GENERALIZED MODELS FOR DETERMINING OPTIMAL NUMBER OF MINIMAL REPAIRS BEFORE REPLACEMENT

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Journal of the Operations Research Society of Japan Vo!. 24, No. 4, December 1981 1981 The Operations Research Society of Japan GENERALIZED MODELS FOR DETERMINING OPTIMAL NUMBER OF MINIMAL REPAIRS BEFORE REPLACEMENT Toshio Naagawa Meijo University (Received January 31, 1981; Revised June 9, 1981) Abstract This paper summarizes the results of the basic model of Maabe and Morimura, in which a unit is replaced at -th failure. The two generalized models are considered: A) A unit has two types of failures and is replaced at type 2 failure or -th type 1 failure. B) A system has two types of units and is replaced at unit 2 failure or -th unit 1 failure. The expected cost rates for each model are obtained, using the calculus method of probability. It is shown that the optimal numbers * which minimize the cost rates are given by unique solutions of equations under certain conditions. 1. Introduction Maabe and Morimura [4, 5, 6] proposed the new replacement where a unit is replaced at -th failure, and discussed the optimum policy. Morimura [7] also considered several extended models. This is a modification of periodic replacement model introduced by Bar10w and Hunter [1], in which a unit is replaced at time T and undergoes only minimal repair at failures between periodic replacements. This policy is useful in the case where the total operating time of a unit is not recorded or it is much time and costly to replace a unit in operation. Therefore, this could be used in maintaining a complex systedl with many equipments of the same type. This paper summarizes the nown re.su1ts of the basic model of Maabe and Morimura and maes additional discussions. Further, the following two ge!nera1ized models are considered: A) A unit has two types of failures when it fails: Type 1 failure occurs with probability a and is corrected with minimal repair, and whereas type 2 failure occurs with probability 1 - a and a unit has to be replaced. If the -th type 1 failure occurs before type 2 failure, then a unit is replaced 325

326 T. Naagawa preventively. B) A system has two types of units: When unit 1 fails, it undergoes minimal repair, and when unit 2 fails, a system has to be replaced. If unit 1 fails at times before unit 2 failure, then a system is replaced preventively. We derive the expected cost rates for each model and obtain the optimal numbers * to minimize the, cost rates, when the hazard rate is monotone increasing. Other useful di.scussions of results are further made. 2. Basic Model The unit is replaced at the time of -th failure ( = 1, 2,... ) after its installation and undergoes only minimal repair at failures between replacements. Assume that the unit has a failure time distribution F(t) with finite mean 1.1 and has a density j'(t). Then, the hazard rate (or the failure rate) - _ t is r(t) = j'(t)/f(t) and the cumulative hazard is R(t) = fo r(u)du, which has a relation F(t) = exp[-r(t)], where F(t) = 1 - F(t). It is further assumed that the hazard rate r(t) is continuous, monotone increasing, and remains undisturbed by minimal repair. Thus, there exists the limit of r(oo) = lim r(t), t-- which may be possibly infinity. Let cl be the cost of minimal repair and c 2 be the cost of replacement. Then, if the times for repair and replacement are negligible, then the expected cost rate is, from [4], (1) C() ( - l)c l + c 2 I: f~ Pj(t)dt ( 1, 2,... ), where p. (t) = {[R(t)]j/j!}e-R(t) (j = 0,1,...), which represents the probability that j failures occur in the interval [0, t]. J We see the optimal number * which minimizes C() in (1) when r(t) is continuous and monotone increasing. Theorem 1. If there exists a solution * which satisfies (2) L() :: c 2 /c l and L() < c/c l ( 1, 2,... ), it is unique and minimizes C(), where (3) L(),[ f 0 P(t)dt I: f~ Pj(t)dt - ( - 1) 00 0 ( 1, 2,... ), ( 0).

