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On the optimal degree of imperfect repair
Reliability Engineering and System Safety 138 (2015) 54–58
Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Short communication
On the optimal degree of imperfect repair Maxim Finkelstein a,b,n a b
Department of Mathematical Statistics, University of the Free State, 339, Bloemfontein 9300, South Africa ITMO University, 49 Kronverkskiy prospekt, St. Petersburg, 197101, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2014 Received in revised form 8 November 2014 Accepted 15 January 2015 Available online 30 January 2015
A simple cost-wise comparison between the minimal and perfect repair of a system is discussed first using a relevant example. The main focus of this note, however, is on imperfect (general) repair. The best repair for our system in this case is defined as the one that corresponds to the optimal level (extent) of repair actions that minimize the long-run expected cost per unit of time. This complex optimization problem is considered for a specific imperfect repair model (Kijima II), using the developed earlier asymptotic approach to the corresponding virtual age modelling. It is shown that the optimal solution exists when the failure rate of a system tends to infinity as t tends to infinity and the corresponding cost function decreases sufficiently fast. An example illustrating the optimization procedure is considered. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Perfect repair Minimal repair Imperfect repair Virtual age Expected cost
1. Introduction Traditionally, reliability modeling of repairable items deals mostly with two types of repair. The perfect or ideal repair brings an item to ‘as good as new’ state. Therefore, the sequence of operating times form the renewal process, whereas the sequences of operating and repair times under standard assumptions form the alternating renewal process. The most common realization of the perfect repair in practice is the replacement of the failed item with a new identical one. The minimal repair, on the contrary, brings the system to a state (defined in statistical terms) it had prior the failure (see, e.g., [20] and [10]), which is also loosely called ‘as bad as old’. It is well known that in the latter case the corresponding sequence of lifetimes is described by the nonhomogeneous Poisson process (NHPP) with the rate equal to the failure rate that corresponds to the lifetime of our item. A natural example of the minimal repair is when the failed item is replaced by the identical one that was operating for the same time in the same conditions but did not fail and, therefore, could be considered as statistically identical. During the last few decades a considerable attention in reliability literature was devoted to various models of imperfect repair, which is neither perfect, nor minimal and is usually intermediate between the described above two types of maintenance actions. Probably one of the first models of this kind was the BrownProschan model [5], when each failure with probability p is
n Correspondence address: Department of Mathematical Statistics, University of the Free State, 339, Bloemfontein 9300, South Africa. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ress.2015.01.010 0951-8320/& 2015 Elsevier Ltd. All rights reserved.
perfectly repaired and with probability 1 p, only minimally. This model was later generalized in numerous publications ([2,10] and references therein). In order to compare different maintenance actions, the basis for comparison should be chosen. A manufacturer (if he, for instance, provides a warranty) or the user are obviously interested in minimizing the operational costs of repairable items. This characteristic for perfectly repaired items is often defined as a stationary one via the concept of the renewal reward theory ([22], pp 133–135) as the long-run expected cost per unit of time (cost rate), i.e., the mean cost incurred at the renewal cycle/ duration of the renewal cycle. Numerous optimal maintenance policies were discussed in the literature minimizing this metric. The most popular strategy considers the setting when an item is perfectly repaired either upon failure or on attaining age T, whichever comes first. Then the optimal T minimizing the expected cost rate is obtained. The other classic strategy considers replacements at periodic instants of time T; 2T; 3T; ::: and minimal repairs in-between. Then again, optimal period minimizing the cost rate is obtained. Those are well known cost driven important optimal decisions focused on preventive maintenance strategies (see e.g., [21,13,25]). On the other hand, it is sometimes of interest to compare ‘pure’ (without PM) perfect and minimal repair in terms of operational costs. This comparison, strictly speaking, is ill-posed as the induced stochastic processes of repair have a different nature for each case. However, some simple useful reasoning can be still presented and this is done in Section 2 as a prelude to the main results of this short note in Section 3, where the case of imperfect repair is considered. The problem of finding an optimal in some sense repair action becomes much more interesting for the models of imperfect (general) repair as it turns now to be not about the comparison of two
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
options, but about finding an optimal extent or degree of repair. The minimal and perfect repairs, in this case are just boundary specific cases of the imperfect repair. As it was mentioned above, there are numerous models and approaches to the problem of imperfect repair. However, the most appealing (at least, for the author) and analytically tractable are the models based on the virtual age concept, specifically, on the Kijima's models (see [7–11,14–16,24], to name a few). The main feature of this approach that makes it tractable is that the repair does not change the baseline distribution function that describes a lifetime of a repairable item; it only changes the initial (virtual) age after each repair. The latter, for simplicity, is assumed to be instantaneous. Therefore, the virtual age of a repairable item after the nth consecutive repair can be explicitly recursively obtained. As our reasoning is aimed at minimizing operational costs, certain cost structures and the corresponding assumptions should be added to the proposed stochastic model. For instance, the expected cost of repair of an item that had failed at age t and was minimally repaired should be given or the cost of imperfect repair as a function of the extent of repair should be also described. Our short note is organized as follows. In Section 2 we compare operational costs for minimal and perfect repair in a certain interval of time. The main focus is on imperfect repair modelling in Section 3. Finally some brief concluding remarks are given in Section 4.
