optimal replacement last with continuous and discrete policies

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IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 3, SEPTEMBER 2014 1

Optimal Replacement Last WithContinuous and Discrete PoliciesXufeng Zhao, Toshio Nakagawa, and Ming J. Zuo, Senior Member, IEEE

Abstract—This paper proposes age and periodic replacementlast models with continuous, and discrete policies. That is, anoperating unit is replaced preventively at time of operation asa strategic policy, or at a number of working cycles to satisfysuccessive job completion, whichever occurs last. Such policiesare named as replacement last, and their expected cost rates andoptimal policies are obtained. However, the focus of this paperis to compare replacement last with replacement first policies,which are formulated under the classical assumption of whicheveroccurs first. From the points of cost and performability, differentcomparative methods for continuous and discrete optimizationsare demonstrated to determine in what cases we should adoptreplacement last rather than replacement first. All theoreticaldiscussions in this paper are made analytically, and are computednumerically.

Index Terms—Replacement, minimal repair, replacement last,maintenance, working cycle.

ACRONYMS AND ABBREVIATIONS

CR corrective replacement

PR preventive replacement

RL replacement last

RF replacement first.

NOTATIONS

distribution of variable working cycles ,where

statistical expectation of

-fold Stieltjes convolution of , where

failure distribution of the operating unit

Manuscript received February 24, 2013; revised December 21, 2013;accepted February 26, 2014. This work was supported in part by NationalNatural Science Foundation of China71371097; Qatar National Research Fun-dunder Grant No. NPRP 09-774-2-297; and Natural Sciences and EngineeringResearch Council of Canadaunder Grant No. 104966-2010. Associate Editor:L. Walls.X. Zhao is with the Graduate School of Management and Information Sci-

ences, Aichi Institute of Technology, Toyoa 470-0392, Japan, with the School ofEconomics andManagement, Nanjing Tech University, Nanjing 211816, China,and also with the Department of Mechanical and Industrial Engineering, QatarUniversity, Doha 2713, Qatar (e-mail: [email protected]).T. Nakagawa is with the Graduate School of Management and Information

Sciences, Aichi Institute of Technology, Toyota 470-0392, Japan (e-mail: [email protected]).M. J. Zuo is with the Department of Mechanical Engineering, University of

Alberta, Edmonton, AB T6G 2G8, Canada (e-mail: [email protected]).Digital Object Identifier 10.1109/TR.2014.2337811

statistical expectation of

failure rate of

cumulative hazard function of

replacement done at time of operation

replacement done at number of workingcycles

corrective replacement cost at failure

preventive replacement cost at time or atnumber

minimal repair cost at failure

expected replacement cost rates of the standardage and periodic replacements

optimal times minimizing

expected replacement cost rates of age andperiodic replacement last

expected replacement cost rates of age andperiodic replacement first


optimal numbers minimizing


optimal numbers minimizing

I. INTRODUCTION

M OST operating units are repaired or replaced when theyhave failed. Replacement policies done after failure, and

before failure are called corrective replacement (CR), and pre-ventive replacement (PR), respectively. It may require muchtime and suffer higher production losses to repair a failed unitor to replace it with a new one, so it is necessary to have suitablePR strategies to prevent failures. Recently, a methodical surveyof maintenance policies in reliability theory was done [1]; andpublished books [2]–[7] collected many maintenance models,providing theory, and their applications in industrial systems.However, it has been assumed in all the above models

that maintenance activities are scheduled under the classicalassumption of repairing at whichever triggering event occursfirst [8]. That is, each unit is maintained preventively at somethreshold of a measurement such as age, periodic time, usagenumber, damage level, etc.; or at failure, whichever occurs first.These policies would be reasonable when a single preventive

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2 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 3, SEPTEMBER 2014

maintenance policy is used, but it may cause frequent main-tenance jobs when several combined policies are scheduled[9]. On the other hand, maintenance is performed more easilyat a planned time , but such a strict fashion would be im-practical when the unit is operating for some successive jobs,because the sudden suspension of any job may suffer losses ofproduction to different degrees ([8], p. 72). When a job has avariable working cycle, it would be better to do maintenanceafter the job is completed even though the maintenance timehas arrived ([1], p. 245). A replacement policy scheduled at thefirst completion of some working cycle over a planned timewas modeled [10]. Inspection policies considering a variableworking cycle were observed in [11].By considering the above two aspects in maintenance mod-