Optimal Number of Minimal Repairs 327 Proof :We suppose that C(O) = 00 for simplicity. Then, a necessary condition that there exists a finite * minimizing C() is that a * satisfies C(+1).:: C() and C() < C() ( = 1, 2,...). Thus, these inequalities give (2). Further, L(+1) - L() > 0, since J~ P(t)dt is a decreasing function of from Lemma 5 of [7, p. 103] or (i) of Appendix, when r(t) is monotone increasing. Thus, a solution to (2) must be unique. We may compute only a minimum * such that L() > czlc if it exists and 1 (4) JooO P * (t)dt -1 < C( *) < - J~ P, *(t)dt J( Further, it is easily seen that L(l) > 1. Thus, if cl.:: c then * 1, viz., 2 the unit should be replaced at the first failure. From (i) of Appendix, we also have that 1im J~ P(t)dt = l/r(oo), where -- Thus, we can give the following upper limit of l/r(oo) = whenever r(oo) = *: Theorem 2. suppose that r(oo) > c2/(~c1)' Then, there exists a unique minimum which satisfies (5) ( 1, 2,... ), and * <. Proof: We easily have (6) ( = 1, 2,... ), since J~ P(t;)dt is decreasing in. Further, from the assumption that r(oo) > c2/(~c1)' there exists a unique minimum such that ~/J; P(t)dt ~ c 2 /c 1 ' i.e., J~ P(t;)dt ~ ~(c1/c2)' Hence, from the inequality of (6), we have * <. If r(oo) > c2/(~c1) then a solution to (2) also exists. For example, sup- - i3 pose that the failure time has a Weibu11 distribution, i.e., F(t) = exp(-t ) for i3 > 1. Then, r(oo) = 00 and from (6),

328 T. Naagawa 00 1 r(+l/a) fo P(t)dt = S r(+1) E f~ j=o P.(t)dt J r (+l/a) r () Thus, there exists a unique * which minimizes C() and it is, from (2), (7) 1 C z [-- (- - 1)] + 1, a - 1 Cl where [ x ] denotes the greatest integer contained in x. Until now, we have assumed that the times for repair and replacement are negligible. In reality, it requires some time to mae a repair or a replacement, and probably, it may be much smaller than an operating time of the unit. Let Tl be the random variable denoting the repair time and T2 denoting the replacement time. Further, let cl(t l ) be the repair cost which includes all costs incurred due to repair and unit failure, and c 2 (T 2 ) be the replacement cost. (8) C() Then, in a similar way of obtaining (1), we easily have ( - l)e[c l (T l )] + E[c 2 (T 2 )].E f~ P.(t)dt + ( - 1)].11 + ].12 J=O J ( - 1, 2,. ), where ].Ii = E[T ] (i i = 1, 2). When cl(t l ) = Cl + cotl and c (T ) 2 2 = c 2 + c O T2' C() agrees with the equation (3.2) of [7]. Further, when cl(t ) l = Tl and c 2 (T 2 ) = T 2, 1 - C() represents the limiting efficiency, i.e., the steadystate availability, which is given by (2.3) of [7]. In this case, Theorems 1 & Z are rewritten as: Theorem If there exists a solution * which satisfies (9) L() > E[c 2 (T 2 )]/E[c l (T l )] and L() < E[c 2 (T 2 )]/E[c l (T l )] it is unique and minimizes C(), where (10) L().E f; p.(t)dt + ( - 1)].11 + ].12 ~_J _ f~ P(t)dt + ].11 [ ( = 1, 2,... ), - ( - 1) ( 1, 2,... ) o ( = 0). Theorem 2'. Suppose that l/r(oo) < (].I + ].I2){E[C l (T l )]/E[C 2 (T 2 )]} - Then, there exists a unique minimum which satisfies ].11'