55
Reliability-wise, it is often manifested by the increasing failure rate. Additionally, we assume, that deterioration of gain is modeled by a positive, decreasing function qðtÞ, whereas the hard failure mechanism is ‘independent’ of this process (see [12] for more details). Assume now that all failures are instantaneously minimally repaired and that the cost of minimal repair at time t is C M ðtÞ, which is usually (but not necessarily) an increasing function of time t (it is often more costly to repair minimally a more worn out item than a younger one). It is well known and easily follows from the properties of the NHPP that the cumulated expected cost of minimally repaired on each failure item in ½0; tÞ, E½C M ð0; tÞ is Z t E½C M ð0; tÞ ¼ C M ðuÞλðuÞdu ð3Þ 0
and thus for the comparison with (1) we should consider Rt C M ðuÞλðuÞdu E½C M ð0; tÞ ¼ 0 ; t t where, λðuÞ, as previously, denotes the failure rate of our item (see, e.g., [3]). When the additional gain characterizes operation of our item, as discussed above, (3) is modified to Z t E½C MQ ð0; tÞ ¼ C M ðuÞλðuÞdu Q ðtÞ; ð4Þ 0
2. Comparing operational costs for perfect and minimal repair Consider first, the case of perfect repair. Let T f be the time to failure of our repairable item with a finite expectation μ E½T f o 1 and absolutely continuous cumulative distribution function (cdf) FðtÞ ¼ P ½T f r t. Denote the corresponding survival function by FðtÞ ¼ 1 FðtÞ, the probability density function (pdf) by f ðtÞ and the failure rate by λðtÞ. We assume that repair is instantaneous, and therefore, the operation of the item can be described by the corresponding renewal process. By perfect repair we mean the replacement, and denote its cost by C P . Perfect repair brings the system to ‘as good as new’ state. Therefore, we will implicitly assume that an item at t ¼ 0 is better in some reasonable stochastic sense than the used one. In this note, we will consider mostly the aging items with increasing failure rate (IFR) or increasing failure rate in average (IFRA), which obviously agrees with this assumption. Define the operational cost cP as the long-run expected cost per unit of time. Using the renewal reward argument ([22], pp 133–135) it is just the ratio of C P and the expected duration of the renewal cycle, i.e., cP ¼
CP
ð1Þ
μ
If we additionally assume (as, e.g. for ‘production’ and power generation systems) that our item that is operable in ½0; xÞ, produces a gain (profit) defined in monitory units by some cumulative Rt function Q ðtÞ ¼ 0 qðuÞdu, where qðtÞ is the corresponding gain in the unit interval of time, then (1) is modified to R1 CP Q ðuÞf ðuÞdu mean cost on the cycle 0 cPQ ¼ mean ¼ μ length ofthe cycle Z 1 1 ¼ CP F ðuÞqðuÞdu ; ð2Þ
μ
0
where, as usual, in cost optimization problems, the costs are set to be positive, whereas the negative sign is assigned to the gain; R μ ¼ 01 FðuÞdu and qðtÞ is restored to its initial value qð0Þ after each repair. As we are looking at deteriorating (aging) items, the corresponding assumptions should be imposed on the gain function Q ðtÞ as well. Degradation is a complex multidimensional process.
where we assume that the repair is minimal with respect to Q ðtÞ as well and does not change its value. Perfect and minimal repairs, in fact, describe different stochastic processes for arriving failures; the renewal process (for perfect repair) that can be characterized by stationary rate as in (1) and nonstationary NHPP with a cumulative characteristic (3) (for minimal repair). A natural way for comparison, is to convert (1) also to the cumulative form. Thus, we should compare expected total costs for some relevant period of time, i.e., to compare E½C M ð0; tÞ with cP t. It can be a mission time or the whole useful period, for instance. The following simple but meaningful example performs this comparison for the specific case. Example 1. Assume for simplicity that C M ðuÞ C M o C p . The assumption of a constant cost of minimal repair, in fact, is widely adopted in practice and in optimal maintenance modelling as well. Rt Let us compare C M 0 λðuÞdu C M ΛðtÞ;where ΛðtÞ is the cumulated failure rate and C P t=μ. Thus, for instance, the minimal repair can be more cost effective in ½0; tÞ than the perfect one if Rt λðuÞdu CP : ð5Þ 4 0 t CM μ The right hand side of this inequality is increasing for IFRA distributions (and obviously, for IFR distributions). In fact, the IFRA distributions are defined as distributions with increasing ΛðtÞ=t. Thus depending on the values of C P ; C M ; μ there can be 3 situations: 1. λð0Þ 4 C P =C M μ. Then perfect repair is less costly for t A ½0; 1Þ. This can be seen after observing that the limit limt-0 ΛðtÞ=t ¼ λð0Þ. Rt 2. λð0Þ o C P =C M μ and limt-1 0 λðuÞdu=t ¼ c 4 0 as, e.g., for Erlangian distribution with parameter λ for which limt-1 λðtÞ ¼ λ ¼ c. Then the minimal repair is more cost effective for t A ½0; 1Þ if c o C P =C M μ, whereas if c 4 C P =C M μ, it is more cost effective in ½0; t c Þ and less cost effective than the perfect repair in ½t c ; 1Þ, Rt where t c is uniquely obtained from the equation C P =C M μ ¼ 0c λðuÞdu=t c . Rt 3. λð0Þ o C P =C M μ and limt-1 0 λðuÞdu=t ¼ 1 as, e.g., for the Weibull distribution with increasing failure rate. Then similar to the previous case, the minimal repair is more cost effective in ½0; t c Þ and less cost effective than the perfect one in ½t c ; 1Þ.
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M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
Thus there are situations when the minimal repair is preferable. Remark 1. Adding the gain function Q ðtÞ, obviously can affect the comparison of cPQ t and E½C MQ ð0; t r Þ (see (2) and (4), accordingly), as compared with the ‘no gain case’. When the gain rate is constant, qðtÞ ¼ q (Q ðtÞ ¼ qt), there is no change in comparison, as the R1 additional terms are equal, i.e., μt 0 F ðuÞqðuÞdu ¼ qt. Assume, for simplicity that qðtÞ is decreasing to 0 as t increases. Then, taking into account that Q ð0Þ ¼ 0, t
μ t
μ
Z
1
0
Z 0
F ðuÞqðuÞdu o Q ðtÞ;
t A ½0; t q Þ;
F ðuÞqðuÞdu 4 Q ðtÞ;
t A ½t q ; 1Þ;
1
which follows because the derivative of the left hand side is constant; Q 0 ðtÞ ¼ qðtÞ is decreasing and Z 1 1 Q 0 ð0Þ ¼ qð0Þ 4 F ðuÞqðuÞdu:
μ
0
R1 (thus, the line t μ 1 0 F ðuÞqðuÞduand the increasing function Q ðtÞshould intersect at some t q 4 0). The latter inequality follows by using the mean value theorem for the integral. The value of t q is obtained from the equation Z t 1 F ðuÞqðuÞdu ¼ Q ðtÞ;
μ
0
and the solution exists due to our assumptions. Thus, as follows from inequalities above, when performance of a system is characterized by the gain, the perfect repair case ‘acquires’ less gain than the minimal repair case for t A ½0; t q Þ and vice versa for t A ½t q ; 1Þ. We will proceed with the main topic of this note.
