The unrevealed failures of a system are detected only by inspection. In this work, an inspection policy along with a maintenance procedure for multiunit systems with dependent times to failure is presented. The existence of an optimum policy is also discussed

Badía Blasco, F. G.

Berrade Ursúa, M. D.

Campos, Clemente A.

UPCommons. Portal del coneixement obert de la UPC

QU¨ESTIIO´, vol. 26, 1-2, p. 247-258, 2002MAINTENANCE POLICY UNDER MULTIPLEUNREVEALED FAILURESF. G. BAD´IA BLASCOM. D. BERRADE URS ´UACLEMENTE A. CAMPOSUniversidad de Zaragoza The unrevealed failures of a system are detected only by inspection. In thiswork, an inspection policy along with a maintenance procedure for multi-unit systems with dependent times to failure is presented. The existence ofan optimum policy is also discussed.Keywords: Maintenance, reliability, replacement, optimum policyAMS Classification (MSC 2000): 90B25, 60K10, 62N05* Departamento de Me´todos Estadı´sticos. Centro Polite´cnico Superior de Ingenieros. Universidad de Zara-goza. Marı´a de Luna, 3. 50015 Zaragoza.Corresponding author: Clemente A. Campos. Centro Polite´cnico Superior de Ingenieros. Universidad deZaragoza. Marı´a de Luna, 3. 50015 Zaragoza. SPAIN. E-mail address: C.Campos@posta.unizar.es. Pho-ne: +34 976 761 884. Fax: +34 976 761 861.– Received November 2001.– Accepted April 2002.2471. INTRODUCTIONThe classical age and block replacement policies are only useful when the failures ofa unit are detected as soon as they occur. However, the failures of spare units or in asystem during the stand-by mode, may remain undiscovered until the next demand ofactivity, causing important availability losses. Periodic inspection or testing is the onlyway to fight against failures of this sort (see Lewis (1987)).The works due to Nakagawa and Yasui (1991), Vaurio (1995, 97, 99) and Berrade(1999) deal with unrevealed failures; all of them consider maintenance policies forsingle-unit systems. On the contrary, Ebrahimi (1997) provides an extension of the agereplacement policy for multi-unit systems under revealed failures.In this work we provide a new inspection policy to detect the occurrence of failures in asystem, otherwise being unrevealed, along with a maintenance procedure to improve itsreliability. This policy can be carried out in multi-unit systems with an only type of fai-lure, or in single-unit systems with multiple «hidden» causes of failure. Many systemsof this sort can be found in practice: the fluid levels in a car are checked periodicallyso that an eventual leakage of oil, water, etc can be observed. The inspection policiesare concerned not only with engineering systems: the periodic tests carried out to detectdisorders of health in humans constitute the most common example of such procedure.We assume the possibility of dependent times to failure between units or types of failurewhich is a realistic assumption when the occurrence of a type of failure is likely tohave an effect on the probability of an event of another type. We characterize optimumpolicies so as to minimize the cost per unit of time for an infinite time span, thus,limt ∞ C tt with C t the cost incurred in 0  t.Under this policy, the unit is periodically checked and replaced, hence, the maintenanceprocess yields a renewal process. In fact, it corresponds to the particular class of therenewal-reward processes with the term cost being used instead of reward. The proper-ties of the renewal reward processes ensure that the previous cost function converges,with probability 1, to the next oneQ T   E C τE τA cycle, denoted τ, is the time span between two consecutive renewals of the system(see Ross (1996)) with C τ denoting its associated (random) cost. From now on Q T ,depending on the time between consecutive inspections, T , will denote the objectivefunction.In section 2, the