 3
421 Define the Initial Population of the Maintenance Matrix119880 According to the multifactor and 2-level orthogonalexperimental design in order to cover widely define theinitial population of the maintenance matrix 119880 The initialpopulation should be filtered so as to make the convergencefaster Moreover the number of the aircraft needs to berepaired in the population which should be less than 119897
considering the dispatched requirements and the cost of themaintenance The relationship among those factors is shownas
119898
sum
119894=1
[
[
119899
prod
119895=1
(1 minus 119906119894119895)]
]
ge 119898 minus 119897 (20)
The Scientific World Journal 5
Yes
Yes
No
YesNo
No
Initializeupdate the U Heuristic rule
Solve the SAdjust the U
Meet themaintenance
resourcesconstrains
Meet the missionrisk constrains
Apply penalty function
Selection crossover and mutation
Meet therequirement for ending
the iterationOutput result
Figure 3 Improved strategies for genetic algorithm
422 Solve the ISS Matrix 119878 It is necessary to find a set offeasible solutions which meet the constraint of the ability ofISS 119877119860and the maintenance time 119877
119861on the basis of a certain
119880 The following heuristic rules can be used in order toreduce the amount of computation increasing the efficiencyof solving
Step 1 According to formula (13) an initial value of the 119878
can be given with the certain 119880 If prod119899119895=1
(1 minus 119906119894119895) = 1 the
aircraft 119894 need not be repaired and all 119904119894119902(119902 = 1 2 119896) =
0 otherwise the aircraft 119894 needs to be repaired Then thedetermining condition is described as sum119896
119902=1119904119894119902
= 1 and 119904119894119902
isin
0 1
Step 2 Consider that there are119910 aircrafts need not berepaired Remove 119910 rows which stand for these aircraftsThen a new (119898 minus 119910) times 119896 matrix 119878
1015840 which represents the newrelationship between the ISS and the aircraft that needs to berepaired can be built as the reduced cycle matrix of 119878
Step 3 Themaintenance time 119879119898119894of the aircraft needs to be
repaired in matrix 1198781015840 which can be obtained by formula (5)
Then the average maintenance time AMT of the ISSs is givenby AMT = sum
119898minus119910
119894=1119879119898119894119896 It can be determined not meet the
time constrain if max(119879119898119894) gt 1199051minus 1199050orAMT gt 119905
1minus 1199050 then
turn to Step 6 Otherwise turn to Step 4
Step 4 Initialize thematrix 1198781015840 and set 119904119894119902
= 0 (119894 = 1 2 119898minus
119910 119902 = 1 2 119896)
Step 5 Set the value of the matrix 1198781015840 from the first row to the
119896th row The method of the 119902th is described as follows
(a) Calculate the value of |119879119898119894
minus AMT| (119894 =
1 2 119898-119910) If the aircraft 119911 makes the|119879119898119911minus AMT| = min |119879119898
119894minus AMT| then 119904
119911119902= 1
Furthermore the aircraft can be selected in randomif there is more than one aircraft that meets thisformula
(b) Remove the line in which the aircraft 119911 is in to builda new reduced cycle matrix 119878
1015840 Update the remainingmaintenance time 119879119866
119902= 1199051minus 1199050minus 119879119898119911of the site 119902
(c) Compare the 119879119866119902and the 119879119898
119894for the 119878
1015840 If theformula ldquomin(119879119898
119894) | 119894 = 119900 le 119879119866
119902rdquo can be met by a
parameter 119900 form a new reduced cycle matrix and set119904119900119902
= 1 Moreover the remaining available referencetime should be updated as 119879119866
119902= 119879119866119902(119887) minus119879119898
119900 This
work should be repeated until the min (119879119898119894) gt 119879119866119902
then turn to the (119902 + 1)th row
Step 6 The matrix 119880 should be adjusted if the constraintsof resource maintenance cannot be met Consider that themaintenance cost should be as low as possible and therequirement of the mission risk should be satisfied theelements which 119906
119894119895= 1 should find 119901
119894119895(119886) corresponded and
the min119901119894119895(119886) then set 119906
119894119895= 0 Return to Step 1 and repeat
after finishing the update for the 119880 until meeting constrains119877119860and 119877