Optimal Number of Minimal Repairs 329 and * <. Theorems l' & 2' are easily proved because L() is monotone increasing in and (12) 3. Two Types of Failures We consider the unit with two types of failures [3]. When the unit fails, type 1 failure occurs with probability a and is removed by minimal repair, and type 2 failure occurs with probability 1 - a and is removed by replacement. Type 1 failure is a minor failure which is easily restored to the same operating state before failure by minimal repai.r, and whereas type Z failure incurs a total breadown. The unit is replaced at the times of type Z failure or -th type 1 failure, whichever occurs first. Then, the expected number of minimal repairs (Le., type 1 failures) before replacement is (13) ( - l)a + E (j j=l 1)a j - 1 (1 - a) Thus, the expected cost rate is easily given by (14) C(;I:X) Cl [(a - a )/(1 - a)] + c 2 -l E a j f~ P j (t)dt (a - a )/(1 - a). ( 1, 2,... ). When a = 1, C(;l) is equal to (1) and the optimal policy is discussed in Section 2. When::. = 0, C(;O) = cz/\l, which is constant for all, and hence, the unit is replaced only at type Z failure. Therefore, we need only discuss the optimal policy in case of 0 < a < 1. To simplify equations, we denote \la 00 - a == fo [F(t)] dt. When a = 1, \la \l which is the mean time to failure of the unit. Theorem 3. (i) Suppose that 0 < a < 1 and 1'(00) > [clc 1 + (1 - a)cz]/(\l1_ac1). exists a finite and unique *(a).which satisfies Then, there (15) ( 1, 2,... ), where

JJO T. Naagawa (16) -l [ ri L f~ Pj(t)dt a - a L(;a) f~ P(t)dt 1 - a - 0 ( 1, 2,. ), ( = 0).. * (ii) If 1'(00).::: [ac l + (1 - a)c2]/()1l_acl) then /( (a) -+- 00, and the resulting expected cost is (17) lim C(;a) -- Proof: c 1 [a/(1 - a)] + c 2 )11-0. The inequalities C(+l;a) :: C(;a) and C(;a) < C(-l;a) imply (15). Further, it is easily seen that L(;a) is an increasing function of, and hence, lim L(;a) ~l 1'(00) - [0./(1 - a)]. Thus, in a similar way of -a obtaining T~-:C;rem 1, if y.(oo) > [ac 1 + (1 - a)c2]/()1l_acl) then a solution to (15) exists and it is unique from the monotonicity of L(;a). On the other hand, if/(oo).::: [ac l + (1 - a)c2]/()1l_acl) then L(;a) < c 2 /c l for all, and hence, -+- 00, and from (14), we have (17). Further, note that if 1'(t) is monotone increasing to infinity then a finite *(a) exists uniquely. It is easily seen that dl(;o.)/do. > 0 for all. Thus, we have the following theorem: Theorem 4. If 1'(00) > [o.c + (1 - o.)c2]/()1l_o.cl) for 0.::: a < 1 then l *(a) is decreasing in a, and hence, - :: *(a) > *, where both *- and exist and are given in (2) and (5), respectively. 4. Two Types of Units Consider a system with two types of units which operate statistical independently. When unit 1 fails, it undergoes minimal repair instantaneously and begins to operate again. When unit 2 fails, the system is replaced without repairing unit 2. Unit 1 has a failure time distribution F(t), the hazard rate 1'(t) and the cumulative hazard R(t), which have the same assumptions as the basic model. Unit 2 has a failure time distribution G(t) with finite mean A. Suppose that the system is replaced at the times of unit 2 failure or -th unit 1 failure, whiehever occurs first. Then, the mean time to replacement is