maintenance policy is described along with the conditions that guaran-tee the existence of an optimum policy for a series system (Theorem 1) and a parallelsystem (Theorem 2). In addition, we consider multivariate distributions which are natu-248ral extensions of the exponential random variable: the bivariate Gumbel distribution, aswell as the Marshall-Olkin model. These distributions allow the dependence betweencomponents and, provided they hold, we obtain particular conditions for the existenceof an optimum policy. Finally, section 3 contains some examples involving the previousdistributions which illustrate the theoretical results.2. THE MODELThe system consists of n units which are periodically checked at times kT , k  1 2    .In addition each unit, had it failed or not, is replaced by a new one. Times of inspectionsand replacements are assumed to be negligible.The following notation will be handled:  Xi, is unit i’s random time to failure  i  1 2      n  µi, Fi, fi, Ri and ri are, respectively, the mean value, cumulative distribution function,density function, reliability function and failure rate of X i,  i  1 2      n  T  : optimum replacement timeLet c1 be the cost of the replacement when no failure has occurred and c 2i,  i  1 2      n the corresponding cost if the unit i has failed. In practice, it is usual to consider c 1 c2i. In addition, we will take into account the cost derived from the down-time whichrepresents the losses incurred while the failure of the system remains undiscovered. Itcan be due to defective production, unavailability of the system when it is needed, etc.Let also consider the ordered times to failure of the units, thus, X 1 X 2      X n. Inaddition, µ i, F i, f i, R i, and r i are the mean value, distribution function, densityfunction, reliability function and failure rate corresponding to X i. Finally, the multiva-riate distribution and reliability functions of the system are, respectivelyF x1 x2      xn  P X1  x1 X2  x2      Xn  xnR x1 x2      xn  P X1  x1 X2  x2      Xn  xnIn what follows, we will analyze the maintenance policy for series and parallel systems,and we will begin with the former.CASE I. The system works if all the component work.In this case, the time to failure of the system is given byX 1  min X1 X2      Xn249being R 1 t  R t       t its corresponding reliability function. Now, cdi is the costderived from the down-time when component i fails.The next result provides the expression of the cost function.Proposition 1 Consider a n-out-of-n system whose failures are only detected by ins-pection. Under the maintenance policy previously described, the following results holdEC τ  c1R 1 T n∑i1c2iFi T n∑i1cdi  T0Ri udun∑i1cdiT(1)Q T  n∑i1cdi c1R 1 T ∑ni1 c2iFi T ∑ni1 cdi T0 Ri uduT(2)ProofAt times kT ,  k  1 2       the system is renewed, hence, the mean length of a cycle isEτ  T . The cost of a cycle under this maintenance procedure isC τ  c1I X1  T X2  T      Xn  T n∑i1c2iI Xi  T n∑i1cdi T min Xi T where I  represents the indicator function. The expectation of the expression above,leads to (1).Both Eτ and (1) provide Q T  given in (2).  The following theorem gives a sufficient condition which guarantees the existence ofan optimum policy.Theorem 2 Consider the same conditions given in Proposition 1, under the next ine-quality(3)n∑i1c2i n∑i1cdiµi  i  1 2      nthere exists T   in  0 ∞ minimizing Q T  in (2). Moreover, T   is one of the roots of thefollowing equationT c1 f 1 T n∑i1c2i fi T n∑i1cdiRi T  c1R 1 T n∑i1c2iFi T n∑i1cdi  T0Ri udu 0(4)250ProofQ T  can be expressed asQ T  n∑i1cdi a T Twherea T   c1R 1 T n∑i1c2iFi T n∑i1cdi  T0Ri udua T  is a continuous function in 0 ∞ verifying a 0  c 1  0. From (3), it is derivedthat a ∞  ∑ni1 c2i ∑ni1 cdiµi  0. Note that the two limiting cases, T  0 andT  ∞, correspond, respectively, to a continuous inspection and no inspection at all.Therefore, there exists T0 in  0 ∞ such that a T0 0, hence, Q T0∑ni1 cdi Q ∞.In addition, Q 0 ∞, so, there exists T1, 0 T1  T0, satisfying Q T  infx0T1 Q xfor all T  T1.Q T  is a continuous function, therefore it has a minimum, T  , in T1 ∞. Moreover,next inequalityQ T1 Q ∞ Q T0leads to T1  T   ∞. Therefore, T   is also the minimum in 0 ∞, and the result holds.Finally, by differentiation of Q T , (4) is obtained.  