Research Article Condition-Based Maintenance Strategy for ...downloads.hindawi.com/journals/mpe/2015/494162.pdfResearch Article Condition-Based Maintenance Strategy for Production

Research ArticleCondition-Based Maintenance Strategy for Production SystemsGenerating Environmental Damage

L Tlili1 M Radhoui2 and A Chelbi1

1Ecole Nationale Superieure drsquoIngenieurs de Tunis Universite de Tunis CEREP Tunis Tunisia2Ecole Nationale drsquoIngenieurs de Carthage Universite de Carthage CEREP Tunis Tunisia

Correspondence should be addressed to A Chelbi anischelbiplanettn

Received 21 December 2014 Revised 10 March 2015 Accepted 12 March 2015

Academic Editor Elena Benvenuti

Copyright copy 2015 L Tlili et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider production systems which generate damage to environment as they get older and degradeThe system is submitted toinspections to assess the generated environmental damage The inspections can be periodic or nonperiodic In case an inspectionreveals that the environmental degradation level has exceeded the critical level 119880 the system is considered in an advanceddeterioration state and will have generated significant environmental damage A corrective maintenance action is then performedto renew the system and clean the environment and a penalty has to be paid In order to prevent such an undesirable situation alower threshold level 119871 is considered to trigger a preventive maintenance action to bring back the system to a state as good as newat a lower cost and without paying the penalty Two inspection policies are considered (periodic and nonperiodic) For each one ofthem a mathematical model and a numerical procedure are developed to determine simultaneously the preventive maintenance(PM) threshold 119871lowast and the inspection sequence which minimize the average long-run cost per time unit Numerical calculationsare performed to illustrate the proposedmaintenance policies and highlight their main characteristics with respect to relevant inputparameters

1 Introduction

Because of the degradation of the environment in the worldand the growing public pressure governments imposedseveral constraining and penalizing measures to companieswhose production processes potentially generate any formof environmental damage These measures have been takenglobally (Kyoto Protocol in 1997) and nationally Moreoverthe companiesrsquo greenhouse gas emissions (primarily CO

2) are

severely limited and penalized financially These constraintsled decision-makers to establish and implement effectiveenvironmental sustainability policies considering environ-mental and cost criteria

Numerous efforts have emerged to address the envi-ronment issues related to environmental impact assessmentwithin the company A set of qualitative tools such as FMEA[1] Fault Tree Analysis (FTA) [2] and HAZOP [3] are usedas well as quantitative models [4ndash6] It is clearly establishedthat preventive maintenance plays a crucial role in meetingthe environmental legislation requirements by limiting and

preventing equipment degradation which may lead to envi-ronmental damage Martorell et al [7] proposed integratedmulticriteria decision-making (IMCDM) in order to achievebetter scores in reliability availability maintainability safetyand cost (RAMS+C) at nuclear power plants The proposedmethodology integrates among others the risk to envi-ronment The IMCDM is formulated as a multiobjectiveoptimization where the decision variables are preventivemaintenance interval surveillance test interval and max-imum allowed outage time Vassiliadis and Pistikopoulos[8] developed an optimization framework which takes intoaccount the environmental risks and the operability charac-teristics at the early stage of process design This frameworkpermits the identification of optimal preventive maintenanceschedules

The excess of environmental damages generated by pro-duction systems is in numerous situations caused indirectlyby the deterioration of those systems (eg degradationandor corrosion yielding leakages wear causing excess ofenergy consumption etc) For instance in a nuclear power

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 494162 12 pageshttpdxdoiorg1011552015494162

2 Mathematical Problems in Engineering

plant the degradation of the mechanical shaft seal of therefrigeration compressor induces toxic refrigerant leakages[9] Yan andHua [10] showed that the degradation ofmachinetools causes an increase of energy consumption which canbe converted to carbon emission by referring to the standarddeveloped by the Climate Registry

Therefore when it is technically possible condition-based monitoring repair and maintenance are the mostappropriate activities to be adopted It allows monitoring thedegradation level and the resulting environmental damage inorder to take the appropriate preventive actions and limit therisk of penalties and sometimes catastrophes In the literaturetwo approaches are proposed to monitor the degradationcontinuous monitoring [11 12] and inspection [13] In manysituations in practice the continuous monitoring of theenvironmental damage generated by production systems iscostly and sometimes difficult to perform (eg gas emissionswastes etc) The sequential inspection can be more effectiveIt is widely used Numerous efforts have been conducted todeal with the determination of inspection sequences for agiven critical threshold of system deterioration optimizinga certain objective like maintenance total cost or systemrsquosavailability [14] Chelbi and Ait-Kadi [15] studied a systemsubject to deteriorationThey determined optimal inspectionsequences that minimize the expected total cost per time unitover an infinite time span Castanier et al [16] introduced acondition-based maintenance policy for a repairable systemsubject to a continuous-state gradual deterioration moni-tored by sequential nonperiodic inspections They proposeavailability and cost models where the inspection sequenceis derived

Golmakani and Fattahipour [17] proposed a cost-effectiveage-based inspection scheme (with nonconstant inspectionintervals) for a single unit system Their model is based onthe control limit policy proposed by Makis and Jardine [18]to determine the optimal replacement threshold

Recently Chouikhi et al [9] developed a maintenanceoptimization model taking into account environmental dete-rioration They considered a single-unit system subject torandom deterioration which impacts the quality of the envi-ronment The inspection sequence is optimized in order tominimize the long-run average maintenance cost

To make condition-based maintenance more effectiveboth the inspection sequence and the alarm threshold level ofdegradation need to be optimized In this context Grall et al[19] studied a system subject to a random deterioration pro-cess They developed a model based on a stationary processto determine both the preventive maintenance threshold andinspection dates thatminimize the average long-run cost rateOptimization of alarm threshold and sequential inspectionscheme were also the aim of the work of Jiang [20] Theydeveloped an algorithm to that purpose