eling, we propose a replacement policy based on the approach ofwhichever triggering event occurs last; i.e., the unit is replacedbefore failure at a planned time , or at a random working cycle, whichever occurs last [9]. However, the unit usually executes

jobs with a limited number of successive intervals, e.g., numberof working cycles, so that it is difficult to specify the quality

of maintenance policies without considering the factor of ex-ecuted jobs [12]. The combined age and periodic replacementpolicies, and cumulative damagemodels that are scheduled withworking times under the classical assumption whichever trig-gering event occurs first, have been discussed [10], [13], [14].However, the newly proposed whichever triggering event oc-curs last policy has not been well studied, and specifically notwhether it holds for these models with working cycles, whichbecomes the main purpose in this paper.As a study of the whichever triggering event occurs last

policy, this paper takes up age and periodic replacement modelswith continuous and discrete policies, i.e., the unit is replacedpreventively at time of operation, or at number of workingcycles, whichever occurs last, and first, which are called re-placement last (RL), and replacement first (RF), respectively.The continuous -policy is scheduled strategically, and thediscrete -policy is made to satisfy the executed jobs. Weobtain expected cost rates of RL and their optimal policies,i.e., optimal and are discussed analytically. To comparethe optimized results of RL with those of RF, we give directlyoptimal and for RF [13], [14]. Comparisons betweenand are made in detail by using the method proposed in

[9]; however, the comparative methods between and

need to be explored. We show that RL gives more flexibility toensure job completions, and to determine whether RL is betterthan RF from the points of cost and performability. Finally, wesummarize the work of this paper, and give potential applica-tions of RL techniques in the Conclusions.

II. AGE REPLACEMENT

We assume that an operating unit works for jobs successively,and each job has a variable working cycleaccording to a statistically independent and identical distribu-tion with finite mean . The unit de-teriorates with its operating time, and has a failure time ac-cording to a general distribution with finitemean , which is statistically independent of random variables. In addition, denote

, which presents the -fold Stieltjes convo-lution of with itself, and for . Let

be the failure rate of , where is adensity function of for any function ,and it is also assumed that increases strictly fromto .

A. Optimization of Replacement Last

Suppose that, when the operating unit fails, its failure is im-mediately detected, and then CR is done. Following PR policies,the unit is replaced before failure at time of op-eration, or at a number of working cycles,whichever occurs last, which is called replacement last (RL).Then, the probability that the unit is replaced before failure attime or at working cycles is shown in (1) at the bottom ofthe page, and the probability that the unit is replaced at failure isshown in (2) at the bottom of the page. Note that the sum of (1)and (2) is unity. Then, the mean time to replacement is shownin (3) at the bottom of the next page.We consider the problem of minimizing the expected replace-

ment cost per unit of time for an infinite time span. Two replace-ment costs are introduced: is the CR cost at failure, and isthe PR cost at time or at number , where could beestimated reasonably in practice. Therefore, the expected costrate is shown in (4) at the bottom of the next page.Equation (4) includes some basic replacement models. When

the unit is replaced only at time ,

(1)

(2)


ZHAO et al.: OPTIMAL REPLACEMENT LAST WITH CONTINUOUS AND DISCRETE POLICIES 3

(5)

which agrees with ([8], p. 87) for the standard age replacement.When it is replaced only at number , see (6) at the bottom ofthe page. When it is replaced only at failure,

(7)

We next find the optimal for a fixed ,and for a fixed which minimizein (4). Differentiating with respect to , and set-ting it equal to zero, see (8) at the bottom of the page. De-noting , and denoting the left-hand side of (8)by , see the first unnumbered equation at the bottomof the page. Thus, increases strictly with fromthe second unnumbered equation at the bottom of the page.to

.

Therefore, if , then there existsa finite, unique which satisfies (8), and theresulting cost rate is

(9)

If , then , and the resultingcost rate is given by (7).In addition, a finite increases with because

decreases with . From (9), if optimal minimizingin (4) exist simultaneously, it would be at ,

where is given by a solution of the equation

(10)

Next, forming the inequality ,see (11) at the bottom of the next page, where we seethe first unnumbered equation at the bottom of the next

(3)

(4)

(6)

(8)



page. Denote the left-hand side of (11) as .If increases strictly with from to

, then increasesstrictly with from

to .When , i.e.,

, and the failure rate increases strictly with, we prove in the Appendix that

increases strictly with , and converges to as .Therefore, if , then there existsa unique, minimum which satisfies (11).When is given by (5); and when, the resulting cost rate is shown in (12) at the bottom of the

page. If , then , and theresulting cost rate is given by (7).

