Choice of the best time scale for preventive maintenance in heterogeneous environments

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 98 (1997) 64-74

Theory and Methodology

Choice of the best time scale for preventive maintenance in heterogeneous environments I. Gertsbakh a,*, Kh.B. Kordonsky h a Ben Gurion University of the Negev, Department of Mathematics and Computer Science, P.O. Box 563, 84105 Beer-Sheva, Israel b Brookline, MA, USA

Received January 1994; revised July 1995

Abstract

A principal feature of the model considered in this paper is the presence of several time scales for lifetime measurements when similar objects operate in heterogeneous conditions. A typical example are lifetimes of an aircraft module which can be measured in the total time in air and in the number of flights. A family of lifetimes can be generated by considering a linear combination of the above two principal time scales. We consider a problem of finding an optimal maintenance period for an equipment operating during a single maintenance/replacement cycle, in heterogeneous environmental conditions. Our main concern is finding the time scale which provides the maximal value of the return (cost) functional. We consider two "principal" time scales (the operation time and the total number of shocks) and show numerically that the optimal linear combination of these two scales has also the minimal coefficient of variation (c.v.) of system lifetime. We develop some general theory to connect the optimality in terms of the return functional with the optimality in terms of c.v. @ 1997 Elsevier Science B.V. Keywords: Optimal time scale; Mixtures; Coefficient of variation; Optimal maintenance

1. I n t r o d u c t i o n

Typical models considered in the theory of Preventive Maintenance (PM) deal with objects whose lifetime cumulative distribution function (cdf) F ( t ) is completely specified, see e.g. Block et al. (1988), Gertsbakh (1977), and Valdez-Florez and Feldman (1989). One o f the major difficulties in implementing these PM models lies in the fact that similar objects (cars, aircraft engines, etc.) operate in heterogeneous field conditions. For example, jet engines are used by different companies in different operation regimes, * Corresponding author.

i.e. in flights of various duration and altitude, and in various atmospheric conditions. Speaking in formal terms, this means that the cdf of an engine lifetime is a mixture of several cdf's. It is difficult to work with mixtures: often, they do not possess monotone failure rate, they have a complicated analytic form, and--this might be the worse--the probabilistic weights of the mixture components are known very approximately. This situation is quite well understood by maintenance and reliability engineers. Their way of handling heterogeneity is, in fact, introducing different time scales for lifetime measurements (Kordonsky et al., 1983; Kordonsky, 1984; Kordonsky and Gertsbakh, 1993; Neese, 1984). The life of a car engine can be

0377-2217/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. SSDI 0377-2217 (95) 00251-0

L Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

measured in the total number of operation hours, in total calendar time in use, in the number of warmups, in mileage, etc. After studying lifetime distributions in various time scales, the nomination of PM actions is done usually for each particular part of the system in the most appropriate time scale. For example, PM of aircraft undercarriage is done, say each 4,000 flights (landings), while the checkups of the fuselage are scheduled and carried out in calendar time, say each 20,000 flight hours. This paper deals with formal aspects of finding the optimal time scale for PM. To the best of our knowledge, this is one of the first trials in this direction ( Gertsbakh and Kordonsky, 1994). Let us mention one aspect of the connection between the PM and the time scale. As soon as there are at least two different time scales, there is always a possibility to create infinitely many new time scales simply by considering linear combinations of the existing scales. For example, one can measure lifetime and plan PM in a "combined" time T where T = H -- 0.5L, H being the number of takeoffs and L being the number of operation hours. For each system, an appropriate return functional is used as the measure of PM efficiency. The goal of our study is to find that time scale which is the best within a defined class of time scales, with respect to a specified return functional and to a specified family of cdf's. The paper is organized as follows. In Section 2, we formalize the notion of Operational Conditions (OC), introduce a return functional for PM and consider an example of finding optimal PM in a fixed time scale. In Section 3 we introduce a damage accumulation model, and derive from it two time scales. Then we extend the example of Section 2 to this situation and compare optimal PM rules in these scales. We also define a family of time scales generated as a linear combination of our "principal" time scales. We find out for our example the best linear combination of the time scales which optimizes the PM return functional over the whole family of time scales. In parallel, we calculate the coefficient of variation (c.v.) of system lifetime in various scales and discover numerically that the best time scale for PM has also the smallest c.v. Section 4 is devoted to the optimal choice of a time scale for a return functional which is a generalization of the functional considered in the examples. We assume here that for heterogeneous environ-

65

ment, the return functional is the mean of return functionals derived for fixed operational conditions, and that the only information known is the c.v. for each "principal" time scale in the family. Then we demonstrate that the time scale with minimal c.v. has the property that it optimizes the return functional in minimax sense: it guarantees the maximal cost under the best choice of the PM period and for the worst mixture of the lifetime distributions within a given class. Summing up, this paper has two distinct but connected lines of exposition. The first one deals with a specific return functional and is concerned with its optimization b y a Proper choice of the time scale. The second line (Section 4 ) d e a l s with a more general return (cost) function and is concerned with its minimax-type properties.

