Optimal maintenance decisions over bounded and unbounded horizons

Transcription

1 Optimal maintenance decisions over bounded and unbounded horizons Report Jan M. van Noortwijk Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics

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3 Optimal Maintenance Decisions over Bounded and Unbounded Horizons z Jan M. van Noortwijk x January 9, 1995 Abstract Since maintenance of hydraulic structures is expensive, a decision model has been developed to enable optimal maintenance decisions to be determined. To overcome the usual lack of deterioration data, we can apply a decision-theoretic approach that is initialized with expert judgment and can be updated with actual data. Three related cost-based criteria can be used to compare decisions over unbounded time-horizons: long-term average costs, discounted costs, and equivalent average costs. These costs can be determined using renewal theory, and depend on the distribution of the state of deterioration. The decision-maker should choose that maintenance decision for which the expected costs are minimal. The approach can be applied in many elds of engineering to solve problems in life cycle costing and maintenance optimisation. Key words decision theory, discounting, life cycle costing, maintenance optimisation, renewal theory. z This work was partly supported by Delft Hydraulics, The Netherlands, under Project No. Q1210. The author gratefully acknowledges helpful comments from Roger M. Cooke, Matthijs Kok, Tim J. Bedford and J.J. Schoorl. x Delft University of Technology, Faculty of Mathematics & Computer Science, P.O. Box 5031, 2600 GA Delft, The Netherlands. 1

4 1 Introduction This paper presents a maintenance decision model to make maintenance decisions against minimal costs. Although many mathematical models for maintenance optimisation have been published, only a few of them have been applied (see Dekker [3]). The main reasons for this poor applicability are that the models in question require much data, which is often not present in real applications, and that they are static, in the sense that they cannot be updated with actual data. To overcome these problems, statistical decision theory oers decision-makers the means for encoding a priori beliefs about the deterioration process in a prior distribution and for updating this distribution with new deterioration data using Bayes' theorem. As Mazzuchi & Soyer [11] note, only a few maintenance optimisation models are Bayesian in nature. Van Noortwijk, Cooke & Misiewicz [14] have argued that deterioration can best be characterized by the generalized gamma process. Accordingly, it is possible to dene discrete-time deterioration processes for which the increments of deterioration are identically distributed exponential random quantities that are conditionally independent when the value of the average deterioration is given. Given this average, important probabilistic properties can be expressed explicitly. By denition, each time a maintenance action is carried out, a structure is brought back to its desired condition. Hence, maintenance can best be modelled using renewal theory. Moreover, since maintenance actions will occur at dierent points in time, Wagner [17, Ch. 11] recommends the use of three cost-based criteria to compare decisions over unbounded time-horizons: long-term average costs, discounted costs, and equivalent average costs. The decision-maker should choose that maintenance decision for which the expected costs are minimal with respect to the distribution of the state of deterioration. Hence, the decision model is a combination of renewal theory and statistical decision theory with Bayesian updating. Although the model was developed with hydraulic engineering in mind, it can be applied to many other elds of engineering to solve problems in life cycle costing and maintenance optimisation. The purpose of this paper is to describe the basic properties of the maintenance decision model itself. Later papers will report on applications of the model in three case studies currently in progress. The paper is organized as follows. On the basis of a denition of maintenance in Section 2, we formulate maintenance problems in a decision-theoretic format in Section 3. In Section 4 three cost-based criteria are introduced to compare maintenance decisions. Conclusions are presented in Section 5. The theorem needed to obtain the cost-based criteria is given in an appendix. 2