In the same context of condition-based policies optimiz-ing alarm threshold levels and inspection schedules we focusin this paper on production systems whose degradation gen-erates directly (or indirectly) environmental damage A con-dition-based maintenance strategy is proposed for such sys-tems considering two threshold levels related to the amountof environmental damage a critical one 119880 that is known

and which yields a significant penalty in case it is exceededand a lower one 119871 to be determined In case inspectionreveals the latter is exceeded a preventive maintenanceaction is undertaken avoiding the penalty We propose a newmodeling approach based on the fact that the degradationprocess is modeled by the Wiener process (discussed inSection 2) Thus the first hitting time and the remaininguseful life of the system conditional to its environmentaldegradation level are considered Two types of inspectionpolicy periodic and nonperiodic can be used to reveal thelevel of degradation We develop a mathematical model anda numerical procedure for each policyThe objective for bothstrategies is to determine the threshold level 119871lowast and theinspection sequence which minimize the total expected costper time unit over an infinite time horizon

The remainder of this paper is organized as follows Themain notations used in themathematicalmodel are presentedin Notations Other notations will be introduced through-out the following sections The next section is devoted tothe modeling of the environmental degradation processThe detailed problem description and the definition of theassumptions are given in Section 3The fourth section is ded-icated to the modeling of the periodic inspection policy andan optimization algorithm is developed to find the optimalinspection period and the PM threshold level In Section 5we extend the model presented in Section 4 consideringnonperiodic inspections We develop a numerical procedureallowing the generation of a nearly optimal solution andwe discuss obtained numerical results Concluding remarkstogether with some indications about extensions currentlyunder consideration are provided in the last section

2 Analysis of the Degradation Process

Stochastic processes are appropriate tomodel the degradationprocess involving independent increments Many stochasticprocesses have been studied in the literature (for more detailssee [21 22]) that is Gamma process [23] andWiener process[24]

The Wiener process (also called Brownian motion pro-cess) had firstly been used to model the irregular motionof the pollen particles floating in water The mathematicaltheory of Brownian motion was discussed in detail inWienerrsquos dissertation in 1918 and in papers that followedThetheory was completed by Levy Ito McKean and others [25]Moreover the Wiener process has been adopted for manyapplications for example finance [26] wind energy [27]electrical devices [28] demography linguistics employmentservice and chemical reactions [29] In addition it has beenfound useful to analyze degradation data [30 31]

The Wiener process as a degradation model is basedon the consideration that the degradation increment in aninfinitesimal time interval might be viewed as an additivesuperposition of a large number of small external effects[32] Moreover it is flexible in incorporating random effectsand explanatory variables that take in consideration the het-erogeneities commonly observed in degradation problemsIt is typically used for modeling degradation processes in

Mathematical Problems in Engineering 3

a random environment and where the degradation increaseslinearly with time

Suppose that the degradation process 119883(119905) 119905 gt 0 obeysa Wiener process and is written as

119883(119905) = 120582119905 + 120590119882 (119905) (1)

where 120582 is the drift coefficient 120590 gt 0 is the diffusion coef-ficient and119882(119905) is the standard Brownian motion

119883(119905) has the following properties [33]

(a) 119883(0) = 0 almost surely(b) For any time sequence 119905

119894 119894 = 1 2 119899 with 0 lt 119905

1lt

1199052lt sdot sdot sdot lt 119905

119899 the random increments 119883(119905

1) 119883(1199052minus

1199051) 119883(119905

119899minus 119905119899minus1) where 119883(119905

119895minus1minus 119905119895) = 119883(119905

119895minus1) minus

119883(119905119895) are independent and any 119883(119905) minus 119883(119903) (119905 gt

0 119903 gt 0) follows a normal distribution119873(0 1205902|119905minus 119903|)(c) The paths of 119883(119905) are continuous with probability

one

The Wiener process is used in modeling degradation pro-cesses and has shown some advantages regarding the mathe-matical properties As an example the inverse Gaussian (IG)distribution is used to formulate analytically the first hittingprocess of the Wiener process when the degradation path ismonotonic and gradual [30]

The failure time is defined as the first hitting time (FHT)which is the time from present time to the instant 119879

119880at

which the degradation first hits a critical level 119880 Based onthe concept of FHT the failure time can be defined as

119879119880= inf 119905 ge 0 | 119883 (119905) ge 119880 (2)

Furthermore the FHT or failure time of the process to athreshold 119880 follows the inverse Gaussian distribution witha pdf [24]

119891119879119880(119905) =

119880

radic212058711990531205902exp(minus(119880 minus 120582119905)

2

21205902119905) (3)

and the cdf is given by [24]

119865119879119880(119905) = Φ(

minus119880 + 120582119905

120590radic119905) + exp(2120582119880

1205902)Φ(

minus119880 minus 120582119905

120590radic119905) (4)

where Φ denotes the standard normal cdfWe define the remaining useful life (RUL) of a deteriorat-

ing system associated with the degradation process given thatthe degradation level119883(119905) is 119871 as

119879119904= 119879119880minus 119879119871= inf 119904 ge 0 | 119883 (119905 + 119904) ge 119880 if 119871 lt 119880 (5)

otherwise 119879119904= 0

Based on property (b) of the Wiener process the degra-dation increments are independent hence if 119871 lt 119880 119879

119904can

be written as follows

119879119904= inf 119904 ge 0 | 119883 (119905) + 119883 (119904) ge 119880

= inf 119904 ge 0 | 119883 (119904) ge 119880 minus 119871

(6)

Thus the remaining useful life (RUL) distribution is derivedutilizing the property that the sum of Gaussian variables isGaussian again then the pdf of the RUL of 119879

119904from time zero

is expressed as follows [34 35]

119891119879119880minus119879119871

(119905) =119880 minus 119871

radic212058711990531205902exp(minus(119880 minus 119871 minus 120582119905)

2

21205902119905) (7)

And the cdf of the RUL is given by

119865119879119880minus119879119871

(119905) = Φ(minus (119880 minus 119871) + 120582119905

120590radic119905) + exp(2120582 (119880 minus 119871)

1205902)Φ

sdot (minus (119880 minus 119871) minus 120582119905

120590radic119905)

(8)

This (RUL) distribution will be used in the cost model whichwill be developed below