119861
423 Deal with the Constrain of the Mission Risk Somematrix 119880 which is initial or got by adjusting crossover andmutation may not meet the requirement of the mission riskconstrain The penalty method can be used in the methodfollowed to solve this problem
The energy function for every 119880 can be written as
119864 (119880 119878) = 119862 (119880 119878) + 119865 (119880 119878) sdot 119872119879 (21)
where 119865(119880 119878) is the vector of the penalty function and the119865119894(119880 119878) = max0 119892
119894(119880 119878) while 119872 which is the penalty
factor vector is a large positive number
Step 1 (fitness function design) The fitness function is givenas follows in order to minimize the objective function
119891 (119880 119878) = 1 minus
119864 (119880 119878) minus 119864min119864max minus 119864min
(22)
where 119864max and 119864min are the maximum and the minimumvalues of the energy function in the population
Step 2 (selection crossover and mutation) Proportionalselection single-point crossover and the basic alleles can beused in solving this problem
This problem can be dealt with by some method writtenin the article [24ndash26] in order to avoid the premature and thestalling that appear in the genetic algorithms
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 5
Yes
Yes
No
YesNo
No
Initializeupdate the U Heuristic rule
Solve the SAdjust the U
Meet themaintenance
resourcesconstrains
Meet the missionrisk constrains
Apply penalty function
Selection crossover and mutation
Meet therequirement for ending
the iterationOutput result
Figure 3 Improved strategies for genetic algorithm
422 Solve the ISS Matrix 119878 It is necessary to find a set offeasible solutions which meet the constraint of the ability ofISS 119877119860and the maintenance time 119877
119861on the basis of a certain
119880 The following heuristic rules can be used in order toreduce the amount of computation increasing the efficiencyof solving
Step 1 According to formula (13) an initial value of the 119878
can be given with the certain 119880 If prod119899119895=1
(1 minus 119906119894119895) = 1 the
aircraft 119894 need not be repaired and all 119904119894119902(119902 = 1 2 119896) =
0 otherwise the aircraft 119894 needs to be repaired Then thedetermining condition is described as sum119896
119902=1119904119894119902
= 1 and 119904119894119902
isin
0 1
Step 2 Consider that there are119910 aircrafts need not berepaired Remove 119910 rows which stand for these aircraftsThen a new (119898 minus 119910) times 119896 matrix 119878
1015840 which represents the newrelationship between the ISS and the aircraft that needs to berepaired can be built as the reduced cycle matrix of 119878
Step 3 Themaintenance time 119879119898119894of the aircraft needs to be
repaired in matrix 1198781015840 which can be obtained by formula (5)
Then the average maintenance time AMT of the ISSs is givenby AMT = sum
119898minus119910
119894=1119879119898119894119896 It can be determined not meet the
time constrain if max(119879119898119894) gt 1199051minus 1199050orAMT gt 119905
1minus 1199050 then
turn to Step 6 Otherwise turn to Step 4
Step 4 Initialize thematrix 1198781015840 and set 119904119894119902
= 0 (119894 = 1 2 119898minus
119910 119902 = 1 2 119896)
Step 5 Set the value of the matrix 1198781015840 from the first row to the
119896th row The method of the 119902th is described as follows
(a) Calculate the value of |119879119898119894
minus AMT| (119894 =
1 2 119898-119910) If the aircraft 119911 makes the|119879119898119911minus AMT| = min |119879119898
119894minus AMT| then 119904
119911119902= 1
Furthermore the aircraft can be selected in randomif there is more than one aircraft that meets thisformula
(b) Remove the line in which the aircraft 119911 is in to builda new reduced cycle matrix 119878
1015840 Update the remainingmaintenance time 119879119866
119902= 1199051minus 1199050minus 119879119898119911of the site 119902
(c) Compare the 119879119866119902and the 119879119898
119894for the 119878
1015840 If theformula ldquomin(119879119898
119894) | 119894 = 119900 le 119879119866
119902rdquo can be met by a
parameter 119900 form a new reduced cycle matrix and set119904119900119902
= 1 Moreover the remaining available referencetime should be updated as 119879119866
