Extended great deluge algorithm for the imperfect ...liacs.leidenuniv.nl/~csnaco/SWI/papers/Great deluge...extended great deluge algorithm is proposed. This method has the advantage

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Reliability Engineering and System Safety 93 (2008) 1658–1672

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Extended great deluge algorithm for the imperfect preventivemaintenance optimization of multi-state systems

Nabil Nahas�, Abdelhakim Khatab, Daoud Ait-Kadi, Mustapha Nourelfath

Mechanical Engineering Department, Faculty of Science and Engineering, Interuniversity Research Center on Enterprise Networks,

Logistics and Transportation (CIRRELT), Universite Laval, Quebec, Canada

Received 2 March 2007; received in revised form 25 December 2007; accepted 20 January 2008

Available online 14 February 2008

Abstract

This paper deals with preventive maintenance optimization problem for multi-state systems (MSS). This problem was initially

addressed and solved by Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng

Syst Saf 2000;67:193–203]. It consists on finding an optimal sequence of maintenance actions which minimizes maintenance cost while

providing the desired system reliability level. This paper proposes an approach which improves the results obtained by genetic algorithm

(GENITOR) in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf

2000;67:193–203]. The considered MSS have a range of performance levels and their reliability is defined to be the ability to meet a given

demand. This reliability is evaluated by using the universal generating function technique. An optimization method based on the

extended great deluge algorithm is proposed. This method has the advantage over other methods to be simple and requires less effort for

its implementation. The developed algorithm is compared to than in Levitin and Lisnianski [Optimization of imperfect preventive

maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203] by using a reference example and two newly generated

examples. This comparison shows that the extended great deluge gives the best solutions (i.e. those with minimal costs) for 8 instances

among 10.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Multi-state systems; Preventive maintenance; Universal generating function; Extended great deluge algorithm

1. Introduction

Redundancy and maintenance are methods used toimprove system reliability. In the existing literature, toguarantee a given required system reliability level undercost constraint, researchers solve the problems of redun-dancy optimization or maintenance optimization eitherjointly or separately.

This paper deals with preventive maintenance optimiza-tion of multi-state systems. A system is called a multi-statesystem (MSS) if it is capable of assuming a range ofperformance levels, varying from perfect functioning to

atter r 2008 Elsevier Ltd. All rights reserved.

1.006

or.

[email protected] (N. Nahas),

laval.ca (A. Khatab),

val.ca (D. Ait-Kadi),

mc.ulaval.ca (M. Nourelfath).

complete failure. Due to their applicability in manyindustrial areas, the maintenance of MSS has received agrowing attention in the existing literature. Gurler andKaya [2] develop a maintenance policy for a multi-component system where the lifetime assigned to eachcomponent is given by several stages. An approximateapproach based on renewal theory is proposed in order toderive the long-run average cost function per unit time,which is optimized by numerical methods. In [3], understate-deteriorating assumption, Hsieh and Chiu develop anoptimal maintenance policy for a multi-state deterioratingstandby production system. A component that ensures thesystem to be operational deteriorates during the produc-tion process. At a given deteriorating state, this componentis replaced by a standby component and sent to themaintenance service center. The optimal maintenancepolicy is determined by means of the number of thestandby components and the optimal state where the

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replacement of the deteriorating components should beperformed. In the work of Su and Chang [4], a model ofMSS with state-dependent cost is considered. The statespace of the system is partitioned into two subsets: the firstallows to represent all states of normal operations whilethe second is characterized by the single failure state. Aperiodic maintenance model is developed and the optimalcycle time of maintenance actions is determined over aspecific finite horizon. In [5], Lam and his coauthorsconsider monotone process model for a multi-statedegenerative system with one working state and k failurestates. More recently, in [6], Lam considers a one-component MSS for which the state space is characterizedby k working states and l failure states. For such a system,a monotone process maintenance model is studied. Undersome assumptions, it is shown that this model can beapplied to multi-state deteriorating system and multi-stateimproving system. A replacement policy N is adopted,which is based on the failure number of the system. Ananalytical approach is used to determine the optimalreplacement policy. In [7], Zhang et al. also consider amulti-state deteriorating system with k failure states andone working state. Replacement policy N is exploited andthe optimal replacement time is derived to maximize thelong-run average profit per time unit. In [8], Levitin andLisnianski generalize the replacement schedule optimiza-tion problem to MSS. In [8], an MSS is given as a set ofcomponents in a series–parallel configuration. The systemas well as its components are characterized by variousperformance levels and the system reliability is defined asthe ability of the system to meet a demand. The optimalnumber of component replacements corresponds to thatwhich ensure the desired level of the system reliability levelby minimizing the sum of maintenance cost and the costof unsupplied demand. A genetic algorithm is adopted asan optimization technique. Within this kind of MSS,Levitin and Lisnianski [1] solve the preventive maintenanceoptimization problem. Components of the system arecharacterized by their corresponding hazard function andthe preventive maintenance actions may have the ability toreduce the effective age of components. A geneticalgorithm is used to derive, for a given system lifetime,the optimal sequence of maintenance actions that ensurethe desired system reliability level. Another existingsolution technique of this problem is ant colony optimiza-tion [9]. However, the best-published results have beenprovided by genetic algorithm [1].

As MSS applicability is becoming more and moreimportant, while requiring short development schedulesand very high reliability, it is becoming increasinglyimportant to develop efficient solutions to preventivemaintenance optimization problem for MSS. By exploitingthe model proposed in [1], to solve the imperfect preventivemaintenance optimization problem for MSS, this paperpresents an efficient algorithm inspired from the extendedgreat deluge metaheuristic [10]. This algorithm performswell and is competitive with that proposed in [1]. This is

demonstrated by the solutions found by the proposedalgorithm.The remainder of this paper is organized as follows. In

the next section, MSS reliability definition and estimationare presented. Section 3 addresses the preventive main-tenance model. Section 4 presents the MSS model and theproblem formulation. The optimization method is given inSection 5. Numerical results are presented in Section 6.Conclusion is given in Section 7.