Optimal Number of Minimal Repairs 331 (18) E i; tp '-1 (t)dg(t) + J~ tg(t)p_1 (t)r(t)dt j=l J where G(t) == 1 - G(t), and the expected ::lumber of minimal repairs before replacement is (19) E (j - l)j; Pj_l(t)dG(t) + ( - l)f; G(t)P_l(t)r(t)dt j=l 1<-1 E J=l Thus, the expected cost rate ia (20) C(;:l) -2 cl E J~ G(t)p.(t)r(t;)dt + c J 2 E J~ G(t)p j (t)dt ( 1, 2,... ). When G(t) 1 for t > 0, C(;G) is equal to (1), and whe3 G(t) 1 for t < T and for t > 'r, this corresponds to the model of "Policy IV" of Morimura I: 7]. Assume th.'it the hazard rate of unit 2 is h(t) == a density of G(t). g(t) /G(t), where g(t) is Theorem 5. Suppose that h(t) is continuous and increasing. Then, if there exists a solution * which satisfies (21) ( 1, 2,... ), it is unique and minimizes C(;G), where (22) L(;(;) - J; G(t)P_1(t)r(t)dt.E J; G(t)Pj(t)dt J=O J~ G(t)P (t)dt -2 E J; G(t)Pj(t)r(t)dt ( 1, 2,... ), o ( = 0). Proof: The inequalities C(+1;G) > ~(;G) and C(;G) < C(-l;G) give (21). Further, 00 _ [J~ G(t)P(t)r(t)dt (23) L(+1 ;G) - L(;G) =.: J 0 G(t)pj(t)dt -"--00---...:..:=----- J-O J 0 G(t)P+1 (t)dt

332 T. Naagawa _ I~ G~t~_l(t)r(t)dtl 10 G(t)P(t)dt since I~ > 0, G(t)p(t)r(t)dtII~ G(t)P+l(t)dt is an increasing function of from (ii) of Appendix, when h(t) is increasing. it is unique. (24) Further, we have, from (ii) of Appendix, lim L(;G) -- A [r(oo) + h(oo)] - I; G(t)r(t)dt. Thus, if a solution to (21) exists, Thus, if r(t) + h(t) is monotone increasing and r{oo) + h(oo) > (l/a) [(c 2 /c l ) + I; G(t)r(t)dt] then there exists a finite and unique * which satisfies (21). For example, suppose that h(t) is monotone increasing and F(t) is exponential, i.e., F(t) = 1 - exp(-t/i1). Then, if h(oo) > c/(ac l ) then a solution to (21) exists uniquely. In Sections 3 & 4, 'Ne have assumed that the replacement costs for the both cases of the -th type 1 (unit 1) failure and type 2 (unit 2) failure are the same. In reality, it may be different from each other. We suppose that c 2 is the replacement cost of the -th type 1 (unit 1) failure and c 3 is the replacement cost of the type 2 (unit 2) failure. in (14) and (20) are rewritten as, respectively, (25) C(;a) cl [(a - a)/(l - a)] + c a 2 + -l E aji~ Pj(t)dt Then, the expected cost rates c 3 (1 - a ) -.2 cl E I~ G(t)Pj(t)r(t)dt + c 2 I; G(t)P_l(t)r(t)dt (26) C(;G) -l E I~ G(t)PJ.(t)dt We can discuss the optimal numbers which minimize C(;a) and C(;G) by a similar method, although we omit here. 5. Conclusions and Examples We have considered the basic model, where the unit is replaced at: -th

Optimal Number of Minimal Repairs JJJ failure, and the two generalized models which include the basic model as a special case. We have discussed the optimal numbers * which minimize the expected cost rates of each model. * is given by a minimum such that It has been shown in the theorems that L(; ) ~ ci/c l if it exists. Finally, we compute the optimal number (Cl) which minimizes the expected cost C(;Cl) in (Z5) when F(t) = exp(-t f3 ) for f3 > 1. When C z = c ' it is shown 3 from Theorems 3 and 4 that *(Cl) exists uniquely and is decreasing in Cl for o < Cl < 1. Further, when Cl 1, *(Cl) is given in (7). If cl + (c - cz) (1-3 Cl) * > 0 then (Cl) is given by a minimum value such that (1 - Cl)r(+l) -l Cl j r(j+l/f3) L + Cl > r (+l/f3) r (j+l) It is easily seen that *(Cl) is small when cl/c or c /c for C > cl is large. i 3 Z z Conversely, if cl + (c - cz) (1 - Cl).:: I) then (Cl) -+ 00. 3 ' Table 1 shows the optimal number (Cl) for the probability Cl of type 1 failure and the ratio of cost c to cost C when we assume f3 3 z = Z and cl/cz = 0.1. It is of great interest that *(Cl) is increasing in Cl for c~ > cz' however, is decreasing for c 3.:: cz. We can explain the reason why (Cl) is increasing in Cl for c 3 /c Z : If c 3 > C z then the cost of replacement for type 1 failure is cheaper than that for type Z failure and the number of its failures increases with Cl, and hence, *(Cl) is large when Cl is large. This situation Table 1. Variation in the optimal number *(Cl) for the probability Cl of type 1 failure and the ratio of c 3 to cz' where f3 = Z and cl/cz = 0.1. 0.8 0.9 1.0 l.z 1.5 Z.O 3.0 0.1 00 30 6 Z 1 1 O.Z 00 Z7 6 3 1 1 0.3 ZZO Z4 6 3 Z 1 0.4 11Z ZZ 7 3 Z 1 0.5 64 ZO 7 4 Z 1 0.6 Z88 39 17 7 4 Z 1 0.7 64 25 15 8 5 3 Z 0.8 26 17 13 8 6 4 Z 0.9 14 12 11 9 7 5 4 1.0 10 10 10 10 10 10 10

