Failure and Maintenance 2005

European Journal of Operational Research 161 (2005) 183–190

www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

Failure replacement and preventive maintenancespare parts ordering policy

Timothy S. Vaughan *

Department of Management and Marketing, University of Wisconsin-Eau-Claire, Eau Claire, WI 54702-4004, USA

Received 11 September 2002; accepted 2 June 2003

Available online 14 November 2003

Abstract

This paper addresses inventory policy for spare parts, when demand for the spare parts arises due to regularly

scheduled preventive maintenance, as well as random failure of units in service. A stochastic dynamic programming

model is used to characterize an ordering policy which addresses both sources of demand in a unified manner. The

optimal policy has the form ðsðkÞ; SðkÞÞ, where k is the number of periods until the next scheduled preventive main-

tenance operation. The nature of the ðsðkÞ; SðkÞÞ policy is characterized through numeric evaluation. The efficiency of

the optimal policy is evaluated, relative to a simpler policy which addresses the failure replacement and preventive

maintenance demands with separate ordering policies.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Inventory; Failure replacement; Preventive maintenance; Dynamic programming

1. Introduction

The problem of providing an adequate yet effi-

cient supply of spare parts, in support of mainte-

nance and repair of plant and equipment, is an

especially vexing inventory management scenario.

Spare parts for plant and equipment may be veryexpensive, and thus costly to keep in inventory.

Nonetheless, spares must be on hand when needed,

in order to avoid costly plant shutdown or

equipment unavailability. In the face of this com-

bination of high cost and high criticality, random

* Tel.: +1-715-836-4408.

E-mail address: [email protected] (T.S. Vaughan).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.06.026

failure of units in service typically generates a low-

volume, intermittent demand process.

In addition to random failure replacement, de-

mand for spare parts may also arise under a policy

of regularly scheduled shutdown and preventive

maintenance for the larger system in which the

parts are used. In contrast to the low volumefailure replacement demand, preventive mainte-

nance may present a ‘‘lumpy demand’’ scenario in

which a larger number of units are required at a

known point in time.

This paper is directed toward an efficient

ordering policy recognizing both preventive

maintenance requirements, and requirements due

to random failure of units in service. The premiseis that greater efficiency will be realized by

ed.

mail to: [email protected]

184 T.S. Vaughan / European Journal of Operational Research 161 (2005) 183–190

addressing these two sources of demand in a uni-fied manner, relative to the use of separate order-

ing policies for the two demand streams.

This scenario is closely related to a system of

‘‘age replacement’’ and associated inventory

ordering policies. Under age replacement, a ran-

domly failing item is replaced upon reaching some

specified age T , or upon failure, whichever occurs

first. Barlow and Proschan (1965) originally stud-ied the problem in terms of finding the best

replacement age T . Subsequently, Kaio and Osaki

(1978, 1981) examined the situation where spares�inventory ordering policy and replacement age are

jointly optimized. Park and Park (1986) extended

this model to the case of random lead time. Kabir

and Al-Olayan (1994, 1996) further extended the

analysis to the case of multiple units in service, andthe possibility of holding more than a single unit of

inventory.

Whereas the age replacement literature assumes

units are replaced upon reaching a certain number

of periods in service, the current paper is addressed

toward a scenario where the system in question is

shut down for preventive maintenance or overhaul

at regularly scheduled intervals. The timing of thepreventive replacement is not necessarily dictated

by the age of any specific unit, but rather, on the

requirements of the larger system in which the unit

is used. As such, we treat the time between pre-

ventive maintenance operations as a given

parameter, the determination of which is beyond

the scope of the present investigation.

We assume a system having n identical parts inservice (see Walker, 1997). Random failure of

units in service generates intermittent, single-unit

demands between preventive maintenance periods.

At preventive maintenance, all units in service are

inspected, and a decision is made as to which ones

should be proactively replaced. Providing suffi-

cient inventory in support of preventive mainte-

nance is of paramount importance, so as not todelay a schedule of inter-related preventive main-

tenance activities.

We should note that the present investigation is

directed toward the case of spare parts in support

of plant maintenance and equipment, rather than

spare parts in support of equipment at a number

of geographically dispersed customer sites, or in

support of product service or warranty repair. Thispaper is also directed toward the case where the

parts in question are not repairable. Kennedy et al.