2. Heterogeneous environment. Cost functional and an example of optimal PM for a fixed time scale Denote b y / 2 the set of all possible operational conditions (OC). A member of this set is denoted by Q, and it will be assumed that a probabilistic measure PQ is defined on g2. One can think that the operating system "chooses" the OC Q with probability dPQ. Let F(tIQ) be the system lifetime cdf under Q. The underlying cdf is the following mixture:

FE(t) = f F(t IQ) dPQ.

(1)

J2

It will be assumed in this section that there is only a single tim e scale, say operation or calendar time. Denote by B(z, F ]Q) the average return (cost) accumulated by the system operation under Q when the PM policy is characterized by the parameter z (for instance, z might be the period of planned replacement). Suppose that we can find the optimal parameter z0 (Q) maximizing B ( z, F [ Q ). Then the maximal return is expressed as Bmax(F)

= f B(zo(Q), F[ Q) dPO.

(2)

Suppose, as it is usually done in practice, that a single PM parameter z0 is chosen for the whole assembly

66

L Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

of operating objects. Then the return functional for randomly chosen Q will be defined as

z

B ( z , F ] Q ) = c ~ / ( 1 - F ( t I Q ) ) dt 0

B(zo, F, X2) = / B(zo, F I Q) dPQ.

(3)

- [ C + D . F ( z [Q)] z

12

Obviously, B ( zo, F,, Y2) ~ Bmax(F). The standard approach to finding the optimal PM parameter is fitting a single distribution to the lifetime data (which in fact are generated by the mixture ( 2 ) ) , see e.g. Gertsbakh and Kordonsky (1969). This approach is justified by the absence of exact knowledge of F ( t I Q) for each Q, and by a "fuzzy" knowledge of the weights PQ. Besides, the set of observed lifetimes is usually rather small. E x a m p l e 1 (Finding an optimal PM parameter for a mixture of lognormal distributions). Suppose that for each Q ~ ~ the lifetime is lognormal (with o- = 0.5) :

F(tiQ)=@(Int--o.a-(Q)),

(4)

where ~ ( x ) is the standard normal cdf. Here only the location parameter a(Q) depends on OC. This model might be realistic for fatigue failures (Gertsbakh and Kordonsky, 1969). Suppose that a(Q) itself is random and is uniformly distributed in the interval [a0 - 0.5, a0 + 0.5]. Then a0+0.5

I o(In a) da.

(5)

a0--0.5

The graph of this cdf is similar to a lognormal one, and a very large sample is needed to discriminate between (5) and a lognormal distribution. Suppose that it is decided to fit the latter to the experimental data. By equating the first two initial moments of the mixture to the corresponding moments of the lognormal distribution, we find the fitted distribution: F ( t ) = @( (In t - a0)/0.577).

(6)

Suppose that our system operates on the interval [0, z ], and at time t = z it undergoes a PM (replacement). We introduce the following return functional (for fixed Q ) :

-/3 f F(tlO) dt.

(7)

0

Here ce is the income in $ per unit of operational time, /3 is the penalty in $ per unit of nonoperational time. Note that the term f o ( 1 - F(t t Q)) dt is the average up-time on the interval [ 0, z ] and the second integral is the average down time on the same interval. (Without loss of generality, it can be assumed that a = 1.) At time t = z, the equipment is "docked" and inspected, which costs Co. If the inspection reveals failure, which happens with probability F ( z [Q), it will be eliminated for a cost Cr. Otherwise (no failure discovered), the equipment undergoes a checkup and maintenance, and this costs Cp. Thus the extra costs for inspecting and repairing the equipment and paid a t t = z are

C(z) = Co + F(z I Q)Cr + (1 - F(Z ]Q) )Cp. They have the form of the second term in (7): set C = C 0 + C p and D = Cr - Cp. Such type of return functional is appropriate if it is decided to take into account only the income and the cost accumulated during the first PM cycle, e.g. during producer's warranty period [0, z ]. The functional of type (7) is appropriate for an equipment which cannot be monitored continuously, say an equipment operating in arctic areas or on a satellite whose mission interval [0, z] cannot be interrupted by inspections. It is often customary to include into the functional also the income and costs/losses accumulated during the subsequent operation/maintenance cycles. One might say that (7) reflects an alternative, "myopic" approach to counting the income and costs. The difficulty with including the "future" income and cost into the functional is that they depend on the future cdf of the lifetime which is usually unknown and which can differ significantly from the original cdf, due to maintenance, technological changes, aging; etc. The functional (7) is adequate also for a "single-mission" equipment which permits at most one failure.

L Gertsbakh, Kh.B. Kordonsky/EuropeanJournalof OperationalResearch98 (1997) 64-74 The optimal PM period can be found from the equation

dB(z, F[ Q) dz

=0,

(8)

which can be easily solved numerically. For the cdf (4), (7) takes the form:

B(z, F I Q )

-[C+D.q~(lnz-:(Q))].