5 2 The concept of maintenance In introducing the concept of maintenance, we follow Kelly [10, Ch. 2] and Van Noortwijk [12] who focus on mechanical and hydraulic engineering, respectively. The life cycle of a hydraulic structure can be divided into four phases: design, building, use, and demolition. During its use, it is necessary to maintain and, sometimes, to improve a structure. 2.1 The purpose of maintenance Each structure has to fulll certain requirements, such as safety, environment, transport, and recreation. As soon as a structure fails to meet these requirements it should be repaired, preferably against minimal costs. When focussing on safety only, a distinction can be made between a structure's resistance and the applied stress. In that case, a failure may be dened as the event at which the resistance drops below the stress. We dene maintenance as a combination of actions carried out to return a structure to, or restore it to, its desired condition. Inspections, repairs, and replacements are possible maintenance actions. The event that the structure ceases to meet its desired condition, or its requirements, is called a failure. 2.2 Types of maintenance Kelly [10, Ch. 2] distinguishes two types of maintenance: corrective maintenance (after failure) and preventive maintenance (before failure). Corrective maintenance should be chosen if the costs arising from failure are low; preventive maintenance if these costs are high. We can choose between two preventive maintenance strategies: time-based maintenance carried out at regular intervals of time, operation, or use, and condition-based maintenance carried out at times determined by inspecting or monitoring a structure's condition. Time-based maintenance can be applied if uncertainty about the time to failure is low. Condition-based maintenance can be relied on if the deterioration is inspectable or monitorable. By contrast, a structure should not simply be maintained but redesigned or seriously improved when its cost of failure is high and when, moreover, the rate of its deterioration is very uncertain and its condition cannot be inspected or monitored. Within hydraulic engineering we mainly have expensive condition-based maintenance. During a structure's life cycle there are mainly two phases in which it is most economical to apply maintenance optimisation techniques: the design phase and the use phase. In the design phase, one should obtain an optimum balance between building costs and maintenance costs. In the use phase, one should minimize the costs of inspection, repair, replacement, and failure. Problems of nding a good balance be- 3

6 tween initial and future costs belong to the area of life cycle costing; these problems are treated by Flanagan et al. [6]. 3 The maintenance decision model Since condition-based maintenance is often applied, we introduce a probabilistic model based on the structure's condition. However, due to unpredictable deterioration, this condition is uncertain and degrades in time. To make maintenance decisions dealing with this uncertainty, we can use statistical decision theory, which we introduce following the treatment of DeGroot [2, Ch. 8]. 3.1 Maintenance decisions with minimal costs In this section, the general structure of maintenance decision problems is specied. Let us consider a decision problem in which the decision-maker must choose a decision d from all possible decisions D, with the consequences of decision d depending on the unknown value of the state of deterioration (for example, the average deterioration per unit time). The set of all possible decisions d is called the decision space D and the set of all possible values of the state of deterioration is called the parameter space. Let us assume that there exists a probability distribution P on the parameter space assigning a probability to every event belonging to a -eld of subsets of. The distribution P may represent a priori beliefs about the state of deterioration. Let L(; d) be the loss when the decision-maker chooses decision d and when the value of is. In maintenance decision problems the loss function often represents monetary losses such as costs of maintenance. For any decision d 2 D, the expected loss, or risk, is given by (P; d) = Z L(; d) dp () = E(L(; d)): The decision-maker should choose, if possible, a maintenance decision d for which the risk with respect to the probability distribution P on the state of deterioration is minimal. The Bayes risk is dened to be (P ) = (P; d ) = inf (P; d): d2d Any decision d whose risk (P; d ) equals the Bayes risk is called an optimal decision or a Bayes decision with respect to the distribution P. If a maintenance decision were to be taken without any observation, the optimal decision is with respect to the prior density (). However, as soon as we observe deterioration data, for example, and assuming we have a likelihood function l(j), 4