(2002) provide a recent survey of literature on

spare parts inventories.

The remainder of the paper is organized as fol-

lows. In Section 2 a stochastic dynamic program-

ming model is formulated, implicitly defining the

optimal policy for the scenario at hand. The char-acteristics of the optimal policy, examined through

numeric evaluation of the dynamic programming

model, are discussed in Section 3. Section 4 pre-

sents a comparison between the cost of the optimal

policy vs. a that of a simpler policy which treats the

two sources of demand separately. Summary and

possible extensions are discussed in Section 5.

2. Model development

Assume a system containing n identical com-

ponents. Each component fails according to a

constant failure rate k, such that the time to failurefor each component is exponentially distributed

with mean 1=k periods, independent of theremaining n� 1 units. The total number of part

failures during any given period then has Poisson

distribution with mean kn. Failed units must be

replaced from existing inventory, or expedited.

Scheduled preventive maintenance occurs every

T periods, during which the n units in service are

inspected, and some or all are replaced in a pro-

active manner. We assume the number of unitsreplaced is binomial with parameters n and p. Aswith the failure replacement, demands due to

preventive maintenance must be satisfied through

existing inventory or expedited.

We desire an optimal ordering policy which

accounts for both sources of demand as the pre-

ventive maintenance period approaches. Toward

this end, we will develop a stochastic dynamicprogramming characterization of the optimal

ordering policy.

For the derivations to follow it will be conve-

nient to adopt a common notation to represent

both the random failures and preventive mainte-

nance demands. Define DðtÞ as the number of units

required during period t. Without loss of general-

T.S. Vaughan / European Journal of Operational Research 161 (2005) 183–190 185

ity, assume period 0 is a preventive maintenanceperiod. Following dynamic programming conven-

tion, we will number the periods backwards in

time, thus period 1 is the period immediately prior

to the scheduled preventive maintenance in period

0, etc. Following this approach, periods 0; T ;2T ; . . . represent successively earlier preventive

maintenance periods, generating binomial ðn; pÞdemands on the inventory system. Any period t forwhich t mod T 6¼ 0 is an operational period, during

which the random failure process generates Pois-

son-distributed demand. In general, then, we have:

PtðdÞ ¼ P ðDðtÞ ¼ dÞ ¼ e�knðknÞd=d!;d ¼ 0; 1; 2; . . . ; t ¼ 1; 2; 3; . . . ; tmodT 6¼ 0;

ð1Þ

PtðdÞ ¼ P ðDðtÞ ¼ dÞ ¼n

d

� �pdð1� pÞðn�dÞ

;

d ¼ 0; 1; 2; . . . ; n; t ¼ 0; T ; 2T ; . . . ð2Þ

Let CH equal the carrying cost per unit, per

period, and CO the fixed cost of placing an order for

any quantity greater than zero units. Let B1 rep-

resent the penalty cost associated with inventory

shortage in the case of random part failure between

preventive maintenance periods. Similarly, let B2

represent the penalty cost associated with inven-

tory shortage during the preventive maintenance

operation. In both cases, the penalty includes the

cost of emergency ordering and expediting, as well

as a penalty associated with process downtime or

costly delay of the preventive maintenance project.

As above, it will be convenient to adopt a common

notation for the shortage penalty:

BðtÞ ¼ B1; t ¼ 1; 2; 3; . . . ; tmodT 6¼ 0; ð3Þ

BðtÞ ¼ B2; t ¼ 0; T ; 2T ; . . . ð4ÞLet xðtÞ represent the inventory on hand at the

beginning of period t. Let QðtÞ represent the

number of units ordered at the beginning of period

t, in response to observing xðtÞ units on hand.

These units will not be received into inventory

until the end of period t, i.e., after a one-period

lead time. The length of an operational period is

then implicitly assumed to be equal to the order

lead time. The preventive maintenance period isnot necessarily of equal length.

Inventory shortage results if period t demandDðtÞ is greater than xðtÞ. The shortage in any given

period is then ½DðtÞ � xðtÞ�þ, where ½y�þ ¼maxðy; 0Þ. Any shortages that occur during periodt are expedited, and any outstanding order quan-

tity QðtÞ is reduced by the number of units expe-

dited. Ending inventory is then ½xðtÞ � DðtÞþQðtÞ�þ. Under the expediting assumption, period

t � 1 begins with n units in operation.