3 Problem Description

We consider a production system assimilated to a single unitthat causes a random amount of damage to environment asit gets older and degrades It is assumed that the environ-mental degradation process is modeled by Wiener processThe value of the environmental degradation level (damage)can be known (measured) only by inspection Hence thesystem is submitted to inspections to assess the generatedenvironmental damage In case an inspection reveals thatthe environmental degradation level has exceeded a criticallevel119880 (generally known and fixed by legislation) the systemis considered in advanced deterioration state and will havegenerated significant environmental damage subject to apenalty incurred from the instant at which the critical level119880 has been exceeded A CM action is then performed torenew the systemand clean the environment and the resultingpenalty has to be paid In order to lower the chances tobe in such an undesirable situation a lower threshold level119871 has to be considered to trigger a PM to bring back thesystem to a state as good as new at a lower cost and withoutpaying the penalty (Figure 1) In case inspection reveals thatthe environmental degradation level is lower than 119871 nomaintenance action is done

The following working assumptions are considered

(i) The degradation of the system induces the degrada-tion of the environment

(ii) Inspections are perfect and their duration is negligi-ble

(iii) The durations of PM and CM actions are also negligi-ble

(iv) After each inspection only one of the three followingevents is possible do nothing perform a PM actionor perform a CM action

(v) Both PM and CM actions renew or bring back thesystem to a state as good as new

4 Mathematical Problems in Engineering

Environmental degradation level

PM zone

Time

L

t

X(t)

CM

U

1205791 1205792 1205793 1205794

Figure 1 The condition-based maintenance strategy

(vi) All costs related tomaintenance inspection and envi-ronmental penalty are considered as average costsThey are known and constant

(vii) The resources necessary for the achievement of main-tenance actions are always available

4 Periodic Inspection Optimization Model

In this section we assume that periodic inspections areperformed at times 119894120591 119894 = 1 2 Our objective is to deter-mine simultaneously the optimal PM threshold level 119871lowast aswell as the interinspection period 120591lowast that minimizes the aver-age long-run cost rate

By using classical renewal arguments the total averagecost per time unit can be expressed over a renewal cycle119878 defined as the period between consecutive maintenanceactions In fact as previously stated the system is consideredto be as good as new after both maintenance actions (PM orCM)

Hence the expression of the long-run average cost perunit of time is given by

1198641198621

(119871 120591) =

119864 [1198621

119879(119871 120591)]

119864 [1198791

cycle (119871 120591)] (9)

where 119864[1198621

119879] represents the expected total cost incurred

within a cycle and 119864[1198791cycle] is the expected cycle lengthThe following analysis will lead to the expression of the

average long-run cost per time unit

41 Expected Total Cost within a Cycle The expected totalcost during a cycle 119864[1198621

119879] can be expressed as follows

119864 [1198621

119879(119871 120591)] = 119862

1198881198751

119888+ 1198621199011198751

119901+ 119862119894119864 [1198731

] + 119862env119864 [1198791

120577]

(10)

The three first terms are the PM and CM and inspectionaverage costs during a cycle respectivelyThe last term repre-sents the average penalty cost related to excess environmentaldamage during a cycle

The analytical expressions of these different componentsof the total expected cost are developed below

PM zone

Time

L

Environmental degradation level

t

X(t)

U

TL

TU

(i minus 1)120591 i120591

Figure 2 Example of cycle that ends by CM action

(a) The Probability 1198751119888That the Cycle Ends with a Corrective

Maintenance Action Recall that a CM action is undertakenin case following an inspection the measured amount ofenvironment degradation is found higher than the criticalknown level 119880 Let us consider any inspection interval ](119894 minus1)120591 119894120591] The probability PR1

119888(119894) that this inspection interval

ends with a CM action can be expressed as follows (Figure 2)

PR1119888(119894) = 119875 (119894 minus 1) 120591 lt 119879

119871lt 119879119880le 119894120591

= int

119894120591

(119894minus1)120591

119865119879119880minus119879119871

(119894120591 minus 119910) 119891119879119871(119910) 119889119910

(11)

Hence 1198751119888is given by

1198751

119888=

infin

sum

119894=1

PR1119888(119894)

=

infin

sum

119894=1

int

119894120591

(119894minus1)120591

119865119879119880minus119879119871

(119894120591 minus 119910) 119891119879119871(119910) 119889119910

(12)

(b) The Probability 1198751119901That the Cycle Ends with a Preventive

Maintenance Action The PM action is performed wheneverthe measured environmental degradation 119883(119905) is foundbetween 119871 and 119880 (119871 le 119883(119905) lt 119880) (Figure 3) Therefore theprobability PR1

119901(119894) that the inspection interval ](119894 minus 1)120591 119894120591]

ends with a PM action can be expressed as

PR1119901(119894) = 119875 (119894 minus 1) 120591 lt 119879

119871le 119894120591 lt 119879

119880

= int

119894120591

(119894minus1)120591

(1 minus 119865119879119880minus119879119871

(119894120591 minus 119910)) 119891119879119871(119910) 119889119910

(13)

Mathematical Problems in Engineering 5

PM zone

L

Environmental degradation level

X(t)

U

Time t

TL

TU

(i minus 1)120591 i120591

Figure 3 Example of cycle that ends by PM action

Thus

1198751

119901=

infin

sum

119894=1

PR1119901(119894)

=

infin

sum

119894=1

int

119894120591

(119894minus1)120591

(1 minus 119865119879119880minus119879119871

(119894120591 minus 119910)) 119891119879119871(119910) 119889119910

(14)

(c) The Expected Number of Inspections during a Cycle Theexpected number of inspections during a cycle 119864[1198731] isgiven by

119864 [1198731

] =

infin

sum

119894=1

119894119875 119873 = 119894 (15)

where 119875119873 = 119894 is the probability of having a total of 119894 inspec-tions within a cycle

Performing 119894 inspections within a cycle is equivalent tothe fact that at the last inspectionwhich is carried out at 119894120591 (119894thinspection) the observed environmental degradation levelhas reached either the critical level 119880 or the PM thresholdlevel 119871 Consequently 119875119873 = 119894 is given by

