400 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-28, NO. 5, DECEMBER 1979

A Modified Block Replacement PolicyUsing Less Reliable Items

Toshiro Tango q CAICB.The Tokyo Metropolitan Institute of Medical 6 (CA - CB)lCp.Science, Tokyo pi s-expected life of the item i, i = A, B.

T, T* planned replacement interval and its oQptimal solution.Key Words-Block replacement policy, Weibull distribution v, v* replacement-by-item-B interval and its optimal solu-

tion, 0 6 v < T.Reader Aids- B(T) [CAMA (T) + Cp I/T; s-expected unit cost under BRP.

Purpose: Widen state of the art T* optimum T under BRP.Special math needed: Probability, renewal theory bResults useful to: Reliability theoreticians L(T, v) mA(T-v)(CBQ(v)-CA).

C(T, v) s-expected unit cost under the modified BRP.Abstract-A modified block replacement policy is examined with

the purpose of excluding the wastefulness caused by the property that Other standard notation is given in "Information for Readersalmost new items might be sometimes replaced at planned replacement & Authors" at the rear of each issue.times under the ordinary block replacement policy. This modifiedpolicy can be stated as follows: 1) Operating items are replaced by newitems at times kT (k = 1, 2, ...), 2) If operating items fail in [(k -I)T, 3. MODEL AND ASSUMPTIONSkT - v), they are replaced by new items, and in [kT - v, kl), they arereplaced by less reliable items than new items. Numerical comparisons While under the ordinary BRP, operating items are replacedbetween the proposed policy and the ordinary block replacement instantly at failure and at fixed points of time kT (k = 1, 2,policy are carried out for Weibull distribution. ... ated as follows:...), EBRP [4] was stated as follows:

1. INTRODUCTION i) Operating items are replaced by items A at times kT,ii) If operating items fail in [(k -1)T, kT - v), they are re-

One of the most elementary and important replacement placed by items A, and if in [kT- v, kT), they are replaced bypolicies is the well-known block replacement policy (BRP). used items A.Under BRP, operating items are replaced at failure and atplanned replacement times by new items. Therefore, BRP has The used item was defined as the one that has been exchangedbeen recognized as the replacement policy of practical value, earlier after reaching the age T. Thus the renewal function ofin that it is unecessary to keep detailed history on times of the used item generally becomes very complicated function offailures and age of currently operating item. It has, however, T, which often makes it difficult to obtain the optimal solu-been modified and extended in several directions [1-4] tion of EBRP even by computer. In order to overcome thisbecause of its central drawback that almost new items might kind of difficulty and also generalize EBRP, the followingbe removed at planned replacement times. strategy ii' was substituted for the above ii, i.e.,

The purpose of this paper is to propose another modifica- ii') If operating items fail in [(k -1)T, kT - v), they aretion of BRP so as to generalize the policy presented in [4] replaced by items A, and if in [kT - v, k7), they are replaced(called EBRP) and to remove the practical infeasibility oc- by items B, 0 < v S T.curred in obtaining the optimal solution for the general typeof failure distributions in those modified policies [1-4]. In this modified model, items B should be cheaper and thusNumerical cost comparisons between BRP and this modified less durable than items A, thus we assume throughout thisBRP for Weibull distribution are made to examine the im- paper that:provement rate. a) rA (t) t t, rB(t) t t, i.e., IFR,

2. NOTATIONb) rA (t)

6rB(t) for all t > 0,

2. NOTATION ~~~C)CA -CP.<CB<CA.Fi(t), ri(t) Cdf and failure rate of life time of the item i, i =

A, B. 4 U LTm,{t), Ms(t) renewal density and renewal function of the item 4 EUT

t, i =A, B. The s-expected cost rate under the modified BRP is:mAB(t), MAB(t) renewal density and renewal function for a

delayed renewal process with initial Cdf FA (t) and C(T, v) = B(T) + (liT) fV L(T, x)dx, for 0 S v . T,remaining FB(t).0

Q(t) 1 +MB(t) -MAB(t). which is another derivation different from the case of EBRPC1 cost of a failure replacement by the item i, i = A, B. [4] .Cp1 cost of a planned replacement by the item A. The following lemma and theorem are obtained.

0018-9529/79/1200-400$00.7501979 IEEE

A MODIFIED BLOCK REPLACEMENT POLICY USING LESS RELIABLE ITEMS 401

Lemma 1. the same shape parameter ,B = 1.6 but different scale param-eters XA = 1.0 and XB = 1.4. From assumption c, we have

b) Q(0) A1 <CA(ICB ( O <v <1, and if CB- CA then 6 - 0, while if CB -+ CA -

c) Q(v) > 1, for all v >0. CB then 6 -+ 1. Therefore 6 can be a measure of the relatived) Q(v) = 1, foB + o(1). location of CB to other costs. The value of q, on the other

hand, means the significance of "failure" in service.