7 we can use Bayes' theorem to update the prior density of. This results in the conditional density of when the observation is given, called the posterior density: l(j)() (j) = R l(j~ )( ) ~ d ~ = l(j)() : p() We call l(j) the likelihood function. As can be found in DeGroot [2, Ch. 8], the optimal maintenance decision is now given by the Bayes decision with respect to the posterior distribution of when is given. Hence, the statistical decision problem changes only in the sense that the prior distribution is replaced with the posterior distribution. 3.2 Deterioration as a stochastic process Since the main aim of this paper is obtaining the loss function L(; d), we only briey illustrate how to derive the likelihood function of the increments of deterioration. The standard way to model deterioration, according to the literature, is to describe it in terms of a Brownian motion with drift (see Karlin & Taylor [9, Ch. 7], Hontelez, Burger & Wijnmalen [8], and Pettit [15]). However, this model implies the existence of periods in which the condition of the structure actually improves and, moreover, it gives rise to computational diculties when applied to decision problems. Therefore, Van Noortwijk, Cooke & Misiewicz [14] have argued that a deterioration process with linear drift can best be characterized by increments that are nonnegative, exchangeable, and isotropic, for every uniform time-partition. For an innite sequence of such increments, the corresponding stochastic process is the so-called generalized gamma process. For this monotone jump process, we may nd a unique uniform timepartition in time-intervals of length for which the joint probability density function of the increments of deterioration is a mixture of conditionally independent exponentials when the average rate of deterioration is given. Let D i be the increment of deterioration in unit time ((i? 1); i], i = 1; : : : ; n, then Z 1 ( ) ny 1 p( 1 ; ; n ) = 0 exp? Z 1 i ny dp () = l( i j) dp () 0 for ( 1 ; ; n ) 2 IR n + and zero otherwise. The random quantity describes the uncertainty about the limiting average deterioration lim N!1 ( P N D i)=n, where the average converges with probability one if E(D 1 ) < 1. With respect to the units of time ((i? 1); i], i 2 IN, we may regard deterioration as a discrete-time stochastic process (implying discrete renewal times). For notational convenience, let the unit time length be equal to 1. Moreover, since we are equipped with the exponential likelihood function, we can express various probabilistic properties, such as the probability of failure in unit time i, in explicit form conditional on the average deterioration. See Van Noortwijk, Cooke & Kok [13] for some illustrations. 5

8 4 Maintenance decisions over unbounded horizons In this section, we use renewal theory to obtain the monetary losses L(; d) when the decision-maker chooses maintenance decision d and when the the state of deterioration is, for both bounded and unbounded time-horizons. We consider three cost-based criteria to compare maintenance decisions over unbounded time-horizons: long-term average costs, discounted costs, and equivalent average costs. 4.1 Types of cost-based critera Wagner [17, Ch. 11] gave two reasons to make decisions over unbounded instead of bounded time-horizons. First, in making repeated investment decisions it is better to employ an unbounded horizon model than to simply ignore the future. Second, as we shall see later, the mathematical models are less complex, while they still provide reasonable answers in practice. However, since maintenance costs over an unbounded time-horizon are innite in most cases, we need models that can handle an innite accumulation of costs. Wagner [17, Ch. 11] distinguishes three cost-based criteria to compare decisions over unbounded horizons: 1. The long-term average costs per unit time, which are determined by simply averaging the costs. 2. The discounted costs, which are based on the assumption that the utility of money decreases in time. The present discounted value of the costs c n in unit time n is assumed to be n c n, where = [1+(r=100)]?1 is the discount factor per unit time and r% is the real (ination free) discount rate per unit time. Since 0 < < 1, this actually means that \a dollar today is worth more than a dollar a year from today". Hence, the higher the discount rate, the better it will be to postpone expensive maintenance actions. 3. The equivalent average costs, which relate the two notions of average and discounted costs. These three cost-based criteria will be discussed in more detail in the following subsections. 4.2 Long-term average costs In view of the denition of maintenance, we can best model it by a renewal process, where the renewals are the actions restoring a structure to its desired condition. After each renewal we start, in a statistical sense, all over again. An introduction to renewal theory can be found in Karlin & Taylor [9, Ch. 5], and Taylor & Karlin [16, Ch. 7]. 6