119875 119873 = 119894 = PR1119888(119894) + PR1

119901(119894) (16)

zy

Time0 t(i minus 1)120591 i120591TL

TU T1120577

Figure 4 The duration of generation of excess amount of damage

As a result using (11) and (13) we have

119864 [1198731

] =

infin

sum

119894=1

119894 int

119894120591

(119894minus1)120591

119865119879119880minus119879119871

(119894120591 minus 119910) 119891119879119871(119910) 119889119910

+

infin

sum

119894=1

119894 int

119894120591

(119894minus1)120591

(1 minus 119865119879119880minus119879119871

(119894120591 minus 119910)) 119891119879119871(119910) 119889119910

119864 [1198731

] =

infin

sum

119894=1

119894 int

119894120591

(119894minus1)120591

119891119879119871(119910) 119889119910

(17)

Then

119864 [1198731

] =

infin

sum

119894=1

119894 (119865119879119871(119894120591) minus 119865

119879119871((119894 minus 1) 120591)) (18)

(d) The Average Time 119864[1198791120577] of Generation of Excess Amount

of Environmental Damage during a Cycle Let us consider theinspection instant 119894120591 119894 = 1 2 If 119894120591 ge 119879

119880 the system will

have generated an excess amount of damage to environmentduring a period 1198791

120577(119894) = 119894120591minus119879

119880 Let 119864[1198791

120577] denote the average

time between the instant when the amount of environmentdegradation exceeds the critical level 119880 and the moment ofthe inspection that reveals it (Figure 4)

119864[1198791

120577] is expressed as follows

119864 [1198791

120577] =

infin

sum

119894=1

119875 (119894 minus 1) 120591 lt 119879119871lt 119879119880le 119894120591 | 119879

120577= 119894120591 minus 119879

119880

=

infin

sum

119894=1

int

119894120591

(119894minus1)120591

int

119911

(119894minus1)120591

(119894120591 minus 119911) 119865119879119880minus119879119871

(119911 minus 119910)119891119879119871(119910) 119889119910 119889119911

(19)

42 The Expected Renewal Cycle Length The cycle is con-sidered to be the interval between consecutive maintenanceactivities either PM or CM Therefore the expected cyclelength is given as follows

119864 [1198791

cycle (119871 120591)] =infin

sum

119894=1

119894120591PR1119888(119894) +

infin

sum

119894=1

119894120591PR1119901(119894) (20)

Using (11) and (13) we obtain the following expression

119864 [1198791

cycle (119871 120591)] =infin

sum

119894=1

119894120591 int

119894120591

(119894minus1)120591

119891119879119871(119910) 119889119910 (21)

119864 [1198791

cycle (119871 120591)] =infin

sum

119894=1

119894120591 (119865119879119871(119894120591) minus 119865

119879119871((119894 minus 1) 120591)) (22)

6 Mathematical Problems in Engineering

Table 1 Input data

119862119888($) 119862

119901($) 119862

119894($) 119862env ($week) 119880 (Tons)

900 500 100 10000 10

Hence the average long-run cost rate function 1198641198621(119871 120591)can be obtained by combining (12) (14) (18) (19) and (22)

It is expressed below as a function of the decision variableswhich are the inspection period 120591 and the PM threshold value119871

1198641198621

(119871 120591)

= [119862119901

infin

sum

119894=1

(119865119879119871(119894120591) minus 119865

119879119871((119894 minus 1) 120591))

+ (119862119888minus 119862119901)

infin

sum

119894=1

int

119894120591

(119894minus1)120591

119865119879119880minus119879119871

(119894120591 minus 119910) 119891119879119871(119910) 119889119910

+ 119862119894

infin

sum

119894=1

119894 (119865119879119871(119894120591) minus 119865

119879119871((119894 minus 1) 120591)) + 119862env

sdot

infin

sum

119894=1

int

119894120591

(119894minus1)120591

int

119911

(119894minus1)120591

(119894120591 minus 119911) 119865119879119880minus119879119871

(119911 minus 119910) 119891119879119871(119910) 119889119910 119889119911]

sdot (

infin

sum

119894=1

119894120591 (119865119879119871(119894120591) minus 119865

119879119871((119894 minus 1) 120591)))

minus1

(23)

The following simple numerical iterative procedure has beenused to obtain the optimal condition-based maintenancepolicy This procedure looks for the optimum values of thePM threshold level 119871lowast and the interinspection interval 120591lowast for119871 isin (0 119880] and 120591 isin [120591min 120591max]

43 Numerical Example Consider a production system sub-ject to continuous CO

2gas emissions (it could be any other

source of environmental damage) We suppose that theamount of environmental damage generated 119883(119905) follows aWiener process with drift coefficient 120582 and diffusion coef-ficient 120590

The following input parameters of the problem have beenarbitrarily chosen The parameters of the processrsquos pdf are120582 = 13 and 120590 = 035 The costs of maintenance actionsinspection cost environmental penalty cost and the criticallevel of damage to environment are presented in Table 1

Hence the pdf of the FHT for the critical level 119871 (3) iscomputed by

119891119879119871(119905) =

119871

radic0091205871199053exp(minus(119871 minus 119905)

2

009119905) (24)

The cdf (4) is given by

119865119879119871(119905) = Φ(

minus119871 + 119905

015radic119905) + exp( 2119871

0045)Φ(

minus119871 minus 119905

015radic119905) (25)

Start

End

Compute and store

Yes

Yes

No

No

Input data Cc Cp Ci Cenv 120582 120590 U

L = L + ΔL

120591 = 120591 + Δ120591

Initialize L ΔL 120591 Δ120591 120591min 120591max

Compute EC1(L 120591998400)

Has 120591maxbeen reached

Has U beenreached

Compute EC1(L 120591998400) for each 120591

and store the minimum EC1(L 120591lowast)

EC1(Llowast 120591lowast) = min(EC1(L 120591lowast)) foreach L

Figure 5 Numerical procedure for the periodic strategy

The cdf of the remaining useful life (RUL) 119879119880minus 119879119871(8) is ex-

pressed as follows

119865119879119880minus119879119871

(119905) = Φ(10 minus 119871 minus 119905

015radic119905) + exp(20 minus 2119871

0045)Φ

sdot (minus (10 minus 119871) minus 119905

015radic119905)