Following the dynamic programming approach

of Bellman (1957), we will define the optimal pol-

icy using a recursive equation. Define TCtðxðtÞÞ asthe minimum total expected cost of the inventory

system over periods t; t � 1; t � 2; . . . ; 0, if xðtÞunits are on hand at the beginning of period t. Wecan recursively express TCtðxðtÞÞ as follows:TCtðxðtÞÞ

¼ minQðtÞ

COdðQðtÞÞ(

þX1d¼0

½BðtÞ½d � xðtÞ�þ

þ CH½xðtÞ � d þ QðtÞ�þ

þ TCt�1ð½xðtÞ � d þ QðtÞ�þÞ�PtðdÞ);

t ¼ 1; 2; 3; . . . ð5Þ

The function dðqÞ ¼ 0 if q ¼ 0, while dðqÞ ¼ 1 if

q > 0. Thus, the term COdðQðtÞÞ applies the

ordering cost CO if an order is placed for any

quantity greater than zero. The summation term

represents an expected cost over all possible DðtÞ,using the appropriate PtðdÞ ¼ P ðDðtÞ ¼ dÞ for theperiod at hand. Here the appropriate shortage

penalty is applied to the shortage quantity½d � xðtÞ�þ, carrying costs are levied on ending

inventory ½xðtÞ � d þ QðtÞ�þ, and period t � 1 is

begun with ½xðtÞ � d þ QðtÞ�þ units on hand.

The boundary condition can be written as

TC0ðxð0ÞÞ ¼Xn

d¼xð0Þþ1Bð0Þðd � xð0ÞÞP0ðdÞ; ð6Þ

i.e., the expected penalty cost if the preventive

maintenance operation scheduled for period 0 is

begun with xð0Þ units on hand.

0

1

2

3

4

5

6

1112131415161718191k = # of Periods Until Preventive Maintenance

s(k)S(k)

Fig. 1. Optimal ðsðktÞ; SðktÞÞ ordering policy n ¼ 5, k ¼ 0:001,

CO ¼ 25, CH ¼ 0:5, B1 ¼ 1000.


3. Policy characterization

Analytical evaluation of the optimal policy is

difficult, given the non-stationary nature of the

problem at hand. We will therefore examine the

results of numerical characterization of the opti-

mal policy, derived through simple backward

evaluation of the recursive relationship given in (5)and (6). The optimal policy was evaluated over all

32 combinations of the following parameter levels:

n ¼ 5, 20; k ¼ 0:001, 0.01; CO ¼ 25, 50; CH ¼ 0:5,2; B1 ¼ 300, 1000. These parameters are varied

relative to a constant B2 ¼ 1000. The preventive

maintenance requirements have binomial ðn; pÞdistribution, with p ¼ 0:7. While not required by

the model, the motivation for B2 PB1 applied hereis that shortage during preventive maintenance

upsets the entire maintenance schedule, possibly

idling a number of related preventive maintenance

activities.

At each t, TCðxðtÞÞ was evaluated over xðtÞ ¼0; 1; 2; . . . ; 2n. The time between preventive main-

tenance periods is T ¼ 100 in all cases. The

recursive evaluation was extended backwardsthrough t ¼ 499, thus the evaluation represents

five successive preventive maintenance intervals.

In every case, the optimal policies covering

t ¼ ½299–200�, [399–300], and [499–400] were

identical. Thus, once the analysis extends far en-

ough back to escape end-of-horizon effects, the

optimal policy becomes ‘‘cyclical-stationary’’ in

the sense that the policy depends on t only throughtmodT , i.e. the number of periods remaining untilthe next preventive maintenance period. The pol-

icy covering periods (99–0), and occasionally

periods (199–100), displayed minimal differences

from the ‘‘cyclical-stationary’’ policy. These end-

of-horizon effects are primarily characterized by

some nervousness in the SðktÞ parameter discussedbelow, in the periods immediately prior to pre-ventive maintenance.