2. MSS reliability estimation by using universal moment

generating functions

Consider an MSS composed of a number of failure-prone components. Each component is characterized by arange of performance levels from complete failure up toperfect functioning. The entire system is assumed to have K

different states corresponding to different output perfor-mance levels. Within MSS, reliability is related to theability of the system to satisfy, at a given time t, therequired performance level (demand). According to [11],MSS reliability is given as follows:

Rðt;W Þ ¼ PrfGðtÞXW g, (1)

where GðtÞ represents the output performance of the systemat time t; and W is the required performance level.Let Gk be the output performance level of the system kth

state and qkðtÞ be the probability PrfGðtÞ ¼ Gkg (for k ¼

1; . . . ;KÞ; it follows that the output performance distribu-tion (OPD) of the system can be determined on the basis oftwo sets G and q such that

G ¼ fGk : 1pkpKg (2)

and

q ¼ fqkðtÞ : 1pkpKg. (3)

According to Eqs. (1)–(3), the MSS reliability can beexpressed as the probability that the system is confined,during the time interval ½0; t�; in those states for which theoutput performance level is greater than or equal to thedemand W . That is

Rðt;W Þ ¼ PrfGðtÞXW g ¼X

GkXW

qkðtÞ. (4)

To evaluate the MSS reliability, several approaches havebeen developed in the literature. However, when dealingwith optimization problems, such as the one studied in thispaper, it is important to have an efficient and fastprocedure to estimate the MSS reliability. A procedurebased on the universal z-transform as a modern mathema-tical technique is developed in [12] and widely exploitedwithin MSS performance measure evaluation. In theliterature, the universal z-transform is also called universalmoment generating function (UMGF) and denoted asu-function or z-function. More details on the usage of theUMGF in MSS performance measure evaluation can be

ARTICLE IN PRESSN. Nahas et al. / Reliability Engineering and System Safety 93 (2008) 1658–16721660

found in [13–15]. The present paper exploits this techniquefor which a brief review is given below.

Consider a discrete random variable X which can take K

possible values x1; . . . ;xK with probabilities q1ðtÞ; . . . ; qK ðtÞ;i.e. PrfX ðtÞ ¼ xkg ¼ qkðtÞ; for k ¼ 1; . . . ;K : The UMGF ofX is given as a polynomial Uðt; zÞ such that

Uðt; zÞ ¼XK

k¼1

qkðtÞzxk . (5)

Dealing with MSS reliability evaluation, the polynomialUðt; zÞ represents all possible system states where xk

represents the output performance level Gk assigned tothe kth state and qkðtÞ is the probability of being in thatstate. Accordingly, the UMGF of the system OPD can bewritten as

Uðt; zÞ ¼XK

k¼1

qkðtÞzGk . (6)

From the above equation, the MSS reliability is given by

Rðt;W Þ ¼ PrfGðtÞXW g ¼ CðUðt; zÞ;W Þ

¼XK

k¼1

CðqkðtÞzGk�W Þ, (7)

where C is a function defined as

CðqkðtÞzGk�W Þ ¼

qkðtÞ if GkXW ;

0 otherwise:

�

To evaluate the MSS reliability of a series–parallel systemwhere each component is characterized by its own UMGF,two basic composition operators are used. The UMGFcorresponding to the entire system reliability is thenobtained by using simple algebraic operations on indivi-dual UMGF of components. These operations definitionstake into account both the physical nature of MSSperformance and the interactions between MSS compo-nents. For a simple illustration and without loss ofgenerality, let us consider the case of an MSS composedof two components Comp1 and Comp2 characterized,respectively, by the UMGF u1ðt; zÞ and u2ðt; zÞ such that

u1ðt; zÞ ¼Xn1i¼1

qiðtÞzgi (8)

and

u2ðt; zÞ ¼Xn2i¼1

piðtÞzf i . (9)

In the above equations, parameters gi and f i represent therespective output performance levels of the two compo-nents. The terms n1 and n2 are numbers of possible states ofthe components, while qiðtÞ and piðtÞ are the instantaneousprobabilities corresponding, respectively, to the outputperformance levels gi and f i. For these components, the

composition operators take the following form:

Owðu1ðt; zÞ; u2ðt; zÞÞ ¼ Ow

Xn1i¼1

qiðtÞzgi ;Xn2i¼1

piðtÞzf i

!

¼Xn1i¼1

Xn2j¼1

qiðtÞpjðtÞzwðgi ;f jÞ, (10)

where the function w is determined with respect to thephysical nature of the system performance measure and tothe nature of the interaction of system components. In thispaper, system (components) performance is the productiv-ity or capacity. In this case, the total capacity ofcomponents connected in parallel is given as the sumof all component capacities, i.e. the function w is definedas a sum of its arguments. Therefore, the UMGF Upðt; zÞ;corresponding to the MSS composed of componentsComp1 and Comp2 in parallel, can be obtained as

Upðt; zÞ ¼ Owðu1ðt; zÞ; u2ðt; zÞÞ ¼Xn1i¼1

Xn2j¼1

qiðtÞpjðtÞzgiþf j

¼ u1ðt; zÞu2ðt; zÞ. (11)

However, the total capacity of components connected inseries corresponds to the minimum of all componentcapacities. In this case the function w is defined as aminimum of its arguments. Therefore, the UMGF U sðt; zÞ,corresponding to the MSS composed of componentsComp1 and Comp2 in series, can be obtained as

U sðt; zÞ ¼ Owðu1ðt; zÞ; u2ðt; zÞÞ ¼Xn1i¼1

Xn2j¼1

qiðtÞpjðtÞzminðgi ;f jÞ.