9 A discrete renewal process fn(n) : n 2 INg is a nonnegative integer-valued stochastic process that registers the successive renewals in the time-interval (0; n], where the interoccurrence times are nonnegative, independent, identically distributed, discrete random quantities. Hence, 0 N(n) n. Suppose the renewal times T 1 ; T 2 ; : : : ; are conditionally independent of the state of deterioration, at the time when the decisionmaker chooses decision d, and that they have the distribution PrfT k = ig = p i (; d), i 2 IN, where P 1 p i (; d) = 1. Moreover, let c i (; d) 0 be the costs involved in the event that the renewal time T k = i occurs, where i 2 IN, 2, and d 2 D. For notational convenience, let ~c(; d) = P 1 c i (; d)p i (; d) for all 2 and d 2 D. As was briey described at the end of the previous section, for the generalized gamma process the renewal probabilities p i (; d) and their associated costs c i (; d) can be determined in explicit form. Van Noortwijk, Cooke & Kok [13] studied two types of renewal times: times to failure (corrective maintenance), and times to failure or preventive repair, whichever comes rst (condition-based maintenance). We derive three probabilistic properties connected with the renewal process. First, the expected number of maintenance actions to be taken in time-duration (0; n]: the so-called renewal function M(n) = E(N(n)). This function follows from the expected maintenance costs C(n; ; d), over a bounded time-horizon (0; n], that solve the equation C(n; ; d) = nx p i (; d)[c i (; d) + C(n? i; ; d)] (1) for n 2 IN and C(0; ; d) 0, for all 2 and d 2 D. To obtain this equation, we condition on the values of the rst renewal time T 1 and apply the law of total probability. The costs associated with occurrence of the event T 1 = i are c i (; d) plus the expected additional costs during the interval (n? i; n], i = 1; : : : ; n. Second, since the dierences C(i; ; d)? C(i? 1; ; d), i = 1; : : : ; n, satisfy the discrete renewal equation (see Feller [4, Ch. 12 & 13] and Karlin & Taylor [9, Ch. 3]), we can prove that the expected long-term average costs per unit time are C(n; ; d) lim n!1 n = P 1 c i(; d)p i (; d) P 1 ip i (; d) = C(; d) (2) (see the Appendix for the proof). Let a renewal cycle be the time-period between two renewals, then we recognize the numerator as the expected cycle costs and the denominator as the expected cycle length. Third, if c i (; d) 1 for all i in Eq. (1), then the expected long-term average number of renewals per unit time is: M(n; ; d) lim n!1 n = 1 P 1 ip i(; d) ; where the denominator is the mean life time when we regard p i (; d) as the probability of failure in unit time i, i 2 IN. 7

10 Furthermore, it can easily be seen that the sequence of functions fc(n; ; d)=ng is dominated by the function ~c(; d). If E(~c(; d)) and E(C(n; ; d)) exist for all n, then we may interchange the order of the operations of expectation and passing to the limit, lim E (C(n; ; d)=n) = E lim C(n; ; d)=n = E (C(; d)) n!1 n!1 for all d 2 D, using Lebesgue's Theorem of Dominated Convergence. 4.3 Discounted costs To obtain the expected discounted costs over a bounded time-horizon, we determine a recursive formula similar to that for the nondiscounted costs in Eq. (1). Again, we condition on the values of the rst renewal time T 1 and apply the law of total probability. In this case, however, we want to account for the present value of the renewal costs c i (; d) plus the expected additional discounted costs in time-interval (n? i; n], i = 1; : : : ; n. Hence, the expected present discounted value of the costs for a bounded time-horizon (0; n] can be written as C (n; ; d) = nx i p i (; d) [c i (; d) + C (n? i; ; d)] (3) for n 2 IN, and C (0; ; d) 0, for all 2, d 2 D, and 0 < < 1. With the dierences C (i; ; d)? C (i? 1; ; d), i = 1; : : : ; n, we can reformulate the above recursive formula as a discrete renewal equation. By using Feller's proof [4, Ch. 13] we can obtain the expected discounted costs over an unbounded time-horizon lim C (n; ; d) = n!1 P 1 i c i (; d)p i (; d) 1? P 1 i p i (; d) = C (; d) (4) for 2, d 2 D, and 0 < < 1 (see the Appendix for the proof). To illustrate: suppose p m (; d) 1, p i (; d) 0 for all i 6= m, and c i (; d) c, i 2 IN, then we have the present value of an innite stream of identical costs at times m; 2m; 3m; : : : being equal to m c=(1? m ) for 0 < < 1. We recognize the numerator of C (; d) as the discounted cycle costs, while the denominator can be interpreted as the probability that the renewal process terminates due to discounting. Such a renewal process is called a terminating renewal process since innite interoccurrence times can cause the renewals to cease. The interoccurrence times Z 1 ; Z 2 ; : : :, of our imaginary terminating renewal process have the distribution PrfZ k = ig = i p i (; d), i 2 IN, and PrfZ k = 1g = 1? P 1 i p i (; d). The expected number of imaginary \discounted renewals" over an unbounded time-horizon is lim M (n; ; d) = n!1 P 1 i p i (; d) 1? P 1 i p i (; d) = PrfZ k < 1g PrfZ k = 1g 8

