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RAIRO. RECHERCHE OPÉRATIONNELLE B. VISCOLANI Random inspection schedules with n

RAIRO. RECHERCHE OPÉRATIONNELLE B. VISCOLANI Random inspection schedules with non- decreasing intensity RAIRO. Recherche opérationnelle, tome 26, no 3 (1992), p. 269-283 <http://www.numdam.org/item?id=RO_1992__26_3_269_0> © AFCET, 1992, tous droits réservés. L’accès aux archives de la revue « RAIRO. Recherche opérationnelle » implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/legal.php). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce ﬁchier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Recherche opérationnelle/Opérations Research (vol 26, n° 3, 1992, p. 269 à 283) RANDOM INSPECTION SCHEDULES WITH NON-OECREASING INTENSITY (*) by B. VISCOLANÏ (*) Comrmmieated by S. OSAKI Abstract. — The problem of minimizing expected cost until détection of failure is addressed here, using random checking (inspection) schedules. Non-homogeneous Poisson checking processes with continuons non-decreasing intensity are investigated. An approximation of the original problem, in the form of an optimal control problem, is discussed. An optimal solution is proved to exist and then it is characterized, using Poniryagin's Maximum Principle. Keywords : Reliability; random checking schedules; non-hornogeneous Poisson process; optimal control. Résumé. - Nous considérons le problème de minimiser le coût espéré jusqu'à la détection d'une panne, en employant des politiques d'inspection aléatoires. Nous étudions des processus d'inspection de Poisson non homogènes avec intensité continue non-décroissante. Nous discutons une approxima- tion du problème original, qui a la forme d'un problème de contrôle optimal Nous prouvons l'existence d'une solution optimale et la caractérisons par le Principe du Maximum de Pontryagin. Mots clés : Fiabilité; politiques d'inspection aléatoires; processus de Poisson; contrôle optimal. 1. INTRODUCTION We consider a variant of the classical problem of "minimizing expected cost until détection of failure", which is relevant for situations in which a human being, who may want his failure to remain undetected as long as possible, is the subject of the possible failure. The original problem, concern- ing an indus trial System subject to random failures, has been treated by Barlow, Hunter and Proschan ([1]; [2], pp. 108-116), who obtained optimal deterministic checking schedules. In their formulation, an event (failure) can (*) Received September 1991. The research was supportedby M.U.R.S.T. and C.N.R.-G.MA.F.A. (*) Dipartimento di Matematica Applicata ed Informatica, Universita' degli Studi di Venezia, Dorsoduro 3825/E, 1-30123 Venezia, Italie. Recherche opérationnelle/Opérations Research, 0399-0559/92/03 269 15/$ 3.50 © AFCET-Gauthier-Villars 2 7 0 B. VISCOLANI occur at a random time and its occurrence has relevant conséquences for the (industrial) system under study. Hence it is important to detect the failure as soon as possible and this can be done by checking (inspecting) the state of the system from time to time. Other authors have addressed the same problem [10,7,8,9], with the purpose of finding approximations of optimal inspection schedules. Now, if a human being is the subject of the possible failure, and if his interests conflict with those of the system, then from the system viewpoint it is désirable that the failing subject be unable to foresee the inspection times. Therefore deterministic checking schedules are no longer useful. As an example, let us consider the problem of inspecting the behavior of a taxpayer, who may try to avoid paying taxes on the revenues from a certain economie activity. We may assume that this illégal behavior begins possibly at a random epoch and continues, unless it is detected by an inspection. In some cases (e. g. "scontrino fiscale" in Italy) we may assume also that the illegality which has occurred at a certain time may be detected at that time only: afterward no trace of it can be observed, although a real loss of taxes has occurred. If the taxpayer knew the time of an inspection before it takes place, then he would resumé paying the tax just before the inspection. In this way, one would pay the taxes only on a minor part of his activity. Therefore a random inspection schedule is needed here and the loss of taxes associated with an illégal behavior will increase (linearly perhaps) with the delay in detecting it after it begins. In view of this sort of application, we consider stochastic checking schedules and search for approximations of optimal ones. The following assumptions define the system