Theorem 1. ACKNOWLEDGMENT

a) A sufficient condition for v* = T is Q(v) < CA /CB for all The author is extremely thankful to the referees for theirv >0. valuable comments and suggestions.

b) If Q(v) t v and MA/IMB > CAICB, then a unique v* < Texists References

[11 M. Berg, B. Epstein, "A modified block replacement policy,"For more details, see the Supplement [51 . Naval Res. Logist. Quart., vol 23, 1976, pp 15-24.

[2] B. R. Bhat, "Used item replacement policy," J. Appl. Prob.,5. NUMERICAL EXAMPLE vol 6, 1969, pp 309-318.

[3] D.R. Cox, Renewal Theory, Methuen, London, 1963.Table 1 presents an example of cost comparisons using [4] T. Tango, "Extended block replacement policy with used items,"

Weibull distributions weif(Xt, ,B), where FA(t) and FB(t) have J. Appl. Prob., vol 15, 1978, pp 560-572.[5] Supplement: NAPS document No. 03471-C; 5 pages in this

Table 1 Supplement. For current ordering information, see "Informa-

Costacomparionsbetween (Tj*) and C( v*)fortion for Readers & Authors" in a current issue. Order NAPSCost comparisons between B(Tb) and C(T*, v*) for document No. 03471, 18 pages. ASIS-NAPS; Microfiche Publi-Weibull distributions where fixed parameters are cations; P.O. Box 3513, Grand Central Station; New York, NY

j = 1.6 XA = 1.0 and XB = 1.4. 10017 USA.

q 6 T* v* T* C/B(%) BIOGRAPHYToshiro Tango; Dept. of Clinical Epidemiology; The Tokyo Metro-

7.5 0.1 0.45 0.08 0.45 0.15 politan Institute of Medical Science; 3-18-22, Honkomagome, Bunkyo-7.5 0.5 0.45 0.25 0.45 2.02 ku, Tokyo 113 JAPAN.7.5 1.0 0.45 0.45 0.45 6.14 Toshiro Tango was born in Japan on March 4, 1950. He received

his BS degree in Industrial Engineering from the University of Electro-

8.0 0.1 0.43 0.08 0.43 0.14 Communications, Tokyo, and MS degree in Industrial Engineering from8.00.10.43 0.08 0.43 0.14 ithe Tokyo Institute of Technology, Tokyo, in 1973, and 1975. His

8.0 0.5 0.43 0.24 0.43 1.90 interests include reliability, probability, statistics and their applications8.0 1.0 0.43 0.42 0.42 5.73 to clinical medicine. He has worked to the present in the area of bio-

statistics at the Tokyo Metropolitan Institute of Medical Science. He is

8.5 0.1 0.41 0.08 0.41 0.14 currently engaged in developing statistical package for medical science.8.5 0.5 0.41 0.23 0.41 1.79 Manuscript TR78-123 received 1978 October 3; revised 1979 April 9.8.5 1.0 0.41 0.40 0.41 5.35

Manuscripts Received For information, write to the author at the address listed; do NOT write to the Editor.

"Methods for the probabilistic analysis of noncoherent "Predicting the service distribution in manpower sys-fault trees", Dr. G. Apostolakis; 5532 Boelter Hall; tems", P.-C.G. Vassiliou; Statistics Unit, Dept. of Math-UCLA; Los Angeles, CA 90024 USA. ematics; University of Ioannina; Ioannina, GREECE.

"Generation of fault trees for noncoherent systems", "Decomposition method for computing the reliability ofMaylin H. Dittmore (82211); c/o Technical Information complex input-output systems by network expressions",Dept.; Lawrence Livermore Laboratory; Livermore, CA Hayao Nakazawa; 4 Gooto 432 Shin-Sakuragaoka-Dan-94550 USA. chi; Imaicho Hodogayaku, Yokohama City 240 JAPAN.

"Stochastic behaviour of a 2-tunit standby system with "Determination of optimal sample size for a non-fault analysis", Roshan Lal; Deptt of Statistics; Institute replacement life test", Doris Grosh; Dept. of Industrialof Social Sciences; Agra University; Agra 282 005 Engineering; Durland Hall, Kansas State University;INDIA. Manhattan, KS 66506 USA.

"State-transition Monte Carlo method for evaluating "Reliability analysis of a fluidized-bed boiler for a coal-unreliability of large, repairable systems", Mlr. Hiromit- fueled power plant", Thomas G. Woo; 5 Marlyn Rd.;su Kumamoto; Dept. of Precision Mech.; Faculty of Medfield, MA 02052 USA.Engrg.; Kyoto University; Kyoto 606 JAPAN.

"The 2-unit redundant system, and the method of"A new life distribution", Dr. Balbir Dhillon; Business stages", Douglas P. Wiens; Dept. of Mathematics &cAdministration, TG Div.; Ontario Hydro; 700 University Statistics; University of Calgary; 2920 24-th Ave., NW;Avenue; Toronto, Ontario M5G 1X6 CANADA. Calgary, Alberta T2N 1N4 CANADA.