(12)

On the basis of the above composition operators, theUMGF of the entire MSS can be straightforwardly isderived.

3. Imperfect preventive maintenance model

In this section, the imperfect preventive maintenancemodel is given in accordance with that of Levitin andLisnianski [1]. This model is given on the basis of the agereduction concept initially introduced by Nakagawa [16].According to this concept, the effective age of a givencomponent is reduced when PM action is performed onthis component. Following Martorel et al. [17], the effectiveage tj assigned to a component j for which PM actions areperformed at chronological times ðtj1 ; tj2 ; . . . ; tjnj

Þ is givensuch that

tjðtÞ ¼ tþj ðtjiÞ þ ðt� tji

Þ, (13)

where tjiototjiþ1

, i ¼ 0; . . . ; nj and

tþj ðtjiÞ ¼ eiðtjðtji

ÞÞ ¼ eiðtjðtji�1Þ þ ðtji

� tji�1ÞÞ. (14)

In the above equation tþj ðtjiÞ represents the age of

component j immediately after the ith PM action whilethe parameter ei 2 ½0; 1� is the age reduction coefficient


corresponding to the ith PM action. In this model it isassumed that tjð0Þ ¼ 0 and tj0 ¼ 0. Regarding the valuestaken by a given age reduction coefficient ei, two particularcases may be distinguished. The first case corresponds toei ¼ 1 and means that the ith PM action has no effect onthe age of a component (the component status becomes asbad as old), while the second case is where ei ¼ 0 andmeans that the component age is reset to the null value(replacement).

For a given component j; its corresponding hazardfunction can be written as

h�j ðtÞ ¼ hjðtjðtÞÞ þ hj0 , (15)

where hj0 represents the initial constant hazard rate ofcomponent j and hjðtÞ is the hazard function of componentj in the case where that component does not receive anyPM action. Accordingly, between the ith and ði þ 1Þth PMactions, the reliability rjðtÞ of component j is given fortjiptptjiþ1

by

rjðtÞ ¼ exp �

Z tj ðtÞ

tþjðtjiÞ

h�j ðxÞdx

!

¼ expðHjðtþj ðtjiÞÞ �Hjðtþj ðtÞÞÞ, (16)

where Hj is the accumulated hazard function of component j.Minimal repairs are performed on components that fail

between PM actions. The cost induced by minimal repairsis function of components failure rates. Following the workof Boland [18], for a given component j, the expectedminimal repair cost CMj

in an interval ½0; t� is as follows:

CMj¼ cj

Z t

0

hjðxÞdx. (17)

According to the above equation, the total minimal repaircost CMj

assigned to a component j which undergoespreventive maintenance actions at chronological timesðtj1 ; tj2 ; . . . ; tjnj

Þ is given such that

CMj¼ cj

Xnj

i¼0

Z tj ðtjiþ1Þ

tþjðtjiÞ

hjðxÞdx

!

¼ cj

Xnj

i¼0

ðHjðtjðtjiþ1ÞÞ �Hjðtþj ðtji

ÞÞ, (18)

where, by definition, tj0 ¼ 0 and tjnjþ1¼ T .

Fig. 1. Series–parallel MSS

4. MSS model and problem formulation

As in [1], let us consider an MSS composed of a numberof series subsystems each consisting of different compo-nents arranged in parallel. An example of such a system isgiven in Fig. 1 taken from [1]. Each system component j ischaracterized by its nominal performance (production) rateGj , hazard function hjðtÞ and its minimal repair cost cj. Inthis paper, it is assumed that all components mayexperience only two possible states: functioning state withnominal performance and failure state. All components aremutually independent and the MSS is assumed to satisfythe demand W �.A list of possible PM actions is available for a given

MSS. For each PM action n is assigned the cost CpðnÞ of itsimplementation, the age reduction coefficient eðnÞ and thenumber of element affected eðnÞ. It is assumed that the timein which a given component undergoes either preventivemaintenance or minimal repair is negligible if compared tothe time elapsed between consecutive actions. The lifetimeT of the system is partitioned into Y intervals with possiblydifferent durations yy, for y ¼ 1; . . . ;Y . Preventive main-tenance actions are performed at the end of intervalswhenever the reliability Rðt;W �Þ becomes lower than therequired reliability level R�.Let V be the vector corresponding to the sequence of PM

actions performed so that to maintain MSS reliability at agiven desired reliability level. Each time the PM is necessaryto improve system reliability, the action to be performed isdefined by the next number from V. Whenever a chosen PMaction ni is insufficient to maintain MSS reliability to therequired level, the next action niþ1 should be performed atthe same time. It follows that, for a given vector V, the totalnumber nj and chronological times ðtj1 ; tj2 ; . . . ; tjnj

Þ of PMactions are determined for each component j. In theparticular case where nj ¼ 0, i.e. eðniÞaj for all ni 2 V,the minimal repair cost CMj

assigned to component j andgiven by Eq. (18) where tj0 ¼ 0 and tjnjþ1

¼ tj1 ¼ T .For a given vector V, the cost CpðVÞ of PM actions and

the cost CMðVÞ of minimal repairs are given by thefollowing equations:

CpðVÞ ¼XN

i¼1

CpðniÞ (19)

structure of Example 1.


and

CMðVÞ ¼XJ

j¼1

cj

Xnj

i¼0

ðHjðtjðtjiþ1ÞÞ �Hjðtjðtji

ÞÞÞ

!. (20)

From the above equations, it follows that the total costCtotalðVÞ of all maintenance actions is given byCtotalðVÞ ¼ CpðVÞ þ CMðVÞ. Therefore, the imperfect main-tenance optimization problem consists on finding theoptimal vector V� which minimizes the total maintenancecost while providing the required MSS reliability level.Thus we have to solve the following mathematicalprogramming model:

V� ¼ argfCtotalðVÞ�!minjRðV; t;W �ÞXR�; 0ptpTg.