119902= 119879119866119902(119887) minus119879119898
119900 This
work should be repeated until the min (119879119898119894) gt 119879119866119902
then turn to the (119902 + 1)th row
Step 6 The matrix 119880 should be adjusted if the constraintsof resource maintenance cannot be met Consider that themaintenance cost should be as low as possible and therequirement of the mission risk should be satisfied theelements which 119906
119894119895= 1 should find 119901
119894119895(119886) corresponded and
the min119901119894119895(119886) then set 119906
119894119895= 0 Return to Step 1 and repeat
after finishing the update for the 119880 until meeting constrains119877119860and 119877
119861
423 Deal with the Constrain of the Mission Risk Somematrix 119880 which is initial or got by adjusting crossover andmutation may not meet the requirement of the mission riskconstrain The penalty method can be used in the methodfollowed to solve this problem
The energy function for every 119880 can be written as
119864 (119880 119878) = 119862 (119880 119878) + 119865 (119880 119878) sdot 119872119879 (21)
where 119865(119880 119878) is the vector of the penalty function and the119865119894(119880 119878) = max0 119892
119894(119880 119878) while 119872 which is the penalty
factor vector is a large positive number
Step 1 (fitness function design) The fitness function is givenas follows in order to minimize the objective function
119891 (119880 119878) = 1 minus
119864 (119880 119878) minus 119864min119864max minus 119864min
(22)
where 119864max and 119864min are the maximum and the minimumvalues of the energy function in the population
Step 2 (selection crossover and mutation) Proportionalselection single-point crossover and the basic alleles can beused in solving this problem
This problem can be dealt with by some method writtenin the article [24ndash26] in order to avoid the premature and thestalling that appear in the genetic algorithms
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 The Scientific World Journal
Table 1 The RULs of the LRM
Number 1 2 3 4 5 6 7 8 9 10LRMA (25 77) (19 44) (29 83) (29 91) (13 38) (20 57) (21 63) (20 62) (9 25) (21 59)LRMB (28 74) (3 07) (29 66) (15 33) (28 85) (2 05) (23 56) (6 13) (2 05) (10 28)LRMC (4 10) (9 29) (5 12) (25 57) (24 71) (26 79) (23 61) (22 72) (3 08) (29 91)LRMD (28 75) (17 56) (30 79) (5 12) (29 79) (29 71) (12 35) (1 03) (25 64) (2 06)
Simulate the annealing stretching for fitness before select-ing the operator as follows
119891119894=
119890119891119894119879
sum119873
119895=1119890119891119895119879
119879 = 1198790times 119888119892minus1
0 lt 119888 lt 1 (23)
where 119873 is the size of the population and 119892 is the geneticalgebra while119879
0is the initial temperature and119891
119894is the fitness
of the ith individual119875119890and 119875
119891can be defined as (24) in order to make the
crossover and mutation probability changing dynamic withthe fitness which means that if the fitness of each individualis consistent 119875
119890and 119875
119891will increase otherwise they will
decrease
119875119890=
1198961(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198962
1198911015840lt 119891avg
119875119891=
1198963(119891max minus 119891
1015840)
(119891max minus 119891avg)1198911015840ge 119891avg
1198964
1198911015840lt 119891avg
(24)
where 119891max and 119891avg are the maximum fitness and the averagefitness in the populations and the 119891
1015840 is the maximum fitnessof the parent The 119896
1 1198962 1198963 1198964are all constant
5 Case Study
Consider a fleet containing 10 aircrafts and each aircraftincludes 4 LRM (A B C D) of which life can be predictedAssume the RUL following Gaussian distributions 119873(120583 120590
2)
and the mean 120583 and the variance 1205902 are given in Table 1
The mission requires dispatch 8 aircrafts one hour laterand lasting two hours 119875
119904119897should be below the 10minus8 while 119875
119904119897
should be below 10minus6Assume there are 3 ISSs of which ability of the mainte-
nance are the same being in charge of all aircraftsrsquo mainte-nance The maintenance time and cost of each LRM is givenin Table 2
Consider that there are 100 individuals in populationsand one of these individuals is described as follows
119880 =
[
[
[
[
1 1 0 1 0 0 0 1 1 1
0 1 0 0 0 1 1 0 0 0
1 0 1 0 1 1 0 0 1 0