334 T. Naagawa would reflect a real world. On the other hand, if c ~ c then it is not good 3 2 to replace the unit often before type 2 failure, however, the total cost of minimal repairs for type 1 failure increases as the number of its failures does with a. Thus, it may be better to replace the unit preventively at some number when a is large. small enough. * Evidently, (a) is rapidly increasing when cl is Acnowledgement The author wishes to than Dr. Kowada of Nagoya Institute of Technology for useful suggestions and help full comments. References [1] Bar1ow, R.E. and Hunter, L.C.: Optimum Preventive Maintenance Policies. Operations Research, Vo1.9, No.9 (1960), 90-100. [2] Bar1ow, R.E. and Proschan, F.: Mathematical Theory of Reliability. John Wi1ey & Sons, New Yor, 1965. [3] Beiche1t, F. and Fischer, K.: General Failure Model Applied to Preventive ~ffiintenance Policies. IEEE Trans. on Reliability, Vo1.R-29, No.1 (1980), 39-41. [4] Maabe, H. and Morimura, H.: A New Policy for Preventive Maintenance. J. Operations Res. Soc. Japan, Vo1.5, No.3 (1963), 110-124. [5] Maabe, H. and Morimura, H.: On Some Preventive Maintenance Policies. J. Operations Res. Soc. Japan, V~1.6, No.1 (1963), 17-47. [6] Maabe, H. and Morimura, H.: Some Considerations on Preventive Maintenance Policies with Numerical Analysis. J. Operations Res. Soc. Japan, Vol. 7, No.4 (1965), 154-17l. [7] Morimura, H.: On Some, Preventive Maintenance Policies for IFR. J. Operation Res. Soc. Japan, Vol.12, No.3 (1970), 94-124. [8] Pars, K.S.: Optimal Number of Minimal Repairs before Replacement. IEEE Trans. on Reliability, Vol.R-28, No.2 (1979), 137-140. Appendix We show the following two results which are useful for proving the theorems, when the hazard rate' r(t) is continuous and monotone increasing: (i) f; P(t)dt is decreasing in and Hm f; P(t)dt = l/r(oo), where l/r(oo) --

Optimal Number oj'minimal Repairs 335 whenever r(oo) = 00. (ii) If h(t) is continuous and increasing then o ( - 1)! J~ [R~~)] G(t)F(t)dt Joo JR(t)] G(t)f(t)dt is increasing in and Joo [R(t)] G(t)f(t)dt, 0 ( - 1)! I ~m -- J~ [Rg)] G(t)F(t)dt 1. Proof of (i) Using the relation (AI) t [R(u)]-l J O ( _ 1)! r(u)du [R(t) ]! ( = 1, 2,... ), and the assumption that r(t) is monotone increasing, we have OO _[R(t) ) -R(t)dt Joo [Jt [R(u) ]-l ()d J -R(t)dt (A2) JO! - e = 0 0 ( _ l)! r u u e -l < Jooo [Jt [R(u)] o ( - l)! dujf(t)dt 00 [R(t)J - 1 R(t) = J o ( _~ e- dt. Thus, J~ P(t)dt is decreasing in. Further, (A3) [R(t)] : -R(t) J~! e dt = Joo [R( 2..L ~ dt o! r(t) > _1_ J'" [R~~)] f(t)dt 1 - r(oo) 0 = r(oo) On the other hand, for any T E (0, 'JO), Thus, 00 [R(t)] e-r(t)dt J 0 ~:;-<! c...l- = JT lbit)] o! ~ J~ [R~~)] -R(t)dt + _l_ e r(t) (A4) 00 [R(t)] e-r(t)dt < 1 limjo! -r(t) --