(9)

Replacing F(t] Q) by P(t) in this formula, one can obtain the functional for the fitted cdf (6). The functional (7) is linear with respect to the cdf F(t] Q). Therefore, the functional for the mixture FE(t) is the corresponding mixture of functionals B ( z, F [ Q). There are no principal difficulties in finding the optimal z-values for F(t) = F ( t ) and for F(t) = Fe(t). Explicit solutions can be obtained, for example, for c~ = /3 = 1 and C = D = 0. Then the optimal value of the return functional B(z, F) is max z B(z, F) = ea° •0.787. It is attained, according to (8), at z0 = ea°This value of the PM period is optimal with respect to the cdf F. In reality, the true cdf is the mixture Fe(t). Therefore, the true value of the functional B(z, F e) which corresponds to Z0 equals to B(go,Fe). We omit tedious calculations and present the final result: B(~o, F e) = ea° -0.644, which is considerably less than B (Z0, ie). This phenomenon may happen: replacing one cdf by another one results in a considerable change in the value of the PM period and reduces the return functional. To conclude this example, let us imagine a situation when for each member of the whole population of operating systems we know exactly for each Q the corresponding F ( t I Q), Then for each system we can choose the optimal PM period according to (8). This will produce the maximal return Bmax, see (1). We omit technical details and present the final result: Bmax(F) = ea° • 0.826.

67

3. Introducing diffei~nt time scales

3.1. Damage accumulation model in two different time scales In this section we introduce a failure model which leads to two different time scales. We describe a situation in which the lifetimes in both these scales are negatively correlated. A set of OC's is introduced and the PM scheme is considered in both time scales. It will be demonstrated that one of these scales has a clear advantage over the other. It will be shown also that an appropriate linear combination of these scales can provide even better results. We need some new notation. Time scales will be denoted by script capitals/2, ~ , T, etc. Random variables (lifetimes) will be denoted by italic capitals L, H, T, and lower case italics is used to denote nonrandom variables. If there is a need to stress that a certain quantity ce is defined in a particular time scale T, we will write it with a subscript, like otT-. Let us consider the following simple failure model in time scale /2. Unit shocks appear according to a Poisson flow with intensity A, each single shock leads to a unit damage with probability q, q << 1. Failure appears if the total number of damages reaches the critical number m. It is supposed that there is a set g2 of OC's, and that for a particular Q E g2, model parameters depend on Q: A(Q), q(Q). Obviously, the lifetime L in/2 for a fixed Q is Gamma(m, A(Q)q(Q) ). One may think of 12 as of a "regular" operation (calendar) time scale. Another time scale is obtained by means of a shock counter. (Landing of an aircraft is a good example of a shock.) Each single shock leads to a single damage with probability q. This creates another time scale denoted by ~ , the"landing scale". Under our assumption that q is small, the geometric distribution of the landings needed to cause a single damage can be closely approximated by an exponential distribution with parameter q. The lifetime H is also GammaCm, q(Q)), where Q denotes the particular OC. We assume that m is large enough and that the Gamma distributions inf. and 7-/can be approximated by the Normal distribution, see e.g. Gertsbakh and Kordonsky (1969), Kordonsky and Friedman (1976), Lawless (1982). The corresponding means and standard deviations are

I. Gertsbakh. Kh.B. Kordonsky/European

68

Table 1 Lifetime parameters

for various Qi

Qi

MQi)

q(Qi). lo3

E[LIQil

dL,I Qil

ElHl Qil

glHI Qil

Ql

1.oooo 0.7857 0.6562 0.5000 0.3958 0.346 1 0.2857

4.167 4.545 4.762 5.000 5.263 5.555 6.250

6000 7000 8000 10000 12000 13000 14000

1200 1400 1600 2000 2400 2600 2800

6000 5500 5250 5000 4750 4500 4000

1200 1100 1050 1000 950 900 800

Q2 Q3 Q4 Q5 Q6 Q7

loooo rT-----

E[L I Ql = m/(NQMQ)), 4L 1Ql = fi/(UQ>q(Q>>,

E[HI dHl

Ql =mls(Q)v Ql

Journal of Operational Research 98 (1997) 64-74

(10)

= v'+dQ).

SOOO-

The cdf’s are: x-E[LlQl

F~(xlQl=@

4LlQl Y -E[HlQl

h(~lQ>=@

dHlQ1

b%

>’

>.