The ‘‘cyclical-stationary’’ optimal policy has the

form ðsðktÞ; SðktÞÞ, where kt ¼ tmodT is the num-

ber of periods at time t until the next scheduled

preventive maintenance. Thus QðtÞ ¼ 0 if xðtÞ >sðktÞ, while QðtÞ ¼ SðktÞ � xðtÞ if xðtÞ6 sðktÞ.

The optimal policy is obviously non-stationary

in the narrower sense that the ordering parameters

depend on kt. As the preventive maintenance de-mand becomes sufficiently distant, however, the

optimal policy becomes concerned only with

the failure replacement demand process. The

ðsðktÞ; SðktÞÞ policy thus converges to a stationary

ðs; SÞ policy as kt becomes large. For clarity in

subsequent discussion, we will refer to this policy

as ðsFR; SFRÞ, where the FR subscript refers to

failure replacement. As the stationary ðs; SÞ policyhas been widely studied and is well understood, we

will focus here on the deviation from the ðsFR; SFRÞpolicy as kt approaches 0.

Fig. 1 (n ¼ 5, k ¼ 0:001, CO ¼ 25, CH ¼ 0:5,B1 ¼ 1000) displays one of the most predominant

and intuitive behaviors of the optimal ðsðktÞ; SðktÞÞpolicy. (The shaded regions in Figs. 1–5 are

bounded above by SðktÞ and below by sðktÞ.) Inthis case sðktÞ ¼ 0 until kt ¼ 1, at which point sð1Þjumps to 4. This essentially guarantees that an

order will be placed in period 1 to cover the pre-

ventive maintenance demand, if such an order has

not been placed already. This increase in sðktÞnever occurs prior to period 1 in any of the cases

evaluated. Intuitively, the model suggests there is

no point in forcing an order to cover preventivemaintenance demand any sooner than necessary.

SðktÞ, on the other hand, increases from SFR ¼ 1

to SðktÞ ¼ 5 at kt ¼ 13. The intuition here is that if

an order is to be placed (as governed by the sðktÞparameter discussed above) the SðktÞ parameter

should at some point recognize the preventive

0

1

2

3

4

5

6

1112131415161718191

k = # of Periods Until Preventive Maintenances(k)S(k)


CO ¼ 50, CH ¼ 0:5, B1 ¼ 1000.

0

1

2

3

4

5

6

7


S(k) s(k)


CO ¼ 25, CH ¼ 2, B1 ¼ 1000.


maintenance demand. In Fig. 1, the order-up-to

level accounts for preventive maintenance demand

as early as 13 periods prior to the preventivemaintenance itself.

Fig. 2 (n ¼ 5, k ¼ 0:001, CO ¼ 50, CH ¼ 0:5,B ¼ 1000) provides a useful contrast to Fig. 1, in

that CO has increased from 25 to 50 with all other

parameters held constant. The distinguishing

characteristic, again common to all scenarios, is

that the order-up-to level begins to account for

preventive maintenance demand at larger kt as CO

increases.

Fig. 3 (n ¼ 5, k ¼ 0:01, CO ¼ 25, CH ¼ 2,

B ¼ 1000) demonstrates that when sFR > 0, the

optimal ðsðktÞ; SðktÞÞ policy may actually implydecreasing sðktÞ for some number of periods priorto the eventual spike at kt ¼ 1. This has the effect

of forgoing an order that would have been placed

under a larger sðktÞ. In Fig. 3, as the inevitable

order to cover preventive maintenance demand

draws near, the policy responds to the probability

that a current inventory xðtÞ ¼ 1 will suffice until

receipt of that order. In short, the number ofperiods of exposure to the lower inventory levels

does not merit the cost of a replenishment

order, given that a subsequent replenishment order

will soon follow. At larger kt the ‘‘window of

exposure’’ to stock-out is large enough that an

order is justified when xðtÞ ¼ 1.