(21)

5. Optimization method

5.1. The extended great deluge

The extended great deluge is a local search metaheuristicrecently introduced by Burke et al. [10]. Like other localsearch methods, the extended great deluge iterativelyrepeats the replacement of a current solution s by a newone s�, until some stopping condition has been satisfied.The new solution is selected from a neighborhood NðsÞ.The mechanism of accepting or rejecting the candidatesolution from the neighborhood is different from othermethods. In the extended great deluge approach, thealgorithm accepts a solution whose objective function valueis more than or equal (for the maximization problems) tothe upper limit L, which is monotonically increased duringthe search by DL [10].

The initial value of ceiling (L) is equal to the initial costfunction f ðsÞ and only one input parameter DL has to bespecified. In [10], the authors applied successfully theextended great deluge on exam timetabling problem anddemonstrated that it outperformed well-known best resultsfound by other metaheuristics, such as simulated annealingand tabu search. In [19], Nahas et al. exploit this method asa new local search approach for solving the bufferallocation problem in unreliable production lines. In [20],the authors solve the redundancy allocation problem byan hybridization of the ant colony and the extendedgreat deluge metaheuristics. In [21], the authors havedeveloped a two-phase extended great deluge method tosolve efficiently the dynamic layout problem. In [22],Khatab et al. exploit such an algorithm to solve theselective maintenance optimization problem for series–parallel systems.

The extended great deluge algorithm is an extension ofthe ‘‘great deluge’’ method which was introduced as analternative to simulated annealing. Extended great delugeand simulated annealing algorithms share the characteristicthat they may both accept worse candidate solutions thanthe current one. The difference is in the acceptance

criterion of worse solutions. The simulated annealingmethod accepts configurations which deteriorate the obj-ective function only with a certain probability. Theextended great deluge algorithm incorporates both theworse solution acceptance (of the ‘‘great deluge’’ algo-rithm) if the solution fitness is less than or equal to somegiven upper limit L, i.e. ðf ðs�ÞXLÞ, and the well-known hillclimbing rule ðf ðs�ÞXf ðsÞÞ. The introduction of thedynamic parameter has an important effect on the search.As explained in [10], the decreasing of L may be seen as acontrol process, which drives the search toward a desirablesolution. Note finally that extended great deluge algorithmhas the advantage to require only one parameter ðDLÞ tobe tuned.

5.2. Extended great deluge for the preventive maintenance

optimization problem

In order to adapt the extended great deluge algorithm tothe optimal preventive maintenance problem, it is neces-sary to define an adequate solution representation andspecify the type of the adopted neighborhood. The searchproceeds iteratively from one feasible solution to anotherby moves in the neighborhood. An overview of thisalgorithm is given in Fig. 2.

5.2.1. Solution representation

As in [1], the solution is represented by a vector S withfinite length which contains numbers generated between 1and P, where P is the total number of possible PM actions.Only the first N positions will be used to represent themaintenance plan. The remaining positions affect thesolutions only when N is increased. At each iteration,the first N elements of S can be modified by using theneighborhood mechanism. Furthermore, the value of N

can be changed at each iteration depending on the solutionfeasibility. A given solution is feasible if the correspondingN is less than or equal to the length of the vector S. Toconstruct the initial feasible solution, the followingprocedure is proposed.

5.2.2. Initial solution construction

Step 1 Set the length of S to a constant number Nmax.Step 2 Generate randomly the elements of S from the setf1; . . . ;Pg.Step 3 Set N ¼ 1.Step 4 Calculate the objective function and theconstraint values by using N first elements (i.e. PMactions) of S.Step 5 if N ¼ Nmax and S is not feasible then go tostep 2.Step 6 If NoNmax and S is not feasible then set N ¼

N þ 1 and go to step 4.

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Fig. 2. Extended great deluge algorithm for the MSS maintenance optimization problem.

N. Nahas et al. / Reliability Engineering and System Safety 93 (2008) 1658–1672 1663

To define the appropriate neighborhood, several structureswere investigated. The following procedure is adopted toprovide the neighbor solution.

5.2.3. Neighboring solution

Step 1 Choose randomly a number x from the interval½0; 1�.Step 2� If xX0:5 then choose randomly an element SðiÞ with1pipNmax: Then, exchange the content of SðiÞ withthat of ðSðiÞ þ 1Þth or the ðSðiÞ � 1Þth PM action whichis randomly selected.� If xo0:5 then choose randomly two elements SðiÞ andSðjÞ with 1pi; jpNmax and exchange the contents of SðiÞand SðjÞ.

5.2.4. Solution decoding procedure

The following procedure has been proposed by [1] toevaluate the objective function value. We present it asdescribed in [1]. Suppose that S ¼ fs1; . . . ; sf g is thesolution to be evaluated.

(1)
Define for all MSS components ð1pjpJÞ effectiveages tj ¼ �y1, Hjðtþj Þ ¼ 0. Assign 0 to chronol-ogical time t, the interval number y, the number ofPM actions performed m and total maintenance costCtotal.
(2)
Increment the interval number y by 1. Incrementchronological time t and ages tj of all the systemcomponents ð1pjpJÞ by yy.
(3)
Calculate HjðtjÞ for each MSS component ð1pjpJÞ.

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Table 1

N. Nahas et al. / Reliability Engineering and System Safety 93 (2008) 1658–16721664

(4)
Parameters of systems components: case of the MSS of Example 1
Calculate reliability rjðtjÞ ¼ exp½Hjðtþj Þ �HjðtjÞ� for allsystem components ð1pjpJÞ.

Component (j) Gj lj g h0 cj
(5)
j j

1 0.4 0.05 1.8 0.0001 0.9

Using the UMGF approach, define MSS OPD andcalculate system reliability Rðt;W �Þ for a given demandW �.