The particular case of time to failure exponentially distributed is analyzed in the follo-wing result.Proposition 3 Consider the same conditions given in Proposition 1, and X i, exponen-tially distributed with failure rate λi, i  1 2      n. If the next two conditions holdλi cdic2i  i  1 2      n(5)r2 1 T  r 1 T (6)with r 1 T  being the derivative of r 1 T . Then, T   is the only root of (4).ProofDenote A T  the left-hand side in equation (4). In this case, the derivative of A T  isexpressed asdA T dT  T c1r2 1 T  r 1 T R 1 T n∑iiλ2i eλiTc2icdiλi251The hypotheses (5) and (6) assure that the expression above is positive and, therefore,A T  has an only root.  The multivariate extensions of exponential distribution arise easily to model the time tofailure when the maintenance of multi-unit systems is studied. We consider the follo-wing bivariate extensions1. The bivariate Gumbel distribution (Gumbel (1960))The corresponding bivariate reliability function is given byR x y  ex λ1λ3yλ2y  0 λ3  λ1λ2  λ1 λ2  0  x y  0This distribution has exponential marginal distributions, however, the failure rate defi-ned asr x y f  x yR x yis not a constant, unlike the univariate exponential.2. The Marshall-Olkin model (Marshall-Olkin (1966))Marshall and Olkin derived the joint reliability distributionR x y  eλ1xλ2yλ3 max xy  λ1 λ2  0  λ3  0  x y 0so as to describe a two-component system subject to shocks following a Poisson pro-cess. An important property of this distribution is that X  min T1 T2, Y  min T2 T3where T1, T2 and T3 are independent exponential random variables. It follows thatmin X  Y  is also exponentially distributed. In addition,  X  Y  is the only distributionto have the so-called no-aging propertyP X  x t Y  y t  X  t Y  t  P X  x Y  y x y  t  0which means that the remaining lifetime, given that both components have survived upage t, does not depend on t.In the former models, both X and Y are exponentially distributed. Hence, by directapplication of Proposition 2, the following corollaries are derivedCorollary 4 Consider a two-out-of-two system whose time to failure is distributed asthe bivariate Gumbel model. Under the following conditionsλi cdic2i  i  1 2 λ1 λ22  2λ3T   is the only root in equation (4).252ProofThe reliability function and the failure rate corresponding to X 1  min X  Y  are, res-pectivelyR 1 T   e λ1λ2Tλ3T2r 1 T    λ1 λ22λ3TThe hypotheses of Proposition 2 are satisfied and the result follows.  Corollary 5 Consider a two-out-of-two system whose time to failure is distributed asthe Marshall-Olkin model. T  , is the only root of equation (4), providedλi λ3 cdic2i  i  1 2ProofX and Y are exponentially distributed with parameters λ 1λ3 and λ2λ3, respectively.Besides, the reliability function of X 1  min X  Y  and its failure rate are respectivelygiven byR 1 T   e λ1λ2λ3Tr 1 T   λ1 λ2 λ3again, Proposition 2 leads to the result.  Next, the particular case of independence between times to failure of the components isconsidered.Proposition 6 Let X1 X2      Xn be independent random variables, and such thatαi cdic2i  i  1 2      n(7)with αi  sup0t∞ ri t. Then, T   is the only root of (4).ProofDenoting A T  the left-hand side of (4), its derivative can also be expresseddA T dT  T c1r2 1 T R 1 T n∑i1c2iri T Ri T  T c1r 1 T R 1 T n∑i1ri T  fi T c2icdiri T 253The independence of  X1 X2      Xn leads toR 1 T  ∏ni1 Ri T , r 1 T   ∑ni1 ri T , r 1 T   ∑ni1 ri T hencedA T dT  Tc1n∑i0ri T 	2n∏i1Ri T n∑i1ri T Ri T c2i c1∏j iR j T 	 Tri T  fi T c2icdiri T Usually, it is assumed c1  c2i, i  1 2      n. This fact along with the hypotheses givenin (7), prove that the derivative of A T  is positive and the result holds.  