Fig. 4 (n ¼ 5, k ¼ 0:01, CO ¼ 50, CH ¼ 0:5,B ¼ 1000) displays another common characteristicof the optimal policy, in that SðktÞ decreases fromSFR for some interval prior to the eventual in-

crease. As in Fig. 3, the policy is responding to the

fact that a subsequent order must surely be placed

no later than kt ¼ 1. As such, the order-up-to level

need not cover the same number of periods� de-mand than it would if there were no impending

order to cover preventive maintenance require-ments. The SðktÞ parameter is essentially attempt-

ing to avoid carrying large inventory into a period

that will experience an order driven by preventive

maintenance requirements. Interestingly, this

behavior reflects logic similar to that underlying

the Wagner–Whitin algorithm for deterministic

process lot-sizing (Wagner and Whitin, 1958).

Finally, Fig. 5 (n ¼ 20, k ¼ 0:01, CO ¼ 25,CH ¼ 0:5, B ¼ 1000) displays a policy in which all

these effects are present simultaneously. The gen-

eral impression is that of a step-wise ‘‘draw down’’

of the ordering parameters, prior to the eventual

increase to cover preventive maintenance demand.

First intuition may suggest sðktÞ and Sðkt) willboth steadily increase as kt decreases, representinga gradual build up of inventory in anticipation ofthe upcoming preventive maintenance demand.

The basic counter-argument to this logic is that

that timing of the preventive maintenance demand

is known. As such, there is no need for a larger sparameter until the latest possible moment. In-

deed, the s parameter may actually decrease for

reasons discussed within the context of Fig. 3. The

0

1

2

3

4

5

6

7

8

9

1112131415161718191

k = # of Periods Until Preventive Maintenances(k)S(k)


CO ¼ 50, CH ¼ 0:5, B1 ¼ 1000.

0

5

10

15

20

25


s(k)S(k)


CO ¼ 25, CH ¼ 0:5, B1 ¼ 1000.


S parameter may decrease as well, for reasons

discussed within the context of Fig. 4. Eventually,

of course, SðktÞ must increase to cover the pre-

ventive maintenance demand, while sðktÞ increasesat kt ¼ 1 to force such an order if one has not yet

occurred ‘‘naturally’’.

4. Cost comparison

To characterize the cost savings afforded by the

optimal ðsðktÞ; SðktÞÞ policy, we will compare the

cost to that of a simpler policy which addresses

the failure replacement and preventive mainte-

nance demands separately. Under this policy, theðsFR; SFRÞ policy is used to satisfy failure replace-

ment demand through all periods t in which kt 6¼ 1.

An optimal single period ordering policy is used at

kt ¼ 1 to provide for the preventive maintenance

demand. This policy is given by the optimal ðs; SÞat t ¼ 1 of the backward evaluation of (5) and (6),

and will be denoted as ðsPM; SPMÞ. In general, then,we have:

ðs0ðktÞ; S0ðktÞÞ ¼ ðsFR; SFRÞ; kt 6¼ 1; ð7Þ

ðs0ðktÞ; S0ðktÞÞ ¼ ðsPM; SPMÞ; kt ¼ 1: ð8Þ

Let TC0tðxðtÞÞ represent the expected cost of fol-

lowing this policy through periods t; t � 1;t � 2; . . . ; 0. We then have:

TC00ðxð0ÞÞ ¼ TC0ðxð0ÞÞ ð9Þ

and

TC01ðxð1ÞÞ ¼ TC1ðxð1ÞÞ; ð10Þ

where TC0ðxð0ÞÞ and TC1ðxð1ÞÞ are given by (6) and(5), respectively.

Let Q0ðtÞ ¼ S0ðktÞ � xðtÞ for xðtÞ6 s0ðktÞ, and

Q0ðtÞ ¼ 0 for xðtÞ > s0ðktÞ. For tP 2 the cost of the

simpler policy is then given by

TC0ðxðtÞÞ

¼ COdðQ0ðtÞÞ þX1d¼0

½BðtÞ½d � xðtÞ�þ

þ CH½xðtÞ � d þ Q0ðtÞ�þ

þ TC0t�1ð½xðtÞ � d þ Q0ðtÞ�þÞ�PtðdÞ: ð11Þ

As kt ! 1 the ratio TC0tðxðtÞÞ=TCtðxðtÞÞ ! 1, as

the portion of total cost affected by ðsðktÞ;SðktÞÞ 6¼ ðsFR; SFRÞ is diminished. Similarly, any