2 0.4 0.05 1.8 0.0001 0.9
(6) 3 0.4 0.05 1.8 0.0 0.9
4 0.4 0.07 1.2 0.0003 0.8

5 0.6 0.01 1.5 0.0 0.5

6 1.3 0.01 1.8 0.00007 2.4

7 0.6 0.02 1.8 0.0 1.3

8 0.5 0.008 2.0 0.0001 0.4

9 0.4 0.02 2.1 0.0 0.7

10 1.0 0.034 1.6 0.0 1.2

11 1.0 0.008 1.9 0.0004 1.9

Table 2

Parameters of preventive maintenance actions (Example 1)

If Rðt;W �ÞoR�, increment m by 1 and define the PMaction to be performed at time t as v ¼ sm,

add the cost of PM CpðvÞ to Ctotal.Determine the cost of minimal repairs of component

eðvÞ in the interval between previous and current PM asCeðvÞ½HeðvÞðtþeðvÞÞ �HeðvÞðteðvÞÞ� and add this value toCtotal.

Modify the age teðvÞ of component eðvÞ by multiplyingit by the age reduction coefficient eðvÞ.

Calculate the new value of HeðvÞðtþeðvÞÞ for modifiedage teðvÞ.

Recalculate the reliability of component eðvÞ andreturn to step 5.

PM action ðuÞ eðuÞ eðuÞ C ðuÞ
(7) If Rðt;W �ÞXR� and toT , return to step 2. p
(8)
1 1 1.00 2.2
2 1 0.56 2.9

3 1 0.00 4.1

4 2 1.00 2.2

If Rðt;W �ÞXR� and tXT , evaluate the costs ofminimal repairs during the last interval for all thecomponents ð1pjpJÞ as cj½HjðtjÞ �Hjðtþj Þ� and addthese costs to Ctotal.

5 2 0.56 2.9

6 2 0.00 4.1

7 3 1.00 2.2

8 3 0.56 2.9

9 3 0.00 4.1

10 4 0.76 3.7

11 4 0.00 5.5

12 5 1.00 7.3

13 5 0.60 9.0

14 5 0.00 14.2

15 6 0.56 15.3

16 6 0.00 19.0

17 7 0.75 4.3

18 7 0.00 6.5

19 8 0.80 5.0

20 8 0.00 6.2

21 9 1.00 3.0

22 9 0.65 3.8

23 9 0.00 5.4

24 10 1.00 8.5

25 10 0.70 10.5

26 10 0.00 14.0

27 11 1.00 8.5

28 11 0.56 12.0

29 11 0.00 14.0

6. Numerical experiments

In this section, experiments are conducted to compareour algorithm and the genetic algorithm GENITORproposed in [1]. In the present work, we limit ourselvesto compare the two algorithms only on the basis of thenumber of evaluated solutions and the solutions qualityrather than computational time. In our opinion, sucha comparison is more viable since it is very difficultto compare computation times using different computingsystems, programming language compilers, coding techni-ques, etc. Two experiments are proposed where thefirst one exploits the MSS proposed by Levitin andLisnianski [1], while the second one uses two other largerMSS examples. Algorithms are implemented by usingMATLAB software tool on a 1.8GHz Pentium 4processor.

6.1. The first experiment

In this experiment, the MSS example studied iscomposed of J ¼ 11 components (Fig. 1). The reliabilityfunction of each component is given by a Weibull intensityfunction hðtÞ ¼ lggðtðtÞÞg�1 þ h0; where l and g are,respectively, the scale and shape parameters. The accumu-lated hazard function has the form HðtÞ ¼ ðltðtÞÞg þ h0tðtÞ.

For each component j ðj ¼ 1; . . . ; 11Þ productivity Gj ,scale lj and shape gj parameters, hazard constant h0j

andminimal repair cost cj , as originally presented in [1], arereproduced in Table 1. From this table, one can see that allcomponents are with increasing failure rate ðg41Þ.Preventive maintenance actions list assigned to the studiedMSS is given by Table 2. The MSS lifetime T and the

interval duration yy are the same as those given in [1], i.e.T ¼ 25 years and yy ¼ 0:125 years.Before dealing with the comparison of the two stochastic

methods, it is interesting to show how such methodsoperate. To this end, the total cost versus the number ofiterations is given in Figs. 3 and 4, respectively, for theextended great deluge and the GENITOR methods. InFig. 3, each point represents the accepted total cost at agiven iteration. Results shown in Fig. 4 are those obtainedfor the first cycle of the GENITOR algorithm. Both Figs. 3


and 4 are derived in the case where the required demandlevel W � and reliability level R� are fixed to 0:8 and 0:9,respectively. As it can be seen in Fig. 3, the search processevolves according to a degraded ceiling toward theconvergence point which is obtained after nearly about11 000 iterations. The search process of the GENITORalgorithm starts with a set of solutions represented by an

Fig. 3. Accepted solutions with the extended great deluge algorithm (MSS

of Example 1, Rðt;W �ÞX0:9).

Fig. 4. Evolution of solutions for one cycle of the GENITOR algorithm

(MSS of Example 1, Rðt;W �ÞX0:9).

Fig. 5. Influence of DL on the extended

initial population. According to Fig. 4, this algorithmconverges before 2000 reproductions.In what follows, we will analyze the influence of DL, and

then we will compare the two methods.