(11) 0

Example 2. The set 0 = (Qt,. . . , Q7), each Qi appears with probability l/7. The corresponding values of A(Q) and q(Q) are presented in Table 1. An assumption is made that the failure probabilities increase when the Poisson flow rate decreases, according to the principle that less frequent shocks are more severe. The last four columns of Table 1 give the mean values and standard deviations of the lifetimes in both scales computed according to ( 10). The mutual location of the means for various Qi, together with their standard deviations on the (L, IFI) plane is shown on Fig. 1. It is easy to see that Cov[ L, H] < 0. The true lifetimes in & and ‘7-Lare the following mixtures:

- dA(Qi)q(Qi) Ff(X)=$Dxfi(WQi>s(Qi>)-l

>-’

1=l

(12) The fitted lifetime distribution for each time scale will be a Normal distribution whose first two moments

0

5000

10000

15000

20000

Fig. 1. The location of conditional means E[ L 1Qi], E[ H 1Qi] on the (Is, H) plane. The sides of the triangles designate the corresponding standard deviations. K* is the optimal scale with the smallest C.V.

coincide with the corresponding Note also that

moments

of ( 12).

Remark 1. It is important to clarify the connection of our Example 2 to a general situation with two time scales C and I-L (e.g. operation hours and takeoffs) considered in Introduction. Operation hours and takeoffs create two “independent” wear accumulation processes which have a joint (possibly additive) effect on a fatigue failure, while our example represents only one type of wear mechanism: accumulation of damages caused by shocks. Statistically, however, the location of failure points on the (L, H) plane for fixed operational conditions Q is quite similar for both cases: a scatter with positive covariation between L and H.

I. Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

69

Table 2 Parameters of preventive maintenance in/~ and 7-I scales Scale

z0

B(zo, F E)

Zo

B(ZO, F)

B(~o, F E)

L 7-/

3952 2390

3378 4316

3108 2243

2050 3627

2091 3991

The mixture of such scatters when several OC's are considered may create statistically equivalent pictures with overall negative correlation between L and H times. This negative correlation explains, as we will see below, the advantages of introducing an appropriate linear combination of £ and 7-l.

3.2. Optimal preventive maintenance in £ and 7-[ time scales We will use the functional (7) to find out the optimal PM in both scales 12 and 7-/. There is a problem of defining and comparing incomes (costs) in different time scales. Suppose that we use (7) in the £ time scale (operation hours) for which the cost coefficients a and/3 are known. They have the dimension of $ per unit of £ time. If we want to write the same functional for another time scale 7-/, these coefficients must be recomputed so that they acquire the dimension o f $ per unit of 7-/time. To do this, note that the same object lives on the average in the £ scale E [ L I Q] units of time (under OC Q), while under the same Q in the 7-/scale the average lifetime is E[H] Q]. Thus, a unit of/2 time is equivalent to E [ H I Q ] / E [ L I Q ] units of 7-t time. For example, the average aircraft life in flight hours is, say 50,000, and during its life the aircraft does 10,000 landings. Then one flight hour is equivalent to 0.2 landings, and 1 S/hour should be equivalent to 5 S/landing. This leads to the following formulas:

a ~ ( Q ) - E [ L I Q ] cec,

E[HI a]

E[LI O] /3L. ~7~(O) - E[HI a]

(13)

E x a m p l e 2 (continued). Let us carry out the computations by assuming that for all Q, otc = 1,/3L = 99, C = 100, D = 1000. (C and D have already the desired dimension of $.) Table 2 shows the results of

calculating the optimal PM parameters in both time scales, z0 is the optimal PM period which corresponds to the "true" mixture distribution, B (z0, F E) is the corresponding maximal return. Z0 is the optimal PM period for the fitted cdf F, with the return equal to B(~0, F ) . Finally, B(Z0, F E) is the actual return for the PM with period ~0 which is applied when the true cdf is F E. Important is the conclusion that the 7-/scale is better than the £ scale: it provides considerably higher return values. This might be explained by the fact that the conditional means E[H[ Qi] are less scattered around the general mean than the corresponding E [ L IQi] values.

3.3. Linear combinations of £ and 7-[ scales If there are two different time scales, it is always possible to create a new scale which is their linear combination: 35 =/2 + g . 7-/, g / > 0 .

(14)

By the definition, (14) means that the lifetime in the 35 scale is obtained according to the formula K = L + gH, where L and H are the lifetimes in the £ and 7-/ scales, respectively. Eq. (14) defines a family of time scales by varying g E [0, cx~). What time scale in this family is the best one? In Kordonsky (1984) and Kordonsley and Gertsbakh (1993), a heuristic principle was suggested that the best time scale is the scale which has the smallest coefficient of variation (c.v.) of the corresponding lifetime. It is a matter of routine calculation to derive that the minimal c.v. is attained at g = g* given by the following formula:

g,

E[H]Var[L] - E[L]Cov[H, L] E[ L ] Var[ H] -- E[ H] Cov[ H, L ] "

(15)

Note that g. ~> 0 if the covariance between L and H is negative. If g* by (15) is negative then either £ or 7-/is optimal.

1. Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

70

Example2(completed). All terms in

(15) are known except for C o v [ H , L ] . Note that it can be written as 7 1 C o v [ L , H ] = ~ ~ E [ L . H I Qi] - E[L]E[H], i=1 (16) where E[H] 1

= ~

~i7=1E[HIQi]

and E[L]

=

7

7 ~ i = I E [ L I Q i ] , see Table 1 and Fig. 1. Using conditioning on the number of shocks needed to have a single damage, it is easy to derive that E[ L.H[ Qi] = m/ h(Qi)q2(Qi) +m2 / A(Qi)q 2 (Qi) m/A(Qi)q(Qi). For a numerical example we took m = 25. Calculations show that g* = 4.75. It is instructive to interpret (14) geometrically. The time scale given by the formula K~ = (L ÷ g * H ) / ( 1 + (g,)2)V2 is nothing but a projection of a two-dimensional random variable (L, H) on the axis which is collinear to the vector ( 1, g*). The lifetime K~ differs from K* = L + g*H by a multiple which does not affect the c.v. One could conclude from Fig. 1 that the lifetime K* has smaller c.v. than L or H. Indeed, calculations show that c . v . [ H ] = 0 . 2 4 , c . v . [ L ] = 0.36, and c.v.[K*] = 0.20. Another interesting feature of the time scale/C* is that the subpopulations corresponding to different Qi are practically indistinguishable in this scale and look the same, see also a similar example in Kordonsky and Gertsbakh (1993). Now compute the PM parameters for the time scale /C*. For each fixed g, we assume that the cdf of K = L + gH is a mixture of Normal distributions with parameters E[ K I Qi] = E[ L [Qi] + gE[ H [Qi] and V a r [ K ] Qi] = War[L I Qi] -}- 2 g C o v [ L , HI Qi] qg2War[H I Qi]. The functional becomes a linear combination of functionals corresponding to ai, i = 1 . . . . . 7. Return (cost) reduction to a common time scale £ was done using (13), i.e. we put oz~(Qi ) = o~£E[ L [ ai] / E [ K[ Qi] = otz2/ ( 1 + gA(Oi) ). The following results were established. The maximal return for the mixture is B(zo, F~:. ) = 4,558, z0 = 3155. The maximal return for the fitted cdf in the/C* scale is B(Z0, Pr* ) = 4,731. The true return when the optimal PM period go is applied to F~. is equal to B (z0, F~. ) = 4, 531. Therefore, the/C* scale is better than both £~ and 7-[ scales (compare with Ta-

ble 2). Numerical analysis reveals that g = g* = 4.75 guarantees the highest value of max B(zo, F E) in the whole family of lifetimes (14). This is a remarkable coincidence: the time scale with the smallest lifetime c.v. is the best in terms of the PM return functional. It is also interesting to observe that the c.v. decreases from 0.36 to 0.20 when g grows from 0 to 4.75 (on the grid with a step of 0.025) and increases when g increases beyond 4.75. The functional B (z0, F~. ) has an opposite behaviour: it grows monotonically from 3,379 to 4,558 when g grows from 0 to 4.75. Afterwards, the functional monotonically decreases to the value 4,316.

Remark 2. The fact that a linear combination o f / 2 and ~ scales is better than any of the single time scales might seem counterintuitive. Indeed, for each fixed Q both time scales in Example 2 are in fact proportional to each other and it seems that nothing could be gained by considering their linear combination. This is true if we have only one OC Q. A new situation arises when several OC's are present. The mixture of lifetimes in the £ scale is rather flat, see Fig. 1. The mixture in the 7-/scale has a smaller c.v. The c.v. becomes minimal when the lifetimes are projected on the/C*-axis, see Fig. 1. All mean values are projected onto a very narrow region on this axis and this explains the decrease in the c.v. The decrease of c.v. alone guarantees a good performance of the PM. At the same time, due to the mixture, the overall covariance between L and H becomes negative, while for each fixed Q it is positive. The presence of a mixture of several distributions typically has a negative effect on the efficiency of the PM in some time scales and may have a positive effect in some other time scales. Somehow this fact attracted little attention in reliability theory and practice.

4. Best time scale for preventive maintenance

4.1. Introduction. The family of lifetime distributions What in practice is a quite reliable rule of t h u m b - take the time scale with the smallest c . v . - - i s not so easy to justify formally for general PM scheme. In this section, we will specialize the form of the return functional B(z, F), restrict ourselves to a particular

I. Gertsbakh, Kh.B. Kordonsky/ European Journal of Operational Research 98 (1997) 64-74

family of lifetime distributions, and compare PM in different time scales using a minimax-type criterion. Suppose we have two principal time scales /2 and ~ , and we introduce the class of time scales as T =/2 + g . 7-{, g / > 0 .

=f B(z, FT-(tlQ))dPe.