3. Imperfect repair We have briefly considered two marginal types of repair: the perfect and minimal repair. The perfect repair starts the age of the item again from 0, whereas the minimal repair does not change it. It is natural to consider in this framework the intermediate value of age, which is in our approach is closely related to the notion of virtual age (see the references in the Introduction). On the other hand, the virtual age concept directly or indirectly was used in the optimal preventive maintenance (with imperfect repair) modeling (see [4,6,14,18,19,25,26] among others). Let our item, which lifetime, as previously, is described by the cdf FðtÞ starts operating at t ¼ 0 and fails at time y. The instantaneous repair brings back its age not to 0 (perfect repair) and does not keep it as it was (minimal repair), but changes it to some intermediate level 0 o x o y. It means that after the instantaneous repair the cdf of the item's lifetime is Fðtj xÞ ¼ ðFðx þ tÞ FðtÞÞ=FðxÞ, retaining the same ‘shape’ of the cdf. The new initial age x is often called the virtual (effective) age. Then the process of imperfect repairs continues and on each cycle there is an initial virtual age and the same governing cdf FðtÞ. In order to specify this process, the rule, in accordance with which the repair decreases the age of an item should be specified. The most popular and suitable for stochastic analysis model is Kijima's virtual age model (Kijima II): After each repair the virtual age of an item x is decreased in accordance with the function qx; 0 r q r 1. When q ¼ 0, it is a perfect repair and when q ¼ 1, it is a minimal one. We do not consider here the repair that is worse than minimal (q 4 1) that can also occur sometimes in practice (e.g., introducing new bugs when eliminating previous ones during the repair of software).
Let X i ; i ¼ 1; 2; ::: be the inter-arrival times for the corresponding point process of failures (repairs), whereas xi be their realizations. After the first imperfect repair, at the beginning of the second cycle, the virtual age v1 is q x1 , after the second repair the virtual age is v2 ¼ qðqx1 þ x2 Þ ¼ q2 x1 þ qx2 ; …; and after the nth repair the virtual age is vn ¼ qn x1 þ qn 1 x2 þ … þ qxn ¼
n 1 X
qn i xi þ 1 :
ð6Þ
i¼0
For convenience, in what follows, we shall call the described point process the ‘virtual age process’ and the corresponding inter-arrival times-the cycles of the virtual age process. This process is completely defined by the cdf FðtÞ and parameter q. In order to proceed with our comparisons, we must specify a reasonable cost structure for this model. Assume that the cost of imperfect repair at any cycle of the described process is defined by the function CðqÞ, meaning that it depends only on the extent of repair and not on other factors. This is, of course, a simplification but it allows for dealing effectively with a rather complex optimization problem. It somehow corresponds to our assumption in Example 1 that the cost of minimal repair is constant, which is a practical assumption adopted, for instance, in numerous optimal maintenance models. Thus CðqÞ is a decreasing function in q A ½0; 1and C M ¼ Cð1Þ r CðqÞ r Cð0Þ ¼ C P :
ð7Þ
Consider the first cycle of the virtual age process. The expected cost per unit of time for this cycle is CðqÞ=μ. Thus, as follows from (1) and (7), CðqÞ=μ o C P =μ for 0 o q r 1 and if we are concerned only about one cycle, minimal repair is the best choice, however the one cycle reasoning here makes no sense and only is used as a supplementary step. For the described above virtual age process, we must obviously exclude the minimal repair as an option. What is happening at the subsequent (non-identically distributed) cycles of this process? The conditional expected cost per unit of time for, e.g., the ith cycle is now CðqÞ=μi ðvi Þ, where
μi ðvi 1 Þ ¼
Z
Z
1
Fðuj vi 1 Þdu ¼ 0
0
1
Fðu þ vi 1 Þ Fðvi 1 Þ
¼ μ1 ð0Þ ¼ μ:
du; i ¼ 2; 3; :::; μ1 ðv0 Þ ð8Þ
However, (8) (except the first cycle) is written for realizations of virtual age at the start of each cycle and we need to obtain the unconditional values of the mean cycles lengths. We will apply now mathematical results obtained in [9,10] to the problem under consideration. Although the proofs in these references are rather cumbersome, the applications to be considered are well and clearly motivated by the corresponding monotonicity and limiting properties to be discussed now. Denote by V i ; i ¼ 0; 1; 2; ::: the random virtual age that corresponds to vi , by θi ðvÞ its cdf and by F i ðtÞ the cdf of the ith inter-arrival time (cycle duration in our terminology) for the described virtual age process. It follows from [9] that if the baseline cdf FðtÞ is IFR, then V i is a stochastically increasing in isequence of random variables and limi-1 θi ðvÞ ¼ θL ðvÞ
ð9Þ
meaning that it stochastically (in distribution) converges to a limit random variable V L with the cdf θL ðvÞ. Moreover, the following result immediately follows from Theorem 4 and Corollary 3 in [9]. Proposition 1. Let the baselineFðtÞbe IFR. Then the sequence of inter-arrival times in the described virtual age process is stochastically decreasing to a random variable with a
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
limiting distribution F L ðtÞ, i.e., Z Z 1 1 exp limi-1 F i ðtÞ ¼ F L ðtÞ ¼ 0
v
vþt
λðuÞdu dðθL ðvÞÞ: ð10Þ
Relationships (9) and (10) result in the following corollary: Corollary 1. Let assumptions of Proposition 1 hold. Then the sequence of the mean inter-arrival times is decreasing to a limit, μL Z 1 Z 1 limi-1 μi ¼ limi-1 F i ðuÞdu ¼ F L ðuÞdu ¼ μL : ð11Þ 0
0
These theoretical results have a very clear and important meaning in our context: In accordance with the renewal reward processes reasoning ([22], pp 132–135), the expected long-run cost per unit of time cq , for our virtual age process is defined as cq ¼
CðqÞ
μL ðqÞ
:
As indicated by this expression, the limiting μL is obviously also a function of q, i.e., μL ðqÞ and it is proved in [10] (see also [16]) that when FðtÞ A IFR, it is a decreasing function of q A ½0; 1Þ Moreover, if limt-1 λðtÞ ¼ 1 (as, e.g., for the Weibull distribution with increasing failure rate), similar to the minimal repair case, limq-1 μL ðqÞ ¼ 0:
ð12Þ
In the previous section, we were comparing two options, now we have a continuum of them and, therefore, the corresponding optimization problem can be defined: to find optimal qn that satisfies cqn ¼ min
CðqÞ
μL ðqÞ
;
0 rq o1:
equal to veq ðqÞ, however, in expectation. Indeed the integral in (15) defines the mean remaining lifetime for the item that started operation with virtual age veq ðqÞ. Thus we can conclude that Fðuj veq ðqÞÞ in (15) can be considered as an approximation of F L ðuÞ in (11). Rearranging (15), Z 1 1 ð16Þ Fðuj veq ðqÞÞdu ¼ veq ðqÞ 1 : q 0 As the left hand side of this equation (for each fixed q) is decreasing in veq ðqÞ(for the IFR distributions) and the right hand side is increasing as a linear function of veq ðqÞ, it has the unique solution. Thus the left hand side of (16) can be considered as an approximation for μL ðqÞin (13). A simple computational procedure is as follows. For the given cdf FðtÞ(e.g., the Weibull distribution with increasing failure rate as in Example 2), eq. (16) is solved numerically with respect toveq ðqÞ for a reasonable step in q A ð0; 1Þ. The right hand side of (16) is (approximately) equal to μL ðqÞ in (13). Then, for the given cost structure (14), CðqÞ=μL ðqÞ is analyzed and its minimum is obtained. Example 2. Let FðtÞ ¼ expf t 2 g; therefore, limt-1 λðtÞ ¼ limt-1 2t ¼ 1. Then, as can be seen from Fig. 1, as expected, 1=μL ðqÞ MðqÞ is increasing in q A ð0; 1Þ. Let C P ¼ 1; C M ¼ 0:3; α ¼ 2. Then the resulting graph for the expected long-run cost per unit of time, cq is given in Fig.2. We see that the optimal degree of repair is ‘well-pronounced’ and qn 0:58. In Fig. 3 two more graphs are added for the same cdf. The left one is for C P ¼ 1; C M ¼ 0:15; α ¼ 2 and the right one is for C P ¼ 1; C M ¼ 0:15; α ¼ 3. As
ð13Þ
The function CðqÞ is decreasing in q A ½0; 1 (see (7)). Assume a rather flexible functional form [23]: CðqÞ ¼ C M þ ΔC PM ð1 qÞα ; α 40;
ð14Þ
where ΔC PM ðC P C M Þ. We see that larger values of α increase j C 0 ðqÞj when q-0. Proposition 2. Let the cdf of an item,FðtÞ be IFR and limt-1 λðtÞ ¼ 1. Then if j C 0 ðqÞj , q-0 is sufficiently large there exists an optimal value qn A ð0; 1Þ that minimizes the long-run expected cost per unit of time for the described imperfect repair model. The proof is obvious, as in accordance with (7) and (12), limq-1 CðqÞ=μL ðqÞ ¼ 1, whereas limq-0 ðCðqÞ=μL ðqÞÞ0 limq-0 ðC 0 ðqÞ μL ðqÞ μ0L ðqÞCðqÞÞ. Thus when j C 0 ðqÞj is sufficiently large, the optimal value exists (e.g., when ΔC PM or α in (14) are sufficiently large, limq-0 ðCðqÞ=μL ðqÞÞ0 is negative and thus cqn initially decreases). Proposition 2 provides a theoretical proof of existence of the optimal degree of imperfect repair that minimizes the long-run expected cost per unit of time. It can be found numerically, for instance, by means of simulation of the virtual age process (covering the range 0 r q o 1 with a reasonable step), then computing μL ðqÞ from (11) and, finally, obtaining the optimal value from (13). This is rather computationally intensive. On the other hand, an approximate meaningful and much simpler computationally approach can be used. In fact, this approach is based on the limiting property (9). Let veq for each q define an ‘equilibrium value’ of virtual age, i.e., Z 1 q veq ðqÞ þ Fðuj veq ðqÞÞdu ¼ veq ðqÞ ð15Þ
Fig. 1. The function MðqÞ.
0
meaning that if an item starts some cycle having an initial virtual age veq ðqÞ, then after the next repair its virtual age will be again
57
Fig. 2. The function cq for C P ¼ 1; C M ¼ 0:3; α ¼ 2
58
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
Fig. 3. The function cq for C P ¼ 1; C M ¼ 0:15; α ¼ 2 (left) and C P ¼ 1; C M ¼ 0:3; α ¼ 3(right).
expected, and also follows from Proposition 2, the shape of the graph is very sensitive to the ratio C P =C M and to the value of α Direct simulation of the optimal solution in (13) for the considered examples shows a relatively good approximation of our ‘quasioptimal’ approach; however, we plan to conduct theoretical investigations of the accuracy of approximation in the future research. 4. Concluding remarks In this short note, we first have performed a simple comparison of long-run expected costs per unit of time for perfect and minimal repairs. The peculiarity of this comparison is in the different nature of point processes governing these two models. The perfect repair corresponds to the renewal process, whereas the minimal repair corresponds to the NHPP. Therefore, we compare, in fact, the cumulated costs for some interval of time and consider the illustrative Example 1, where all options are clearly seen. Our main effort is focused in imperfect (general repair). The best repair for our system in this case is the one that corresponds to the optimal level (extent) of repair actions that minimize the long-run expected costs per unit of time. It turns out that it is a rather complex problem even for the specific imperfect repair model (Kijima II) that was considered. However, we were able to use and modify our previous asymptotic results for the virtual age and cycles durations for this model and to suggest an innovative approximate procedure for defining the optimal solution. Our results show and the considered example illustrates it, that when the failure rate of an item is increasing to infinity, e.g., according the Weibull law and the cost function decreases with the sufficiently large absolute value of the corresponding derivative (when q-0), the optimal degree of repair exists (and it is not the perfect or minimal repair). These results can be definitely useful in practice for systems with different degrees of repair. Acknowledgments The author is grateful to 3 anonymous referees, whose detailed comments were extremely helpful for the revision. This work was
supported by the NRF (National Research Foundation of South Africa) Grant IFR2011040500026.