6.1.1. Influence of DL

As it is mentioned before, the extended great delugealgorithm has the advantage to require the tuning of onlyone parameter (i.e. DL). In order to demonstrate theinfluence of DL value on the algorithm performance, therequired demand level W � and reliability level R� are set,respectively, to 0:8 and 0:9; while DL value is varied from0:008 to 0:1: For each value of DL, 10 executions of thealgorithm are performed. The stopping criterion is satisfiedwhenever either the value of L is strictly less than thecurrent best total cost, the maximum number of iterationsis reached, or no solution has been accepted after themaximum number of iterations which is fixed to 15 000.The results obtained are reported in Fig. 5 which depicts,for each value of DL, the average total cost, the best totalcost and the number of evaluated solutions. This figureshows that the low is the value of DL, the best is thesolution provided by the extended great deluge algorithm,and large is the number of evaluated solutions. The bestsolution is obtained when DL is fixed to 0.008 or 0.01, whilethe number of evaluated solutions is found to be about20 000 and 15 000 for DL ¼ 0:008 and 0.01, respectively.Therefore, in all experiments, the parameter DL of theextended great deluge algorithm has been set to 0.01.

6.1.2. Comparison results of the two methods

In what follows, the extended great deluge and theGENITOR algorithms are compared on the basis of resultsgiven by four instances. To each instance is assigned avalue of the required demand level W � and a value of therequired reliability level R�. These values are the same asthose proposed by Levitin and Lisnianski [1]. Within thisalgorithm, the stopping criterion adopted is satisfied eitherif the value of L is strictly less than the current best totalcost obtained, or if the maximum number of iterations is

great deluge algorithm performance.

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Table 6

The best maintenance plan obtained for Rðt; 0:8Þ40:95 (Example 1)

t PM action Component affected Rpðt; 0:8Þ

11:750 6 2 0:96814:000 9 3 0:95614:875 23 9 0:95816:125 16 6 0:98720:375 3 1 0:97122:875 17 7 0:96124:000 4 2 0:959

Table 7


t PM action Component affected Rpðt; 1Þ


reached, or no solution has been accepted after 15 000iterations. The maximum number of iterations is fixed to30 000. For each instance, its corresponding best total costobtained by the extended great deluge algorithm is given inTable 3. This table gives also the average and the maximumof the best total cost found among 10 runs, as well as thestandard deviation of the 10 final best total costs. FromTable 3, it can be seen that the standard deviationcorresponding to each instance is relatively low. Conse-quently, the proposed method is robust and credible.

For the sake of comparison, the best total costs obtainedby our algorithm and those obtained by the GENITORalgorithm [1] are given in Table 4. It is found that theextended great deluge algorithm outperformed the GENI-TOR algorithm. For the four instances considered, the bestmaintenance plans obtained by the proposed algorithm aregiven in Tables 5–8.

10:625 18 7 0:98213:625 3 3 0:96216:000 9 2 0:95717:000 15 6 0:97518:500 10 7 0:98320:875 12 5 0:97021:625 27 11 0:964

6.2. The second experiment

In this experiment, we consider two examples of largeMSS. The MSS of Example 2 (Fig. 6) is composed of J ¼

15 components, while that of Example 3 is composed ofJ ¼ 19 components (Fig. 7). Components 1–15 of Example3 are the same of those of Example 2. For each component

Table 3

Total cost obtained by the extended great deluge algorithm (Example 1)

ðW ;R�Þ Cmaxtotal Cav

total Cmintotal

Std. dev

ð0:8; 0:9Þ 34:9208 34:3115 33:4664 0:5237ð0:8; 0:95Þ 52:1565 50:6503 49:0058 0:9974ð1:0; 0:9Þ 51:4228 50:0948 49:9271 0:4677ð1:0; 0:95Þ 82:9834 81:4930 80:5207 0:8070

Table 4

Comparison of the extended great deluge algorithm with the GENITOR

algorithm: case of the MSS of Example 1

ðW ;R�Þ GENITOR Extended great deluge Improvement (%)

ð0:8; 0:9Þ 34:824 33:466 3:89ð0:8; 0:95Þ 51:581 49:005 4:99ð1:0; 0:9Þ 51:301 49:927 2:67ð1:0; 0:95Þ 82:625 80:520 2:54

Table 5



14:250 8 3 0:94916:875 3 1 0:93119:250 6 2 0:90920:125 23 9 0:92322:500 27 11 0:92024:125 7 3 0:910

Table 8



7:750 18 7 0:98210:750 5 2 0:96211:750 3 1 0:95612:375 16 6 0:97214:250 9 3 0:96215:250 18 7 0:98118:125 10 4 0:96919:500 27 11 0:96320:375 6 2 0:96121:250 3 1 0:95521:625 18 7 0:97323:750 21 9 0:95624:250 7 3 0:960

j (j ¼ 1; . . . ; 15) productivity Gj , scale lj and shape gj

parameters, hazard constant h0jand minimal repair cost cj

are reported in Table 9. Values of these characteristicsassigned to each additional component j (j ¼ 16; . . . ; 19) ofExample 3 are given in Table 10. Preventive maintenanceactions corresponding to component j (j ¼ 1; . . . ; 15) isgiven in Table 11 while that of component j (j ¼ 16; . . . ; 19)is given in Table 12. For both MSS examples, the lifetime T

as well as the interval duration yy are the same as inExample 1, i.e. T ¼ 25 years and yy ¼ 0:125 years.The GENITOR algorithm proposed in [1] has been

implemented. Several values have been tested for the lengthF of the integer string representing solutions, the size Ns ofpopulation, the number Nrep of reproductions within a

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Fig. 6. Series–parallel MSS structure of Example 2.

Fig. 7. Series–parallel MSS structure of Example 3.