(18)

We are interested i n finding such time scale To which would provide the largest value of maxz B(z, FT, g2) over the set of all time scales (17). Let us fix a time scale T in the class (17) and introduce a family of lifetime distributions for this particular time scale. To simplify the notation, we omit the subscript indicating the time scale. Let us assume that for each Q E g2, the lifetime has the cdf F (t [ Q) = Fo(t/c(Q)). Here c(Q) is the scale parameter, and F0 (t) is some "standard" distribution. Suppose that Q is selected from a finite set g2 with probability p(Q), so that the lifetime is the following mixture:

F(t) = ~ p ( Q ) F ( t

I Q).

(19)

QEg2

We assume that the s e t / 2 has at least two elements. It is easy to see that

E[ H I o ] _ c~(O) E[L[ Q] c£(Q)

maximal value at some x0 > 0. It is assumed that ~g ( z ) decreases monotonically when z moves to the left or to the right away of x0. Then the class of return functions we will be dealing with is of the following form:

(17)

Denote by B(z, FT, ~0) the return functional for the maintenance period z in time scale 7-. The corresponding cdf of the lifetime is FT. Then let z0 (FT-, g2) be the optimal choice of z in time 7-, and let maxz B(z, FT,/2) = B0(FT). Suppose that for OC Q the corresponding lifetime is FT- (t I Q). Then, for a random choice of Q the return functional will be defined as

B(z, FT-,g2)

71

(20)

Indeed, E[H[ Q] = f~'~( 1 -FT-t(t] Q) ) dt = f o ( 1 Fo(t)) dt - cT-t(Q). In words: for fixed Q, the ratio of lifetime mean values equals to the ratio of scale coefficients.

4.2. The return functional Suppose that g~(z) is a unimodal differentiable function defined for z ~> 0 and attaining its unique

B(z,F, d2) = Z

~b(c(Q))gt(z/c(O))P(Q)'

(21)

QcO

where ~b(-) is a positive function. Let us demonstrate how (21) can be specified to become applicable to several practical situations. First, note that (21) is a generalization of the previously introduced functional (7). Indeed, (7) can be written as

B(z, FIQ)

[z

= c(Q) ~

c-~ +

c(Q)

J

z/c(Q)

- (ot + /3) f

Fo(t) dt 1.

(22)

0

Therefore, the linear combination of B (z, F I Q) with weights p(Q) has the form (21). Let us omit simple calculations showing that the expressions for the functionals in time scales /2 and 7-(, BL and BT~, differ only by the c-coefficients in the arguments of q~-functions. As a second example, let us consider system stationary availability for age replacement (Gertsbakh, 1977, Chapter 2). The system is replaced at the failure or at the prescribed time z, whatever happens first. The renewal after :failure lasts a ( f ) , the replacement lasts a (r). The stationary availability A ( z, F ) is

A(z,F) f o ( 1 - F(t)) dt .

f o ( 1 - F(t) ) dt + a ( f ) F ( z ) + a(r)F(z) (23) This formula was investigated in Gertsbakh (1977) and it was demonstrated that typically it has a unique maximum. Now write i t for the time scale ~ and OC Q and note that FT~(tIQ) = Fo(t/c~(O)). It is assumed that a ( f ) / E [ H I Q ] = 131, and a(r)/E[H[ Q] =/32, where/31,/32 remain the same in any time scale and for any Q. Then, after simple algebra and using (20), we arrive at the expression

72

I. Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74 z/c~(Q)

A(z, FT~,Q)=(

f

Therefore,

(t-Fo(t))dt)

( c . v . [ H ] ) 2 = E[H2]/E2[H] - 1

0

~~aEs?P(O)c~(O) tzO - - (~QEop(Q)c~(Q) ) 2 (/x°) 2

z/cT~(Q)

x (

/

( 1 - Fo(t))dt+ txo(fllFo(z/cT~(Q))

1.

(27)

0

+ fl2~'o(z/cT-t(Q) ) )) -1

(24)

It is of the desired type ~ (z/cT-t), and the corresponding linear combination has ~b(.) = 1. Let us define the return functional for the whole system as the average of return functionals over the set Q E 12. For example, for the 7-/ scale, the functional will be a linear combination of A(z, FT~,Q)functionals with the weights p ( Q ) , Q E 12. Very popular in preventive maintenance are functionals of type g~*(z) = [a + F ( z ) ] / f o ( 1 F(t) ) dr, which express costs in $ per unit time for the age replacement, see e.g. Barlow and Proschan (1960), Block et al. (1988), Gertsbakh (1977)• Typically, g~* has a unique minimum point• By considering ~F ( z ) = _gt* ( z ) the problem is reduced to finding the corresponding maximum point• Let us check the situation with the dimension. Suppose that the OC is Q and the time scale is ~ . Then the dimension of gt ( z ) is $ per unit of ~ time. Dividing by E[ L 1Q] and multiplyingby E[ H I Q ], the dimension is brought to $ per unit of/2 time. By invoking (20), the functional becomes [ 1/cc (Q) ] • gt (z/cT~ (Q) ). Then the linear combination will have the form (21) with ~b(cc ( Q ) ) = 1/cL(Q). Thus again the functionals in/2 and 7-[ scales differ only by the c-coefficients of the g~-functions.