(26)

Given the above input parameters we applied the numericalprocedure of Figure 5 with Δ119871 = 1 ton of CO

2gas emissions

Δ120591 = 1 week 120591min = 1 week and 120591max = 12 weeks Theobtained optimal solution is presented in Table 2

Mathematical Problems in Engineering 7

Table 2 The obtained optimal solution for periodic inspection policy

119871lowast

(Ton) 120591lowast (week) 119875

1

119888() CM cost ($) 119875

1

119901() PM cost ($) 119864[119873

1

]Inspectioncost ($) 119864[119879

1

120585]

Environmentalpenalty cost ($) 119864[119879

1

cycle]1198641198621

($week)2 7 1776 160 8224 412 108 108 002 200 71 12394

Table 3 Effect of 119862env

119862env 120591lowast

119871lowast

1198751

119888() CM cost ($) 119875

1

119901() PM cost ($) 119864[119873

1

] Inspection cost ($) 119864[1198791

120585]

Environmentalpenalty cost ($)

119864[1198791

cycle]

(week)1198641198621

($week)0 7 9 6143 553 3857 193 144 144 71 0 1006 884710000 7 2 1776 160 8224 412 108 108 002 200 71 12394100000 5 2 043 387 9957 498 100 100 7 10minus3 700 5 26037

Table 4 Effect of 119862119894

119862119894

120591lowast

119871lowast

1198751

119888() CM cost ($) 119875

1

119901() PM cost ($) 119864[119873

1

] Inspection cost ($) 119864[1198791

120585]

Environmentalpenalty cost ($)

119864[1198791

cycle]

(week)1198641198621

($week)0 1 8 021 2 9979 499 665 0 8 10minus6 8 10minus2 665 7535100 7 2 1776 160 8224 412 108 108 002 200 71 123941000 9 2 9525 857 475 24 100 1000 113 11300 9 146456

From the results presented in Table 2 the maintenancecrew has to inspect the system every 7 weeks If an inspectionreveals that the environmental damage level is between 2and 10 tons of CO

2 the maintenance crew has to undergo

immediately a preventive maintenance action The totalaverage cost per time unit of this strategy is 12394 $weekThe optimal strategy preconizes to performmore PM actions(1198751119901= 8224) than CM actions (1198751

119888= 1776)

In what follows while keeping the original combinationof input parameters the unitary cost of environmentalpenalty119862env the inspection cost119862119894 and the CM actions costsare varied respectively in order to investigate their influenceon the optimal solutionThe obtained results are presented inTables 3 4 and 5

From Table 3 we can notice that as the environmentalpenalty cost increases the inspection period remains ratherconstant whereas the PM threshold level is reduced enablinga PM action to be taken earlier to reduce the probabilityto generate excess CO

2emissions and be penalized These

results confirm the relevance of adopting a condition-basedmaintenance policy

From Table 4 we can clearly see that for a relatively lowinspection cost interinspection interval is rather low the PMthreshold level is relatively high and the cycle is most likely tofinish with a PM action

In case of costly inspections interinspection interval ishigh (less frequent inspections) the PM threshold level isreduced and the cycle ismost likely to finish with a correctiveaction and an important penalty

Finally Table 5 shows the variation of the cost of a cor-rective maintenance action One can see that as this costincreases the proposed policy suggests undergoing ratherfrequent inspections and encourages PM actions (higher

probability 1198751119901) This is similar to the effect of costly environ-

mental penalty

5 Nonperiodic Inspection Optimization Model

In this section it is assumed that the system is submittedto nonperiodic inspections to assess the generated environ-mental damage Let Θ = (120579

1 1205792 120579

119873 ) denote the

inspection instants sequence The purpose is to determinesimultaneously the optimal PM threshold value 119871lowast as wellas the inspection instants sequence Θlowast that minimizes theaverage long-run cost rate

The expression of the expected total cost per time unit isdeveloped as follows

(i) The probability that the cycle ends with a correctivemaintenance action is

1198752

119888=

infin

sum

119894=1

int

120579119894

120579119894minus1

119865119879119880minus119879119871

(120579119894minus 119910)119891

119879119871(119910) 119889119910 (27)

(ii) The probability that the cycle ends with a preventivemaintenance action is

1198752

119901=

infin

sum

119894=1

int

120579119894

120579119894minus1

(1 minus 119865119879119880minus119879119871

(120579119894minus 119910)) 119891

119879119871(119910) 119889119910 (28)

(iii) The expected number of inspections during a cycle is

119864 [1198732

] =

infin

sum

119894=1

119894 (119865119879119871(120579119894) minus 119865119879119871(120579119894minus1)) (29)

8 Mathematical Problems in Engineering

Table 5 Effect of 119862119888

119862119888

120591lowast

119871lowast

1198751

119888() CM cost ($) 119875

1

119901() PM cost ($) 119864[119873

1

]Inspectioncost ($) 119864[119879

1

120585]

Environmentalpenalty cost ($)

119864[1198791

cycle]

(week)1198641198621

($week)600 8 4 6747 405 3253 163 100 100 00262 262 8 11625900 7 2 1776 160 8224 412 108 108 002 200 71 123941100 5 2 043 387 9957 498 100 100 7 10minus3 70 5 13437

(iv) The average time of generation of excess amount ofenvironmental damage during a cycle is

119864 [1198792

120577] =

infin

sum

119894=1

int

120579119894

120579119894minus1

int

119911

120579119894minus1

(120579119894minus 119911) 119865

119879119880minus119879119871(119911 minus 119910) 119891

119879119871(119910) 119889119910 119889119911

(30)

(v) The expected renewal cycle length is

119864 [1198792

cycle (119871 Θ)] =infin

sum

119894=1

120579119894(119865119879119871(120579119894) minus 119865119879119871(120579119894minus1)) (31)

By collecting (27) (28) (29) (30) and (31) the expectedlong-run rate cost is written as follows

1198641198622

(119871 Θ)

= [119862119901

infin

sum

119894=1

(119865119879119871(120579119894) minus 119865119879119871(120579119894minus1))