Table 9

Parameters for MSS components of Example 2

Component (j) Gj lj gj h0jcj

1 1.3 0.050 1.5 0.00020 0.5

2 1.4 0.050 1.4 0.00020 0.6

3 1.5 0.050 1.4 0.00000 0.6

4 1.3 0.040 1.2 0.00010 0.6

5 1.4 0.030 1.1 0.00000 0.6

6 1.5 0.030 1.2 0.00009 1.1

7 1.6 0.020 1.5 0.00000 1.0

8 1.5 0.008 1.5 0.00010 0.4

9 1.3 0.020 1.2 0.00020 0.7

10 1.3 0.030 1.1 0.00010 1.2

11 1.3 0.009 1.2 0.00040 1.9

12 1.4 0.008 1.2 0.00010 0.4

13 1.7 0.002 2.0 0.00000 0.9

14 1.4 0.034 1.9 0.00000 1.2

15 1.5 0.008 1.0 0.00040 1.3

Table 10

Parameters for MSS components of Example 3: component j ¼ 16; . . . ; 19

Component (j) Gj lj gj h0jcj

16 1.3 0.020 1.2 0.00020 0.7

17 1.3 0.030 1.1 0.00010 1.2

18 1.3 0.009 1.2 0.00040 1.9

19 1.4 0.008 1.2 0.00010 0.4

Table 11

Parameters of preventive maintenance actions: MSS of Example 2

PM

action ðuÞeðuÞ eðuÞ CpðuÞ PM

action ðuÞeðuÞ eðuÞ CpðuÞ

1 1 0.60 2.0 18 9 1.00 3.0

2 1 0.00 2.5 19 9 0.65 3.8

3 2 0.65 2.0 20 9 0.00 5.4

4 2 0.00 4.0 21 10 1.00 6.5

5 3 0.65 2.5 22 10 0.70 7.5

6 3 0.00 4.0 23 10 0.00 10

7 4 0.80 2.5 24 11 1.00 6.5

8 4 0.00 4.5 25 11 0.56 7.0

9 5 1.00 3.3 26 11 0.00 9.0

10 5 0.70 5.0 27 12 0.55 3.0

11 5 0.00 7.2 28 12 0.00 4.2

12 6 0.75 2.3 29 13 0.55 3.8

13 6 0.00 3.0 30 13 0.00 5.4

14 7 0.65 4.3 31 14 0.65 7.5

15 7 0.00 6.5 32 14 0.00 10

16 8 0.80 2 33 15 1.00 3.0

17 8 0.00 3.2 34 15 0.90 6.5

35 15 0.00 9.0

Table 12

Parameters of preventive maintenance actions: MSS of Example 3,

component j ¼ 16; . . . ; 19

PM action ðuÞ eðuÞ eðuÞ CpðuÞ

36 16 1.00 3.0

37 16 0.65 3.8

38 16 0.00 5.4

39 17 1.00 6.5

40 17 0.70 7.5

41 17 0.00 10

42 18 1.00 6.5

43 18 0.56 7.0

44 18 0.00 9.0

45 19 0.55 3.0

46 19 0.00 4.2


cycle, and for the maximum number Nc of cycles. For theMSS of Example 2, parameters F , Ns and Nrep areset, respectively, to 35, 90 and 2000, while for the MSSof Example 3, these parameters are F ¼ 50, Ns ¼ 100 andNrep ¼ 2500: For both MSS examples parameter Nc hasbeen set to 50.

Concerning the extended great deluge algorithm, themost appropriate value assigned to the length Nmax ofsolution vector S is 35 and 50 for MSS of Examples 2and 3, respectively.

ARTICLE IN PRESS

Fig. 8. Total cost versus the number of evaluated solutions (MSS of Example 2).


Three different reliability constraints are used tocompare the two algorithm. In the case of the MSS ofExample 2 these constraints are Rðt; 0:7ÞX0:7, Rðt; 0:7ÞX0:80 and Rðt; 0:7ÞX0:85, while in the case of the MSS ofExample 3, the reliability constraint considered areRðt; 0:8ÞX0:75, Rðt; 0:8ÞX0:80 and Rðt; 0:8ÞX0:85.

Each algorithm was tested by performing 10 trials. Anexample of the convergence curves of the two algorithms isshown in Figs. 8 and 9 for MSS Examples 2 and 3,respectively. These figures show the total cost versus thenumber of evaluated solutions. By 30 000 and 40 000evaluations of objective function, respectively, for Exam-ples 2 and 3, the lowest cost has levelled out. These numberof evaluations are used to assess the performance of thealgorithms.

The results obtained after 10 trials are given in Tables 13and 14, where for each considered instance, the best totalcost obtained is indicated in bold.

For the MSS of Example 2 with reliability constraintsRðt; 0:7ÞX0:70 and Rðt; 0:7ÞX0:80, the GENITOR algo-rithm outperforms the extended great deluge, except in oneaverage total cost. However, in case of reliability constraintRðt; 0:7ÞX0:85; the extended great deluge algorithm out-performs the GENITOR algorithm.For the MSS of Example 3, all the best solutions (i.e.

those with minimal costs) are obtained by the extendedgreat deluge. On the other hand, in the three consideredcases, the extended great deluge gives better solutions intwo cases for average total costs, and in one case formaximal total costs. We remark also that:

(1)
The standard deviation values of the results obtainedby the GENITOR algorithm were generally lessthan those of the extended great deluge (for thefive reliability constraints). However, in most cases(four among six), the solutions average values

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Fig. 9. Total cost versus the number of evaluated solutions (MSS of Example 3).

Table 13

Comparison of the extended great deluge algorithm with the GENITOR algorithm: case of the MSS of Example 2 (10 trials)

ðW�;R�Þ GENITOR Extended great deluge

Cmaxtotal Cav

total Cmintotal

Std. dev Cmaxtotal Cav

total Cmintotal

Std. dev

ð0:7; 0:7Þ 44:863 44:137 42:664 0:73 45:824 44:073 42:680 1:03ð0:7; 0:8Þ 60:898 59:082 58:180 0:84 61:742 60:092 58:239 1:15ð0:7; 0:85Þ 73:113 72:074 71:145 0:67 72:664 71:788 70:913 0:57

Table 14

Comparison of the extended great deluge algorithm with the GENITOR algorithm: case of the MSS of Example 3 (10 trials)

ðW�;R�Þ GENITOR Extended great deluge

Cmaxtotal Cav

total Cmintotal

Std. dev Cmaxtotal Cav

total Cmintotal

Std. dev

ð0:8; 0:75Þ 64.030 62.853 61.905 1.04 64.755 63.008 61:523 1.19

ð0:8; 0:8Þ 74:967 74:367 73:978 0:37 74:520 73:549 72:764 0:62ð0:8; 0:85Þ 91:680 90:504 89:099 1:22 91:993 90:305 88:018 1:55


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Fig. 10. Average number of evaluated solutions: a comparison.