CASE II. The system works if at least one component works.The time to failure in a parallel system is given by the longer lifetime, thusX n  max X1 X2      Xnwith µ n  EX n. Its reliability function isR n x  1P X1  x X2  x      Xn  x  1F x x      xNow, we consider a unique cd representing the cost derived while the failure of thewhole components is not detected. First, the cost function is obtained.Proposition 7 Consider a 1-out-of-n system whose failures are only detected by ins-pection. Under the maintenance policy previously described, the following results holdEC τ  c1R 1 T n∑i1c2iFi T  cdT  cd  T0R n udu(8)Q T   cd c1R 1 T ∑ni1 c2iFi T  cd T0 R n uduT(9)ProofIn Proposition 1, it was proved that Eτ  T . The cost of a cycle isC τ  c1I X1  T X2  T      Xn  T n∑i1c2iI Xi  T  cdT minX n T254The expected value of the expression above is given by (8), therefore, (9) holds.  The following result provides a sufficient condition for the existence of an optimumpolicy.Theorem 8 Consider the same conditions of Proposition 4, if(10) µ n ∑ni1 c2icdthen, there exists T  , 0  T    ∞ minimizing Q T  in (9). Moreover, T   is one of theroots of the following equationT c1 f 1 T n∑i1c2i fi T  cdR n T  c1R 1 T n∑i1c2iFi T  cd  T0R n udu 0(11)ProofQ T  can be also expressed asQ T   cd  a T Twherea T   c1R 1 T n∑i1c2iFi T  cd  T0R n uduClearly, a 0  c1  0 and, from hypothesis (10), it is obtained a ∞  ∑ni1 c2i cdµ n  0. Hence, there exists T0, 0  T0  ∞ such that Q T0  cd  Q ∞. Inaddition Q 0 ∞, and the same strategy used in Theorem 1 shows the existence of T  ,finite, minimizing Q T .On the other hand, the roots of the derivative of Q T  should verify (11).  Remark 1 Conditions (3) and (10) mean that an optimum policy exists whenever themean cost incurred for the down-time is greater than the cost derived from the preven-tive maintenance.  2553. EXAMPLESWe present two examples that aim at illustrating the results presented in this work. Thefirst of them refers to a series system following a bivariate Gumbel distribution and thesecond deals with a parallel system whose time to failure is distributed as the Marshall-Olkin model.The periodic tests which are carried out so as to detect disorders in the blood pressureand the cholesterol level, correspond to the case of a series system. No sooner does oneof the causes appear than the health disorder arises. In general if the cholesterol levelis high so does the blood pressure, that is to say, both turn out to be related and mayindicate the occurrence of a more serious illness. The disorders in the blood pressureand the cholesterol level are unrevealed failures as, apart from the corresponding tests,there is no other way to detect them.Consider now an engineering system with two spare units being available. Both can bemodeled by a parallel system and should be tested periodically so as to detect its failureswhich may occur even while they are not in use. The failures may be dependent if theunits are stored under the same conditions, hence a failed unit makes more likely thefailure of the second spare component.Table 1E X 1 λ1 λ2 λ3 T  1 QT  1  T  2 QT  2 2849 98 104 2 104 108 12 92 1 56 15 82 1 282263 39 104 3 104 108 11 19 1 80 14 16 1 461868 104 4 104 108 10 01 2 02 12 93 1 561586 3 104 5 104 108 9 14 2 21 11 97 1 571376 104 6 104 108 8 46 2 39 11 20 1 811332 84 104 6 104 2 108 8 46 2 39 11 20 1 811295 53 104 6 104 3 108 8 46 2 39 11 20 1 81787 44 5 104 5 104 2 107 7 08 2 87 8 18 2 49285 103 2 103 106 4 09 5 01 5 01 4 11158 63 103 5 103 106 2 90 7 10 3 80 5 47152 30 103 5 103 2 106 2 90 7 10 3 80 5 