GROUP REPLACEMENT POLICY FOR A MAINTAINED COHERENT SYSTEM

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1 Jurnal f the Operatins Research Sciety f Japan Vl. 25, N. 3, September The Operatins Research Sciety f Japan GROUP REPLACEMENT POLICY FOR A MAINTAINED COHERENT SYSTEM Mamru Ohashi Anan Technical Cllege (Received June 8,98; Final May 20,982) Abstract In this paper we cnsider a cherent system cnsisting f n cmpnents. Ea..:h cmpnent is repaired upn failure. This maintained cherent system is mnitred cntinuusly, and based upn the results f mnitring a decisin must be made as t whether r nt t replace the system. We shw that the ptimal grup replacement plicy has a mntne prperty withut regard t the values f failure and repair rates. Further we discuss sufficient cnditins fr the replacement f the maintained cherent system.. Intrductin This paper deals with a cherent system cnsisting f n cmpnents. Each cmpnent is subject t randm failure. Upn failure the cmpnent is repaired and recvers its functining perfectly. The maintained cherent system has been investigated by Barlw and Prschan [], Chiang and Niu [2], and Rss [5]. In particular, Rss prved that the distributin f the time t first system failure has NBU (i.e., new better than used) prperty when all cmpnents are initially up at time zer, and have expnential uptime distributins with parameter Ai fr i=,2,...,n and dwntime distributins with parameter ~i fr i=,2,..,n. Thus effrts t replace the maintained cherent system befre system failure may be advantageus. On the ther hand, when cmpared with individual repair upn failed cmpnents, the grup replacement may cause t thrwaway sme cmpnents which are in gd perating cnditin. Hwever, we can expect t btain a large discunt n the purchase price and ther ecnmic attendant t large-scale undertakings. In this paper we cnsider a replacement prblem fr a cherent system with repairable cmpnents. The abve system is mdeled by a cntinuus time Markv decisin prcess. The system is mnitred cntinuusly, and based upn the 228

2 Grup Replacement Plicy 229 results f mnitring a decisin must be made as t whether r nt t replace the maintained cherent system. The bj,~ctive f this paper is t study the structure f the ptimal grup replacement plicy minimizing the expected ttal discunted cst fr the maintained cherl~nt system. We shw that the ptimal grup replacement plicy has a mntne prperty withut regard t the values f failure rate A. and repair rate ~.. Further we discuss sufficient cnditins ~ ~ fr the replacement f the maintained cherent system. Finally t illustrate the ptimal grup replacement plicy, a numerical example is presented. 2. Prblem Frmulatin Cnsider a maintained cherent system. The system cnsists f n cmpnents and n repair facilities. Each f its cmpnents is either up r dwn, indicating whether it is functining r nt, and acts independent f the behaviur f ther cmpnents. When the ith cmpnent ges up [dwn], it remains up [dwn] fr expnentially distributed time with parameter A. [~.] and then ges dwn ~ ~ [up]. Let uptimes and dwntimes be independent. We suppse that the system state at any time depends n cmpnent states thrugh a cherent structure functin ~ (see Barlw and Prschan []). Let l x.(t)j ~ ~O if the ith cmpnent is up at time t, therwise, fr any ien={l,2,...,n}. Then the evlutin f the state f the ith cmpnent is described by the stchastic prcess {X. (t), t~o}. ~ Let and let HX(t) )=~: if the system is up at time t, therwise, then the state f the system is summarized by binary n-vectr X(t) with a state space S and the actual state f the maintained cherent system is described by the stchastic prcess {~(X(t)), t~o}. In the present paper we are interested in a grup replacement prblem fr the abve system. decisin is made t replace t~e At each time epch tet=[o,), bserving the state X(t), a maintained cherent system, r t keep it. We assume that the time needed t replace the system is expnential with parameter ~O. Let n(x)ed={o,l} represent the decisin made fr the system at any time t, Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