B. Comparison With Replacement First

We first compare the above policy with a conventional re-placement, which is well-known in reliability theory. Supposethat the unit is replaced before failure at time ofoperation, or at working cycles, whicheveroccurs first, which is called replacement first (RF). The modeland its optimization were discussed in [13]. We give the ex-pected cost rate and its optimal policy without proof, and focuson comparisons between RL and RF.The expected replacement cost rate is

(13)

Clearly,

(14)

Differentiating with respect to , and setting itequal to zero for a fixed ,

(15)

Denoting the left-hand side of (15) as , thenincreases strictly with from 0 to the second unnumbered equa-tion at the bottom of the page.Therefore, if , then a finite,

unique which minimizes in (13)exists, and the resulting cost rate is

(16)

If , then , and the resultingcost rate is given by (6).In addition, a finite decreases with because

increases with . From (16), the optimal minimizingin (13) is , where is given by a solution

of (10).From the inequality for a

fixed ,

(17)

where

Denote the left-hand side of (17) as . Ifincreases strictly with to ,

(11)

(12)



then increases strictly with to the first unnumberedequation at the bottom of the page.It was proved in ([1], p. 86) that, when , if

the failure rate increases strictly with , then

increases strictly with , and converges to as .Therefore, if , a unique, minimum

which satisfies (15) exists, and the re-sulting cost rate is shown in (18) at the bottom of the page. If

, then , and the resultingcost rate is given by (5).1) Comparison Result 1: Using the comparative method pro-

posed in [9], we first compare with as follows. Sup-pose that failure function increases strictly from 0 to ,i.e., , and . Then, there exist both unique

and which satisfy (8) and(15). Compare the left-hand sides of (8) and (15) by denoting thesecond unnumbered equation at the bottom of the page. Clearly,we see the third unnumbered equation at the bottom of the page.Thus, there exists a finite, unique which sat-isfies .

From (15), denote that

(19)

From the above discussions, the following comparison resultscan be given.1. If , then ; RLshould be adopted because, from (9) and (16), we obtain

, which means RL could savemore replacement cost than RF.

2. If , then , i.e., RFshould be adopted.

3. If , then RL is the same as withRF.

Furthermore, let be an optimal replace-ment time which minimizes in (5). Then, is a finite,unique solution of (10), and the resulting cost rate is

(20)

It is easy to show in the equation at the bottom of the next page,that is, increases with from , and decreases withto . It can be also shown that decreases with , i.e.,

increases strictly with . In other words, RL would showmore superior cases than RF as becomes smaller.

(18)



TABLE IOPTIMAL , AND FOR WHEN , AND

In addition, denoting the left-hand side of (10) as ,

It follows that and . From (9), (16), and (20),we obtain , and ;that is, the standard age replacement should be adopted.Suppose that failure time could be estimated from historical

data in practice as aWeibull distribution, and random working cycles follow an exponential distri-

bution . We give the values of parameters, and within a reasonable range for the fol-

lowing numerical illustrations. Table I presents optimal, and which satisfy (8), (15), (19), and (10) for

and , when , and .• Both and increase with . Whenincreases, PR should be advanced to prevent a higher CRcost. In other words, the unit can work longer asbecomes smaller. In addition, RL is much better than RFwhen becomes smaller, especially for small . Forexample, when , and , i.e.,

is much less than .• For a given , when

, and RF is better than RL. Conversely,when , andRL is better than RF. For example, when

, and hence,for , i.e., , and

for ,i.e., . That is, even though CR cost is much

higher than PR cost , there still exist some cases whenRL is better than RF.

• Optimal increases with from decreases withto , and RL is better than RF as becomes smaller.

When is small enough, the unit has to be replacedmainlyat random working cycles for RF, and mainly at total op-erating time for RL. Conversely, when is large enough,the unit has to be replacedmainly at total operating time forRF, and mainly at random working cycles for RL. For ex-ample, when , RF is always done at , andRL is always done at which is greater than 0.1; when

, RF is mainly done at , and RL is mainly doneat .