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Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Short communication
On the optimal degree of imperfect repair Maxim Finkelstein a,b,n a b
Department of Mathematical Statistics, University of the Free State, 339, Bloemfontein 9300, South Africa ITMO University, 49 Kronverkskiy prospekt, St. Petersburg, 197101, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2014 Received in revised form 8 November 2014 Accepted 15 January 2015 Available online 30 January 2015
A simple cost-wise comparison between the minimal and perfect repair of a system is discussed first using a relevant example. The main focus of this note, however, is on imperfect (general) repair. The best repair for our system in this case is defined as the one that corresponds to the optimal level (extent) of repair actions that minimize the long-run expected cost per unit of time. This complex optimization problem is considered for a specific imperfect repair model (Kijima II), using the developed earlier asymptotic approach to the corresponding virtual age modelling. It is shown that the optimal solution exists when the failure rate of a system tends to infinity as t tends to infinity and the corresponding cost function decreases sufficiently fast. An example illustrating the optimization procedure is considered. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Perfect repair Minimal repair Imperfect repair Virtual age Expected cost
1. Introduction Traditionally, reliability modeling of repairable items deals mostly with two types of repair. The perfect or ideal repair brings an item to ‘as good as new’ state. Therefore, the sequence of operating times form the renewal process, whereas the sequences of operating and repair times under standard assumptions form the alternating renewal process. The most common realization of the perfect repair in practice is the replacement of the failed item with a new identical one. The minimal repair, on the contrary, brings the system to a state (defined in statistical terms) it had prior the failure (see, e.g., [20] and [10]), which is also loosely called ‘as bad as old’. It is well known that in the latter case the corresponding sequence of lifetimes is described by the nonhomogeneous Poisson process (NHPP) with the rate equal to the failure rate that corresponds to the lifetime of our item. A natural example of the minimal repair is when the failed item is replaced by the identical one that was operating for the same time in the same conditions but did not fail and, therefore, could be considered as statistically identical. During the last few decades a considerable attention in reliability literature was devoted to various models of imperfect repair, which is neither perfect, nor minimal and is usually intermediate between the described above two types of maintenance actions. Probably one of the first models of this kind was the BrownProschan model [5], when each failure with probability p is
n Correspondence address: Department of Mathematical Statistics, University of the Free State, 339, Bloemfontein 9300, South Africa. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ress.2015.01.010 0951-8320/& 2015 Elsevier Ltd. All rights reserved.
perfectly repaired and with probability 1 p, only minimally. This model was later generalized in numerous publications ([2,10] and references therein). In order to compare different maintenance actions, the basis for comparison should be chosen. A manufacturer (if he, for instance, provides a warranty) or the user are obviously interested in minimizing the operational costs of repairable items. This characteristic for perfectly repaired items is often defined as a stationary one via the concept of the renewal reward theory ([22], pp 133–135) as the long-run expected cost per unit of time (cost rate), i.e., the mean cost incurred at the renewal cycle/ duration of the renewal cycle. Numerous optimal maintenance policies were discussed in the literature minimizing this metric. The most popular strategy considers the setting when an item is perfectly repaired either upon failure or on attaining age T, whichever comes first. Then the optimal T minimizing the expected cost rate is obtained. The other classic strategy considers replacements at periodic instants of time T; 2T; 3T; ::: and minimal repairs in-between. Then again, optimal period minimizing the cost rate is obtained. Those are well known cost driven important optimal decisions focused on preventive maintenance strategies (see e.g., [21,13,25]). On the other hand, it is sometimes of interest to compare ‘pure’ (without PM) perfect and minimal repair in terms of operational costs. This comparison, strictly speaking, is ill-posed as the induced stochastic processes of repair have a different nature for each case. However, some simple useful reasoning can be still presented and this is done in Section 2 as a prelude to the main results of this short note in Section 3, where the case of imperfect repair is considered. The problem of finding an optimal in some sense repair action becomes much more interesting for the models of imperfect (general) repair as it turns now to be not about the comparison of two
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
options, but about finding an optimal extent or degree of repair. The minimal and perfect repairs, in this case are just boundary specific cases of the imperfect repair. As it was mentioned above, there are numerous models and approaches to the problem of imperfect repair. However, the most appealing (at least, for the author) and analytically tractable are the models based on the virtual age concept, specifically, on the Kijima's models (see [7–11,14–16,24], to name a few). The main feature of this approach that makes it tractable is that the repair does not change the baseline distribution function that describes a lifetime of a repairable item; it only changes the initial (virtual) age after each repair. The latter, for simplicity, is assumed to be instantaneous. Therefore, the virtual age of a repairable item after the nth consecutive repair can be explicitly recursively obtained. As our reasoning is aimed at minimizing operational costs, certain cost structures and the corresponding assumptions should be added to the proposed stochastic model. For instance, the expected cost of repair of an item that had failed at age t and was minimally repaired should be given or the cost of imperfect repair as a function of the extent of repair should be also described. Our short note is organized as follows. In Section 2 we compare operational costs for minimal and perfect repair in a certain interval of time. The main focus is on imperfect repair modelling in Section 3. Finally some brief concluding remarks are given in Section 4.