3 230 M.Ohashi where n(x)=o means t replace the system and n(x)=l means t keep it. As the cst rate r(x,n(x» assciated with the maintained cherent system, we cnsider the fllwing cst rate. At time t when the state is X and decisin n(x)=l is made n the maintained cherent system, then the cst is incurred at the rate r(x,l)=p(l-~(x»+ Er., where P is the system dwn cst rate, r. (ien) is iec (X).-.- O the repair cst rate f the ith cmpnent and CO(X) dentes the set f currently failed cmpnents. On the ther hand, when decisin n(x)=o is made, then the cst is incurred at the rate r(x,o)=r, where R is the replacement cst rate (i.e., R/~O is the expected replacement cst). The bjective is t investigate the structure f the ptimal grup replacement plicy minimizing the expected ttal discunted cst with discunt factr a>o. Nw let V (x) be the minimum expected ttal discunted cst when a We begin n n fr the state f the system is X(0)=x=(x l,x 2,,x ) at the beginning. by deriving the weak infinitesimal peratr A f 'the prcess {X (t); t~o} T n ned. Fr a functin f in the dmain f An we have A f(x)=lim t-l[e [f(x(t»]-f(x)] n NO x = E ~. (f(l.,x)-f(x»+ EA. (f(o.,x)-f(x» fec(x).-.- iec l (x).-.- ~O (f(ll) -f(x» n(x)"'l, n(x)=o, where C.(x)={ilx.=j, ien}, (j=o,l), (.,x)=(xl,.,x. l' 'x.+l,,x), and J n ll=(l,.,l). Of great imprtance t us is Dshi's frmula (see Dshi [3]) (2.) av (x)=inf{r(x,n(x»+a V (x)}. a ned n a In the fllwing sectin the structural prperties f the ptimal grup replacement plicy minimizing the expected ttal discunted cst are characterized. It is shwn that a mntnic plicy is ptimal and sufficient cnditins fr replacing the whle system are presented. Here we ntice that the existp.nce f a statinary plicy minimizing the expected ttal discunted cst is guaranteed, since all csts are bunded and the actin space is finite. 3. Structure f Optimal Grup Replacement Plicy In this sectin we discuss an ptimal grup replacement plicy fr a maintained cherent system. We can find the ptimal grup replacement plicy by slving the functinal equatin (2.). We cannt btain, hwever, a slutin as a functin f the parameters in the mdel. S sme prperties n the ptimal Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

4 Grup Replacement Plicy 23 plicy and the crrespnding ptimal cst f the functinal equatin (2.) are discussed. The fllwing lemmas shw the strueture f ptimal expected ttal discunted cst functin, and they are used in the prf f therems which present the structural prperties f the ptimal grup replacement plicy. Lemma 3.. The minimum expected ttal discunted cst V (x) is a nna. increasing functin f its cmpnent variables x., ien. t- Prf: The functinal equatin (2.) can be written as V (x)=min {P(l-Ijl(x»+ E (P.+~.v (l.,x»+ E A.V (.,x) Cl. C () t- t- Cl. t-. C ( ) t- Cl. t-e 0 t- X t-e x (3.) +(A- E ~.- E A.)V (x)}/(a+a.), iec 0 0c) t- iecl(x) t- Cl. {R+~OV () +(\-~0) V (x)} / (\+a.), Cl. (~ where A is any value larger than max{~o,max{ E ~.+ E A.+ max A.}}. XES ieco(x) t- iec l (x) t- ieco(x) t- We can calculate by using the successiv(~ apprximatin technique: V (x;n+)=min {P(l-CP(x»+ E (T'.+~.V (l.,x;n»+ E LV (O.,x;n) Cl. ieco(x) t- t. a. t. iec l (x) t. a. t. (3.2) +(A-:~ ~.- E L)V (x;n)}/(a+a.), ieco(x) t- iec l (x) t- a. {R+~OV 0.,( ;n) +(A-~O) V. (x;n) } / (A+a.), where V (x;o)=o fr all XES. Then prving the mntnicity f V (x) is carried a. a. ut by using the mathematical inductin. Fr n=l the result fllws easily frm the prperties f the structure functin cp and the definitins f CO(x) and Cl (x). Suppse the result is true fr :sme n. At the n+l-th stage, if the ptimal decisin is t keep the maintained cherent system fr (O.,X)ES, ien, t- then V (O.,x;n+)-V (l.,x;n+)~[p(l-ijl(o.,x»+ E (T'.+~.V (O.,l.,x;n» a. t. a. t. - t. C (0 ) J J a. t. J JE 0 i'x + E LV (O.,O.,x;n)+(A- E ~.- E L)V (O.,x;n)] jecl(oi'x) J a. t. J jeco(oi'x) J jecl(oi'x) J a. t. /(A+a.] - [P(l-CP(l.,x»+ E (T'.+~.V (.,.,x;n»+ E A. t. jeco(li'x) J J a. t. J JEC l (li'x) J 'V (l.,o.,x;n)+(\- E ~.- E A.)V (l.,x;n)]/[\+a.] Cl. t. J ( ) J C ( ) J Cl. - JEC O i'x JE i'x Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