From now on we assume that the set /'2 is finite and that the scale parameters are ordered: c(Q1) c(Q2) ~< -.. ~< C(QN), N ~> 2. (The subscript indicating the time scale is omitted.) We need a property of the c.v. about its behaviour when the smallest c(Q1) decreases and the largest c(QN) increases. Claim 1. The expression

~Nl p(Qi)c2(Qi) f(C(Ol) . . . . . C(ON) ) = (~_~NIp(Oi)c(Oi) )2 (28)

increases monotonically as c( Q1) -+ 0 and C( QN ) --+ e~. Its supremum is at least as large as 1/p(QN). Proof. It easy to check that Of/oc(al) < 0 and that Of/OC(QN) > 0. Thus the supremum will be attained as c(Q1) is zero and c(QN) goes to infinity• In the latter case, l i m f = 1/p(QN). [] We would prefer the time scale To over the time scale ~ if

maxB(z, F%, 12) > maxB(z, FN, 12). z>0

z>0

(29)

We say that the time scale To is optimal in the strong sense if

max B(z, F%, 12) ~ max B(z, FT-, 12) 4•3. Comparing cost functionals in different time scales Let us derive the expression for the c.v. of FT~(t). L e t / z ° and/x ° be the first and the second initial moments of Fo(t). Then it is easy to check that

E[H] = ~ p(Q)c~(Q)tx °,

(25)

QEs2

E [ H 2]

Z p ( O ) c 2 ( Q ) t z °. QE/2

(26)

z>0

z>0

for any T in the family (17). In practice, we rarely may have an analytic form of FT- for any T, and thus finding the best scale is a solvable problem only if we have a model providing FT- in a closed form. More realistic is an assumption that our knowledge about the cdf is restricted to its coefficient of variation. Suppose that the lifetime in the scale T has a c.v. equal to ~T. Then denote by S(~T) the set of all scale parameters (c(Q1) C(QN) ) which provide the given value of the c.v.: . . . . .

I. Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

S(8~r) = (cT-(Ql) . . . . . cT(QN): c.v.[F~-(t) ] = & - ) .

(30)

Definition 1. Time scale To is called optimalin a wide sense if for any time scale 7- in the family (17)

Now fix one particular scale 7- and consider the set S(~T). Define the following sequence of vectors:

(c~-,n = 1,2 . . . . ): (bn(Q1),c%(Q2) . . . . . c % ( Q N - I ) , b n ( Q N ) , n = 1,2 . . . . ),

min max B (z, F%, ~2)

73

(33)

~(6%) z>0

(31)

>~ rain max B (z, F T , / 2 ) . S(6~-) z>0

This definition can be explained as follows. Suppose, we look for the best PM parameter z in some time scale 7" to maximize the return functional. The "nature" is playing against us by changing the scale coefficients c( Q ) , Q E g2 within the class S(67-) and thus minimizes our gain. Then the worst possible time scale in the class S(6%) can not be worse then the worst possible time scale in S(67-). The main result of this section is the following

Theorem 1. Suppose that (i) the time scale To has the smallest c.v. in the class (17); (ii) for each 7", the corresponding c.v.[T] < Oz. 0)2 (3//P(QN) -- 1) °"5, where 3/= ]%2/t]J.1 Then To is optimal in a wide sense in the class (17). .

The proof is based on the property of the c.v. stated in Claim 1 and on the following: Claim2. Suppose that the smallest c(Q1) in qt(z/c(Q1) is replaced by a positive b(Q1) such that b(Q1) < c(Q1), and the largest C(QN) in ~F(z/C(QN) is replaced by b(QN) > c(QN). Then maxz>0 B ( z, E O) can only decrease under this replacement. The role of (ii) is to guarantee that there is a possibility to apply the operation of increasing C(QN) and decreasing c(Q1): the actual c.v.(T) must be below the value (3//P(QN) -- 1) 0"5, where 3/=/z~/(/z~) 2. The proof of Claim 2 is given in Appendix A.

Proof of Theorem 1. Suppose that the min max of the left-hand side of (31 ) is attained at z* and for the set c(TO) E S(3%) of scale coefficients c(TO): c%(Q1) ~< c%(Q2) <<.... <<.cT-o(QN). (32)

where bn(Q1) -+ 0, bn(QN) -+ ~ as n -+ c~. According to Claim 1 and (ii) of Theorem 1 it is always possible to find b*(Q1) and b*(QN) such that b* (Q1) < c% ( Q I ) , b* (QN) > c% (QN) and the vector (b* (Q1), C7~ ( Q 2 ) . . . . . c % ( Q N - 1 ), b* (QN))

s(6~-). By Claim 2, max B(z, Fzr, g2) cannot exceed max B (z, F%, ~ ) , which proves (31) for any fixed time scale 7" and thus for the whole class (17). []

5. Conclusion Using several "parallel" time scales for planning preventive maintenance is quite common in engineering practice. Probabilistic modeling of problems arising in the search for the "best" time scale is useful both for engineering practice and for reliability theory. This paper covers a rather narrow class of problems related to the theory of preventive maintenance in various time scales. This class is restricted by the form of the cost functionals and by the type of lifetime distributions considered. Further efforts and research are needed in this area to provide a comprehensive coverage of the problem of choosing the best time scale for preventive maintenance. The authors will be satisfied if this paper will attract the attention of specialists in the statistical reliability theory to the problem of planning preventive maintenance in various time scales.

