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

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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

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Applied Computational Intelligence and Soft Computing

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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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