5 232 M Ohashi ~[P(<P<.,x)-<P<O.,x»+!'.+ E l.(v (O.,.,x;n)-V (.,.,x;n» C ( ) J a..- J a..- J JE 0 i'x ;:,,0. + E L(V (O.,O.,x;n)-V (.,O.,x;n» C ( ) J a..- J a..- J JE i'x On the ther hand, even if the ptimal decisin is t replace the maintained +(A-A.- E.- E A.)(V (O.,x;n)-V (l.,x;n»]/[a+a.].- jec ( ) J C ( ) J a..- a..- O i'x JE i'x cherent system, then V (O.,x;n+l)-V (l.,x;n+l»o is prved similarly t the = abve. Thus fr each n, V (x;n) is a nnincreasing functin f its cmpnent a. variables x., ien. Then frm the successive apprximatin technique, as.- lim V (x;n)=v (x), a. a. n-- Va.(x) is a nnincreasing functin f each f its cmpnent variables xi' ien. Lemma 3.2. The minimum expected ttal discunted cst V (x) is nt larger a. than RI.. The abve lemma is easily prved by the functinal equatin (3.). the structure prperties f the ptimal grup replacement plicy fr a maintained cherent system are characterized. Therem 3.. Next If all cmpnents are perating, then the ptimal decisin is t keep the maintained cherent system. Prf: Fllws directly frm the functinal equatin (3.) and Lemma 3.2. Therem 3.2. The ptimal grup replacement plicy TI*(x) is a nndecreasing functin f each f lts cmpnent variables x., ien..- The functinal equatin (2.) can be written as Prf: V (x)=min {P(l-<p(x»+ E (!'.+. V (.,x»+ E A. V (0.,x) a..c() c().-a..-.-e 0 X.-E x (3.3) +(A- E.- E A.)V (x)}/(a+a.), ieco(x).- iec l (x).- a. {R+llOVa. () } (0.+ 0 ). Ntice that the latter quantity des nt cntain variable x. Frm Lemma 3. and the abve fact, the re sui t is easily btained. The fllwing Therems 3.3 and 3.4 are cncerned with sufficient cnditins fr the replacement f the maintained cherent system. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

6 Grup Replacement Plicy 233 Therem 3.3. If ~~ L ~.+ L A. and R<P(l-~(x»+ L r. fr XES, v-ieco(x) - iec (x) - = ieco(x) - then the ptimal decisin is t replace the maintained cherent system. Prf: The functinal equatin (2.. ) can be written as v (x)=min {P(l-Hx»+ L (l?+~.v (l.,x»+ L LV (.,x)) a ieco(x) - ~ a - iec (x) - a ~ (3.4) I{a+ L ~.+ LA.}, ieco(x) - iec (x) ~ {R+~O Va () } I {a+~o}' The result is shwn by cmparing the terms in functinal equatin (3.4). Thus [V (x)]l-[v (x)]o={p(l-~(x»+ L (r.+~.v (l.,x»+ L A.V (O.,x)} a a :EC O (X) - ~ a - iec (x) 'l- a ~ I{a+ L ~.+ L A.}-{R+~OV () }/{a+~o} icco(x) ~ iec (x) ~ a >[l/{a+ L ~.+ L A.}-l/{a+~O}]R = iec O (;C) ~ iec (x) - +[{ z ~.+ L A.}/{a+ L ~.+ LA.} iec O (x) ~ iec (x) ~ iec O (x) - iec (x) ~ -~0/{(+~0} ]V a () >[l/{a+ L ~.+ L A.}-l/{a+~O}]aV () = ieco~c) ~ iec (x) - a =0. +[{ Z ~.+ L A.}/{a+ L ~.+ LA.} ieco(x) ~ iec (x) - ieco(x) ~ iec (x) ~ -~Ol {(+~0} ]V a () The first inequality fllws frm the assumptins and Lemma 3.. inequality is true frm Lemma 3.2. The last Therem 3.4. If ~O< L ~.+ L A. and R/~~[P(l-~(x»+ L r.] ieco(x) ~ iec (x) 'l- ieco(x) - L ~.+ L A.] fr XES, then the ptimal decisin is t replace the ieco(x) ~ iec (x) - maintained cherent system. Prf: The result is prved similarly t Therem 3.3. Remark. The results f this sectin remain valid even when we extend the cst rate P(l-<p(x»+ L r. t the general cst rate r(x,l), where r(x,l) ieco(x) ~ Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