11 for 2, d 2 D, and 0 < < 1. For notational convenience, let ~c (; d) = P 1 i c i (; d)p i (; d). The sequence of functions fc (n; ; d)g is dominated by the function ~c (; d)=(1? ). If E(~c (; d)) and E(C (n; ; d)) exist for all n, then we can interchange the order of the operations of expectation and passing to the limit, lim n!1 E (C (n; ; d)) = E lim C (n; ; d) n!1 = E (C (; d)) for d 2 D and 0 < < 1, using Lebesgue's Theorem of Dominated Convergence. Note 1 Using a terminating renewal argument (see Feller [5, Ch. 6]), the expression for the expected total discounted costs can also be obtained when the renewal times are nonnegative and real-valued. Let F (t) be the cumulative distribution function of the renewal time T and let c(t) be the costs associated with a renewal at time t. Then, the expected total discounted costs can be written as C = R 1 0 R t c(t) df (t) 1 1? R 1 0 t df (t) =? 0 t c(t) df (t) log() R 1 0 t F(t) dt ; where F (t) = 1? F (t), and 0 < < 1 is the discount factor. This result generalizes the work of Berg [1] and Fox [7] who studied age and block replacement policies with discounting. 4.4 Equivalent average costs To determine the relation between the notions of average and discounted costs, we construct a new innite stream of identical costs with the same present discounted value as the discounted costs. This can easily be achieved by dening an innite stream of costs appearing at times i = 0; 1; 2; : : :, which are all equal to (1? )C (; d). Using the geometric series, we can write P 1 i=0 i (1? )C (; d) = C (; d) for 2, d 2 D, and 0 < < 1. We call (1? )C (; d) the equivalent average costs. As tends to 1, from below, the equivalent average costs approach the long-term average costs per unit time: lim "1 (1? )C (; d) = C(; d); for 2 and d 2 D, using L'H^opital's rule. Furthermore, using the inequality (1? )C (; d) ~c(; d), and the existence of E(~c(; d)) and E(C (; d)), for all 0 < < 1, 2, and d 2 D, we may interchange the order of the operations of expectation and passing to the limit, lim "1 E((1? )C (; d)) = E lim "1 (1? )C (; d) 9 = E (C(; d))

12 for all d 2 D, using Lebesgue's Theorem of Dominated Convergence. 4.5 Choice of cost-based criteria There are two life cycle phases of a hydraulic structure in which it is most economical to perform maintenance optimisation: the design phase and the use phase. In the design phase, we are interested in an optimum balance between initial and future costs, which is the area of life cycle costing. Denote the initial building costs bringing a structure to its desired condition by c 0 (d). Then, the monetary losses over an unbounded horizon are the sum of the initial costs and the discounted future costs, L (; d) = c 0 (d) + C (; d); when the decision-maker chooses decision d, the state of deterioration is, and the discount factor is. In the design phase, we cannot use the criterion of average costs per unit time, L(; d) = C(; d); because the initial costs are averaged out over an unbounded time-horizon: its contribution to the average costs is completely ignored, since lim n!1 c 0 (d)=n = 0. In the use phase, the costs of inspection, repair, replacement, and failure should be minimized. For an unbounded time-horizon, we may use both the average and the discounted cost criterion to optimize condition-based maintenance. A preference for one or the other depends on the application and cannot be given in general. If high failure cost is included in the decision model, we suggest applying the discounted cost criterion, since we now have to nd an optimum balance between initial preventive maintenance costs and future failure cost. However, in a realistic maintenance decision model, the cost of failure due to postponed preventive maintenance should grow fast enough to outweigh the discounting. If failure cost is either small or is excluded in the decision model, the average cost criterion should be used. 10