Acknowledgements The authors wish to thank the third referee for a valuable discussion and numerous suggestions aimed at improving the exposition.

L Gertsbakh, Kh.B. Kordonsky/European Journal of Operational Research 98 (1997) 64-74

74

Appendix A. Proof of Claim 2 Let 0 < bl < cl <~ .. • <~ CN-1 ~ CN < bN. Define N

Therefore, (A.1) is true also for z E 12, and thus it is true for the whole positive semiaxis z > 0.Similar arguments hold for the case when CN is replaced by bN. []

B ( Z,C) = ~-~piq~( ci)~Ir( z/ci), i=1

References

and N--1

B(Z, c, b) = P l q~(Cl )~lb'(Z/bl) q - Z

Piq~(ci)~b'(z/ci)

i=2

+ pNC~(CN)'I~(z/CN). It has to be proved that maxB(z,c) z>0

>~ m a x B ( z , c , b ) . z>0

(A.1)

Denote by x0 the maximum point of 9S(x) and by

z0 (a) the maximum point of qt ( z / a ) . Obviously, 0 < z 0 ( b l ) < Z0(Cl) ~ Z0(C2) ~ " ' "

<~ z o ( c s ) < zo(blv).

(A.2)

Consider first the case when bN = CN, i.e. only cl is replaced by bl in the argument of q~. All ~-functions decrease as z moves to the left of z0 (Cl) or to the right of ZO(CN). Thus max z B ( z , c) is attained at some Zl E [ Z o ( C I ) , Z O ( C N ) ] = 11. Fix any z* E 11 and consider 3B (z, c, b) lob1 = - P l q~( c l ) qt,(Z/bl ) ( z / b 2) at z = z*. It is positive and thus B ( z * , c , b ) increases as bl increases. Therefore, for z* E 11, B ( z * , c , b ) <~ B ( z *, c) and (A. 1 ) i s true. Denote R ( z, c2 . . . . . CN) = ~N2Pi(b(Ci)~'(Z/Ci). L e t Z E [ z o ( b l ) , Z o ( c l ) ] = 12. R is increasing in z in I2. Then

max B ( z , c, b) z El2

<~ m a x p l q ~ ( c l ) ~ ( z / b l ) Z>0

+ m a x R ( Z , Cl . . . . . CN) zEI2

= m a x Pl t~(C1 ) 1/'r(Z/C1) ~- R(ZO(Cl ) , c2 . . . . . z>0 N

= ~pi4,(ci)~(zO(Cl)/Ci) ~ m~B(z,c). i=l

CN)

Barlow, R.E., and Proschan, E, "Optimum preventive maintenance policies", Operations Research 8 (1960) 90-100. Block, H.W., Borges, W.S., and Savits, T.H., "A general age replacement model with minimal repair", Naval Reseacrh Logistics 35 (1988) 365-372. Gertsbakh, I., Models of Preventive Maintenance, North-Holland, Amsterdam, 1977. Gertsbakh, I., Statistical Reliability Theory, Marcel Dekker, New York, 1989. Gertsbakh, I., and Kordonsky, Kh.B., Models of Failure, SpringerVerlag, Berlin, 1969. Gertsbakh, I., and Kordonsky, Kh.B., "Best time scale for age replacement", International Journal of Reliability, Qualilty and Safety Engineering (1994) 219-229. Kordonsky, Kh.B., "Linear calculus of degradation time", in the Coll. Reliability and Quality Control 11 (1984) 6-18 (in Russian). Kordonsky, Kh.B., and Friedman, Ja., "Problems of probabilistic description of fatigue damages", Industrial Laboratory 7 (1976) 829-847. Kordonsky, Kh.B., and Gertsbakh, I., "Choice of the best time scale for reliability calculations", European Journal of Operational Research 65 (1993) 235-246. Lawless, J.E, Statistical Models and Methods for Lifetime Data Analysis, Wiley, New York, 1982. Neese, W.A., "Use of expanded aids in engine health monitoring on the CF-6-80 engine for 310 Airbus", SAE Technical Paper Session (1984) 504-134. Valdez-Florez, C., and Feldman, R.M., "A survey of preventive maintenance models for stochastically deteriorating single-unit systems", Naval Research Logistics 36 (1989) 419-446.