336 T. Naagawa Therefore, combining (A3) and (A4), we have (AS) 00 [R(t») e-r(t)dt 1 lim 10! = l' (00), -- since T is arbitrary. 2. Proof of (ii) Integrating by parts, we have [R(t) ] [R(t) ] [R(t)] I~ ( - 1)! G(t)f(t)dt,= I~ -,zr G(t)f(t)dt + I~ -! F(t)g(t)dt. First, we show that (A6) I~ [R(t) { G(t)f(t)dt I~ [R(t») G(t)j7(t)dt is increasing in when 1'(t) is monotone increasing. Let q(t) _ I~ [R(t»)+1 G(t)f(t)dt I~ [R(t») G(t)F(t)dt [R(t») G(t)f(t)dt I~ [R(t))+1 G(t)F(t)dt. Then, it is easily seen that (An q(o) = 0, q~ (T) > o. Thus, q(t) > 0 for all T > 0, and hence, (A6) is increasing in. Similarly, (AB) I~ [R(t») F(t)g(t)dt I~ [R(t») F(t)"(;(t)dt is also increasing, which may be possibly constant. Therefore, from (A6) and 00 [R(t») [R(t)] (AB), we have that 10 (~ G(t)f(t)dtll~! G(t)F(t)dt is i.ncreasing in. Next, we show that (A9) 100 [R(t») C(t)f(t)dt lim 0 -- I; [R(t) ]C(t)F(t)dt = 1'(00), Evidently,

Optimal Number of Minimal Repairs 337 (Ala) I~ [.R(t)] C(t)f(t)dt ::: 1'(00). I; [R(t)] G(t)F(t)dt On the other hand, for any T E (0, 00), we have (All) I~ [.R(t)] C(t)f(t)dt T - 00 10 [R(l;)] G(t)f(t)dt + IT [R(t)] C(t)f(t)dt I~ [R(t)] C(t)P(t)dt = I~ [R(l;) ] C(t)P(t)dt + I; [R(t) / G(t)P(t)dt 1'(T)I; [R(t)] C(t)l?(t)dt > - I~ [R(t) ] C(t)P(t)dt + I; [R(t) / G(t)P(t)dt 1'(T) Further, the bracet of the denominator is, for T < T I, I~ [R(t)] G(t)P(t)dt [R(T) ]I~ G(t)P(t)dt (AI2) < -.::-...,~---- I; [R(t) ] C(t)P(t)dt - I; [R(t) ] C(t)P(t)dt I I~ C(t)P(t)dt < - [R(TI)/R(T)] I; C(t)P(t)dt I Thus, from (Ala), (All), and (AI2), we have I~ [R(t)] G(t)f(t)dt (AI3) 1'(00) > lim > r(t) - +oo I~ [R(t) { G(t)P(t)dt - ' + 0 as + 00. which imply (A9) since T is arbitrary. In a similar way, (AI4) I; [R(t)] P(t)g(t)dt lim 00 = h(oo). +oo I [R(t)] P(t)G(t)dt o Therefore, combinig (A9) and (Al4), we complete the proof. Toshio NAKAGAWA: Department of Mathematics, Faculty of Science and Engineering, Meijo University, Tenpau-cho, Tenpau-u, Nagoya, 468, Japan.

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