+ (119862119888minus 119862119901)

infin

sum

119894=1

int

120579119894

120579119894minus1

119865119879119880minus119879119871

(120579119894minus 119910)119891

119879119871(119910) 119889119904

+ 119862119894

infin

sum

119894=1

119894 (119865119879119871(120579119894) minus 119865119879119871(120579119894minus1)) + 119862env

sdot

infin

sum

119894=1

int

120579119894

120579119894minus1

int

119911

120579119894minus1

(120579119894minus 119911) 119865

119879119880minus119879119871(119911 minus 119910) 119891

119879119871(119910) 119889119910 119889119911]

sdot (

infin

sum

119894=1

120579119894(119865119879119871(120579119894) minus 119865119879119871(120579119894minus1)))

minus1

(32)

Due to the complexity of the analytical model a numericalprocedure is developed in the next section in order to finda nearly optimal solution (119871lowast Θlowast) for any instance of theproblem

51 Numerical Procedure The developed procedure is basedon the Nelder-Mead algorithm It is presented in Figure 6followed by the description of the use of Nelder-Mead algo-rithm at one of the steps Basically the proposed numericalprocedure consists in searching for each value of 119871 the bestinspection sequence Θ using the Nelder-Mead algorithm[36] It is one of the best known algorithms for multidimen-sional unconstrained optimization without derivatives and itis especially appreciated for its robustness its simplicity its

low use of memory (few variables) and its short computingtime [37]

It consists in an iterative procedure comparing the valuesof the objective function at the (119899 + 1) vertices and movinggradually to a quasioptimal point Its movement is achievedby using four operations known as reflection expansioncontraction and shrink The steps of Nelder-Mead algorithmapplied to our model are described as follows

Step 1 Sort the simplex vertices according to the functionvalue at that point

1198641198622

(Θ1) le 119864119862

2

(Θ2) le sdot sdot sdot le 119864119862

2

(Θ119899+1) (33)

We refer to Θ1as the best vertex and to Θ

119899+1as the worst

vertex

Step 2 Compute the centroidΘ using all points exceptΘ119899+1

the worst point Consider Θ = sum

119899

119894=1Θ119894

Step 3 Compute the reflection point

Θ119903= Θ + 120572 (Θ minus Θ

119899+1) (reflection) (34)

Step 31 If1198641198622(Θ1) le 119864119862

2

(Θ119903) le 119864119862

2

(Θ119899) replaceΘ

119899+1with

Θ119903 go to Step 1

Step 32 If 1198641198622(Θ119903) le 119864119862

2

(Θ1) then compute the expansion

point

Θ119890= Θ + 120573 (Θ

119903minus Θ) (expansion) (35)

and evaluate if 1198641198622(Θ119890) lt 119864119862

2

(Θ119903) then replace Θ

119899+1with

Θ119890 else replace Θ

119899+1with Θ

119903 go to Step 1

Step 33 If 1198641198622(Θ119899) le 119864119862

2

(Θ119903) lt 119864119862

2

(Θ119899+1) compute the

outside contraction point

Θoc = Θ119899+1 + 120574 (Θ minus Θ119899+1) (outside contraction) (36)

If 1198641198622(Θ119899) le 119864119862

2

(Θ119903) then use Θoc and reject Θ

119899+1 go to

Step 1 Else go to Step 4

Step 34 If 1198641198622(Θ119903) ge 119864119862

2

(Θ119899+1) compute the inside

contraction point

Θic = Θ119899+1 minus 120574 (Θ minus Θ119899+1) (inside contraction) (37)

If 1198641198622(Θic) lt 1198641198622

(Θ119899+1) then use Θic and reject Θ

119899+1 go to

Step 1 Else go to Step 4

Mathematical Problems in Engineering 9

Start

using Nelder-Mead method andcompute the corresponding

End

Compute and store

and the corresponding

Compute and store

each Land the corresponding

Yes

Yes

No

No

Input data Cc Cp Ci Cenv 120582 120590 U

Initialize L ΔLNNmax

L = L + ΔL

L lt U

N = N+ 1

N le Nmax

Find the best

EC2(L Θlowast)

EC2 lowast (L Θlowast) = min(EC2(L Θlowast)) with

EC2 lowast (L Θlowast) = min(EC2(L Θlowast)) for

i = 1 2 N

Θlowast = (1205791 1205792 120579N)

Θlowast = (1205791 1205792 120579N)

Θlowast = (1205791 1205792 120579N)

Figure 6 Numerical procedure for the nonperiodic strategy

Step 4 Compute new 119899 vertices keeping only the best oneΘ1 Consider

Θ119894= Θ1+ 120578 (Θ

119894minus Θ1) forall2 le 119894 le 119899 + 1 (shrink) (38)

go to Step 1

The convergence of the algorithm is achieved when thestandard deviation of the objective function at the (119899 + 1)vertices is smaller than a specific value 120576 that is

radic1

119899 + 1

119899+1

sum

119894=1

[1198641198622(Θ119894) minus 1198641198622]

2

lt 120576 (39)

The standard values of Nelder-Mead parameters arechosen to be 120572 = 1 120573 = 2 120574 = 05 and 120578 = 05 [37 38]

In the numerical procedure described in Figure 6 thefollowing notations were adopted

(i) Δ119871 increment of environmental degradation(ii) 119873 number of inspections(iii) 119873max maximum number of inspections

52 Numerical Example (Nonperiodic Inspections) We con-sider the same example of Section 43 with the same inputdata The pdf and cdf of 119879

119871given by (24) and (25) respec-

tively and the cumulative function of the RUL 119879119880minus 119879119871given

by (26) are consideredGiven the above input parameters we applied the proce-

dure of Figure 6 with Δ119871 = 1 119873 = 1 and 119873max = 10 Theobtained nearly optimal solution is 119871lowast = 2 tons of CO

2gas

emissions and Θlowast = 66 71 74 (see Table 6)Note that the first inspection instant is 66 weeks We give

only the first three values in this particular example becausethe cycle average duration is 67 weeks and the averagenumber of inspections per cycle is 109 Therefore havingmore than three inspections per cycle is nearly impossible

PM action should be performed whenever an inspec-tion reveals that the environmental degradation level hasexceeded 2 tons of CO