long-run comparison of the two policies is con-

founded by the number of periods comprising the

preventive maintenance interval. To eliminate this

effect from the comparison, we will evaluate the

relative performance of the two policies condi-tionally, under various ðt; xðtÞÞ in the interval

where ðsðktÞ; SðktÞÞ 6¼ ðsFR; SFRÞ.Table 1 displays the maximum value over all xðtÞ

of the ratio TC0tðxðtÞÞ=TCtðxðtÞÞ for the 32 scenarios

evaluated, at t ¼ 2; 3, and 10. As can be seen, the

optimal policy may generate considerable cost

Table 1

Cost comparison of optimal ðsðktÞ; SðktÞÞ policyScenario n k CO CH B1 Max. TC0

10ðxð10ÞÞ=TC10ðxð10ÞÞ

Max. TC03ðxð3ÞÞ

=TC3ðxð3ÞÞMax. TC0

2ðxð2ÞÞ=TC2ðxð2ÞÞ

1 5 0.001 25 0.5 300 1.3971 1.7121 1.7745

2 5 0.001 25 0.5 1000 1.1231 1.5343 1.6276

3 5 0.001 25 2 300 1.0000 1.0000 1.0000

4 5 0.001 25 2 1000 1.0086 1.3773 1.4806

5 5 0.001 50 0.5 300 1.6334 1.8394 1.8752

6 5 0.001 50 0.5 1000 1.3847 1.7127 1.7793

7 5 0.001 50 2 300 1.0000 1.0000 1.0000

8 5 0.001 50 2 1000 1.2277 1.5842 1.6639

9 5 0.01 25 0.5 300 1.1149 1.4233 1.4956

10 5 0.01 25 0.5 1000 1.1556 1.5855 1.7036

11 5 0.01 25 2 300 1.0153 1.1104 1.2594

12 5 0.01 25 2 1000 1.1732 1.4618 1.5365

13 5 0.01 50 0.5 300 1.3538 1.6264 1.6870

14 5 0.01 50 0.5 1000 1.415 1.7671 1.8394

15 5 0.01 50 2 300 1.0469 1.3282 1.4661

16 5 0.01 50 2 1000 1.0583 1.2721 1.3507

17 20 0.001 25 0.5 300 1.0205 1.2827 1.4253

18 20 0.001 25 0.5 1000 1.0084 1.0974 1.2461

19 20 0.001 25 2 300 1.0165 1.2169 1.2784

20 20 0.001 25 2 1000 1.0016 1.0021 1.0727

21 20 0.001 50 0.5 300 1.0574 1.5126 1.6248

22 20 0.001 50 0.5 1000 1.0280 1.3235 1.4555

23 20 0.001 50 2 300 1.1555 1.4047 1.4572

24 20 0.001 50 2 1000 1.0080 1.1081 1.2525

25 20 0.01 25 0.5 300 1.0437 1.1554 1.3380

26 20 0.01 25 0.5 1000 1.0650 1.2681 1.4441

27 20 0.01 25 2 300 1.0208 1.0324 1.1769

28 20 0.01 25 2 1000 1.0303 1.1581 1.2782

29 20 0.01 50 0.5 300 1.1207 1.4210 1.5672

30 20 0.01 50 0.5 1000 1.1469 1.5131 1.6501

31 20 0.01 50 2 300 1.0364 1.1987 1.3595

32 20 0.01 50 2 1000 1.0197 1.0300 1.1321


savings, in the event that the inventory trajectoryprovides an ordering opportunity shortly prior to

period 1. (The minimal end-of-horizon effects in-

cluded in this comparison are unavoidable, given

the structural difference between the optimal vs.

sub-optimal policies, coupled with our interest in a

conditional rather than a long-run cost compari-

son. These end-of-horizon effects tend to work in

favor of the myopic sub-optimal policy.)In general, the cost savings tend to be greatest

at small CH and large CO. Given the source of

the optimal policy�s advantage, this is under-

standable. The advantage of the optimalðsðktÞ; SðktÞÞ policy lies in properly recognizing

the preventive maintenance demand, and includ-

ing those units in an order placed within some

proper interval prior to the preventive mainte-

nance. The savings associated with this ability to

economize on ordering cost are naturally greatest

when CO is large. Conversely, since the optimal

policy implies ordering the preventive mainte-nance units earlier than under the simpler policy,

the advantage of doing so is greatest when CH is

small.