and its behavior. The system starts working at time 0 and the first system failure occurs at a time T, where T is a positive random variable with probability distribution function F and density ƒ: J = f(s) ds, Jo The first failure is relevant if and only if it occurs by a fîxed fînite time tx >0. The density ƒ is a continuous function on [0, tx] and 0<F(tl)<L (2) Thus the event of a failure occurring in [0, f J is possible, but not certain. Inspecting the state of the system (which can be either "failed" or "working") has a constant cost c0, takes a negligible time and does not influence the system performance. Moreover, let l(x) be the cost due to the delay x from a failure to its détection, where l(x) is assumed to be a strictly increasing, Recherche opérationnelle/Opérations Research RANDOM INSPECTION SCHEDULES 271 concave and continuously differentiable function. If Te[0, tx] and if X is the failure détection delay, then T+ X is the time of discovery of the first failure and we stop the checking process immediately after it. If T>t±, then the failure need not be detected and the checking process ends on the first check after the instant tv Thus the "détection delay loss' is (3) where 1£ is the indicator function of the event E. A checking (or inspection) schedule S={yk:k^l}y 0<yk<yk+u * * 1 , (4) is an increasing séquence of points in time. In these terms, the final time of the checking process is yM, where M=M(S,T) = min{k:yk>T A tuykeS}, (5) where A A B — min(A,B). If T^tt, then yM is the time of détection of the first failure, yM = T+X, where X is the détection delay. Otherwise, if T>tl9 the failure is unimportant and yM can very well be less than T, or also more than T, as it is when T^tv The éxpected total cost resulting from the inspections and the possible (first) failure is then (6) and we want to détermine an inspection schedule S={yk} minimizing it. In the following, we dénote by à {f) the derivative of any function a(t), depending on the time /, a(t) = da(t)/dt. Moreover, we dénote by l'(x) the derivative of the function l(x% l'(x) = dl(x)/dx. 2. NON-DECREASING INTENSITY POISSON CHECKING PROCESSES Let N(t) dénote the Poisson process with intensity n(t)^0, and dénote by Sp(n) the checking schedule SP(n)={Yl9Y2t...,Yk,...}, (7) where Yk is the occurrence time of the fc-th event concerning N(t). Then N(t) is the number of checks during the interval [0, f]. We will refér to N(t) as to the Poisson checking process (PCP). The éxpected number of checks in the vol. 26, n* 3, 1992 272 B. VISCOLANÏ interval (f, t + s] is (jee [4], p.48) E[N(t + s}-N(t)]= f 5 n(w)dw, t, s^O. (8) Here, we restrict our attention to a special family of monotonie intensity fonctions. Assume that the Poisson checking process intensities are conti- nuous, non-decreasing, piecewise continuously differentiable. Assume further that the derivative of the intensities is bounded in [0, tx] and null in [f x, - + • oo). The conditional expected number of inspections is, by (5), (9) whereas the conditional expected détection delay loss is, by (3), \ \ (10) There is no simple way of expressing (10), but the foliowing Lemma gives us a useful upper bound. LEMMA 2.1: If N(t) is a time-dependent Poisson checking process with non-decreasing intensity n(t), if T is the first failure time, independent of all checking times, then E[X\T\^\jn{T), Moreover, if Z(x) is an increasing and concave function, then .£[/(X)| 71^/(l/n(7)). (11) Proof: The occurrence of the first check after t at a time y dépends only on the intensity n(w)y we(t,y], whereas it is independent of the occurence of other checks in the interval (0, t]. Thus the delay of the first check after an arbitrary time instant t>0, independent of { Yk}9 has the same distribution as an inter-checking interval which starts from the time t. Hence its mean is E[X\T=t]= exp - n(w)dw \dx^ exp(-n(t)x)dx=lfn(t). Jo L Jt J Jo Now, since l(x) is concave, Jensen's inequality ([5], p. 153) implies that E [l (X) \T\<Ll(E[X\ T\) and hence the thesis foliows, because / is increasing. • Notice that the smaller «(/J — n(t), the tighter the above inequality. This is interesting in particular in those applications, where a physical bound to the increasing rate of the inspection intensity is present. In the special case Recherche opérationnelle/Opérations Research RANDOM INSPECTION SCHEDULES 273 in which the intensity is constant, i.e. when the PCP is homogeneous, the equality sign holds in (11). From Lemma 2.1, we obtain that the expected total cost satisfies the inequality + f'Tco \'n(w)dw+l(l/n(i))\nt)dt. (12) Jo L Jo J After uploads/Science et Technologie/control-optimo 1 .pdf