2gas emissions By adopting this

strategy it would cost in average a total of 10046 $weekIn comparison with the periodic inspection policy one

can notice that in the case of nonperiodic inspections theprobability that the cycle ends with a CM action and theaverage period of excess CO

2gas emissions are lower This

indicates that with the nonperiodic inspection policy PMactions are more likely to be performed in order to avoid theexceeding of the critical level and therefore the reduction ofthe emission of excess damage to environment

Moreover the nonperiodic inspection policy has thelowest average cost rate due to the reduction of the expectedCM cost as well as the environmental penalty cost Henceeven with a nearly optimal solution the nonperiodic inspec-tion policy is more economical than the periodic inspectionpolicyThis can be explained by the fact that in the case of thenonperiodic strategy sequential inspections are scheduledin accordance with the evolution of environmental damagegeneration

6 Conclusion

In this paper we have considered a condition-based main-tenance policy for production systems which degrade asthey get older and generate environmental damage We have

10 Mathematical Problems in Engineering

Table 6 The obtained nearly optimal solution for the nonperiodic inspection policy

119871lowast

(Ton)Θlowast

(week)1198752

119888

()CM cost

($)1198752

119901

()PM cost

($) 119864[1198732

]Inspection cost

($) 119864[1198792

120585]

Environmentalpenalty cost ($)

119864[1198792

cycle]

(week)1198641198622

($week)2 66 71 74 627 5641 9373 46867 109 109 00039 39 67 10046

proposed a new modeling approach based on the fact thatthe degradation process is modeled by the Wiener processThus the first hitting time and the remaining useful life ofthe system conditional to its environmental degradation levelare considered Moreover two types of inspection policiesperiodic and nonperiodic can be used to reveal the level ofenvironmental damage and act consequently According tothe observed amount of environmental degradation at eachinspection one decides to undertake or not maintenanceactions (PM or CM) on the system

CM action is undertaken following inspections thatreveal the exceeding of a known critical level of environmen-tal degradation In such situation an environmental penaltyis incurred due to the excess amount of environmental degra-dation generated To prevent such event a lower thresholdlevel has to be considered to trigger a PM action to renewthe system at a lower cost and without paying the penaltyThis lower threshold level and the inspection schedule wereconsidered as the decision variables

For the two proposed inspection policies the totalexpected cost per time unit has beenmathematicallymodeledand the optimal inspection schedules andPM threshold levelswere derived using two numerical procedures

The developed condition-based preventive maintenancemodels permit highlighting the role of preventive mainte-nance in reducing environmental damage and its conse-quences that could be caused by the degradation of produc-tion systems The proposed models can be relatively easilyused by decision-makers in the perspective of implementingan effective green maintenance

This work can be improved in several ways First it wouldbe of interest to consider situations in which maintenanceactions durations are not negligible and where resources arenot always immediately available to perform maintenanceactions Moreover in many real situations production sys-tems may not generate only a single damage to environmentbut several kinds of damages at the same time at different rateswith different impacts It would be interesting to investigatethese issues taking the present model as a start

Notations

119880 Critical level of environmentaldegradation

119871 PM threshold level of environmentaldegradation

119862119901 Preventive maintenance (PM) action cost

119862119888 Corrective maintenance (CM) action cost

119862env Penalty cost per time unit relatedto excess environmental damageincurred once the critical level 119880is exceeded

119862119894 Inspection cost

119879119880 The time at which the

environmental damage of thesystem exceeds the critical level 119880for the first time

119879119871 The time at which the

environmental damage of thesystem exceeds the PM thresholdlevel 119871 for the first time

1198791

cycle(1198792

cycle) The system renewal cycle lengthwithin the periodic (nonperiodic)inspection policy a cycle is thetime between consecutivemaintenance actions (eitherpreventive or corrective)

119891119879119880(119905) 119865119879119880(119905) Probability density function (pdf)

and cumulative distributionfunction (cdf) associated with 119879

119880

119891119879119871(119905) 119865119879119871(119905) pdf and cdf associated with 119879

119871

1198731

(1198732

) Discrete random variableassociated with the total numberof inspections during a cyclewithin the periodic (nonperiodic)inspection policy

120579119894 The th inspection instant

Θ = (1205791 1205792 120579

119873 ) Inspection instants sequence

1198751

119901(1198752

119901) Probability that the cycle ends

with a PM action within theperiodic (nonperiodic)inspection policy

1198751

119888(1198752

119888) Probability that the cycle ends

with a CM action within theperiodic (nonperiodic)inspection policy

119864[1198731

](119864[1198732

]) The expected number ofinspections during a cycle in thecase of the periodic(nonperiodic) inspection policy

119879119894

120577 Period during which excess of

damage is generated between theinstant when the amount ofenvironment degradation exceedsthe critical level 119880 and themoment of the inspection thatreveals it (119894 = 1 for the periodicinspection policy 119894 = 2 for thenonperiodic inspection policy)

Mathematical Problems in Engineering 11

1198641198621

(119871 120591) The long-run average cost perunit of time corresponding to theperiodic inspection policy

1198641198622

(119871 Θ) The long-run average cost perunit of time corresponding to thenonperiodic inspection policy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R K Sharma D Kumar and P Kumar ldquoPredicting uncertainbehavior of industrial system using FMmdasha practical caserdquoApplied Soft Computing Journal vol 8 no 1 pp 96ndash109 2008

[2] M Cepin and B Mavko ldquoFault tree developed by an object-based method improves requirements specification for safety-related systemsrdquo Reliability Engineering and System Safety vol63 no 2 pp 111ndash125 1999

[3] T C McKelvey ldquoHow to improve the effectiveness of hazard ampoperability analysisrdquo IEEETransactions onReliability vol 37 no2 pp 167ndash170 1988

[4] P R S da Silva and F G Amaral ldquoAn integrated methodologyfor environmental impacts and costs evaluation in industrialprocessesrdquo Journal of Cleaner Production vol 17 no 15 pp1339ndash1350 2009