7 234 M.Ohashi is a nnincreasing functin f each f its cmpnent variables xi' i N with r(ll,l)=o. Remark 2. We ntice that the mntne prperty f the ptimal plicy keeps true irrespective f failure and repair rates, but it is f curse that the actual plicy n(x) depends n the values f failure and repair rates. 4. Numerical Example In this sectin we cnsider the s-called bridge structure system shwn in Figure. T illustrate the ptimal grup replacement plicy f the preceding sectin, we give a numerical example. The failure and repair rates f cmpnents are given in Table. The repair cst rates f cmpnents are als given in Table. The system dwn cst rate, replacement cst rate, and replacement rate are P=S.O, R=lO.O, and ~0=, respectively. Then we btain Figure. The bridge structure system. Table. Failure rates, repair rates, and cst rates. A ~ r U l 0..0 U 2 0. U U Us 0. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

8 Grup Replllcement Plicy 235 Table 2. The ptimal replacement plicy with P=5.0, R=0.0, 0 =, and (X'=0.05. N III c/>(x) (x) l V(x) the ptimal grup replacement plicy fr the maintained cherent system by the value iteratin methd. Als t illustrate the results f Therems 3.3 and 3.4, the values M (X)= E.+ Eh., M (x)=p(-c/>(x»+ Er., and ieco(x) ~ iec (x) ~ 2 ieco(x) ~ M 2 (x)/m (x) are cmputed. The results f these cmputatins are given in Table 2 in the case f discunt factr a=0.05. N. 4, 6,, 3, 8, and 25 satisfy the cnditin f Therem 3.3, and N., 2, 3, 5, 7, 9, 7, and 2 satisfy the cnditin f Therem 3.4. But N. S, 0, 2, 4, 5, 9, 20, 22, 26, 27, and 29 dn't satisfy these cnditins, while the ptimal decisin is t replace the maintained cherent system. This shws that the cnditins f Therems 3.3 and 3.4 are nt necessary fr replacing the system. N. 0, 2, 4, 5, 9, 20, 22, 26, and 27 shw that a preventive replacement is ptimal. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

9 236 M. Ohashi 5. Cnclusin In this paper we have been examined the structure f the ptimal grup replacement plicy fr the maintained cherent system. We shwed that the ptimal grup replacement plicy minimizing the expected ttal discunted cst is a mntne plicy withut regard t the failure and repair rates. Further we discussed the sufficient cnditins fr the grup replacement f the maintained cherent system. It is a future prblem t find the structure f the ptimal grup replacement plicy in the case where a grup replacement plicy in the presence f fixed csts fr turning n r turning ff repair facilities. Acknwledgment The authr is grateful t the referees fr their valuable cmments n this paper. References [] Barlw, R.E. and Prschan, F.: Statistical Thery f Reliability and Lif Testing: Prbability Mdels. Hlt, Rinehart and Winstn, New Yrk, 975. [2] Chiang, D.T. and Niu, S.C.: On the Distributin f Time t First System Failure. J. Appl. Prb., Vl.7 (980), [3] Dshi, B.T.: Cntinuus Time Cntrl f Markv Prcesses n an Arbitrary State Space: Discunted Rewards. Ann. Statist., Vl.4 (976), [4] Rss, S.M.: Applied Prbability Mdels with Optimizatin Applicatins. Hlden-Day, San Francesc, 970. [5] Rss, S.M.: On the Time t First Failure in Multicmpnent Expnential Reliability Systems. Stchastic Prcesses Appl., Vl.4 (976), [6] Sivazlian, B.D. and Mahney, J.F.: Grup Replacement f a Multicmpnent System which is Subject t Deteriratin nly. Adv. Appl. Prb., Vl.lO (978), Mamru OHASHI: Anan Technical Cllege, Minbayashi, An an, Tkushima, 744, Japan. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.