Table 15

The best maintenance plan: case of the MSS of Example 2

Rðt; 0:7ÞX0:7 Rðt; 0:7ÞX0:8 Rðt; 0:7ÞX0:85

t PM action Rp t PM action Rp t PM action Rp

08:375 3 0:778 6:750 2 0:848 5:875 3 0:88609:625 2 0:772 7:875 7 0:850 6:625 2 0:88111:000 7 0:795 8:875 3 0:866 7:500 12 0:88912:625 3 0:752 10:125 12 0:848 8:375 7 0:89313:500 12 0:773 11:000 6 0:848 9:375 15 0:91014:625 6 0:764 12:125 15 0:897 10:625 9 0:88015:875 15 0:836 14:000 30 0:815 11:250 6 0:94118:125 29 0:736 14:375 2 0:844 13:250 3 0:90018:750 9 0:814 15:375 3 0:822 14:250 30 0:87020:375 2 0:808 15:875 7 0:904 14:750 12 0:89122:125 3 0:773 17:750 9 0:886 15:500 2 0:89323:375 7 0:807 19:250 5 0:854 16:500 3 0:868

– – – 20:250 2 0:826 17:000 7 0:904– – – 20:875 14 0:853 18:250 5 0:867– – – 21:875 12 0:870 18:750 18 0:865– – – 23:000 3 0:857 19:125 15 0:923– – – 24:000 7 0:854 20:625 9 0:907– – – – – – 21:625 3 0:902– – – – – – 22:625 2 0:878– – – – – – 23:250 12 0:905– – – – – – 24:375 5 0:877


obtained by the extended great deluge algorithm arebetter.

(2)
The average number of evaluated solutions necessaryto converge for the GENITOR algorithm is clearlyless than those of the extended great deluge algorithm(see Fig. 10).
The best maintenance plans obtained either by theGENITOR or by the extended great deluge algorithmsare shown in Tables 15 and 16. In Table 15, the first andthe second maintenance plans are due to the GENITORalgorithm while the third is derived by the extended great

deluge algorithm. However, all maintenance plans inTable 16 are obtained by the extended great delugealgorithm developed in this paper.

7. Conclusion

In this paper, we applied an algorithm based on theextended great deluge metaheuristic to solve the imperfectpreventive maintenance problem for multi-state systems.In order to prove its efficiency, we compared the extendedgreat deluge algorithm with the genetic algorithm [1]. Theresults obtained using the extended great deluge method to

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Table 16

The best maintenance plan: case of the MSS of Example 3

Rðt; 0:8Þ40:75 Rðt; 0:8Þ40:8 Rðt; 0:8Þ40:85

t PM action Rp t PM action Rp t PM action Rp

7:375 3 0:811 6:625 7 0:833 5:750 2 0:8788:500 12 0:793 7:250 5 0:860 6:500 5 0:8779:250 9 0:791 8:375 2 0:850 7:125 12 0:89010:000 1 0:834 9:500 9 0:842 8:000 15 0:90411:375 5 0:804 10:375 15 0:873 9:125 4 0:88712:375 15 0:844 11:625 3 0:872 10:125 9 0:91413:875 7 0:820 13:000 13 0:859 11:500 17 0:86915:000 3 0:826 14:125 2 0:834 11:875 7 0:89616:125 30 0:776 14:875 30 0:815 12:750 6 0:90516:500 2 0:813 15:250 6 0:844 14:000 2 0:87417:750 17 0:785 16:375 7 0:857 14:625 13 0:88118:375 12 0:815 17:625 3 0:821 15:500 7 0:86319:375 9 0:796 18:125 18 0:815 15:875 14 0:90120:125 5 0:840 18:500 17 0:845 16:875 3 0:87921:625 14 0:804 19:375 15 0:861 17:500 30 0:88922:375 3 0:813 20:500 9 0:862 18:375 18 0:86223:375 7 0:807 21:500 36 0:827 18:625 9 0:88524:250 1 0:792 22:000 2 0:857 19:375 36 0:867

– – – 23:000 3 0:51 19:750 5 0:887– – – 23:875 7 0:868 20:500 15 0:892– – – – – – 21:375 12 0:875– – – – – – 21:875 2 0:897– – – – – – 22:875 7 0:883– – – – – – 23:625 3 0:888– – – – – – 24:500 16 0:874


solve the problem, are clearly encouraging. The mainadvantage of the proposed method is that it requires thesetting of only one parameter that may correspond toa search time. Another important characteristic of thismethod is that it found the best minimal cost solutions for8 instances among 10 (the 4 cases of Example 1 from [1], 1case among 3 in Example 2 and the 2 cases of Example 3).

The current study makes a comparison between theextended great deluge algorithm and the genetic algorithmwhich is widely proven to be very efficient in solving thisproblem. While the extended great deluge approach wasshown to produce within an acceptable amount of timebetter results than genetic algorithm, future researchshould address the extended great deluge algorithm ascompared to other heuristic techniques, such as tabusearch, variable neighborhood search, GRASP and beamsearch, to name a few.

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Extended great deluge algorithm for the imperfect ...liacs.leidenuniv.nl/~csnaco/SWI/papers/Great deluge...extended great deluge algorithm is proposed. This method has the advantage

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