• From the RL definition, there exists some waiting timeuntil replacement if optimal is much larger than thegiven , e.g., when and

is more than 10 times larger than. However, another 9 working cycles could be oper-

ated during this waiting time. From such a consideration,RL would be absolutely better than RF when is smallenough.

2) Comparison Result 2: We next compare with asfollows. From the Appendix, it is easy to show that

Differentiating with respect to , see the equation atthe bottom of the next page. Then, increases strictlywith , and

which increases strictly with from 0. Then,

(21)



TABLE IIOPTIMAL AND FOR WHEN , AND

where is given in the left-hand side of (10).From the above results, there exists a finite, unique

which satisfies , and, where is given in (10). Thus, we obtain the following

results about .1. If , then , and .2. If , then , and

.It is easy to show from ([1], p. 86) that

Thus, we have in (21), and obtain the fol-lowing results about .1. If , then , and

.2. If , then , and

.From above discussions, the following comparison results

can be given.1. If , then , and , i.e.,RL should be adopted, not only because

, but also because RF degradesinto the standard age replacement where we cannot set anynumber of working cycles.

2. If , then , and , i.e., RFshould be adopted, and the reasons are the same as thosegiven above.

3. If , then , and , i.e.,the standard age replacement should be adopted, and theexpected cost rate is given by (5).

Table II presents optimal and which satisfy (11) and(17), and which satisfies , when

, and . This result indicates severalother observations.• If finite and exist, both of them increase with

. As becomes smaller, especially for

small , RL shows more superior cases than RF does, be-cause most of but the are finite. For example,when and , i.e.,

while .• , which is computed by , is lessthan in Table I. We can find that, when a given ,RL should be adopted. For example, when , whichis less than while forall . When , e.g.,which is greater than for

, and RF should be adopted for all cases as. However, there exist particular cases when the

standard age replacement policy in (5) should be adoptedfor . However, compared to Tables I andII, the differences between and are very small. Forexample, when , when

, and the standard policy isadopted.

• Compute the mean time to failure. When a given is too small, e.g., , to

prevent frequent replacement cost, RL should be adoptedto let the unit operate as long as possible. When is largeenough, e.g., , RF should be adopted to preventhigh failure cost. When is given in a reasonable range,e.g., , we can find that, from Table II, thecost rate plays an important role in selecting whichis better between RL and RF.

III. PERIODIC REPLACEMENT

A. Optimization of Replacement Last

Suppose that the unit is replaced at time ofoperation, or at working cycles, whicheveroccurs last; and undergoes minimal repair ([1], p. 95) at eachfailure between replacements. The other assumptions are thesame as those in Section II. Then, the expected number of min-imal repairs before replacement is shown in (22) at the bottom



of the page, where is called the cumulativehazard function that represents the expected number of failuresin . The mean time to replacement is

(23)

Therefore, the expected cost rate is

(24)

where is the minimal repair cost at failure, and is givenin (4).Similarly, (24) also includes some basic replacement models.

When the unit is replaced only at time ,

(25)

which agrees with ([8], p. 97) for the standard periodic replace-ment. When it is replaced only at number ,

(26)

When only minimal repair is done at failure,

(27)

We find optimal for a fixed , and optimalfor a fixed , that minimize in

(24), respectively. Differentiating with respect to ,and setting it equal to zero, see (28) at the bottom of the page.Denoting the left-hand side of (28) to be , see the un-numbered equation at the bottom of the page.Therefore, if , then there exists a finite,

unique which satisfies (28), and the resultingcost rate is

(29)

If , then , and the resulting costrate is given by (27).In addition, a finite increases with because

decreases with . From (29), if the optimal min-imizing in (24) exists simultaneously, it would be

, where is given by a solution of the equation

(30)

Next, forming the inequality ,see (31) at the bottom of the page, where

When , and the failure rate increasesstrictly with , we prove in the Appendix that

(22)

(28)

/

(31)



increases strictly with , and converges to as .Denoting the left-hand side of (31) asincreases strictly with , and

Therefore, if , then there exists a unique,minimum which satisfies (31). When

is given by (25); and when , theresulting cost rate is

(32)

If , then , and the resulting costrate is given by (27).