Finally, note that scenarios 3 and 7 in Table 1represent a special case (small kn, large CH, small

B1) in which no inventory is carried for the pur-

pose of failure replacement under the ðsFR; SFRÞpolicy. In these cases, the optimal ðsðktÞ; SðktÞÞpolicy is identical to the simpler policy.

5. Conclusion

This paper has presented a dynamic program-

ming characterization for a spare parts ordering

policy. The situation investigated is that in which

demand for spare parts arises from two sources,

i.e. random failure of units in service, and bulk

replacement at regularly scheduled preventive

maintenance intervals. The optimal policy isshown to have the form ðsðktÞ; SðktÞÞ, where kt isthe number of periods until the next preventive

maintenance operation.

The optimal ðsðktÞ; SðktÞÞ policy is generally

characterized by SðktÞ increasing to cover the

preventive maintenance demand some number of

periods prior to the preventive maintenance

period. This increase in SðktÞ occurs farther inadvance of the preventive maintenance period, the

larger is the fixed cost per order, and the smaller is

the carrying cost per unit. The sðktÞ parameter

increases in the period prior to the preventive

maintenance operation, to a level which forces an

order to cover the preventive maintenance de-

mand.

Prior to their respective increases, sðktÞ and/orSðktÞ may decrease from their values under the

stationary failure replacement policy. This draw-

down of the two ordering parameters is in recog-

nition of the impending order that must be placed

prior to the preventive maintenance operation.

The appropriate magnitude and duration of the

alternative ðsðktÞ; SðktÞÞ depend highly on the de-

mand and cost parameters. Nonetheless, a simplebackward evaluation of the recursive relationship

presented in this paper provides an optimal policy

for a given parameter set.

Extensions of this paper include the case of lead

time greater than one period, and more impor-

tantly, the case of a random lead time. Intuitively,

this would cause the increase in sðktÞ to occur prior

to kt ¼ 1, and may impact the policy in other waysas well. Another possible extension would be to

assume an increasing failure rate for units in ser-

vice, rather than the constant failure rate implied

by the exponential distribution. This could have

the effect of prescribing lower inventory levels

immediately after preventive maintenance, when

presumably a larger number of new units are in

operation. Unfortunately, these extensions willprobably render an intractable model under the

dynamic programming approach, and will require

some other means of evaluation.

Acknowledgements

The author would like to thank the two anon-ymous reviewers for their constructive comments,

which helped to greatly improve the original

manuscript.

References

Barlow, R.E., Proschan, F., 1965. Mathematical Theory of

Reliability. Wiley, New York.

Bellman, R.E., 1957. Dynamic Programming. Princeton Uni-

versity Press, Princeton, NJ.

Kabir, A.B.M.Z., Al-Olayan, A.S., 1994. Joint optimization of

age replacement and continuous review spare provisioning

policy. International Journal of Operations and Production

Management 14 (7), 53–69.

Kabir, A.B.M.Z., Al-Olayan, A.S., 1996. A stocking policy for

spare part provisioning under age based preventive replace-

ment. European Journal of Operational Research 90 (1),

171–181.

Kaio, N., Osaki, S., 1978. Optimum ordering policies with lead

time for an operating unit in preventive maintenance. IEEE

Transactions on Reliability 27, 270–271.

Kaio, N., Osaki, S., 1981. Optimum planned maintenance

policies with lead time. IEEE Transactions on Reliability 30,

79.

Kennedy, W.J., Patterson, J.W., Fredendall, L.D., 2002. An

overview of recent literature on spare parts inventories.

International Journal of Production Economics 76, 201–

215.

Park, Y.T., Park, K.S., 1986. Generalized spare ordering

policies with random lead time. European Journal of

Operational Research 23 (3), 320–330.

Wagner, H.M., Whitin, T.M., 1958. Dynamic version of the

economic lot size model. Management Science 5, 89–96.

Walker, J., 1997. Base stock level determination for ‘‘insurance

type’’ spares. International Journal of Quality and Reliabil-

ity Management 14 (6), 569–574.

Failure and Maintenance 2005

Documents

Transcript of Failure and Maintenance 2005