[5] S K Stefanis A G Livingston and E N Pistikopoulos ldquoMin-imizing the environmental impact of process plants a processsystemsmethodologyrdquo Computers amp Chemical Engineering vol19 supplement 1 pp 39ndash44 1995

[6] B Gjorgiev D Kancev andM Cepin ldquoA newmodel for optimalgeneration scheduling of power system considering generationunits availabilityrdquo International Journal of Electrical Power andEnergy Systems vol 47 no 1 pp 129ndash139 2013

[7] S Martorell J F Villanueva S Carlos et al ldquoRAMS+Cinformed decision-making with application to multi-objectiveoptimization of technical specifications and maintenance usinggenetic algorithmsrdquo Reliability Engineering and System Safetyvol 87 no 1 pp 65ndash75 2005

[8] C G Vassiliadis and E N Pistikopoulos ldquoMaintenance-basedstrategies for environmental risk minimization in the processindustriesrdquo Journal of Hazardous Materials vol 71 no 1ndash3 pp481ndash501 2000

[9] H Chouikhi A Khatab and N Rezg ldquoA condition-basedmaintenance policy for a production system under excessiveenvironmental degradationrdquo Journal of Intelligent Manufactur-ing vol 25 no 4 pp 727ndash737 2014

[10] J Yan and D Hua ldquoEnergy consumptionmodeling for machinetools after preventive maintenancerdquo in Proceedings of the IEEEInternational Conference on Industrial Engineering and Engi-neering Management (IEEM rsquo10) pp 2201ndash2205 IEEE MacaoChina December 2010

[11] V A Kopnov ldquoOptimal degradation processes control by two-level policiesrdquo Reliability Engineering and System Safety vol 66no 1 pp 1ndash11 1999

[12] X Zhou L Xi and J Lee ldquoReliability-centered predictivemaintenance scheduling for a continuously monitored systemsubject to degradationrdquo Reliability Engineering amp System Safetyvol 92 no 4 pp 530ndash534 2007

[13] B Castanier A Grall and C Berenguer ldquoA condition-basedmaintenance policy with non-periodic inspections for a two-unit series systemrdquo Reliability Engineering and System Safetyvol 87 no 1 pp 109ndash120 2005

[14] A Chelbi and D Ait-Kadi ldquoInspection strategies for randomlyfailing systemsrdquo inHandbook of Maintenance Management andEngineering chapter 13 pp 303ndash335 Springer London UK2009

[15] A Chelbi and D Ait-Kadi ldquoAn optimal inspection strategy forrandomly failing equipmentrdquoReliability Engineering and SystemSafety vol 63 no 2 pp 127ndash131 1999

[16] B Castanier C Berenguer and A Grall ldquoA sequentialcondition-based repairreplacement policy with non-periodicinspections for a system subject to continuous wearrdquo AppliedStochasticModels in Business and Industry vol 19 no 4 pp 327ndash347 2003

[17] H R Golmakani and F Fattahipour ldquoAge-based inspectionscheme for condition-basedmaintenancerdquo Journal of Quality inMaintenance Engineering vol 17 no 1 pp 93ndash110 2011

[18] V Makis and A K S Jardine ldquoOptimal replacement in the pro-portional hazards modelrdquo Information Systems and OperationalResearch vol 30 no 1 1992

[19] A Grall C Berenguer and L Dieulle ldquoA condition-basedmaintenance policy for stochastically deteriorating systemsrdquoReliability Engineering amp System Safety vol 76 no 2 pp 167ndash180 2002

[20] R Jiang ldquoOptimization of alarm threshold and sequentialinspection schemerdquo Reliability Engineering and System Safetyvol 95 no 3 pp 208ndash215 2010

[21] M Grigoriu Stochastic Calculus Applications in Science andEngineering Birkhauser Boston Mass USA 2002

[22] F C Klebaner Introduction To Stochastic Calculus With Appli-cations Imperial College Press 3rd edition 2012

[23] M Abdel-Hameed ldquoA gammawear processrdquo IEEE Transactionson Reliability vol 24 no 2 pp 152ndash153 1975

[24] R Chhikara and L Folks The Inverse Gaussian DistributionTheory Methodology and Applications Marcel Dekker NewYork NY USA 1989

[25] S Sato and J Inoue ldquoInverse Gaussian distribution and itsapplicationrdquo Electronics and Communications in Japan Part IIIFundamental Electronic Science vol 77 no 1 pp 32ndash42 1994

[26] R C Merton ldquoOn the role of the Wiener process in financetheory and practice the case of replicating portfoliosrdquo in TheLegacy of Norbert Wiener A Centennial Symposium vol 60American Mathematical Society 1997

[27] W E Bardsley ldquoNote on the use of the inverse Gaussiandistribution for wind energy applicationsrdquo Journal of AppliedMeteorology vol 19 no 9 pp 1126ndash1130 1980

[28] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[29] T Z Fahidy ldquoApplying the inverse Gaussian distribution tothe assessment of chemical reactor performancerdquo InternationalJournal of Chemistry vol 4 no 2 2012

[30] X Wang and D Xu ldquoAn inverse gaussian process model fordegradation datardquo Technometrics vol 52 no 2 pp 188ndash1972010

[31] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

12 Mathematical Problems in Engineering

[32] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[33] Z Brzezniak and T Zastawniak Basic Stochastic Process ACourse Through Exercise Springer London UK 2002

[34] D R Cox and H D Miller The Theory of Stochastic ProcessesMethuen and Company London UK 1965

[35] C Guo W Wang B Guo and X Si ldquoA maintenance opti-mization model for mission-oriented systems based onWienerdegradationrdquo Reliability Engineering amp System Safety vol 111pp 183ndash194 2013

[36] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquoThe Computer Journal vol 7 no 4 pp 308ndash3131965

[37] O Roux D Duvivier G Quesnel and E Ramat ldquoOptimizationof preventive maintenance through a combined maintenance-production simulation modelrdquo in Proceedings of the Interna-tional Conference on Industrial Engineering and Systems Man-agement Montreal Canada May 2009

[38] L Han and M Neumann ldquoEffect of dimensionality on theNelder-Mead simplex methodrdquo Optimization Methods andSoftware vol 21 no 1 pp 1ndash16 2006

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