55
Reliability-wise, it is often manifested by the increasing failure rate. Additionally, we assume, that deterioration of gain is modeled by a positive, decreasing function qðtÞ, whereas the hard failure mechanism is ‘independent’ of this process (see [12] for more details). Assume now that all failures are instantaneously minimally repaired and that the cost of minimal repair at time t is C M ðtÞ, which is usually (but not necessarily) an increasing function of time t (it is often more costly to repair minimally a more worn out item than a younger one). It is well known and easily follows from the properties of the NHPP that the cumulated expected cost of minimally repaired on each failure item in ½0; tÞ, E½C M ð0; tÞ is Z t E½C M ð0; tÞ ¼ C M ðuÞλðuÞdu ð3Þ 0
and thus for the comparison with (1) we should consider Rt C M ðuÞλðuÞdu E½C M ð0; tÞ ¼ 0 ; t t where, λðuÞ, as previously, denotes the failure rate of our item (see, e.g., [3]). When the additional gain characterizes operation of our item, as discussed above, (3) is modified to Z t E½C MQ ð0; tÞ ¼ C M ðuÞλðuÞdu Q ðtÞ; ð4Þ 0
2. Comparing operational costs for perfect and minimal repair Consider first, the case of perfect repair. Let T f be the time to failure of our repairable item with a finite expectation μ E½T f o 1 and absolutely continuous cumulative distribution function (cdf) FðtÞ ¼ P ½T f r t. Denote the corresponding survival function by FðtÞ ¼ 1 FðtÞ, the probability density function (pdf) by f ðtÞ and the failure rate by λðtÞ. We assume that repair is instantaneous, and therefore, the operation of the item can be described by the corresponding renewal process. By perfect repair we mean the replacement, and denote its cost by C P . Perfect repair brings the system to ‘as good as new’ state. Therefore, we will implicitly assume that an item at t ¼ 0 is better in some reasonable stochastic sense than the used one. In this note, we will consider mostly the aging items with increasing failure rate (IFR) or increasing failure rate in average (IFRA), which obviously agrees with this assumption. Define the operational cost cP as the long-run expected cost per unit of time. Using the renewal reward argument ([22], pp 133–135) it is just the ratio of C P and the expected duration of the renewal cycle, i.e., cP ¼
CP
ð1Þ
μ
If we additionally assume (as, e.g. for ‘production’ and power generation systems) that our item that is operable in ½0; xÞ, produces a gain (profit) defined in monitory units by some cumulative Rt function Q ðtÞ ¼ 0 qðuÞdu, where qðtÞ is the corresponding gain in the unit interval of time, then (1) is modified to R1 CP Q ðuÞf ðuÞdu mean cost on the cycle 0 cPQ ¼ mean ¼ μ length ofthe cycle Z 1 1 ¼ CP F ðuÞqðuÞdu ; ð2Þ
μ
0
where, as usual, in cost optimization problems, the costs are set to be positive, whereas the negative sign is assigned to the gain; R μ ¼ 01 FðuÞdu and qðtÞ is restored to its initial value qð0Þ after each repair. As we are looking at deteriorating (aging) items, the corresponding assumptions should be imposed on the gain function Q ðtÞ as well. Degradation is a complex multidimensional process.
where we assume that the repair is minimal with respect to Q ðtÞ as well and does not change its value. Perfect and minimal repairs, in fact, describe different stochastic processes for arriving failures; the renewal process (for perfect repair) that can be characterized by stationary rate as in (1) and nonstationary NHPP with a cumulative characteristic (3) (for minimal repair). A natural way for comparison, is to convert (1) also to the cumulative form. Thus, we should compare expected total costs for some relevant period of time, i.e., to compare E½C M ð0; tÞ with cP t. It can be a mission time or the whole useful period, for instance. The following simple but meaningful example performs this comparison for the specific case. Example 1. Assume for simplicity that C M ðuÞ C M o C p . The assumption of a constant cost of minimal repair, in fact, is widely adopted in practice and in optimal maintenance modelling as well. Rt Let us compare C M 0 λðuÞdu C M ΛðtÞ;where ΛðtÞ is the cumulated failure rate and C P t=μ. Thus, for instance, the minimal repair can be more cost effective in ½0; tÞ than the perfect one if Rt λðuÞdu CP : ð5Þ 4 0 t CM μ The right hand side of this inequality is increasing for IFRA distributions (and obviously, for IFR distributions). In fact, the IFRA distributions are defined as distributions with increasing ΛðtÞ=t. Thus depending on the values of C P ; C M ; μ there can be 3 situations: 1. λð0Þ 4 C P =C M μ. Then perfect repair is less costly for t A ½0; 1Þ. This can be seen after observing that the limit limt-0 ΛðtÞ=t ¼ λð0Þ. Rt 2. λð0Þ o C P =C M μ and limt-1 0 λðuÞdu=t ¼ c 4 0 as, e.g., for Erlangian distribution with parameter λ for which limt-1 λðtÞ ¼ λ ¼ c. Then the minimal repair is more cost effective for t A ½0; 1Þ if c o C P =C M μ, whereas if c 4 C P =C M μ, it is more cost effective in ½0; t c Þ and less cost effective than the perfect repair in ½t c ; 1Þ, Rt where t c is uniquely obtained from the equation C P =C M μ ¼ 0c λðuÞdu=t c . Rt 3. λð0Þ o C P =C M μ and limt-1 0 λðuÞdu=t ¼ 1 as, e.g., for the Weibull distribution with increasing failure rate. Then similar to the previous case, the minimal repair is more cost effective in ½0; t c Þ and less cost effective than the perfect one in ½t c ; 1Þ.
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M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
Thus there are situations when the minimal repair is preferable. Remark 1. Adding the gain function Q ðtÞ, obviously can affect the comparison of cPQ t and E½C MQ ð0; t r Þ (see (2) and (4), accordingly), as compared with the ‘no gain case’. When the gain rate is constant, qðtÞ ¼ q (Q ðtÞ ¼ qt), there is no change in comparison, as the R1 additional terms are equal, i.e., μt 0 F ðuÞqðuÞdu ¼ qt. Assume, for simplicity that qðtÞ is decreasing to 0 as t increases. Then, taking into account that Q ð0Þ ¼ 0, t
μ t
μ
Z
1
0
Z 0
F ðuÞqðuÞdu o Q ðtÞ;
t A ½0; t q Þ;
F ðuÞqðuÞdu 4 Q ðtÞ;
t A ½t q ; 1Þ;
1
which follows because the derivative of the left hand side is constant; Q 0 ðtÞ ¼ qðtÞ is decreasing and Z 1 1 Q 0 ð0Þ ¼ qð0Þ 4 F ðuÞqðuÞdu:
μ
0
R1 (thus, the line t μ 1 0 F ðuÞqðuÞduand the increasing function Q ðtÞshould intersect at some t q 4 0). The latter inequality follows by using the mean value theorem for the integral. The value of t q is obtained from the equation Z t 1 F ðuÞqðuÞdu ¼ Q ðtÞ;
μ
0
and the solution exists due to our assumptions. Thus, as follows from inequalities above, when performance of a system is characterized by the gain, the perfect repair case ‘acquires’ less gain than the minimal repair case for t A ½0; t q Þ and vice versa for t A ½t q ; 1Þ. We will proceed with the main topic of this note.