B. Comparison With Replacement First

Suppose that the unit is replaced at time ofoperation, or at working cycles, whicheveroccurs first; and it undergoes minimal repair at each failure be-tween replacements [14]. Then, the expected cost rate is

(33)

Clearly, the same result as that in (14) can also be derived forperiodic RL and RF.Differentiating with respect to , and setting it

equal to zero for a fixed ,

(34)

Denoting the left-hand side of (34) as , see the un-numbered equation at the bottom of the page.Therefore, if , then there exists a finite,

unique which satisfies (34), and the resultingcost rate is

(35)

If , then , and the resulting costrate is given by (26).In addition, decreases with because in-

creases with . Therefore, from (35), the optimalminimizing in (33) is , where is givenby a solution of (30).From the inequality ,

(36)

where

It was proved in ([15], p. 160) that, when ,if the failure rate increases strictly with , then

increases strictly with , and converges to as .Denoting the left-hand side of (36) to beincreases strictly with to

Therefore, if , a unique, minimumwhich satisfies (36) exists, and the re-

sulting cost rate is shown in (37) at the bottom of the page. If, then , and the resulting cost

rate is given by (25).1) Comparison Result 1: Suppose that the failure rate

increases strictly from 0 to .Then, there exist both unique and

which satisfy (28) and (34). Denote the equationat the bottom of the next page, which increases with from

to . Thus, there exists a finite,unique which satisfies .From (34), denote

(38)

(37)



TABLE IIIOPTIMAL AND FOR WHEN , AND

Therefore, the following comparison results can be given.1. If , then , i.e., RL should beadopted.

2. If , then , i.e., RF should beadopted.

3. If , then RL is the same with RF.Let be an optimal replacement time which

minimizes in (25). Then, is a finite, unique solution of(30), and the resulting cost rate is

(39)

It is easy to show that increases with from , anddecreases with to , because

Furthermore, increases strictly with , becausedecreases with , i.e., RL would showmore superior cases thanRF as becomes smaller.In addition, denoting the left-hand side of (30) by ,

which follows that and , i.e., we adoptthe standard periodic replacement rather than RL or RF. Suchcomparative results are similar to those in Section 2.2.1.Suppose that , and .

Table III presents optimal , and whichsatisfy (28), (34), (38), (30), for and , when

, and . This shows that optimal ,and their comparison results have the same properties as thosein Table I.2) Comparison Result 2: It is easy to show from results in

the Appendix that

which increases strictly with , because

Thus, we have

increases strictly with from 0, and

(40)

where is given in the left-hand side of (30).From the above results, there exists a finite, unique

which satisfies , and ,where is given by (30). Therefore, we obtain the followingresults about .



TABLE IVOPTIMAL AND FOR WHEN AND

1. If , then , and .2. If , then , and .Because in (40), we obtain the followingresults about .1. If , then , and .2. If , then , and .From the above discussions, similar comparison results with

those in Section 2.2.2 can be given, according to the relation-ships among , and that follow.1. If , then and , i.e., RL isadopted.

2. If , then and , i.e., RF isadopted.

3. If , then and , i.e., thestandard periodic replacement is adopted.

Table IV presents optimal and which satisfy (31) and(36), and which satisfies , when

, and . This result also shows that optimal, and their comparison results have the same properties

as those in Table II.

IV. CONCLUSION

We have proposed age, and periodic RL models with con-tinuous, and discrete variables. The operating unit is replacedpreventively at time of operation, or at working cycles,whichever occurs last. Such combined policies have been sched-uled to satisfy a strategic plan of replacement and operationfor jobs with working cycles. Expected replacement costrates, and their optimal policies have been discussed analyti-cally, and numerically. Comparisons between optimal , and

have been focused to determine analyticallyin what cases RL would be better than RF. In Sections 2.2.1and 3.2.1, optimal and have been compared by using thecomparative method in [9]. We found that three cases could bedivided to determine whether RL is better or not by comparingthe relative size of or , and .It has also been found that small plays an important role inadopting RL to satisfy the job completion. In Sections 2.2.2 and3.2.2, a new comparative method between and has beenexplored. There also exist three cases to determine whether RLshould be adopted or not; however, we need to compare two pa-rameters and with a given .We showed that RL gives more flexibility than RF to ensure

the job completion for models with and .