0 0 0 1 0 0 1 1 0 1
]
]
]
]
119879
(25)
Set up the 1198961= 1198962= 097 119896
3= 1198964= 002 The result is
described in Figure 4 after 250 iterations
Table 2 The maintenance time of the LRM
LRM A B C DMaintenance time 20min 25min 116min 166minMaintenance cost 23482 2843 12973 10092
0 50 100 150 200 2501
2
3
4
5
6
7
Y o
bjec
tive f
unct
ion
X number of generations
times104
Figure 4 Result of calculation
The total cost of the maintenance is 144393 and themission risk is 895 lowast 10minus07 which meet the requirement
Then optimal maintenance program can be written asTable 3
119880 =
[
[
[
[
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1
]
]
]
]
119879
119878 =[
[
1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 1
]
]
119879
(26)
Then the optimal scheme of aircraft CBM and dispatch-ing are described completely in Table 3 where the elementsin the table such as LRMc LRMBD are the LRMs that needto be repaired There are six aircrafts and eight LRMs thatneed to be repaired and the numbers of the aircrafts that needdispatch are 1 3 4 5 6 7 8 and 10
6 Conclusion
This paper researches optimization decision method foraircraft fleet CBM oriented to mission success considering
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 7
Table 3 The optimal scheme for aircraft fleet CBM and dispatching
Aircraft number 1 2 3 4 5 6 7 8 9 10ISS 1 LRMC LRMB ISS 2 LRMC LRMBD ISS 3 LRMD LRMBD
Dispatch Yes No Yes Yes Yes Yes Yes Yes No Yes
prognostics uncertainty and the resource constrain TheCBM and dispatch process of fleet is analyzed the modelingmethod and an improved genetic algorithm for the problemare given and the method is verified by case about fleet with10 aircrafts
Themain advantages of thismethod are shown as follows(1) The alternative strategy sets for single aircraft are
defined then the optimization problem of fleetCBM is translated to the combinatorial optimizationproblem of single aircraft strategy The relationshipbetween maintenance strategy and mission risk isestablished and the problem becomes easier to solve
(2) This paper used the RUL distribution which has themaximum information and the highest in prognos-tics It has more accurate description of the uncer-tainty compared with others
(3) The optimization decision with risk for fleet CBMis realized The fleet mission risk is quantitativelyassessed and the optimal CBM strategy for fleet couldsatisfy the requirement of lowest maintenance costand acceptable risk
This paper presents a theoretical approach for fleet CBMconsidering prognostics uncertainty Some factors have beensimplified such as the cost of risk the consequences ofrisk mission the effect of the CBM process form ability ofmaintenance personnel and the effect of random failuresThefocus of further work is a more detailed and comprehensivemodel considering all above factors
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Sun S Zeng R Kang and M G Pecht ldquoBenefits and chal-lenges of system prognosticsrdquo IEEE Transactions on Reliabilityvol 61 no 1 pp 323ndash335 2012
[2] B Sun S Liu L Tong L Shunli and F Qiang ldquoA cognitiveframework for analysis and treatment of uncertainty in prog-nosticsrdquoChemical Engineering Transactions vol 33 pp 187ndash1922013
[3] I Lopez and N Sarigul-Klijn ldquoA review of uncertainty in flightvehicle structural damage monitoring diagnosis and controlchallenges and opportunitiesrdquo Progress in Aerospace Sciencesvol 46 no 7 pp 247ndash273 2010
[4] J Fang M Xiao Y Zhou and Y Wang ldquoOptimal dynamicdamage assessment and life prediction for electronic productsrdquo
Chinese Journal of Scientific Instrument vol 32 no 4 pp 807ndash812 2011
[5] I Barlas G Zhang et al Confidence Metrics and UncertaintyManagement in Prognosis MARCON Knoxville Tenn USA2003
[6] B P Leao and J P P Gomes ldquoImprovements on the offlineperformance evaluation of fault prognostics methodsrdquo in Pro-ceedings of the IEEE Aerospace Conference IEEE ComputerSociety pp 1ndash6 2011
[7] B P Leao T Yoneyama G C Rocha and K T FitzgibbonldquoPrognostics performancemetrics and their relation to require-ments design verification and cost-benefitrdquo in Proceedings ofthe International Conference on Prognostics and HealthManage-ment (PHM rsquo08) October 2008