47147 02 103 5 103 3 106 2 90 7 10 3 80 5 47140 71 1 1 103 5 103 4 106 2 88 7 17 3 75 5 5547 81 0 01 0 01 105 1 59 13 51 1 84 11 8146 04 0 01 0 01 2 105 1 59 13 51 1 84 11 8142 14 0 01 0 01 5 105 1.59 13 51 1 84 11 8128 50 0 01 0 02 104 1 30 16 58 1 60 13 7422 63 0 02 0.02 104 1 13 19 60 1 30 17 212561. Series system with Gumbel distributionSuppose that c1  10, c21  75, c22  35, and cd1  400. We assume two different costsfor the down-time due to failure in the second component: c d2  400 or cd2  200, beingT  1 and T  2 their corresponding optimum inspection times.Table 1 contains the optimum inspection times as well as the corresponding optimumcost for different mean times to failure of the system. The smaller EX 1 is, the smallerT  ; on the contrary, Q T   increases. In addition T  1  T  2 and Q T  1  Q T  2 , whichmeans that the higher the down-time cost is, the more frequent the inspection and,therefore, the more expensive the total cost.2. Parallel System with the Marshall-Olkin modelCosts: c1  10, c21  75, c22  35 and cd  400 (T  3 ) or 200 (T  4 ).Table 2 shows that the optimum inspection time, T  , is non-monotonic when the meantime to failure decreases. Moreover, T  3  T  4 and Q T  3   Q T  4 ; therefore, the con-clusions derived are similar to those in the previous example.Table 2E X n λ1 λ2 λ3 T  3 QT 3  T  4 QT  4 12150 104 105 7446 108 25 92 0 79 36 67 0 5611070 104 104 2967 108 40 71 0 50 57 37 0 3610632 104 104 3422 108 37 97 0 54 53 55 0 3810414 104 2 104 1229 108 60 05 0 34 83 16 0 2410371 104 2 104 1276 108 59 11 0 34 81 93 0 2510334 104 2 104 1317 108 58 32 0 35 80 90 0 2510290 104 2 104 1366 108 57 42 0 35 79 71 0 263213 5 104 2 104 1507 107 18 09 1 15 25 52 0 831215 103 103 1975 107 15 19 1 40 21 14 1 031058 103 2 103 1055 107 18 15 1 14 24 47 0 871053 103 2 103 1103 107 17 90 1 16 24 18 0 881048 103 2 103 1160 107 17 62 1 18 23 84 0 891041 103 2 103 123 106 17 30 1 21 23 45 0 91968 1 1 103 2 103 1358 107 16 56 1 27 22 48 0 96158 0 01 5 103 2421 106 4 34 5 51 6 06 4 21154 0 01 5 103 2628 106 4 19 5 71 5 86 4 35152 0 01 6 103 2030 106 4 62 5 18 6 40 3 98122 0 01 0 01 1976 106 4 47 5 33 6 13 4 1477 0 02 0 01 5142 106 2 96 8 63 4 13 6 72257ACKNOWLEDGEMENTWe would like to thank an associate editor of Qu¨estiio´ and the two anonymous refe-rees, for helpful comments that led to improvement of the paper. We also thank Ce´sarBerrade for his helpful assistance.REFERENCESBerrade, M. D. (1999). «Te´cnicas Estadı´sticas en Fiabilidad: Propiedades de Edad ba-jo Mixturas y Mantenimiento ´Optimo de Sistemas». PhD-Thesis, Universidad deZaragoza.Ebrahimi, N. (1997). «Multivariate Age Replacement». Journal of Applied Probability,34, 1032-1040.Gumbel, E. J. (1960).«Bivariate Exponential Distribution». Journal of American Sta-tistical Association, 55, 698-707.Lewis, E. E. (1987). Introduction to Reliability Engineering, Wiley, New York.Marshall, A. W. and Olkin, I. (1966). «A Multivariate Exponential Distribution». Jour-nal of American Statistical Association, 62, 30-44.Nakagawa, T. and Yasui, K. (1991). «Periodic-Replacement Models with Threshold Le-vels». IEEE Transactions on Reliability, 40, 395-397.Ross, S. (1996). Stochastic Processes, 2nd edition, John Wiley & Sons, New York.Vaurio, J. K. (1995). «Optimization of Test and Maintenance Intervals Based on Riskand Cost». Reliability Engineering and System Safety, 49, 23-26.Vaurio, J. K. (1997). «On Time-Dependent Availability and Maintenance Optimizationof Standby Units under Various Maintenance Policies». Reliability Engineering andSystem Safety, 56, 79-89.Vaurio, J. K. (1999). «Availability and Cost Functions for Periodically Inspected Pre-ventively Maintained Units». Reliability Engineering and System Safety, 63, 133-140.258

Maintenance policy under multiple unrevealed failures

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