We consider a bivariate PR policy where two replacementcandidates are scheduled at times and , then RL, and RFcan be described as replacements are done before failure at

, and at , respectively. Not only that,but the RL method could be jointly used with RF, e.g., we re-place the unit at before time , and atafter has arrived. That is, we can let the unit operate for jobsas long as possible before time , and replace it as early aspossible after time . The RL discussed in this paper wouldhave practical significance when RF is not adequate.Two potential applications of the whichever occurs last ap-

proach have been indicated in maintaining electronic systemsof naval ships under battle and non-battle statuses [9], and inperforming incremental backup schemes when the full backupthreshold is not so high [16]. Now, we give a new applicationof RL in maintaining a Database System by considering thefollowings points. (i) The Database Engine periodically issuescheckpoints on each database to keep data security; however, weneed to guarantee atomicity, consistency, isolation, durability(ACID) properties for any database transaction when it is pro-cessing series of operations [17], [18]. (ii) Processing timefor every transaction depends on the amount of data, and can besupposed as a random variable, so that the strict periodic settingsmay increase the possibility of transaction faults, which wouldleave the database state unchanged [19]. (iii) If the checkpointsare set up by two triggers under the whichever occurs first ap-proach, another problem occurs, where frequent settings wouldincrease the system’s load. (iv) To reduce the fault possibility,and lighten the system load, this paper could provide program-mers with a new thinking of issuing checkpoints, when the ap-proaches of whichever triggering event occurs first and last areused.

APPENDIX

A. Proof A

When , i.e.,, and the failure rate increases strictly with

, we prove that

increases strictly with , and converges to as .



Proof: Letting the first equation at the bottom of the page,it is easy to show that , and see the secondequation at the bottom of the page, because increasesstrictly. Thus, decreases strictly with to 0, and hence,

, which proves that increases strictly with.Next, from the assumption that increases, for any,

On the other hand, for any such that , seethe third equation at the bottom of the page, where we see thefourth equation at the bottom of the page. Thus,

Because and are arbitrary,

which completes the proof that converges toas .

B. Proof B

When , and the failure rate increasesstrictly with , we prove that

increases strictly with , and converges to as .Proof: The proof procedure is similar to the previous. Let-

ting the last equation at the bottom of the page, it is easy toshow that , and see the first equation at thebottom of the next page. Thus, increases strictly with. Furthermore, for any ,



On the other hand, for any such that , seethe second equation at the bottom of the page, where we see thethird equation at the bottom of the page. Thus,

Because and are arbitrary,

which completes the proof.

ACKNOWLEDGMENT

The authors would like to thank Associate Editor Dr. LesleyWalls, managing editor Dr. J. W. Rupe, and two referees fortheir constructive comments and suggestions which improvedand enriched the presentation of the paper.

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Xufeng Zhao received his B.S. degree in information engineering in 2006, andM.S. degree in system engineering in 2009, both from Nanjing University ofTechnology, China; and his Ph.D. degree in 2013 from Aichi Institute of Tech-nology, Japan.He is now a researcher at Aichi Institute of Technology, Japan, and is also

cooperating with Nanjing Tech University, China, and Qatar University, Qatar.He is currently interested in reliability theory and maintenance policies of sto-chastic systems, shock models, and their applications in computer science. Hehas published in peer reviewed journals such as European Journal of Opera-tional Research, IEEE TRANSACTIONS ON RELIABILITY, and Reliability Engi-neering & System Safety. He is the author or coauthor of five book chaptersfrom Springer, Wiley, and World Scientific, and he is now editing two booksfrom Springer.

Toshio Nakagawa received B.S.E. and M.S. degrees from Nagoya Institute ofTechnology in 1965 and 1967, respectively; and a Ph.D. degree from KyotoUniversity in 1977.

He worked as a Research Associate at Syracuse University for two yearsfrom 1972 to 1973. He is now a Honorary Professor with Aichi Institute ofTechnology, Japan. He has published 5 books from Springer, and about 200journal papers. His research interests are in optimization problems in operationsresearch andmanagement science, and analysis for stochastic and computer sys-tems in reliability and maintenance theory.

Ming J. Zuo received M.Sc. and Ph.D. degrees in industrial engineering fromIowa State University, Ames, IA, USA.He is Professor in the Department of Mechanical Engineering, University

of Alberta, Canada. His research interests include system reliability analysis,maintenance planning and optimization, condition monitoring, and prognosisand health management.Prof. Zuo is an Associate Editor of IEEE TRANSACTIONS ON RELIABILITY,

and Department Editor of IIE Transactions. He is a Fellow of the Institute ofIndustrial Engineers, the Engineering Institute of Canada, and International So-ciety of Engineering Asset Management.

optimal replacement last with continuous and discrete policies

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