[8] A Saxena J Celaya B Saha S Saha and K Goebel ldquoEval-uating prognostics performance for algorithms incorporatinguncertainty estimatesrdquo in Proceedings of the IEEE AerospaceConference March 2010
[9] I A Raptis andG Vachtsevanos ldquoAn adaptive particle filtering-based framework for real-time fault diagnosis and failureprognosis of environmental control systemsrdquo in Proceedings ofthe Prognostics and Health Management 2011
[10] L Tang J Decastro G Kacprzynski K Goebel and GVachtsevanos ldquoFiltering and prediction techniques for model-based prognosis and uncertainty managementrdquo in Proceedingsof the Prognostics and System Health Management Conference(PHM rsquo10) January 2010
[11] B Saha and K Goebel ldquoUncertainty management for diagnos-tics and prognostics of batteries using Bayesian techniquesrdquo inProceedings of the IEEE Aerospace Conference (AC rsquo08) March2008
[12] G Xuefei H Jingjing J Ratneshwar et al ldquoBayesian fatiguedamage and reliability analysis using Laplace approximationand inverse reliability methodrdquo in Proceedings of the Prognosticsand Health Management Society Conference (PHM Society rsquo11)2011
[13] L Tang G J Kacprzynski K Goebel and G VachtsevanosldquoMethodologies for uncertaintymanagement in prognosticsrdquo inProceedings of the IEEE Aerospace Conference March 2009
[14] A Coppe R T Haftka and N-H Kim ldquoLeast squares-filteredBayesian updating for remaining useful life estimationrdquo inProceedings of the 51st AIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference April 2010
[15] M Orchard G Kacprzynski K Goebel B Saha and GVachtsevanos ldquoAdvances in uncertainty representation andmanagement for particle filtering applied to prognosticsrdquo inProceedings of the International Conference on Prognostics andHealth Management (PHM rsquo08) October 2008
[16] M L Neves L P Santiago and C A Maia ldquoA condition-based maintenance policy and input parameters estimation fordeteriorating systems under periodic inspectionrdquo Computersand Industrial Engineering vol 61 no 3 pp 503ndash511 2011
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
8 The Scientific World Journal
[17] P A Sandborn and C Wilkinson ldquoA maintenance planningand business case development model for the applicationof prognostics and health management (PHM) to electronicsystemsrdquo Microelectronics Reliability vol 47 no 12 pp 1889ndash1901 2007
[18] Q Feng H Peng and D W Coit ldquoA degradation-basedmodel for joint optimization of burn-in quality inspection andmaintenance a light display device applicationrdquo InternationalJournal of Advanced Manufacturing Technology vol 50 no 5-8pp 801ndash808 2010
[19] B Wu Z Tian and M Chen ldquoCondition based maintenanceoptimization using neural network based health conditionpredictionrdquo Quality and Reliability Engineering Internationalvol 29 no 8 pp 1151ndash1163 2013
[20] K T Huynh A Barros and C Berenguer ldquoMaintenancedecision-making for systems operating under indirect condi-tion monitoring value of online information and impact ofmeasurement uncertaintyrdquo IEEETransactions onReliability vol61 no 2 pp 410ndash425 2012
[21] R FlageDWCoit J T Luxhoslashj andTAven ldquoSafety constraintsapplied to an adaptive Bayesian condition-based maintenanceoptimization modelrdquo Reliability Engineering and System Safetyvol 102 pp 16ndash26 2012
[22] Q Feng S Li and B Sun ldquoA multi-agent based intelligentpredicting method for fleet spare part requirement applyingcondition based maintenancerdquo in Proceedings of the 5th Inter-national Conference onMultimedia Information Networking andSecurity pp 808ndash811 IEEE Computer Society 2013
[23] Q Feng S Li and B Sun ldquoAn intelligent fleet condition-basedmaintenance decision making method based on multi-agentrdquoInternational Journal of Prognostics and Health Managementvol 3 no 1 pp 1ndash11 2012
[24] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[25] P Vasant ldquoA novel hybrid genetic algorithms and pattern searchtechniques for industrial production planningrdquo InternationalJournal of Modeling Simulation and Scientific Computing vol3 no 4 pp 1ndash19 2012
[26] P Vasant ldquoHybrid mesh adaptive direct search genetic algo-rithms and line search approaches for fuzzy optimizationproblems in production planningrdquo Intelligent Systems ReferenceLibrary vol 38 pp 779ndash799 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014