13 5 Conclusions It is possible to make optimal maintenance decisions by applying a decision-theoretic approach in the design phase or in the use phase. We can use three cost-based criteria to compare decisions over unbounded horizons: long-term average costs, discounted costs, and equivalent average costs (where the last notion combines the rst two). In the design phase, only the discounted cost criterion can be used for the purpose of life cycle costing. In the use phase, both the average cost criterion and the discounted cost criterion, whichever is most suitable, can be applied for maintenance optimisation. The decision-maker should choose that maintenance decision for which the expected costs are minimal with respect to the prior distribution of the state of deterioration. In the light of actual deterioration data, the decision-maker can revise the initial, optimal, decision by simply replacing the prior distribution with the posterior distribution. 11

14 Appendix Theorem 1 (The discrete renewal theorem.) Let fp i g, fu i g, fb i g be sequences indexed by i = 0; 1; 2; : : : with p i 0 for all i, and P 1 i=0 jb i j < 1. Suppose the renewal equation u n = b n + P n p i=0 iu n?i (5) is satised for n = 0; 1; 2; : : : by a bounded sequence fu i g of real nonnegative numbers. P Then, (a) if 1 i=0 p i = 1, p 1 > 0, lim u n = n!1 P 1 i=0 b i P1 i=0 ip i ; and (b) if P 1 i=0 p i < 1. Hence, u n converges to 0, as n! 1, at such a rate that P lim n n!1 i=0 u i = P 1 b i=0 i 1? P 1 p : i=0 i Proof: See Feller [4, Ch. 13] and Karlin & Taylor [9, Ch. 3]. 2 Corollary 1 Let u i C(i; ; d)? C(i? 1; ; d), p i p i (; d), b i c i (; d)p i (; d) for all i = 1; 2; : : :, 2, d 2 D, and u 0 p 0 b 0 0, then the recurrence relation (1) can be used to obtain the renewal equation (5). With Cauchy's rst limit theorem, the expression for the long-term average costs (2) follows from part (a) of the discrete renewal theorem. Corollary 2 Let u i C (i; ; d)?c (i?1; ; d), p i i p i (; d), b i c i (; d) i p i (; d) for all i = 1; 2; : : :, 2, d 2 D, 0 < < 1, and u 0 p 0 b 0 0, then the recurrence relation (3) can be used to obtain the renewal equation (5). The expression for the discounted costs (4) follows from part (b) of the discrete renewal theorem. 12

15 References [1] Menachem Berg. A marginal cost analysis for preventive replacement policies. European Journal of Operational Research, 4:136{142, [2] Morris H. DeGroot. Optimal Statistical Decisions. McGraw-Hill Book Company, [3] Rommert Dekker. Applications of maintenance optimisation models: A review and analysis. Technical Report 9228/A, Erasmus University Rotterdam, The Netherlands, May [4] William Feller. An Introduction to Probability Theory and its Applications; Volume 1. John Wiley & Sons, [5] William Feller. An Introduction to Probability Theory and its Applications; Volume 2. John Wiley & Sons, [6] Roger Flanagan, George Norman, Justin Meadows, and Graham Robinson. Life Cycle Costing; Theory and Practice. BSP Professional Books, [7] Bennett Fox. Age replacement with discounting. Operations Research, 14:533{537, [8] Jan A.M. Hontelez, Helen H. Burger, and Diederik J.D. Wijnmalen. Optimum condition-dependent maintenance policies for deteriorating systems with partial information. Reliability Engineering and System Safety (to appear), [9] Samuel Karlin and Howard M. Taylor. A First Course in Stochastic Processes; Second Edition. Academic Press, Inc., [10] Anthony Kelly. Maintenance Planning and Control. Butterworth & Co., [11] Thomas A. Mazzuchi and Rek Soyer. Adaptive Bayesian replacement strategies. Technical report, The George Washington University, Washington, D.C., U.S.A., [12] Jan M. van Noortwijk. Preventief onderhoud van waterbouwkundige constructies; probleeminventarisatie [Preventive maintenance of hydraulic structures; inventory of problems]. Technical Report Q1210, Delft Hydraulics, Delft, The Netherlands, June [13] Jan M. van Noortwijk, Roger M. Cooke, and Matthijs Kok. A Bayesian failure model based on isotropic deterioration. European Journal of Operational Research (to appear), 82(2), March

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