3. Imperfect repair We have briefly considered two marginal types of repair: the perfect and minimal repair. The perfect repair starts the age of the item again from 0, whereas the minimal repair does not change it. It is natural to consider in this framework the intermediate value of age, which is in our approach is closely related to the notion of virtual age (see the references in the Introduction). On the other hand, the virtual age concept directly or indirectly was used in the optimal preventive maintenance (with imperfect repair) modeling (see [4,6,14,18,19,25,26] among others). Let our item, which lifetime, as previously, is described by the cdf FðtÞ starts operating at t ¼ 0 and fails at time y. The instantaneous repair brings back its age not to 0 (perfect repair) and does not keep it as it was (minimal repair), but changes it to some intermediate level 0 o x o y. It means that after the instantaneous repair the cdf of the item's lifetime is Fðtj xÞ ¼ ðFðx þ tÞ FðtÞÞ=FðxÞ, retaining the same ‘shape’ of the cdf. The new initial age x is often called the virtual (effective) age. Then the process of imperfect repairs continues and on each cycle there is an initial virtual age and the same governing cdf FðtÞ. In order to specify this process, the rule, in accordance with which the repair decreases the age of an item should be specified. The most popular and suitable for stochastic analysis model is Kijima's virtual age model (Kijima II): After each repair the virtual age of an item x is decreased in accordance with the function qx; 0 r q r 1. When q ¼ 0, it is a perfect repair and when q ¼ 1, it is a minimal one. We do not consider here the repair that is worse than minimal (q 4 1) that can also occur sometimes in practice (e.g., introducing new bugs when eliminating previous ones during the repair of software).
Let X i ; i ¼ 1; 2; ::: be the inter-arrival times for the corresponding point process of failures (repairs), whereas xi be their realizations. After the first imperfect repair, at the beginning of the second cycle, the virtual age v1 is q x1 , after the second repair the virtual age is v2 ¼ qðqx1 þ x2 Þ ¼ q2 x1 þ qx2 ; …; and after the nth repair the virtual age is vn ¼ qn x1 þ qn 1 x2 þ … þ qxn ¼
n 1 X
qn i xi þ 1 :
ð6Þ
i¼0
For convenience, in what follows, we shall call the described point process the ‘virtual age process’ and the corresponding inter-arrival times-the cycles of the virtual age process. This process is completely defined by the cdf FðtÞ and parameter q. In order to proceed with our comparisons, we must specify a reasonable cost structure for this model. Assume that the cost of imperfect repair at any cycle of the described process is defined by the function CðqÞ, meaning that it depends only on the extent of repair and not on other factors. This is, of course, a simplification but it allows for dealing effectively with a rather complex optimization problem. It somehow corresponds to our assumption in Example 1 that the cost of minimal repair is constant, which is a practical assumption adopted, for instance, in numerous optimal maintenance models. Thus CðqÞ is a decreasing function in q A ½0; 1and C M ¼ Cð1Þ r CðqÞ r Cð0Þ ¼ C P :
ð7Þ
Consider the first cycle of the virtual age process. The expected cost per unit of time for this cycle is CðqÞ=μ. Thus, as follows from (1) and (7), CðqÞ=μ o C P =μ for 0 o q r 1 and if we are concerned only about one cycle, minimal repair is the best choice, however the one cycle reasoning here makes no sense and only is used as a supplementary step. For the described above virtual age process, we must obviously exclude the minimal repair as an option. What is happening at the subsequent (non-identically distributed) cycles of this process? The conditional expected cost per unit of time for, e.g., the ith cycle is now CðqÞ=μi ðvi Þ, where
μi ðvi 1 Þ ¼
Z
Z
1
Fðuj vi 1 Þdu ¼ 0
0
1
Fðu þ vi 1 Þ Fðvi 1 Þ
¼ μ1 ð0Þ ¼ μ:
du; i ¼ 2; 3; :::; μ1 ðv0 Þ ð8Þ
However, (8) (except the first cycle) is written for realizations of virtual age at the start of each cycle and we need to obtain the unconditional values of the mean cycles lengths. We will apply now mathematical results obtained in [9,10] to the problem under consideration. Although the proofs in these references are rather cumbersome, the applications to be considered are well and clearly motivated by the corresponding monotonicity and limiting properties to be discussed now. Denote by V i ; i ¼ 0; 1; 2; ::: the random virtual age that corresponds to vi , by θi ðvÞ its cdf and by F i ðtÞ the cdf of the ith inter-arrival time (cycle duration in our terminology) for the described virtual age process. It follows from [9] that if the baseline cdf FðtÞ is IFR, then V i is a stochastically increasing in isequence of random variables and limi-1 θi ðvÞ ¼ θL ðvÞ
ð9Þ
meaning that it stochastically (in distribution) converges to a limit random variable V L with the cdf θL ðvÞ. Moreover, the following result immediately follows from Theorem 4 and Corollary 3 in [9]. Proposition 1. Let the baselineFðtÞbe IFR. Then the sequence of inter-arrival times in the described virtual age process is stochastically decreasing to a random variable with a
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
limiting distribution F L ðtÞ, i.e., Z Z 1 1 exp limi-1 F i ðtÞ ¼ F L ðtÞ ¼ 0
v
vþt
λðuÞdu dðθL ðvÞÞ: ð10Þ
Relationships (9) and (10) result in the following corollary: Corollary 1. Let assumptions of Proposition 1 hold. Then the sequence of the mean inter-arrival times is decreasing to a limit, μL Z 1 Z 1 limi-1 μi ¼ limi-1 F i ðuÞdu ¼ F L ðuÞdu ¼ μL : ð11Þ 0
0
These theoretical results have a very clear and important meaning in our context: In accordance with the renewal reward processes reasoning ([22], pp 132–135), the expected long-run cost per unit of time cq , for our virtual age process is defined as cq ¼
CðqÞ
μL ðqÞ
:
As indicated by this expression, the limiting μL is obviously also a function of q, i.e., μL ðqÞ and it is proved in [10] (see also [16]) that when FðtÞ A IFR, it is a decreasing function of q A ½0; 1Þ Moreover, if limt-1 λðtÞ ¼ 1 (as, e.g., for the Weibull distribution with increasing failure rate), similar to the minimal repair case, limq-1 μL ðqÞ ¼ 0:
ð12Þ
In the previous section, we were comparing two options, now we have a continuum of them and, therefore, the corresponding optimization problem can be defined: to find optimal qn that satisfies cqn ¼ min
CðqÞ
μL ðqÞ
;
0 rq o1:
equal to veq ðqÞ, however, in expectation. Indeed the integral in (15) defines the mean remaining lifetime for the item that started operation with virtual age veq ðqÞ. Thus we can conclude that Fðuj veq ðqÞÞ in (15) can be considered as an approximation of F L ðuÞ in (11). Rearranging (15), Z 1 1 ð16Þ Fðuj veq ðqÞÞdu ¼ veq ðqÞ 1 : q 0 As the left hand side of this equation (for each fixed q) is decreasing in veq ðqÞ(for the IFR distributions) and the right hand side is increasing as a linear function of veq ðqÞ, it has the unique solution. Thus the left hand side of (16) can be considered as an approximation for μL ðqÞin (13). A simple computational procedure is as follows. For the given cdf FðtÞ(e.g., the Weibull distribution with increasing failure rate as in Example 2), eq. (16) is solved numerically with respect toveq ðqÞ for a reasonable step in q A ð0; 1Þ. The right hand side of (16) is (approximately) equal to μL ðqÞ in (13). Then, for the given cost structure (14), CðqÞ=μL ðqÞ is analyzed and its minimum is obtained. Example 2. Let FðtÞ ¼ expf t 2 g; therefore, limt-1 λðtÞ ¼ limt-1 2t ¼ 1. Then, as can be seen from Fig. 1, as expected, 1=μL ðqÞ MðqÞ is increasing in q A ð0; 1Þ. Let C P ¼ 1; C M ¼ 0:3; α ¼ 2. Then the resulting graph for the expected long-run cost per unit of time, cq is given in Fig.2. We see that the optimal degree of repair is ‘well-pronounced’ and qn 0:58. In Fig. 3 two more graphs are added for the same cdf. The left one is for C P ¼ 1; C M ¼ 0:15; α ¼ 2 and the right one is for C P ¼ 1; C M ¼ 0:15; α ¼ 3. As
ð13Þ
The function CðqÞ is decreasing in q A ½0; 1 (see (7)). Assume a rather flexible functional form [23]: CðqÞ ¼ C M þ ΔC PM ð1 qÞα ; α 40;
ð14Þ
where ΔC PM ðC P C M Þ. We see that larger values of α increase j C 0 ðqÞj when q-0. Proposition 2. Let the cdf of an item,FðtÞ be IFR and limt-1 λðtÞ ¼ 1. Then if j C 0 ðqÞj , q-0 is sufficiently large there exists an optimal value qn A ð0; 1Þ that minimizes the long-run expected cost per unit of time for the described imperfect repair model. The proof is obvious, as in accordance with (7) and (12), limq-1 CðqÞ=μL ðqÞ ¼ 1, whereas limq-0 ðCðqÞ=μL ðqÞÞ0 limq-0 ðC 0 ðqÞ μL ðqÞ μ0L ðqÞCðqÞÞ. Thus when j C 0 ðqÞj is sufficiently large, the optimal value exists (e.g., when ΔC PM or α in (14) are sufficiently large, limq-0 ðCðqÞ=μL ðqÞÞ0 is negative and thus cqn initially decreases). Proposition 2 provides a theoretical proof of existence of the optimal degree of imperfect repair that minimizes the long-run expected cost per unit of time. It can be found numerically, for instance, by means of simulation of the virtual age process (covering the range 0 r q o 1 with a reasonable step), then computing μL ðqÞ from (11) and, finally, obtaining the optimal value from (13). This is rather computationally intensive. On the other hand, an approximate meaningful and much simpler computationally approach can be used. In fact, this approach is based on the limiting property (9). Let veq for each q define an ‘equilibrium value’ of virtual age, i.e., Z 1 q veq ðqÞ þ Fðuj veq ðqÞÞdu ¼ veq ðqÞ ð15Þ
Fig. 1. The function MðqÞ.
0
meaning that if an item starts some cycle having an initial virtual age veq ðqÞ, then after the next repair its virtual age will be again
57
Fig. 2. The function cq for C P ¼ 1; C M ¼ 0:3; α ¼ 2
58
M. Finkelstein / Reliability Engineering and System Safety 138 (2015) 54–58
Fig. 3. The function cq for C P ¼ 1; C M ¼ 0:15; α ¼ 2 (left) and C P ¼ 1; C M ¼ 0:3; α ¼ 3(right).
expected, and also follows from Proposition 2, the shape of the graph is very sensitive to the ratio C P =C M and to the value of α Direct simulation of the optimal solution in (13) for the considered examples shows a relatively good approximation of our ‘quasioptimal’ approach; however, we plan to conduct theoretical investigations of the accuracy of approximation in the future research. 4. Concluding remarks In this short note, we first have performed a simple comparison of long-run expected costs per unit of time for perfect and minimal repairs. The peculiarity of this comparison is in the different nature of point processes governing these two models. The perfect repair corresponds to the renewal process, whereas the minimal repair corresponds to the NHPP. Therefore, we compare, in fact, the cumulated costs for some interval of time and consider the illustrative Example 1, where all options are clearly seen. Our main effort is focused in imperfect (general repair). The best repair for our system in this case is the one that corresponds to the optimal level (extent) of repair actions that minimize the long-run expected costs per unit of time. It turns out that it is a rather complex problem even for the specific imperfect repair model (Kijima II) that was considered. However, we were able to use and modify our previous asymptotic results for the virtual age and cycles durations for this model and to suggest an innovative approximate procedure for defining the optimal solution. Our results show and the considered example illustrates it, that when the failure rate of an item is increasing to infinity, e.g., according the Weibull law and the cost function decreases with the sufficiently large absolute value of the corresponding derivative (when q-0), the optimal degree of repair exists (and it is not the perfect or minimal repair). These results can be definitely useful in practice for systems with different degrees of repair. Acknowledgments The author is grateful to 3 anonymous referees, whose detailed comments were extremely helpful for the revision. This work was
supported by the NRF (National Research Foundation of South Africa) Grant IFR2011040500026.
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